Bholes
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Abhipsit MishraBholes
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Abhipsit MishraGeometry and physics of black holes
Lecture notes
IAP, Paris, March-April 2016
CP3, UCL, Louvain-la-Neuve, November 2016
DIAS-TH, BLTP, Dubna, May 2017
Les Houches, July 2018
CSGC, IITM, Chennai, January 2022
ENS, Paris, May-June 2023
AEI, Potsdam, December 2023
IHP, Paris, March 2024
ENS, Paris, April-June 2025
Éric Gourgoulhon
Laboratoire d’étude de l’Univers et des phénomènes eXtrêmes
CNRS, Observatoire de Paris - PSL, Sorbonne Université
eric.gourgoulhon@obspm.fr
https://relativite.obspm.fr/blackholes
— DRAFT —
version of 29 May 2025
Corrections and comments are welcome
2
Preface
These notes correspond to lectures given
• at Institut d’Astrophysique de Paris (France) in March-April 2016, within the framework
of the IAP Advanced Lectures:
https://www.iap.fr/vie_scientifique/cours/cours.php?nom=cours_iap&annee=2016
• at the Centre for Cosmology, Particle Physics and Phenomenology in Louvain-la-Neuve
(Belgium) in November 2016, within the framework of the Chaire Georges Lemaître:
https://uclouvain.be/fr/instituts-recherche/irmp/chaire-georges-lemaitre-2016.html
• at the Bogoliubov Laboratory of Theoretical Physics, in Dubna (Russia) in May 2017, within
the framework of the Dubna International Advanced School of Theoretical Physics:
http://www.jinr.ru/posts/lecture-course-geometry-and-physics-of-black-holes/
• at the Summer School Gravitational Waves 2018, taking place at Les Houches (France) in
July 2018:
https://www.lkb.upmc.fr/gravitationalwaves2018/
• remotely at the School on Black Holes and Gravitational Waves organized at the Centre for
Strings, Gravitation and Cosmology of the Indian Institute of Technology Madras, Chennai
(India) in January 2022:
https://physics.iitm.ac.in/~csgc/events/sbhgw
• at the École Normale Supérieure, Paris (France) in May-June 2023, as part of the PSL
graduate programs in Physics and in Astrophysics:
https://relativite.obspm.fr/blackholes/paris23
• at the Albert Einstein Institute, Potsdam (Germany) in December 2023:
https://relativite.obspm.fr/blackholes/aei23
• at the Institut Henri Poincaré, Paris (France) in March 2024, within the program Quantum
and classical fields interacting with geometry:
https://relativite.obspm.fr/blackholes/ihp24
• at the École Normale Supérieure, Paris (France) in April-June 2025, as part of the PSL
graduate program in Physics:
https://relativite.obspm.fr/blackholes/paris25
4
In complement to these notes, one may recommend various monographs devoted to black
holes: O’Neill (1995) [391], Heusler (1996) [275], Frolov & Novikov (1998) [202], Poisson (2004)
[416], Frolov & Zelnikov (2011) [204], Bambi (2017) [30], Chruściel (2020) [119], Grumiller &
Sheikh-Jabbari (2022) [246] and King (2023) [316], as well as review articles by Carter (1987)
[101], Wald (2001) [505], Chruściel (2002, 2005) [117, 118] and Chruściel, Lopes Costa & Heusler
(2012) [123] and recent monographs with various chapters about black holes: Andersson (2020)
[14] and Shibata (2016) [456]. In addition, let us point out other lecture notes on black holes:
Hawking (1994) [265, 268], Townsend (1997) [483], Compère (2006, 2019) [136, 138], Dafermos
and Rodnianski (2008) [149], Deruelle (2009) [161], Andersson, Bäckdahl & Blue (2016) [13]
and Reall (2020) [426].
The history of black holes in theoretical physics and astrophysics is very rich and fascinating.
It is however not discussed here, except in some small historical notes. The interested reader is
referred to Nathalie Deruelle’s lectures [161], to Kip Thorne’s book [479], to Carter’s article
[105] and to Jean Eisenstaedt’s articles [180, 182].
The web pages associated to these notes are
https://relativite.obspm.fr/blackholes
They contain supplementary material, such as the SageMath notebooks presented in Ap-
pendix D.
I warmly thank Cyril Pitrou for having organized the Paris 2016 lectures, Fabio Maltoni and
Christophe Ringeval for the Louvain-la-Neuve ones, Anastasia Golubtsova and Irina Pirozhenko
for the Dubna ones, Bruce Allen, Marie-Anne Bizouard, Nelson Christensen and Pierre-François
Cohadon for the Les Houches ones, Chandra Kant Mishra for the Chennai ones, Jean-François
Allemand for the Paris 2023 and 2025 ones, Masaru Shibata and Karim Van Aeslt for the Potsdam
ones and Dietrich Häfner, Frédéric Hélein and Michał Wrochna for the Paris 2024 ones.
Besides, I am deeply indebted to Imène Belahcene, Jack Borthwick, Brandon Carter, Marc
Casals, Udit Narayan Chowdhury, Stéphane Collion, Xiangyang Chen, Sumit Dey, Jean Eisen-
staedt, Romain Gervalle, David Hirondel, Ted Jacobson, Michel Le Bellac, Alexandre Le Tiec,
Jean-Philippe Nicolas, Jordan Nicoules, Micaela Oertel, Paul Ramond, Nicolas Seroux, Irène
Urso and Frédéric Vincent for spotting mistakes, correcting typos and making nice suggestions
in preliminary versions of the text.
These notes are released under the
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License
Contents
I Foundations 11
1 General framework 13
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Worldlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Quantities measured by an observer . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Einstein equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 The concept of black hole 1: Horizons as null hypersurfaces 25
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Black holes and null hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Geometry of null hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Evolution of the expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3 The concept of black hole 2: Non-expanding horizons and Killing horizons 57
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Non-expanding horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3 Killing horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Bifurcate Killing horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4 The concept of black hole 3: The global view 91
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Conformal completion of Minkowski spacetime . . . . . . . . . . . . . . . . . 92
4.3 Conformal completions and asymptotic flatness . . . . . . . . . . . . . . . . . 100
4.4 Black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5 Stationary black holes 121
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Stationary spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.3 Mass and angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4 The event horizon as a Killing horizon . . . . . . . . . . . . . . . . . . . . . . 152
5.5 The generalized Smarr formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.6 The no-hair theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6 CONTENTS
II Schwarzschild black hole 181
6 Schwarzschild black hole 183
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.2 The Schwarzschild-(anti-)de Sitter solution . . . . . . . . . . . . . . . . . . . . 183
6.3 Radial null geodesics and Eddington-Finkelstein coordinates . . . . . . . . . . 189
6.4 Black hole character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7 Geodesics in Schwarzschild spacetime: generic and timelike cases 207
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.2 Geodesic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.3 Timelike geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
8 Null geodesics and images in Schwarzschild spacetime 233
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
8.2 Main properties of null geodesics . . . . . . . . . . . . . . . . . . . . . . . . . 234
8.3 Trajectories of null geodesics in the equatorial plane . . . . . . . . . . . . . . . 241
8.4 Asymptotic direction from some emission point . . . . . . . . . . . . . . . . . 260
8.5 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9 Maximal extension of Schwarzschild spacetime 281
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
9.2 Kruskal-Szekeres coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
9.3 Maximal extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
9.4 Carter-Penrose diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
9.5 Einstein-Rosen bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
9.6 Physical relevance of the maximal extension . . . . . . . . . . . . . . . . . . . 318
III Kerr black hole 321
10 Kerr black hole 323
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
10.2 The Kerr solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
10.3 Extension of the spacetime manifold through ∆ = 0 . . . . . . . . . . . . . . . 334
10.4 Principal null geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
10.5 Event horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
10.6 Global quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
10.7 Families of observers in Kerr spacetime . . . . . . . . . . . . . . . . . . . . . . 354
10.8 Maximal analytic extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
10.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
CONTENTS 7
11 Geodesics in Kerr spacetime: generic and timelike cases 373
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
11.2 Equations of geodesic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
11.3 Main properties of geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
11.4 Timelike geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
11.5 Circular timelike orbits in the equatorial plane . . . . . . . . . . . . . . . . . . 418
11.6 Going further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
12 Null geodesics and images in Kerr spacetime 443
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
12.2 Main properties of null geodesics . . . . . . . . . . . . . . . . . . . . . . . . . 443
12.3 Spherical photon orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
12.4 Black hole shadow and critical curve . . . . . . . . . . . . . . . . . . . . . . . 479
12.5 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
13 Extremal Kerr black hole 505
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
13.2 Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
13.3 Maximal analytic extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
13.4 Near-horizon extremal Kerr (NHEK) geometry . . . . . . . . . . . . . . . . . . 523
13.5 Going further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
IV Dynamical black holes 539
14 Black hole formation 1: dust collapse 541
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
14.2 Lemaître-Tolman equations and their solutions . . . . . . . . . . . . . . . . . . 542
14.3 Oppenheimer-Snyder collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
14.4 Observing the black hole formation . . . . . . . . . . . . . . . . . . . . . . . . 568
14.5 Going further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
15 Black hole formation 2: Vaidya collapse 583
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583
15.2 The ingoing Vaidya metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583
15.3 Imploding shell of radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
15.4 Configurations with a naked singularity . . . . . . . . . . . . . . . . . . . . . 602
15.5 Going further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
16 Evolution and thermodynamics of black holes 617
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
16.2 Towards the first law of black hole dynamics . . . . . . . . . . . . . . . . . . . 617
16.3 Evolution of the black hole area . . . . . . . . . . . . . . . . . . . . . . . . . . 635
16.4 The laws of black hole dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 642
16.5 Hawking radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
8 CONTENTS
16.6 Black hole thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650
16.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
17 Black hole thermodynamics beyond general relativity 655
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655
17.2 Diffeomorphism-invariant theories of gravity . . . . . . . . . . . . . . . . . . 655
17.3 Wald entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666
17.4 Covariant phase space formalism . . . . . . . . . . . . . . . . . . . . . . . . . 670
17.5 First law for diffeomorphism-invariant theories . . . . . . . . . . . . . . . . . 684
17.6 What about the second law? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
18 The quasi-local approach: trapping horizons 695
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695
18.2 Trapped surfaces and singularity theorems . . . . . . . . . . . . . . . . . . . . 695
18.3 Trapping horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707
V Appendices 709
A Basic differential geometry 711
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
A.2 Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
A.3 Pseudo-Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 721
A.4 The three basic derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726
A.5 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734
B Geodesics 739
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739
B.2 Definition and first properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 739
B.3 Existence and uniqueness of geodesics . . . . . . . . . . . . . . . . . . . . . . 744
B.4 Geodesics and extremal lengths . . . . . . . . . . . . . . . . . . . . . . . . . . 750
B.5 Geodesics and symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757
B.6 Geodesics and curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759
C Kerr-Schild metrics 763
C.1 Generic Kerr-Schild spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . 763
C.2 Case of Kerr spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765
D SageMath computations 773
D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773
D.2 Minkowski spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774
D.3 Anti-de Sitter spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774
D.4 Schwarzschild spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774
D.5 Kerr spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778
D.6 Evolution and thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 781
CONTENTS 9
E Gyoto computations 783
E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783
E.2 Image computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783
F On the Web 785
Bibliography 787
Index 819
10 CONTENTS
Part I
Foundations
Chapter 1
General framework
Contents
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Worldlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Quantities measured by an observer . . . . . . . . . . . . . . . . . . . . 20
1.5 Einstein equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.1 Introduction
This chapter succinctly presents the spacetime framework used in these lectures (Sec. 1.2)
and recalls useful basic concepts, such as worldlines of particles and observers (Sec. 1.3 and
1.4). In an important part of these lectures (but not always!), we shall assume that the theory
of gravitation is general relativity; this means that the spacetime metric obeys the Einstein
equation, which is recalled in Sec. 1.5.
This chapter is by no means an introduction to relativistic gravity. In this respect, we
recommend the textbooks [87, 109, 163, 257, 371, 464, 499], as well as [162, 229, 337] for
the French-speaking reader. The reader might also find useful to start by taking a look at
Appendix A, which recaps the concepts from differential geometry employed here.
1.2 Spacetime
1.2.1 General settings
In these lectures we consider a n-dimensional spacetime, i.e. a pair (M , g), where M is a
smooth real manifold of dimension n ≥ 2 and g is a Lorentzian metric on M . In some parts, n
will be set to 4 — the standard spacetime dimension — but we shall consider spacetimes with
n ̸= 4 as well.
14 General framework
Figure 1.1: A smooth manifold M : the infinitesimal vector dx connects the nearby points p and q and thus
can be thought as a displacement within the manifold, while the finite vector v does not correspond to any
displacement within the manifold and “lives” in the tangent space Tp M .
The precise definition and basic properties of a smooth manifold are recalled in Appendix A.
Here let us simply say that, in loose terms, a manifold M of dimension n is a “space” (precisely
a topological space) that locally resembles Rn , i.e. can be described by a n-tuple of coordinates
(x0 , . . . , xn−1 ). However, globally, M can be very different from Rn , in particular regarding its
topology. It could also happen that many coordinate systems are required to cover M , while a
single one is obviously sufficient for Rn .
The smooth structure endows the manifold with the concept of infinitesimal displace-
ment vectors dx, which connect infinitely close points of M (cf. Fig. 1.1 and Sec. A.2.3 of
Appendix A). However, for finitely separated points, there is no longer the concept of connect-
ing vector (contrary for instance to points in Rn ). In other words, vectors on M do not live in
the manifold but in the tangent spaces Tp M , which are defined at each point p ∈ M . Each
Tp M is a n-dimensional vector space, which is generated for instance by the infinitesimal
displacement vectors along the n coordinate lines of some coordinate system. Unless explicitly
specified, we assume that M is an orientable manifold (cf. Sec. A.3.4).
The full definition of the metric tensor g is given in Sec. A.3 of Appendix A. At each point
p ∈ M , g induces a (non positive definite) scalar product on Tp M , which we shall denote by a
dot:
∀(u, v) ∈ Tp M × Tp M , u · v := g(u, v). (1.1)
Given a coordinate system (xα ), the components of g with respect to it are the n2 scalar fields
(gαβ ) such that [cf. Eq. (A.40)]
g = gαβ dxα dxβ , (1.2)
where dxα is the differential 1-form of xα (cf. Sec. A.2.4) and the notation dxα dxβ stands for
the symmetric tensor product defined by Eq. (A.39). The scalar square ds2 of an infinitesimal
displacement vector dx = dxα ∂α is expressed in terms of the components (gαβ ) by the line
element formula
ds2 := g(dx, dx) = gαβ dxα dxβ . (1.3)
This formula follows from Eq. (1.2) via the identity ⟨dxα , dx⟩ = dxα [Eq. (A.18)].
1.2 Spacetime 15
Figure 1.2: A Lorentzian manifold (M , g): at each point, the metric tensor g defines privileged directions:
those lying in the null cone at p.
Remark 1: In this book, we shall use the bilinear form identity (1.2) rather than the line element (1.3) to
present a metric (cf. Box. 3.2 of MTW [371] for a discussion). It turns out to be more convenient when
various metric tensors are involved, the notation ds2 being then ambiguous.
The fact that the signature of g is Lorentzian, i.e.
sign g = (−, +, . . . , + ), (1.4)
| {z }
n − 1 times
implies that at each point p ∈ M , there are privileged directions, along which the line element
(1.3) vanishes; they form the so-called null cones or light cones (cf. Fig. 1.2). The null cones
constitute an absolute structure of spacetime, independent from any observer. A vector at a
point p ∈ M that is either timelike or null (cf. Sec. A.3.2) is said to be causal. It lies necessarily
inside the null cone at p (timelike vector) or along it (null vector).
1.2.2 Time orientation
When dealing with black hole spacetimes, it is very important to have clear concepts of “past”
and “future”. Therefore, we assume that the spacetime (M , g) is time-orientable, i.e. that it
is possible to divide continuously all causal (i.e. timelike or null) vectors into two classes, the
future-directed ones and the past-directed ones. More precisely, at each tangent space Tp M , we
may split the causal vectors in two classes by declaring that two causal vectors belong to the
same class iff they are located inside or onto the same nappe of the null cone at p. This defines
an equivalence relation on causal vectors at p, with two equivalence classes. The spacetime
(M , g) is then called time-orientable iff some choice of an equivalence class can be performed
continuously over all M . The vectors belonging to the chosen equivalence class are called
future-directed and the other ones past-directed.
As a characterization of future-directed causal vectors, we shall often use the following
lemmas:
16 General framework
Lemma 1.1: scalar product of a timelike vector with a causal vector
Let (M , g) be a time-orientable spacetime and u a future-directed timelike vector. For any
null or timelike (nonzero) vector v, we have necessarily g(u, v) ̸= 0 and
g(u, v) < 0 ⇐⇒ v is future-directed (1.5a)
g(u, v) > 0 ⇐⇒ v is past-directed. (1.5b)
Proof. Without any loss of generality, we may assume that u is a unit vector: g(u, u) = −1.
Let then (ei )1≤i≤n−1 be a family of n − 1 unit spacelike vectors such that (u, e1 , . . . , en−1 )
is an orthonormal basis of Tp M . We may expand v on this basis: v = v 0 u + v i ei . We have
necessarily v 0 ̸= 0, otherwise v = v i ei would be a spacelike vector, which is excluded by
hypothesis. Moreover, the time-orientation of v is the same as that of u iff v 0 > 0. Since
g(u, v) = −v 0 , this establishes (1.5a) and (1.5b).
Lemma 1.2: scalar product of a null vector with a causal vector
Let (M , g) be a time-orientable spacetime and u a future-directed null vector. For any
null or timelike vector v, we have
g(u, v) < 0 ⇐⇒ v is not collinear with u and is future-directed (1.6a)
g(u, v) = 0 ⇐⇒ v is collinear with u (and thus null) (1.6b)
g(u, v) > 0 ⇐⇒ v is not collinear with u and is past-directed. (1.6c)
Proof. Without any loss of generality, we may find an orthonormal basis (eα )0≤α≤n−1 of Tp M
such that u = e0 + e1 , where the timelike unit vector e0 is future-directed since u is. Let us
expand v on this basis: v = v 0 e0 + v i ei , with v 0 ̸= 0 since v is not spacelike. We have then
g(u, v) = −v 0 + v 1 . Now, since v is null or timelike, g(v, v) ≤ 0, which is equivalent to
n−1
X
0 2
(v ) ≥ (v i )2 . (1.7)
i=1
This implies |v 0 | ≥ |v 1 |. If |v 0 | > |v 1 |, then v cannot be collinear with u (since this would
imply v 0 = v 1 ) and g(u, v) = −v 0 + v 1 ̸= 0, with a sign identical to that of −v 0 . If |v 0 | = |v 1 |,
Eq. (1.7) implies v 2 = v 3 = · · · = v n−1 = 0. We have then either v = v 0 (e0 + e1 ) = v 0 u or
v = v 0 (e0 − e1 ). In the first case, v is collinear with u and g(u, v) = 0. In the second case,
g(u, v) = −2v 0 ̸= 0. To summarize, the only case where g(u, v) = 0 is v being collinear with
u. This establishes (1.6b). In all the other cases, v is not collinear with u and the sign of g(u, v)
is that of −v 0 . Since v is future-directed if v 0 > 0 and past-directed if v 0 < 0, this establishes
(1.6a) and (1.6c).
Two useful properties are immediate consequences of the above lemmas. From Lemma 1.1,
we get
1.3 Worldlines 17
Figure 1.3: Worldlines of a massive particle (L ) and of a massless one (L ′ ).
Property 1.3: no causal orthogonality to a timelike vector
A timelike vector cannot be orthogonal to a timelike vector or to a null vector.
From the part (1.6b) of Lemma 1.2, we get
Property 1.4: orthogonality of two null vectors
Two null vectors are orthogonal if, and only if, they are collinear.
1.3 Worldlines
1.3.1 Definitions
In relativity, a particle is described by its spacetime extent, which is a smooth curve, L say,
and not a point. This curve is called the particle’s worldline and might be thought of as the set
of the “successive positions” occupied by the particle as “time evolves”. Except for pathological
cases (tachyons), the worldline has to be a causal curve, i.e. at any point, a tangent vector
to L is either timelike or null. This reflects the impossibility for the particle to travel faster
than light with respect to any local inertial frame. The dynamics of a simple particle (i.e. a
particle without any internal structure nor spin) is entirely described by its 4-momentum or
energy-momentum vector 1 , which is a vector field p defined along L , tangent to L at each
point and future-directed (cf. Fig. 1.3).
One distinguishes two types of particles:
1
When n ̸= 4, energy-momentum vector is definitely a better name than 4-momentum!
18 General framework
• the massive particles, for which L is a timelike curve, or equivalently, for which p is a
timelike vector:
g(p, p) = p · p < 0; (1.8)
• the massless particles, such as the photon, for which L is a null curve, or equivalently,
for which p is a null vector:
g(p, p) = p · p = 0. (1.9)
In both cases, the mass of the particle is defined by2
√
m= −p · p. (1.10)
Of course, for a massless particle, we get m = 0.
1.3.2 Geodesic motion
If the particle feels only gravitation, i.e. if no non-gravitational force is exerted on it, the
energy-momentum vector must be a geodesic vector, i.e. it obeys
∇p p = 0 , (1.11)
or, in index notation,
pµ ∇µ pα = 0. (1.12)
This implies that the worldline L must be a geodesic of the spacetime (M , g) (cf. Appendix B).
Remark 1: The reverse is not true, i.e. having L geodesic and p tangent to L does not imply (1.11),
but the weaker condition ∇p p = α p, with α a scalar field along L . In this case, one says that p is a
pregeodesic vector (cf. Sec. B.2.2 in Appendix B).
For massive particles, Eq. (1.11) can be derived from a variational principle, the action being
simply the worldline’s length τ as given by the metric tensor:
s
Z B Z λB
dx dx
S= dτ = −g , dλ (1.13)
A λA dλ dλ
(cf. Sec. B.4.2 for details). For photons, Eq. (1.11) can be derived from Maxwell equations within
the geometrical optics approximation (see e.g. Box 5.6 of Ref. [418]), with the assumption that
the photon energy-momentum vector is related to the wave 4-vector k by
p = ℏk. (1.14)
2
Unless specified, we use geometrized units, for which G = 1 and c = 1.
1.3 Worldlines 19
1.3.3 Massive particles
For a massive particle, the constraint of having the worldline L timelike has a simple geomet-
rical meaning: L must always lie inside the light cones of events along L (cf. Fig. 1.3). The
fundamental link between physics and geometry is that the proper time τ of the particle is
nothing but the metric length along the worldline, increasing towards the future:
(1.15)
p p
dτ = −g(dx, dx) = −gµν dxµ dxν ,
where dx is an infinitesimal future-directed3 displacement along L .
The particle’s 4-velocity is defined as the derivative vector u of the parametrization of L
by the proper time:
dx
u := . (1.16)
dτ
By construction, u is tangent to L and is a unit timelike vector:
u · u = −1. (1.17)
For a simple particle (no internal structure), the 4-momentum p is tangent to L ; it is then
necessarily collinear to u. Since both vectors are future-directed, Eqs. (1.10) and (1.17) lead to
p = mu . (1.18)
1.3.4 Massless particles (photons)
For a massless particle, Eq. (1.15) would lead to dτ = 0 since the displacement dx would be a
null vector. There is then no natural parameter along a null geodesic. However, one can single
out a whole family of them, called affine parameters. As recalled in Appendix B, an affine
parameter along a null geodesic L is a parameter λ such that the associated tangent vector,
dx
v := , (1.19)
dλ
is a geodesic vector field: ∇v v = 0. In general, the tangent vector associated to a given
parameter fulfills only ∇v v = α v, with α a scalar field along L (cf. Remark 1 above).
The qualifier affine arises from the fact any two affine parameters λ and λ′ are necessarily
related by an affine transformation:
λ′ = aλ + b, (1.20)
with a and b two constants. Given that the photon energy-momentum vector p is a geodesic
vector [Eq. (1.11)], a natural choice of the affine parameter λ is that associated with p:
dx
p= . (1.21)
dλ
This restricts the transformations (1.20) between the affine parameters to a = 1.
3
Cf. Sec. 1.2.2.
20 General framework
Figure 1.4: Orthogonal decomposition of the energy-momentum vector p of a particle with respect to the
4-velocity uO of an observer O, giving birth to the energy E and linear momentum P as measured by O.
1.4 Quantities measured by an observer
In the simplest modelization, an observer O in the spacetime (M , g) is described by a timelike
worldline LO that is equipped with an orthonormal basis (eα ) at each point, such that e0
is future-directed and tangent to LO and (eα ) varies smoothly along LO (see e.g. Sec. 13.6
of Ref. [371] or Chap. 3 of Ref. [228] for an extended discussion). The vector e0 is then the
4-velocity of O and the vectors (e1 , e2 , e3 ) form an orthonormal basis of the 3-dimensional
local rest space of O. (eα ) is called the observer’s frame.
Let us suppose that the observer O encounters a particle at some event A. Geometrically,
this means that the worldline L of the particle intersects LO at A. Then, the energy E and
the linear momentum P of the particle, both measured by O, are given by the orthogonal
decomposition of the particle’s energy-momentum vector p with respect to LO (cf. Fig. 1.4):
p = EuO + P , with uO · P = 0, (1.22)
where uO = e0 is the 4-velocity of observer O. By taking the scalar product of Eq. (1.22) with
uO , we obtain the following expressions for E and P :
E = −uO · p (1.23)
P = p + (uO · p) uO . (1.24)
The scalar square of Eq. (1.22) leads to
p · p = E 2 uO · uO +2E uO · P +P · P , (1.25)
|{z} | {z } | {z }
−m2 −1 0
where we have used Eq. (1.10) to let appear the particle’s mass m. Hence we recover Einstein’s
relation:
E 2 = m2 + P · P . (1.26)
1.4 Quantities measured by an observer 21
An infinitesimal displacement dx of the particle along its worldline is related to the energy-
momentum vector p by
dx = p dλ, (1.27)
where λ is the affine parameter along the particle’s worldline whose tangent vector is p [cf.
Eq. (1.21) for a massless particle and Eqs. (1.16) and (1.18) with λ := τ /m for a massive particle].
Substituting (1.22) for p in (1.27), we get the orthogonal decomposition of dx with respect to
LO :
dx = Edλ uO + dλ P . (1.28)
O’s proper time elapsed during the particle’s displacement is the coefficient in front of uO :
dτO = Edλ and the particle’s displacement in O’s rest frame is the part orthogonal to uO :
dX = dλ P . By definition, the particle’s velocity with respect to O is
dX dλ P
V := = . (1.29)
dτO Edλ
Hence the relation
P =EV . (1.30)
By combining with (1.22), we get the following orthogonal decomposition of the particle’s
4-momentum:
p = E (uO + V ) . (1.31)
Relations (1.26), (1.31) and (1.30) are valid for any kind of particle, massive or not. For a
massive particle, the energy-momentum vector p is related to the particle’s 4-velocity u via
(1.18). Inserting this relation into (1.23), we obtain
E = Γm , (1.32)
where
Γ := −uO · u (1.33)
is the Lorentz factor of the particle with respect to the observer. If we depart from units with
c = 1, Eq. (1.32) becomes the famous relation E = Γmc2 . Furthermore, combining (1.30) and
(1.32) yields the familiar relation between the linear momentum and the velocity:
P = Γm V . (1.34)
Finally, inserting (1.32) and (1.34) into (1.26) leads to the well-known expression of the Lorentz
factor in terms of the velocity:
Γ = (1 − V · V )−1/2 . (1.35)
If we divide Eq. (1.22) by m and use Eqs. (1.18), (1.32) and (1.34) to express respectively p/m,
E/m and P /m, we get the following orthogonal split of the particle’s 4-velocity u with respect
to observer O:
u = Γ (uO + V ) . (1.36)
For a massless particle (photon), inserting (1.30) into the Einstein relation (1.26) with m = 0
yields
V · V = 1. (1.37)
22 General framework
This means that the norm of the velocity of the massless particle with respect to O equals the
speed of light c (= 1 in our units). For a photon associated with a monochromatic radiation,
the wave 4-vector k admits the following orthogonal decomposition:
k = ω (uO + V ) , (1.38)
where ω = 2πν, with ν being the radiation frequency as measured by observer O. In view of
Eq. (1.31) with p = ℏk [Eq. (1.14)], we get the Planck-Einstein relation:
E = hν . (1.39)
1.5 Einstein equation
1.5.1 General form
Saying that gravitation in spacetime (M , g) is ruled by general relativity amounts to de-
manding that the spacetime dimension fulfills n ≥ 3 and the metric g obeys the Einstein
equation:
1
R − R g + Λ g = 8πT , (1.40)
2
where R is the Ricci tensor of g, R is the Ricci scalar of g (cf. Sec. A.5.3 in Appendix A), Λ is
some constant, called the cosmological constant, and T is the energy-momentum tensor of
matter and non-gravitational fields. If we let appear the Einstein tensor G := R − (R/2)g
[Eq. (A.111)], the Einstein equation is recast as
G + Λ g = 8πT . (1.41)
Remark 1: The case n = 2 has been excluded since the Einstein equation would no longer involve the
spacetime curvature, given that the Einstein tensor G is identically zero for any metric g if n = 2 (the
trace of Eq. (A.112) in Appendix A yields R = (R/2)g). The exclusion of n = 2 also follows by noticing
that the Einstein-Hilbert action, which gives birth to Eq. (1.40) for n ≥ 3, is proportional to the Euler
characteristic of M for n = 2 (by virtue of the Gauss-Bonnet theorem), the Euler characteristic being a
topological invariant independent of g.
By taking the trace of (1.40) with respect to g, it is easy to show that the Einstein equation
(1.40) is equivalent to
2 1
R= Λ g + 8π T − Tg , (1.42)
n−2 n−2
where T := g µν Tµν is the trace of T with respect to g.
Remark 2: The spacetime dimension n explicitely appears in the variant (1.42) of the Einstein equation,
but not in the original version (1.40). Notice as well that Eq. (1.42) would be ill-posed for n = 2 (cf.
Remark 1).
1.5 Einstein equation 23
The vacuum Einstein equation with cosmological constant is Eq. (1.42) with T = 0:
2
R= Λ g. (1.43)
n−2
In the mathematical literature, a solution to Eq. (1.43) is called an Einstein metric. The special
case Λ = 0 is called the vacuum Einstein equation:
R=0. (1.44)
It thus corresponds to the vanishing of the Ricci tensor. Solutions of Eq. (1.44) are sometimes
called Ricci-flat metrics.
Taking the covariant divergence of the Einstein equation (1.41) and invoking the contracted
Bianchi identity (A.110) leads to
→
−
∇· T =0, (1.45)
→
−
where T in the type-(1, 1) tensor associated by metric duality to T [cf. Eq. (A.48)]. In index
notation, the above equation reads [cf. Eq. (A.67)]
∇µ T µα = 0.
Equation (1.45) is often referred to as the equation of energy-momentum conservation.
1.5.2 Electrovacuum Einstein equation
An electromagnetic field in the spacetime (M , g) is a 2-form F (cf. Sec. A.2.6) such that any
particle of mass m > 0, electric charge q and 4-velocity u is subject to the 4-acceleration
q→− q
∇u u = F (., u) ⇐⇒ uµ ∇µ uα = F αµ uµ . (1.46)
m m
Moreover, in standard electromagnetism, F is governed by the Maxwell equations:
↠
dF = 0 and ∇· F = µ0 j, (1.47)
↠
where dF is the exterior derivative of F (cf. Sec. A.4.3), F in the type-(2, 0) antisymmetric
tensor (bivector) associated by metric duality to F [cf. Eqs. (A.49) and (A.50b)], µ0 is a constant
called the vacuum permeability and j is the electric current density vector field, which
describes the distribution of electric charges in spacetime. In view of formula (A.90c) for the
exterior derivative and of definition (A.67) for the divergence operator, the Maxwell equations
(1.47) can be expressed in terms of components with respect to some coordinates (xα ) as
∂Fβγ ∂Fγα ∂Fαβ
+ + =0 and ∇µ F αµ = µ0 j α . (1.48)
∂xα ∂xβ ∂xγ
Remark 3: Thanks to an identity valid for the divergence of any antisymmetric type-(2, 0) tensor field,
the second Maxwell equation can be written in terms of partial derivatives only:
1 ∂ √
−gF αµ = µ0 j α , (1.49)
√ µ
−g ∂x
24 General framework
where g stands for the determinant of the components (gαβ ) of the metric tensor g with respect to the
coordinates (xα ).
Remark 4: The second Maxwell equation can be expressed in terms of differential forms, as the first one,
by introducing the (n − 2)-form ⋆F (n being the spacetime dimension) and the (n − 1)-form ⋆j, which
are respectively the Hodge dual of the 2-form F and the Hodge dual of the 1-form j (the metric dual of
the vector field j, cf. Sec. A.3.3). We shall define the Hodge dual of a p-form in Chap. 5 [Eq. (5.38)]; here,
let us simply state the Maxwell equations in terms of it and of the exterior derivative d:
dF = 0 and d ⋆F = µ0 ⋆j. (1.50)
A source-free electromagnetic field is a 2-form F that obeys the Maxwell equations (1.47)
with j = 0:
↠
dF = 0 and ∇· F = 0. (1.51)
The energy-momentum tensor T of a source-free electromagnetic field F is
1 1
Tαβ = µ µν
Fµα F β − Fµν F gαβ , (1.52)
µ0 4
the trace of which with respect to g is
4−n
T := g µν Tµν = Fµν F µν . (1.53)
4µ0
Remark 5: T = 0 only for n = 4 (the standard spacetime dimension).
The electrovacuum Einstein equation is the Einstein equation (1.42) with Λ = 0 and T
given by Eq. (1.52); in view of Eq. (1.53), it writes
8π 1
Rαβ = Fµα F µβ − Fµν F µν gαβ . (1.54)
µ0 2(n − 2)
A solution to the Einstein-Maxwell system is a triplet (M , g, F ) such that (M , g) is a n-
dimensional spacetime, F is a source-free electromagnetic field on M and (g, F ) obeys the
the electrovacuum Einstein equation (1.54).
Chapter 2
The concept of black hole 1: Horizons as
null hypersurfaces
Contents
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Black holes and null hypersurfaces . . . . . . . . . . . . . . . . . . . . 25
2.3 Geometry of null hypersurfaces . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Evolution of the expansion . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.1 Introduction
In this chapter, we shall start from a naive “definition” of a black hole, as a region of spacetime
from which no particle can escape, and we shall convince ourselves that the black hole boundary
— the so-called event horizon — must be a null hypersurface (Sec. 2.2). We shall then study the
properties of these hypersurfaces (Secs. 2.3 and 2.4). The precise mathematical definition of a
black hole will be given in Chap. 4.
2.2 Black holes and null hypersurfaces
2.2.1 A first definition of black holes
Given a n-dimensional spacetime (M , g) as presented in Chap. 1 (with n ≥ 2), a naive definition
of a black hole, involving only words, could be
A black hole is a localized region of spacetime from which neither massive particles nor
massless ones (photons) can escape.
There are essentially two features in this definition: localization and inescapability. Let us for a
moment focus on the latter. It implies the existence of a boundary, which no particle emitted
26 The concept of black hole 1: Horizons as null hypersurfaces
Figure 2.1: The three causal types of hypersurfaces: spacelike (left), timelike (middle) and null (right).
in the black hole region can cross. This boundary is called the event horizon and is quite
often referred to simply as the horizon. It is a one-way membrane, in the sense that it can be
crossed from the black hole “exterior” towards the black hole “interior”, but not in the reverse
way. The one-way membrane must be a hypersurface of the spacetime manifold M , for it has
to divide M in two regions: the interior (the black hole itself) and the exterior region. Let us
recall that a hypersurface is an embedded submanifold of M of codimension 1 (cf. Sec. A.2.7
in Appendix A).
2.2.2 The event horizon as a null hypersurface
To discuss further which hypersurface could act as a black hole boundary, one should recall
that, on a Lorentzian manifold (M , g), a (smooth) hypersurface Σ can locally be classified in
three categories. The classification, called the causal type, depends on the signature of metric
induced by g on Σ — the induced metric being nothing but the restriction g|Σ of g to vector
fields tangent to Σ. The hypersurface Σ is said to be
• spacelike iff g|Σ is positive definite, i.e. iff sign g|Σ = (+, · · · , +) (n − 1 plus signs), i.e.
iff (Σ, g|Σ ) is a Riemannian manifold;
• timelike iff g|Σ is a Lorentzian metric, i.e. iff sign g|Σ = (−, +, · · · , +) (1 minus sign
and n − 2 plus signs), i.e. iff (Σ, g|Σ ) is a Lorentzian manifold;
• null iff g|Σ is degenerate1 i.e. iff sign g|Σ = (0, +, · · · , +) (1 zero and n − 2 plus signs).
All these definitions are local, i.e. apply to a point p ∈ Σ. Of course, it may happen that Σ has
not the same causal type among all its points.
The causal type of the hypersurface Σ can also be deduced from any normal vector2 n to it
(cf. Fig. 2.1):
• Σ spacelike ⇐⇒ n timelike;
1
Cf. Sec. A.3.1 in Appendix A for the definition of a degenerate bilinear form; the degeneracy implies that the
bilinear form g|Σ is not, strictly speaking, a metric on Σ.
2
The definition of a vector normal to a hypersurface is recalled in Sec. A.3.5 of Appendix A.
2.2 Black holes and null hypersurfaces 27
1 2
Figure 2.2: A timelike hypersurface is a two-way membrane: L1→2 is a timelike worldline from Region 1 to
Region 2, while L2→1 is a timelike worldline from Region 2 to Region 1.
1
Figure 2.3: A spacelike hypersurface is a one-way membrane: L1→2 is a timelike worldline from Region 1 to
Region 2, while there is no timelike or null worldline from Region 2 to Region 1.
• Σ timelike ⇐⇒ n spacelike;
• Σ null ⇐⇒ n null.
These equivalences are easily proved by considering a g-orthogonal basis adapted to Σ.
Remark 1: Null hypersurfaces have the distinctive feature that their normals are also tangent to them.
Indeed, by definition, the normal n is null iff n · n = 0, which is nothing but the condition for n to be
tangent to Σ.
A timelike hypersurface is a two-way membrane: it divides (locally) the spacetime in two
regions, 1 and 2 say, and a future-directed timelike or null worldline can cross it from Region 1 to
Region 2, or from Region 2 to Region 1 (see Fig. 2.2). On the contrary, a spacelike hypersurface
is a one-way membrane: a future-directed timelike or null worldline, which is constrained to
move inside the light cones, can cross it only from Region 1 to Region 2, say (see Fig. 2.3). A
null hypersurface is also a one-way membrane (see Fig. 2.4). At most, a null worldline that is
28 The concept of black hole 1: Horizons as null hypersurfaces
2 1
Figure 2.4: A null hypersurface is a one-way membrane: L1→2 is a timelike worldline from Region 1 to
Region 2, while there is no timelike or null worldline from Region 2 to Region 1.
not going from Region 1 to Region 2 must stay on the hypersurface; an example of such null
worldline is the one depicted in Fig. 2.4 as the thin black line tangent to the normal n.
The limit case between two-way membranes (timelike hypersurfaces) and one-way ones
being null hypersurfaces, it is quite natural to select the latter ones for the black hole boundary,
rather than spacelike hypersurfaces. This choice will be fully justified in Chap. 4, where we
shall see that the generic definition of a black hole implies that its boundary (the event horizon)
is a null hypersurface as soon as it is smooth (Property 4.11). Note however that in Chap. 18,
we shall see that spacelike hypersurfaces, called dynamical horizons, are involved in quasi-local
approaches to black holes.
2.3 Geometry of null hypersurfaces
Since smooth black hole event horizons are null hypersurfaces, let us examine the geometrical
properties of such objects. We shall denote the hypersurface under study by H , for horizon,
but the results of this section and of Sec. 2.4 are valid for any null hypersurface.
2.3.1 Hypersurfaces as level sets
As any hypersurface, H can be locally considered as a level set: around any point of H , there
exists an open subset U of M (possibly U = M ) and a smooth scalar field u : U → R such
that
∀p ∈ U , p ∈ H ⇐⇒ u(p) = 0 (2.1)
and
du ̸= 0 on H , (2.2)
where du is the differential of the scalar field u (cf. Sec. A.2.4): du = (∂u/∂xα ) dxα in any
coordinate chart (xα ). Condition (2.2) ensures that H is a regular hypersurface (an embedded
submanifold, cf. Sec. A.2.7); without it, H could be self-intersecting.
2.3 Geometry of null hypersurfaces 29
Figure 2.5: Null hyperplane H of equation t − x = 0 in Minkowski spacetime. The dimension along z has
been suppressed, so that H is pictured as a 2-plane.
Example 1 (null hyperplane): A very simple example of null hypersurface is a null hyperplane of the
4-dimensional Minkowski spacetime. The Minkowski spacetime is defined by M = R4 with g being a
flat Lorentzian metric. Natural coordinates are Minkowskian coordinates (t, x, y, z), i.e. coordinates
with respect to which the metric components are gαβ = diag(−1, 1, 1, 1). The scalar field
u(t, x, y, z) = t − x (2.3)
defines then a null hyperplane H by u = 0 (cf. Fig. 2.5).
Example 2 (light cone): Another simple example of null hypersurface, still in the 4-dimensional
Minkowski spacetime, is the future sheet H of a light cone, also called future light cone. Note that we
have to take out the cone apex from H , in order to have a regular hypersurface. In the Minkowskian
coordinates (t, x, y, z), the choice of the “retarded time”
p
u(t, x, y, z) = t − x2 + y 2 + z 2 (2.4)
defines a future light cone H by u = 0 and t > 0 (cf. Fig. 2.6).
Example 3 (Schwarzschild horizon): Let us consider the 4-dimensional spacetime (M , g) with M
diffeomorphic to R4 and equipped with a coordinate system (xα ) = (t, r, θ, φ) (t ∈ R, r ∈ (0, +∞),
θ ∈ (0, π) and φ ∈ (0, 2π)) such that the metric tensor takes the form3
2m 4m 2m
g =− 1− dt2 + dt dr + 1 + dr2 + r2 dθ2 + r2 sin2 θ dφ2 , (2.5)
r r r
where m is a positive constant. We shall see in Chap. 6 that (M , g) is actually a part of Schwarzschild
spacetime, described in coordinates different from the standard Schwarzschild-Droste ones, (t̄, r, θ, φ)
3
Formula (2.5) follows the notation introduced in Eq. (1.2); see Sec. A.3.1 in Appendix A for an extended
discussion, in particular Eq. (A.39). When reading the metric components (gαβ ) from (2.5), keep in mind that
cross terms involve a factor 2: the 4m/r coefficient of dt dr implies that gtr = 2m/r.
30 The concept of black hole 1: Horizons as null hypersurfaces
Figure 2.6: Future sheet H of the light cone of equation t − x2 + y 2 + z 2 = 0 in Minkowski spacetime.
p
The dimension along z has been suppressed, so that H looks 2-dimensional, whereas it is actually 3-dimensional.
say, the link between the two being t = t̄ + 2m ln |r/(2m) − 1|. The present coordinates are called the
ingoing Eddington-Finkelstein coordinates and have the advantage over the standard ones to be
regular on the event horizon, which is located at r = 2m. Indeed, the metric components (2.5) remain
finite when r → 2m, as do those of the inverse metric, which are
− 1 + 2m 2m
r r 0 0
2m
1 − 2m 0 0
αβ
g = r r
. (2.6)
1
0 0 r 2 0
1
0 0 0 r2 sin2 θ
Let us consider the scalar field defined on M by
r r−t
u(t, r, θ, φ) = 1 − exp . (2.7)
2m 4m
It is then clear that the hypersurface u = 0 is the 3-dimensional “cylinder” H of equation r = 2m (cf.
Fig. 2.7). We shall see below4 that H is indeed a null hypersurface.
2.3.2 Null normals
Let ℓ be a vector field normal to H . Since H is a null hypersurface, ℓ is a null vector:
ℓ · ℓ = 0. (2.8)
Moreover, we choose ℓ to be future-directed (cf. Sec. 1.2.2).
Remark 1: As a consequence of (2.8), there is no natural normalization of ℓ, contrary to the case of
timelike or spacelike hypersurfaces, where one can always choose the normal to be a unit vector (scalar
4
This should be obvious to the experienced reader, since a normal 1-form to H is dr and from Eq. (2.6),
g µν
∂µ r ∂ν r = g rr = 0 on H .
2.3 Geometry of null hypersurfaces 31
Figure 2.7: Schwarzschild horizon H introduced in Example 3; The figure is drawn for θ = π/2 and is based
on coordinates (t, x, y) related to the ingoing Eddington-Finkelstein coordinates (t, r, θ, φ) by x = r cos φ and
y = r sin φ.
square equal to 1 or −1). It follows that there is no unique choice of ℓ. At this stage, any rescaling
ℓ 7→ ℓ′ = αℓ, with α some strictly positive (to preserve the future orientation of ℓ) scalar field on H ,
yields a normal vector field ℓ′ as valid as ℓ.
The null normal vector field ℓ is a priori defined on H only and not at points p ̸∈ H .
However, it is worth to consider ℓ as a vector field not confined to H but defined in some
open subset of M around H . In particular this would permit to define the spacetime covariant
derivative ∇ℓ, which is not possible if the support of ℓ is restricted to H . Following Carter
[102], a simple way to achieve this is to consider not only a single null hypersurface H , but
a foliation of M (in the vicinity of H ) by a family of null hypersurfaces, such that H is an
element of this family. Without any loss of generality, we may select the value of the scalar
field u defining H to label these hypersurfaces and denote the family by (Hu ). The null
hypersurface H is then nothing but the element H = Hu=0 of this family [Eq. (2.1)]. The
vector field ℓ can then be viewed as being defined in the part of M foliated by (Hu ), such that
at each point in this region, ℓ is null and normal to Hu for some value of u.
Example 4: The scalar field u introduced in Example 1 (null hyperplane) does define a family of null
hypersurfaces (Hu ). A counter-example would be u(t, x, y, z) = (t − x)(1 + x2 ), since u = a does not
define a null hypersurface except for a = 0. Similarly, the scalar fields u of Example 2 (light cone) and
Example 3 (Schwarzschild horizon) do define a family of null hypersurfaces (Hu ). In the latter example,
this would not have been the case for the simpler choice u(t, r, θ, φ) = r − 2m. Some of these null
hypersurfaces are represented in Fig. 2.8
Obviously the family (Hu ) is nonunique but all geometrical quantities that we shall intro-
duce hereafter do not depend upon the choice of the foliation Hu once they are evaluated at
H.
32 The concept of black hole 1: Horizons as null hypersurfaces
15
u=0 u= −1 u= −2
10
t/m
0 u=1
u=2
-5
-10
0 2 4 6 8 10 12 14
r/m
Figure 2.8: Hypersurfaces Hu defined by u = const for the Schwarzschild horizon (Example 3).
Since H is a hypersurface where u is constant [Eq. (2.1)], we have, by definition,
→
−
∀v ∈ Tp M , v tangent to H ⇐⇒ ∇v u = 0 ⇐⇒ ⟨du, v⟩ = 0 ⇐⇒ ∇u · v = 0, (2.9)
→
−
where ∇u is the gradient of the scalar field u, i.e. the vector field that is the metric-dual of
the 1-form du (cf. Sec. A.3.3); in a coordinate chart (xα ), its components are given by
∂u
∇α u = g αµ ∇µ u = g αµ . (2.10)
∂xµ
→
−
Property (2.9) means that ∇u is a normal vector field to H . By uniqueness of the normal
direction to a hypersurface, it must then be collinear to ℓ. Therefore, there must exist some
scalar field ρ such that
→
−
ℓ = −eρ ∇u . (2.11)
→
−
We have chosen the coefficient linking ℓ and ∇u to be strictly negative (minus an exponential).
This is always possible by a suitable choice of the scalar field u. The minus sign ensures that in
the case of u increasing toward the future, ℓ is future-directed, as the following example shows:
Example 5 (null hyperplane): We deduce from expression (2.3) for u in Example 1 that du = dt − dx.
→
−
The gradient vector field obtained by metric duality is ∇u = −∂t − ∂x . Choosing for simplicity ρ = 0,
formula (2.11) yields
ℓ = ∂t + ∂x . (2.12)
This vector field is depicted in Fig. 2.5.
2.3 Geometry of null hypersurfaces 33
Example 6 (light cone): Regarding Example 2, we have, given expression (2.4) for u,
x y z p
du = dt − dx − dy − dz, with r := x2 + y 2 + z 2 .
r r r
Choosing for simplicity ρ = 0 in (2.11), we get the normal
x y z
ℓ = ∂t + ∂x + ∂y + ∂z . (2.13)
r r r
This vector field is depicted in Fig. 2.6.
Example 7 (Schwarzschild horizon): We deduce from expression (2.7) for u in Example 3 that
1 (r−t)/(4m) h r r i
du = e − 1− dt − 1 + dr .
4m 2m 2m
The corresponding gradient vector field, is computed from (2.10) via expression (2.6) for g αµ :
→
− 1 (r−t)/(4m) h r r i
∇u = e − 1+ ∂t + 1 − ∂r .
4m 2m 2m
This time, we do not chose ρ = 0 but rather select ρ so that ℓt = 1:
1 t−r r
eρ = − t ⇐⇒ ρ = − ln 1 + + ln(4m). (2.14)
∇u 4m 2m
Equation (2.11) leads then to
r − 2m
ℓ = ∂t + ∂r . (2.15)
r + 2m
Given the metric (2.5), we check that g(ℓ, ℓ) = 0. Since ℓ ̸= 0, this proves that all hypersurfaces Hu ,
and in particular H , are null. The vector field ℓ is depicted on H in Fig. 2.7 and in all M in Fig. 2.11.
2.3.3 Null geodesic generators
Frobenius identity
Let us take the metric dual of Eq. (2.11); it writes ℓ = −eρ du = −eρ ∇u, or, in index notation,
ℓα = −eρ ∇α u. Taking the covariant derivative, we get
∇α ℓβ = −eρ ∇α ρ∇β u − eρ ∇α ∇β u = ∇α ρ ℓβ − eρ ∇α ∇β u
Antisymmetrizing and using the torsion-free property of ∇ (i.e. ∇α ∇β u − ∇β ∇α u = 0, cf.
Eq. (A.68)), we get
∇α ℓβ − ∇β ℓα = ∇α ρ ℓβ − ∇β ρ ℓα . (2.16)
In the left-hand side there appears the exterior derivative dℓ of the 1-form ℓ [cf. Eq. (A.91b)],
while one recognizes in the right-hand side the exterior product of the two 1-forms dρ and ℓ.
Hence we may rewrite (2.16) as
dℓ = dρ ∧ ℓ . (2.17)
This reflects the Frobenius theorem in its dual formulation (see e.g. Theorem B.3.2 in Wald’s
textbook [499] or Theorem C.2 in Straumann’s textbook [464]): the exterior derivative of the
1-form ℓ is the exterior product of some 1-form (here dρ) with ℓ itself if, and only if, ℓ defines
hyperplanes that are integrable in some hypersurface (H in the present case).
34 The concept of black hole 1: Horizons as null hypersurfaces
Geodesic generators
Let us contract the Frobenius identity (2.16) with ℓ:
ℓµ ∇µ ℓα − ℓµ ∇α ℓµ = ℓµ ∇µ ρ ℓα − ℓµ ℓµ ∇α ρ. (2.18)
|{z}
0
Now, since ℓ is a null vector, ℓµ ∇α ℓµ = ∇α (ℓµ ℓµ ) − ℓµ ∇α ℓµ = −ℓµ ∇α ℓµ , from which we get
ℓµ ∇α ℓµ = 0. (2.19)
Hence (2.18) reduces to
ℓµ ∇µ ℓα = κ ℓα , (2.20)
with
κ := ℓµ ∇µ ρ = ∇ℓ ρ. (2.21)
The metric dual of (2.20) is
∇ℓ ℓ = κ ℓ . (2.22)
This equation implies that the field lines of ℓ are geodesics (cf. Property B.5 in Appendix B). To
demonstrate this, we note that a rescaling
ℓ 7→ ℓ′ = αℓ (2.23)
with α a positive scalar field can be performed to yield a geodesic vector field ℓ′ , i.e. a vector
field that obeys5 Eq. (B.1):
∇ℓ′ ℓ′ = 0. (2.24)
Proof. Equations (2.23) and (2.22) imply
∇ℓ′ ℓ′ = α (∇ℓ α + κα) ℓ. (2.25)
Hence, since α > 0,
d
∇ℓ′ ℓ′ = 0 ⇐⇒ ∇ℓ ln α = −κ ⇐⇒ ln α = −κ,
dλ
where λ is the parameter
R associated
to ℓ along its field lines L : ℓ = dx/dλ|L . Therefore,
λ
setting α(λ) = exp − 0 κ(λ̄)dλ̄ along each field line L ensures that ℓ′ = αℓ fulfills
Eq. (2.24).
Because of (2.24), the field lines of ℓ′ are null geodesics and ℓ′ is the tangent vector to
them associated with some affine parameter λ′ . On the other side, if κ ̸= 0, ℓ is not a geodesic
vector field and therefore cannot be associated with some affine parameter. For this reason
the quantity κ is called the non-affinity coefficient of the null normal ℓ (cf. Sec. B.2.2 in
Appendix B).
Since ℓ is collinear to ℓ′ , it obviously shares the same field lines, which have just been
shown to be null geodesics. Hence we may state:
5
A vector field that obeys the weaker condition (2.22), with κ possibly nonzero, is called a pregeodesic vector
field, cf. Sec. B.2.2.
2.3 Geometry of null hypersurfaces 35
Property 2.1: geodesic generators of a null hypersurface
Any null hypersurface H is ruled by a family of null geodesics, called the null geodesic
generators of H , and each vector field ℓ normal to H is tangent to these null geodesics.
Remark 2: The above result is not trivial: while it is obvious that the field lines of the normal vector
field ℓ are null curves that are tangent to H , one must keep in mind that not all null curves are null
geodesics. For instance, in Minkowski spacetime, the helix defined in terms of some Minkowskian
coordinates (xα ) = (t, x, y, z) by the parametric equation xα (λ) = (λ, cos λ, sin λ, 0) is a null curve, i.e.
it has a null tangent vector at each point, but it is not a geodesic, given that all geodesics of Minkowski
spacetime are straight lines.
As a by-product of Eq. (2.25), we get the behavior of the non-affinity coefficient under a
rescaling of the null normal:
ℓ′ = αℓ =⇒ κ′ = ακ + ∇ℓ α . (2.26)
Example 8 (null hyperplane): It is clear on expression (2.12) for ℓ that the covariant derivative ∇ℓ
vanishes identically. In particular ∇ℓ ℓ = 0. Equation (2.22) then implies κ = 0, which is in agreement
with Eq. (2.21) and the choice ρ = 0 performed in Example 5. The null geodesic generators of H are the
straight lines defined by t = x, y = y0 and z = z0 for some constants (y0 , z0 ) ∈ R2 . They are depicted
as green lines in Fig. 2.5. Either t or x can be chosen as affine parameters of these generators.
Example 9 (light cone): From expression (2.13) for ℓ and the fact that ∇β ℓα = ∂β ℓα in the Minkowskian
coordinates (t, x, y, z), we get
0 0 0 0
2 2
0 y +z
1 −xy −xz (α = row index;
α
∇β ℓ = 3 (2.27)
r 0 β = column index).
−xy x 2 + z2 −yz
0 −xz −yz 2
x +y 2
We obtain then ℓµ ∇µ ℓα = 0. From Eq. (2.22), we conclude that κ = 0, which is in agreement with
Eq. (2.21) and the choice ρ = 0 performed in √ Example 6. The null geodesic generators of H are
the half-lines defined by x = at, y = bt, z = 1 − a2 − b2 t, with t > 0 and (a, b) ∈ R2 such that
a2 + b2 ≤ 1. They are depicted as green lines in Fig. 2.6. Since from (2.13) ∇ℓ t = 1 and κ = 0, λ = t is
an affine parameter along these null geodesic generators.
Example 10 (Schwarzschild horizon): The covariant derivative of the vector field ℓ as given by (2.15)
is (cf. Sec. D.4.1 for the computation)
m m 3r+2m
r2 r2 r+2m
0 0
m r−2m m 3r2 −4m(r+m)
r2 r+2m r2 (r+2m)2 0 0 (α = row index;
α
∇β ℓ = (2.28)
β = column index).
r−2m
0 0 r(r+2m) 0
r−2m
0 0 0 r(r+2m)
36 The concept of black hole 1: Horizons as null hypersurfaces
Contracting with ℓβ , we obtain
4m 4m(r − 2m) 4m
∇ℓ ℓ = 2
∂t + 3
∂r = ℓ.
(r + 2m) (r + 2m) (r + 2m)2
Hence, for any Hu , κ = 4m/(r + 2m)2 . On H (r = 2m), we get
1
κ= . (2.29)
4m
This value agrees with κ = ∇ℓ ρ [Eq. (2.21)] and the choice (2.14) made for ρ. Contrary to Examples 8
and 9, κ does not vanish; hence t, which is a parameter of the null geodesic generators associated with ℓ
(since ∇ℓ t = 1 by virtue of (2.15)), is not an affine parameter. The null geodesic generators are depicted
as vertical green lines in Fig. 2.7.
2.3.4 Cross-sections
Let us now focus on the first aspect of the black hole definition given in Sec. 2.2.1: localization.
This feature is crucial to distinguish a black hole boundary from other kinds of null hypersur-
faces. For instance the interior of a future null cone in Minkowski spacetime is a region from
which no particle may escape, but since the null cone is expanding indefinitely, particles can
travel arbitrarily far from the center. Therefore, a null cone does not define a black hole. A
key parameter is hence the expansion of null hypersurfaces, which we shall discuss in the next
section, after having introduced cross-sections.
Let H be a null hypersurface of a n-dimensional spacetime (M , g) with n ≥ 3. We define
a cross-section of H as a submanifold S of H of dimension n − 2 such that (i) the null
normal ℓ is nowhere tangent to S and (ii) each null geodesic generator of H intersects
S at most once. If each null geodesic generator intersects S exactly once, we shall say
that S is a complete cross-section (cf. Fig. 2.9).
Notation: Indices relative to a cross-section will range from 2 to n − 1 and will be denoted
by a Latin letter from the beginning of the alphabet: a, b, etc.
To encompass the idea that an event horizon H delimitates some region of spacetime, we
shall assume that its cross-sections are closed manifolds, i.e. are compact without boundary6 .
The simplest example is the sphere, more precisely the (n − 2)-dimensional sphere Sn−2 , as
illustrated in Fig. 2.9. This example is the one relevant for standard 4-dimensional stationary
black holes, thanks a topology theorem that we shall discuss in Sec. 5.2.3. But for n > 4, other
compact topologies, like that of a torus, are allowed for cross-sections of a stationary black
hole.
A first important property of cross-sections, independently of whether they are closed or
not, is:
6
Scrictly speaking, a closed manifold is simply a compact manifold, since a topological manifold, as defined
in Sec. A.2.1, has no boundary. Indeed, the concept of manifold with boundary requires a separate definition (cf.
Sec. A.2.2), so that a manifold with boundary is not a manifold. Hence, we may consider the terminology closed
manifold as stressing the no-boundary aspect of a compact manifold.
2.3 Geometry of null hypersurfaces 37
Figure 2.9: The null hypersurface H and two (complete) cross-sections S and S ′ . The green curves represent
some null geodesic generators, with the null normal ℓ tangent to them.
Property 2.2: spacelike character of cross-sections
Any cross-section S of a null hypersurface is spacelike, i.e. all vectors tangent to S are
spacelike.
The spacelike character of S follows from
Lemma 2.3: tangent vectors to a null hypersurface
Every nonzero tangent vector v to a null hypersurface is either spacelike or null. Moreover,
if it is null, v is tangent to a null geodesic generator, i.e. v is normal to the hypersurface.
Proof. Tangent vectors to a null hypersurface H are by definition vectors v such that g(ℓ, v) =
0, where ℓ is the normal to H . Since ℓ is null, it follows then from Property 1.3 (Sec. 1.2.2)
that v cannot be timelike. Besides, if v is null, Property 1.4 (Sec. 1.2.2) implies that it must be
collinear to ℓ.
Proof of Property 2.2. Let p ∈ S and v ∈ Tp M be a nonzero tangent vector to S . The above
lemma implies that v is either spacelike or tangent to the null geodesic generator L going
through p, but then L would be tangent to S , which is not allowed, given the definition of a
cross-section. We conclude that v is necessarily spacelike, which proves that S is a spacelike
submanifold.
Example 11 (light cone): The future sheet H of the Minkowski-spacetime light cone considered
in Examples 2, 6 and 9 has the topology R × S2 since we have excluded the cone apex from H . The
38 The concept of black hole 1: Horizons as null hypersurfaces
complete cross-sections are then compact and it natural to choose them as the spheres defined by t = t0
for some positive constant t0 : S = {p ∈ H , t(p) = t0 }. That S is a 2-dimensional sphere in the
hyperplane t = t0 is clear on its equation in terms of the Minkowskian coordinates (t, x, y, z):
S : t = t0 and x2 + y 2 + z 2 = t20 ,
which follows immediately from u = 0 [cf. Eq. (2.4)]. Moreover, this equation shows that the radius of
the sphere is t0 .
Example 12 (Schwarzschild horizon): The 3-dimensional cylinder H introduced in Example 3 has the
topology R × S2 (cf. Fig. 2.7). Since it is defined by r = 2m in terms of the ingoing Eddington-Finkelstein
coordinates (t, r, θ, φ), a natural coordinate system on H is xA = (t, θ, φ). Moreover, we have seen
that the coordinate t is the (non-affine) parameter of the null geodesics generating H associated with
the null normal ℓ. As in Example 11, a natural choice of cross-section is a sphere defined by t = t0 for
some constant t0 : S = {p ∈ H , t(p) = t0 }. The equation of S in terms of the coordinates (t, r, θ, φ)
is then
S : t = t0 and r = 2m.
Note that xa = (θ, φ) constitutes a coordinate system on S .
Example 13 (binary black hole): Some cross-sections of the event horizon H in numerically generated
binary black hole spacetimes are displayed in Figs. 4.21 and 4.22 of Chap. 4.
Let us denote by q the metric induced by g on a cross-section S , i.e. the bilinear form
defined at any point p ∈ S by
∀(u, v) ∈ Tp S × Tp S , q(u, v) := g(u, v). (2.30)
Saying that S is spacelike (Property 2.2) is equivalent to saying that q is positive definite,
i.e. q(v, v) ≥ 0 for all v ∈ Tp S , with q(v, v) = 0 ⇐⇒ v = 0. In other words, (S , q) is a
Riemannian manifold (cf Sec. A.3.2 in Appendix A).
Example 14 (Schwarzschild horizon): The metric induced by g on the cross-section S of the
Schwarzschild horizon defined in Example 12 is readily obtained by setting t = const = t0 and
r = const = 2m in Eq. (2.5), given that (xa ) = (θ, φ) is a coordinate system on S :
q = 4m2 dθ2 + sin2 θ dφ2 . (2.31)
An important consequence of S being spacelike is (cf. Fig. 2.10):
Property 2.4: orthogonal complement of a cross-section tangent space
At each point p of a cross-section S of a null hypersurface H , the tangent space Tp S
has an orthogonal complement Tp⊥ S , i.e. a timelike 2-dimensional vector plane such
that (i) the tangent space to M at p, Tp M , is the direct sum of Tp S and Tp⊥ S :
Tp M = Tp S ⊕ Tp⊥ S (2.32)
2.3 Geometry of null hypersurfaces 39
Figure 2.10: The tangent space Tp S to the cross-section S and its 2-dimensional orthogonal complement
Tp⊥ S . Only the dimensionality of the latter is respected in the figure: S and Tp S are depicted as 1-dimensional
objects, while they are truly (n − 2)-dimensional ones.
and (ii) every vector in Tp⊥ S is orthogonal to S .
Proof. Using the Gram-Schmidt process to construct a g-orthogonal basis of Tp M , starting
form a q-orthogonal basis of Tp S , one naturally ends up with a complenent of Tp S that is
2-dimensional and such that sign g|Tp⊥ S = (−, +) (timelike signature), in order to ensure that
the signature of g is (−, +, . . . , +).
Since S ⊂ H , the null normal ℓ to H is orthogonal to any vector tangent to S , which
implies ℓ ∈ Tp⊥ S . One can supplement ℓ by another null vector to form a basis of Tp⊥ S :
Property 2.5: complement null normal to a cross-section
At each point p of a cross-section S of a null hypersurface H with future-directed null
normal ℓ, there exists a unique future-directed null vector k transverse to H such that
k · ℓ = −1 and Tp⊥ S = Span (ℓ, k) . (2.33)
Proof. As a timelike plane, Tp⊥ S has two independent null directions, which can be seen as
the two intersections of the null cone at p with the 2-plane Tp⊥ S (cf. Fig. 2.10). Let us denote
by k a future-directed null vector in the null direction of Tp⊥ S that is not along ℓ. By a proper
rescaling k 7→ αk, we may choose k so that k · ℓ = −1 (cf. Lemma 1.2). Since ℓ and k are
non-collinear vectors of Tp⊥ S and dim Tp⊥ S = 2, they constitute a basis of Tp⊥ S .
Remark 3: While ℓ is independent of the choice of the cross-section S of H through p, k does depend
on S .
40 The concept of black hole 1: Horizons as null hypersurfaces
Property 2.6: extension of q
The bilinear form q, which has been defined only on Tp S via (2.30), can be extended to all
vectors of Tp M by setting
q := g + ℓ ⊗ k + k ⊗ ℓ , (2.34)
or, in index notation,
qαβ := gαβ + ℓα kβ + kα ℓβ . (2.35)
It obeys
∀(u, v) ∈ Tp M × Tp M , q(u, v) = q(u∥ , v ∥ ), (2.36)
where u∥ and v ∥ are the tangent parts of u and v to S , which are uniquely defined by the
direct sum (2.32):
u = u∥ + u⊥ and v = v ∥ + v ⊥ , with u∥ , v ∥ ∈ Tp S , u⊥ , v ⊥ ∈ Tp⊥ S . (2.37)
Proof. Thanks to (2.33), we may write the orthogonal decompositions (2.37) as
u = u∥ + u0 ℓ + u1 k and v = v ∥ + v 0 ℓ + v 1 k.
Using ℓ · ℓ = 0, k · k = 0 and ℓ · k = −1, we have then u · v = u∥ · v ∥ − u0 v 1 − u1 v 0 , so that
q defined by (2.34) obeys
q(u, v) = u · v + (ℓ · u)(k · v) + (k · u)(ℓ · v)
= u∥ · v ∥ − u0 v 1 − u1 v 0 + u1 v 0 + u0 v 1 = u∥ · v ∥ .
Example 15 (light cone): In continuation with Example 11, the null vector k orthogonal to the sphere
S and obeying k · ℓ = −1 is
1 x y z
k = ∂t − ∂x − ∂y − ∂z .
2 2r 2r 2r
Evaluating q via (2.34), given expression (2.13) for ℓ, we get the following components of q with respect
to the Minkowskian coordinates (xα ) = (t, x, y, z):
0 0 0 0
y 2 +z 2
0
r2
− xy
r2
− xz
r2
qαβ = .
x2 +z 2
0
− xy r2 r2
− yz
r 2
x2 +y 2
0 − xz r2
− yz
r2 r2
If we consider the spherical coordinates (xα ) = (t, r, θ, φ) deduced from the Minkowskian ones via the
standard formulas: x = r sin θ cos φ, y = r sin θ sin φ and z = r cos θ, the components of q become
2.3 Geometry of null hypersurfaces 41
r
2 4 6 8
Figure 2.11: Null vector fields ℓ (green) and k (red) corresponding to Example 16 (Schwarzschild horizon). The
plot is a 2-dimensional slice θ = const and φ = const of the spacetime M , with t and r labelled in units of m.
Note that since k diverges at r = 0 [cf. Eq. (2.39)], it is not represented there.
instead
0 0 0 0
0 0 0 0
qαβ = . (2.38)
0 0 r2 0
0 0 0 r2 sin2 θ
and we recognize in qab = diag(r2 , r2 sin2 θ) the standard metric on the 2-sphere of radius r.
Example 16 (Schwarzschild horizon): For the Schwarzschild horizon case, we deduce from the metric
(2.5) and the expression (2.15) for ℓ that the null vector k obeying k · ℓ = −1 and orthogonal to the
sphere S introduced in Example 12 is
1 m 1 m
k= + ∂t − + ∂r . (2.39)
2 r 2 r
The vector field k is depicted in Fig. 2.11. We have (cf. the notebook D.4.1)
2m − r 1 m 1 m
ℓ= dt + dr and k=− + dt − + dr, (2.40)
2m + r 2 r 2 r
so that Eq. (2.34) leads to the following components of q in terms of the ingoing Eddington-Finkelstein
42 The concept of black hole 1: Horizons as null hypersurfaces
coordinates xα = (t, r, θ, φ):
0 0 0 0
0 0 0 0
qαβ = . (2.41)
0 0 r2 0
0 0 0 r2 sin2 θ
The pair (ℓ, k) forms a null basis of Tp⊥ S [cf. Eq. (2.33)]. One can construct from it an
orthonormal basis (n, s) as follows:
n = 1ℓ + k ℓ = n+s
2
⇐⇒ (2.42)
s = 1 ℓ − k. k = 1 (n − s) .
2 2
Since ℓ · ℓ = 0, k · k = 0 and ℓ · k = −1, it is easy to check that:
n · n = −1, s · s = 1 and n · s = 0. (2.43)
In other words, (n, s) is an orthonormal basis of the Lorentzian plane (Tp⊥ S , g).
If we substitute (2.42) for ℓ and k in Eq. (2.34), we get
q = g + n ⊗ n − s ⊗ s. (2.44)
Property 2.7: orthogonal projector onto a cross-section
The metric duala of the bilinear form q defined by Eq. (2.34), i.e. the tensor of type (1, 1)
defined by
→
−q := Id + ℓ ⊗ k + k ⊗ ℓ , (2.45)
or, in index notation,
q αβ := δ αβ + ℓα kβ + k α ℓβ , (2.46)
is nothing but the orthogonal projector onto the cross-section S :
∀v ∈ Tp M , →
−
q (v) = v ∥ . (2.47)
a
See Eq. (A.48) of Appendix A for the arrow notation.
Proof. This follows readily from the decomposition v = v ∥ + v 0 ℓ + v 1 k used above.
In particular, we have
→
−
q (ℓ) = 0 and →
−
q (k) = 0. (2.48)
2.3.5 Expansion along the null normal
One defines the expansion of the cross-section S along the vector field ℓ as follows.
2.3 Geometry of null hypersurfaces 43
Figure 2.12: Lie dragging of the surface S along ℓ by the small parameter ε. S is drawn as a 1-dimensional
submanifold, while it is actually a (n − 2)-dimensional one, n being the spacetime dimension.
Given an infinitesimal parameter ε ≥ 0, take a point p ∈ S and displace it by the
(infinitesimal) vector εℓ, thereby getting a nearby point pε (cf. Fig. 2.12). Since ℓ is tangent
to H and p ∈ H , we have pε ∈ H . By repeating this for each point in S , keeping the
value of ε fixed, we define a new (n − 2)-dimensional surface, Sε say (cf. Fig. 2.12). One
says that Sε is obtained from S by Lie dragging along ℓ by the parameter ε. Note that
Sε=0 = S . Since pε ∈ H for every p ∈ S , we have Sε ⊂ H . Because the null direction
ℓ is transverse to Sε by construction, it follows that Sε is spacelike (cf. Lemma 2.3 in
Sec. 2.3.4). At each point p ∈ S , the expansion of S along ℓ is defined from the rate of
change θ(ℓ) of the areaa δA of a surface element δS of S around p:
1 δAε − δA
θ(ℓ) := lim . (2.49)
ε→0 ε δA
In the above formula, δAε stands for the area of the surface element δSε ⊂ Sε that is
obtained from δS by Lie dragging along ℓ by the parameter ε (cf. Fig. 2.12).
a
We are using the words area and surface even if n − 2 ̸= 2, i.e. even if n ̸= 4, being aware that for n = 3
the words length and line would be more appropriate, as well as volume for n ≥ 5.
Remark 4: The reader may wonder why the expansion is not denoted by something like θ(ℓ) (S ), since
its definition depends explicitly on S . We shall show below that, because H is a null hypersurface,
θ(ℓ) is actually independent of the choice of the cross-section S .
In formula (2.49), the area δA is measured with respect to the metric q of S and similarly
the area δAε is measured by the Riemannian metric induced by g on Sε . For instance, if the
surface element δS ⊂ S is a (n−2)-dimensional parallelogram delimited by some infinitesimal
44 The concept of black hole 1: Horizons as null hypersurfaces
displacement vectors dx(2) , . . ., dx(n−1) , the area of δS is
δA = Sϵ(dx(2) , . . . , dx(n−1) ), (2.50)
where Sϵ is the Levi-Civita tensor associated with the metric q of S (cf. Sec. A.3.4 in Ap-
pendix A). The latter is connected to the Levi-Civita tensor of g by the following property.
Property 2.8: area (n − 2)-form of a cross-section
The Levi-Civita tensor Sϵ associated with the metric q of S is expressible in terms of the
spacetime Levi-Civita tensor ϵ as
S
ϵ = ϵ(n, s, . . .) = ϵ(k, ℓ, . . .) , (2.51)
or, in index notation,
S
ϵα1 ···αn−2 = nµ sν ϵµνα1 ···αn−2 = k µ ℓν ϵµνα1 ···αn−2 ,
where (n, s) is the orthonormal basis of Tp⊥ S associated to the null basis (ℓ, k) by
Eq. (2.42).
Proof. We note that ϵ(n, s, . . .) defines a fully antisymmetric (n − 2)-linear form on Tp S .
Since the space of such forms is 1-dimensional (for dim Tp S = n − 2), there exists necessarily
some proportionality factor a such that ϵ(n, s, . . .) = a Sϵ. Now ϵ(n, s, dx(2) , . . . , dx(n−1) ) is
the volume of the n-parallelepiped constructed on the vectors n, s, dx(2) , . . . , dx(n−1) . Given
that n and s are unit-length vectors for the metric g, we have actually
ϵ(n, s, dx(2) , . . . , dx(n−1) ) = δA.
This implies that a = 1, thereby establishing the first equality in Eq. (2.51). The second
equality follows by substituting (2.42) for n and s in (2.51) and using the multilinearity and
antisymmetry of ϵ.
In the vicinity of S , let us consider a spacetime coordinate system (xα ) = (u, v, x2 , . . . , xn−1 )
that is adapted to S and ℓ in the sense that H is (locally) the set u = 0, S is the set
(u, v) = (0, 0) and
ℓ = ∂v . (2.52)
Then, from the very definition of the Lie dragging of S along ℓ, Sε is the set (u, v) = (0, ε)
and (xa ) = (x2 , . . . , xn−1 ) can be viewed as a coordinate system7 on both S and Sε . Let us
choose the n − 2 infinitesimal displacement vectors in (2.50) along the coordinate lines of
this system: dx(a) = dxa ∂a (no summation on a). Then expression (2.50) for the area of δS
becomes
δA = dx2 · · · dxn−1 Sϵ(∂2 , . . . , ∂n−1 ) = dx2 · · · dxn−1 Sϵ2···(n−1)
√
δA = q dx2 · · · dxn−1 , (2.53)
7
Let us recall that according to the convention stated in Sec. 2.3.4, Latin indices from the beginning of the
alphabet, a, b, etc., range from 2 to n − 1.
2.3 Geometry of null hypersurfaces 45
where we have used the multilinearity of Sϵ and Eq. (A.52) for the components of the Levi-
Civita tensor Sϵ, q standing for the determinant of the metric q with respect to the coordinates
(xa ). By the very definition of the Lie dragging, the surface element δSε on Sε is defined by
the same values of the coordinates (xa ) as δS. In particular, the coordinate increments dx2 , . . .,
dxn−1 on Sε take the same values as on S . Therefore, the area of δSε is
(2.54)
p
δAε = q(ε) dx2 · · · dxn−1 ,
where q(ε) stands for the determinant of the components with respect to the coordinates (xa )
of the metric qε induced by g on Sε . Since Sε is spacelike (cf. above), qε is positive definite, so
that q(ε) > 0. In view of (2.53)-(2.54), the definition (2.49) of the expansion of S along ℓ can
be rewritten as p p
1 q(ε) − q(0)
θ(ℓ) = lim p .
ε→0 ε q(0)
We recognize the derivative of the function ε 7→ ln q(ε) = 1/2 ln q(ε) at ε = 0:
p
1 d
θ(ℓ) = ln q. (2.55)
2 dε
Given that Sε is deduced from S by Lie dragging along ℓ and ε is the value of the coordinate
v associated with ℓ [cf. Eq. (2.52)], we may rewrite this formula as the Lie derivative of ln q
along ℓ:
1
θ(ℓ) = Lℓ ln q . (2.56)
2
Example 17 (light cone): For the light cone in Minkowski spacetime, it is easy to evaluate θ(ℓ) by
means of the spherical coordinates introduced in Example 15, since these coordinates are adapted
to the surface S , the metric of S being q = r2 dθ2 + r2 sin2 θ dφ2 [cf. Eq. (2.38)]. We have then
q = det(qab ) = r4 sin2 θ. Moreover, the coordinate v can be chosen as v = t − t0 since t is an (affine)
parameter associated with ℓ (cf. Example 9). Given that t = r on H , we have ε = v = r − t0 , so that
Eq. (2.55) yields
1 d 1 d
θ(ℓ) = ln q = (4 ln r + 2 ln sin θ) ,
2 dr 2 dr
i.e.
2
θ(ℓ) = . (2.57)
r
Example 18 (Schwarzschild horizon): As above, we have q = r4 sin2 θ [cf. Eq. (2.5)], so that Eq. (2.56)
yields
1 1 ∂ ∂ r − 2m ∂ 2 r − 2m
θ(ℓ) = Lℓ ln q = ℓµ µ ln q = ln(r2 sin θ) + ln(r2 sin θ) = , (2.58)
2 2 ∂x ∂t
| {z } r + 2m ∂r r r + 2m
0
where we have used the components ℓµ read on (2.15). The above expression is valid for any hypersurface
of the family Hu . For the case of the Schwarzschild horizon, r = 2m and Eq. (2.58) yields a vanishing
expansion:
θ(ℓ) = 0. (2.59)
46 The concept of black hole 1: Horizons as null hypersurfaces
Note that for large r, Eq. (2.58) yields θ(ℓ) ∼ 2/r, i.e. we recover the flat spacetime result (2.57), which
is consistent with the fact that for large r, Hu is close to a Minkowskian light cone (cf. Fig. 2.8). Note
also that Eq. (2.58) yields θ(ℓ) < 0 for r < 2m and θ(ℓ) > 0 for r > 2m. These expansion values are
in agreement with what can be inferred from Fig. 2.11, since r is directly related to the area of the
cross-sections of H : A = 4πr2 from Eq. (2.41) and ℓ points towards decreasing (resp. increasing) values
of r for r < 2m (resp. r > 2m).
Using the general law of variation of a determinant, as given by Eq. (A.73) in Appendix A,
Eq. (2.56) can be rewritten as
1
tr Q−1 × Lℓ Q ,
θ(ℓ) =
2
where Q is the matrix representing the components of q with respect to the coordinates
(xa ) = (x2 , . . . , xn−1 ): Q = (qab ). Since Q−1 = (q ab ), there comes
1 ab
θ(ℓ) = q Lℓ qab . (2.60)
2
The Lie derivative along ℓ of the metric q of the cross-section S that appears in this formula is
defined as follows. As in Sec. A.4.2 of Appendix A, let us denote by Φε the smooth map S → H
that corresponds to the displacement of points of S by some infinitesimal quantity ε along
ℓ. Using the notations of Fig. 2.12, we have then pε = Φε (p), qε = Φε (q) and Sε = Φε (S ).
The Lie derivative along ℓ of q is then the field Lℓ q of bilinear forms on S defined by the
following action on any pair of vectors (u, v) tangent to S at the same point p:
1
Lℓ q (u, v) := lim [qε (Φε∗ u, Φε∗ v) − q(u, v)] , (2.61)
ε→0 ε
where, as above, qε is the metric induced by g on the (n − 2)-surface Sε deduced from S by
Lie dragging along ℓ by the quantity ε and Φε∗ u (resp. Φε∗ v) is the tangent vector to Sε at
Φε (p) that is the pushforward of u (resp. v) by the map Φε (cf. Sec. A.2.8).
Remark 5: Since q is nothing but the metric induced by the spacetime metric g on cross-sections of
H , we may rewrite the above formula as
1h i
Lℓ q (u, v) := lim g|Φε (p) (Φε∗ u, Φε∗ v) − g|p (u, v) . (2.62)
ε→0 ε
One may wonder about the link between the Lie derivative Lℓ q defined by Eq. (2.61), which
is a tensor field on S , and the Lie derivative along ℓ of the spacetime extension q introduced
by Eq. (2.34). For the sake of clarity, let us denote here the latter by q̄. More precisely, we
may
S consider that q̄ is a field defined in some neighborhood of the portion of H sliced by
ε S ε via Eq. (2.34), with k defined at each point p ∈ S ε as the unique null vector of T ⊥
p S ε
obeying ℓ · k = −1. Let u and v be vector fields on H that are tangent to the cross-sections
Sε . Applying the bilinear form Lℓ q̄ to them and using the Leibniz rule to expand Lℓ [q̄(u, v)]
yields
Lℓ q̄ (u, v) = Lℓ [q̄(u, v)] − q̄ (Lℓ u, v) − q̄ (u, Lℓ v) . (2.63)
2.3 Geometry of null hypersurfaces 47
Now, since u and v are tangent to Sε , we may write q̄(u, v) = q(u, v). Moreover, by the very
definition of the Lie derivative of a vector field (cf. Sec. A.4.2) and the fact that the cross-sections
Sε are Lie-dragged along ℓ, the vectors Lℓ u and Lℓ v are also tangent to Sε . Therefore, we
have q̄ (Lℓ u, v) = q (Lℓ u, v) and q̄ (u, Lℓ v) = q (u, Lℓ v) as well. Thus, we may rewrite
(2.63) as
Lℓ q̄ (u, v) = Lℓ [q(u, v)] − q (Lℓ u, v) − q (u, Lℓ v) .
The right-hand side is identical to what would be obtained by expressing Lℓ q (u, v) via the
Leibniz rule. Hence we conclude that Lℓ q̄ (u, v) = Lℓ q (u, v). Since this identity holds
for a pair (u, v) of vectors tangent to Sε , we may express it for any pair of vectors, i.e. not
necessarily tangent to Sε by introducing the orthogonal projector → −q onto Sε [cf. Eq. (2.45)]:
Lℓ q̄ (→
−
q (u), →
−
q (v)) = Lℓ q (→
−
q (u), →
−
q (v)), (2.64)
or, in index notation,
Lℓ q̄µν q̄ µα q̄ νβ = Lℓ qab q̄ aα q̄ bβ .
Taking the trace with respect to g, we get Lℓ q̄µν q̄ µσ q̄ νσ = Lℓ qab q̄ aσ q̄ bσ . Now, since q̄ is
symmetric and →−
q is a projector, q̄ µσ q̄ νσ = q̄ µσ q̄ σν = q̄ µν . Similarly, q̄ aσ q̄ bσ = q̄ ab . Hence
q̄ µν Lℓ q̄µν = q̄ ab Lℓ qab = q ab Lℓ qab ,
where the second equality follows from q̄ ab = q ab . Hence we may rewrite (2.60) as
1 µν
θ(ℓ) = q Lℓ qµν . (2.65)
2
Note that we have dropped the bar over q, i.e. we revert to the previous notation.
Substituting (2.34) for qµν , and using the Leibniz rule, we get
1 µν 1
θ(ℓ) = q (Lℓ gµν + Lℓ ℓµ kν + ℓµ Lℓ kν + Lℓ kµ ℓν + kµ Lℓ ℓν ) = q µν Lℓ gµν ,
2 2
where the last equality follows from q µν ℓµ = 0 and q µν kµ = 0 [Eq. (2.48)]. Using the Killing
expression (A.87) of the Lie derivative of g: Lℓ gµν = ∇µ ℓν + ∇ν ℓµ , and the symmetry of q µν ,
we arrive at
θ(ℓ) = q µν ∇µ ℓν . (2.66)
We can transform this relation further by expressing q µν via (2.34):
θ(ℓ) = (g µν + ℓµ k ν + k µ ℓν ) ∇µ ℓν = ∇µ ℓµ + k ν ℓµ ∇µ ℓν +k µ ℓν ∇µ ℓν
| {z }
κℓν
1
= ∇µ ℓµ + κ k ν ℓν + k µ ∇µ (ℓν ℓν ) = ∇µ ℓµ − κ,
|{z} 2 | {z }
−1 0
where we have used respectively the properties (2.22), (2.19) and (2.33). Denoting the divergence
of ℓ by ∇ · ℓ = ∇µ ℓµ , we may write
θ(ℓ) = ∇ · ℓ − κ . (2.67)
48 The concept of black hole 1: Horizons as null hypersurfaces
Figure 2.13: Two cross-sections S and S ′ through the same point p of H .
Remark 6: Contrary to θ(ℓ) or κ, the quantity ∇ · ℓ depends a priori on the extension of ℓ outside H
(cf. the discussion in Sec. 2.3.2). For Eq. (2.67) to hold, we have supposed that ℓ remains null outside H ,
so that k µ ∇µ (ℓν ℓν ), which is a derivative in a direction transverse to H , could be set to zero in the
computation leading to (2.67).
Example 19 (light cone): ∇ · ℓ is easily computed by taking the trace of (2.27) and we have κ = 0 (cf.
Example 9), so that (2.67) yields
2(x2 + y 2 + z 2 ) 2
θ(ℓ) = = .
r3 r
Hence we recover the result obtained in Example 17.
Example 20 (Schwarzschild horizon): Here also, ∇ · ℓ is easily computed by taking the trace of (2.28):
m m 3r2 − 4m(r + m) r − 2m 2(r2 + 2mr − 4m2 )
∇·ℓ= + + 2 = .
r2 r2 (r + 2m)2 r(r + 2m) r(r + 2m)2
Given the value κ = 4m/(r + 2m)2 found in Example 10, formula (2.67) leads to
2(r2 + 2mr − 4m2 ) − 4mr 2(r2 − 4m2 ) 2 r − 2m
θ(ℓ) = 2
= 2
= .
r(r + 2m) r(r + 2m) r r + 2m
Hence we recover the result (2.58).
We notice that the right-hand side of (2.67) is independent of the explicit choice of the
cross-section S : clearly both ∇ · ℓ and κ depends only on the null normal ℓ of H . This
justifies the notation θ(ℓ) , which does not refer to S (cf. Remark 4 in page 43). This can be
understood geometrically as follows. Let p ∈ H be a point where one would like to evaluate
θ(ℓ) . Let S and S ′ be two distinct cross-sections of H going through p (cf. Fig. 2.13). Let q be
2.3 Geometry of null hypersurfaces 49
a point of S infinitely close to p and let q ′ be the point of S ′ located on the same null geodesic
−
→
generator as q, i.e. qq ′ = εℓ, with ε infinitely small. Let dx (resp. dx′ ) be the infinitesimal
vector connecting p to q (resp. p to q ′ ). We have then dx′ = dx + εℓ, the scalar square of
which is
dx′ · dx′ = dx · dx + 2ε dx 2
| {z· ℓ} +ε |{z}
ℓ · ℓ,
0 0
where we have used the fact that ℓ is normal to any tangent vector to H , such as dx and ℓ
itself. Hence dx′ · dx′ = dx · dx. In other words, the lengths of all segments from p do not
depend on the cross-section in which they are taken, provided their second end lies on the
same null geodesic generator of H . It follows that all infinitesimal surfaces δS that (i) contain
p and (ii) are enclosed in a tube made of null geodesic generators have the same area δA. Hence
the expansion θ(ℓ) at p does not depend on the choice of δS. We conclude:
Property 2.9: independence of the expansion from the cross-section
The expansion θ(ℓ) at a point p of a null hypersurface H does not depend on the choice of
the cross-section through p. It depends only on the null normal ℓ of H . Accordingly, from
now on, we shall call θ(ℓ) the expansion of the null hypersurface H along ℓ.
The dependency of the expansion on ℓ is given by:
Property 2.10: rescaling of the expansion upon a change of null normal
Two null normals ℓ and ℓ′ to a null hypersurface H are necessarily collinear: ℓ′ = αℓ,
where α is a non-vanishing scalar field on H . We have then the following scaling law:
ℓ′ = αℓ =⇒ θ(ℓ′ ) = αθ(ℓ) . (2.68)
Proof. This follows immediately from expression (2.56) for θ(ℓ) , given that the metric q is
independent of ℓ and Lαℓ ln q = αLℓ ln q. One can also get the result from Eq. (2.66): θ(ℓ′ ) =
q µν ∇µ (αℓν ) = αq µν ∇µ ℓν = αθ(ℓ) , since q µν ℓν = 0 [Eq. (2.48)].
Remark 7: The reader may check that the rescaling laws (2.26) and (2.68) for respectively κ and θ(ℓ)
are compatible with expression (2.67) for θ(ℓ) , given that ∇ · ℓ′ = α∇ · ℓ + ∇ℓ α.
Let us gather all the expressions of the expansion θ(ℓ) obtained so far:
1 δAε − δA 1 1
θ(ℓ) = lim = Lℓ ln q = q µν Lℓ qµν = q µν ∇µ ℓν = ∇ · ℓ − κ , (2.69)
ε→0 ε δA 2 2
with the reminder that the last equality is valid insofar as the vector field ℓ is null in some
entire open neighborhood of H (and not only on H ), as stressed in Remark 6.
50 The concept of black hole 1: Horizons as null hypersurfaces
2.3.6 Deformation rate and shear tensor
Let us consider a cross-section S of the null hypersurface H . The deformation rate Θ of S
along ℓ is defined from the Lie derivative along ℓ of the induced metric q on S [cf. Eq. (2.61)]
as
1− ∗
Θ := → q Lℓ q , (2.70)
2
where →−q ∗ stands for the action of the orthogonal projector →−
q onto S on the bilinear form
Lℓ q. This action extends Lℓ q, which is defined a priori only for vectors of Tp S by Eq. (2.61),
to all vectors of Tp M , for any p ∈ S , via
∀(u, v) ∈ Tp M × Tp M , → −q ∗ Lℓ q (u, v) := Lℓ q →
−
q (u), →
− (2.71)
q (v) .
Since q is symmetric, it is clear from the above definition that Θ is a field of symmetric bilinear
forms. Thanks to the identity (2.64), we can use for q the spacetime extension (2.34) and write
the index-notation version of the definition (2.70) as
1
Θαβ = q µα q νβ Lℓ qµν . (2.72)
2
Substituting gµν + ℓµ kν + kµ ℓν for qµν [Eq. (2.34)], we get
1 1
Θαβ = q µα q νβ (Lℓ gµν + kν Lℓ ℓµ + ℓµ Lℓ kν + ℓν Lℓ kµ + kµ Lℓ ℓν ) = q µα q νβ Lℓ gµν ,
2 2
where the last equality follows from q µα ℓµ = 0 and q µα kµ = 0 [Eq. (2.48)]. Now, using the
Killing expression (A.87) and the Frobenius identity (2.16), we may write
Lℓ gµν = ∇µ ℓν + ∇ν ℓµ = 2∇µ ℓν + ∇ν ρ ℓµ − ∇µ ρ ℓν .
Given that q µα ℓµ = 0, we obtain
Θαβ = q µα q νβ ∇µ ℓν . (2.73)
Let us substitute (2.45) for the projector →
−
q:
Θαβ = (δ µα + ℓµ kα + k µ ℓα ) δ νβ + ℓν kβ + k ν ℓβ ∇µ ℓν .
Expanding and simplifying (in particular via ℓν ∇µ ℓν = 0) yields
∇α ℓβ = Θαβ + ωα ℓβ − ℓα k µ ∇µ ℓβ , (2.74)
where we have let appear the 1-form ω defined by
ωα := −k µ ∇ν ℓµ Πνα = −k µ ∇α ℓµ − k µ k ν ∇µ ℓν ℓα . (2.75)
Here Π is the projector onto H along k:
Παβ = δ αβ + k α ℓβ . (2.76)
2.3 Geometry of null hypersurfaces 51
Indeed, for any vector v tangent to H , one has Π(v) = v since Παµ v µ = v α + k α ℓµ v µ with
ℓµ v µ = 0, while Π(k) = 0 since Παµ k µ = k α + k α ℓµ k µ = k α − k α = 0. Consequently, the
action of ω on tangent vectors to H takes a simple form:
∀v ∈ Tp H , ⟨ω, v⟩ = −k · ∇v ℓ. (2.77)
In particular, for v = ℓ, we get ⟨ω, ℓ⟩ = −k · ∇ℓ ℓ. Given that ∇ℓ ℓ = κℓ [Eq. (2.22)] and
k · ℓ = −1 [Eq. (2.33)], we arrive at the simple formula:
⟨ω, ℓ⟩ = κ . (2.78)
Remark 8: Thanks to the projector Π involved in its definition, the 1-form ω does not depend on
the extension of the vector field ℓ away from H . The same property holds for Θ. On the contrary,
the tensor field ∇ℓ, which appears in the left-hand side of formula (2.74) and in the last term of its
right-hand side, depends on the extension of ℓ away from H .
By comparing (2.65) and (2.72), we notice that the trace of Θ is nothing but the expansion
θ(ℓ) :
θ(ℓ) = g µν Θµν = q µν Θµν = Θµµ . (2.79)
The trace-free part of Θ is called the shear tensor of S along ℓ:
1
σ := Θ − θ(ℓ) q , (2.80)
n−2
or, in index notation:
1
σαβ = Θαβ − θ(ℓ) qαβ . (2.81)
n−2
Remark 9: The 1/(n − 2) factor arises from the trace of q, which is n − 2. This follows immediately
from q being a metric tensor on the (n − 2)-dimensional manifold S ; this can also be recovered from
the spacetime extension (2.34) of q: q µµ = δ µµ + 2ℓµ k µ , with δ µµ = n and ℓµ k µ = −1.
By construction, we have σ µµ = g µν σµν = q µν σµν = 0. Note that, as q, the tensor fields Θ
and σ are tangent to S , in the sense that
∀v ∈ Tp⊥ S , Θ(v, .) = σ(v, .) = 0 , (2.82)
with the important special cases v = ℓ and v = k.
Example 21 (light cone): Let us consider the light cone in Minkowski spacetime described in terms
of the spherical coordinates introduced in Example 15. Since the coordinates (t, θ, φ) are adapted to
the vector field ℓ (i.e. the θ and φ are constant along the field lines of ℓ on H and ℓ = ∂/∂t in these
coordinates, in other words, ℓα = (1, 0, 0)), we have [cf. formula (A.85) in Appendix A]
∂ ∂
Lℓ qab = qab = qab ,
∂t ∂r
52 The concept of black hole 1: Horizons as null hypersurfaces
where the second equality follows from t = r on H . Given that qab = diag(r2 , r2 sin2 θ) [cf. Eq. (2.38)],
we obtain
2r 0
Lℓ qab = = 2 qab .
0 2r sin2 θ r
Hence (2.70) yields
1
Θ= q.
r
Taking the trace, we get immediately θ(ℓ) = 2/r, i.e. we recover the result of Examples 17 and 19. From
(2.80), we get a vanishing shear:
σ = 0.
Example 22 (Schwarzschild horizon): The Lie derivative of q, as given by Eq. (2.41), along ℓ is (cf.
Appendix D for the computation):
r − 2m 2 r − 2m
Lℓ q = 2r dθ2 + sin2 θ dφ2 = q.
r + 2m r r + 2m
Since →
−
q ∗ q = q, Eq. (2.70) yields
r − 2m
Θ= q.
r(r + 2m)
This formula is valid for any hypersurface of the Hu family. For the specific case of the Schwarzschild
horizon H , r = 2m and it reduces to
Θ = 0. (2.83)
Property 2.11: rescaling of the deformation rate and shear upon a change of null
normal
Two null normals ℓ and ℓ′ to a null hypersurface H are necessarily collinear: ℓ′ = αℓ,
where α is a non-vanishing scalar field on H . The deformation rates Θ and Θ′ and shear
tensors σ and σ ′ of a given cross-section S of H along respectively ℓ and ℓ′ obey the
following scaling law:
ℓ′ = αℓ =⇒ Θ′ = αΘ and σ ′ = ασ. (2.84)
Proof. Equation (2.73) yields Θ′ αβ = q µα q νβ ∇µ (αℓν ) = αq µα q νβ ∇µ ℓν + q µα q νβ ℓν ∇µ α =
αΘαβ + 0 since q νβ ℓν = 0 [Eq. (2.48)]. The result for the shear tensor follows then from
the definition (2.80) along with the scaling law (2.68) for the expansion.
Remark 10: The scaling law (2.84) shows explicitely that the deformation rate Θ and shear tensor
σ depend on the choice of the null normal ℓ to H . Both tensors also depend on the choice of the
cross-section S of H , contrary to the expansion θ(ℓ) (Property 2.9).
2.4 Evolution of the expansion 53
2.4 Evolution of the expansion
2.4.1 Null Raychaudhuri equation
Let us derive an evolution equation for the expansion θ(ℓ) ; it is quite natural to consider the
evolution along the null generators of H , i.e. to evaluate the quantity ∇ℓ θ(ℓ) , all the more
that ℓ is by hypothesis future-directed. The starting point is the contracted Ricci identity
[Eq. (A.107)] applied to ℓ:
∇µ ∇α ℓµ − ∇α ∇µ ℓµ = Rµα ℓµ ,
where Rµα stands for the Ricci tensor of g. Substituting Eq. (2.74) for ∇α ℓµ and θ(ℓ) + κ for
∇µ ℓµ = ∇ · ℓ [cf. Eq. (2.67)] yields
∇µ (Θαµ + ωα ℓµ − ℓα k ν ∇ν ℓµ ) − ∇α θ(ℓ) + κ = Rµα ℓµ .
Expanding the left-hand side and using again Eqs. (2.67) and (2.74) results in
∇µ Θµα + ℓµ ∇µ ωα − ∇α θ(ℓ) + κ + θ(ℓ) + κ ωα − Θαµ k ν ∇ν ℓµ
− [ωµ k ν ∇ν ℓµ + ∇µ (k ν ∇ν ℓµ )] ℓα = Rµα ℓµ . (2.85)
The above relation is a 1-form identity that has interesting consequences. In Chap. 3, we shall
apply it to tangent vectors to a cross-section in order to get the so-called zeroth law of black
hole mechanics. Here we shall instead apply it to the normal vector field ℓ (i.e. contract it with
ℓα ); since ωα ℓα = κ [Eq. (2.78)] and ℓα ℓα = 0, we get
ℓν ∇µ Θµν + ℓν ℓµ ∇µ ων − ℓµ ∇µ θ(ℓ) + κ + κ θ(ℓ) + κ = Rµν ℓµ ℓν . (2.86)
Now, using Θµν ℓν = 0 [Eq. (2.82)] and Eq. (2.74) to express ∇µ ℓν , we can write
ℓν ∇µ Θµν = ∇µ (Θµν ℓν ) − Θµν ∇µ ℓν = −Θµν ∇µ ℓν = −Θµν (Θµν + ωµ ℓν − ℓµ k σ ∇σ ℓν )
| {z }
0
= −Θµν Θµν .
On the other side, thanks to Eqs. (2.78) and (2.22), we have
ℓν ℓµ ∇µ ων = ℓµ ∇µ (ων ℓν ) − ων ℓµ ∇µ ℓν = ℓµ ∇µ κ − κ2 .
|{z} | {z }
κ κℓν
Accordingly Eq. (2.86) simplifies to
−Θµν Θµν − ℓµ ∇µ θ(ℓ) + κθ(ℓ) = Rµν ℓµ ℓν . (2.87)
The first term in the left-hand side can be evaluated by expressing Θ in terms of θ(ℓ) and the
shear tensor σ [Eq. (2.80)]:
µν 1 µν 1 µν
Θµν Θ = σµν + θ(ℓ) qµν σ + θ(ℓ) q
n−2 n−2
2 1 1
= σµν σ µν + θ(ℓ) q µν σµν + 2
θ(ℓ) qµν q µν = σab σ ab + θ2 .
| {z } n − 2 | {z } (n − 2) 2 | {z } n − 2 (ℓ)
σab σ ab 0 n−2
Hence Eq. (2.87) leads to:
54 The concept of black hole 1: Horizons as null hypersurfaces
Property 2.12: null Raychaudhuri equation
Any normal vector field ℓ to a null hypersurface H of a n-dimensional spacetime (M , g)
obeys the null Raychaudhuri equation:
1
∇ℓ θ(ℓ) = κθ(ℓ) − θ2 − σab σ ab − R(ℓ, ℓ) . (2.88)
n − 2 (ℓ)
If ℓ is future-directed, this is an evolution equation for θ(ℓ) along the null generators of H .
Remark 1: Actually Eq. (2.88) is a particular case of what is generally called the null Raychaudhuri
equation, namely the case where the vorticity of the vector field ℓ vanishes. This appends because ℓ is
hypersurface-orthogonal, being normal to H . The general case regards a generic congruence of null
geodesics, i.e. a family of null geodesics, one, and exactly one, through each point of M . A null vector
field ℓ tangent to the geodesics of the congruence has a priori some vorticity w and a term +wab wab
must be added to the right-hand side of Eq. (2.88) (see e.g. Eq. (4.35) of Ref. [266]).
Remark 2: Since θ(ℓ) is a scalar field on H , ∇ℓ θ(ℓ) can be replaced by the Lie derivative Lℓ θ(ℓ) in the
left-hand side of the Raychaudhuri equation.
If the spacetime (M , g) is ruled by general relativity, i.e. if g obeys Einstein equation (1.42),
we may express the term involving the Ricci tensor in terms of the total energy-momentum
tensor T :
2 1
(2.89)
R(ℓ, ℓ) = Λ g(ℓ, ℓ) +8π T (ℓ, ℓ) − T g(ℓ, ℓ) = 8πT (ℓ, ℓ).
n − 2 | {z } n − 2 | {z }
0 0
The null Raychaudhuri equation becomes then
1
∇ℓ θ(ℓ) = κθ(ℓ) − θ2 − σab σ ab − 8πT (ℓ, ℓ). (2.90)
n − 2 (ℓ)
Remark 3: The cosmological constant Λ does not appear in the null Raychaudhuri equation (2.90),
despite the latter involves the Einstein equation with Λ [Eq. (1.42)].
Remark 4: We have stressed above that the shear tensor σ does depend on the choice of a cross-section
of H , in addition to ℓ (cf. Remark 10 on p. 52). The null Raychaudhuri equation (2.88) shows that the
“shear square” σab σ ab does not depend on the choice of a cross-section, for all the other terms in (2.88)
are independent of it.
Example 23 (light cone): Let us check the null Raychaudhuri equation on the light cone in Minkowski
spacetime. From Example 9, we have κ = 0, while from Example 21, we have σ = 0, hence σab σ ab = 0.
Moreover, the Ricci tensor of Minkowski spacetime vanishes identically. The null Raychaudhuri equation
(2.88) reduces then to
1 2
∇ℓ θ(ℓ) = − θ(ℓ) ,
2
2.4 Evolution of the expansion 55
where we have set n = 4. Now, from Example 17, we have θ(ℓ) = 2/r. Since, in the present case
∇ℓ θ(ℓ) = Lℓ θ(ℓ) = ∂θ(ℓ) /∂r = −2/r2 , we conclude that the null Raychaudhuri equation is satisfied
(as it should!).
Example 24 (Schwarzschild horizon): For the Schwarzschild horizon H , the null Raychaudhuri
equation is trivially satisfied, i.e. each of its terms vanishes identically: θ(ℓ) = 0 on H [Eq. (2.59)],
which implies ∇ℓ θ(ℓ) = 0 since ℓ is tangent to H , σ = 0 since Θ = 0 [Eq. (2.83)] and the Ricci tensor
of the metric (2.5) is zero (cf. the notebook D.4.1).
2.4.2 Null convergence condition
If one chooses the null normal ℓ associated to an affine parametrization of the null generators
of H , then κ = 0 and the null Raychaudhuri equation (2.88) reduces to
1
∇ℓ θ(ℓ) = − θ2 − σab σ ab − R(ℓ, ℓ). (2.91)
n − 2 (ℓ)
The first term in the right-hand side is manifestly non-positive. The same property holds for
the second term, since
σab σ ab ≥ 0. (2.92)
To prove (2.92), it suffices to consider a q-orthonormal basis of Tp S . Since q is a Riemannian
metric, the components of its inverse in that basis are q ab = diag(1, . . . , 1) = δ ab . Hence
σ ab = q ac q bd σcd = δ ac δ bd σcd = σab , so that
n−1 X
X n−1
σab σ ab
= (σab )2 ≥ 0. (2.93)
a=2 b=2
Regarding the last term in the right-hand side of Eq. (2.91), it is non-positive if the Ricci tensor
obeys the so-called null convergence condition:
R(ℓ, ℓ) ≥ 0 for any null vector ℓ . (2.94)
If general relativity is assumed, i.e. if the Einstein equation (1.40) is fulfilled, we have R(ℓ, ℓ) =
8πT (ℓ, ℓ) [Eq. (2.89)], where T is the total energy-momentum tensor of the matter and non-
gravitational fields. Accordingly, the null convergence condition is equivalent to the so-called
null energy condition:
T (ℓ, ℓ) ≥ 0 for any null vector ℓ , (2.95)
GR
where the index ‘GR’ stands for general relativity. The null energy condition (2.95) is a pretty
weak physical requirement: it is satisfied by
• vacuum: T = 0;
• any “reasonable” matter model, such as a perfect fluid with a proper energy density ρ
and pressure p obeying ρ + p ≥ 0 (8 );
8
Indeed, for the perfect-fluid energy-momentum tensor T = (ρ+p)u⊗u+pg, one has T (ℓ, ℓ) = (ρ+p)(u·ℓ)2
with (u · ℓ)2 ≥ 0.
56 The concept of black hole 1: Horizons as null hypersurfaces
• any electromagnetic field9 ;
• any massless scalar field [266, 321];
• “dark energy” modeled by T = − 8π
Λ
g.
Note also that the null energy condition is implied by the so-called weak energy condition,
which states that
T (u, u) ≥ 0 for any timelike vector u. (2.96)
The null energy condition follows from the weak energy condition by continuity. Selecting for
u the 4-velocity of an observer, we see that the weak energy condition has a simple physical
interpretation: the energy density ε = T (u, u) as measured by any observer is non-negative,
hence the name energy condition.
If the null convergence condition holds, all the terms in the right-hand side of the reduced
Raychaudhuri equation (2.91) are non-positive and we may conclude:
Property 2.13: non-increase of the expansion
If the null convergence condition (2.94) is fulfilled on a null hypersurface H — for general
relativity, this is equivalent to demanding that the null energy condition (2.95) holds on H
—, then the expansion θ(ℓ) along any null normal ℓ associated to an affine parametrization
of H ’s null generators cannot increase along ℓ, i.e. it obeys
∇ℓ θ(ℓ) ≤ 0. (2.97)
This property justifies the name convergence condition given to R(ℓ, ℓ) ≥ 0: if θ(ℓ) < 0 in
some part of H , so that nearby null generators are converging, Property 2.13 shows that the
null generators cannot subsequently diverge.
9
This is readily seen from the electromagnetic energy-momentum tensor (1.52), which yields T (ℓ, ℓ) =
µ−1
0 E · E where E α := F αµ ℓµ . Thanks to the antisymmetry of F , the vector E obeys E · ℓ = 0. It follows from
Property 1.3 that E cannot be timelike. Hence E · E ≥ 0 and T (ℓ, ℓ) ≥ 0.
Chapter 3
The concept of black hole 2:
Non-expanding horizons and Killing
horizons
Contents
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Non-expanding horizons . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3 Killing horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Bifurcate Killing horizons . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.1 Introduction
Having discussed in depth the geometry of null hypersurfaces in Chap. 2 we move forward to
distinguish a null hypersurface representing a black hole event horizon from, let us say, that
representing a mere future light cone. We do it here for black holes in equilibrium. Indeed, for
such objects, it is quite natural to assume a vanishing expansion. This leads us to the concept
of non-expanding horizon (Sec. 3.2). A special kind of these objects is that of Killing horizons,
which occur in spacetimes that are globally stationary (Secs. 3.3 and 3.4). Actually, we shall see
in Chap. 5 that, under some rather general hypotheses, the event horizon of a black hole in
equilibrium must be a Killing horizon.
3.2 Non-expanding horizons
3.2.1 Motivation and definition
In Chap. 2, the null hypersurfaces have been introduced as boundaries of black holes, from
the “no-escape” aspect of the naive definition given in Sec. 2.2.1. To enforce the “localized”
58 The concept of black hole 2: Non-expanding horizons and Killing horizons
facet of the definition, we could demand that the cross-sections are closed (compact without
boundary1 ) and have a constant area, i.e. a vanishing expansion. Hence the definition:
A non-expanding horizon is a null hypersurface H having the topology
H ≃ R×S, (3.1)
where S is a closed manifold of dimension n − 2, and such that the expansion of H along
any null normal ℓ vanishes identically:
θ(ℓ) = 0. (3.2)
Remark 1: Given the scaling law (2.68), if θ(ℓ) = 0 for some normal ℓ, then θ(ℓ′ ) = 0 for any other
normal ℓ′ . Hence the definition of a non-expanding horizon does not depend on the choice of the null
normal.
The above definition captures only the event horizon of black holes in equilibrium, to be
discussed in detail in Chap. 5. For a black hole out of equilibrium, one has generically θ(ℓ) > 0,
as we shall see in Chap. 16.
Example 1 (Schwarzschild horizon): In view of Eq. (2.59), we may assert that the Schwarzschild
horizon considered in Examples 3, 7, 10, 12, 16, 18, 20, 22 and 24 of Chap. 2 is a non-expanding horizon.
Example 2 (null hyperplane and light cone as counter-examples): The null hyperplane and light
cone in Minkowski spacetime considered in the examples of Chap. 2 are excluded by the above definition,
having non-compact cross-sections (null hyperplane) or nonzero expansion (light cone).
Historical note : The concept of non-expanding horizon has been introduced by Petr Hájíček in 1973
under the name of totally geodesic null hypersurface [251] or perfect horizon [252, 253]. The terminology
non-expanding horizon is due to Abhay Ashtekar, Stephen Fairhurst and Badri Krishnan in 2000 [25]
(see also [23]).
3.2.2 Invariance of the area
Given a complete cross-section S of H , the area of S , with respect to the spacetime metric
g, is [cf. Eqs. (2.50) and (2.53)]
√
Z Z
S
A= ϵ(dx(2) , . . . , dx(n−1) ) = q dx2 · · · dxn−1 , (3.3)
S S
where xa = (x2 , . . . , xn−1 ) is a coordinate system on S and q is the determinant with respect
to these coordinates of the Riemannian metric q induced by g on S .
1
Cf. the discussion in Sec. 2.3.4.
3.2 Non-expanding horizons 59
Property 3.1: area of a non-expanding horizon
The area A does not depend on the choice of the complete cross-section S of H . For this
reason, A is called the area of the non-expanding horizon H .
Proof. Let S ′ be a second complete cross-section of H . Let us assume first that S ′ ∩ S = ∅.
Then each null geodesic generator L of H intersects S and S ′ in two distinct points and
it is possible to chose a parameter λ of L such that λ = 0 on S and λ = 1 on S ′ . Let
ℓ = dx/dλ|L be the associated tangent vector. We may then say that the cross-section S ′
is deduced from S by the Lie dragging of S along ℓ by the parameter increase δλ = 1.
More precisely, we may consider that S ′ is deduced from S by a continuous deformation,
represented by a 1-parameter family (Sλ ) of cross-sections such that S0 = S and S1 = S ′ .
Associated with this family is a real-valued function λ 7→ A(λ) given the area of each element
Sλ . By the very definition of the expansion along ℓ [Eq. (2.49)], we have then
Z
dA
= θ(ℓ) δA.
dλ Sλ
If H is a non-expanding horizon, then θ(ℓ) = 0 and it follows that A(λ) is a constant function.
Hence the area of S ′ is equal to that of S . If S and S ′ have a non-empty intersecting part,
the argument can be repeated on each of the non-intersecting parts, given that the area of the
part S ∩ S ′ is obviously the same for S and S ′ .
Example 3 (Schwarzschild horizon): The area of the Schwarzschild horizon is readily computed from
√
the metric (2.31): q = 4m2 dθ2 + sin2 θ dφ2 , yielding q = 4m2 ; we get
A = 16πm2 . (3.4)
3.2.3 Trapped surfaces
If there exists some natural concept of outer/inner region with respect to H , for instance the
outer region being the one having some asymptotically flat end, and if the transverse null
normals k to cross-sections point to the inner region, then the property θ(ℓ) = 0 means that any
cross-section S of the non-expanding horizon H is a marginally outer trapped surface
(often abridged as MOTS). This definition is due to Hawking [261], an outer trapped surface
would be one for which θ(ℓ) ≤ 0.
The MOTS definition is related to, but distinct from, the definition of a marginally trapped
surface by Penrose [405]: a (n − 2)-dimensional submanifold S of M is a trapped surface
iff (i) S is closed (i.e. compact without boundary), (ii) S is spacelike and (iii) the two systems
of null geodesics emerging orthogonally from S towards the future converge locally at S , i.e.
they have negative expansions:
θ(ℓ) < 0 and θ(k) < 0, (3.5)
60 The concept of black hole 2: Non-expanding horizons and Killing horizons
Figure 3.1: Trapped surface (left): δA(ℓ) (ℓ)
ε < δA and untrapped surface (right): δAε > δA, both surfaces having
(k)
δAε < δA.
where the expansion along k is defined in the same way as that along ℓ [cf. Eq. (2.69)]:
(k)
1 δAε − δA 1 1
θ(k) := lim = Lk ln q = q µν Lk qµν = q µν ∇µ kν , (3.6)
ε→0 ε δA 2 2
(k)
δAε being the area of the surface element that is deduced from the surface element of area
δA on S by the Lie dragging along k by a parameter ε (cf. Fig. 3.1). The limit case θ(ℓ) = 0
and θ(k) < 0 correspond to the so-called marginally trapped surface. In flat spacetime
(Minkowski), given any closed spacelike surface, one has θ(ℓ) > 0 and θ(k) < 0 (cf. Fig. 3.1
right), so there is no trapped surface. We shall discuss trapped surfaces in more detail in
Sec. 18.2.1.
Cross-sections of a non-expanding horizon are usually marginally trapped surfaces (cf. the
example below). However, there exist some pathological situations for which θ(k) > 0 at some
points of S [217].
Example 4: Let us consider a cross-section S of the Schwarzschild horizon as defined in Example 12 of
Chap. 2. Computing q µν ∇µ kν from the components kν given by (2.40), we get (cf. the notebook D.4.1 for
the computation) θ(k) = −(r + 2m)/r2 . In particular, on S (r = 2m), θ(k) = −1/m. Hence θ(k) < 0.
Since we had already θ(ℓ) = 0 [cf. Eq. (2.59)], we conclude that S is a marginally trapped surface.
This could also have been inferred from Fig. 2.11, since according to the metric (2.41), the area of the
cross-sections of H is nothing but 4πr2 and k points to decreasing values of r, while, on H , ℓ points
to a fixed value of r.
3.2.4 Vanishing of the deformation rate tensor
If H is a non-expanding horizon, we may set θ(ℓ) = 0 in the null Raychaudhuri equation (2.88);
it reduces then to
σab σ ab + R(ℓ, ℓ) = 0, (3.7)
where R is the Ricci tensor of the metric g. Let us assume that the null convergence condition
discussed in Sec. 2.4.2 holds: R(ℓ, ℓ) ≥ 0 [Eq. (2.94)]. Then, given the property σab σ ab ≥ 0
3.2 Non-expanding horizons 61
[Eq. (2.92)], Eq. (3.7) implies both σab σ ab = 0 and
R(ℓ, ℓ) = 0 . (3.8)
The identity σab σ ab = 0 is possible only if each of the terms σab in the sum (2.93) is zero. Hence
we have necessarily σ = 0. Since we had already θ(ℓ) = 0 (H non-expanding), this implies
that the full deformation rate tensor Θ vanishes identically [cf. Eq. (2.80)]. Moreover, given the
scaling law (2.84), this holds for any null normal ℓ. We can thus conclude:
Property 3.2: invariance of the cross-section metric along the null generators
Provided that the null convergence condition (2.94) holds — which occurs in general
relativity if the null energy condition (2.95) holds —, the deformation rate of any cross-
section S of a non-expanding horizon H along any null normal ℓ is identically zero:
1− ∗
Θ := →q Lℓ q = 0 . (3.9)
2
In other words, the whole metric q (and not only the area form Sϵ, as a mere θ(ℓ) = 0 would
suggest) of any cross-section S is invariant along the geodesic generators of H .
Example 5 (Schwarzschild horizon): We had already noticed that, for the Schwarzschild horizon,
Θ = 0 [Eq. (2.83) in Example 22 of Chap. 2].
3.2.5 Induced affine connection
Since H is a null hypersurface, the “metric” g|H induced on it by the spacetime metric g is
degenerate. As a consequence, there is a priori no unique affine connection on H associated
with g|H . However, when H is a non-expanding horizon and the null convergence condition
holds on H , so that Θ = 0 (Property 3.2), the spacetime Levi-Civita connection ∇ induces a
unique torsion-free connection H ∇ on H . Indeed, let u and v be two vector fields on H . We
have, using (2.74) to express ∇ν ℓµ in terms of Θνµ :
ℓµ uν ∇ν v µ = uν ∇ν (ℓµ v µ ) − v µ uν ∇ν ℓµ = − Θνµ v µ uν − ων uν ℓµ v µ +v µ uν ℓν k σ ∇σ ℓµ = 0.
|{z} |{z} |{z} |{z}
0 0 0 0
Hence ℓ is orthogonal to the vector field ∇u v. It follows immediately that ∇u v is tangent to
H . We conclude:
Property 3.3: induced affine connection
Let X(H ) be the space of vector fields tangent to the non-expanding horizon H . Under
62 The concept of black hole 2: Non-expanding horizons and Killing horizons
the hypotheses of Property 3.2, the operator
H
∇ : X(H ) × X(H ) −→ X(H )
(3.10)
H
(u, v) 7−→ ∇u v := ∇u v,
is well-defined, i.e. H ∇u v belongs to X(H ). Moreover, this operator fulfills all the
properties of an affine connection (cf. Sec. A.4.1), since ∇ does. We naturally call H ∇ the
affine connection induced on H by ∇.
As a consequence of the identity H ∇u v = ∇u v for tangent vector fields to H , (H , H ∇)
is a totally geodesic submanifold of (M , g), i.e. any geodesic of (H , H ∇) is also a geodesic
of (M , g) (cf. the historical note on p. 58).
Property 3.4: horizon-intrinsic derivative of the null normal
Under the hypotheses of Property 3.2, the covariant derivative of the null normal ℓ with
respect to the affine connection H ∇ takes the form
H
∇ℓ = ℓ ⊗ H ω , (3.11)
where H ω is the tensor field on H defined as the restriction of the 1-form ω introduced
in Sec. 2.3.6 to tangent vectors to H [cf. Eq. (2.77)]. In other words, H ω is the pullback
ι∗ ω of ω by the inclusion map ι : H → M (cf. Sec. A.2.8).
Proof. By definition of the covariant derivative with respect to an affine connection [cf. Eq. (A.62)
in Appendix A], H ∇ℓ is a tensor field of type (1, 1) on H , the action of which on any pair
(a, u) formed by a 1-form a on H and a vector field u on H is H ∇ℓ(a, u) = ⟨a, H ∇u ℓ⟩.
Let ā be the extension of a to a 1-form on M defined by ā := a ◦ Π, where Π is the projector
onto H along k [Eq. (2.76)]. Since ∇u ℓ is tangent to H , we have ⟨a, H ∇u ℓ⟩ = ⟨ā, ∇u ℓ⟩. In
view of (3.10) and using Eq. (2.74) to express ∇ℓ, we then get
H
∇ℓ(a, u) = ⟨ā, ∇u ℓ⟩ = āµ uν ∇ν ℓµ = āµ uν Θµν +ων ℓµ − ℓν k ρ ∇ρ ℓµ
|{z}
0
= āµ ℓ ων u − ℓν u āµ k ∇ρ ℓ = ⟨ā, ℓ⟩ ⟨ω, u⟩ = ⟨a, ℓ⟩ ⟨H ω, u⟩.
µ ν ν ρ µ
|{z}
0
Given the definition of a tensor product, this proves Eq. (3.11).
A priori the 1-form H ω depends upon the choice of the cross-section S of H , via the vector
k involved in Eq. (2.77): ⟨H ω, v⟩ = −k · ∇v ℓ for any v ∈ Tp H . Formula (3.11) shows that for
a non-expanding horizon, this is not the case: H ω is a quantity intrinsic to H and to the value
of ℓ on H . Moreover, under a change of null normal, ℓ 7→ ℓ′ = αℓ, H ω remains constant up
to the addition of an exact 1-form: H ω ′ = H ω + d ln α. The 1-form H ω is usually called the
connection 1-form or rotation 1-form of (H , ℓ) [23, 230]; the term rotation stems from the
angular momentum of a non-expanding horizon being given by an integral involving H ω, as
we shall see in Chap. 18.
3.3 Killing horizons 63
Figure 3.2: Group action of G on M .
3.2.6 Going further
See Refs. [27, 230, 297, 26] for more about non-expanding horizons, in particular for a subclass
of them called isolated horizons.
3.3 Killing horizons
A special kind of non-expanding horizons, which is of primordial importance for the theory of
stationary black holes, is that of Killing horizons with closed-manifold cross-sections. Defining
a Killing horizon requires the concepts of 1-dimensional group of isometries and Killing vector,
which we discuss first.
3.3.1 Spacetime symmetries
Symmetries of spacetime, such as stationarity and axisymmetry, are described in a coordinate-
independent way by means of a group acting on the spacetime manifold M . Through this
action, each element of the group displaces points within M and one demands that the metric
g is invariant under such displacement. More precisely, given a group G, a group action of G
on M is a map2
Φ : G × M −→ M
(3.12)
(g, p) 7−→ Φ(g, p) =: Φg (p)
such that (cf. Fig. 3.2)
2
Do no confuse the generic element g of group G with the metric tensor g.
64 The concept of black hole 2: Non-expanding horizons and Killing horizons
• ∀p ∈ M , Φe (p) = p, where e is the identity element of G;
• ∀(g, h) ∈ G2 , ∀p ∈ M , Φg (Φh (p)) = Φgh (p), where gh stands for the product of g by
h according to G’s group law.
The orbit of a point p ∈ M is the set {g(p), g ∈ G} ⊂ M , i.e. the set of points which are
connected to p by some transformation belonging to G. One says that p is a fixed point of the
group action if its orbit is reduced to {p}.
An important class of group actions are those for which G is a 1-dimensional Lie group, i.e.
a so-called “continuous group” (actually a differentiable group). Then around e, the elements
of G can be labelled by a parameter t ∈ R, such that gt=0 = e. It is then common to use the
shorthand notation
Φt := Φgt . (3.13)
If G is a 1-dimensional Lie group, the orbit of a given point p ∈ M under the group action is
then either {p} (when p is a fixed point of the group action) or a curve Lp of M . In the latter
case, t is a natural parameter along Lp (cf. Fig. 3.3). The tangent vector ξ to Lp corresponding
to that parameter is called the generator of the group G associated with the t-parametrization
of G. At each point of the orbit Lp , it is given by (cf. Sec. A.2.3 and Fig. 3.3)
dx
ξ= . (3.14)
dt Lp
We can extend ξ to a vector field on all M , by varying p over M and setting ξ = 0 at fixed
points of the group action. For any p ∈ M and dt infinitesimal, the (infinitesimal) vector
connecting p to Φdt (p) is then dx = dt ξ (cf. Fig. 3.3), so that we may state:
Property 3.5: infinitesimal transformations as displacements along the generator
A 1-dimensional group action limited to infinitesimal transformations of parameter dt
around the identity (dt = 0) amounts to displacements of points of M by the infinitesimal
vector dt ξ, where ξ is the generator of the group.
Given a spacetime (M , g) and a 1-dimensional Lie group G, one says that G is a symmetry
group or a isometry group of (M , g), or equivalently that (M , g) is invariant under the
action of G, iff there is an action Φ of G on M such that for any value of the parameter t of G,
Φt is an isometry of (M , g), i.e. Φt preserves the “scalar products” (and hence the “distances”)
in the following sense: for any p ∈ M and any pair of points (q, r) infinitely close to p, one has
g|Φt (p) (dx′ , dy ′ ) = g|p (dx, dy), (3.15)
−−−−−−−→
for the infinitesimal displacement vectors dx := → −
pq, dy := → −
pr, dx′ := Φt (p)Φt (q) and
−−−−−−−→
dy ′ := Φt (p)Φt (r) (cf. Sec. 1.2). Now, by definition, dx′ is nothing but the pushforward of the
vector dx ∈ Tp M to the tangent space TΦt (p) M by the map Φt (cf. Sec. A.2.8), and similarly
dy ′ is the pushforward of dy by Φt :
dx′ = Φt∗ (dx) and dy ′ = Φt∗ (dy).
3.3 Killing horizons 65
Figure 3.3: Orbit Lp of a point p under the action Φ of a 1-dimensional Lie group, parameterized by t ∈ R.
The tangent vector ξ = dx/dt|Lp is the group generator associated with this parameter.
By rescaling by infinitely small parameters (using the bilinearity of g), it is clear that (3.15)
holds for finite vectors as well, so that we may say that Φt is an isometry of (M , g) iff
∀p ∈ M , ∀(u, v) ∈ (Tp M )2 , g|Φt (p) (Φt∗ u, Φt∗ v) = g|p (u, v), (3.16)
where Φt∗ u (resp. Φt∗ v) is the pushforward of the vector u ∈ Tp M (resp. v ∈ Tp M ) to the
tangent space TΦt (p) M by Φt (cf. Sec. A.2.8). Given the definition (A.33) of the pullback of a
multilinear form, we may reexpress the isometry condition (3.16) in terms of the pullback of g
by Φt :
Φ∗t g = g. (3.17)
This is equivalent to the vanishing of the Lie derivative of g along the generators of G:
Property 3.6: characterization of continuous spacetime isometries
A 1-dimensional Lie group G is a symmetry group of the spacetime (M , g) iff the Lie
derivative of the metric tensor along a generator ξ of G vanishes identically:
Lξ g = 0 . (3.18)
The vector field ξ is then called a Killing vector of (M , g), Eq. (3.18) being equivalent to
the so-called Killing equation:
∇α ξβ + ∇β ξα = 0 . (3.19)
Proof. According the definition (A.82) of the Lie derivative, we have
1 ∗
Lξ g := lim
(Φt g − g) .
t→0 t
The isometry condition (3.17) implies then Lξ g = 0. The reverse is true by integration. The
equivalence between Eqs. (3.18) and (3.19) immediately results from expression (A.87) for the
Lie derivative of g.
66 The concept of black hole 2: Non-expanding horizons and Killing horizons
Property 3.7: isometry in adapted coordinates
In terms of the components gαβ of g with respect to coordinates (xα ) = (t, x1 , . . . , xn−1 )
adapted to the Killing vector ξ, i.e. such that ξ = ∂t , the isometry condition (3.18) is
equivalent to
∂gαβ
= 0. (3.20)
∂t
t is then called an ignorable coordinate.
Proof. This is a direct consequence of the identity (A.85).
3.3.2 Definition and examples of Killing horizons
A Killing horizon is a connected null hypersurface H in a spacetime (M , g) admitting a
Killing vector field ξ such that, on H , ξ is normal to H .
Thus the existence of a Killing horizon requires that the spacetime (M , g) has some
continuous symmetry (usually stationarity), namely that it is invariant under the action of a
1-parameter group, as described in Sec. 3.3.1. Since the normal to a null hypersurface is tangent
to its null geodesic generators and a Killing vector is tangent to the orbits of the isometry
group, a definition equivalent to the above one is:
A Killing horizon is a connected null hypersurface H whose null geodesic generators
are orbits of a 1-parameter group of isometries of (M , g).
Its immediately follows from the definition that the Killing vector ξ is non-vanishing on
H (a normal to a hypersurface cannot vanish) and is null on H :
ξ|H ̸= 0 and ξ · ξ|H = 0. (3.21)
We shall see in Chap. 5 that in a stationary spacetime, under some rather generic hypotheses,
a (connected part of a) black hole event horizon must be a Killing horizon.
Example 6 (null hyperplane as a translation-Killing horizon): Let us consider the null hyperplane
of Minkowski spacetime H discussed in Examples 1, 5 and 8 of Chap. 2. H is defined by the equation
t = x. The vector field
ξ := ∂t + ∂x (3.22)
is a Killing vector of Minkowski spacetime: ξ is the generator of translations in the direction ∂t + ∂x ,
and these translations constitute a 1-dimensional subgroup of the Poincaré group — the symmetry group
of Minkowski spacetime. We note that ξ coincides with the null vector ℓ defined by Eq. (2.12). Since ℓ is
normal to H , we conclude immediately that H is a Killing horizon with respect to ξ.
Example 7 (null hyperplane as a boost-Killing horizon): Let us consider the same null hyperplane
H as above, but with another Killing vector of Minkowski spacetime:
ξ := x∂t + t∂x . (3.23)
3.3 Killing horizons 67
Figure 3.4: Null half-hyperplanes H + and H − as Killing horizons for the Killing vector field ξ = x∂t + t∂x
generating Lorentz boosts in Minkowski spacetime. The green lines are the null geodesic generators of H , while
the thick black line (actually a 2-plane) marks the location where ξ vanishes.
This vector is the generator of the 1-parameter group of Lorentz boosts in the (t, x) plane. On H we
have (cf. Fig. 3.4):
H H
ξ = t(∂t + ∂x ) = t ℓ,
H
where ℓ is the null normal to H defined by Eq. (2.12) and the notation = means that the equality holds
only on H . We conclude that ξ is a normal to the null hypersurface H as soon as t ̸= 0. Therefore, we
may split H \ {t = 0} in two open half-hyperplanes:
H + := {p ∈ H , t(p) > 0} and H − := {p ∈ H , t(p) < 0}, (3.24)
so that each of them is a Killing horizon with respect to ξ (cf. Fig. 3.4).
Example 8 (null hyperplane as a null-rotation-Killing horizon): Another example of Killing
horizon is still provided by the null hyperplane H considered above, but this time with the Killing
vector
ξ := y(∂t + ∂x ) + (t − x)∂y . (3.25)
This vector is indeed the generator of null rotations leaving the plane Span(ℓ, ∂z ) strictly invariant (cf.
e.g. Sec. 6.4.5 of Ref. [228]), ℓ being the null normal of H defined by Eq. (2.12). These null rotations
form a 1-dimensional subgroup of the Lorentz group, and thereby a symmetry group of Minkowski
spacetime. It is also immediate to check that the vector defined by (3.25) obeys Killing equation (3.19).
On H , t − x = 0, so that (3.25) reduces to
H H
ξ = y(∂t + ∂x ) = y ℓ.
It follows that ξ is a null normal to H as soon as y ̸= 0. We may then split H \ {y = 0} in two open
half-hyperplanes:
H1 := {p ∈ H , y(p) < 0} and H2 := {p ∈ H , y(p) > 0},
68 The concept of black hole 2: Non-expanding horizons and Killing horizons
Figure 3.5: Null half-hyperplanes H1 and H2 as Killing horizons for the Killing vector field ξ = y(∂t + ∂x ) +
(t − x)∂y generating null rotations in Minkowski spacetime. The green lines are the null geodesic generators of
H , while the thick black line (actually a 2-plane) marks the location where ξ vanishes.
each of them being a Killing horizon with respect to ξ (cf. Fig. 3.5).
Example 9 (light cone as a counter-example): The future light cone introduced in Example 2 of
Chap. 2 is not a Killing horizon of Minkowski spacetime: it is invariant under the action of the Lorentz
group, but its null generators are not invariant under the action of a single 1-dimensional subgroup of
the Lorentz group. Actually the future light cone is an example of a more general structure, which Carter
has termed a local isometry horizon [89, 92]: a null hypersurface that is invariant under some group
G of isometries (here: the Lorentz group) and such that each null geodesic generator is an orbit of some
1-dimensional subgroup of G, this subgroup being not necessarily the same from one null generator to
the next (here: using Minkowskian spherical coordinates (t, r, θ, φ), the null geodesic generator through
the point of coordinates (1, 1, θ0 , φ0 ) is the orbit of this point under the subgroup of boosts in the plane
(θ, φ) = (θ0 , φ0 )). A Killing horizon is a local isometry horizon for which dim G = 1.
Example 10 (Schwarzschild horizon): Given the expression (2.15) for the null normal ℓ of the family
of hypersurfaces Hu and the fact that the Schwarzschild horizon H is defined by r = 2m, we have
H
ℓ = ∂t . (3.26)
Now the vector field ∂t is clearly a Killing vector of metric g as given by (2.5), since none of the metric
components gαβ depends upon t. Hence (3.26) shows that the Schwarzschild horizon is a Killing horizon.
By the way, Eq. (3.26) was our motivation for the choice of the null normal ℓ performed in Example 7 of
Chap. 2.
Historical note : The concept of Killing horizon has been introduced by Brandon Carter in 1966
[88, 89] and developed in an article published in 1969 [92]. The properties of Killing horizons have been
studied in detail by Robert H. Boyer, in an article prepared posthumously from his notes by J. Ehlers and
J.L. Stachel and published in 1969 [71], leading to the concept of bifurcate Killing horizon, to be discussed
in Sec. 3.4 (cf. the historical note on p. 87).
3.3 Killing horizons 69
3.3.3 Killing horizons as non-expanding horizons
Let us consider a cross-section S of a Killing horizon H with respect to a Killing vector ξ
H
and let us select the null normal ℓ to coincide with ξ on H : ℓ = ξ. Since ξ is an isometry
generator, it is pretty obvious that the metric q induced by g on S will not evolve when Lie
dragged along ξ and hence that the deformation rate Θ of S along ξ, as defined by Eq. (2.70),
vanishes identically. One can establish this rigorously from expression (2.73) for Θ. Indeed,
without any loss of generality, we may express the null vector field ℓ in a neighborhood of H
as ℓ = ξ + uw where u is a scalar field defining H as the level set u = 0, i.e. a scalar field
obeying Eqs. (2.1)-(2.2). Then Eq. (2.73) leads to
u q µα q νβ ∇µ wν + q µα ∇µ u q νβ wν = q µα q νβ ∇µ ξν ,
Θαβ = q µα q νβ ∇µ ξν + |{z}
| {z }
0 0
where the identifications with zero hold on S , the second one because u is constant (being
equal to zero) on S . Now thanks to the Killing equation (3.19), q µα q νβ ∇µ ξν is an antisymmetric
tensor field, while Θαβ is symmetric (cf. Sec. 2.3.6). It follows that Θαβ = 0. In view of the
scaling law (2.84), we can extend this result to the deformation rate of S along any null normal
to H and therefore state:
Property 3.8: vanishing of the deformation rate of cross-sections of a Killing
horizon
On a Killing horizon H , the deformation rate Θ of any cross-section along any null normal
ℓ is identically zero:
Θ=0. (3.27)
This implies that the expansion of H along any null normal ℓ vanishes:
θ(ℓ) = 0. (3.28)
In view of the definition of a non-expanding horizon (cf. Sec. 3.2.1), an immediate corollary is
Property 3.9: Killing horizons as non-expanding horizons
Any Killing horizon with closed-manifold cross-sections is a non-expanding horizon.
Remark 1: Θ vanishes for any Killing horizon [Eq. (3.27)], while to get the same result on a generic non-
expanding horizon, one has to assume that the null convergence condition holds on H (Property 3.2).
3.3.4 Expressions of the non-affinity coefficient
Let κ be the non-affinity coefficient (cf. Sec. 2.3.3 and B.2.2) of the null normal ℓ coinciding
with the Killing vector ξ on a Killing horizon H . According to the definition (2.22), we have
H
∇ξ ξ = κ ξ. (3.29)
70 The concept of black hole 2: Non-expanding horizons and Killing horizons
H
Let us consider the metric dual of this relation, i.e. ξ µ ∇µ ξα = κ ξα , and use the Killing equation
(3.19) as ∇µ ξα = −∇α ξµ ; we get
H
ξ µ ∇α ξµ = −κ ξα .
Now ξ µ ∇α ξµ = 1/2 ∇α (ξµ ξ µ ). Hence
H
∇α (ξµ ξ µ ) = −2κ ξα . (3.30)
Since ξµ ξ µ = ξ · ξ is a scalar field, we may replace the covariant derivative by the differential:
H
d(ξ · ξ) = −2κ ξ . (3.31)
Remark 2: If u := ξ · ξ is a regular scalar field in the vicinity of H , in the sense that du ̸= 0, it is
not surprising to have du proportional to ξ on H . Indeed, if du ̸= 0, the hypersurface H can be
→
−
considered as the level set u = 0 for ξ is null on H (cf. Sec. 2.3.1). It follows that the gradient ∇u
H
must be collinear to the normal ξ to H (cf. Sec. 2.3.2) or equivalently du = α ξ. Equation (3.31) simply
shows that the proportionality factor is α = −2κ.
Another interesting relation is obtained from the Frobenius theorem applied to ξ. Indeed,
since on H , ξ is normal to a hypersurface (H ), the Frobenius theorem in its dual formulation
(see e.g. Theorem B.3.2 in Wald’s textbook [499] or Theorem C.2 in Straumann’s textbook
[464]) states that there exists a 1-form a such that
H
dξ = a ∧ ξ, (3.32)
or equivalently
H
∇α ξβ − ∇β ξα = aα ξβ − aβ ξα . (3.33)
Remark 3: In the case of the vector ℓ, which is normal to H by definition, the Frobenius identity is
H
Eq. (2.16): ∇α ℓβ − ∇β ℓα = ∇α ρ ℓβ − ∇β ρ ℓα . Since ℓ = ξ, we may write
H
∇α ℓβ − ∇β ℓα = ∇α ρ ξβ − ∇β ρ ξα .
But in general, ∇α ℓβ ̸= ∇α ξβ on H , since ℓ and ξ do not coincide outside H . Accordingly, one cannot
identify the left-hand side of the above equation with the left-hand side of Eq. (3.33), so that the 1-form
a is not ∇ρ.
Thanks to the Killing equation (3.19), we may reshape (3.33) to
H
2∇α ξβ = aα ξβ − aβ ξα . (3.34)
Contracting this relation with ξ, we get
H
2ξ µ ∇µ ξα = aµ ξ µ ξα − ξµ ξ µ aα .
|{z}
H
=0
3.3 Killing horizons 71
In view of Eq. (3.29), the left-hand side of this equation is 2κξα . Hence we obtain
H
aµ ξ µ = 2κ. (3.35)
Besides, taking the square of (3.34) leads to
H
4∇µ ξν ∇µ ξ ν = (aµ ξν − aν ξµ ) (aµ ξ ν − aν ξ µ )
H H
= aµ aµ ξν ξ ν − aµ ξ µ aν ξ ν − aν ξ ν aµ ξ µ +aν aν ξµ ξ µ = −8κ2 ,
|{z} |{z} |{z} |{z} |{z} |{z}
H 2κ 2κ 2κ 2κ H
=0 =0
where we have used Eq. (3.35). Hence
H 1
κ2 = − ∇µ ξν ∇µ ξ ν . (3.36)
2
This is an explicit expression of κ in terms of the Killing vector field ξ. However, in actual
calculations, it is generally preferable to employ formula (3.31) to evaluate κ, because it does
not involve the computation of any covariant derivative, contrary to formula (3.36).
3.3.5 The zeroth law of black hole dynamics
We are going to derive a result of great importance for black hole physics, namely the non-
affinity coefficient κ discussed above is constant on a Killing horizon, provided some mild
energy condition holds.
H
Let us denote by ℓ the null normal to H that coincides with the Killing vector field: ℓ = ξ.
The vector field ℓ is then a symmetry generator on H , which implies
Lℓ κ = 0. (3.37)
This means that κ is constant along any field line of ℓ (i.e. any null geodesic generator of H ).
It could however vary from one field line to another. To prove that this is not the case, let us
consider a complete cross-section S of H and show that κ is constant on S . The starting
point is applying the 1-form identity (2.85) to a generic vector field v tangent to S , i.e. contract
(2.85) with v α . Since ℓµ v µ = 0, we get
v ν ∇µ Θµν +v ν ℓµ ∇µ ων −v µ ∇µ θ(ℓ) + κ + θ(ℓ) + κ ωµ v µ −Θµν v µ k σ ∇σ ℓν = Rµν ℓµ v ν . (3.38)
Now, since H is a Killing horizon, we have Θ = 0 [Eq. (3.27)] and in particular θ(ℓ) = 0, so
that many terms in the above equation vanish. The term ∇µ Θµν requires some special care
→
−
though, because it potentially involves derivatives of Θ in directions transverse3 to H . Let us
then introduce in the vicinity of S a spacetime coordinate system (xα ) = (u, v, x2 , . . . , xn−1 )
3
Recall that, in our setting, tensor fields are extended beyond H by considering H as the element u = 0
of a family (Hu )u∈R of null hypersurfaces (cf. Sec. 2.3.2). We cannot assume that each Hu is a Killing horizon
→
−
(because typically the scalar ξ · ξ vanishes on a single hypersurface, not on an open subset of M ), so Θ is a priori
not zero outside H .
72 The concept of black hole 2: Non-expanding horizons and Killing horizons
adapted to S , as the one introduced in Sec. 2.3.5 [cf. Eq. (2.52)], namely a coordinate system such
→
−
that S is the set (u, v) = (0, 0). Then the components of Θ with respect to (xα ) necessarily
→
−
verify Θ0α = 0 and Θ1α = 0. Indeed, since Θ is a tensor field tangent to S , on which u
and v are constant, we have Θµα ∇µ u = 0 and Θµα ∇µ v = 0 with ∇µ u = ∂µ u = δ 0µ and
∇µ v = ∂µ v = δ 1µ by definition of (xα ). Expressing the covariant derivative via Eq. (A.64), we
get
∇µ Θµν = ∂a Θaν + Γµµσ Θσν − Γσµν Θµσ ,
where the sum of the partial derivatives has been limited to the indices a ∈ {2, . . . , n − 1}
→
−
since Θ0ν = Θ1ν = 0. Given that the term ∂a Θaν involves only the variation of Θ in directions
→
−
tangent to S , we conclude that ∇µ Θµν = 0 if Θ = 0 on S . Hence, for a Killing horizon,
Eq. (3.38) reduces to
(ℓν ∇ν ωµ + κ ωµ )v µ − v µ ∇µ κ = Rµν ℓµ v ν . (3.39)
The terms in the parentheses are related to the Lie derivative of ω along ℓ; indeed formula
(A.86) gives:
Lℓ ωµ = ℓν ∇ν ωµ + ων ∇µ ℓν = ℓν ∇ν ωµ + ων Θµν + ωµ ℓν − ℓµ k σ ∇σ ℓν
= ℓν ∇ν ωµ + κωµ − ων ℓµ k σ ∇σ ℓν ,
where we have expressed ∇µ ℓν via Eq. (2.74) and have used Θµν = 0 and ων ℓν = κ [Eq. (2.78)].
Given that ℓµ v µ = 0, Eq. (3.39) becomes
⟨Lℓ ω, v⟩ − ∇v κ = R(ℓ, v). (3.40)
It is judicious to express Lℓ ω in terms of Lℓ H ω, where H ω is the restriction of ω to tangent
vectors to H (pullback to H ) introduced in Sec. 3.2.5. The reason is that, contrary to ω, H ω is
a geometric quantity intrinsic to H and ℓ, independent of k and hence of the cross-section S
(cf. Property 3.4 and the discussion below it). It has therefore to obey the spacetime symmetry
H
generated by ξ = ℓ, i.e. one has
Lℓ H ω = 0. (3.41)
Now, using the Leibniz rule twice, we get
⟨Lℓ ω, v⟩ = Lℓ ⟨ω, v⟩ − ⟨ω, Lℓ v⟩ = ⟨Lℓ H ω, v⟩,
| {z } | {z }
⟨H ω,v⟩ ⟨H ω,Lℓ v⟩
where the idendities ⟨ω, v⟩ = ⟨H ω, v⟩ and ⟨ω, Lℓ v⟩ = ⟨H ω, Lℓ v⟩ hold by the very definition
of H ω, since v and Lℓ v are tangent4 to H . It follows then from (3.41) that ⟨Lℓ ω, v⟩ = 0;
hence the first term in Eq. (3.40) vanishes identically and there remains only:
∇v κ = −R(ℓ, v). (3.42)
4
Lℓ v is tangent to H for both ℓ and v are tangent to H .
3.3 Killing horizons 73
To go further, we shall assume the null dominance condition [421], namely that there
exists a scalar field f such that
→
−
the vector W := − G(ℓ) − f ℓ is zero or future-directed (null or timelike)
. (3.43)
for any future-directed null vector ℓ
→
−
In the above equation, G stands for the type-(1, 1) tensor associated by metric duality to the
Einstein tensor G [Eq. (A.111)], so that the expression of W in index notation is [cf. Eq. (A.50a)]
W α := −Gαµ ℓµ − f ℓα . (3.44)
Note that the null dominance condition implies the null convergence condition (2.94) since
R
R(ℓ, ℓ) = R(ℓ, ℓ) − g(ℓ, ℓ) +f g(ℓ, ℓ) = G(ℓ, ℓ) + f g(ℓ, ℓ) = −W · ℓ ≥ 0, (3.45)
2 | {z } | {z }
0 0
the inequality holding because both W and ℓ are future-directed (cf. Lemma 1.2).
If gravity is described by general relativity, i.e. if the metric g fulfills the Einstein equation
(1.41), then the null dominance condition with f = Λ is equivalent to the null dominant
energy condition:
→
−
The vector W := − T (ℓ) is zero or future-directed (null or timelike)
, (3.46)
for any future-directed null vector ℓ
GR
where T is the energy-momentum tensor of matter and non-gravitational fields. By continuity,
the null dominant energy condition is implied by the standard dominant energy condition:
→
−
The vector W := − T (u) is zero or future-directed (null or timelike)
. (3.47)
for any future-directed timelike vector u
Physically, the dominant energy condition states that, with respect to any observer (represented
by its 4-velocity u, which is future-directed timelike), the energy of matter and non-gravitational
fields does not move faster than light (see Ref. [104] for an extended discussion).
Remark 4: While the name null convergence condition for R(ℓ, ℓ) ≥ 0 [Eq. (2.94)] is standard in the
literature (e.g. [266, 452, 454]), the name null dominance condition for (3.43) is not standard. We are
using it to distinguish from the null dominant energy condition (3.46), which a condition on the matter
energy-momentum tensor, while (3.43) is a pure geometrical identity, independent of the Einstein
equation. In this way, null dominance condition is on the same footing as null convergence condition.
Coming back to Eq. (3.42), we note that its right-hand side is nothing but the scalar product
of the vector W defined by Eq. (3.43) with v:
→
− R
W · v = −( G(ℓ) + f ℓ) · v = −R(ℓ, v) + − f |{z}
ℓ · v = −R(ℓ, v).
2
0
74 The concept of black hole 2: Non-expanding horizons and Killing horizons
If we assume the null dominance condition, the null convergence condition holds, so that
R(ℓ, ℓ) = 0 on H [Eq. (3.8)]. Then, according to Eq. (3.45), ℓ · W = −R(ℓ, ℓ) = 0. This
implies that the vector W is tangent to H . The latter being a null hypersurface, W must
then be either collinear to ℓ or spacelike (cf. Lemma 2.3 in Sec. 2.3.4). Now, according to the
null dominance condition (3.43), W cannot be spacelike. We conclude that W is collinear
to ℓ. Consequently, we have W · v = 0. Hence the right-hand side of Eq. (3.42) vanishes
identically and we are left with ∇v κ = 0. Since v is a generic vector field tangent to S and
S is connected (for it is a complete cross-section of a Killing horizon, which is connected by
definition), this implies that κ is constant over S . Given that κ is constant along each null
geodesic generator of H , this completes the demonstration that κ is constant over H . More
precisely, we have established the following property:
Property 3.10: zeroth law of black hole dynamics
If the null dominance condition (3.43) is fulfilled on a Killing horizon H — which is
guaranteed in general relativity if the null dominant energy condition (3.46) holds —, then
the non-affinity coefficient κ of the null normal coinciding with the Killing vector ξ defining
H is constant over H :
κ = const. (3.48)
In the context of Killing horizons, the non-affinity coefficient κ is called the horizon’s surface
gravity, for a reason to be detailed in Sec. 3.3.7, and the result (3.48) is known as the zeroth
law of black hole dynamics. More precisely, the zeroth law — to be discussed in detail in
Chap. 16 — states that the surface gravity of a black hole in equilibrium is constant and we shall
see in Chap. 5 that (any connected part of) the event horizon of a black hole in equilibrium is a
Killing horizon.
Remark 5: The standard proof of the zeroth law, based on taking the covariant derivative of Eq. (3.29)
[40, 101, 499] or of Eq. (3.31) [275] and on an identity expressing the second order derivative of a Killing
vector in terms of the Riemann tensor [Eq. (3.83) below], is quite long (see e.g. pp. 333-334 of Wald’s
textbook [499] and the remark at the end of Sec. 5.5.2 of Poisson’s textbook [416]). The proof presented
above is shorter but it relies on the concept of induced affine connection on a non-exanding horizon
(Sec. 3.2.5) and the associated rotation 1-form H ω (cf. Property 3.4). This proof is actually adapted from
a proof presented by Damour [151, 152] (cf. historical note below); see also Ref. [22] for a related proof
regarding isolated horizons.
Remark 6: The constancy of κ on a Killing horizon can also be proved without the null dominance
condition, but at the price of additional hypotheses: either the Killing horizon H is part of a so-called
bifurcate Killing horizon, as we shall see in Sec. 3.4.3 (Property 3.16) or the spacetime is axisymmetric,
in addition to be stationary, and the two Killing vectors associated with stationarity and axisymmetry
are orthogonal to (n − 2)-dimensional surfaces [96, 422].
Example 11 (null hyperplane as a translation-Killing horizon): For the null hyperplane H
considered in Example 6 as a Killing horizon with respect to the translation group along its normal, we
have κ = 0, as already noticed in Example 8 of Chap. 2, which is obviously constant over H .
3.3 Killing horizons 75
Example 12 (null hyperplane as a boost-Killing horizon): Let us consider each of the null half-
hyperplanes H + and H − of Example 7, which are Killing horizons with respect to the boost Killing
vector ξ = x∂t + t∂x . On H + , the future-directed null normal coinciding with this Killing vector is
ℓ+ = t ℓ, ℓ being the geodesic null normal defined by ℓ := ∂t + ∂x [cf. Eq. (2.12)]. Using κℓ = 0 and
the scaling law (2.26), we get the non-affinity coefficient of ℓ+ as κ+ = ∇ℓ t = ∂t t + ∂x t, i.e.
κ+ = 1.
On H − , ξ is past-directed (cf. Fig. 3.4). Sticking to future-directed null normals, we shall then consider
H − as a Killing horizon with respect to the Killing vector field −ξ. The future-directed null normal
coinciding with −ξ on H − is then ℓ− = −t ℓ, from which we deduce the non-affinity coefficient of ℓ− :
κ− = ∇ℓ (−t) = ∂t (−t) + ∂x (−t), i.e.
κ− = −1.
We check that κ+ (resp. κ− ) is constant over the Killing horizon H + (resp. H − ), in agreement with
Property 3.10.
Example 13 (null hyperplane as a null-rotation-Killing horizon): In Example 8, we have introduced
the Killing horizons H1 and H2 with respect to the null-rotation Killing vector ξ = y(∂t +∂x )+(t−x)∂y
of Minkowski spacetime. On H1 , ξ is past-directed (cf. Fig. 3.5), so that we shall actually consider H1 as
a Killing horizon with respect to the Killing vector field −ξ. The future-directed null normal coinciding
with −ξ on H1 is then ℓ1 = −y ℓ. Since it is clearly constant along the null geodesic generators of
H1 , we have ∇ℓ1 ℓ1 = 0, hence the associated non-affinity coefficient vanishes: κ1 = 0. On H2 , ξ
is future-directed (cf. Fig. 3.5) and the null normal coinciding with it is ℓ2 = y ℓ, whose non-affinity
coefficient is κ2 = 0.
Example 14 (Schwarzschild and Kerr horizons): We have found in Example 10 of Chap. 2 [cf.
Eq. (2.29)] that on a Schwarzschild horizon κ = 1/(4m), which is clearly constant. But this last feature
is rather trivial since the Schwarzschild horizon is spherically symmetric, so that no dependence of κ on
θ nor φ could have been expected. A much less trivial example is that of the event horizon of a Kerr
black hole, which we shall discuss in Chap. 10. This horizon is only axisymmetric, so that a priori κ
could depend on θ. But it does not, as we shall see in Sec. 10.5.4:
√
m2 − a2
κ= √ ,
2m(m + m2 − a2 )
where (m, a) are the two constant parameters of the Kerr solution. Note that for a = 0, we recover the
Schwarzschild value: κ = 1/(4m).
Example 15 (Cubic Galileon black hole as a counter-example): It has been found recently that the
surface gravity of rotating stationary black holes in a scalar-tensor theory of gravity known as the cubic
Galileon is not constant [241]. This evades the zeroth law (Property 3.10) because the null dominance
condition is not satisfied by these solutions.
Historical note : The constancy of κ for a Killing horizon has been proven by Stephen Hawking
in his lecture at the famous 1972 Les Houches Summer School [261] (p. 43). It has also been proven
without requiring the null dominance condition, but assuming axisymmetry and orthogonal transitivy
(cf. Remark 6 on p. 74) by Brandon Carter in his lecture at the same summer school [96] (Theorem 8,
p. 167). A third proof of the constancy of κ, using the null dominance condition, has also been given in
76 The concept of black hole 2: Non-expanding horizons and Killing horizons
1973 by James Bardeen, Brandon Carter and Stephen Hawking in their seminal article The Four Laws of
Black Hole Mechanics [40]. Yet another proof has been provided in 1979 by Thibault Damour [151, 152],
who developed a fluid-bubble approach to the dynamics of an event horizon. In Damour’s framework,
the pullback of the 1-form −ω/(8π) to the cross-section S is considered as the momentum surface
density of the fluid bubble and Eq. (3.38) is turned into a 2-dimensional Navier-Stokes equation (see e.g.
Sec. 6.3 of Ref. [230] for details), where κ/(8π) plays the role of the bubble’s surface pressure, so that it
must constant in equilibrium. As mentioned in Remark 5, the proof presented above is derived from
Damour’s one.
3.3.6 Classification of Killing horizons
Since κ is constant on a Killing horizon H (assuming the null dominance condition), we may
use it to classify Killing horizons in two categories, depending whether κ vanishes or not:
• if κ = 0, the Killing vector ξ is a geodesic vector on H and H is called a degenerate
Killing horizon;
• if κ ̸= 0, ξ is only a pregeodesic vector on H (cf. Sec. B.2.2) and H is called a non-
degenerate Killing horizon.
Example 16 (Killing horizons in Minkowski spacetime): In Minkowski spacetime, the null hyper-
plane as a translation-Killing horizon (Example 11) and the two half-hyperplanes as null-rotation-Killing
horizons (Example 13) are degenerate Killing horizons, while the two half-hyperplanes as boost-Killing
horizons (Example 12) are non-degenerate.
Example 17 (Schwarzschild and Kerr horizons): From the values of κ given in Example 14, we see
that the Schwarzschild horizon and the Kerr horizon for a < m are non-degenerate Killing horizons,
while the Kerr horizon for a = m is a degenerate one.
The next example regards the anti-de Sitter spacetime and will play some role in the study
of the extremal (a = m) Kerr black hole in Chap. 13.
Example 18 (Poincaré horizon in AdS4 ): The 4-dimensional anti-de Sitter spacetime (AdS4 ) is
(M , g) with M ≃ R4 and g is the metric whose components in the so-called global static coordinates
(τ, r, θ, φ) are given by
dr2
2 2 2 2 2 2 2
(3.49)
g=ℓ −(1 + r ) dτ + + r dθ + sin θ dφ ,
1 + r2
where ℓ is a positive constant. Note that τ spans R, r spans (0, +∞), while (θ, φ) are standard spherical
coordinates on S2 : θ ∈ (0, π) and φ ∈ (0, 2π). The metric (3.49) is a solution to the vacuum Einstein
equation (1.43) with the negative cosmological constant Λ = −3/ℓ2 . Using the so-called conformal
coordinates (τ, χ, θ, φ) with χ := arctan r ∈ (0, π/2), one gets
ℓ2
−dτ 2 + dχ2 + sin2 χ dθ2 + sin2 θ dφ2 . (3.50)
g= 2
cos χ
3.3 Killing horizons 77
3 3
2 2
1 1
P
τ
0 0
τ
1 1
2 2
3 3
1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5
xχ xχ
Figure 3.6: Left: 2-dimensional slice (x, y) = (0, 0) of the Poincaré patch MP of anti-de Sitter spacetime plotted
in terms of the (global) coordinates τ and xχ := χ cos φ (pale yellow region). From Eq. (3.53), (x, y) = (0, 0)
implies θ = π/2 and φ = 0 or π, so that xχ = χ in the right half of the plot (φ = 0) and xχ = −χ in the left half
(φ = π). The plotted slice of MP is thus spanned by the Poincaré coordinates (t, u). The red lines are curves
of constant u, i.e. integral curves of the coordinate vector field ∂t = ξ, with u increasing from 0 to +∞ from
the right to the left of the diagram. The grey lines are curves of curves of constant t, i.e. integral curves of the
coordinate vector field ∂u , with t increasing from −∞ to +∞ from the bottom to the top of the diagram. The
Poincaré horizon H is depicted in green. The two connected components H− and H+ of H appear as straight
line segments, since for θ = π/2 and φ ∈ {0, π}, u = 0 ⇐⇒ cos τ = ± sin χ (+ for φ = 0 and − for φ = π).
Note that the curves of constant u tend to H when u → 0, in agreement with the characterization of H by
u = 0. The curves of constant t tend to H when t → ±∞, which is expected as well since the first line of
Eq. (3.53) and Eq. (3.52) imply that u → 0 for t → ±∞. Right: Killing vector field ξ defined by Eq. (3.58) on AdS4 ;
ξ is timelike everywhere, except on the Poincaré horizon H , where it is null (and normal, and thus tangent, to
H ). [Figures generated by the notebook D.3.1]
78 The concept of black hole 2: Non-expanding horizons and Killing horizons
Note that (χ, θ, φ) spans one half5 of the hypersphere S3 . Yet another set of coordinates commonly
used in AdS4 is Poincaré coordinates (t, x, y, u). They cover only a subpart MP of M (cf. Fig. 3.6),
usually called the Poincaré patch and defined by
MP : u>0 and − π < τ < π, (3.51)
where u is the following scalar field on M :
ℓ(cos τ − sin χ sin θ cos φ)
u := . (3.52)
cos χ
On MP , the Poincaré coordinates (t, x, y, u) are related to the conformal coordinates (τ, χ, θ, φ) by6
ℓ sin τ
t = cos τ −sin χ sin θ cos φ
x = ℓ sin χ sin θ sin φ
cos τ −sin χ sin θ cos φ
(3.53)
ℓ sin χ cos θ
y =
cos τ −sin χ sin θ cos φ
u = ℓ(cos τ −sin χ sin θ cos φ)
.
cos χ
The last line is simply (3.52) restricted to MP , the scalar field u being viewed there as one of the Poincaré
coordinates. The metric components with respect to Poincaré coordinates7 are (cf. the notebook D.3.1
for the computation):
u2 ℓ2
g = 2 −dt2 + dx2 + dy 2 + 2 du2 . (3.54)
ℓ u
The Poincaré horizon is the hypersurface H bounding the Poincaré patch MP in M . In view of the
definition (3.51) of the latter, H appears to be the level set u = 0:
H : u=0 and − π < τ < π, (3.55)
Note that H is not included in MP , so that the Poincaré coordinates are not defined on H (except
for u, they actually diverge in the vicinity of H ). Note also that H has two connected components:
H− , where τ ∈ (−π, 0), and H+ , where τ ∈ (0, π), since u = 0 cannot be achieved for τ = 0, as a
consequence of formula (3.52) and | sin χ| < 1 on M . The Poincaré horizon is depicted in Fig. 3.6. Its
normal is given by the gradient of u, so that the vector field defined in all M by k := ∇u
⃗ is normal to
H on H . Given expressions (3.50) and (3.52) for respectively g and u, the components k α = g αµ ∂µ u
of k with respect to conformal coordinates are
1 cos θ cos φ sin φ
k= sin τ cos χ∂τ + (cos τ sin χ − sin θ cos φ) ∂χ − ∂θ + ∂φ . (3.56)
ℓ tan χ tan χ sin θ
The scalar square of k is k · k = kµ k µ = k µ ∂µ u; we obtain
u2
k·k = . (3.57)
ℓ2
5
It would span the whole hypersphere if χ would run in all of (0, π), instead of being limited to (0, π/2).
6
See e.g. Ref. [45].
7
The name Poincaré coordinates stems from a variant of these coordinates obtained by using z := ℓ2 /u instead
of u, so that g = ℓ2 −dt2 + dx2 + dy 2 + dz 2 /z 2 , which is similar to the metric of the Poincaré half-space
model of the hyperbolic space H4 , except for the signature (−, +, +, +) instead of (+, +, +, +).
3.3 Killing horizons 79
H
Hence k · k = 0, which implies that H is a null hypersurface.
Since the metric components (3.54) do not depend on t, the vector field ξ := ∂t is a Killing vector of
(MP , g). By inverting the Jacobian matrix associated with the change of coordinates (3.53), we get the
expression of ξ = ∂t in terms of conformal coordinates:
1 − cos τ sin χ sin θ cos φ sin τ cos χ sin θ cos φ sin τ cos θ cos φ sin τ sin φ
ξ= ∂τ − ∂χ − ∂θ + ∂φ .
ℓ ℓ ℓ sin χ ℓ sin χ sin θ
(3.58)
A priori, ξ is defined on MP only, but the right-hand side of the above expression is regular on all M .
Hence, we may use (3.58) to define ξ as a vector field on all M . It is depicted in the right panel of
Fig. 3.6. By analytical continuation, it is immediate that ξ obeys the Killing equation (3.19) everywhere
and not only on MP (cf. the notebook D.3.1 for an explicit check). Hence ξ is a Killing vector of the
entire anti-de Sitter spacetime (M , g). By comparing Eqs. (3.56) and (3.58), we get
sin τ u
ξ= k + 2 (cos τ cos χ ∂τ − sin τ sin χ ∂χ ) . (3.59)
cos χ ℓ
H
Since u = 0 [Eq. (3.55)], it follows immediately that, on H , ξ is collinear to the null normal to
H
H : ξ = (sin τ / cos χ) k, with sin τ ̸= 0 (given that τ ̸= 0 on H ). Hence, ξ is normal to the null
hypersurface H . We therefore conclude that each of the connected components H− and H+ of the
Poincaré horizon H is a Killing horizon with respect to ξ. Let us evaluate the non-affinity coefficient κ
of ξ on H± via formula (3.31). First of all, we compute the scalar square of ξ by noticing that on MP ,
ξ = ∂t , so that ξ · ξ = gtt ; with gtt read on Eq. (3.54), we get
u2
ξ·ξ =− . (3.60)
ℓ2
By means of the global components (3.50) and (3.58) of respectively g and ξ, one checks that formule
(3.60) holds in all M . Given that the right-hand side is negative wherever u ̸= 0, it follows that ξ is
timelike everywhere on M , except on H . Furthermore, formula (3.60) results in d(ξ · ξ) = −2ℓ−2 u du.
H H
Since u = 0, this implies d(ξ · ξ) = 0, so that formula (3.31) yields κ = 0. We conclude that the two
connected components of the Poincaré horizon of AdS4 are degenerate Killing horizons.
3.3.7 Interpretation of κ as a “surface gravity”
In this section, we assume that H is a non-degenerate Killing horizon, i.e. that κ ̸= 0. Let
p ∈ H and v ∈ Tp M be a vector transverse to H , i.e. not tangent to H . According to
Eq. (3.31), we have
∇v (ξ · ξ) = −2κ ξ · v.
The right-hand side of this expression does not vanish, because κ ̸= 0 and ξ · v ̸= 0 (since v is
not tangent to H ). Hence we get ∇v (ξ · ξ) ̸= 0. In other words, the derivative of the scalar
square ξ · ξ along any direction transverse to H does not vanish. Since ξ · ξ = 0 on H , we
conclude that, in the vicinity of H , ξ · ξ < 0 on one side of H and ξ · ξ > 0 on the other side:
80 The concept of black hole 2: Non-expanding horizons and Killing horizons
Property 3.11
In the vicinity of a non-degenerate Killing horizon H , the Killing vector field ξ is timelike
on one side of H , null on H and spacelike on the other side.
Let us focus on the side of H where ξ is timelike. There we define the “norm” of ξ by
(3.61)
p
V := −ξ · ξ .
We have V > 0 and the square of the gradient of V provides a new expression for κ:
κ2 = lim ∇µ V ∇µ V , (3.62)
H
where limH stands for the limit as one approaches H from the timelike side, which implies
V → 0.
Proof. Let us consider the twist 3-form ω defined by
ω := ξ ∧ dξ (3.63)
or, using index notation,
ωαβγ := ξα (dξ)βγ + ξβ (dξ)γα + ξγ (dξ)αβ
= ξα (∇β ξγ − ∇γ ξβ ) + ξβ (∇γ ξα − ∇α ξγ ) + ξγ (∇α ξβ − ∇β ξα ) . (3.64)
Killing equation (3.19) enables us to simplify each term inside parentheses in (3.64), yielding
ωαβγ = 2 (ξα ∇β ξγ + ξβ ∇γ ξα + ξγ ∇α ξβ ) . (3.65)
The “square” of ω is then
ωµνρ ω µνρ = 4 ξµ ∇ν ξρ ξ µ ∇ν ξ ρ + ξµ ∇ν ξρ ξ ν ∇ρ ξ µ + ξµ ∇ν ξρ ξ ρ ∇µ ξ ν
+ξν ∇ρ ξµ ξ µ ∇ν ξ ρ + ξν ∇ρ ξµ ξ ν ∇ρ ξ µ + ξν ∇ρ ξµ ξ ρ ∇µ ξ ν
µ ν ρ ν ρ µ ρ µ ν
+ξρ ∇µ ξν ξ ∇ ξ + ξρ ∇µ ξν ξ ∇ ξ + ξρ ∇µ ξν ξ ∇ ξ .
Now in the first line,
ξµ ∇ν ξρ ξ µ ∇ν ξ ρ = ξµ ξ µ ∇ν ξρ ∇ν ξ ρ = −V 2 ∇ν ξρ ∇ν ξ ρ = −V 2 ∇µ ξν ∇µ ξ ν (3.66)
and (using Killing equation (3.19))
1
ξµ ∇ν ξρ ξ ν ∇ρ ξ µ = ξµ ∇ρ ξ µ ξ ν ∇ν ξρ = −ξµ ∇ρ ξ µ ξ ν ∇ρ ξν = − ∇ρ V 2 ∇ρ V 2 = −V 2 ∇ρ V ∇ρ V.
4
(3.67)
3.3 Killing horizons 81
Actually, we notice that each line is made of one term of type (3.66) and two terms of type
(3.67). Hence
ωµνρ ω µνρ = −12V 2 (∇µ ξν ∇µ ξ ν + 2∇µ V ∇µ V ) . (3.68)
On H , each of the terms inside parentheses in Eq. (3.64) can be expressed thanks to the
Frobenius identity (3.33):
H
ωαβγ = ξα (aβ ξγ − aγ ξβ ) + ξβ (aγ ξα − aα ξγ ) + ξγ (aα ξβ − aβ ξα ) .
We notice that all terms in the right-hand side canceal two by two, yielding
H
ωαβγ = 0. (3.69)
Equation (3.69) is actually nothing but a variant of Frobenius theorem, expressing the fact that
the vector field ξ is hypersurface-orthogonal on H (see e.g. Eq. (B.3.6) in Wald’s textbook
[499], taking into account that ωαβγ = 6ξ[α ∇β ξγ] ). Let us evaluate the gradient of the square
(3.68) and take the limit on H :
µνρ µνρ 2 µ ν µ
∇α ωµνρ ω + ω µνρ ∇ α ω = −12 ∇ α V ∇ ξ ∇ ξ +2∇ V ∇ V
|{z} |{z} | {z } | µ ν{z } µ
→0 →0 →2κξα →−2κ2
V 2 ∇α (∇µ ξν ∇µ ξ ν + 2∇µ V ∇µ V ) ,
−12 |{z}
→0
H
where we have used Eq. (3.31) in the form ∇α V 2 = 2κξα , as well as expression (3.36) of κ2 .
Hence we are left with
κ ∇µ V ∇µ V − κ2 ξα −→ 0 on H .
Now, by the very definition of a Killing horizon, ξα ̸= 0 on H . Moreover, H being a non-
degenerate Killing horizon, we have κ ̸= 0 as well. The above limit is then equivalent to
(3.62).
In the region where ξ is timelike, the vector field
1
u := ξ (3.70)
V
is a future-directed unit timelike vector field. It is future-directed because by convention8 ξ is
future-directed null on H and by continuity this orientation must be preserved in the region
where ξ is timelike. The unit vector field u can be then considered as the 4-velocity of an
observer O, whose worldline of is a field line of ξ, i.e. an orbit of the isometry group generated
by ξ. One may call O a stationary observer since the spacetime geometry is not changing
along its worldline. The 4-acceleration of O is
a := ∇u u
= ∇V −1 ξ V −1 ξ = V −1 ∇ξ V −1 ξ = V −1 −V −2 (∇ξ V ) ξ + V −1 ∇ξ ξ .
8
Were ξ past-directed, we could always consider the Killing field −ξ instead.
82 The concept of black hole 2: Non-expanding horizons and Killing horizons
Now, since ξ is a symmetry generator, ∇ξ V = 0. This can be shown explicitly by means of
Killing equation (3.19):
p 1 1 µ ν
∇ξ V = ξ µ ∇µ ( −ξν ξ ν ) = − √ ξ µ
∇µ (ξν ξ ν
) = − ξ ξ ∇µ ξν = 0.
2 −ξν ξ ν V | {z }
0
We have thus
1
∇ξ ξ.
a= (3.71)
V2
Thanks to Killing equation (3.19), we may rewrite this relation as
1 µ 1 1 1
aα = 2
ξ ∇µ ξα = − 2 ξ µ ∇α ξµ = − 2 ∇α (ξµ ξ µ ) = ∇α V 2 = ∇α ln V,
V V 2V 2V 2
hence
→
−
a = ∇ ln V. (3.72)
The norm of a, which is always a spacelike vector (since u · u = −1 implies u · a = 0), is
√ 1p
a := a·a= ∇µ V ∇µ V . (3.73)
V
Given the result (3.62), we get an expression of κ involving a:
κ = lim V a , (3.74)
O→H
where O → H means that the limit is achieved by choosing the worldline of observer O
arbitrarily close to H . Since V → 0 as one approaches H , it follows that
lim a = +∞. (3.75)
O→H
This means that the acceleration felt by observer O (the “gravity”) diverges as O approaches
H . In that sense, the physical surface gravity of H is infinite. But Eq. (3.74) shows that the
rescaled acceleration V a remains finite as one approaches H , and tends to κ. It is this quantity
that is named the surface gravity of the Killing horizon H .
Remark 7: As stressed above, the surface gravity κ is not the actual gravity a measured locally, i.e. by an
observer at rest with respect to H and infinitely close to it. However, κ can be interpreted as a physical
force (per unit mass) measured by a distant observer, at least in the special case of a Schwarzschild
black hole, for which ξ is timelike in the entire region outside the Killing horizon9 . In this case, one can
identify κ with the magnitude of the force exerted by an observer “at infinity” to hold in place a particle
of unit mass close to H by means of an infinitely long massless string (see e.g. Sec. 5.2.4 of Poisson’s
textbook [416]).
9
This is not true for a rotating Kerr black hole: ξ becomes null at some “light-cylinder” outside H and is then
spacelike away from it, cf. Eq. (10.68), where ξ is denoted by χ.
3.4 Bifurcate Killing horizons 83
Figure 3.7: Bifurcate Killing horizon H1 ∪ H2 with respect to the Killing vector field ξ; S is the bifurcation
surface. L1 and L2 are null geodesic generators of respectively H1 and H2 , which cross each other at the point
p ∈ S.
3.4 Bifurcate Killing horizons
We extend here the study of Killing horizons by introducing the concept of bifurcate Killing
horizon, which is particularly important for black hole physics.
3.4.1 Definition and first properties
Let (M , g) be a n-dimensional spacetime endowed with a Killing vector field ξ. A bifurcate
Killing horizon is the union
H = H1 ∪ H2 , (3.76)
such that
• H1 and H2 are two null hypersurfaces;
• S := H1 ∩ H2 is a spacelike (n − 2)-surface;
• each of the sets H1 \ S and H2 \ S has two connected components, which are
Killing horizons with respect to ξ.
The (n − 2)-dimensional submanifold S is called the bifurcation surface of H .
Hence we may say that a bifurcate Killing horizon is formed by four Killing horizons, H1+ ,
H1− , H2+ and H2− say, which are glued together at the bifurcation surface S (cf. Fig. 3.7), in
such a way that
H1 = H1− ∪ S ∪ H1+ and H2 = H2− ∪ S ∪ H2+
84 The concept of black hole 2: Non-expanding horizons and Killing horizons
are null hypersurfaces.
A first property of bifurcate Killing horizons is
Property 3.12: vanishing of the Killing vector on the bifurcation surface
The Killing vector field vanishes on the bifurcation surface of a bifurcate Killing horizon:
ξ|S = 0 . (3.77)
Proof. Let p ∈ S and let us assume that ξ|p ̸= 0. Let L1 (resp. L2 ) be the null geodesic
generator of H1 (resp. H2 ) that intersects S at p (cf. Fig. 3.7). By definition of a Killing horizon,
ξ is tangent to L1 ∩ H1+ and to L1 ∩ H1− , i.e. to L1 \ {p}. If ξ|p ̸= 0, then by continuity, ξ
is a (non-vanishing) tangent vector field all along L1 . Similarly, ξ is tangent to all L2 . At their
intersection point p, the geodesics L1 and L2 have thus a common tangent vector, namely ξ|p .
The geodesic uniqueness theorem (Property B.10 in Appendix B) then implies L1 = L2 , so
that L1 ⊂ H1 ∩ H2 = S . But since S is spacelike and L1 is null, we reach a contradiction.
Hence we must have ξ|p = 0.
Remark 1: Having a Killing vector field that vanishes somewhere (here S ) is not the sign of any
pathology: it simply means that the points of S are fixed points of the isometries generated by ξ, since
setting ξ = 0 in Eq. (3.14) leads to dx = 0, i.e. to Φdt (p) = p.
Remark 2: Contrary to what the name may suggest, a bifurcate Killing horizon is not a Killing horizon,
for the latter, as defined in Sec. 3.3.2, is a regular (i.e. embedded) hypersurface of M (cf. Sec. A.2.7 in
Appendix A), while the union of two hypersurfaces is not in general a hypersurface. Moreover on a
Killing horizon, the Killing vector field is nowhere vanishing [cf. Eq. (3.21)], while on a bifurcate Killing
horizon, it is vanishing at the bifurcation surface.
Example 19 (Lorentz-boost bifurcate Killing horizon): Let us consider Minkowski spacetime and
the Killing vector given by Eq. (3.23): ξ := x∂t + t∂x , namely the generator of Lorentz boosts in the
(t, x) plane. Let us take for H1 the null hyperplane t = x considered in Example 7 and denoted there by
H . The two half-hyperplanes defined by Eq. (3.24) are then the Killing horizons H1+ and H1− . The
union H1 ∪ H2 , where H2 is the null hyperplane t = −x is a bifurcate Killing horizon with respect
to ξ, with the 2-plane (t, x) = (0, 0) as bifurcation surface (cf. Fig. 3.8). Note that on H1 , the Killing
vector ξ points away from S , while on H2 , it points towards S , as in the generic figure 3.7.
3.4.2 Non-degenerate Killing horizons and Boyer’s theorem
Let us consider a Killing horizon H with respect to some Killing vector ξ, such that the surface
gravity κ is constant over H . According to the zeroth law (Property 3.10), this is guaranteed if
the null dominance condition is fulfilled. In what follows, we focus on the case where κ ̸= 0,
i.e. H is a non-degenerate Killing horizon (cf. Sec. 3.3.6). In this case, the parameter t of the
null geodesic generators of H associated to ξ is not affine (for κ is the non-affinity coefficient
of ξ). However, we may rescale ξ to get an affine parameter, at the price of loosing the Killing
vector feature:
3.4 Bifurcate Killing horizons 85
Figure 3.8: Bifurcate Killing horizon H1 ∪ H2 with respect to the Killing vector ξ = x∂t + t∂x generating
Lorentz boosts in the plane (t, x) of Minkowski spacetime. The dimension along z having been suppressed, the
bifurcation surface S appears as a line, while it is actually a 2-plane.
Property 3.13: affine parametrization of a non-degenerate Killing horizon
Let H be a Killing horizon (with respect to a Killing vector ξ) of constant nonzero surface
gravity κ. Let t be the parameter of the null geodesic generators L of H associated to ξ
(ξ = dx/dt along L ). The null vector field ℓ defined on H by
ℓ = e−κt ξ ⇐⇒ ξ = eκt ℓ (3.78)
is a geodesic vector field and the affine parameter associated to it is
eκt
λ= + λ0 , (3.79)
κ
where λ0 is constant along a given geodesic generator.
Proof. We have
∇ℓ ℓ = ∇e−κt ξ e−κt ξ = e−κt ∇ξ e−κt ξ = e−κt ∇ξ e−κt ξ + e−κt ∇ξ ξ = 0.
| {z } | {z }
de−κt /dt κξ
Hence ℓ is a geodesic vector. Besides, along any null generator of H , one has [cf. Eq. (A.8)]
dλ
= ξ(λ) = eκt ℓ(λ) = eκt ,
dt |{z}
1
which, once integrated, yields Eq. (3.79).
86 The concept of black hole 2: Non-expanding horizons and Killing horizons
Let us assume κ > 0 and consider a null geodesic generator L of H ; L can be param-
eterized by t, the corresponding tangent vector being ξ. When t spans the whole interval
(−∞, +∞), Eq. (3.79) implies that λ spans the interval (λ0 , +∞) only. Since λ is an affine pa-
rameter of L , this means that L is an incomplete geodesic (cf. Sec. B.3.2). Moreover, Eq. (3.78)
leads to
ξ → 0 when t → −∞ (κ > 0). (3.80)
In other words, the Killing vector field ξ vanishes and the null geodesic L stops at the “edge”
of H corresponding to t → −∞. If there is no obstacle (spacetime singularity or spacetime
edge10 ), L can be extended to λ ∈ (−∞, λ0 ], giving rise to a complete null geodesic L˜. This
operation can be performed for all the null geodesic generators of H and we have the freedom
to choose the same value of λ0 in Eq. (3.79) for all of them. In this process, one gives birth to
a null hypersurface, H˜ say, which contains H . Let S ⊂ H˜ be the set of points of affine
parameter λ = λ0 along all the extended null geodesics L˜. S is clearly a cross-section of
H˜ (cf. Sec. 2.3.4); it is then a spacelike (n − 2)-dimensional surface. Assuming that ξ is
future-directed on H , S constitutes the past boundary of H , i.e. the boundary corresponding
to t → −∞. Since ξ is a smooth vector field on M , Eq. (3.80) implies that ξ vanishes on S . In
other words, S is a set of fixed points for the isometry group generated by ξ (cf. Remark 1
above). Let us denote by H − the subset of H˜ generated by the segments λ < λ0 of the null
geodesics L˜: H − = H˜ \ (H ∪ S ). H − is clearly a null hypersurface. Since S is spacelike
and (n − 2)-dimensional, there are, at each point p ∈ S , only two null directions normal to
S (cf. Sec. 2.3.4). One of them is along ℓ. The set of all null geodesics departing from S along
the other null direction forms a null hypersurface, H2+ say, in the future of S and another
null hypersurface, H2− say, in the past of S . By studying the behavior of a Killing vector field
around the set of its fixed points (here S ), Boyer [71] has shown that in the current setting
(i.e. S spacelike), ξ acts locally as the generator of Lorentz boosts in Minkowski spacetime
and S is the bifurcation surface of a bifurcate Killing horizon similar to that of Example 19 (cf.
Fig. 3.8). More precisely, Boyer proved the following:
Property 3.14: Boyer’s theorem (Boyer 1969 [71])
A Killing horizon H with respect to a Killing vector ξ is contained in a bifurcate Killing
horizon if, and only if, H contains at least one null geodesic orbit of the isometry group
that is complete as an orbit, i.e. such that the group parameter t associated to ξ takes all
values in R, but that is incomplete and extendable as a geodesic.
That H contains an orbit that is an incomplete geodesic is guaranteed by κ ̸= 0, as
we have just seen. It follows that H − , H2+ and H2− are three Killing horizons, so that
H ∪ H − ∪ H2+ ∪ H2− ∪ S is a bifurcate Killing horizon.
If κ < 0, we see from Eq. (3.79) that while t spans the whole interval (−∞, +∞), the affine
parameter λ spans the interval (−∞, λ0 ) only. Moreover, Eq. (3.78) leads to
ξ → 0 when t → +∞ (κ < 0). (3.81)
10
A spacetime edge is generally not a genuine obstacle for extending an incomplete geodesic: it simply indicates
that the spacetime itself needs to be extended so that the geodesic becomes complete, cf. Remark 3 below.
3.4 Bifurcate Killing horizons 87
The reasoning developed for κ > 0 can be applied mutatis mutandis, leading to a bifurcate
Killing horizon with a bifurcation surface S that is the future boundary of H . Hence we
conclude:
Property 3.15: non-degenerate Killing horizons and bifurcate Killing horizons
The null geodesic generators of a non-degenerate Killing horizon H are incomplete; if
they can be extended, H is contained in a bifurcate Killing horizon, the bifurcation surface
of which is the past (resp. future) boundary of H if κ > 0 (resp. κ < 0).
Remark 3: It could be that the generators of H cannot be extended simply because the spacetime
(M , g) is not been chosen “large enough”. I. Rácz and R.M. Wald [422] have shown that under some
mild hypotheses (in particular that (M , g) is globally hyperbolic), one can extend (M , g) to a larger
spacetime with a bifurcate Killing horizon containing H .
Remark 4: For a degenerate Killing horizon, the problem of extension disappears, since t is then an
affine parameter of the null generators. Consequently if t spans the whole interval (−∞, ∞), the null
generators are complete geodesics. One can still have ξ → 0 at some boundary of H , but this is a
null boundary, not a spacelike one, and it does not correspond to a bifurcation surface. An example is
the Killing horizon with respect to a null-rotation Killing vector in Minkowski spacetime, exhibited as
Examples 8 and 13, p. 67 and 75 respectively (cf. Fig. 3.5): ξ = 0 on the null 2-plane of equation t = x,
y = 0.
Historical note : The concept of bifurcate Killing horizons has been introduced by Robert H. Boyer
(1932-1966), a young American mathematical physicist who had just been appointed to the University of
Liverpool. Sadly, Boyer was killed, among 14 victims, in a mass murder that occurred in the University
of Texas at Austin on 1 August 1966. His last notes, containing the definition of a bifurcate Killing
horizon and the proof of Property 3.14, have been turned into an article by Jürgen Ehlers and John
Stachel, which has been published in 1969 [71].
3.4.3 Zeroth law for bifurcate Killing horizons
For a Killing horizon that is part of a bifurcate Killing horizon, one can establish the constancy
of the surface gravity κ (the “zeroth law”) by means of the vanishing of the Killing vector ξ at
the bifurcation surface, without requiring the null dominance condition as for a generic Killing
horizon in Property 3.10:
Property 3.16: zeroth law for bifurcate Killing horizons
The surface gravity is a nonzero constant over any Killing horizon that is part of a bifurcate
Killing horizon:
κ = const ̸= 0. (3.82)
The proof makes use of the following lemma:
88 The concept of black hole 2: Non-expanding horizons and Killing horizons
Lemma 3.17: second derivative of a Killing vector
For any Killing vector ξ, the following identity, often called the Kostant formula, holds:
∇γ ∇β ξ α = Rαβγµ ξ µ , (3.83)
where Rαβγµ is the Riemann curvature tensor of g. The contraction on the indices α and γ
lets the Ricci tensor appear:
∇µ ∇α ξ µ = Rαµ ξ µ . (3.84)
Proof. Consider the covariant version of the Ricci identity (A.98): ∇α ∇β ξγ − ∇β ∇α ξγ =
Rγµαβ ξ µ and call it equation (1). Perform two successive cyclic permutations of the indices
(α, β, γ), leading to equations (2) and (3), say. Then consider the linear combination (1) −
(2) − (3). The left-hand side of it reduces to 2∇γ ∇β ξα thanks to the Killing equation (3.19),
while the right-hand side is (Rγµαβ − Rαµβγ − Rβµγα )ξ µ . Now, using the antisymmetries
(A.100) and (A.103) of the Riemann tensor and then the cyclic symmetry (A.101), one has
−Rαµβγ − Rβµγα = −Rµαγβ − Rµβαγ = Rµγβα . This turns the right-hand side into (Rγµαβ +
Rµγβα )ξ µ = 2Rγµαβ ξ µ = 2Rαβγµ ξ µ , the last equality resulting from (A.104). This establishes
Eq. (3.83). Finally, let us note that Eq. (3.84) can be proved directly, without appealing to the
cyclic property of the Riemann tensor: it suffices to use the the contracted Ricci identity (A.107):
∇µ ∇α ξ µ − ∇α ∇µ ξ µ = Rµα ξ µ = Rαµ ξ µ and the fact that ∇µ ξ µ = 0 as a consequence of Killing
equation (3.19).
Proof of Property 3.16. Let H be a Killing horizon that is part of a bifurcate Killing horizon Hˆ
with respect to a Killing vector ξ. Let v be a generic non-vanishing vector field tangent to H .
Let us evaluate the derivative of κ along v from expression (3.36):
1
2κ∇v κ = − v ρ ∇ρ (∇µ ξν ∇µ ξ ν ) = −v ρ ∇ρ ∇µ ξν ∇µ ξ ν .
2
Using Eq. (3.83) and the antisymmetry property (A.103) to set −Rνµρσ = Rµνρσ , we get
2κ∇v κ = v ρ Rµνρσ ξ σ ∇µ ξ ν . (3.85)
H
If the vector v is chosen to be the null normal ℓ = ξ, one gets immediately 2κ∇ℓ κ = 0, given
that Rµνρσ ξ ρ ξ σ = 0 [Eq. (A.100)]. Then κ = 0 or ∇ℓ κ = 0. In both cases, this means that
κ must be constant along a given null generator L of H . We thus recover the result (3.37).
There remains to prove that κ does not vary from one generator to the other. For this, we
shall use that H belongs to the bifurcate Killing horizon Hˆ . By virtue of Boyer’s theorem
(Property 3.14), one has κ ̸= 0, for if κ would be zero along L , the latter would be a complete
geodesic and H could not be part of a bifurcate Killing horizon. The bifurcation surface S of
Hˆ is part of the topological closure of H (cf. Fig. 3.7, with e.g. H1+ standing for H ), so it is
meaningfull to consider the limit of Eq. (3.85) to S for a vector field v on H that tends to a
S
nonzero vector field tangent to S . Given that ξ = 0 and ∇ξ remains finite at S for ξ is a
S
smooth vector field on M , the limit yields (taking into account κ ̸= 0) ∇v κ = 0. It follows that
3.5 Summary 89
κ (or more precisely the limit to S of H ’s surface gravity κ) is uniform over S . Each null
generator L of H is an incomplete geodesic, which once extended to a complete geodesic,
intersects S at single point. The constancy of κ on S implies then that κ cannot vary from
one null generator to the other. κ is thus uniform over H .
Example 20 (Lorentz-boost bifurcate Killing horizon): Let us consider the bifurcate Killing horizon
Hˆ of Minkowski spacetime discussed in Example 19 and illustrated in Fig. 3.8. The corresponding Killing
vector is the Lorentz-boost generator ξ := x∂t + t∂x . An easy computation yields ∇ξ ξ = t∂t + x∂x .
On the components H1+ and H1− of Hˆ (cf. Fig. 3.8), one has t = x, so that ξ = t(∂t + ∂x ) and
∇ξ ξ = t(∂t + ∂x ). Hence ∇ξ ξ = ξ and κ = 1 there. On the components H2+ and H2− of Hˆ (cf.
Fig. 3.8), one has t = −x, so that ξ = t(−∂t + ∂x ) and ∇ξ ξ = t(∂t − ∂x ). Hence ∇ξ ξ = −ξ and
κ = −1 there. We conclude that κ is constant over any of the four Killing horizons constituting Hˆ .
Remark 5: As the above example shows, the surface gravity is not uniform over all the bifurcate Killing
horizon Hˆ ; it keeps the same value only on the subsets H − ∪ H + and H − ∪ H + .
1 1 2 2
Historical note : The proof of the zeroth law for Killing horizons that are part of a bifurcate Killing
horizon (Property 3.16) has been first presented by Bernard Kay and Robert Wald in 1991 [310]; these
authors mentioned that the proof had been suggested to them by Robert Geroch. The proof appears also
in the notes from a lecture given by Wald at a school held in Erice in May 1991 [501]. It is also presented
in Poisson’s textbook [416].
3.5 Summary
Here is an inheritance diagram summarizing the main results of this chapter. The vertical
arrow means “is a”, i.e. the item at the bottom of an arrow is a special case of the item at the
top of the arrow. “NCC” stands for “Null Convergence Condition” [Eq. (2.94)], “NDC” for “Null
Dominance Condition” [Eq. (3.43)] and “BKH“ for “Bifurcate Killing Horizon” (Sec. 3.4.1).
90 The concept of black hole 2: Non-expanding horizons and Killing horizons
Null hypersurface
null geodesic generators
∇ℓ ℓ = κℓ
Non-expanding horizon
closed-manifold cross-sections
θ(ℓ) = 0
area independent of the cross-section
NCC =⇒ Θ = 0
=⇒ induced affine connection
Killing horizon
with closed-manifold cross-sections
Θ=0
NDC =⇒ κ = const (Zeroth Law 1)
part of BKH =⇒ κ = const ̸= 0 (Zeroth Law 2)
Chapter 4
The concept of black hole 3: The global
view
Contents
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Conformal completion of Minkowski spacetime . . . . . . . . . . . . 92
4.3 Conformal completions and asymptotic flatness . . . . . . . . . . . . 100
4.4 Black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.1 Introduction
Having attempted in Chaps. 2 and 3 to characterize a black hole by the local properties of
its boundary, we turn now to the general definition of a black hole. As it could have been
anticipated from the naive “definition” given in Sec. 2.2.1, the mathematically meaningful
definition of a black hole cannot be local: it has to take into account the full spacetime structure,
in particular its future asymptotics. Indeed, to conclude firmly that a particle has escaped from
a given region, one has to wait until the “end of time” to make sure that the particle will never
be back...
In this chapter, we therefore consider the global spacetime picture to arrive at the general
definition of a black hole in Sec. 4.4. This amounts to focusing on the spacetime asymptotics,
which can be seen as the region where the “distant observers” live and may, or may not, receive
light signal from some “central region”. This far-away structure is best described in terms of the
so-called conformal completion, which brings the spacetime infinity(ies) to a finite distance in
another manifold — a technique initiated by R. Penrose [403, 404] (see Refs. [198] and [382] for
a review). We start by investigating the conformal completion of the simplest of all spacetimes:
Minkowski spacetime.
92 The concept of black hole 3: The global view
t
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
r
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Figure 4.1: Lines of constant null coordinates u (solid) and v (dashed) in terms of the coordinates (t, r).
4.2 Conformal completion of Minkowski spacetime
In this section (M , g) is the 4-dimensional Minkowski spacetime, i.e. M is a smooth manifold
diffeomorphic to R4 and g is the metric tensor whose expression in terms of some global
coordinates (t, x, y, z) implementing the diffeomorphism to R4 (i.e. Minkowskian coordinates)
is
g = −dt2 + dx2 + dy 2 + dz 2 . (4.1)
4.2.1 Finite-range coordinates on Minkowski spacetime
Since we would like to deal with the “far” region, it is natural to introduce r := x2 + y 2 + z 2
p
and the associated spherical coordinates (t, r, θ, φ), which are related to the Minkowskian ones
by
x = r sin θ cos φ
y = r sin θ sin φ (4.2)
z = r cos θ.
The coordinates (t, r, θ, φ) span R × (0, +∞) × (0, π) × (0, 2π); they do not cover the whole
manifold M as a regular chart (cf. Sec. A.2.1 of Appendix A), but only M \ Π, where Π is the
closed half hyperplane defined by y = 0 and x ≥ 0. Once expressed in terms of the spherical
coordinates, the Minkowski metric (4.1) takes the form
g = −dt2 + dr2 + r2 dθ2 + sin2 θ dφ2 . (4.3)
4.2 Conformal completion of Minkowski spacetime 93
U
1.5
1.0
0.5
u
6 4 2 0.5 2 4 6
1.0
1.5
Figure 4.2: The arctangent function mapping R to (−π/2, π/2).
Let us introduce the null coordinate system (u, v, θ, φ) where u and v are respectively the
retarded and advanced time defined by (cf. Fig. 4.1)
u=t−r t = 1 (v + u)
2
⇐⇒ (4.4)
v =t+r r = 1 (v − u).
2
One has then du dv = dt2 − dr2 and the metric tensor (4.3) takes the shape
1
g = −du dv + (v − u)2 dθ2 + sin2 θ dφ2 . (4.5)
4
The coordinates (u, v) span the half part of R2 defined by u < v. In order to have coordinates
within a finite range, let us consider their arctangents (cf. Fig. 4.2):
U = arctan u u = tan U
⇐⇒ (4.6)
V = arctan v v = tan V.
Given that arctan is a monotonically increasing function (cf. Fig. 4.2), the coordinates (U, V )
span the half part of (−π/2, π/2) × (−π/2, π/2) defined by U < V :
π π π π
− <U < , − <V < , and U < V. (4.7)
2 2 2 2
Since
dU dV sin(V − U )
du = , dv = and tan V − tan U = ,
cos2 U cos2 V cos U cos V
the Minkowski metric (4.5) is expressed in terms of the coordinates (U, V, θ, φ) as1
1 2 2 2 2
(4.8)
g= −4 dU dV + sin (V − U ) dθ + sin θ dφ .
4 cos2 U cos2 V
1
See also the SageMath notebook D.2.1.
94 The concept of black hole 3: The global view
Remark 1: The retarded/advanced times u and v have the dimension of a time, or of a length in the
c = 1 units used here. Therefore, one should introduce some length scale, ℓ0 say, before taking their
arctangent and rewrite (4.6) as
U = arctan(u/ℓ ) u = ℓ tan U
0 0
⇐⇒
V = arctan(v/ℓ0 ) v = ℓ0 tan V.
The coordinates (U, V ) are dimensionless and a global factor ℓ20 should be introduced in the right-hand
side of Eq. (4.8). However, the length scale ℓ0 plays no essential role, so that, to keep simple notations, it
is omitted in what follows. In other words, we are using units for which ℓ0 = 1.
4.2.2 Conformal metric
In the right-hand side of (4.8), the terms in square brackets defines a metric g̃ such that
g̃ = Ω2 g , (4.9)
where Ω is the scalar field M → R+ obeying
Ω = 2 cos U cos V (4.10a)
2
=√ √ (4.10b)
u2 + 1 v 2 + 1
2
=p p . (4.10c)
(t − r)2 + 1 (t + r)2 + 1
We notice on (4.10b) and (4.10c) that the function Ω never vanishes on M , so that the bilinear
form g̃ defined by (4.9) constitutes a well-behaved metric on M . Moreover, since Ω2 > 0, g̃
has the same signature as g, i.e. (−, +, +, +). The expression of g̃ is deduced from (4.8) and
(4.10a):
g̃ = −4 dU dV + sin2 (V − U ) dθ2 + sin2 θ dφ2 . (4.11)
In view of (4.9), one says that the metric g̃ is conformal to the metric g, or equivalently,
that the metrics g and g̃ are conformally related, or that g̃ arises from g via a conformal
transformation. The scalar field Ω is called the conformal factor.
A key property of a conformal transformation is to preserve orthogonality relations, since
(4.9) clearly implies, at any point p ∈ M ,
∀(u, v) ∈ Tp M × Tp M , g̃(u, v) = 0 ⇐⇒ g(u, v) = 0.
In particular, null vectors for g̃ coincide with null vectors for g:
∀ℓ ∈ Tp M , g̃(ℓ, ℓ) = 0 ⇐⇒ g(ℓ, ℓ) = 0.
Consequently the light cones of (M , g) and (M , g̃) are identical, which implies that (M , g)
and (M , g̃) have the same causal structure. Moreover, since Ω2 > 0, the spacelike and timelike
characters of vectors is preserved as well:
∀v ∈ Tp M , v spacelike for g̃ ⇐⇒ v spacelike for g
(4.12)
v timelike for g̃ ⇐⇒ v timelike for g.
4.2 Conformal completion of Minkowski spacetime 95
It follows that a curve L is timelike (resp. null, spacelike) for g̃ iff L is timelike (resp. null,
spacelike) for g. Similarly, a hypersurface Σ is timelike (resp. null, spacelike) for g̃ iff Σ is
timelike (resp. null, spacelike) for g.
What about geodesics? Let us first recall that a null curve is not necessarily a null geodesic
(cf. Remark 2 on p. 35 and Appendix B), so that one cannot deduce from the above results that
conformal transformations preserve null geodesics. However, this turns out to be true:
Property 4.1: null geodesics preserved by conformal transformations
Let g and g̃ be two Lorentzian metrics on a manifold M that are conformally related:
g̃ = Ω2 g. A smooth curve L in M is a null geodesic for g̃ iff L is a null geodesic for g;
any affine parameter λ̃ of L as a g̃-geodesic is then related to any affine parameter λ of
L as a g-geodesic by
dλ̃
= a Ω2 , (4.13)
dλ
where a is a constant.
Sketch of proof. Write explicitly the geodesic equation [Eq. (B.10)] and express the Christoffel
symbols of g̃ in terms of those of g and the derivatives of Ω (see e.g. Appendix D of Wald’s
textbook [499] for details).
On the contrary, conformal transformations preserve neither timelike geodesics nor space-
like ones.
The coordinates (U, V ) are of null type; let us consider instead the “time+space” coordinates
(τ, χ) defined by2
τ =V +U U = 1 (τ − χ)
2
⇐⇒ (4.14)
χ=V −U V = 1 (τ + χ).
2
Given (4.7), the range of these new coordinates is
0<χ<π and χ − π < τ < π − χ. (4.15)
In other words, if we draw the Minkowski spacetime in the (τ, χ) plane, it takes the shape of a
half-diamond, as depicted in Fig. 4.3.
By combining (4.4) (4.6) and (4.14), we get the link between (t, r) and (τ, χ):
sin τ
t = cos τ + cos χ
τ = arctan(t + r) + arctan(t − r)
⇐⇒ (4.16)
χ = arctan(t + r) − arctan(t − r) sin χ
r=
.
cos τ + cos χ
We may use these relations to draw the lines t = const and r = const in Fig. 4.3.
The expression of the conformal factor in the coordinates (τ, χ, θ, φ) is easily deduced from
(4.10a) and (4.14):
Ω = cos τ + cos χ. (4.17)
2
Notice the similarity with (4.4) up to some 1/2 factors.
96 The concept of black hole 3: The global view
τ
i+
3
2 +
i0 χ
0.5 1.0 1.5 2.0 2.5 3.0
−
2
3
i−
Figure 4.3: Conformal diagram of Minkowski spacetime. Constant-r curves are drawn in red, while constant-t
ones are drawn in grey. [Figure generated by the notebook D.2.1]
4.2.3 Conformal completion
The expression of the conformal metric in terms of the coordinates (τ, χ, θ, φ) is easily deduced
from that in terms of (U, V, θ, φ) as given by (4.11):
g̃ = −dτ 2 + dχ2 + sin2 χ dθ2 + sin2 θ dφ2 . (4.18)
Restricting to a τ = const hypersurface, i.e. setting dτ = 0, we recognize the standard metric
of the hypersphere S3 in the hyperspherical coordinates (χ, θ, φ). Moreover, we notice that
the full metric (4.18) is perfectly regular even if we relax the condition on τ in (4.15), i.e. if we
let τ span the entire R. We may then consider the manifold
E = R × S3 (4.19)
and g̃ as the Lorentzian metric on E given by (4.18). The Lorentzian manifold (E , g̃) is nothing
but the Einstein static universe, also called the Einstein cylinder, a static solution to
the Einstein equation (1.40) with Λ > 0 and some pressureless matter of uniform density
ρ = Λ/(4π). We have thus an embedding3 of Minkowski spacetime into the Einstein cylinder
(cf. Fig. 4.4):
Φ : M −→ E (4.20)
3
Cf. Sec. A.2.7 of Appendix A
4.2 Conformal completion of Minkowski spacetime 97
Figure 4.4: Two views of the Einstein cylinder E , with the conformal embedding of Minkowski spacetime M
in it. Each view represents a 2-dimensional cut of E at θ = π/2 and φ = 0 (one half of the plotted cylinder) or
φ = π (the other half). As a standard minor abuse, we authorize here φ = 0 as a valid coordinate value, while we
had excluded it from the definition of the chart (t, r, θ, φ) on M given in Sec. 4.2.1. The S3 sections of E are then
depicted as horizontal circles, which are made of two half-circles corresponding to φ = 0 and φ = π respectively,
with χ running from 0 to π on both on them. The plotted piece of M is then the 2-dimensional cut (y, z) = (0, 0)
of M , which is a timelike plane spanned by the coordinates (t, x), with x ≥ 0 on the φ = 0 half cylinder and
x ≤ 0 on the φ = π half one. The red curves are the same constant-r curves as in Fig. 4.3, while the black curves
are the same constant-t curves as those drawn in grey in Fig. 4.3. [Figure generated by the notebook D.2.1]
and this embedding is a conformal isometry from (M , g) to (Φ(M ), g̃). In the following, we
shall identify Φ(M ) and M , i.e. use the same symbol M to denote the subset of E that is the
image of M via the embedding (4.20).
Since E and M have the same dimension, M is an open subset of E . Its (topological)
closure M in E is (cf. Figs. 4.3 and 4.4)
M = M ∪ I + ∪ I − ∪ i0 ∪ i+ ∪ i− , (4.21)
where
• I + is the hypersurface of E defined by τ = π − χ and 0 < τ < π ⇐⇒ V = π/2 and
−π/2 < U < π/2;
• I − is the hypersurface of E defined by τ = χ − π and −π < τ < 0 ⇐⇒ U = −π/2
and −π/2 < V < π/2;
• i0 is the point of E defined by τ = 0 and χ = π ⇐⇒ U = −π/2 and V = π/2;
• i+ is the point of E defined by τ = π and χ = 0 ⇐⇒ U = π/2 and V = π/2;
• i− is the point of E defined by τ = −π and χ = 0 ⇐⇒ U = −π/2 and V = −π/2.
98 The concept of black hole 3: The global view
Figure 4.5: Conformal factor Ω as a function of (τ, χ) [cf. Eq. (4.17)]. Only the part above the yellow horizontal
plane (Ω = 0) is physical. [Figure generated by the notebook D.2.1]
Note that since the coordinates (τ, χ, θ, φ) are singular at χ = 0 and χ = π, each of the
above conditions (τ, χ) = (0, π), (τ, χ) = (π, 0) and (τ, χ) = (−π, 0) do define a point and
not a 2-surface of E , in the same way as θ = 0 defines a point (the North pole) on S2 , while
θ = π/4 defines a circle. It is customary to pronounce I as “scri”, for script i. Note that in
the above definitions, we have extended the coordinates (U, V ) to E by the transformations
(4.14); (U, V, θ, φ) can be then considered as a chart on E with (U, V ) spanning the infinite
strip 0 < V − U < π of R2 and the metric g̃ on E being given by expression (4.11).
Remark 2: As we shall detail below (Sec. 4.3.1), it is precisely because Ω vanishes at the topological
boundary of M in E (cf. Fig. 4.5),
M \ M = I + ∪ I − ∪ i0 ∪ i+ ∪ i− , (4.22)
that the conformal transformation (4.9) brings the infinity of Minkowski spacetime to a finite distance.
Remark 3: On S3 , the hyperspherical coordinates (χ, θ, φ) are singular at χ = 0 and χ = π, so that
setting χ = 0 (or χ = π) defines a unique point of S3 , whatever the value of (θ, φ). Note also that the
vertical left boundary of the diamond drawn in Fig. 4.3, i.e. the segment defined by τ ∈ (−π, π) and
χ = 0, is not a part of the boundary of M but merely reflect the coordinate singularity at χ = 0, in the
same way that the left vertical boundary of Fig. 4.1 is not a boundary of Minkowski spacetime but is
due to the coordinate singularity at r = 0. Note by the way that χ = 0 implies r = 0 via (4.16).
Let
I := I + ∪ I − (4.23)
and
M˜ := M ∪ I . (4.24)
M˜ is naturally a smooth manifold with boundary4 and its boundary is I :
∂ M˜ = I . (4.25)
4
Cf. Sec. A.2.2 for the precise definition.
4.2 Conformal completion of Minkowski spacetime 99
τ
i+
3
2 +
i0 χ
0.5 1.0 1.5 2.0 2.5 3.0
−
2
3
i−
Figure 4.6: Null radial geodesics in the conformal diagram of Minkowski spacetime. The solid green lines are
null geodesics u = const for 17 values of u uniformly spanning [−8, 8], while the dashed green lines are null
geodesics v = const for 17 values of v uniformly spanning [−8, 8]. [Figure generated by the notebook D.2.1]
Remark 4: Because the closure M is self-intersecting at the point i0 (cf. Fig. 4.4), it is not a manifold
with boundary: no open neighborhood of i0 is homeomorphic to a neighborhood of H4 = R3 × [0, +∞),
as the definition of a manifold with boundary would require, cf. Sec. A.2.2. At the points i+ and i− , M
can be considered as a topological manifold with boundary, but not as a smooth manifold with boundary.
Hence, the three points i0 , i+ and i− are excluded from the definition of the manifold with boundary
M˜.
The hypersurface I + is the location of M˜ where all radial null geodesics terminate, while
I − is the location of M˜ where all these geodesics originate (cf. Fig. 4.6). For this reason I + is
called the future null infinity of (M , g) and I − the past null infinity of (M , g). On the
other side, any timelike geodesic of (M , g) originates at i− and ends at i+ (cf. Fig. 4.3), while
any spacelike geodesic of (M , g) originates at i0 and terminates there (after having completed
a closed path on S3 , cf. Fig. 4.4). The point i+ is then called the future timelike infinity of
(M , g), i− the past timelike infinity of (M , g) and i0 the spacelike infinity of (M , g).
Property 4.2
I + and I − are null hypersurfaces of (M˜, g̃).
Proof. Since I + is defined by V = π/2, a valid coordinate system on I + is (U, θ, φ) with U
100 The concept of black hole 3: The global view
spanning (−π/2, π/2). The metric induced by g̃ on I + is easily obtained by setting V = π/2
in Eq. (4.11):
g̃|I + = cos2 U dθ2 + sin2 θ dφ2 . (4.26)
It appears clearly that the signature of this metric is (0, +, +), i.e. it is degenerate; hence I + is
a null hypersurface of (M˜, g̃). Similarly, I − being defined by U = −π/2, a valid coordinate
system on I − is (V, θ, φ) with V spanning (−π/2, π/2) and the metric induced by g̃ on I −
is obtained by setting U = −π/2 in Eq. (4.11):
g̃|I − = cos2 V dθ2 + sin2 θ dφ2 . (4.27)
Again, it is clearly degenerate, so that I − is null hypersurface of (M˜, g̃).
The null character of I + and I − appears also clearly in the conformal diagrams of Figs. 4.3
and 4.6, since I + and I − are straight lines of slope ±1 in these diagrams.
Historical note : The idea of using a conformal transformation to treat infinity as a boundary “at
a finite distance” has been put forward by Roger Penrose in 1963 [403] and expanded in 1964 in the
seminal paper [404], where Penrose constructed the conformal completion of Minkowski spacetime as a
part of the Einstein cylinder. In particular, Fig. 3 of Ref. [404] is equivalent to Fig. 4.4.
4.3 Conformal completions and asymptotic flatness
Having investigated the asymptotic structure of Minkowski spacetime via a conformal comple-
tion, let us use the latter to define spacetimes that “look like” Minkowski spacetime asymptoti-
cally. A first step is the concept of conformal completion.
4.3.1 Conformal completion
A spacetime (M , g) admits a conformal completion at infinity iff there exists a Lorentzian
manifold with boundary (M˜, g̃) (cf. Sec. A.2.2 for the definition) equipped with a smooth
non-negative scalar field Ω : M˜ → R+ such that
1. M˜ = M ∪ I , with I := ∂ M˜ (the manifold boundarya of M˜);
2. on M , g̃ = Ω2 g;
3. on I , Ω = 0;
4. on I , dΩ ̸= 0.
I is called the conformal boundary of (M , g) within the conformal completion (M˜, g̃).
a
As stressed in Remark 5 in Sec. A.2.2, the set ∂ M˜ is the boundary of M˜ as a manifold with boundary; it
is not the boundary of M˜ as a topological space, the latter being ∅.
4.3 Conformal completions and asymptotic flatness 101
Condition 1 expresses that M has been endowed with some boundary. A rigorous formulation
of it would be via an embedding Φ : M → M˜, as in Eq. (4.20), so that M˜ = Φ(M ) ∪ I .
However, as above, we identify Φ(M ) with M and therefore simply write M˜ = M ∪ I .
Conditions 2 and 3 express that the boundary of M , which “lies at an infinite distance” with
respect to g, has been brought to a finite distance with respect to g̃. Indeed, in terms of length
elements [cf. Eq. (1.3)], condition 2 implies
1
ds2 = 2
ds̃2
Ω
with 1/Ω2 → +∞ as one approaches I (condition 3). Finally, condition 4 ensures that I is a
regular hypersurface of M˜. It is of course fulfilled by Minkowski spacetime, as we can check
graphically on Fig. 4.5: the graph of Ω has no horizontal slope at I .
Remark 1: The statement that (M˜, g̃) is a Lorentzian manifold with boundary implies that g̃ is smooth
everywhere on M˜, including at the boundary I .
Remark 2: Since g̃ is a metric, it is by definition non-degenerate and condition 2 implies that Ω cannot
vanish on M . Being non-negative, we have necessarily Ω > 0 on M .
Remark 3: The conformal boundary I is not part of the physical spacetime M , but only of the
conformal completion M˜.
Remark 4: One often speaks about conformal compactification instead of conformal completion, but in
general M˜ is not a compact manifold. For instance, the completion M˜ of Minkowski spacetime defined
by Eq. (4.24) is not compact, because the points i+ , i− and i0 have been omitted in the construction of
M˜.
Example 1 (conformal completion of AdS4 spacetime): The 4-dimensional anti-de Sitter spacetime
(M , g) has been introduced in Example 18 of Chap. 3. The metric tensor expressed in the conformal
coordinates (τ, χ, θ, φ) is given by Eq. (3.50):
ℓ2
−dτ 2 + dχ2 + sin2 χ dθ2 + sin2 θ dφ2 , (4.28)
g= 2
cos χ
where τ ∈ R, χ ∈ (0, π/2), θ ∈ (0, π), φ ∈ (0, 2π) and the constant length scale ℓ is related to the
negative cosmological constant Λ by ℓ2 = −3/Λ. Defining Ω := ℓ−1 cos χ, we notice that a conformal
completion of (M , g) is (M˜, g̃) where (i) M˜ is the part χ ≤ π/2 of the Einstein cylinder5 introduced
in Sec. 4.2.3 and (ii) g̃ is the metric (4.18). The boundary I = ∂ M˜ is then the hypersurface χ = π/2 of
the Einstein cylinder (cf. Fig. 4.7); I is spanned by the coordinates (τ, θ, φ) and its topology is that of a
3-dimensional cylinder: I ≃ R × S2 . We notice that conditions 3 and 4 of the definition of a conformal
completion are satisfied: Ω = ℓ−1 cos χ = 0 at I and dΩ = −ℓ−1 sin χ dχ = −ℓ−1 dχ ̸= 0 at I . The
metric induced by g̃ on I is obtained by setting χ = π/2 in (4.18): −dτ 2 + dθ2 + sin2 θ dφ2 . This
3-metric is clearly Lorentzian, which shows that I is a timelike hypersurface of (M˜, g̃).
The above example shows that I is not necessarily a null hypersurface, as it is for
Minkowski spacetime (cf. Sec. 4.2.3). Actually the causal type of I is determined by the
cosmological constant, as follows:
5
Recall that on the Einstein cylinder the range of χ is (0, π), cf. Eq. (4.15).
102 The concept of black hole 3: The global view
Figure 4.7: Conformal completion of AdS4 spacetime, depicted on the Einstein cylinder. The conformal
boundary I is shown in yellow, red lines are lines χ = const (uniformly sampled in terms of tan χ = sinh ρ),
green curves are radial null geodesics and the purple curve is a radial timelike geodesic, bouncing back and forth
around χ = 0. [Figure generated by the notebook D.3.2]
Property 4.3: causal type of I and sign of the cosmological constant
If the spacetime dimension obeysa n ≥ 3 and g is a solution to the Einstein equation with
a cosmological constant Λ [Eq. (1.40)] and the trace T of the energy-momentum tensor
tends to zero in the vicinity of I (i.e. when Ω → 0), then
• I is a null hypersurface of (M˜, g̃) iff Λ = 0;
• I is a spacelike hypersurface of (M˜, g̃) iff Λ > 0;
• I is a timelike hypersurface of (M˜, g̃) iff Λ < 0.
a
Cf. Remark 1 in Sec. 1.5.
Proof. It follows from g̃ = Ω2 g that the Ricci scalars R̃ and R of respectively g̃ and g are
related by6
2 µν ˜ ˜ µν
Ω R̃ = R − (n − 1) 2Ω g̃ ∇µ ∇ν Ω − n g̃ ∂µ Ω∂ν Ω , (4.29)
where n = dim M and ∇
˜ stands for the Levi-Civita connection of g̃. Using the trace of the
6
This relation is easily established by starting from Eq. (2.30) of Hawking & Ellis’ textbook [266] or Eq. (2.19)
on p. 645 of Choquet-Bruhat’s one [108] and inverting the roles of g̃ and g, thereby substituting Ω−1 for Ω.
4.3 Conformal completions and asymptotic flatness 103
Einstein equation (1.42) to express R, we get
2
˜ µ∇
˜ ν Ω − n g̃ µν ∂µ Ω∂ν Ω
Ω2 R̃ = (nΛ − 8πT ) − (n − 1) 2Ω g̃ µν ∇
n−2
This equation is a priori valid in M = M˜ \ I only. Taking the limit Ω → 0 and assuming
that T → 0 in that limit, we get, by continuity, an identity on I :
I 2
g̃ µν ∂µ Ω∂ν Ω = − Λ. (4.30)
(n − 1)(n − 2)
Since I corresponds to a constant value of the scalar field Ω (Ω = 0), the left-hand side of
this equation is nothing but the scalar square g̃(n, n) of the vector n normal to I defined as
the dual with respect to g̃ of the 1-form dΩ: nα = g̃ αµ ∂µ Ω (remember that by hypothesis 4
in the definition of a conformal completion, dΩ is non-vanishing on I , so that n is a valid
normal vector to I ). Equation (4.30) implies that the sign of g̃(n, n) is the opposite of that of
Λ. Given the link between the causal type of a hypersurface and the causal type of its normal
(cf. Sec. 2.2.2), this completes the proof.
One may distinguish two subparts of the conformal boundary:
Let (M , g) be a time-orientablea spacetime admitting a conformal completion at infinity
(M˜, g̃), with conformal boundary I . One defines the future infinity of (M , g) as the
subset I + of I whose points can be reached from a point in M by a future-directed
causal curve in M˜. Similarly the past infinity of (M , g) is the subset I − of I whose
points can be reached from a point in M by a past-directed causal curve in M˜. If I +
(resp. I − ) is a null hypersurface, it is called the future null infinity (resp. past null
infinity) of (M , g). Furthermore if I = I + ∪ I − , one says that (M˜, g̃) is a conformal
completion at null infinity of (M , g).
a
Cf. Sec. 1.2.2.
Remark 5: The above definitions of I + and I − generalize those given for the Minkowski spacetime
in Sec. 4.2.3. Note that, contrary to the Minkowski case, I + and I − are not null for spacetimes
with a non-zero cosmological constant (cf. Property 4.3). In particular, the following examples exhibit
respectively timelike and spacelike I + and I − .
Example 2 (Future and past infinities of AdS4 spacetime): Let us consider the conformal completion
of AdS4 discussed in Example 1. It is evident from the behavior of radial null geodesics (the green curves
plotted in Fig. 4.7) that any point of I can be connected to M by a future-directed null geodesic as
well as by a past-directed one. It follows that I + = I − = I and both I + and I − are timelike
hypersurfaces.
Example 3 (Conformal completion of dS4 spacetime): The 4-dimensional de Sitter spacetime
is (M , g) with M ≃ R × S3 and g is the metric whose expression in the so-called global coordinates
(t, χ, θ, φ) is
g = ℓ2 −dt2 + cosh2 t dχ2 + sin2 χ dθ2 + sin2 θ dφ2 (4.31)
,
104 The concept of black hole 3: The global view
where ℓ is a positive constant. Note that t spans R while (χ, θ, φ) are standard polar coordinates on
S3 : χ ∈ (0, π), θ ∈ (0, π) and φ ∈ (0, 2π). The metric (4.31) is a solution of the vacuum Einstein
equation (1.43) with the positive cosmological constant Λ = 3/ℓ2 . Using coordinates (τ, χ, θ, φ) with
τ := 2 arctan(tanh(t/2)) ∈ (−π/2, π/2), one gets
ℓ2
−dτ 2 + dχ2 + sin2 χ dθ2 + sin2 θ dφ2 . (4.32)
g= 2
cos τ
Defining Ω := ℓ−1 cos τ = (ℓ cosh t)−1 , we notice that a conformal completion of (M , g) is (M˜, g̃)
where (i) M˜ is the part −π/2 ≤ τ ≤ π/2 of the Einstein cylinder introduced in Sec. 4.2.3 and (ii)
g̃ is the metric (4.18). The boundary I = ∂ M˜ has two connected components: I + , which is the
hypersurface τ = π/2 of M˜, and I − , which is the hypersurface τ = −π/2. Both I + and I −
are spanned by the coordinates (χ, θ, φ) and their topology is that of S3 . We notice that conditions
3 and 4 of the definition of a conformal completion are satisfied: Ω = ℓ−1 cos τ = 0 at I and
dΩ = −ℓ−1 sin τ dτ = ±ℓ−1 dτ ̸= 0 at I . The metric induced by g̃ on I is obtained by setting
τ = ±π/2 in (4.18): dχ + sin χ dθ + sin θ dφ . This 3-metric is clearly Riemannian (this is
2 2 2 2 2
actually the standard round metric of S3 ), which shows that I is a spacelike hypersurface of (M˜, g̃).
This of course agrees with Property 4.3, given that Λ > 0. Finally, it is clear that I + (resp. I − ) matches
the definition of a future (resp. past) infinity given above. We conclude that (M˜, g̃) is a conformal
completion at infinity of de Sitter spacetime.
4.3.2 Asymptotic flatness
Penrose [404, 407] has defined a spacetime (M , g) to be asymptotically simple iff there exists
a conformal completion at infinity (M˜, g̃) of (M , g) such that every null geodesic in M has
two endpoints in I .
The last condition, which is verified by Minkowski spacetime (cf. Fig. 4.6), de Sitter spacetime
and anti-de Sitter spacetime (cf. the null geodesics in Fig. 4.7), is rather restrictive. In particular,
it excludes black hole spacetimes, since, almost by definition, the latter contain null geodesics
that have no endpoint on I + , having only a past endpoint on I − , as far as I is concerned.
To cope with these spacetimes, Penrose has also introduced the following definition [407]: a
spacetime (M , g) is weakly asymptotically simple iff there exists an open subset U of M
and an asymptotically simple spacetime (M0 , g0 ) with an open neighborhood U0 of I0 = ∂ M˜0
in M˜0 such that (U0 ∩ M0 , g0 ) is isometric to (U , g).
Remark 6: For a given weakly asymptotically simple spacetime, there may be different (non overlapping)
regions U satisfying the above property. For instance we shall see in Chap. 10 that there are an infinite
series of them in the (maximally extended) Kerr spacetime.
Finally one says that a spacetime (M , g) is asymptotically flat (or more precisely weakly
asymptotically simple and empty [266]) iff (M , g) is weakly asymptotically simple and the
Ricci tensor of g vanishes in an open neighborhood of I : R = 0.
Example 4: The de Sitter and anti-de Sitter spacetimes are asymptotically simple but are not asymptoti-
cally flat.
Penrose [406] (see also [198]) has shown that if (M , g) is asymptotically simple and empty,
the Weyl tensor of g (cf. Sec. A.5.4) vanishes at I . Since the Ricci tensor is zero, this implies
4.4 Black holes 105
that the full Riemann curvature tensor vanishes at I [cf. Eq. (A.114)], hence the qualifier
asymptotically flat.
The following property holds:
Property 4.4: null I for asymptotically flat spacetimes
The conformal boundary I of an asymptotically flat spacetime (M , g) is a null hypersur-
face of the conformal completion (M˜, g̃).
Proof. Consider Eq. (4.29). Near I , we have R = 0 by the very definition of asymptotic flatness.
I
The limit Ω → 0 results then in g̃ µν ∂µ Ω∂ν Ω = 0, which, following the argument in the proof
on p. 102, implies that I is a null hypersurface.
4.4 Black holes
4.4.1 Preliminaries regarding causal structure
Before we proceed to the precise definition of a black hole, let us introduce some concepts
regarding the causal structure of a given time-orientable spacetime (M , g). For any subset S
of M , one defines
• the chronological future of S as the set I + (S) of all points of M that can be reached
from a point of S by a future-directed timelike curve of nonzero extent;
• the causal future of S as the set J + (S) of all points that either are in S or can be
reached from a point of S by a future-directed causal curve;
• the chronological past of S as the set I − (S) of all points of M that can be reached
from a point of S by a past-directed timelike curve of nonzero extent;
• the causal past of S as the set J − (S) of all points that either are in S or can be reached
from a point of S by a past-directed causal curve.
From the above definitions, one has always S ⊂ J ± (S) and I ± (S) ⊂ J ± (S).
Remark 1: One has not necessarily S ⊂ I ± (S). For instance, if M does not contain any closed timelike
curve, one has S ∩ I ± (S) = ∅ for S = {p} with p being any point of M .
Here are some topological properties of the future and past sets defined above (see e.g. § 6.2
of [266] or Chap. 14 of [390] for proofs):
• I ± (S) is always an open subset7 of M , while J ± (S) is not necessarily a closed subset.
• The interior of J ± (S) is I ± (S):
int J ± (S) = I ± (S). (4.33)
7
This property is a direct consequence of Lemma 4.6 in Sec. 4.4.3 below.
106 The concept of black hole 3: The global view
• Both sets have the same closure:
J ± (S) = I ± (S). (4.34)
• It follows from (4.33) and (4.34) that both sets share the same (topological) boundary:
∂J ± (S) = ∂I ± (S). (4.35)
The subset of the causal future (resp. past) of S formed by points that cannot be connected to
S by a timelike curve is called the future horismos (resp. past horismos) of S and is denoted
by E + (S) (resp. E − (S)):
E + (S) := J + (S) \ I + (S) and E − (S) := J − (S) \ I − (S). (4.36)
The horismos E ± (S) is formed by null geodesics emanating from points in S (cf. Proposi-
tion 4.5.10 of Ref. [266]). One has E ± (S) ⊂ ∂J ± (S). The spacetime (M , g) is said to be
causally simple iff for every compact set K ⊂ M , E ± (K) = ∂J ± (K). This is equivalent to
saying that J + (K) and J − (K) are closed subsets of M .
4.4.2 General definition of a black hole
We are now in position to give the general definition of a black hole. We shall do it for a
spacetime (M , g) that admits a conformal completion at infinity as defined in Sec. 4.3.1 and
such that the future infinity I + is complete: if I + is a null hypersurface, which occurs if
(M , g) is asymptotically flat (cf. Propery 4.4), this means that all the generators of I + are
complete null geodesics8 . The completeness condition is imposed to avoid “spurious” black
holes, such as black holes in Minkowski space (cf. Remark 2 below). The neighborhood of I +
in M˜ can then be considered as the infinitely far region reached by outgoing null geodesics. If
a null geodesic does not reach this region, it can be considered as being trapped somewhere
else in spacetime: this “somewhere else” constitutes the black hole region. More precisely:
Let (M , g) be a time-oriented spacetime with a conformal completion at infinity such that
the future infinity I + is complete; the black hole region is the set of points of M that do
not belong to the causal past of I + (cf. Fig. 4.8):
B := M \ (J − (I + ) ∩ M ) . (4.37)
If B ̸= ∅, one says that the spacetime (M , g) contains a black hole or that (M , g) is a
black hole spacetime.
The black hole region is thus the set of points of M from which no future-directed causal curve
in M˜ reaches I + .
8
Let us recall that a geodesic is complete iff its affine parameters range through the whole of R, cf. Sec. B.3.2 in
Appendix B. In particular, such a geodesic is inextendible.
4.4 Black holes 107
Figure 4.8: The black hole region B defined as the complement of the causal past of the future infinity,
J − (I + ).
Example 5: The Minkowski spacetime contains no black hole, for all future-directed null geodesics
terminate at I + (cf. Fig. 4.6). More generally, any asymptotically simple spacetime contains no black
hole (cf. Sec. 4.3.2).
Example 6: The prototype of a black hole is the Schwarzschild black hole; it will be shown in Sec. 6.4
that the Schwarzschild spacetime contains a region B that fulfills the above definition of a black hole
region.
Remark 2: If we release the assumption of I + -completeness in the above definition, we may end
up with unphysical or “spurious” black holes. For instance, let us consider the conformal completion
of Minkowski spacetime (M , g) resulting from its embedding in the Einstein cylinder (E , g̃), as in
Sec. 4.2.3, keeping the same I − but defining I + as the hypersurface of E given by τ = π − χ and
0 < τ < π/2, instead of 0 < τ < π in Sec. 4.2.3. The manifold with boundary M˜ := M ∪ I + ∪ I − ,
equipped with the Einstein cylinder metric g̃, is then a conformal completion of (M , g) at null infinity.
With such a I + , the black hole region defined by (4.37) is non-empty, as shown in Fig. 4.9.
Remark 3: Some authors (in particular Hawking and Ellis [266]) define a black hole as a connected
component of Σ(τ ) ∩ B, where Σ(τ ) is a spacelike hypersurface that is a slice of the future development
of a partial Cauchy surface9 Σ(0) such that the closure in M˜ of the domain of dependence of Σ(0)
contains I + . According to such a definition, a black hole is a (n − 1)-dimensional object, while the
black hole region B defined above is a n-dimensional object.
If B ̸= ∅, the boundary H of the black hole region is called the future event horizon (or
simply the event horizon when no ambiguity may arise):
H := ∂B . (4.38)
By plugging expression (4.37) for B in the standard identity ∂B = B ∩ M \ B, we get an
equivalent expression for H :
H = M \ (J − (I + ) ∩ M ) ∩ (J − (I + ) ∩ M ) = ∂(J − (I + ) ∩ M ).
9
The concepts of partial Cauchy surface and future development are defined in Sec. 10.8.3.
108 The concept of black hole 3: The global view
Figure 4.9: Spurious black hole region B in Minkowski spacetime resulting from a conformal completion with
an incomplete I + . Compare with Fig. 4.6.
Now, the boundary of J − (I + ) in M˜ is ∂J − (I + ) = ∂(J − (I + ) ∩ M ) ∪ I + , so that
∂J − (I + ) ∩ M = ∂(J − (I + ) ∩ M ); hence
H = ∂J − (I + ) ∩ M . (4.39)
In words: the future event horizon H is the part of the boundary of the causal past of the
future infinity I + that lies in M (cf. Fig. 4.8). Note that thanks to property (4.35), we can
write as well
H = ∂I − (I + ) ∩ M . (4.40)
Example 7: The event horizon of the Schwarzschild black hole (Example 6) is nothing but the
Schwarzschild horizon H considered in the examples of Chaps. 2 and 3.
White hole
By inverting past and future in the black hole definition (4.37), one defines the white hole
region of a spacetime (M , g) with a conformal completion at infinity as the complement
within M of the causal future of the past infinity I − :
W := M \ (J + (I − ) ∩ M ) . (4.41)
The white hole region is thus the set of points of M from which no past-directed causal curve
in M˜ reaches I − . The boundary of white hole region is called the past event horizon:
H − := ∂W = ∂J + (I − ) ∩ M . (4.42)
Remark 4: The name white fountain is sometimes used instead of white hole. Actually, this name
may seem better suited to describe the time symmetric of a black hole: one can fall into a hole, while
one is expelled by a fountain.
4.4 Black holes 109
Example 8: We shall encounter an example of white hole in the maximal extension of Schwarzschild
spacetime, to be discussed in Chap. 9 (cf. Sec. 9.4.4).
The domain of outer communications is the part ⟨⟨M ⟩⟩ of M that lies neither in the
black hole region nor in the white hole one:
⟨⟨M ⟩⟩ := M \ (B ∪ W ) = J − (I + ) ∩ J + (I − ) ∩ M . (4.43)
The last equality, which is a direct consequence of the definitions of B and W , shows that the
domain of outer communications is the set of points from which it is possible to send a signal
to and to receive a signal from arbitrarily far regions. It also follows immediately from the
definitions of the two event horizons that the boundary of the domain of outer communications
is their union:
∂⟨⟨M ⟩⟩ = H ∪ H − . (4.44)
Historical note : The term event horizon has been introduced by Wolfgang Rindler in 1956 [435] in the
context of a single observer moving in some cosmological spacetime. Regarding the name black hole, the
standard story tells that it has been coined by John A. Wheeler in the end of 1967, following a suggestion
shouted from the audience during one of his conferences (cf. the account by Wheeler himself in Chap. 13
of Ref. [520]). However, a recent study [272] reveals that the expression black hole circulated as early
as 1963 at the first Texas Symposium on Relativistic Astrophysics held in Dallas, while discussing the
discovery of quasars, and could have been forged by Robert Dicke in some lecture given in 1961. In
any case, after have been put forward by Wheeler, the term black hole rapidly superseded the previous
names frozen star, collapsed star, or astre occlus (the latter still appearing along black holes in the title of
the proceedings of the famous 1972 Les Houches Summer School [164]). For instance, “black hole” was
used abundantly, albeit with quotes, in a review article by Roger Penrose published in 1969 [408]. In
the same article, Penrose defined the absolute event horizon as ∂J − (I + ) (actually ∂I − (I + ), but both
coincide, cf. Eq. (4.35)), i.e. essentially the identity (4.39), but he did not provide any formal definition of
the black hole region as M \ (J − (I + ) ∩ M ) [Eq. (4.37)]. It seems that the latter appears first in an
“instant-of-time” version (cf. Remark 3) in a seminal article by Stephen Hawking published in 1972 [260].
The expression domain of outer communications has been introduced in 1971 by Brandon Carter [93].
4.4.3 Properties of the future event horizon
Having defined a black hole region in full generality, let us derive the main properties of its
boundary — the future event horizon H .
Property 4.5: the event horizon as an achronal set
A black hole event horizon H is an achronal set, i.e. no pair of points of H can be
connected by a timelike curve of M .
Note that in the definition of an achronal set, it is not demanded that the timelike curve
lies entirely in the set (for instance, the set can be discrete, so that no curve whatsoever lies in
it). Accordingly, an equivalent statement of Property 4.5 is: no timelike curve of M meets H
at more than one point. The proof of Property 4.5 relies on the following lemma:
110 The concept of black hole 3: The global view
Figure 4.10: Lemma 4.6: moving slightly the ends p and q of a timelike curve L necessarily results in another
timelike curve L ′ .
Figure 4.11: Proving that H is achronal.
Lemma 4.6: stability of timelike curves with respect to their ends
One can “move the ends” of any timelike curve “a little bit” and still get a timelike curve.
More precisely, if two points p, q ∈ M are connected by a timelike curve, there exists
a neighborhood U of p and a neighborhood V of q such that any point p′ ∈ U can be
connected to any point q ′ ∈ V by a timelike curve.
Proof. This is more or less evident on a spacetime diagram (cf. Fig. 4.10) and a formal proof
can be found as Lemma 3 in Chap. 14 of O’Neill’s textbook [390].
Proof of Property 4.5. Let us assume the negation of Property 4.5, i.e. that there exist two points
p and q in H that are connected by a timelike curve L , with q lying in the future of p (cf.
4.4 Black holes 111
Figure 4.12: Proving that, for any p ∈ H , I − (p) ⊂ M \ B (left) and I + (p) ⊂ B (right).
Fig. 4.11). Invoking Lemma 4.6, let U and V be the neighborhoods of respectively p and q
within which one can deform L to a timelike curve. Let us choose p′ ∈ U ∩ B (B being the
black hole region) and q ′ ∈ V ∩ J − (I + ). Such a choice is always possible since p and q lie on
the boundary between B and J − (I + ) (cf. Fig. 4.11). Since q ′ ∈ J − (I + ), the timelike curve
linking p′ and q ′ can then be extended to the future in a causal curve L ′ reaching I + . This
implies p′ ∈ J − (I + ), which contradicts p′ ∈ B.
Property 4.7: the event horizon as a manifold of codimension 1
H is a topological manifold of dimension n − 1, n being the spacetime dimension.
Proof. Let us first show that the chronological past of any point p ∈ H , I − (p) (cf. Sec. 4.4.1),
lies entirely in M \ B (the black hole exterior). For any q ∈ I − (p), we have by definition
p ∈ I + (q) and since I + (q) is an open set (cf. Sec. 4.4.1), this means that I + (q) is an open
neighborhood of p (cf. Fig. 4.12 left). Given that p lies on the boundary of B, we have necessarily
I + (q) ∩ (M \ B) ̸= ∅. Let then a ∈ I + (q) ∩ (M \ B) = I + (q) ∩ J − (I + ). By definition,
there exists a future-directed timelike curve from q to a and a future-directed causal curve from
a to I + , hence a future-directed causal curve from q to I + , which proves that q ∈ J − (I + ),
i.e. q ∈ M \ B. We conclude that I − (p) ⊂ M \ B. On the other hand, let us show that the
chronological future of p, I + (p), lies entirely in B. For any q ∈ I + (p), we have p ∈ I − (q)
(cf. Fig. 4.12 right). Since I − (q) is open, it constitutes an open neighborhood of p. Given that
p ∈ ∂B, we have then I − (q) ∩ B ̸= ∅ and we may pick a point a ∈ I − (q) ∩ B. There exists a
future-directed timelike curve from a to q. If q ̸∈ B, there would exist a future-directed causal
curve from q to I + and hence a future-directed causal curve from a to I + , which would
contradict a ∈ B. We conclude that q ∈ B and thus that I + (p) ⊂ B.
For p ∈ H , let us consider a coordinate chart Φ : U → (−1, 1) × V , q 7→ (x0 , x1 , . . . , xn−1 )
where U is an open neighborhood of p and V is an open subset of Rn−1 , such that x0 (p) = 0
and the coordinate vector ∂0 is future-directed timelike. By choosing U sufficiently small, we
may ensure that the slice x0 = −1/2 lies in I − (p) and the slice x0 = 1/2 lies in I + (p) (cf.
Fig. 4.13). Any coordinate line defined by (x1 , . . . , xn−1 ) = (a1 , . . . , an−1 ), where a1 , ..., an−1
are n − 1 constants, is a timelike curve which connects the point (xα ) = (−1/2, a1 , . . . , an−1 )
112 The concept of black hole 3: The global view
Figure 4.13: Coordinate chart (xα ) in the vicinity of p ∈ H , such that ∂0 is future-directed timelike and the
slice x0 = −1/2 (resp. x0 = 1/2) lies in I − (p) (resp. I + (p)). Any coordinate line along which x0 varies, such as
the purple line, is timelike and intersects H at a single point h.
of I − (p) to the point (xα ) = (1/2, a1 , . . . , an−1 ) of I + (p) (cf. the purple curve in Fig. 4.13).
Given that I − (p) ⊂ M \ B, I + (p) ⊂ B and H = ∂B, such a curve necessarily intersects
H . Moreover the intersection is reduced to a single point h ∈ H for the curve is timelike and
H is achronal (Property 4.5). Let us then give the coordinates (y i ) = (a1 , . . . , an−1 ) to h. By
varying (a1 , . . . , an−1 ), we get a homeomorphism from U ∩ H to the open subset V of Rn−1 .
This makes H a topological manifold of dimension n − 1 (cf. Sec. A.2.1).
Remark 5: Generically, the topological manifold H is not a smooth manifold, for it contains some
points (the crossovers defined below) at which it is not differentiable. Actually H is slightly more than
a mere topological submanifold of M : it is a Lipschitz submanifold of M . The latter is intermediate
between a topological submanifold, i.e. a submanifold of class C 0 (continuous), and a differentiable
submanifold of class C 1q . On U ∩ H , the coordinate x0 is a Lipschitz function of the coordinates (y i ):
x0 (y i ) − x0 (y ′ i ) < K i (y − y ) . This follows from the achronal character of H : the points of
′i 2
P i
coordinates (y i ) and (y ′ i ) cannot have a too large separation in terms of x0 , otherwise they would be
timelike separated. Hence, one says that H is a Lipschitz submanifold of M . The notation C 1− (i.e.
a kind of intermediate between C 0 and C 1 ) is generally used to denote Lipschitz submanifolds.
Beside being achronal topological submanifolds of codimension 1, the main property of
black hole horizons is:
Property 4.8: the event horizon ruled by never leaving null geodesics (Penrose 1968
[407])
A black hole event horizon H is ruled by a family of null geodesics, called the generators
of H , that (i) either lie entirely in H or never leave H when followed into the future
from the point where they enter H , and (ii) have no endpoint in the future. Moreover,
there is exactly one generator through each point of H , except at special points where
null geodesics enter H , which are called crossovers. A special case of crossover, called
caustic, is a point where neighboring null geodesics focus and converge while arriving in
4.4 Black holes 113
Figure 4.14: Lemma 4.9: A causal curve L containing a timelike segment (between a and b on the figure) can
be deformed into an entirely timelike curve L ′ with the ends kept fixed (dashed curve).
H.
In particular, once a null geodesic has merged with H (at a point where it may intersect other
null geodesics), it will stay forever on H and will never intersect any other generator. The set
of all crossovers is called the crease set [457, 458, 74].
The following proof of Property 4.8 is adapted from that given in Box 34.1 of MTW [371].
In addition of Lemma 4.6, it relies on the following lemma.
Lemma 4.9
Let L be a causal curve connecting two points p and q of M . If L contains a timelike
segment, then there exists an entirely timelike curve connecting p and q.
Proof. We shall only provide a graphical “proof”, based on the spacetime diagram of Fig. 4.14.
The causal curve L may have parts where it is null (segments pa and bq in Fig. 4.14); these
parts are drawn with the angle of incline θ = ±45◦ with respect to the horizontal direction. If
L contains a timelike segment (such as ab in Fig. 4.14), i.e. a segment with |θ| > 45◦ , it can be
deformed, while keeping the same ends, to a curve with |θ| > 45◦ everywhere, i.e. to a timelike
curve.
Proof of Property 4.8. Let p ∈ H and let U be some convex normal neighborhood10 of p. Since
p lies in the boundary of J − (I + ), it is always possible to consider a sequence of points
(pn )n∈N converging toward p and such that ∀n ∈ N, pn ∈ U ∩ J − (I + ) (cf. Fig. 4.15). Since
pn ∈ J − (I + ), there exists a future-directed causal curve Ln from pn to I + for each n ∈ N.
The neighborhood U being convex, each Ln intersects its boundary ∂U at a unique point, qn
say, in the future of pn : {qn } = Ln ∩ ∂U (cf. Fig. 4.15). Since ∂U is compact, the sequence
10
Basically, a convex normal neighborhood is an open subset U such that any two points of U can be connected
by a geodesic lying entirely in U , cf. Sec. B.3.4. There always exists a convex normal neighborhood around any
point of a pseudo-Riemannian manifold (cf. Proposition 7 p. 130 in O’Neill’s textbook [390]).
114 The concept of black hole 3: The global view
Figure 4.15: Causal curve L connecting p to q obtained as a limit of causal curves in J − (I + ).
(qn )n∈N admits a subsequence, (qf (n) )n∈N say (f being an increasing function N → N), that
converges to some limit point q. Since from any point pf (n) arbitrarily close to p, there is the
causal curve Lf (n) to the point qf (n) arbitrarily close to q, one can show that there exists a
future-directed causal curve L connecting p to q (cf. Fig. 4.15; see e.g. Lemma 6.2.1 of Hawking
& Ellis’ textbook [266] for a precise demonstration).
As the limit of points in J − (I + ), q lies in the closure J − (I + ) = J − (I + ) ∪ H , H being
the boundary of J − (I + ). Let us show by contradiction that actually q ∈ H . If we assume
q ̸∈ H , then necessarily q ∈ J − (I + ). There exists then an open neighborhood V of q such
that V ⊂ J − (I + ) (cf. Fig. 4.16). Let us choose q ′ ∈ V such that q is connected to q ′ via a
timelike curve. We may then extend L to a causal curve L˜ from p to I + via q and q ′ (cf.
Fig. 4.16). Since L˜ contains a timelike segment (between q and q ′ ), we may invoke Lemma 4.9
to deform it into a timelike curve L˜′ between p and I + . Then, by Lemma 4.6, one can “move
the past end” of L˜′ to get a new timelike curve L˜′′ linking an event p′ ∈ B close to p to I +
(dotted curve in Fig. 4.16), which is impossible by the very definition of the black hole region
B. Hence q ∈ H .
The causal curve L connecting p to q cannot be timelike since p and q are both in H ,
which is achronal (Property 4.5). Actually, L cannot even contain a timelike segment: if it
would, then by Lemma 4.9, it could be deformed into a timelike curve between p and q, which
again would contradict the achronal character of H . Hence L is necessarily a null curve.
Moreover, it is a geodesic. Indeed, let us assume it is not. There is then some non-geodesic
null segment of L , ab say. Now, as shown in Sec. B.4.3 of Appendix B, a curve from a to b is
a geodesic iff any of its parametrizations P : [λa , λb ] → M , λ 7→ P (λ) ∈ L is a stationary
point of the action
Z λb
E(a,b) (P ) := g(v, v) dλ,
λa
where v = dx/dλ is the tangent vector associated with P . For the null segment ab of L ,
we have E(a,b) (P ) = 0. Since ab is assumed to be not geodesic, it is not a stationary point
of E(a,b) (P ), which implies that there exists a nearby curve from a to b with E(a,b) (P ) < 0,
4.4 Black holes 115
Figure 4.16: Proving by contradiction that q lies in H .
Figure 4.17: Proving that L lies entirely in H .
i.e. there exists a curve from a to b with some timelike part. It follows that p and q can be
connected by a causal curve with a timelike segment. But this feature has been excluded above.
We conclude that L is a null geodesic.
At this stage, we have shown that given p ∈ H , there exists a future-directed null geodesic
L connecting p to another point q ∈ H . There remains to show that L lies entirely in H .
Let us start by showing that L ⊂ J − (I + ). Let a be a generic point of L between p and
q. Since L is null, there exists a point a′ arbitrarily close to a such that a′ is connected to
q by a future-directed timelike curve (cf. Fig. 4.17). Thanks to Lemma 4.6 and the property
q ∈ J − (I + ), we may find a point q ′ ∈ J − (I + ) close to q such that a′ is connected to q ′ by a
future-directed timelike curve. Since q ′ ∈ J − (I + ), such a curve can be extended to a causal
curve to I + (the dashed curve in Fig. 4.17); hence a′ ∈ J − (I + ). Since a′ is arbitrarily close to
a, we conclude that a ∈ J − (I + ). Then, by repeating the same reasoning as that employed
above for proving q ∈ H , simply replacing q by a, we get that a ∈ H . Since a is a generic
point of L , we conclude that L lies entirely in H .
Given a point p ∈ H , we have thus constructed a future-directed null geodesic L lying
entirely in H and connecting p to another point q ∈ H . One can repeat the construction
from the point q to get another future-directed null geodesic L ′ ⊂ H connecting q to another
116 The concept of black hole 3: The global view
Figure 4.18: Proof of Lemma 4.10.
point q ′ ∈ H . Now L and L ′ must be two segments of the same null geodesic L ∪ L ′ by
the following lemma.
Lemma 4.10
Let q ∈ H . If L ⊂ H is a null geodesic having q as future end point and L ′ ⊂ H is a
null geodesic having q as past end point, then L and L ′ have collinear tangent vectors at
their common point q. It follows that L and L ′ are two segments of a same null geodesic
through q.
Proof of Lemma 4.10. Assume that L and L ′ have non-collinear tangent vectors at q. Then,
in the vicinity of q, one can find a point a ∈ L and a point b ∈ L ′ such that a and b can be
connected by a timelike curve (cf. Fig. 4.18). Since L ⊂ H and L ′ ⊂ H , we have a ∈ H
and b ∈ H and therefore we get a contradiction with H being achronal.
Thanks to Lemma 4.10, we conclude that L ′ extends L to the null geodesic L ∪ L ′ entirely
lying in H . By iterating, we conclude that the null geodesic L through p can be extended
indefinitely into the future. Moreover, it can never leave H . Indeed, if L would leave H at
some point q, by the same procedure used above for p, one could construct a future-directed
null geodesic L ′ ⊂ H starting from q; then Lemma 4.10 would imply that L and L ′ would
have the same tangent at q, which is not compatible with L leaving H at q.
Another direct consequence of Lemma 4.10 is that no two distinct null generators may
intersect at a point p ∈ H , except if their segments in the past of p lie outside H . This
completes the proof of Property 4.8.
Some features of Property 4.8 are illustrated in Fig. 4.19, which displays the null geodesic
generators in a numerical simulation of the head-on collision of two black holes by Matzner et
al. (1995) [359]. Note that new null geodesics enter the event horizon at the “crotch” of the
“pair of pants”.
The head-on black hole merger has been also computed by Cohen et al. (2009) [133], with
an increased numerical accuracy (cf. Fig. 4.20). Cross-sections of the event horizon H (cf.
Sec. 2.3.4) are depicted in Fig. 4.21. The same figure also shows how some null geodesics are
reaching H to become null generators.
4.4 Black holes 117
Figure 4.19: Spacetime diagram of the event horizon corresponding to the head-on merger of two black holes
as computed by Matzner et al. (1995) [359]. The white curves are some null geodesic generators; the left picture is
a zoom of the merger region, with the crease set (source: Fig. 4 of Ref. [359]; ©1995 American Association for the
Advancement of Science).
Figure 4.20: Spacetime diagram showing the event horizon in the head-on merger of two black holes, as
computed by Cohen et al. (2009) [133]. The blue curves are null geodesics that will eventually become null
generators of the event horizon; those arising from regions close to the event horizon are marked by the arrow
and the black ellipse (source: Fig. 15 of Ref. [133]; ©2009 IOP Publishing Ltd).
118 The concept of black hole 3: The global view
Figure 4.21: Cross-sections (at various coordinate times t) of the event horizon H corresponding to the
head-on merger of two black holes as computed by Cohen et al. (2009) [133] and displayed in Fig. 4.20. Each
figure is a 2D cut of a hypersurface Σt defined by a constant value of the coordinate time t, expressed in units
of the sum M of the initial irreducible masses of each black hole (to be discussed in Chap. 16). The whole 3D
hypersurface Σt can be reconstructed by rotation around the collision axis. tCEH (for “Common Event Horizon”)
is the coordinate time at which the cross-section of H becomes a connected 2-surface. The cross-sections of
H are displayed in black, while the green dashed curves denote the set of the intersections with Σt of the null
geodesics that will become null generators of H through the cusps in the “individual” event horizons. The red
and blue dashed curves denotes apparent horizons (to be discussed in Chap. 18). (source: Fig. 1 of Ref. [133]; ©2009
IOP Publishing Ltd).
Finally, Fig. 4.22 shows a cross-section of the event horizon computed by Cohen et al.
(2012) [132] in some inspiralling binary black hole merger. The black hole spacetime itself has
been computed as a solution to the vacuum Einstein equation (1.44) by Scheel at al. [446]; it
corresponds to 16 inspiralling orbits of an equal-mass binary black hole with vanishing initial
spins.
Generically, for a binary black hole merger, the crease set forms a 2-dimensional subset of
the event horizon H and is bounded by the set of caustic points, which forms a 1-dimensional
subset of H [457, 458, 286, 132].
Property 4.11: the event horizon as a null hypersurface
Wherever it is smooth, H is a null hypersurface. Its generators, as defined by Property 4.8,
are then nothing but the null-hypersurface generators as defined in Sec. 2.3.3.
Proof. Let us assume that H is smooth in some open subset U . By Property 4.7, H is then
a smooth hypersurface in U . According to Property 4.8, there is a null geodesic lying in H
through any point of H ∩U . This implies null tangent vectors at any point of H ∩U , so that, in
U , H must be either a null hypersurface or a timelike one. But H is achronal by Property 4.5
4.4 Black holes 119
Figure 4.22: Cross-section of the event horizon H of the inspiralling merger of two black holes as computed
by Cohen et al. (2012) [132]. The x and y axes define the orbital plane. This cross-section is the first connected
one in the slicing of H by surfaces of constant coordinate time t (source: Fig. 2 of Ref. [132]; ©2012 American
Physical Society).
and therefore cannot be timelike. Hence, H is a null hypersurface in U . Finally, there is only
one congruence11 of null geodesics ruling a null hypersurface: the null generators defined in
Sec. 2.3.3. The generators invoked in Property 4.8 have thus to belong to that congruence.
It can be shown that event horizons are smooth almost everywhere: the only location where
they are not differentiable is the crease set, i.e. the set of points where null geodesics cross each
other while entering H to become a null generator (cf. Fig. 4.19).
Remark 6: Properties 4.5 to 4.11 are not specific to black hole horizons: they are actually valid for the
boundary ∂J − (S) of the causal past of any set S ⊂ M , or S ⊂ M˜ (such as I + ) [407, 266, 209, 390].
Indeed, none of the specific features of I + (i.e. I + is part of the boundary of M˜ and each of its
points can reached by a future-directed causal curve originating from M ) has been used in the proofs
of Properties 4.5 to 4.11. These properties are also valid for the boundary ∂J + (S) of the causal future of
any set S, modulo the changes future ↔ past in Property 4.8. A set of the type ∂J − (S) and ∂J + (S) is
called an achronal boundary [266].
Historical note : The properties of generic achronal boundaries, which yield Properties 4.5 to 4.11 in the
particular case of a black hole event horizon H (cf. Remark 6), have been established by Roger Penrose
in a seminal lecture at the Battelle-Seattle Center at the summer of 1967, which has been published in
1968 [407]. Penrose used the term semispacelike boundary instead of achronal boundary. It seems that the
latter has been introduced by Stephen Hawking and George Ellis in 1973 in their famous textbook [266].
11
Let us recall that a congruence on a submanifold S of M is a family of curves tangent to S , one, and
exactly one, through each point of S .
120 The concept of black hole 3: The global view
Chapter 5
Stationary black holes
Contents
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Stationary spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.3 Mass and angular momentum . . . . . . . . . . . . . . . . . . . . . . . 128
5.4 The event horizon as a Killing horizon . . . . . . . . . . . . . . . . . . 152
5.5 The generalized Smarr formula . . . . . . . . . . . . . . . . . . . . . . 162
5.6 The no-hair theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.1 Introduction
Having defined black holes in all generality in Chap. 4, we focus here on the steady state case,
i.e. black holes in stationary spacetimes. We have already discussed non-expanding horizons
and Killing horizons in Chap. 3 as possible models for the event horizon of a steady state black
hole. Actually, we shall see below that (each connected component of) the event horizon of a
black hole in a stationary spacetime has to be a Killing horizon, at least in the framework of
general relativity and within rather general hypotheses. We shall start by defining properly
the concept of stationary spacetime and investigating some first properties of a black hole in
such a spacetime (Sec. 5.2). Then, in Sec. 5.3, we discuss the concepts of mass and angular
momentum in asymptotically flat spacetimes, which are useful to characterize black holes. In
Sec. 5.4, we shall see that if the Killing vector ξ generating stationarity is null on a connected
event horizon H , the latter is a Killing horizon with respect to ξ. If H is non-degenerate
and the electrovacuum Einstein equation holds, this can only occur in a static spacetime
(staticity theorem, Sec. 5.4.1). On the contrary, if ξ is spacelike1 on some parts of H , then,
modulo the electrovacuum Einstein equation and some additional hypotheses, H is still a
Killing horizon, albeit with respect to a Killing vector distinct from ξ (strong rigidity theorem,
Sec. 5.4.2). Section 5.5 is devoted to an important relation between various global quantities
1
The timelike case is excluded for an event horizon is a null hypersurface.
122 Stationary black holes
characterizing a stationary black hole: the Smarr formula. This is the opportunity to investigate
electromagnetic fields on the horizon of a stationary black hole, in particular to define the black
hole’s electric charge and electric potential, both of them being involved in the Smarr formula.
The culmination point of this chapter is Sec. 5.6, which presents the famous no-hair theorem.
Modulo some hypotheses, this theorem stipulates that in 4-dimensional general relativity, all
isolated stationary electrovacuum black holes are necessarily Kerr-Newman black holes; in the
pure vacuum case (the most relevant one for astrophysics), they are Kerr black holes, to be
explored in Part. III.
5.2 Stationary spacetimes
5.2.1 Definitions
A spacetime (M , g) is called stationary iff (i) it is invariant under the action of the
translation group (R, +) and (ii) the orbits of the group action (cf. Sec. 3.3.1) are everywhere
timelike curves or (ii’) (M , g) admits a conformal completion (cf. Sec. 4.3) and the orbits of
the group action are timelike in the vicinity of the conformal boundary I . It is equivalent
to say that there exists a Killing vector field ξ (the generator of the translation group, cf.
Sec. 3.3.1) that is timelike everywhere or at least in the vicinity of I when there exists
a conformal completion. We shall say that (M , g) is strictly stationary iff the Killing
vector field ξ is timelike in all M , i.e. iff property (ii) above is fulfilled.
Remark 1: Some authors (e.g. Carter [96]) call pseudo-stationary the stationary spacetimes that obey
(ii’), keeping the qualifier stationary for the strictly stationary case. As we are going to see, when M
contains a black hole, ξ cannot be timelike everywhere, so only pseudo-stationarity in Carter’s sense is
relevant for such spacetimes. Our terminology, namely keeping the qualifier stationary even for (ii’),
follows that of Choquet-Bruhat [108], Chruściel, Lopes Costa & Heusler [123], Heusler [275] and Wald
[505].
The Killing vector field ξ is a priori not unique: the translation group (R, +) admits the
reparametrization t 7→ t′ = αt, where α is a nonzero constant, which yields the rescalling
ξ 7→ ξ ′ = α−1 ξ of the Killing vector field [cf. Eq. (3.14)]. When the spacetime admits a
conformal boundary I — which is required if a black hole is present, given the definition (4.37)
—, then one selects ξ by demanding that it is future-directed near I and obeys
ξ · ξ → −1 near I . (5.1)
This determines ξ uniquely, since this fixes the rescaling constant α to ±1, with only +1 being
acceptable to keep the time orientation.
If the stationary spacetime is endowed with electromagnetic and/or matter fields, they must
respect the stationarity as well. For the electromagnetic field F (cf. Sec. 1.5.2), this is expressed
by the vanishing of the Lie derivative along ξ:
Lξ F = 0. (5.2)
This condition is similar to Killing-vector property Lξ g = 0 [Eq. (3.18)].
A notion stronger than stationarity is that of staticity:
5.2 Stationary spacetimes 123
A spacetime (M , g) is called static iff (i) it is stationary and (ii) the Killing vector field ξ
generating the stationary action is orthogonal to a family of hypersurfaces (one says that ξ
is hypersurface-orthogonal). The spacetime (M , g) is called strictly static iff moreover
ξ is timelike in all M .
Remark 2: The same comment as in Remark 1 can be made: some authors would call static only
spacetimes that are strictly static according to the above definition.
In loose terms, a spacetime is stationary if “nothing changes with time”, while it is static
if, in addition, “nothing moves”. A prototype of a stationary spacetime that is not static is a
spacetime containing a steadily rotating body, be it a star or a black hole.
5.2.2 Coordinates adapted to stationarity or staticity
In a n-dimensional stationary spacetime (M , g), a coordinate system (xα ) = (x0 , x1 , . . . xn−1 )
is said adapted to stationarity iff the coordinate vector ∂t , where t := x0 , coincides with the
stationary Killing vector ξ. According to Property 3.7, t is then an ignorable coordinate, i.e.
the components gαβ of the metric tensor with respect to (xα ) obey ∂gαβ /∂t = 0 [Eq. (3.20)]. It
follows that
gαβ = gαβ (x1 , . . . xn−1 ). (5.3)
If the spacetime (M , g) is static, a coordinate system (xα ) = (x0 = t, x1 , . . . xn−1 ) is said
adapted to staticity iff it is adapted to stationarity and the hypersurfaces t = const are
orthogonal to the static Killing vector ξ. Given that a normal 1-form to the hypersurfaces
t = const is dt, the coordinates (xα ) are adapted to staticity iff
ξ = ∂t and ξ = W dt, (5.4)
where ξ is the metric dual of ξ (cf. Sec. A.3.3) and W := g(ξ, ξ) = ⟨ξ, ξ⟩. The orthogonality
of ∂t and the hypersurfaces t = const translates to g0i = 0 for i ∈ {1, . . . , n − 1}, the gαβ ’s
being the metric components with respect to the coordinates (xα ). Hence one may write the
metric of a static spacetime of dimension n as
g = W dt2 + gij dxi dxj , (5.5)
where the indices (i, j) range in {1, . . . , n − 1} and W and gij are functions of (x1 , . . . , xn−1 )
only. It is clear that the metric (5.5) is invariant2 in the transformation t 7→ −t. One says that a
static spacetime is time-reflection symmetric.
Example 1 (staticity of anti-de Sitter spacetime): The anti-de Sitter spacetime AdS4 considered in
Example 18 of Chap. 3 is strictly static, with the coordinates (τ, r, θ, φ) being adapted to staticity (hence
their name: global static). Indeed, the metric (3.49) is of the form (5.5) with t = τ and W = −ℓ2 (1 + r2 ).
The strict staticity follows from W < 0, which shows that the Killing vector ξ = ∂t is everywhere
timelike.
2
Would (5.5) have contained a non-vanishing g0i dt dxi term, this would not have been the case.
124 Stationary black holes
Example 2 (staticity of Schwarzschild spacetime): The Schwarzschild spacetime, introduced in
Example 3 of Chap. 2, is static, but not strictly static. The staticity is however not obvious from the
metric components given by Eq. (2.5) because the coordinates (t, r, θ, φ) used there are not adapted
to staticity: the Killing vector ξ = ∂t is not orthogonal to the hypersurfaces t = const, given that
gtr ̸= 0. We shall introduce coordinates adapted to staticity in Chap. 6, namely the Schwarzschild-Droste
coordinates. That the Schwarzschild spacetime is not strictly static can be read directly on Eq. (2.5):
ξ · ξ = gtt = −1 + 2m/r, so that ξ is timelike only in the region r > 2m.
5.2.3 Black holes in stationary spacetimes
Let us consider a spacetime (M , g) that contains a black hole, as defined in Sec. 4.4.2. In partic-
ular, (M , g) admits a future infinity I + . Furthermore, we assume that (M , g) is stationary, as
defined in Sec. 5.2.1. Since (M , g) is invariant under the action of the isometry group (R, +),
so is I + (under some proper extension of ξ to the conformal completion M˜) and therefore
its causal past J − (I + ). Since the event horizon H is the boundary of J − (I + ) inside M
[Eq. (4.39)], we get
Property 5.1: stationary event horizon
The event horizon H of a black hole in a stationary spacetime is globally invariant under
the action of the stationary group (R, +).
The word globally stresses that H is invariant as a whole, not that each point of H is
a fixed point of the group action. Let us assume that H is smooth (this sounds likely in a
stationary context; a rigorous proof can be found in Ref. [120]); it is then a null hypersurface
(Property 4.11 in Sec. 4.4.3). Now, H is globally invariant if, and only if, the generator ξ
of the isometry group is tangent to H . Since a timelike vector cannot be tangent to a null
hypersurface (cf. Lemma 2.3 in Sec. 2.3.4), we conclude:
Property 5.2: stationary Killing vector tangent to the event horizon
In a stationary spacetime containing a black hole, the stationary Killing vector field ξ is
tangent to the event horizon H , which implies that ξ is either null or spacelike on H . Let
H0 be a connected component of H (H0 = H if H is connected). If ξ is null on all H0 ,
one says that H0 is non-rotating, while if ξ is spacelike on some part of H0 , one says
that H0 is rotating.
Since ξ cannot be timelike on H , it follows immediately that a stationary spacetime
containing a black hole cannot be strictly stationary, according to the definition given in
Sec. 5.2.1. We shall discuss in detail the two cases — null or spacelike — allowed for ξ on H in
Sec. 5.4.
It is rather intuitive that the event horizon H of a black hole in a stationary spacetime
must have a vanishing expansion θ(ℓ) along its null normals ℓ. Indeed, if θ(ℓ) were nonzero, the
area of cross-sections would vary when dragged along ℓ and this would define some “evolution”
5.2 Stationary spacetimes 125
along H (e.g. from small areas to larger ones), which would break the invariance of H under
the action of the stationary group. This of course results from H being part of the global
spacetime structure, since not any null hypersurface in a stationary spacetime has a vanishing
expansion: for instance, a future light cone in Minkowski spacetime (which is stationary!) has
θ(ℓ) > 0 (cf. Example 17 on p. 45), but the light cone is not invariant by any time translation
isometry. The rigorous proof that θ(ℓ) = 0 for stationary event horizons can be found in
Hawking & Ellis’ textbook (Proposition 9.3.1 of Ref. [266]). Here, we shall simply state:
Property 5.3: non-expanding horizons for stationary black holes
The event horizon H of a black hole in a stationary spacetime is a null hypersurface of
vanishing expansion:
θ(ℓ) = 0. (5.6)
Moreover, if the cross-sections S of H are closed manifolds such that H has the topology
R × S , H is a non-expanding horizon, according to the definition given in Sec. 3.2.
In dimension 4, one can strongly constrain the topology of the horizon cross-sections:
Property 5.4: topology theorem 1 (Hawking 1972 [260])
Let (M , g) be a 4-dimensional stationary spacetime containing a black hole of event
horizon H . Let S be a connected component of a complete cross-section of H . Let
us assume that (i) S is closed (compact without boundary) and orientable; (ii) the null
dominance condition (3.43) is fulfilled on H for some scalar field f ≥ 0 [for general
relativity with f = Λ this is equivalent to assuming Λ ≥ 0 and the dominant null energy
condition (3.46)] and (iii) when displaced into the black hole exterior along −k (the opposite
of the ingoing null normal k to S ), S becomes a surface with θ(ℓ) > 0. In particular,
condition (iii) holds if θ(k) < 0 and there is no trapped surface (cf. Sec. 3.2.3) in the black
hole exterior. Then the cross-section S has generically the topology of the 2-sphere (i.e.
S is homeomorphic to S2 ). The non-generic case is that of S having the topology of the
2-torus T2 = S1 × S1 ; this can occur only under special circumstances, among which the
metric induced by g on S must be flat.
Proof. Let ℓ be a future-directed null normal to H and k a complementary future-directed
null vector field normal to S , normalized as in Eq. (2.33), i.e. k · ℓ = −1. At each point p ∈ S ,
the pair (k, ℓ) is a basis of the 2-plane Tp⊥ S orthogonal to S (cf. Fig. 2.10), with ℓ tangent to
H and k transverse to it. Morever k points towards the black hole interior, otherwise null
geodesics leaving H along k would enter J − (I + ). As in Sec. 2.3, let us consider that H
is the level set u = 0 of 1-parameter family of hypersurfaces (Hu )u∈R . This extends ℓ in the
vicinity of H via Eq. (2.11) as a null vector field normal to each Hu . By Property 5.3, we have
θ(ℓ) = 0 on S . Let us displace S by a small parameter ϵ > 0 along −k. The expansion along
ℓ of the obtained surface is positive by hypothesis (iii). By taking the limit ϵ → 0, we form the
derivative L−k θ(ℓ) , which must obey L−k θ(ℓ) ≥ 0 since θ(ℓ) = 0 on S and θ(ℓ) > 0 on the
126 Stationary black holes
displaced surface. Given that L−k θ(ℓ) = −Lk θ(ℓ) , we see that hypothesis (iii) implies
Lk θ(ℓ) ≤ 0. (5.7)
A standard identity (cf. e.g. Eq. (3b) of [269], Eq. (3.1) of [68], Eq. (5.1) of [81] or Eq. (36) of
[297]) expresses Lk θ(ℓ) as
1S
Lk θ(ℓ) = − R − SDa Ωa + Ωa Ωa + G(ℓ, k), (5.8)
2
where SR is the Ricci scalar of the Riemannian metric q on S induced by the spacetime
metric g, SD is the Levi-civita connection associated to q, G is the Einstein tensor of the
spacetime metric g and Ω is the 1-form on S that is the pullback of the 1-form ω defined by
Eq. (2.75). We may also view Ω as a spacetime 1-form, defined as the composition of ω with
the orthogonal projector → −q onto S : Ω = ω ◦ → −q . In view of Eq. (2.77), we may then write
⟨Ω, v⟩ = −k · ∇− →
q (v) ℓ, or in index notation, Ω α = −kµ ∇ν ℓµ q να . Besides, using Eq. (3.43), we
→
−
have G(ℓ, k) = G(ℓ) · k = −W · k − f ℓ · k = −W · k + f . Then, integrating (5.8) over the
compact manifold S and setting the integral of the divergence term SDa Ωa to zero since S
has no boundary, we get
S √
1
Z Z h i√
R qd x = 2 a
Ωa Ω −W · k + f −Lk θ(ℓ) q d2 x. (5.9)
2 S S
| {z } | {z } |{z} | {z }
| {z } ≥0 ≥0 ≥0 ≥0
2πχ
In the left-hand side, we have invoked the Gauss-Bonnet theorem to express the integral of
S
R in terms of the Euler characteristic χ of the surface S , using the fact that in dimension 2,
the Ricci scalar SR is twice the Gaussian curvature. The signs of the terms of the integrand in
the right-hand side are justified as follows: Ωa Ωa = q ab Ωa Ωb ≥ 0 because q is a Riemannian
metric, −W · k ≥ 0 by Lemma 1.2 in Sec. 1.2.2, given that k is a future-directed null vector
and W is a future-directed causal vector thanks to the null dominance condition (3.43), f ≥ 0
by hypothesis and −Lk θ(ℓ) ≥ 0 follows from Eq. (5.7). Now, the Euler characteristic χ is a
topological invariant, which is 2 for the sphere S2 , 0 for the torus T2 and −2(g − 1) for a
connected orientable surface of genus g, i.e. with g “holes”. Equation (5.9) yields χ ≥ 0. The
only possibilities for S compact, connected and orientable are S2 (χ = 2) and T2 (χ = 0).
However the torus case is very special. Indeed, setting χ = 0 in Eq. (5.9) implies that each term
in the integrand of the right-hand side vanishes separately: Ωa Ωa = 0, W · k = 0, f = 0 and
Lk θ(ℓ) = 0. The last condition is the marginal case in the inequality (5.7). Moreover, since q
is positive definite, Ωa Ωa = 0 implies Ω = 0, which in turn implies SDa Ωa = 0. In addition,
G(ℓ, k) = −W · k + f = 0. We then deduce immediately from Eq. (5.8) that SR = 0, i.e. the
Ricci scalar of q is identically zero. Given that the Riemann curvature tensor of a 2-dimensional
metric is proportional to its Ricci scalar [cf. Eq. (A.112)], it follows that q is a flat metric.
Remark 3: The topology theorem 1, as stated above, is slightly different from the original version
given by Hawking [260, 261, 266], for it adds the “outermost” hypothesis (iii). In Hawking’s version, (iii)
is deduced from the non-existence of trapped surfaces in the black hole exterior J − (I + ), the latter
property being proved by assumming that the spacetime is globally hyperbolic (Proposition 9.2.8 in
5.2 Stationary spacetimes 127
Ref. [266]). Another difference with Hawking’s version, as stated in Proposition 9.3.2 of Ref. [266], is
that the torus topology is not totally excluded by Property 5.4. However, as discussed in the historical
note below, it seems that the exclusion of the torus in Hawking’s proof requires additional hypotheses,
which are not explicitely stated in Refs. [260, 266].
Remark 4: The topology theorem 1 is not specific to stationary black hole event horizons, for it relies
only on quasilocal properties. Indeed, the proof does not rely on H being the boundary of a black hole
region, if one agrees to define the “interior” region as that pointed towards by k. The only required
property is H being a hypersurface (not even a null one) sliced by spacelike compact surfaces S
with θ(ℓ) = 0 and Lk θ(ℓ) ≤ 0. The theorem is therefore valid for outer trapping horizons [269] and
(tubes of) apparent horizons, as noticed by Hawking himself [261] (p. 34), which are not necessarily null
hypersurfaces and which exist in non-stationary spacetimes. Both concepts of outer trapping horizon
and apparent horizon will be discussed in Chap. 18.
Another version of the topology theorem relies on the null convergence condition, which
is weaker than the null dominance condition (hypothesis (ii) above), and replaces hypothesis
(iii) by other ones, which are more global. It also fully excludes the 2-torus topology:
Property 5.5: topology theorem 2 (Chruściel & Wald 1994 [130])
Let (M , g) be a 4-dimensional asymptotically flat stationary spacetime containing a black
hole of event horizon H . Let us assume that (i) the null convergence condition (2.94) is
fulfilled, (ii) the domain of outer communications ⟨⟨M ⟩⟩ [Eq. (4.43)] is globally hyperbolic,
(iii) ⟨⟨M ⟩⟩ contains an achronal asymptotically flat hypersurface Σ that intersects H
in a compact cross-section S , and (iv) some technical condition is fulfilled (cf. [130] for
the details). Then, ⟨⟨M ⟩⟩ is simply connected and any connected component of S is
homeomorphic to the sphere S2 .
A part U of spacetime is said globally hyperbolic iff it admits a Cauchy surface, i.e. a
spacelike hypersurface Σ such that every inextendible timelike curve of U intersects Σ exactly
once (cf. Sec. 8.3 of Wald’s textbook [499] for more details). We shall not give the proof of
Property 5.5 here; it can of course be found in the original article by Chruściel & Wald [130].
Remark 5: The topology theorems 1 and 2 regard only a connected component of a given horizon complete
cross-section. Generally, the latter is a connected 2-manifold, but there exist 4-dimensional stationary
(actually static) spacetimes containing a black hole, the event horizon of which has disconnected
complete cross-sections: the Majumdar-Papapetrou spacetimes [353, 399, 258]. They are solutions of the
electrovacuum Einstein equation (cf. Sec. 1.5.2) representing an arbitrary number of charged black holes,
which form a static configuration thanks to an exact balance between the gravitational attraction and
the electrostatic repulsion. The Majumdar-Papapetrou black holes will be discussed further in Sec. 5.6.1
[cf. Eq. (5.108)].
A generalization of the topology theorem 1 to spacetimes of dimension n > 4 has been
obtained in 2006 by Galloway & Schoen [210] and Rácz provided a simplified proof of it in 2008
[421]. However, the theorem for n > 4 only says that some invariant of the smooth structure of
128 Stationary black holes
S , called the Yamabe invariant, must be positive for f ≥ 0 (3 ). This result implies that S must
admit metrics of positive scalar curvature; it is less stringent about the topology of S than the
theorem for n = 4. Actually, the higher n, the less constraints on the topology are provided by
the Yamabe invariant. For instance, for n = 5, the cross-sections of the Myers-Perry black holes
[376, 187, 425], which generalize Kerr black holes to n > 4, have the topology of the sphere S3 ,
but the topology S1 × S2 is allowed as well, as demonstrated by the black ring solution found
by Emparan & Reall [186, 187, 425] (see also Sec. 5.3 of Ref. [119]).
Historical note : The topology theorem for the spacetime dimension n = 4 has been formulated first
by Stephen Hawking in 1992 [260] (see also p. 34 of Ref. [261] and Proposition 9.3.2 in Hawking &
Ellis’ textbook [266]). However, in 1993, Gregory Galloway [208] (p. 119) pointed out some limitation
in Hawking’s proof, namely that it cannot exclude the torus topology (χ = 0) for the horizon’s cross-
sections without any extra hypothesis. For instance, for the proof given in Ref. [260], one shall require
that the spacetime is analytic (cf. Remark 4 in Sec. A.2.1), which is a rather strong hypothesis. The proof
presented above is based on that given by Sean Hayward in 1994 [269] for outer trapping horizons (see
also the proof of Theorem 6.3 in Ref. [381]). The theorem obtained by Piotr Chruściel and Robert Wald
in 1994 [130] (Property 5.5) relies on the topological censorship theorem established by John Friedman,
Kristin Schleich and Donald Witt in 1993 [199]. It leads directly to the spherical topology, excluding the
toroidal one.
5.3 Mass and angular momentum
For an asymptotically flat stationary spacetime, containing a black hole or not, there is a
well-defined concept of mass: the Komar mass, which we introduce here (Secs. 5.3.3-5.3.5). For
axisymmetric spacetimes, which are relevant for stationary rotating black holes, there is in
addition the concept of Komar angular momentum, which we shall introduce in Sec. 5.3.6.
5.3.1 Mass and angular momentum of weakly relativistic stationary
systems
We shall call weakly relativistic a n-dimensional spacetime (M , g) such that M is diffeo-
morphic to Rn and the metric tensor can be expressed as
g = f + h, (5.10)
where f is a flat Lorentzian metric on M and h is small in the following sense: the components
of h obey |hαβ | ≪ 1 in any f -Minkowskian coordinates, i.e. coordinates (xα ) on M such
that4 (fαβ ) = η := diag(−1, 1, . . . , 1). In addition, we assume that g is ruled by the Einstein
equation (1.40) with Λ = 0 and an energy momentum tensor T of compact support.
3
In brief, Eq. (5.9) holds for n > 4 as well, except that the integral of the Ricci scalar SR is no longer proportional
to the Euler characteristic of S (no Gauss-Bonnet theorem for dim S ̸= 2 !) but is related to the Yamabe constant
of the conformal class of the metric q.
4
We keep the notation η for the matrix diag(−1, 1, . . . , 1), so that (ηαβ ) stands for the components of f in
Minkowskian coordinates only.
5.3 Mass and angular momentum 129
Given the flat metric f , f -Minkowskian coordinates are not unique: any Poincaré trans-
formation x̃α = Λαµ xµ + cα , where (Λαβ ) is a Lorentz matrix and the cα ’s are constant, leads
to coordinates (x̃α ) which are f -Minkowskian as well. Furthermore, the flat metric f itself,
and hence the decomposition (5.10), is highly nonunique. Indeed any change of coordinates of
the form x′ α = xα + ζ α (x0 , . . . , xn−1 ) where (xα ) are f -Minkowskian coordinates and ζ α are
infinitesimal functions, leads to the following components of the metric tensor with respect to
(x′ α ):
∂xµ ∂xν
g ′ αβ = gµν ′ α ′ β = (ηµν + hµν ) (δ µα − ∂α ζ µ ) δ νβ − ∂β ζ ν ,
∂x ∂x
where we have used xα = x′ α −ζ α and ∂ζ α /∂x′ β = ∂ζ α /∂xσ ×∂xσ /∂x′ β ≃ ∂ζ α /∂xβ =: ∂β ζ α
to the first order in ζ α . Expanding to the first order in ζ α and h, we get
g ′ αβ = ηαβ + hαβ − ∂α ζβ − ∂β ζα ,
where ζα := ηαµ ζ µ . We may recast the above expression as g = f ′ + h′ , where f ′ is the metric
whose components in the coordinates (x′ α ) are ηαβ (hence f ′ is flat, as5 f ) and h′ has the
following components with respect to the coordinates (x′ α ):
h′ αβ = hαβ − ∂α ζβ − ∂β ζα . (5.11)
The above relation can be viewed as expressing some gauge freedom on h. This freedom actually
reflects the freedom in the choice of the flat background metric f in the decomposition (5.10).
Property 5.6: Lorenz gauge for the metric perturbation
The gauge freedom (5.11) can be used to ensure that, in terms of f -Minkowskian coordinates
(xα ), the metric perturbation h fulfills
∂µ η µν h̄αν = 0 , (5.12)
where
1
h̄ := h − h f , (5.13)
2
h standing for the trace of h with respect to f : h := η µν hµν . The choice (5.12) is referred
to as the Lorenz gauge (sometimes Hilbert gauge [464]).
Proof. From Eq. (5.11), we get h′ := η µν h′ µν = h − 2∂µ ζ µ , so that h̄′αβ = h̄αβ − ∂α ζβ − ∂β ζα +
∂µ ζ µ ηαβ . It follows then that ∂µ η µν h̄′αν = ∂µ η µν h̄αν − □f ζα , where □f := η µν ∂µ ∂ν is
the d’Alembertian operator with respect to f . Accordingly, if the Lorenz gauge (5.12) is not
fulfilled, it suffices to solve the d’Alembert equation
□f ζα = ∂µ η µν h̄αν (5.14)
and plug the solution ζα into Eq. (5.11) to get a metric pertubation h′ that obeys the Lorenz
gauge.
5 α
Note that the components of f with respect to the coordinates (x′ ) are not ηαβ but ηαβ − ∂α ζβ − ∂β ζα .
130 Stationary black holes
Remark 1: The Lorenz gauge (5.12) is equivalent to the first-order expansion in h of the relation
defining harmonic coordinates on M , i.e.
√
□g xα = 0 ⇐⇒ ∂µ −g g µα = 0. (5.15)
Remark 2: The Lorenz gauge does not fully specify the pair (f , h). Indeed, the solutions of the
d’Alembert equation (5.14) are nonunique: they depend on initial and boundary data.
A standard computation (see e.g. Chap. 18 of [371] or Chap. 5 of [464], noticing that the
computation is independent of the spacetime dimension n) shows that, at first order in h and
in the Lorenz gauge, the Einstein tensor of g is
1
G = − □f h̄, (5.16)
2
where □f stands for the d’Alembertian operator relative to the metric f : in f -Minkowskian
coordinates (xα ), □f h̄αβ = η µν ∂µ ∂ν h̄αβ .
Let us now assume that (M , g) is a stationary spacetime, with Killing vector ξ. We may
then choose f so that the f -Minkowskian coordinates (xα ) are adapted to stationarity, i.e. x0
is an ignorable coordinate, or equivalently ∂0 = ξ. Let then Σ be a hypersurface x0 = const
and γ the metric induced by f on Σ. The coordinates6 (xi )1≤i≤n−1 form a Cartesian coordinate
system of (Σ, γ): γ = δij dxi ⊗ dxj = (dx1 )2 + · · · + (dxn−1 )2 . The operator □f reduces to
the Laplace operator of γ, ∆ say, and, thanks to property (5.16), the Einstein equation (1.40)
becomes a system of n(n + 1)/2 independent Poisson equations:
∆h̄αβ = −16πTαβ . (5.17)
The solutions are obtained via the Green function of the (n − 1)-dimensional Laplace operator:
Tαβ (x′ )
Z
16π
h̄αβ (x) = dn−1 x′ , (5.18)
(n − 3)Ωn−2 Σ |x − x′ |n−3
where x := (x1 , . . . , xn−1 ), x′ := (x′ 1 , . . . , x′ n−1 ), |x − x′ |2 := n−1 i=1 (x − x ) and Ωn−2 is
i ′i 2
P
the area of the unit sphere Sn−2 ; the latter is given by the formula
Z +∞
2π (p+1)/2
Ωp = , with Γ(u) := tu−1 e−t dt, (5.19)
Γ((p + 1)/2) 0
so that
8
Ω2 = 4π, Ω3 = 2π 2 , Ω4 = π 2 , Ω5 = π 3 , . . . (5.20)
3
Property 5.7: asymptotic metric of a weakly relativistic system
Let (M , g) be a weakly relativistic stationary spacetime of dimension n ≥ 4, with g obeying
the Einstein equation (1.40) with Λ = 0 and an energy momentum tensor T of compact
support (the “source”), where the energy-density dominates over the spatial stresses (weakly
relativistic matter). Within the Lorenz gauge, there exists a coordinate system (xα ) such that
6
Latin indices i, j, k, . . . range in {1, . . . , n − 1}, while Greek ones range in {0, . . . , n − 1}.
5.3 Mass and angular momentum 131
the metric tensor has the following behavior when r := (x1 )2 + · · · + (xn−1 )2 → +∞:
p
M 1
g00 = − 1 + αn n−3 + O n−1 (5.21a)
r r
n − 2 Jij xj
1
g0i = αn n−1 + O n−1 (5.21b)
2 r r
αn M 1
gij = 1 + δij + O n−2 , (5.21c)
n − 3 rn−3 r
where αn := 16π/((n − 2)Ωn−2 ),
Z
M := T00 (x) dn−1 x (5.22)
Σ
and Z
xj T0i (x) − xi T0j (x) dn−1 x, (5.23)
Jij :=
Σ
Σ being any hypersurface x0 = const. The quantities M and Jij are independent of Σ (i.e.
of x0 ) and are called respectively the mass and the angular momentum of the central
source. The coordinates (xα ) correspond to the central source rest-frame and take their
origin at the center of mass, in the sense that
Z Z
n−1
T0i (x) d x = 0 and xi T00 (x) dn−1 x = 0 (5.24)
Σ Σ
The dominance of the energy-density over the spatial stresses is expressed in terms of the
components of T with respect to the coordinates (xα ) as T00 ≫ |Tij |.
Proof. Let (f , h) obeys the Lorenz gauge and (xα ) be some corresponding f -Minkowskian
coordinates. By a Poincaré transformation (cf. p. 129), one can inforce (5.24). Far from the
source, one may expand the term 1/|x − x′ |n−3 in Eq. (5.18) in powers of x′ i /r:
′ 2 !
1 1 xj x′ j |x |
′ n−3
= n−3 1 + (n − 3) +O ,
|x − x | r r r r2
where Einstein’s summation convention is assumed on the repeated index j ∈ {1, . . . , n − 1}.
Accordingly, Eq. (5.18) yields
xj
Z Z
16π 1 ′ n−1 ′ ′j ′ n−1 ′ 1
h̄αβ (x) = Tαβ (x ) d x + n−1 x Tαβ (x ) d x +O n−1 .
Ωn−2 (n − 3)rn−3 Σ r Σ r
(5.25)
For the component 00, the first integral in the right-hand side is nothing but M , as defined by
(5.22), while the second integral vanishes due to the second equation in (5.24). Hence, we get
16π M 1
h̄00 (x) = + O n−1 . (5.26)
(n − 3)Ωn−2 rn−3 r
132 Stationary black holes
Regarding the component 0i of Eq. (5.25), the first integral in the right-hand side vanishes due
to the first equation in (5.24). There remains then
16π xj 8π Jij xj
Z
′j 1 1
h̄0i (x) = ′ n−1 ′
x T0i (x ) d x + O n−1 = + O n−1 , (5.27)
Ωn−2 rn−1 Σ r Ωn−2 rn−1 r
where the second equality follows from the identity
Z
Jij = 2 xj T0i (x) dn−1 x. (5.28)
Σ
To prove it, consider
∂k (xi xj T 0k ) = δ ik xj T 0k + xi δ jk T 0k + xi xj ∂k T 0k = xj T 0i + xi T 0j ,
| {z }
0
where ∂k T 0k = 0 results from the equation of energy-momentum conservation (1.45) special-
ized to stationary spacetimes and expressed at 0th order in h. Integrating over Σ and invoking
the Gauss-Ostrogradsky theorem to set the integral of the divergence term ∂k (xi xj T 0k ) to zero
(since T is has compact support), we get
Z Z
x T (x) d x + xi T 0j (x) dn−1 x = 0.
j 0i n−1
Σ Σ
In view of the definition (5.23) of Jij , the identity (5.28) follows, since T 0i = −T0i at the 0th
order in h.
Let us now reconstruct h from h̄. Taking the trace of Eq. (5.13) with respect to f yields
h̄ := η µν h̄µν = h − (h/2) × n = (2 − n)h/2. Hence we may invert Eq. (5.13) to
h̄
h = h̄ − f with h̄ = −h̄00 + h̄ii . (5.29)
n−2
The part7 h̄ii of the trace h̄ is deduced from Eq. (5.25):
xj
Z Z
16π 1 ′ n−1 ′ ′j ′ n−1 ′ 1
h̄ii (x) = Tii (x ) d x + n−1 x Tii (x ) d x + O n−1 .
Ωn−2 (n − 3)rn−3 Σ r Σ r
The second integral vanishes identically, as it can be seen from the identity
1 2 ij 1
k j ik
∂i x x T − r T = δ ki xj T ik + xk δ ji T ik + xk xj ∂i T ik −r ∂i r T ij − r2 ∂i T ij
2 | {z } |{z} 2 | {z }
0 xi /r 0
j ii j
= x T = x Tii ,
where ∂i T ij = 0 follows from the energy-momentum conservation law (1.45). Hence the
integral of xj Tii over Σ is that of the divergence of a vector field that vanishes outside the
source and the Gauss-Ostrogradsky theorem allows one to set it to zero. We are thus left with
Z
16π 1 1
h̄ii (x) = ′ n−1 ′
Tii (x ) d x + O n−1 . (5.30)
(n − 3)Ωn−2 rn−3 Σ r
7
Note that Einstein’s summation convention is used: h̄ii = h̄11 + · · · + h̄n−1,n−1 .
5.3 Mass and angular momentum 133
Gathering Eqs. (5.26) and (5.30), we get
Z
16π 1 ′ n−1 ′ 1
h̄ = −M + Tii (x ) d x + O n−1 .
(n − 3)Ωn−2 rn−3 Σ r
However, given expression (5.22) for M , the weakly relativistic matter condition T00 ≫ |Tij |
implies that the integral of Tii over Σ is negligible in front of M . We conclude that
1
h̄ = −h̄00 + O n−1 .
r
It follows then from Eq. (5.29) that, up to terms decaying at least as 1/rn−1 ,
h̄00 n−3
h00 = h̄00 + × (−1) = h̄00
n−2 n−2
h0i = h̄0i
h̄00 h̄00 1
hij = h̄ij + δij = δij + O .
n−2 n−2 rn−2
The last equality holds because the 1/rn−3 term in h̄ij , which is proportional to the integral of
Tij over Σ according to Eq. (5.25), is negligible in front of the 1/rn−3 term in h̄00 , given that
T00 ≫ |Tij |. The above three equations, along with the values (5.26) and (5.27) for h̄00 and h̄0i ,
establish formulas (5.21) for the components gαβ = ηαβ + hαβ of the metric tensor.
5.3.2 Asymptotic expression of the metric in stationary asymptotically
flat spacetimes
Let us now consider a generic (not necessarily weakly relativistic) n-dimensional spacetime
(M , g) that is stationary and asymptotically flat, with g being ruled by the Einstein equation
with Λ = 0 and with T vanishing in the asymptotic region. The decomposition g = f +h, with
f flat and h “small” [Eq. (5.10)] still holds in some neighborhood U of infinity. Moreover, one
can still choose the pair (f , h) such that the Lorenz gauge (5.12) is fulfilled and the linearized
Einstein equation in U reduces to ∆h̄ = 0 [Eq. (5.17) with T = 0]. There exists then a
f -Minkowskian coordinate system (xα ) on U such that the solution for g is given by Eq. (5.21)
to the lowest order in 1/r, the difference being that the constants M and Jij can no longer be
expressed by integrals of the energy-momentum tensor, as in Eqs. (5.22) and (5.23). This can be
shown rigorously by using the expansion in powers of 1/r of the generic solution to ∆f = 0
and the Lorenz gauge (to set some terms to zero), cf. Exercice 19.3 of MTW [371] or Sec. 5.7 of
Straumann’s textbook [464] for details. For a configuration that is not assumed to be weakly
relativistic, one should not limit oneself to the linearized Einstein equation. However, going
beyong the first order expansion in h does not spoil the first terms in the expansions (5.21) of
g’s components. This simply adds terms with a higher power of 1/r. For instance, for n = 4,
the second order expansion in h introduces the term −2M 2 /r2 in g00 (see the above references
for details, as well as Refs. [65, 418] for the nonlinear expansion within a post-Newtonian
framework). We may then assert:
134 Stationary black holes
Property 5.8: asymptotic metric of a stationary asymptotically flat spacetime
Let (M , g) be a stationary asymptotically flat spacetime of dimension n ≥ 4, with g
obeying the Einstein equation (1.40) with Λ = 0 and with the energy-momentum tensor
T vanishing in the asymptotic region. Then there exists a coordinate system (xα ) in
the
p asymptotic region such that the metric tensor has the following behavior when r :=
(x1 )2 + · · · + (xn−1 )2 → +∞:
M 1
g00 = − 1 + αn n−3 + O n−2 (5.31a)
r r
n − 2 Jij xj
1
g0i = αn n−1 + O n−1 (5.31b)
2 r r
αn M 1
gij = 1 + δij + O n−2 , (5.31c)
n − 3 rn−3 r
where
16π
αn := , (5.32)
(n − 2)Ωn−2
(Ωn−2 being the area of the unit sphere Sn−2 , cf. Eqs. (5.19)-(5.20)), so that
8 3 16
α4 = 2, α5 = , α6 = , α7 = , (5.33)
3π 2π 5π 2
M is a constant and (Jij ) is a constant antisymmetric (n−1)×(n−1) matrix. Furthermore,
by a suitable SO(n − 1) transformation of the coordinates (xi )1≤i≤n−1 , (Jij ) can be brought
to the following block diagonal form, depending only on p := [(n − 1)/2] numbers J(1) ,
. . ., J(p) (possibly equal to zero):
0 J(1)
−J 0
(1)
(5.34)
Jij =
0 J(2) ,
−J(2) 0
...
with the last raw and the last column containing only zeros if n is even.
The last assertion follows from the fact that any real antisymmetric matrix J is similar to a
matrix J ′ of the type (5.34) via J ′ = P JP −1 , where P is a special orthogonal matrix.
Note that the difference between (5.31) and the asymptotic expansion (5.21) for a weakly
relativistic system is O (1/rn−2 ) in Eq. (5.31a) versus O (1/rn−1 ) in Eq. (5.21a). As discussed
above, this results from nonlinear terms in the expansion of the Einstein equation.
5.3 Mass and angular momentum 135
5.3.3 Komar mass
In Newtonian gravity, the mass M of an isolated body is defined similarly to Eq. (5.22), namely
by the integral of the mass density ρ ∼ T00 over the body. An alternative formula for M is
→
−
provided by Gauss’s law: M is −(4π)−1 times the flux of the gravitational field g = −∇Φ
through any closed surface S surrounding the body, namely
→
−
Z
1
M= ∇Φ · dS, (5.35)
4π S
where Φ is the Newtonian gravitational and dS is the area element vector normal to S . The
above formula is easily derived when S is a sphere of large radius, using spherical coordinates
→
−
(r, θ, φ). Indeed, for large r, one has Φ ∼ −M/r, so that ∇Φ ∼ (M/r2 ) ∂r . Given that
dS = r2 sin θ dθ dφ ∂r and ∂r · ∂r = 1, formula (5.35) follows. This formula is actually not
restricted to a remote sphere, but is valid for any closed surface S surrounding the body (this
follows from the Gauss-Ostrogradsky theorem and Laplace’s equation, ∆Φ = 0, which holds
outside the central body). Gauss’s law (5.35) reflects the “gravitating aspect” of the mass, while
the volume integral (5.22) identifies the mass with the “amount of matter” constituting the
central body. The latter definition would yield M = 0 for any vacuum spacetime (set T = 0 in
Eq. (5.22)) and therefore cannot be used to generalize the concept of mass to strongly relativistic
systems. In particular, one would expect M > 0 for the gravitating mass of Schwarzschild and
Kerr black holes, which are vacuum solutions of the Einstein equation.
Accordingly, Gauss’s law (5.35), rather than the volume integral (5.22), is the good basis
for any attempt to generalize the concept of mass to strongly relativistic spacetimes. Taking a
look at the asymptotic expression (5.21) of the metric tensor of a weakly relativistic statrionary
system, we notice that the mass M appears as the coefficient of the dominant 1/rn−3 term in
the expansion of g00 , so that one could recover it by considering the flux integral of ∂g00 /∂r
over a (n − 2)-dimensional sphere at large r. In order to have a coordinate-invariant definition,
it is more appropriate to consider the 1-form ξ metric-dual to the stationary Killing vector ξ.
Indeed, in any coordinate system adapted to stationarity, i.e. such that the components of ξ are
ξ α = (1, 0, . . . , 0), the components of ξ are ξα = g0α , so that in particular ξ0 = g00 . Instead of
the partial derivative ∂g00 /∂r, a natural coordinate-independent quantity in then the exterior
derivative dξ (cf. Sec. A.4.3). Since the concept of flux is naturally conveyed by the integral of
the Hodge dual (see e.g. Sec. 16.4.7 of Ref. [228]), one arrives at the definition of mass given
below [Eq. (5.36)], named Komar mass. In addition to be coordinate-invariant, it shares the
same property as the Newtonian mass (5.35), namely to be independent from the integration
surface outside the central body, at least within general relativity (Property 5.15 below).
Let (M , g) be an asymptotically flat spacetime of dimension n ≥ 4 that is stationary, with
stationary Killing vector ξ, normalized such that ξ · ξ → −1 near the asymptotic boundary
[Eq. (5.1)]. Given a spacelike closed (n − 2)-surface S ⊂ M , the Komar mass over S is
defined by
n−2
Z
MS := − ⋆(dξ) , (5.36)
16π(n − 3) S
136 Stationary black holes
where
(i) ξ is the 1-form associated to ξ by metric duality (cf. Sec. A.3.3), i.e. the 1-form of
components ξα = gαµ ξ µ ;
(ii) dξ is the exterior derivative of ξ (cf. Sec. A.4.3, especially Eqs. (A.90b) and (A.91b)),
namely the 2-form whose components are
(dξ)αβ = ∂α ξβ − ∂β ξα = ∇α ξβ − ∇β ξα = 2∇α ξβ , (5.37)
the last equality following from the Killing equation (3.19);
(iii) ⋆(dξ) is the (n − 2)-form that is the Hodge dual of the 2-form dξ. The Hodge dual
of any p-form A is defineda as the (n − p)-form ⋆A given by
1 µ1 ...µp
⋆Aα1 ...αn−p := A ϵµ1 ...µp α1 ...αn−p , (5.38)
p!
where ϵ is the Levi-Civita tensor associated with the metric g (cf. Sec. A.3.4);
(iv) the orientation of S (which is required to define the integral of ⋆(dξ) over S ) is
set by the (n − 2)-formb
S
ϵ := ϵ(n, s, ., . . . , . ), (5.39)
| {z }
n−2 slots
where n is any future-directed unit timelike vector field normal to S and s is a
unit spacelike vector field normal to n, orthogonal to n and directed towards the
“exterior” of S in the following sense: assume that S can be embedded into a
spacelike hypersurface Σ that extends towards the asymptotic flat end of (M , g);
then for any point p ∈ S , there exists a curve entirely lying in Σ, starting from p
with tangent vector s|p and extending to the asympotic flat end without crossing S
again.
a
See e.g. Sec. 14.6 of Ref. [464] or Sec. 14.5 of Ref. [228] for an introduction to Hodge duality.
b
See Sec. A.3.4 for the definition of an oriented manifold.
We shall check below (Property 5.11) that the Komar mass (5.36) gives the coefficient M
that appears in the asymptotic expansion (5.31) of the metric tensor.
Remark 3: As the integral of a (n − 2)-form over an oriented (n − 2)-dimensional manifold, formula
(5.36) is well posed. More precisely, on the (n − 2)-dimensional manifold S , the theory of integration
is defined for (n − 2)-forms on S , while ⋆(dξ) is a (n − 2)-form on M . However, any (n − 2)-form ω
on M yields canonically to a unique (n − 2)-form ι∗ ω on the submanifold S , by restricting the action
of ω at each point p ∈ S to vectors tangent to S . One says that ι∗ ω is the pullback of ω to S via the
embedding ι of S in M (cf. Sec. A.2.8). The pullback operator is thus implicit in Eq. (5.36).
Remark 4: The numerical prefactor −(n − 2)/(16π(n − 3)) in the definition (5.36) is adjusted so that
for weakly relativistic isolated objects, the Komar mass coincides with the volume integral of the matter
5.3 Mass and angular momentum 137
energy density, as it will be shown in Property 5.11. For n = 4 (the standard spacetime dimension), this
prefactor is −1/(8π), while for n = 5, it becomes −3/(32π).
Remark 5: The Komar mass is not defined for n ≤ 3. In particular, formula (5.36) is ill-posed for n = 3.
Already in Newtonian gravity, the very concept of gravitational mass is well defined only for n ≥ 4.
Indeed, in spherical symmetry, a solution to the Poisson equation outside the sources (∆Φ = 0) that
decays with the radius r exists only for n ≥ 4: this solution is8 Φ = −K/rn−3 where the constant K is
proportional to the mass M of the central source. For n = 3, the solutions of ∆Φ = 0 are Φ = a ln r + b
(a and b being constant), while for n = 2, they are Φ = ar + b, none of them decaying as r → +∞.
Remark 6: We have given above a heuristic motivation for defining the Komar mass by Eq. (5.36). We
shall see in Chap. 17 a more fundamental approach, which makes the Komar mass appear as the Noether
charge associated with the invariance of the Einstein-Hilbert Lagrangian under the diffeomorphisms
generated by the vector field ξ (cf. Property 17.9).
In view of Eq. (5.38) with p = 2 and the last equality in Eq. (5.37), the Komar mass formula
(5.36) can be written as
n−2
Z
MS = − ∇µ ξ ν ϵµνα1 ...αn−2 , (5.40)
16π(n − 3) S
where ∇µ ξ ν ϵµνα1 ...αn−2 stands for the (n − 2)-form ω defined by
α
ω(u1 , . . . , un−2 ) = ∇µ ξ ν ϵµνα1 ...αn−2 uα1 1 · · · un−2
n−2
for any (n − 2)-tuple of vector fields (u1 , . . . , un−2 ) tangent to S (cf. Remark 3 above).
Instead of integrals of (n − 2)-forms along the (n − 2)-surface S , as in (5.36) and (5.40),
one may express the Komar mass as a flux integral, i.e. the integral of a 2-form contracted with
some “area element”, which is normal to S . Let us introduce the latter first.
Property 5.9: area element normal bivector to a codimension-2 surface
Given a spacelike (n−2)-dimensional surface S ⊂ M , the area element normal bivector
to S is the infinitesimal antisymmetric type-(2,0) tensor field defined on S by
√
dS := (s ∧ n) q dn−2 x, (5.41)
or, in index notation,
√
dS αβ := (sα nβ − nα sβ ) q dn−2 x, (5.42)
where
(i) n and s are unit timelike and spacelike vector fields normal to S obeying the same
properties as in item (iv) of the above definition of the Komar mass; in particular at
each point p ∈ S , (n|p , s|p ) is an orthonormal basis of the timelike plane Tp⊥ S
normal to S (cf. Fig. ??);
8
This is easy to get since in the Euclidean space of dimension n − 1 the Laplace operator is ∆Φ =
dr for spherically symmetric fields Φ(r).
1 d n−2 dΦ
r n−2 dr r
138 Stationary black holes
(ii) s ∧ n stands for the exterior product of s by n: s ∧ n := s ⊗ n − n ⊗ s (hence
formula (5.42));
(iii) dn−2 x := dx1 · · · dxn−2 , where (xa ) = (x1 , . . . , xn−2 ) is a coordinate system on S
such that (∂1 , . . . , ∂n−2 ) is a right-handed basis for the orientation of S defined by
S
ϵ [Eq. (5.39)], i.e. ϵ(n, s, ∂1 , . . . , ∂n−2 ) > 0;
(iv) q := det(qab ) is the determinant w.r.t. (xa ) of the metric q induced on S by the
spacetime metric g.
The bivector dS is independent of the choice of the coordinates (xa ) on S and of the
orthonormal basis (n, s) of Tp⊥ S .
√ √
Proof. The combination q dn−2 x = q dx1 · · · dxn−2 with q := det(qab ) is the volume (area)
element of the Riemannian manifold (S , q); it is thus invariant in any change of coordinates
(xa ) 7→ (x′ a ). Besides, any two orthonormal bases (n, s) and (n′ , s′ ) of Tp⊥ S having the same
orientation and with n and n′ both future-directed are necessarily related by a 2-dimensional
Lorentz boost (cf. Fig. ??):
n′ = cosh ψ n + sinh ψ s
s′ = sinh ψ n + cosh ψ s,
where ψ ∈ R is the boost rapidity. It follows immediately that s′ ∧ n′ = sinh2 ψ n ∧ s +
cosh2 ψ s ∧ n = (cosh2 ψ − sinh2 ψ) s ∧ n = s ∧ n, which shows that the bivector s ∧ n, and
hence dS, does not depend on the choice of the orthonormal basis (n, s).
The area element normal bivector dS appears in the following useful lemma.
Lemma 5.10: flux integral of a 2-form
For any 2-form A defined in the vicinity of a spacelike (n − 2)-dimensional surface S ,
one has Z Z
1
⋆A = Aµν dS µν , (5.43)
S 2 S
where the (n − 2)-form ⋆A is the Hodge dual of A, as defined by Eq. (5.38) with p = 2.
Proof. Using the definition (5.38) with p = 2, we have
Z Z Z
1 µν 1
⋆A = A ϵµνα1 ...αn−2 = A♯ (e(µ) , e(ν) ) ϵ(e(µ) , e(ν) , dx1 , . . . , dxn−2 ),
S 2 S 2 S
where A♯ is the tensor field of components Aαβ , (e(α) ) is an orthonormal tetrad such that
e(0) = n and e(1) = s, (e(α) ) is its dual cobasis and dx1 , . . ., dxn−2 are displacement vectors
forming elementary parallelograms on S ; for instance dxa = dxa ∂a (no summation on a)
for a ∈ {1, . . . , n − 2}. Note that the last equality in the above expression results from the
very definition of the integral of a (n − 2)-form over a (n − 2)-surface. Given the definition of
5.3 Mass and angular momentum 139
the tetrad (e(α) ), (e(2) , . . . , e(n−1) ) is a basis of the tangent space Tp S ; consequently dx1 , . . .,
dxn−2 are linear combinations of e(2) , . . ., e(n−1) . Thanks to the alternate character of ϵ, we
may then restrict the sum over the indices µ and ν to (µ, ν) = (0, 1) and (µ, ν) = (1, 0). Hence
Z Z Z
♯ (0) (1)
⋆A = A (e , e ) ϵ(e(0) , e(1) , dx1 , . . . , dxn−2 ) = A(0)(1) ϵ(n, s, dx1 , . . . , dxn−2 ).
S S S
Now, since (e(α) ) is an orthonormal basis,
A(0)(1) = g (0)(µ) g (1)(ν) A(µ)(ν) = g (0)(0) g (1)(1) A(0)(1) = (−1) × 1 × A(0)(1) = −A(0)(1) ,
with
1
A(0)(1) = A(e(0) , e(1) ) = A(n, s) = Aµν nµ sν = −Aµν sµ nν = − Aµν (sµ nν − nµ sν ).
2
On the other side, ϵ(n, s, . . .), which appears above, is the (n − 2)-form Sϵ defined by Eq. (5.39).
Actually, it is nothing but the volume (area) form of (S , q), i.e. the Levi-Civita tensor of the
metric q (see e.g. Sec. 16.4.3 of Ref. [228]). In particular, Sϵ is independent of the choice of the
orthonormal basis (n, s) of Tp⊥ S . Hence we may write, for dxa = dxa ∂a ,
√
ϵ(n, s, dx1 , . . . , dxn−2 ) = Sϵa1 ...an−2 dxa11 · · · dxn−2
a n−2
= q dx1 · · · dxn−2 .
Gathering the above results and using Eq. (5.42) establishes Eq. (5.43).
Thanks to Lemma 5.10, we may re-express the Komar mass (5.36) as
n−2
Z
MS = − (dξ)µν dS µν . (5.44)
32π(n − 3) S
Using Eq. (5.37), this becomes
n−2
Z
MS =− ∇µ ξν dS µν . (5.45)
16π(n − 3) S
Alternatively, we may express the exterior derivative in terms of partial derivatives (first
equality in Eq. (5.37)) and get
√ √
(dξ)µν dS µν = (∂µ ξν − ∂ν ξµ )(sµ nν − nµ sν ) q dn−2 x = 2∂µ ξν (sµ nν − nµ sν ) q dn−2 x.
Equation (5.44) yields then an expression of MS that can be used for explicit computations:
n−2 √
Z
MS =− ∂µ ξν (sµ nν − nµ sν ) q dn−2 x . (5.46)
16π(n − 3) S
Example 3 (Komar mass in Schwarzschild spacetime): Let us consider the Schwarzschild spacetime
(M , g) introduced in Example 3 of Sec. 2.3. In terms of the ingoing Eddington-Finkelstein coordinates
(t, r, θ, φ) used there, the stationary Killing vector is simply ξ = ∂t . We have n = 4 and let us choose
140 Stationary black holes
S to be a 2-sphere defined by (t, r) = const. The coordinates (xa ) = (x1 , x2 ) spanning S are then
(θ, φ). For the pair of normal vectors to S , we choose n = N −1 ∂t − 2mN/r∂r and s = N ∂r , where
N := (1 + 2m/r)−1/2 . Given the metric (2.5), it is easy to check that g(n, n) = −1, g(n, s) = 0 and
g(s, s) = 1. It is also immediate that g(n, ∂θ ) = g(n, ∂φ ) = 0 and g(s, ∂θ ) = g(s, ∂φ ) = 0, which
proves that n and s are normal to S . Moreover, s points towards the exterior of S (since N > 0) and
(n, s, ∂θ , ∂φ ) is a right-handed basis. All the hypotheses stated below Eq. (5.42) are thus fulfilled and
we may use Eq. (5.46) to evaluate the Komar mass over S . The components ξα of ξ are easily evaluated
via the components (2.5) of g: ξα = gαµ ξ ν = gαµ (∂t )µ = gαt = (−1 + 2m/r, 2m/r, 0, 0). The only
non-zero derivatives ∂µ ξν are then ∂r ξt and ∂r ξr ; they are both equal to −2m/r2 . It follows that the
integrand in Eq. (5.46) is reduced to only two terms:
2m −1 2m
∂µ ξν (sµ nν − nµ sν ) = ∂r ξt (sr nt − nr st ) = −
N × N + 2mN × 0 =− 2 .
r2 r
√
We read on Eq. (2.5) that the metric induced on S is q = r2 dθ2 + r2 sin2 θdφ2 , so that q = r2 sin θ.
Accordingly, Eq. (5.46) leads to
Z Z
1 2m 2 m
MS = − − 2 r sin θ dθ dφ = dθ dφ.
8π S r 4π S
Since the remaining integral is nothing but 4π, we get simply
MS = m. (5.47)
We note that MS does not depend on r, i.e. does not depend on the choice of the sphere S .
More generally, we have:
Property 5.11: Komar mass in the asymptotic expression of the metric
The coefficient M that appears in expression (5.31) of the metric tensor of a stationary
asymptotically flat n-dimensional spacetime
p in some asymptotically Minkowskian coor-
dinates (x ) coincides in the limit r := (x1 )2 + · · · + (xn−1 )2 → +∞ with the Komar
α
mass over the sphere S of constant (x0 , r). In particular, for a weakly relativistic system,
the Komar mass over S is equal to the volume integral of the matter energy density T00 ,
as given by formula (5.22).
Proof. Without any loss of generality, the coordinate system (xα ) leading to the expansion
(5.31) can be chosen to be adapted to stationarity, i.e. such that ∂0 = ξ, where ξ is the stationary
Killing vector. Let us evaluate the Komar mass MS via the integral (5.46), denoting the surface
√ √
element of S by q dn−2 y instead of q dn−2 x to avoid any confusion with the asymptotically
Minkowskian coordinates (xα ). We have ξα = gαµ ξ µ = gαµ δ µ0 = gα0 , with gα0 = g0α given by
formulas (5.31) for large r, so that
M xi xk xi
n−2 Jjk
∂0 ξα = 0, ∂i ξ0 ≃ −(n − 3)αn n−2 and ∂i ξj ≃ αn n−1 δki − (n − 1) 2 .
r r 2 r r
1 n−1
On the other hand, for r → +∞, we have sα ≃ 0, xr , . . . , x r , nα ≃ (1, 0, . . . , 0) and
√ n−2 √
q d y = rn−2 q̄ dn−2 y, where q̄ stands for the determinant of the round metric of the unit
5.3 Mass and angular momentum 141
sphere Sn−2 with respect to the coordinates (y 1 , . . . , y n−2 ). Given that ∂i ξj decays as 1/rn−1 ,
it follows that only the terms involving ∂i ξ0 contribute to the integral (5.46), which becomes
i
√
n−2
Z
x
MS =− ∂i ξ0 × 1 − 0 × 0 rn−2 q̄ dn−2 y
16π(n − 3) S r
i i √ √ n−2
n−2 n−2
Z Z
x x n−2
= αn M q̄ d y = αn M q̄ d y .
16π S |r{zr} 16π S
| {z }
1 Ωn−2
Since αn Ωn−2 = 16π/(n − 2) [cf. Eq. (5.32)], we get MS = M . The statement about a weakly
relativistic system is an immediate consequence of expansion (5.21) being a particular case of
(5.31), given that M in (5.21) has the integral form (5.22).
Volume integral formula for the Komar mass
We are going to derive a volume integral formula for MS , from which one can show the
independence of MS from the choice of S in the vacuum case, thereby generalizing the result
of Example 3. To this aim, we shall make use of the following lemma:
Lemma 5.12: flux integral of the divergence of a 2-form
In a n-dimensional asymptotically flat spacetime (M , g), let Σ be a compact spacelike
hypersurface bounded by two closed (n − 2)-surfaces: an “internal” one, Sint , and an
“external” one, Sext , the exterior direction being that of the asymptotic flat end of (M , g).
Let n be the future-directed unit normal to Σ and dV the normal volume element vector
of Σ defined by [compare Eq. (5.42)]:
√ √
dV α := −nα γ dx1 · · · dxn−1 =: −nα γ dn−1 x, (5.48)
where (x1 , . . . , xn−1 ) stands for a coordinate system on Σ and γ := det(γij ) is the deter-
minant w.r.t. these coordinates of the metric γ induced on Σ by the spacetime metric g, so
√
that γ dx1 · · · dxn−1 is the volume element on Σ. Then, for any 2-form A, we have
Z Z Z
ν
2 ∇ Aµν dV = µ
Aµν dS −µν
Aµν dS µν , (5.49)
Σ Sext Sint
where dS µν is defined by Eq. (5.42) with s pointing to the asymptotic flat end of (M , g)
for both Sext and Sint .
Proof. Let (t, x1 , . . . , xn−1 ) be a local coordinate system adapted to Σ, i.e. such that Σ is the
hypersurface t = 0, and let V be the vector defined by V := A(., n), i.e. Vα := Aαµ nµ . Thanks
to the antisymmetry of A, V is tangent to Σ: n · V = A(n, n) = 0. We have then, using
142 Stationary black holes
again the antisymmetry of A,
√ √
∇ν Aµν dV µ = ∇ν Aνµ nµ γ dn−1 x = (∇ν Vν − Aνµ ∇ν nµ ) γ dn−1 x
√
= (∇µ V µ + Aνµ K νµ + Aνµ nν Dµ ln N ) γ dn−1 x
| {z } | {z }
0 −Vµ
µ i
√
= ∇µ V − V ∂i ln N γ dn−1 x.
In the second line, we have expressed the derivative of n in terms of the extrinsic curvature
tensor K of Σ and of the lapse function N := −n · ∂t , via the identity ∇α nβ = −K αβ −
Dβ ln N nα (see e.g. Eq. (4.24) of Ref. [227]), where Dβ ln N = ∇β ln N + nβ nµ ∇µ ln N . To get
the third line, we have used the symmetry of K to set Aνµ K νµ = 0 and the property Vµ nµ = 0
to write Vµ Dµ ln N = Vµ ∇µ ln N = V µ ∂µ ln N = V i ∂i ln N (i ∈ {1, . . . , n − 1}). Let us now
use formula (A.74) for the divergence of V , along with V 0 = 0 (V is tangent to Σ) and the
√ √
relation −g = N γ between the determinants of the metrics g and γ (see e.g. Eq. (5.55) of
Ref. [227])
1 √ 1 √
∇µ V µ = √ ∂µ −gV µ = √ ∂i N γV i
−g N γ
1 √ i
γV + V i ∂i ln N = Di V i + V i ∂i ln N,
= √ ∂i
γ
where Di V i is the divergence of V considered as a vector field on (Σ, γ), D standing for the
Levi-Civita connection of γ. Hence we get
√
∇ν Aµν dV µ = Di V i γ dn−1 x.
We may then apply the Gauss-Ostrogradsky theorem to V and write the integral of the
divergence of A as
i√ √
Z Z Z
ν µ n−1
∇ Aµν dV = Di V γ d x = σ i Vi q dn−2 x
Σ ZΣ Z ∂Σ
i √ n−2 √
= s Vi q d x − si Vi q dn−2 x,
Sext Sint
where σ is the unit vector normal to the boundary ∂Σ and outward-pointing with respect to
Σ. In the present case, ∂Σ has two connected components: Sint and Sext , with σ = −s on
Sint and σ = s on Sext , hence the last equality. Now, using again the antisymmetry of A,
1
si Vi = sµ Vµ = sµ Aµν nν = Aµν (sµ nν − nµ sν ).
2
√
In view of expression (5.42) for dS µν , we thus get 2si Vi q dn−2 x = Aµν dS µν , so that Eq. (5.49)
follows.
Let us apply Lemma 5.12 to the 2-form A := dξ, i.e. to the 2-form of components Aαβ :=
∇α ξβ − ∇β ξα = 2∇α ξβ [cf. Eq. (5.37)]. The 1-form ∇ν Aµν which appears in the left-hand
5.3 Mass and angular momentum 143
side of Eq. (5.49) is then, up to a sign, the metric dual of a vector field J (ξ) called the Komar
current of ξ: ∇ν Aµν = −J (ξ)µ , where
J (ξ)α := ∇µ (∇µ ξ α − ∇α ξ µ ) . (5.50)
The Komar current is actually a conserved current, independently of the Killing nature of ξ:
Property 5.13: Conservation of the Komar current
For any vector field ξ defined on the spacetime (M , g) (not necessarily a Killing vector
field), the Komar current defined by Eq. (5.50) is conserved:
∇µ J (ξ)µ = 0. (5.51)
Proof. We have, from the definition (5.50), J (ξ)α = −∇ν Aαν with Aαβ := ∇α ξ β − ∇β ξ α .
Let (xα ) be a coordinate system on M . Since A is antisymmetric, its divergence can be
√ √
expressed in terms of partial derivatives as ∇ν Aαν = ∂ν ( −gAαν ) / −g, where g is the
determinant of g with respect to (xα ). Similarly the divergence of J (ξ) can be written as
√ √
∇µ J (ξ)µ = ∂µ ( −gJ (ξ)µ ) / −g. Combining the above formulas leads to ∇µ J (ξ)µ =
√ √
−∂µ ∂ν ( −gAµν ) / −g. Since ∂µ ∂ν = ∂ν ∂µ and Aµν = −Aνµ , we get immediately ∇µ J (ξ)µ =
0.
Thanks to the identities Aαβ = 2∇α ξβ and J (ξ)α = −∇µ Aαµ , Lemma 5.12 yields
Z Z Z
µ µν
− J (ξ)µ dV = ∇µ ξν dS − ∇µ ξν dS µν .
Σ Sext Sint
Using expression (5.45) of the Komar mass, we get
n−2
Z
MSext = MSint + J (ξ)µ dV µ . (5.52)
16π(n − 3) Σ
Thanks to the Killing equation (3.19), expression (5.50) for the Komar current J (ξ) simplifies
to J (ξ)α = −2∇µ ∇α ξ µ . Now, since ξ is a Killing vector, Eq. (3.84) holds: ∇µ ∇α ξ µ = Rαµ ξ µ .
Hence we get
J (ξ)α = −2Rαµ ξ µ . (5.53)
Pluging this relation into formula (5.52) and using expression (5.48) for dV µ , we arrive at the
following property.
Property 5.14: volume integral formula for the Komar mass
Let (M , g) be an asymptotically flat stationary spacetime of dimension n ≥ 4, with
stationary Killing vector ξ. Let Σ be a compact spacelike hypersurface bounded by two
closed (n − 2)-surfaces: an “internal” one, Sint , and an “external” one, Sext , the exterior
direction being that of the asymptotic flat end of (M , g). The Komar masses over Sext
144 Stationary black holes
and Sint are then related by
n−2 √
Z
MSext = MSint + R(ξ, n) γ dn−1 x, (5.54)
8π(n − 3) Σ
√
where R is the Ricci tensor of g, n is the future-directed unit normal to Σ and γ dn−1 x is
the volume element on Σ induced by g. If g obeys the Einstein equation (1.42) with some
energy-momentum tensor T and Λ = 0, then the above formula may be rewritten as
Z
n−2 T √ n−1
MSext = MSint + T (ξ, n) − ξ·n γ d x, (5.55)
n−3 Σ n−2
where T := trg T = g µν Tµν .
Remark 7: If Σ is a compact spacelike hypersurface bounded by a single closed (n − 2)-surface, Sext
say, formulas (5.54) and (5.55) still hold, provided that MSint is replaced by zero. This follows directly
from Eq. (5.49) with the integral on Sint set to zero, since the boundary of Σ, which appears in the
Gauss-Ostrogradsky theorem, is only made of Sext .
Example 4 (Komar mass of a static star): We may use formula (5.55) to check that at the Newtonian
limit, the Komar mass over a surface Sext surrounding a star reduces to the volume integral of the
matter mass density, i.e. to the classical expression of the mass in Newtonian gravity. To this aim, we set
n = 4 and assume that the spacetime is strictly static, i.e. that ξ is timelike and hypersurface-orthogonal
in all M . A static fluid star is then defined by the energy-momentum tensor T being a perfect fluid one:
T = (ρ + p)u ⊗ u + pg, where ρ is the fluid-comoving energy density, p is the fluid pressure and the
fluid 4-velocity u is collinear to the static Killing field ξ:
1
u = ξ with V := −ξ · ξ,
p
V
the value of V being determined to ensure u · u = −1. Let us choose for Σ a hypersurface orthogonal
to ξ with an outer boundary Sext but no inner boundary Sint . Since Σ is orthogonal to ξ, we have
necessarily n = u, so that T (ξ, n) = V T (u, u) = V ρ and ξ · n = V u · u = −V . Given that
T = 3p − ρ, formula (5.55) with n = 4 and MSint = 0 (cf. Remark 7 above) results in
Z
3p − ρ √ 3 √
Z
MSext = 2 Vρ− (−V ) γd x= V (ρ + 3p) γ d3 x.
Σ 2 Σ
Now,√for a weakly relativistic star, |p| ≪ ρ and in coordinates (t, xi ) such that ξ = ∂t , we have
√
V = −ξ · ξ = −gtt ≃ 1 + Φ, where Φ is the Newtonian gravitational potential, i.e. the solution to
the Poisson equation ∆Φ = 4πρ that vanishes at infinity. We may then write
√ 3 √
Z Z
MSext ≃ ρ γd x+ ρΦ γ d3 x. (5.56)
Σ Σ
The first term is the volume integral of the fluid mass density; the second term is negative (since Φ < 0)
and represents the total gravitational potential energy of the star, or equivalently the gravitational binding
energy of the star. Its presence should not be a surprise given the mass-energy equivalence in relativity.
However, at the Newtonian limit, this term can be neglected in front of the first one and formula (5.56)
reduces to the classical expression of mass of a Newtonian body.
Since in vacuum T = 0, an immediate corollary of Property 5.14 is:
5.3 Mass and angular momentum 145
Property 5.15: independence of the Komar mass from the integration surface
Two (n − 2)-surfaces Sext and Sint that are connected by a spacelike hypersurface Σ
in a vacuum region of an asymptotically flat stationary spacetime ruled by the Einstein
equation with Λ = 0 share the same Komar mass:
MSext = MSint . (5.57)
The concept of “total mass” of a stationary and asymptotically flat spacetime is well captured
by the Komar mass at infinity:
Property 5.16: Komar mass at infinity
Let (M , g) be a stationary and asymptotically flat spacetime of dimension n ≥ 4. Let (xα )
be an asymptotically Minkowskian coordinate system (cf. Sec. 5.3.2) such that ∂0 = ξ (the
stationary
p Killing vector). Let Sr be the (n − 2)-surface defined by (x , r) = const, where
0
r := (x1 )2 + · · · + (xn−1 )2 . If the Komar mass MSr converges when r → +∞, we call
M∞ := lim MSr (5.58)
r→+∞
the Komar mass at infinity or total mass of the spacetime (M , g). Moreover, if g obeys
the Einstein equation with Λ = 0 and if the energy-momentum tensor T vanishes for r
larger than some threshold r0 , one has M∞ = MSr for any r ≥ r0 .
Proof. Since for r ≥ r0 , Sr is located in a vacuum region of spacetime, the value of MSr is
independent of r by Property 5.15, hence M∞ = MSr .
Example 5 (Total mass of Schwarzschild spacetime): We have already noticed in Example 3 that in
the Schwarzschild spacetime, the Komar mass MS is equal to the mass parameter m of Schwarzschild’s
metric, whatever the 2-surface S [Eq. (5.47)]. We thus conclude that M∞ = m.
Example 6 (Vanishing total mass of cubic Galileon black holes): In a stationary and asymptotically
flat 4-dimensional spacetime, if dξ decays faster that 1/r2 , then M∞ = 0. This happens for black holes
in a scalar-tensor theory of gravity known as the cubic Galileon, as shown in Sec. 5.1 of Ref. [490].
Actually, for these black holes, the spacetime metric tends to Minkowsky metric as fast as 1/r4 when
r → +∞, which implies that dξ decays as 1/r5 . This rapid convergence to flat space can be interpreted
as a screening of the black hole’s gravity by the surrounding scalar field.
Historical note : The Komar current (5.50) associated to any vector field of spacetime has been
introduced by Arthur Komar in 1959 [319]; he showed that this current is conserved (divergence-free)
[Eq. (5.51)]. In a subsequent study published in 1962 [320], Komar showed that the Komar current
J (ξ) associated to a Killing vector ξ is, up to some factor, the Ricci tensor applied to ξ [Eq. (5.53)] and
that the converved integral representing the flux of J (ξ) through a one-ended hypersurface Σ can
be transformed into a surface integral on the boundary S of Σ, which is nothing but Eq. (5.45) with
n = 4. He proved that such an integral is independent of the surface S as soon as S is located outside
146 Stationary black holes
the sources of the Einstein equation (Property 5.15). The generalization to dimensions n ≥ 4 has been
performed by Robert Myers and Malcolm Perry in 1986 [376] [cf. their Eq. (4.1), where N = n − 1].
5.3.4 N+1 decomposition
Let N := n − 1, n being the spacetime dimension. Given a spacelike hypersurface Σ, with
future-directed unit normal n, the N +1 decomposition of ξ with respect to Σ is the pair
(N, β) where N is a scalar field on Σ and β is a vector field tangent to Σ such that
ξ = N n + β. (5.59)
In other words, β is the orthogonal projection of ξ on Σ and N n is the part of ξ along n. N
and β are uniquely determined by N = −n · ξ and β = ξ + (n · ξ)n. In the language of the
N +1 formalism of general relativity, N is called the lapse function and β the shift vector (cf.
e.g. Ref. [227]).
Similarly, the N +1 decomposition of the energy-momentum tensor T with respect to
the hypersurface Σ is the triplet (E, p, S), where E is a scalar field, p is a 1-form satisfying
⟨p, n⟩ = 0 and S is a symmetric type-(0, 2) tensor field satisfying S(n, .) = 0, such that9
T = En ⊗ n + n ⊗ p + p ⊗ n + S. (5.60)
E, p and S are uniquely determined by
E = Tµν nµ nν , pα = −nµ Tµν γ να , Sαβ = Tµν γ µα γ νβ , (5.61)
where γ αβ = δ αβ + nα nβ stands for the orthogonal projector onto Σ. Physically, E is the
matter energy density measured by the observer O of 4-velocity n, p is the matter momentum
density measured by O and S is the matter stress tensor measured by O.
Property 5.17: N +1 expression of the Komar mass
Under the same hypotheses as Property 5.14, one can rewrite Eq. (5.55) as
Z
1 n−2 √ n−1
MSext = MSint + N E+ S − ⟨p, β⟩ γ d x, (5.62)
Σ n−3 n−3
where S := g µν Sµν = γ ij Sij is the trace of the stress tensor S.
Proof. From Eq. (5.60), the trace of the energy-momentum tensor is T := g µν Tµν = Enµ nµ +
nµ pµ + pµ nµ + S = −E + S, since nµ nµ = −1 and pµ nµ = 0. In view of Eqs. (5.59) and (5.60),
we can then rewrite the integrand in the r.h.s. of Eq. (5.55) as
T S−E N
T (n, ξ)− n·ξ = N T (n, n) + T (n, β) − n·ξ = [(n − 3)E + S]−⟨p, β⟩.
n−2 | {z } | {z } n − 2 |{z} n − 2
E −⟨p,β⟩ −N
This establishes Eq. (5.62).
9
See e.g. Sec. 5.1.2 of Ref. [227].
5.3 Mass and angular momentum 147
Example 7 (Komar mass of a weakly relativistic star): Let us check that formula (5.62) reduces to
(5.22) for a weakly relativistic star in a n-dimensional spacetime. In this case, we have N ≃ 1 and β → 0.
Furthermore, the matter constituting the star shall be weakly relativistic as well, which implies that
the trace S of the stress tensor is negligible in front of the energy density E. Under these hypotheses,
Eq. (5.62) with MSint = 0 yields MSext = M , where M is given by (5.22), given that T00 ≃ E.
5.3.5 Link between the Komar mass and the ADM mass
For generic (i.e. not necessarily stationary) asymptotically flat spacetimes, there exists the
concept of ADM energy-momentum, ADM standing for Arnowitt, Deser and Misner [20]. It
arises from the Hamiltonian formulation of general relativity developed by these authors. For
a spacelike hypersurface Σ that is asymptotically Euclidean, the ADM energy of Σ is defined
as some flux over a closed (n − 2)-sphere S of quantities involving derivatives of the metric
γ induced by g on Σ, in the limit where the coordinate radius of S tends to infinity, while the
3 components of the ADM momentum of Σ are defined by similar integrals with quantities
involving the extrinsic curvature tensor K of Σ (see e.g. [298], Chap. 8 of [227], Sec. 4.3 of
[416], Sec. 3.7 of [464] or Sec. 3.2.1 of [471] for details). The ADM mass of Σ is then defined as
the Minkowskian norm of the ADM energy-momentum. For a vanishing ADM momentum,
the ADM mass coincides with the ADM energy. One may then ask the relation with the Komar
mass if the spacetime is stationary. The answer is very simple:
Property 5.18: equality of Komar and ADM masses (Beig 1978)
If (M , g) is a stationary and asymptotically flat n-dimensional spacetime, its Komar mass
at infinity equals the ADM mass of any spacelike hypersurface Σ that is asymptotically
orthogonal to the stationary Killing vector ξ.
For n = 4, the proof can be found in R. Beig’s article [49] (see also Ref. [28] for a different proof),
while the proof for n > 4 has been obtained by H. Barzegar, P.T. Chruściel and M. Hörzinger
[43].
5.3.6 Komar angular momentum
As a stationary Killing vector gave birth to the Komar mass, an axisymmetric Killing vector
gives birth to an invariant integral quantity: the Komar angular momentum. Before discussing
it, let us first introduce the concept of axisymmetry:
A spacetime (M , g) is called axisymmetric iff (i) it is invariant under the action of the
rotation group SO(2) (or equivalently U(1)) and (ii) the orbits of the group actiona are
everywhere spacelike curves or (ii’) (M , g) admits a conformal completion (cf. Sec. 4.3)
and the orbits of the group action are spacelike in the vicinity of the conformal boundary
I . Since SO(2) is a one-parameter Lie group, its action is generated by a single Killing
vector η, which is unique up to some scaling constant. The orbits of the SO(2) action being
closed curves, we shall fix the scaling constant so that the group parameter φ associated
148 Stationary black holes
with η has the standard periodicity 2π. If non-empty, the set of points of M that are left
invariant by the SO(2) action is called the rotation axis. Equivalently, the rotation axis is
the set of points where the Killing vector η vanishes. Usually, if the dimension of M is
n, the rotation axis is a submanifold of M of dimension n − 2. Moreover, it is a timelike
submanifold in the region where the SO(2) orbits are spacelike.
a
Cf. Sec. 3.3.1 for the concepts of group action and orbit.
Let (M , g) be an axisymmetric spacetime of dimension n ≥ 3, with axisymmetric Killing
vector η. Given a spacelike closed (n − 2)-surface S ⊂ M , the Komar angular momen-
tum over S is defined by
Z
1
JS := ⋆(dη) , (5.63)
16π S
where ⋆(dη) is the Hodge dual of the 2-form dη given by
(dη)αβ = ∂α ηβ − ∂β ηα = ∇α ηβ − ∇β ηα . (5.64)
Remark 8: Besides the change ξ ↔ η, we notice a difference by a factor −(n − 2)/(n − 3) between
the r.h.s. of (5.36) and (5.63). For n = 4, this factor is −2 and is known as Komar’s anomalous factor
(see Ref. [309] for a discussion). Fundamentally, the independence of formula (5.63) from the spacetime
dimension n stems from the fact that the SO(2) action generated by η is effective only in a 2-dimensional
space: the orthogonal complement of the rotation axis.
Remark 9: The Komar angular momentum is well defined for any spacetime dimension n ≥ 3, while
the definition of the Komar mass requires n ≥ 4 (cf. Sec. 5.3.3).
Formulas (5.45) and (5.46) for the Komar mass rely only on the fact that ξ is a Killing vector.
In an axisymmetric spacetime, we may therefore replace ξ by η and the numerical prefactor by
1/(16π) to get the following expressions for the Komar angular momentum:
Z
1
JS = ∇µ ην dS µν (5.65)
16π S
and
√
Z
1
JS = ∂µ ην (sµ nν − nµ sν ) q dn−2 x . (5.66)
16π S
As for the Komar mass (cf. Property 5.11), one can read the Komar angular momentum in
the asymptotic expansion of the metric tensor in appropriate coordinates:
Property 5.19: Komar angular momentum in the asymptotic expression of the
metric
Let (M , g) be an asymptotically flat spacetime of dimension n ≥ 4 that is axisymmetric and
stationary, with axisymmetric Killing vector η. Let (xα ) be an asymptotically Minkowskian
coordinate system in which g admits the expansion (5.31) and that is adapted to the
5.3 Mass and angular momentum 149
axisymmetry in the sense that η = −x2 ∂x1 +x1 ∂x2 (i.e. the rotation axis of the SO(2) action
is the (n − 2)-dimensional surface (x1 , x2 ) = (0, 0)). Then
p the coefficient J(1) = J12 in
the expansion (5.31)-(5.34) of g coincides in the limit r := (x1 )2 + · · · + (xn−1 )2 → +∞
with the Komar angular momentum JS over the sphere S of constant (x0 , r). In particular,
for a weakly relativistic system, JS = J12 , where J12 is given by the volume integral (5.23).
Proof. Let us use formula (5.66) to evaluate JS . From η = −x2 ∂x1 + x1 ∂x2 and the asymptotic
expression (5.31) of g, we get the following components of η with respect to the coordinates
(xα ) in the limit r → +∞: ηα = gαµ η µ = gα1 (−x2 ) + gα2 x1 ≃ (η0 , −x2 , x1 , 0, . . . , 0), with
J(1) x2 2 (−J(1) )x1 1 8πJ(1) (x1 )2 + (x2 )2
2 1 n−2
η0 = −g01 x + g02 x = αn − n−1 x + x = − ,
2 r rn−1 Ωn−2 rn−1
where we have used Eq. (5.32) to express αn . The integrand in Eq. (5.66) is then, taking into
account that ∂0 = 0 (stationarity) and s0 = 0 (s tangent to the hypersurface x0 = const):
∂µ ην (sµ nν − nµ sν ) = ∂i η0 si n0 + ∂i (−x2 )(si n1 − ni s1 ) + ∂i x1 (si n2 − ni s2 )
| {z } |{z}
−δ 2 i δ1i
= n0 si ∂i η0 + 2(s1 n2 − s2 n1 ).
Since si ≃ xi /r and ∂i r = xi /r, we deduce from the above expression of η0 that
8π(n − 3)J(1) (x1 )2 + (x2 )2
si ∂i η0 = .
Ωn−2 rn
In the proof of Property 5.11 for the Komar mass, it was sufficient to consider only ∂i ξ0 and to
express the components of n to the lowest order in 1/r, namely nα ≃ (1, 0, . . . , 0). Here we
have ∂i ηj = O(1), so that we must consider the next order in the spatial components ni of n.
They are given by ni = −β i /N (e.g. Eq. (5.36) in Ref. [227]) where N is the lapse function and
the β i ’s are the components of the shift vector (cf. Sec. 5.3.4). For r large, we may take N ≃ 1
and β i ≃ βi = g0i (Eq. (5.49) in Ref. [227]). Hence n1 ≃ −g01 = −8πJ(1) x2 /(Ωn−2 rn−1 ) and
n2 ≃ −g02 = 8πJ(1) x1 /(Ωn−2 rn−1 ). On the other side, n0 = 1/N ≃ 1. Hence the integrand
becomes
(x1 )2 + (x2 )2
1
x1 x2 x2
µ ν µ ν 8πJ(1) x
∂µ ην (s n − n s ) = (n − 3) +2 × n−1 − × − n−1
Ωn−2 rn r r r r
1 2
8π(n − 1)J(1) (x ) + (x ) 2 2
= .
Ωn−2 rn
Accordingly, formula (5.66) yields
(n − 1)J(1) (x1 )2 + (x2 )2 √ n−2
Z
JS = q d y,
2Ωn−2 S rn
√
where we have denoted the surface element of S by q dn−2 y to avoid any confusion with
the coordinates (xα ). Let us use spherical coordinates (y a )1≤a≤n−2 = (θ1 , θ2 , . . . , θn−3 , φ) on
150 Stationary black holes
S , i.e. coordinates such that all the θk ’s span (0, π), φ spans (0, 2π) and the asymptotically
Minkowskian coordinates (xi )1≤i≤n−1 are expressible as10
x1 = r sin θ1 sin θ2 · · · sin θn−4 sin θn−3 cos φ
x2 = r sin θ1 sin θ2 · · · sin θn−4 sin θn−3 sin φ
x3 = r sin θ sin θ · · · sin θ
1 2 n−4 cos θn−3
x4 = r sin θ1 sin θ2 · · · cos θn−4
..
.
xn−2 = r sin θ1 cos θ2
xn−1 = r cos θ
1
Then (x1 )2 +(x2 )2 = r2 sin2 θ1 · · · sin2 θn−3 . Given that S is the (n−2)-sphere (x0 , r) = const,
√ √
one has, for r large, q ≃ rn−2 q̄, where q̄ is the determinant with respect to the coordinates
(y a ) of the round metric on Sn−2 :
√
q̄ = sinn−3 θ1 sinn−4 θ2 · · · sin2 θn−4 sin θn−3 . (5.67)
It follows that
(n − 1)J(1)
Z
JS = sinn−1 θ1 sinn−2 θ2 · · · sin4 θn−4 sin3 θn−3 dθ1 · · · dθn−3 dφ
2Ωn−2 S
(n − 1)J(1)
= × 2πIn−1 In−2 · · · I3 , (5.68)
2Ωn−2
where we have introduced Z π
Ik := sink θ dθ.
0
Writing sin θ = sin
k k−1
θ sin θ and integrating by part, we get Ik = (k − 1)Ik−2 /k for k ≥ 2.
Hence
n−2 n−3 3 2 2
In−1 In−2 · · · I3 = In−3 × In−4 × · · · × I2 × I1 = In−3 In−4 · · · I1 .
n−1 n−2 4 3 n−1
√
Now, from its definition as the area of Sn−2 , Ωn−2 is the integral of q̄ over the spherical coor-
√
dinates (θ1 , . . . , θn−3 , φ). In view of expression (5.67) for q̄, we get Ωn−2 = 2πIn−3 In−4 · · · I1 .
Equation (5.68) reduces then to JS = J(1) .
The volume integral formula (5.54) for the Komar mass relies only on ξ being a Killing
vector. We may therefore transpose it to the Komar angular momentum:
10
Note that these formulas generalize the standard expression (4.2) with (x, y, z) = (x1 , x2 , x3 ) to n ≥ 4.
5.3 Mass and angular momentum 151
Property 5.20: volume integral formula for the Komar angular momentum
Let (M , g) be an axisymmetric spacetime of dimension n ≥ 3, with axisymmetric Killing
vector η. Let Σ be a compact spacelike hypersurface bounded by two closed (n − 2)-
surfaces: an “internal” one, Sint , and an “external” one, Sext , the exterior direction being
that of the asymptotic flat end of (M , g). The Komar angular momenta over Sext and Sint
are then related by
√
Z
1
JSext = JSint + J (η) · n γ dn−1 x, with J (η)α = −2Rαµ η µ , (5.69)
16π Σ
√
where n is the future-directed unit normal to Σ and γ dn−1 x is the volume element on Σ
induced by g. The vector J (η) is the Komar current of η [compare Eq. (5.53)]. If g obeys
the Einstein equation (1.42) with some energy-momentum tensor T and Λ = 0, then the
above formula may be rewritten as
Z
T √ n−1
JSext = JSint − T (η, n) − η·n γ d x, (5.70)
Σ n−2
where T := trg T = g µν Tµν .
Remark 10: Since the axisymmetry generator η is spacelike, the spacelike hypersurface Σ can be
chosen so that η is tangent to it; one has then η · n = 0 and the second term in the integrand of Eq. (5.70)
vanishes identically. Moreover, if one expresses T via the N +1 decomposition (5.60), one can write the
first term as T (η, n) = −⟨p, η⟩, so that formula (5.70) simplifies to
√
Z
JSext = JSint + ⟨p, η⟩ γ dn−1 x. (5.71)
Σ
As for the Komar mass (Property 5.15), it follows immediately from Eq. (5.70) that
Property 5.21: independence of the Komar angular momentum from the integra-
tion surface
Two (n − 2)-surfaces Sext and Sint that are connected by a spacelike hypersurface Σ in
a vacuum region of an axisymmetric spacetime ruled by the Einstein equation share the
same Komar angular momentum:
JSext = JSint . (5.72)
As for the total mass (cf. Property 5.16), the concept of “total angular momentum” of
an axisymmetric and asymptotically flat spacetime is well captured by the Komar angular
momentum at infinity:
152 Stationary black holes
Property 5.22: Komar angular momentum at infinity
Let (M , g) be an axisymmetric and asymptotically flat spacetime of dimension n ≥ 3.
Let (xα ) be an asymptotically Minkowskian coordinate system (cf. Sec. 5.3.2) such that
the axisymmetric Killing vector η is tangent to the hypersurfaces x0 = const. Let Sr be
the (n − 2)-surface defined by (x0 , r) = const, where r := (x1 )2 + · · · + (xn−1 )2 . If the
p
Komar angular momentum JSr converges when r → +∞, we call
J∞ := lim JSr (5.73)
r→+∞
the Komar angular momentum at infinity or total angular momentum of the space-
time (M , g). Moreover, if g obeys the Einstein equation with Λ = 0 and if the energy-
momentum tensor T vanishes for r larger than some threshold r0 , one has J∞ = JSr for
any r ≥ r0 .
Remark 11: There is no concept of “ADM angular momentum” (due to the so-called supertranslation
ambiguity at spatial infinity, cf. e.g. Sec. 6 of Ref. [523] or Sec. 8.5 of Ref. [227]). Consequently, there is
no point in comparing the Komar angular momentum with an “ADM” counterpart, contrary to what
was done in Sec. 5.3.5 for the mass.
5.4 The event horizon as a Killing horizon
In view of Property 5.2, let us examine successively the only two allowed cases for a connected
component11 H of the event horizon of a stationary black hole: either the Killing vector ξ
is null on all H (Sec. 5.4.1) or ξ is spacelike in some part (possibly all) of H (Sec. 5.4.2). In
both cases, we shall see that, under some hypotheses, H is a Killing horizon; this is immediate
when ξ is null on H , but not trivial otherwise.
5.4.1 Null stationary Killing field on H : the staticity theorem
By Lemma 2.3 (Sec. 2.3.4), if the Killing vector field ξ is null on H , it is necessarily tangent to
the null geodesic generators of H and therefore collinear to the null normals ℓ of H . From
the definition of a Killing horizon (cf. Sec. 3.3.2), we get immediately:
Property 5.23: event horizon as a Killing horizon for ξ null on it
In a stationary spacetime (M , g) containing a black hole, if the stationary Killing vector
ξ is null on the whole of a connected component H of the event horizon, then H is a
11
In the most astrophysically relevant case, namely that of a stationary black hole in a 4-dimensional vacuum
spacetime, the event horizon is a connected hypersurface. However, we shall see that disconnected event horizons
exist for some stationary black holes in 5-dimensional spacetimes; they exist as well in 4 dimensions if one allows
for an electric charge (Majumdar-Papapetrou black holes mentioned in Remark 5 in Sec. 5.2.3).
5.4 The event horizon as a Killing horizon 153
Killing horizon with respect to ξ.
It follows that, on H , the Killing vector field ξ is hypersurface-orthogonal, being normal
to H . If this holds for all the connected components of the event horizon, then, under certain
hypotheses (especially that the electrovacuum Einstein equation is fulfilled), one can go further
and prove that ξ must actually be hypersurface-orthogonal everywhere in M , i.e. that (M , g)
is static, according to the definition given in Sec. 5.2.1:
Property 5.24: staticity theorem (Sudarsky & Wald 1993 [468])
Let (M , g) be an asymptotically flat stationary spacetime of dimension n ≥ 4 that contains
a black hole of event horizon H so that the stationary Killing vector ξ is null on all H .
By Property 5.23, H is a Killing horizon. If H is non-degenerate (i.e. has a non-vanishing
surface gravity κ, cf. Sec. 3.3.6) and g obeys the vacuum or electrovacuum Einstein equation
(cf. Sec. 1.5.2), then the spacetime is static and the domain of outer communications (cf.
Sec. 4.4.2) is strictly static.
Proof. For simplicity, we give only the proof for the case of the vacuum Einstein equation
(1.44), referring to Sudarsky & Wald’s article [468] for the electrovacuum case. Since H is a
non-degenerate Killing horizon, it is contained in a bifurcate Killing horizon (cf. Property 3.15
and Remark 3 on p. 87). Let S be the corresponding bifurcation surface. Thanks to a theorem
by Chruściel & Wald (Theorem 4.2 in Ref. [129]), the domain of outer communications can
be foliated by a family of spacelike hypersurfaces (Σt )t∈R that are asymptotically flat, are
asymptotically orthogonal to ξ, have S as inner boundary, are Lie-dragged12 to each other
by ξ and are maximal. The last property means that the extrinsic curvature tensor K of
each hypersurface Σt has a vanishing trace: K i i = 0, or equivalently, that the unit normal n
to Σt is divergence-free : ∇µ nµ = 0 (cf. Sec. 10.2.2 of Ref. [227]). Let us consider the N + 1
decomposition with respect to Σt introduced in Sec. 5.3.4. The maximality property K i i = 0
allows one to simplify one of the Einstein equations in the N + 1 decomposition, namely the
momentum constraint (see e.g. Chap. 5 of Ref. [227]), which reduces to Dj Kij = 8πpi , where
D is the Levi-Civita connection associated to the metric γ induced by g on Σt and p is the
matter momentum density in the hypersurface Σt (cf. Sec. 5.3.4). Since we consider the vacuum
case, we have p = 0. Hence the momentum constraint reduces to Dj Kij = 0. Contracting
with the shift vector β arising from the orthogonal split (5.59) of ξ, we get
0 = Dj Kij β i = Dj Kij β i − Kij Dj β i = Dj Kij β i − N Kij K ij ,
where the last equality results from the symmetry of Kij and the identity Dj β i +Di β j = 2N K ij ,
which holds for the shift vector β associated with the stationary Killing vector ξ (see e.g.
Eqs. (5.66) and (5.67) in Ref. [227] with ∂γij /∂t = 0). Let us integrate the above equation over
Σt and use the Gauss-Ostrogradsky theorem for the divergence term Dj (Kij β i ), taking into
account that the boundary of Σt is made of the bifurcation sphere S (inner boundary) and the
12
Σt+dt is the image of Σt by the infinitesimal displacement dt ξ.
154 Stationary black holes
sphere “at infinity” S∞ (outer boundary), the latter being a (n − 2)-sphere in Σt of coordinate
radius r → +∞ (r is defined from the asymptotic flatness property); we get
i√ i√ √
Z Z Z
j n−2 j n−2
s Kij β q d x − s Kij β q d x − N Kij K ij γ dn−1 x = 0,
S∞ S Σt
where s is the unit normal to S∞ and S in Σt , oriented towards the asymptotic flat end of
Σt . Now, each of the two surface integrals vanishes identically: the first one because on the
asymptotically flat slice Σt , K decays as O(r−2 ) for r → +∞ and, still in this limit, β → 0
since ξ is asymptotically orthogonal to Σt [cf. the orthogonal decomposition (5.59)]. As for the
integral over S , it is zero because β|S = 0. Indeed, since S is the bifurcation surface of a
bifurcate Killing horizon with respect to ξ (Property 5.23), one has ξ|S = 0 (Property 3.12),
which implies via Eq. (5.59) both N |S = 0 and β|S = 0. We therefore get
√
Z
N Kij K ij γ dn−1 x = 0.
Σt
Given that N > 0 on Σt \ S and Kij K ij ≥ 0 on Σt (for Kij K ij = γ ik γ jl Kij Kkl with
γ being a Riemannian metric), the vanishing of the above integral implies Kij = 0. Now,
K = −(2N )−1 Lm γ (cf. Eq. (4.30) in Ref. [227]), where m := N n is the evolution vector
along the normal to Σt (cf. Sec. 4.3 of Ref. [227]). The vanishing of K implies then
γ̇ := Lm γ = 0 and π̇ := Lm π = 0, (5.74)
where π is the momentum canonically conjugate to γ in the ADM Hamiltonian formulation of
general relativity [20, 196]. π is a tensor density that is related to the extrinsic curvature tensor
√
K by π ij = γ(K kk γ ij − K ij ). Hence, K = 0 implies π = 0 so that π̇ = 0 in Eq. (5.74) is
trivially fulfilled. In the Hamiltonian formulation, the spacetime dynamics is entirely described
by the “time development” (i.e. the variation across the Σt foliation) of (γ, π), which are
the (q, p) pairs of Hamiltonian mechanics. Hamilton’s equations (5.74) shows that (γ, π) do
not evolve at all. This means that m is a Killing vector of the spacetime metric g. Since
m = N n is by construction timelike and hypersurface-orthogonal, this proves that the part of
spacetime foliated by the (Σt )t∈R , i.e. the domain of outer communications, is strictly static.
The connection between m and ξ is given by Eq. (5.59): ξ = m + β. Hence the Killing vectors
ξ and m coincide iff the shift vector β vanishes. If β would not vanish identically, it would be
a Killing vector itself, as the difference ξ − m of two Killing vectors. However, as seen above, β
vanishes at the inner boundary S of Σt and at the “outer boundary”, i.e. for r → +∞. In most
cases, these constraints will result in β = 0 (for instance, they exclude β being a rotational
Killing vector), so that we conclude that m = ξ.
Remark 1: In the original formulation by Sudarsky & Wald (Theorem 2 of Ref. [468]), the non-degeneracy
of H is replaced by H being part of a bifurcate Killing horizon. However, as mentioned in the above
proof, these two properties are essentially equivalent.
Remark 2: Sudarsky & Wald stated the staticity theorem for the dimension n = 4 but, as pointed out
in Refs. [118, 123, 280], the theorem straightforwardly generalizes to n ≥ 4. Actually, the proof given
above13 does not rely on n = 4.
13
The first part of the proof, leading to K = 0, differs from Sudarsky & Wald’s one, which is based on integral
mass formulas.
5.4 The event horizon as a Killing horizon 155
(a) (b)
Figure 5.1: Two equivalent representations of an event horizon H with cross-sections of Sn topology and with
a stationary Killing vector field ξ that is spacelike on H : (a) Representation with the null geodesic generators of
H drawn as vertical lines; two of them are actually depicted, in dark green and light green respectively, with a
null normal ℓ along them; besides, two field lines of ξ (orbits of the stationary group) are depicted, in black and
brown respectively. (b) Representation with the field lines of ξ as vertical lines. The color code is the same as
in (a) and labelled points (a, b, etc.) help to identify the two figures. A few light cones are drawn in each figure;
note that ξ, being spacelike, lies outside of them, while the null normal ℓ is tangent to them. The strong rigidity
theorem (Property 5.25) states that ℓ is collinear to a Killing vector field χ, making H a Killing horizon.
Remark 3: Given the definition of a non-rotating horizon given in Property 5.2, the staticity theorem can
be summarized as follows: an electrovacuum spacetime containing a non-rotating and non-degenerate
black hole is static.
5.4.2 Spacelike stationary Killing field on H : the strong rigidity theo-
rem
When the stationary Killing vector ξ is spacelike on some part of a connected component
H of the event horizon, it obviously cannot be collinear to a null normal ℓ of H . Assuming
that H has cross-sections of spherical topology (as it must in dimension n = 4 according to
Property 5.4), we observe that, with respect to the null geodesic generators of H , the field lines
of ξ form helices, as depicted in Fig. 5.1a. By reciprocity, with respect to the field lines of ξ, the
null geodesic generators form helices as well, as depicted in Fig. 5.1b: observe that Fig. 5.1b
can be obtained from Fig. 5.1a by “untwisting” the field lines of ξ. Since asymptotically the
field lines of ξ are worldlines of inertial observers, Fig. 5.1b leads us to say that H is rotating,
thereby justifying the terminology introduced in Sec. 5.2.3 (Property 5.2).
When the Killing field ξ is not null on H , we cannot say a priori that H is a Killing horizon.
156 Stationary black holes
However, modulo some additional hypotheses, it turns out that H is still a Killing horizon,
albeit with respect to a Killing vector distinct from ξ. This result is due to S.W. Hawking (1972)
[260, 266] and is known as the (strong) rigidity theorem. We give below a modern version of
this theorem, due to Hollands, Ishibashi & Wald (2007) [282] and Moncrief & Isenberg (2008)
[372] (see also Theorem 8.1 p. 470 of Choquet-Bruhat’s textbook [108]).
Property 5.25: strong rigidity theorem
Let (M , g) be a stationary spacetime of dimension n ≥ 4 containing a black hole. Let
H be a connected component of the black hole event horizon. Let us assume that the
stationary Killing vector ξ is spacelike on some parts of H . If
1. M and H are (real) analytic manifolds (cf. Remark 4 in Sec. A.2.1), with g being an
analytic field,
2. g fulfills the vacuum Einstein equation (1.44) or the electrovacuum Einstein equation
(1.54),
3. n = 4 or H has a null geodesic generator that is incomplete,
4. H has compact cross-sections,
5. ξ is transverse to some cross-sections,
then the spacetime (M , g) admits a second Killing vector, χ say, such that ξ and χ
commute: [ξ, χ] = 0 and H is a Killing horizon with respect to χ.
Sketch of proof. The proof of the strong rigidity theorem is pretty long and we give here only a
sketch for n = 4, referring to Hawking & Ellis’ textbook [266] (Proposition 9.3.6) for full details
and to Refs. [282, 280, 372] for n > 4. Let us thus assume n = 4 and that the (electro)vacuum
Einstein equation holds. The null dominant energy condition is then satisfied and, since H is
connected, the topology theorem 1 (Property 5.4) implies that the complete cross-sections of
H have the topology of S2 , or equivalently that the topology of H is R × S2 .
Let us first show that the spacetime stationary action Φ : R × M → M , (t, p) 7→ Φt (p)
induces a SO(2) action Φ̂ on the set G of null geodesic generators of H . Each Φt being an
isometry of (M , g), it maps any null geodesic L of H to a null geodesic Φ̂t (L ) := Φt (L ).
Morever, since Φt leaves H globally invariant (Property 5.1), Φ̂t (L ) ⊂ H , i.e. Φ̂t (L ) ∈ G . Let
check that Φ̂ obeys the laws of a group action: obviously Φ̂0 (L ) = L and for any (t1 , t2 ) ∈ R2
and p ∈ L , we have Φt2 +t1 (p) ∈ Φ̂t2 +t1 (L ) but Φt2 +t1 (p) = Φt2 (Φt1 (p)) ∈ Φ̂t2 (Φ̂t1 (L )) as
well, from which we conclude that Φ̂t2 +t1 (L ) = Φ̂t2 (Φ̂t1 (L )), given that there is only one null
geodesic generator through the point Φt2 +t1 (p). Since the stationary generator ξ is assumed to
be spacelike on some parts of H , the group action is not trivial: there exist some null generators
L for which Φ̂t (L ) ̸= L (when ξ is null on a part of H , it is necessarily tangent to a null
generator L , which is then a fixed point of the group action: ∀t ∈ R, Φ̂t (L ) = L ). Moreover,
the group action Φ̂ is periodic: due to the topology R × S2 of H , there exists a smallest T > 0
5.4 The event horizon as a Killing horizon 157
Figure 5.2: Action Ψ induced by the spacetime stationary action Φ (Killing vector ξ) on a cross-section S
of a rotating event horizon H : for p ∈ S and t ∈ R, Ψt (p) is denoted by p1 on the figure, while p0 = Φ−t (p).
Similarly, for a point q ∈ S close to p, q0 = Φ−t (q) and q1 = Ψt (q). Note that this figure is drawn following the
same convention as Fig. 5.1a, namely the null geodesic generators of H appear as vertical lines.
such that for any null generator L , Φ̂T (L ) = L (cf. Fig. 5.1, where a′ = ΦT (a), b′ = ΦT (b),
c′ = ΦT (c) and d′ = ΦT (d), noticing that each pair (p, p′ ), where p stands for a, b, c or d, lie
on the same null generator). Hence Φ̂ is a SO(2) action on the set G of null generators of H ,
the standard angle parameter φ ∈ [0, 2π) of SO(2) being related to t ∈ [0, T ) by φ = 2πt/T .
Moreover, the SO(2) action must have exactly two fixed points, L1 and L2 say, for G has the
topology of S2 : one can identify G with an arbitrary complete cross-section Sˆ of H (each
L ∈ G intersects Sˆ at a single point and through each p ∈ Sˆ there is a single L ∈ G ) and,
as recalled above, all cross-sections have the topology of S2 .
Let S be a complete cross-section of H such that the stationary Killing vector ξ is nowhere
tangent to S . Let q be the metric induced by g on S . We are going to show that the spacetime
stationary action Φ induces a SO(2) action Ψ : [0, T ) × S → S , (t, p) 7→ Ψt (p) such that
each Ψt is an isometry of (S , q). For t ∈ R and p ∈ S , the point Φ−t (p) lies in H since
H is globally invariant by Φt (Property 5.1). Let then L be the unique null generator of H
through Φ−t (p). Since S is a complete cross-section of H , L intersects S at a unique point,
which we define as Ψt (p) (cf. Fig. 5.2, where L is denoted Φ̂−t (Lp )). In other words, Ψt (p) is
obtained by displacing p by a parameter −t along the field lines of ξ and then going back to S
along a null generator of H . Equivalently, we may write
Ψt (p) = Φ̂−t (Lp ) ∩ S , (5.75)
where Lp is the null geodesic generator through p (cf. Fig. 5.2). Let us check that Ψ is a
1-parameter group action on S , i.e. fulfills Ψ0 (p) = p and Ψt2 +t1 (p) = Ψt2 (Ψt1 (p)) for
158 Stationary black holes
any p ∈ S (cf. Sec. 3.3.1) and (t1 , t2 ) ∈ R2 . The first property is trivial. Regarding the
second one, Ψt1 (p) lies on the same null generator L as Φ−t1 (p) (by definition of Ψt1 ). It
follows that both Φ−t2 (Ψt1 (p)) and Φ−t2 (Φ−t1 (p)) lie on the null generator Φ̂−t2 (L ). But
Φ−t2 (Φ−1 (p)) = Φ−t2 −t1 (p), so that Ψt2 +t1 (p) = Φ̂−t2 (L ) ∩ S = Ψt2 (Ψt1 (p)). Hence Ψ :
R × S → S , (t, p) 7→ Ψt (p) is 1-parameter group action on S . Moreover this group
action inherits the T -periodicity from Φ̂ and thus is a SO(2) action. Indeed, Eq. (5.75) implies
ΨT (p) = Φ̂−T (Lp ) ∩ S = Lp ∩ S = p (see also Fig. 5.2, where Φ−T (p) lies on the same null
generator as p).
Let us now show that for any t ∈ [0, T ), Ψt is an isometry of (S , g). Let p and q be
infinitely close points of S . Because Φ−t is a spacetime isometry, we have
q(→
−
pq, →
−
pq) = g|p (→
−
pq, →
−
pq) = g|p0 (−
p−→ −−→
0 q0 , p0 q0 ), (5.76)
where p0 := Φ−t (p) and q0 := Φ−t (q) (cf. Fig. 5.2). Now, p0 and q0 are two points of S0 :=
Φ−t (S ), which is a complete cross-section of H that has no intersection with S for t ̸= 0,
given that ξ is transverse to S . One can then choose a parametrization λ of the null generators
L of H such that λ = 0 on S0 and λ = 1 on S . Let ℓ = dx/dλ|L be the associated tangent
vector. One may consider that S0 and S are the two ends of a foliation of H by a family
(Sλ )λ∈[0,1] of complete cross-sections (same construction as in the proof of Property 3.1). By
denoting by q the metric induced by g on each of the surfaces Sλ , we have g|p0 (− p−→ −−→
0 q0 , p 0 q0 ) =
q(−p−→ −−→
0 q0 , p0 q0 ). Now, since H is a non-expanding horizon (Property 5.3) and the null energy
condition is fulfilled (since we are considering the (electro)vacuum Einstein equation), the
metric q is preserved along ℓ: → −q ∗ Lℓ q = 0 [Property 3.2]. Since p1 := Ψt (p) and q1 := Ψt (q)
are obtained from p0 and q0 by the same displacement δλ = 1 along ℓ (cf. Fig. 5.2), it follows that
q(−p−→ −−→ −−→ −−→ →
− →− −−→ −−→
0 q0 , p0 q0 ) = q(p1 q1 , p1 q1 ). By combining with Eq. (5.76), we get q(pq, pq) = q(p1 q1 , p1 q1 ),
which proves that Ψt is an isometry of (S , q).
At this stage, we have shown that each cross-section of H admits an isometry group
isomorphic to SO(2), i.e. is axisymmetric. One thus may say that, in addition to be stationary,
H is axisymmetric, although the concept of axisymmetry as the result of the metric invariance
under a group action (cf. Sec. 3.3.1) is ill-defined for H , given that, as a null manifold, H is
not endowed with a proper (non-degenerate) metric tensor.
The major step is to show that there exists a Killing vector field χ of (M , g) that, on
H , is tangent to the null generators; it will then follow that the whole spacetime (M , g)
is axisymmetric, the generator of the SO(2) isometry being T /(2π)(χ − ξ). We first define
χ on H only, by demanding that χ is the tangent vector to the null generators L of H
corresponding to a parameter v of L that is connected to the parameter t of the stationary
action Φ, in the sense that for any point p of a given null generator14 L ,
v(ΦT (p)) = v(p) + T. (5.77)
The above condition is not sufficient to fully specify the parameter v, and hence χ, along L .
We require in addition that the non-affinity coefficient κ of χ is constant along L : ∇χ κ = 0.
→
−
In view of Eq. (2.21), this is equivalent to χ = −eρ ∇u with d2 ρ/dv 2 = 0 along L , where u is
14
Recall that on H , ΦT (p) and p always lie on the same null generator.
5.4 The event horizon as a Killing horizon 159
a scalar field such that H is the hypersurface u = 0. By construction, the vector field χ on H
is invariant under the stationary action, i.e. Lξ χ = 0, so that ξ and χ commute: [ξ, χ] = 0.
Let then define the following vector field on H :
1 2π
η := (χ − ξ) , with ΩH := . (5.78)
ΩH T
By construction, η has closed field lines of period 2π. Indeed, since ξ and χ commute, moving
by a parameter 2π along η is equivalent to moving by a parameter t = −T along the field
lines of ξ and then by a parameter v = T along the field lines of χ; the property (5.77) ensures
that one is back to the starting point. The vector field η vanishes at points where ξ = χ, i.e.
at points where ξ is tangent to some null generator of H . This occurs along null generators
that are fixed points of the group action Φ̂ on G . As discussed above, there are only two such
fixed points, L1 and L2 . These two null generators of H define thus the common “axis” of
all the rotations generated by η. Let C be a curve in H from L1 and L2 , orthogonal to η
and such that the field lines of η from C form a smooth cross-section S of H . Let us then
denote by Sv the cross-section obtained by displacing S by a parameter v along χ. Finally,
let k be the future-directed null vector orthogonal to Sv and complementary to χ, normalized
such that χ · k = −1 (k is transverse to H and is pointing towards the black hole region).
In a neighborhood of H , we introduce a spacetime coordinate system (v, r, θ, φ), named
Gaussian null coordinates, as follows. First of all, we choose spherical coordinates (θ, φ)
on the cross-section S , such that η = ∂φ and θ is a parameter along the “meridional” curve
C . To any point p ∈ H , we assign then the coordinates (v, θ, φ) where v is such that p ∈ Sv
and (θ, φ) are the coordinates of the intersection of S and the null generator L through p.
Let us then consider the null geodesics L ′ of tangent vector k, which are transverse to H
and an affine parameter r along each of them such that, on H , r = 0 and k = −dx/dr. To
any point p ∈ L ′ in the vicinity of H , we assign the coordinates (v, r, θ, φ) such that r is the
affine parameter of p along L ′ and (v, θ, φ) are the coordinates of the intersection of L ′ and
H . We may then extend the vector fields χ, η and k away from H by setting
χ := ∂v , η := ∂φ , k := −∂r .
By means of the (electro)vacuum Einstein equation, one can then show by induction that,
H
∀N ∈ N, Lk · · · Lk Lχ g = 0. (5.79)
| {z }
N times
This is the lengthy part of the computation (see p. 343 of Ref. [266]). Once (5.79) is established,
one can use the analyticity hypothesis to conclude that Lχ g = 0 in all M , i.e. that χ is a
Killing vector of (M , g). Since, on H , χ is normal to H , it follows that H is a Killing horizon
with respect to χ.
Remark 4: For n ̸= 4, the Killing horizon H is necessarily non-degenerate, given that H must have
an incomplete null geodesic generator in that case (third hypothesis of the theorem).
Remark 5: For n = 4, the stationary Killing vector ξ cannot be spacelike on the whole of H : it is
indeed null on the rotation axis, i.e. at the points where η = 0, given that Eq. (5.78) implies ξ = χ for
η = 0.
160 Stationary black holes
Remark 6: In some formulations [282, 266], the hypothesis of ξ being transverse to some cross-sections
is replaced by H lying in the future of the past infinity I − . Actually it can be shown that the latter
implies the former (see e.g. the proof of Proposition 9.3.1 in Ref. [266]).
Remark 7: The strong ridigity theorem may not hold in theories of gravity distinct from general
relativity (hypothesis 2 of Property 5.25). For instance, the so-called disformed Kerr solution [16, 54]
describes a n = 4 stationary black hole in a higher-order scalar-tensor theory (belonging to the so-called
DHOST family, for Degenerate Higher-Order Scalar-Tensor [338]), the event horizon of which does not
seem to be a Killing horizon [16].
Historical note : Stephen Hawking first established the strong rigidity theorem in 1972 [260] for a
spacetime of dimension n = 4 and assuming that there exists a past event horizon (i.e. a white hole
region) in addition to the future event horizon H . The theorem was restated and proved without the
white hole hypothesis in Hawking & Ellis’ famous textbook published in 1973 [266] (Proposition 9.3.6).
An account by Hawking himself about this reformulation can be found in Sec. 8 of Ref. [261]. The
demonstration presented in Ref. [266] relies on the vacuum Einstein equation, but it is noted there
that “similar arguments hold in the presence of matter fields, like the electromagnetic or scalar fields,
which obey well-behaved hyperbolic equations”. Some details in Hawking’s proof have been fixed by
Piotr T. Chruściel in 1997 [116]. The extension to spacetimes of dimension n ≥ 4 has been performed
independently by Stefan Hollands, Akihiro Ishibashi and Robert M. Wald in 2007 [282] and by Vincent
Moncrief and James Isenberg in 2008 [372]. The treatment of the electrovacuum case is done explicitly
in Hollands, Ishibashi & Wald’s work [282].
The strong rigidity theorem relies on the rather strong assumption that the manifolds and
fields are analytic. On physical grounds, it would be desirable to assume only smooth manifolds
and fields. In 1999, H. Friedrich, I. Rácz, and R.M. Wald [200] could remove the analyticity
hypothesis but only towards the interior of the black hole, i.e. they could prove that the vector
χ normal to H can be extended to a Killing vector of the black hole interior. Another partial
success has been achieved in 2014 by S. Alexakis, A.D. Ionescu and S. Klainerman [11], who
proved the strong rigidity theorem without the analyticity assumption, but only for slowly
rotating black holes. See Ref. [288] for a review of the progresses in removing the analyticity
hypothesis.
The strong ridigity theorem has been extended beyond general relativity (assumption 2
in Property 5.25) by Hollands, Ishibashi and Reall in 2023 [281]. These authors have shown
that, under some assumptions, the theorem holds in gravity theories with actions involving
quadratic and higher orders combinations of the curvature tensor.
It turns out that, for each connected component of the even horizon, the Killing vector χ is
expressible as a linear combination of the stationary Killing vector ξ and some Killing vectors
generating axisymmetries:
Property 5.26: axisymmetry of stationary black holes
Under the same hypotheses as in Property 5.25, there exist L Killing vectors η(1) , . . . , η(L)
with 1 ≤ L ≤ [(n − 1)/2], which have closed orbits of period 2π, such that
[ξ, η(i) ] = 0 and [η(i) , η(j) ] = 0 (1 ≤ i, j ≤ L) (5.80)
5.4 The event horizon as a Killing horizon 161
and the Killing vector χ normal to H writes
χ = ξ + Ω(1) η(1) + · · · + Ω(L) η(L) , (5.81)
where Ω(1) , . . ., Ω(L) are L real constants, all of whose ratios are irrational.
For n = 4, one has [(n − 1)/2] = 1, so that necessarily L = 1. Denoting η(1) simply by
η and Ω(1) by ΩH , Eq. (5.81) reduces to
χ = ξ + ΩH η. (5.82)
The constant ΩH is then called the black hole rotation velocity if H coincides with the
whole event horizon (i.e. if the event horizon is connected).
Since it has closed orbits, the isometry group generated by each Killing vector η(i) is SO(2).
Each isometry is thus an axisymmetry, according to the definition given in Sec. 5.3.6. For n = 4,
Eq. (5.82) is equivalent to Eq. (5.78). The constancy of ΩH in Eq. (5.82) can be interpreted
by saying that the event horizon is rotating rigidly (no differential rotation) and justifies the
name rigidity theorem. For n > 4, we refer to the original articles [282, 372] for the proof of
Property 5.26.
Remark 8: Property 5.26 is often considered as the second part of the strong rigidity theorem. In some
statements, it is even incorporated into the strong rigidity theorem (e.g. Proposition 9.3.6 of Ref. [266]).
Remark 9: If the event horizon has various connected components, the Killing vectors η(i) and the
constants Ω(i) may vary from one connected component to the other. In particular, the horizon-
generating Killing vector χ may not be the same for each connected component. For instance, in
dimension n = 5, the black saturn found by H. Elvang and P. Figueras in 2007 [184] is a stationary
black hole, the event horizon of which has two connected components: Hp (the “planet”) and Hr (the
“ring”), with topology R × S3 and R × S1 × S2 respectively. For both Hp and Hr , L = 1 and the
two components share the same axisymmetry Killing vector η(1) , but they may have different angular
(1) (1)
velocities: ΩHp ̸= ΩHr , and hence different normal Killing vectors: χHp ̸= χHr .
In the literature, the strong rigidity theorem is sometimes called simply the rigidity theorem,
without the qualifier strong (e.g. Refs. [108, 280]). The terminology strong rigidity, employed
in Refs. [103, 123, 275], aims to distinguish from the so-called weak rigidity theorem proved
by Carter in 1969 (corollary to Theorem 1 in Ref. [92]; see also [94], Sec. 4.5 of Ref. [101] and
Sec. 6.3 of Ref. [275]). The latter asserts that the event horizon of a black hole in a stationary,
axisymmetric and circular15 spacetime must be a Killing horizon. The strong rigidity theorem is
stronger in that it assumes only the stationarity of the spacetime, the axisymmetry becoming a
consequence (Property 5.26). On the other hand, the weak rigidity theorem is stronger in so far
as it does not rely on the Einstein equation, nor on the assumption of analyticity.
By the very definition of stationarity, the Killing vector field ξ is timelike in the vicinity of
I and I − . If ξ is spacelike on some parts of H , as assumed in this section, by continuity it
+
must be spacelike in some part of the domain of outer communications ⟨⟨M ⟩⟩ near H . The
15
A stationary and axisymmetric spacetime is said to be circular iff the R×SO(2) group action is orthogonally
transitive, i.e. the 2-surfaces generated by the two commuting Killing vectors ξ and η are orthogonal to a family
of (n − 2)-dimensional surfaces [92].
162 Stationary black holes
simplest configuration is when ξ is spacelike in some connected region G ⊂ ⟨⟨M ⟩⟩ around
H , null at the boundary of G and timelike outside G up to I + and I − . The subset G is
called the ergoregion and its boundary E := ∂G the ergosphere. We shall discuss it further in
Chaps. 10 and 11, especially in connection with the so-called Penrose process (Sec. 11.3.2).
5.4.3 Extremal black holes
Properties 5.23 and 5.25, show that, under some rather generic hypotheses (at least as far as
general relativity is concerned, cf. Remark 7), the event horizon of a stationary black hole, if
connected, is a Killing horizon. Now, in Sec. 3.3.6 we have classified Killing horizons in two
categories: the degenerate ones (zero surface gravity) and the non-degenerate ones (nonzero
surface gravity). This justifies the following definition:
A black hole in a stationary spacetime is said to be extremal (resp. non-extremal) if, and
only if, its event horizon H is a degenerate (resp. non-degenerate) Killing horizon.
The surface gravity κ of an extremal black hole is identically zero and the null generators
of its horizon are complete geodesics: the parameter t associated to the stationary Killing ξ is
an affine parameter ranging in the whole of R. Standard examples are the extremal Reissner-
Norström black hole (presented briefly in Sec. 5.6.1) and the extremal Kerr black hole (to be
discussed in Chap. 13).
5.5 The generalized Smarr formula
As shown in the previous section, under some hypotheses, the connected components of the
event horizon of a stationary black hole are Killing horizons; this gives rise to a nice formula
connecting the mass, angular momentum and area of these objects. After having established
the formula in the most general case (Sec. 5.5.1), we shall specialize it to electrovacuum black
holes (Sec. 5.5.2).
5.5.1 General form
Property 5.27: generalized Smarr formula
Let (M , g) be a stationary spacetime of dimension n ≥ 4 (stationary Killing vector ξ) that
contains a black hole, the event horizon of which has K ≥ 1 connected components H1 ,
. . ., HK . We shall assume that each Hk is a Killing horizon with respect to a Killing vector
χk . This is guaranteed with χk = ξ if Hk is non-rotating (ξ null on all Hk ; Property 5.23),
while if Hk is rotating (ξ spacelike on some parts of Hk ), this holds under the hypotheses
P k (i)
of the strong rigidity theorem (Property 5.25), with χk = ξ + Li=1 ΩHk ηHk (i) , where the
(i)
ΩHk are constants and the ηHk (i) are axisymmetric Killing vectors (Property 5.26). Being a
non-expanding horizon, each Hk has a well defined area Ak (Property 3.1). Let κk be the
surface gravity of Hk , i.e. the coefficient such that ∇χk χk = κk χk on Hk [cf. Eq. (3.29)].
5.5 The generalized Smarr formula 163
Let us assume that the null dominance condition (3.43) is fulfilled on Hk or that Hk is
part of a bifurcate Killing horizon; by the zeroth law of black hole dynamics (Property 3.10
or Property 3.16), this implies that κk is constant. Then the Komar mass MHk over any
cross-section of Hk , as defined by Eq. (5.36), obeys
Lk
!
n−2 κk Ak X (i) (i)
MHk = +2 ΩHk JHk , (5.83)
2(n − 3) 4π i=1
(i)
where JHk is the Komar angular momentum with respect to the axisymmetric Killing
vector ηHk (i) over any cross-section of Hk as given by Eq. (5.63). In Eq. (5.83), it is intended
that Lk = 0 if χk = ξ, so that there is no sum over i in that case. Furthermore, the Komar
mass at infinity (cf. Property 5.16) obeys the generalized Smarr formula:
K L
!
k
2(n − 3) √
Z
X κk Ak X (i) (i) 1
M∞ = +2 ΩHk JHk + R(ξ, n) γ dn−1 x , (5.84)
n−2 k=1
4π i=1
4π Σ
where Σ is any asymptotically flat spacelike hypersurface, the inner boundary of which
is a cross-section of the event horizon, i.e. some union of cross-sections of the connected
components Hk , R is the Ricci tensor of g and n is the future-directed unit normal to Σ.
If g obeys the Einstein equation (1.42) with Λ = 0, this formula can be rewritten as
K Lk
!
2(n − 3) X κk Ak X (i) (i)
M∞ = +2 ΩHk JHk
n−2 k=1
4π i=1
Z
T ξ · n √ n−1
+2 T (ξ, n) − γ d x, (5.85)
Σ n−2
where T is the matter energy-momentum tensor.
Proof. Let us evaluate the Komar mass MHk via formula (5.45). If we denote the cross-section
of Hk by Sk and we substitute ξ by its expression arising from Eq. (5.81), we get16
Lk
!
n−2
Z X
MHk = − ∇µ χν − Ω(i) η(i)ν dS µν
16π(n − 3) Sk i=1
"Z Lk
#
n−2 X Z
= − ∇µ χν dS µν − Ω(i) ∇µ η(i)ν dS µν .
16π(n − 3) Sk
| {z } i=1 | Sk {z }
−2κk Ak 16πJH
(i)
k
(i)
The second surface integral being 16πJHk follows directly from Eq. (5.65). As for the first
integral, the value −2κk Ak is obtained by introducing a null vector k normal to Sk such that
16
Here we drop the index k on χ and the label Hk on Ω(i) and η(i) for clarity, given that the integral regards a
single connected component Hk .
164 Stationary black holes
χ · k = −1. At each point p ∈ Sk , the pair (χ, k) is then a null basis of the 2-plane Tp⊥ Sk (cf.
Fig. 2.10 where ℓ stands for χ) and we may rewrite the area element normal bivector dS αβ as17
√
dS αβ = (χα k β − k α χβ ) q dn−2 x. (5.86)
We have then
√ √
∇µ χν dS µν = χµ ∇µ χν k ν − k µ χν ∇µ χν q dn−2 x = 2 χµ ∇µ χν k ν q dn−2 x.
| {z } | {z }
−∇ν χµ κk χν
Since χν k ν = −1, we get
√
∇µ χν dS µν = −2κk q dn−2 x. (5.87)
Given that κk is constant (Property 3.10 or Property 3.16), the integral of the above expression
over Sk reduces to −2κk Ak , by the very definition of the area Ak [cf. Eq. (3.3)]. This establishes
Eq. (5.83).
S Finally, Eq. (5.84) followsP
from Eqs. (5.58), (5.54) and (5.83), once one has noticed that
Sint = K k=1 Sk , so that MSint = k=1 MHk .
K
For vacuum black holes of general relativity, T = 0, the null dominance condition is trivially
fulfilled and the generalized Smarr formula (5.85) simplifies since the integral term disappears.
Example 8 (black saturn): For the black saturn solution discussed in Remark 9, one has n = 5, K = 2,
L1 = L2 = 1 and formula (5.83) reduces to
3 3
MH k = κk Ak + ΩHk JHk , k ∈ {1, 2}, (5.88)
32π 2
where k = 1 corresponds to the “planet” component of the event horizon and k = 2 corresponds to the
“ring” component. Equation (5.88) coincides with the two formulas given in Eq. (3.43) of the discovery
article [184], which were obtained from the explicit expressions of MHk , Ak , JHk , κk and ΩHk in terms
of the solution parameters. For the same solution, the Smarr formula (5.85) with T = 0 is recovered by
combining Eqs. (3.36) and (3.43) of Ref. [184].
It is worth to specialize the generalized Smarr formula to a 4-dimensional spacetime and
to a connected event horizon, by setting n = 4, K = 1 and L1 = 1 (cf. Property 5.26) in
Eqs. (5.83), (5.84) and (5.85):
Property 5.28: 4-dimensional generalized Smarr formula
Let (M , g) be a 4-dimensional stationary spacetime (stationary Killing vector ξ) containing
a black hole with a connected event horizon H . If H is non-rotating (ξ null on all H ), it
is necessarily a Killing horizon with respect to ξ. If H is rotating (ξ partly spacelike on
H ), we shall assumes that the hypotheses of the strong rigidity theorem (Property 5.25)
are fulfilled, so that H is a Killing horizon with respect to the Killing vector χ = ξ + ΩH η
[Eq. (5.82)], where ΩH is the angular velocity of H and η is an axisymmetric Killing
vector. We shall combine these two cases by stating that the former corresponds to ΩH = 0,
H
so that χ = ξ. The surface gravity κ of H is defined by the identity ∇χ χ = κχ [cf.
Eq. (3.29)]. Assuming that the null dominance condition (3.43) is fulfilled on H or that H
17
√ √
This follows readily by setting n = (χ + k)/ 2 and s = (χ − k)/ 2 in formula (5.42).
5.5 The generalized Smarr formula 165
is part of a bifurcate Killing horizon, the zeroth law of black hole dynamics (Property 3.10
or 3.16) leads to κ = const. Then the Komar mass MH over any cross-section of H [cf.
Eq. (5.36)] obeys
κ
MH = A + 2ΩH JH , (5.89)
4π
where A is the area of H (cf. Property 3.1) and JH is the Komar angular momentum
over any cross-section of H [cf. Eq. (5.63)]. Furthermore, the Komar mass at infinity (cf.
Property 5.16) obeys
√
Z
κ 1
M∞ = A + 2ΩH JH + R(ξ, n) γ d3 x , (5.90)
4π 4π Σ
where Σ is an asymptotically flat spacelike hypersurface, the inner boundary of which is a
cross-section of H , R is the Ricci tensor of g and n is the future-directed unit normal to
Σ. If g obeys the Einstein equation (1.42) with Λ = 0, this formula can be rewritten as
√
Z
κ
M∞ = A + 2ΩH JH + (2T (ξ, n) − T ξ · n) γ d3 x, (5.91)
4π Σ
where T is the matter energy-momentum tensor.
Example 9 (Schwarzschild black hole): The Schwarzschild black hole will be discussed in details in
Chap. 6; however, we have sufficiently studied its event horizon in various examples of the preceding
chapters to check that the Smarr formula holds for it. From Example 10 of Chap. 2, we have κ = 1/(4m)
[Eq. (2.29)], while from Example 3 of Chap. 3 we have A = 16πm2 [Eq. (3.4)]. Hence κA/(4π) = m.
Since we have seen in Example 5 of the current chapter that the parameter m is nothing but the Komar
mass at infinity M∞ , we get
κ
M∞ = A. (5.92)
4π
This is nothing but the Smarr formula (5.91) for a static (ΩH = 0) black hole in vaccum (T = 0).
5.5.2 Smarr formula for charged black holes
Beside the vacuum case, the generalized Smarr formulas (5.85) and (5.91) simplifies significantly
for electrovacuum spacetimes, i.e. spacetimes ruled by general relativity and for which T
is the energy-momentum tensor of a source-free electromagnetic field (cf. Sec. 1.5.2). To
perform the computation of the integral involving T , we need first to characterize a stationary
electromagnetic field on a Killing horizon:
Property 5.29: electromagnetic field on a Killing horizon
Let (M , g) be a n-dimensional asymptotically flat spacetime endowed with some Killing
vector field χ and containing a Killing horizon H with respect to χ. Let us assume that
(M , g) is endowed with an electromagnetic field F that respects the symmetry generated
by χ, i.e. that obeys Lχ F = 0. Furthermore let us assume that the electrovacuum Einstein
166 Stationary black holes
equation is fulfilled by (g, F ) (cf. Sec. 1.5.2). Then the pseudo-electric fielda E defined by
E := F (., χ) ⇐⇒ Eα := Fαµ χµ (5.93)
is an exact 1-form:
E = −dΦ, (5.94)
where the scalar potential Φ is expressible in terms of any electromagnetic potential A
that obeys the χ-symmetry (i.e. any 1-form A such that F = dA and Lχ A = 0) by
Φ = −⟨A, χ⟩ + const ⇐⇒ Φ = −Aµ χµ + const. (5.95)
We shall choose the additive constant so that Φ → 0 in the asymptotic flat end of (M , g).
This determines Φ uniquely and we shall call it the Killing electric potential associated
→
−
to χ . Furthermore, on H , the vector field E is collinear to the null normal χ:
→
− H H
E = µ0 σχ ⇐⇒ E α = µ0 σχα , (5.96)
where the scalar field σ defined on H can be interpreted as the effective electric charge
density of any cross-section S of H (Eq. (5.98) below). Finally, the Killing electric potential
Φ is constant on H :
H
Φ = ΦH = const. (5.97)
The constant ΦH is called the electric potential of the Killing horizon H .
a
If χ were a unit timelike vector, E would be a genuine electric field: the one measured by the observer
of 4-velocity χ.
Proof. Let A be some electromagnetic potential associated to F (F = dA) such that Lχ A = 0.
Thanks to the Cartan identity (A.95), we get 0 = Lχ A = χ · dA + d(χ · A) = χ · F +
d⟨A, χ⟩ = −E + d⟨A, χ⟩. This proves Eq. (5.94) with Φ given by Eq. (5.95). Let A′ be another
electromagnetic potential associated to F such that Lχ A′ = 0. We have the change-of-gauge
relation A′ = A + dΨ, where Ψ is a scalar field. Then Lχ dΨ = Lχ A′ − Lχ A = 0 − 0 = 0.
Invoking the commutation relation (A.96), we get d Lχ Ψ = 0. Hence, Lχ Ψ = const. Given
the identity ⟨dΨ, χ⟩ = Lχ Ψ, it follows that ⟨A′ , χ⟩ = ⟨A, χ⟩ + ⟨dΨ, χ⟩ = ⟨A, χ⟩ + const.
This proves that, up to some additive constant, the scalar field Φ defined by Eq. (5.95) does not
depend on the choice of the electromagnetic potential A. Let us now turn to relation (5.96).
→
− →
−
First, it is trivial that E is tangent to H , given that χ · E = ⟨E, χ⟩ = F (χ, χ) = 0 by
antisymmetry of F . Second, given that the Killing horizon H is a non-expanding horizon
and that any electromagnetic field obeys the null energy condition (cf. Sec. 2.4.2), property
H
(3.8) holds: R(χ, χ) = 0. Since χ is null on H , we can invoke the Einstein equation (1.40)
H
to transform this relation into T (χ, χ) = 0, where T is the energy-momentum tensor of the
electromagnetic field. Using expression (1.52) for T , we get
1 H
Fµρ χρ F µσ χσ − Fµν F µν gρσ χρ χσ = 0,
| {z } | {z } 4 | {z }
Eµ Eµ 0
5.5 The generalized Smarr formula 167
→
− → − H →
−
i.e. E · E = 0. Thus E is null vector on H . Being tangent to H , it is then necessarily
collinear to the null normal to H , namely χ, hence there exists a scalar field σ on H such
→
−
that Eq. (5.96) holds. Finally, since on H , E is collinear to H ’s null normal χ, we have
→
−
E · v = 0 for any vector field v tangent to H . This identity can be rewritten as ⟨E, v⟩ = 0 or,
in view of Eq. (5.94), as ⟨dΦ, v⟩ = 0. It follows that the scalar field Φ is constant on H , hence
Eq. (5.97).
The fact that σ is an effective18 surface charge density follows from:
Property 5.30: electric charge of a Killing horizon
Under the same assumptions as in Property 5.29, the electric charge within any cross-section
S of the Killing horizon H is
√
Z Z Z
1 1
QH = ⋆F = Fµν dS = µν
σ q dn−2 x. (5.98)
µ0 S 2µ0 S S
The quantity QH is independent of the choice of the cross-section S and is called the
electric charge of the Killing horizon H .
Proof. The first equality in Eq. (5.98) is the Gauss-law expression of the electric charge (see
e.g. Eq. (8.201) of Ref. [464] or Eq. (18.40) of Ref. [228]), while the second equality results from
Lemma 5.10. Now, using expression (5.86) for dS µν , we have
√ √
Fµν dS µν = Fµν (χµ k ν − k µ χν ) q dn−2 x = (−Eν k ν − k µ Eµ ) q dn−2 x
√ √
= −2µ0 σ χµ k µ q dn−2 x = 2µ0 σ q dn−2 x, (5.99)
| {z }
−1
where we have used successively Eqs. (5.93) and (5.96). This establishes the last equality in
Eq. (5.98). To show that QH is independent of the choice of S , it suffices to consider the
part W of H that is delineated by two cross-sections S and S ′ , as in Fig. 2.9. W is then
a (n − 1)-dimensional manifold with boundary ∂W = S ∪ S ′ and we may apply Stokes’
theorem (A.94) on W to the (n − 2)-form ⋆F ; taking into account the outward orientation of
∂W involved in Stokes formula (A.94), we get
Z Z Z
⋆F − ⋆F = d⋆ F .
S′ S W
| {z } | {z }
µ0 Q′H µ0 QH
Now, thanks to the second source-free Maxwell equation, d ⋆ F = 0 [Eq. (1.50) with j = 0], so
that the right-hand side of the above equation identically vanishes. Hence Q′H = QH .
We are now in position to state:
18
σ is not a genuine surface charge density since there are no electrically charged particles on H , given the
electrovacuum hypothesis (vanishing of the electric n-current density j, cf. Sec. 1.5.2); see Damour’s work [150]
for interpreting σ as the component along χ of an effective surface current 4-vector on H .
168 Stationary black holes
Property 5.31: generalized Smarr formula for charged black holes
Let (M , g) be a stationary spacetime of dimension n ≥ 4, with stationary Killing vector ξ,
endowed with a source-free electromagnetic field F such that (g, F ) obeys the electrovac-
uum Einstein equation (1.54). Let us assume that (M , g) contains a black hole, the event
horizon of which has K ≥ 1 connected components H1 , . . ., HK . Each HkP (1 ≤ k ≤ K)
is assumed to be a Killing horizon with respect to the Killing vector χ = ξ + Li=1 Ω(i) η(i) ,
where 0 ≤ L ≤ [(n − 1)/2], the Ω(i) are constants and the η(i) are axisymmetric Killing
vectorsa . One may have L = 0, i.e. χ = ξ (static configuration). The Smarr formula is then
K L
2(n − 3) X κk Ak 2(n − 3) X
M∞ = + ΦHk QHk +2 Ω(i) J∞
(i)
, (5.100)
n−2 k=1
4π n−2 i=1
(i)
where M∞ is the Komar mass at infinity, J∞ is the Komar angular momentum at infinity
with respect to the axisymmetric Killing vector η(i) , κk is the surface gravity of Hk , Ak is
the area of Hk , ΦHk is the electric potential of Hk , as defined by Property 5.29, and QHk
is the electric charge of Hk , as defined by Property 5.30.
a
Note that contrary to the more general setting of Property 5.27, all the connected components Hk
are assumed to be Killing horizons with respect to the same Killing vector χ; in other words, the rotation
velocities Ω(i) and axisymmetric Killing vectors η(i) are the same for all the Hk ’s.
Proof. The electromagnetic field energy-momentum tensor (1.52) obeys the null dominant
energy condition (3.46) [321], so that the surface gravities κk are constant over each Killing
horizon Hk (cf. Property 3.10). All the hypotheses of Property 5.27 are thus fulfilled and the
Smarr formula (5.84) holds. Thanks to the independence of Ω(i) and η(i) from Hk , we may
permute the sums over k and i in it and write
K L K
2(n − 3) √
Z
X κk Ak X
(i)
X (i) 1
M∞ = +2 Ω JHk + R(ξ, n) γ dn−1 x,
n−2 k=1
4π i=1 k=1
4π Σ
where Σ is an asymptotically flat spacelike hypersurface, of future-directed unit
S normal n to Σ
and of inner boundary a cross-section Sint of the event horizon, i.e. Sint = K k=1 Sk , where
Sk is a cross-section of Hk . Now, according to Eq. (5.69) with Sext going to infinity, we have
K
√
Z
X (i) (i) (i) 1
JHk = JSint = J∞ + R(η(i) , n) γ dn−1 x.
k=1
8π Σ
Hence
K L
2(n − 3) √
X κk Ak Z
X 1
M∞ = +2 Ω(i) J∞
(i)
+ R(χ, n) γ dn−1 x, (5.101)
n−2 k=1
4π i=1
4π Σ
5.5 The generalized Smarr formula 169
where we have used the identity ξ + Li=1 Ω(i) η(i) = χ to collect all the integrals on Σ. Let us
P
evaluate the integral by using the electrovacuum Einstein equation (1.54), which yields
2nµ
1 1
R(χ, n) = σ ν
Fσµ F ν χ − ρσ
Fρσ F χµ . (5.102)
4π µ0 2(n − 2)
We recognize the pseudo-electric field E σ = F σν χν [cf. Eq. (5.93)] in the first term in the
right-hand side; thanks to Eq. (5.94), we may re-express it in terms of the Killing electric
potential Φ:
Fσµ F σν χν = Fσµ E σ = −Fσµ ∇σ Φ = −∇σ ΦF σµ = −∇σ ΦF σµ = ∇σ (ΦFµσ ), (5.103)
where the last but one equality stems from the source-free Maxwell equation ∇σ F ασ = 0
[Eq. (1.48)]. As for the second term in the integrand (5.102), let us rewrite it in terms of some
electromagnetic potential 1-form A that obeys Lχ A = 0 and such that Φ = −⟨A, χ⟩ (i.e.
such that the constant in Eq. (5.95) is zero). Then Fρσ = ∇ρ Aσ − ∇σ Aρ , so that, using the
antisymmetry of F ρσ and again the source-free Maxwell equation ∇σ F σρ = 0,
Fρσ F ρσ χµ = 2∇σ Aρ F σρ χµ = 2 [∇σ (Aρ F σρ χµ ) − Aρ F σρ ∇σ χµ ] .
Now, expressing the property Lχ F = 0 via formula (A.86), we have Lχ F µρ = χσ ∇σ F µρ −
F σρ ∇σ χµ − F µσ ∇σ χρ = 0, from which
Aρ F σρ ∇σ χµ = Aρ F σµ ∇σ χρ − Aρ χσ ∇σ F ρµ = Aρ F σµ ∇σ χρ − ∇σ (χσ Aρ F ρµ ) + F ρµ χσ ∇σ Aρ
= −∇σ (χσ Aρ F ρµ ) + F ρµ (χσ ∇σ Aρ + Aσ ∇ρ χσ ) = −∇σ (Aρ F ρµ χσ ),
| {z }
0
where we have used ∇σ χσ = 0, as implied by the Killing equation for χ, in the first line and
Lχ Aρ = 0 in the second line. Hence, we get
Fρσ F ρσ χµ = 2∇σ Aρ F σρ χµ + Aρ F ρµ χσ = 2∇σ Aρ F ρµ χσ − Aρ F ρσ χµ .
Together with Eq. (5.103), this expression allows us to rewrite Eq. (5.102) as
1 2nµ ν 1
with Ωµν := ΦFµν + χµ Aρ F ρν − χν Aρ F ρµ .
R(χ, n) = ∇ Ωµν
4π µ0 n−2
Note that Ωµν defines a 2-form, since Ωµν = −Ωνµ . It follows then from Lemma 5.12 that
√ n−1 µ√
Z Z Z
1 2 ν n−1 2
I := R(χ, n) γ d x = ∇ Ωµν n γ d x = − ∇ν Ωµν dV µ
4π Σ µ0 Σ µ0 Σ
Z Z
1
= − Ωµν dS µν − Ωµν dS µν
µ0 S∞ Sint
| {z }
0
We have set the integral on S∞ to zero because F decays to zero sufficiently fast at the
asymptotically flat end of Σ (otherwise (M , g) could not be asymptotically flat). Since Sint =
k=1 Sk , we may write
SK
K Z
1 X
I= Ωµν dS µν .
µ0 k=1 Sk
170 Stationary black holes
Now, thanks to the antisymmetry of dS µν ,
2
Ωµν dS µν = ΦFµν dS µν + Aρ F ρν χµ dS µν
n−2
2 √
= ΦFµν dS µν + Aρ F ρν (χµ χµ k ν − χµ k µ χν ) q dn−2 x
n−2 | {z } | {z }
0 −1
2 √
= ΦFµν dS µν + Aρ E ρ q dn−2 x,
n−2
where use has been made of expression (5.86) for dS µν , of the null character of χ on Hk and of
the definition (5.93) of E. Since the latter obeys (5.96), we have Aρ E ρ = µ0 σAρ χρ = −µ0 σΦ.
Hence
µν µν 2µ0 √ n−2
Ωµν dS = Φ Fµν dS − σ qd x .
n−2
Given that Φ is a constant ΦHk on each Hk [Eq. (5.97)], the above expression yields
K K
1 Z √ n−2 2(n − 3) X
Z
X
µν 2
I= ΦHk Fµν dS − σ qd x = ΦHk QHk ,
k=1
µ0 S k n − 2 Sk n − 2 k=1
| {z } | {z }
2QHk QHk
where we have used two expressions of the electric charge of Hk given by Eq. (5.98). In view
of Eq. (5.101), the above value for I proves the Smarr formula (5.100).
Property 5.32: 4-dimensional Smarr formula for a connected charged black hole
For n = 4 (which implies L ≤ 1) and for a connected black hole event horizon H (K = 1)
of surface gravity κ, area A, angular velocity ΩH , electric charge Q and electric potential
ΦH , the electrovacuum Smarr formula (5.100) simplifies to
κ
M∞ = A + 2ΩH J∞ + ΦH Q . (5.104)
4π
Historical note : Formula (5.104) has been first derived in 1972 (published: 1973) by Larry Smarr [459] in
the case of the Kerr-Newman black hole. Actually, Smarr did not obtain it from integral formulas for M∞
and J∞ , as presented here; rather he used an explicit expression of M∞ in terms of A, J∞ and Q, which
holds for the Kerr-Newman solution. After noticing that this expression is a homogeneous function
of degree 1/2 of (A, J∞ , Q2 ), he applied Euler’s homogeneous function theorem to get Eq. (5.104).
The derivation of Smarr formula from generic Komar-type integral formulas for mass and angular
momentum, possibly with some non-vacuum exterior, is due to James Bardeen, Brandon Carter and
Stephen Hawking in 1973 [40]. They obtained Eq. (5.91) [their Eq. (13)]. The general (i.e. not assuming
the Kerr-Newmann metric) n = 4 electrovacuum Smarr formula (5.104) has been derived by Brandon
Carter in 1973 [96] [his Eq. (9.29)]. Actually, the formula obtained by Carter is more general than
Eq. (5.104) since it allows for electric sources (non-vanishing electric 4-current j) and matter in the black
5.6 The no-hair theorem 171
hole exterior (see also Carter’s review articles [99] [Eq. (6.323)] and [101] [Eq. (4.68)]). The generalization
of the Smarr formula to dimensions n ≥ 4 has been obtained by Robert Myers and Malcolm Perry in
1986 [376] for connected event horizons in vacuum: they obtained Eq. (5.85) for K = 1 and T = 0
[their Eq. (4.6), where N := n − 1]. The n ≥ 4 generalization of the electrovacuum Smarr formula
has been obtained by Rabin Banerjee, Bibhas Majhi, Sujoy Modak, and Saurav Samanta in 2010 [32],
but only for electrically charged Myers-Perry black holes, which are approximate solutions for small
rotation velocities and a single angular momentum parameter, i.e. they obtained Eq. (5.100) for K = 1
(connected horizon) and L = 1 (single angular momentum) [their Eq. (66)].
5.6 The no-hair theorem
Arguably, the most beautiful achievement in general relativity is the no-hair theorem. This is a
uniqueness theorem, which basically states that all stationary black holes in our (4-dimensional)
Universe are described by the same rather simple solution to the Einstein equation: the Kerr
metric (or the Kerr-Newman metric if one allows for an electric charge), which depends on only
two numbers: the mass and the angular momentum (plus the electric charge for Kerr-Newman).
The absence of any functional degree of freedom justifies the “no-hair” qualifier. Actually,
there are various uniqueness theorems, depending on various assumptions on the equilibrium
state (static versus rotating), the matter/field content (vacuum, electrovacuum, scalar field,
etc.), the Killing-horizon type of the event horizon (non-degenerate versus degenerate) and the
spacetime dimension. We shall discuss first uniqueness theorems that regard static black holes
in any spacetime dimension (Sec. 5.6.1), before theorems about axisymmetric rotating black
holes in dimension 4 (Sec. 5.6.2) and finally “the” no-hair theorem itself, which regards rotating
black holes in dimension 4 (Sec. 5.6.3).
5.6.1 Uniqueness theorems for static black holes
The uniqueness theorems for static (i.e. non-rotating) black holes orginate from two famous
theorems by Werner Israel at the end of the sixties, regarding respectively vacuum [289] and
electrovacuum configurations [290]. Both were derived in dimension 4 and for non-degenerate
horizons. They have been generalized to higher dimensions and to degenerate horizons; we
present these generalized versions here.
Property 5.33: generalized Israel uniqueness theorem (vacuum)
Let (M , g) be a static and asymptotically flat spacetime of dimension n ≥ 4 containing a
black hole of event horizon H . If
• g fulfills the vacuum Einstein equation (1.44),
• n = 4 or all the connected components of H are non-degenerate (i.e. have non-zero
surface gravity),
then the domain of outer communications ⟨⟨M ⟩⟩ of (M , g) is isometric to the domain of
172 Stationary black holes
outer communications of a n-dimensional Schwarzschild spacetime, also known as a
Schwarzschild-Tangherlini spacetime. This means that there exists a coordinate system
(t, r, θ1 , . . . , θn−3 , φ) on ⟨⟨M ⟩⟩ such that t ∈ R, r ∈ (µ, +∞), θi ∈ (0, π) (1 ≤ i ≤ n − 3),
φ ∈ (0, 2π) and
µ 2 µ −1 2 ◦
g = − 1 − n−3 dt + 1 − n−3 dr + r2 q, (5.105)
r r
◦
where q is the standard round metrica on the unit sphere Sn−2 , which is spanned by
(θ1 , . . . , θn−3 , φ), and the constant µ > 0 is related to the spacetime Komar mass (or ADM
mass, cf. Property 5.18) M by
16πM
µ= . (5.106)
(n − 2)Ωn−2
Ωn−2 is the area of Sn−2 , as given by Eqs. (5.19)-(5.20), so that µ = 2M for n = 4,
µ = 8M/(3π) for n = 5 and µ = 3M/(2π) for n = 6. In particular, the event horizon H
is connected and is a non-degenerate Killing horizon.
◦ ◦
a
for n = 4, q = dθ12 + sin2 θ1 dφ2 ; for n = 5, q = dθ12 + sin2 θ1 (dθ22 + sin2 θ2 dφ2 ); etc.
The proof can be found in Ref. [219], which relies on the assumption that the connected
components of H are non-degenerate Killing horizons. The fact that this assumption is not
necessary for n = 4 has been proven in Ref. [127]. The proof in Ref. [219] relies on the positive
mass theorem. An elementary proof specific to the case n = 4 can be found in Sec. 8.2 of
Straumann’s textbook [464] and well as in Chap. 9 of Heusler’s textbook [275] and in Robinson’s
articles [438] and [439] (appendix). See also Ref. [389] for a recent alternative proof.
Remark 1: The fact that H is connected means that there does not exist any static solution in vacuum
general relativity with “multiple black holes”19 . Intuitively, two or more black holes in vacuum would
attract each other, making a static configuration impossible.
Historical note : The uniqueness theorem 5.33 has been first proven in the case n = 4 by Werner Israel
in 1967 [289] under the additional hypotheses
√ that (i) the spacetime is analytic (cf. Remark 4 in Sec. A.2.1),
(ii) the isosurfaces t = const, V := −ξ · ξ = const are regular and topologically 2-spheres in ⟨⟨M ⟩⟩
(in particular, dV ̸= 0 in ⟨⟨M ⟩⟩), (iii) the event horizon H is connected, (iv) H is non-degenerate
and (v) H ’s cross-sections are topologically 2-spheres. Property (v) actually follows from the topology
theorem for n = 4 (Properties 5.4 and 5.5), which has been established 5 years later by Stephen Hawking
[260]; it can therefore be removed from the hypotheses of the theorem. The hypothesis (ii) has been
shown unnecessary by David Robinson in 1977 [438] (see also the work [373]). The requirement (iii) (H
connected) has been removed as unnecessary by Gary Bunting and Abul Masood-ul-Alam in 1987 [77],
while the requirement (iv) (H non-degenerate) has been removed by Piotr Chruściel, Harvey Reall and
Paul Tod in 2006 [127]. Finally the analyticity hypothesis (i), which was implicit in Israel’s study, has
been removed in 2010 by Piotr Chruściel and Gregory Galloway [121]. The generalization to dimensions
n ≥ 4 has been achieved by Seungsu Hwang in 1998 [287] and by Gary Gibbons, Daisuke Ida and
Tetsuya Shiromizu in 2002 [219]. The generalization (5.105) of Schwarzschild solution to dimensions
n ≥ 4 has been found by Frank Tangherlini in 1963 [473].
19
The quotes indicate that even when H has various connected components, there is formally a single black
hole region in spacetime, albeit not connected.
5.6 The no-hair theorem 173
For the static electrovacuum uniqueness theorem, we shall distinguish the dimension n = 4
from n ≥ 5. Starting by the former, we have:
Property 5.34: generalized Israel uniqueness theorem (n = 4 electrovacuum)
Let (M , g) be a 4-dimensional static and asymptotically flat spacetime endowed with
an electromagnetic field F such that (g, F ) fulfills the electrovacuum Einstein equation
(1.54). If (M , g) contains a black hole, then the domain of outer communications ⟨⟨M ⟩⟩
is isometric to the domain of outer communications of either a Reissner-Nordström black
hole or a Majumdar-Papapetrou black hole.
The Reissner-Nordström black hole is defined by the event horizon being connected
and the existence of a coordinate system (t, r, θ, φ) on ⟨⟨M ⟩⟩ such that t ∈ R, r ∈
(rH , +∞), θ ∈ (0, π), φ ∈ (0, 2π) and
−1
µ0 Q2 + P 2 µ0 Q 2 + P 2
2M 2 2M
g =− 1− + 2
dt + 1 − + 2
dr2
r 4π r r 4π r
2 2 2 2
(5.107a)
+ r dθ + sin θ dφ
µ0 Q
F =− dt ∧ dr − P sin θ dθ ∧ dφ , (5.107b)
4π r2
where M is the Komar mass at infinity (or ADM mass), Q is the black hole electric charge
(cf. Property 5.30) and P is its magnetic monopole charge
q (cf. Remark 2 below). The three
constants M , Q and P must fulfill Q2 + P 2 ≤ 4π M and the lower bound of r on
p
µ0
⟨⟨M ⟩⟩ is rH = M + M 2 − µ0 /(4π)(Q2 + P 2 ).
p
The Majumdar-Papapetrou black hole is defined by (i) the event horizon being
disconnected, with K ≥ 2 connected components (Hk )1≤k≤K , which are all degenerate
Killing horizons, and (ii) the existence of a coordinate system (t, x, y, z) on ⟨⟨M ⟩⟩ such
that
r
µ0 −2
−2 2 2 2
g = −U dt + U (dx + dy + dz ), 2 2
F =± U dU ∧ dt, (5.108a)
4π
K
X µk
U := 1 + p . (5.108b)
(x − x )2 + (y − y )2 + (z − z )2
k=1 k k k
This solution is characterized by K positive constants µk , which can be interpreted as
the masses of each component Hk for widely separated configurations, and 3K constants
(xk , yk , zk ), which are the coordinate locations of the K degenerate Killing horizons Hk .
Note that each (xk , yk , zk ) corresponds to a coordinate singularity of the (t, x, y, z) sys-
tem, so that Hk appears as the curve (t, x, y, z) = (t, xk , yk , zk ), while it is actually a
174 Stationary black holes
hypersurface. The electric charge Qk of Hk is proportional to µk :
r
4π
Qk = ± µk , (5.109)
µ0
where the ± sign is the same as in expression (5.108a) for F ; in particular it must be the
same for all the components Hk , i.e. the charges Qk are either all positive or all negative.
For a proof, see Secs. 9.3 and 9.4 of Heusler’s textbook [275] for the case where all connected
components of the event horizon are assumed to be non-degenerate and Ref. [128] for the
complementary case.
Remark 2: At the level of elementary particles, a magnetic monopole is a particle that generates the
radial magnetic field B = µ0 P/(4πr2 )∂r in its inertial rest frame, where P is the magnetic charge.
Although predicted by some grand unified theories and string theories, no magnetic monopole has
ever been detected and standard electromagnetism postulates that magnetic monopoles do not exist
(see Refs. [360, 423] for reviews). From a formal point of view, magnetic monopoles would render the
Maxwell equations more symmetric, since instead of Eq. (1.50), they would write
dF = µ0 ⋆j m and d ⋆F = µ0 ⋆j, (5.110)
where j m is the 1-form associated to the magnetic current density vector jm , which describes the
distribution of elementary magnetic monopoles. In the electrovacuum black hole framework, we are
considering the source-free Maxwell equations dF = 0 and d ⋆F = 0, which are perfectly symmetric in
terms of F and ⋆F . As the Reissner-Nordström solution (5.107) shows, a black hole with a non-vanishing
magnetic monopole charge can exist even if magnetic monopoles are excluded as elementary particles,
i.e. even if jm = 0.
q
Remark 3: The constraint Q2 + P 2 ≤ 4π µ0 M in the Reissner-Nordström case enforces the existence
p
of a black hole. If it is not fulfilled, Eqs. (5.107) still define a valid solution to the electrovacuum Einstein
equation, but it corresponds to a naked singularity, not to a black hole.
Remark 4: The Majumdar-Papapetrou solution (5.108) with K = 1 (connected event horizon) is still
a valid static solution to the electrovacuum Einstein equation. It is actually a member of the first
family invoked in the uniqueness theorem, namely the Reissner-Nordström one. More pprecisely, it is
an extremal Reissner-Nordström spacetime, i.e. a solution (5.107) for which |Q| = 4π/µ0 M and
P = 0, or equivalently a Reissner-Nordström solution with P = 0 and a degenerate event horizon.
To see this, it suffices to choose (x1 , y1 , z1 ) = (0, 0, 0), to move from coordinates (x, y, z) to spherical
coordinates (r̄, θ, φ) via the standard formulas (in particular r̄2 = x2 + y 2 + z 2 ) and to introduce a new
radial coordinate r := r̄ + µ1 . Then U = r/(r − µ1 ) and the Majumdar-Papapetrou solution (5.108)
coincides with the Reissner-Nordström solution (5.107) with M = µ1 , Q = ± 4π/µ0 µ1 and P = 0.
p
In higher dimensions, the static electrovacuum uniqueness theorem is
Property 5.35: generalized Israel uniqueness theorem (n ≥ 5 electrovacuum)
Let (M , g) be a static and asymptotically flat spacetime of dimension n ≥ 5. Let us assume
that (M , g) is endowed with an electromagnetic field F and contains a black hole of event
horizon H . If
5.6 The no-hair theorem 175
• (g, F ) fulfills the electrovacuum Einstein equation (1.54),
• all the connected components of H are non-degenerate (i.e. have non-zero surface
gravity),
then the domain of outer communications ⟨⟨M ⟩⟩ of (M , g) is isometric to the domain of
outer communications of a n-dimensional Reissner-Nordström spacetime with vanish-
ing magnetic monopole, also known as a Reissner-Nordström-Tangherlini spacetime.
This means that there exists a coordinate system (t, r, θ1 , . . . , θn−3 , φ) on ⟨⟨M ⟩⟩ such that
t ∈ R, r ∈ (rH , +∞), θi ∈ (0, π) (1 ≤ i ≤ n − 3), φ ∈ (0, 2π) and
−1
q2 q2
µ µ ◦
2
g = − 1 − n−3 + 2(n−3) dt + 1 − n−3 + 2(n−3) dr2 + r2 q, (5.111a)
r r r r
µ0 Q
F =− dt ∧ dr, (5.111b)
Ωn−2 rn−2
◦
where q is the standard round metric on the unit sphere Sn−2 , which is spanned by
(θ1 , . . . , θn−3 , φ), Q is the black hole electric charge (cf. Property 5.30) and the constants
µ and q are related to the Komar mass at infinity (or ADM mass) M and to the electric
charge Q by
16πM 8πµ0 Q2
µ= and q2 = , (5.112)
(n − 2)Ωn−2 (n − 2)(n − 3)Ω2n−2
where Ωn−2 is the area of Sn−2 [Eqs. (5.19)-(5.20)]. The constants µ and q must fulfill the con-
1/(n−3)
straint µ > 2q and the lower bound of r on ⟨⟨M ⟩⟩ is rH = µ/2 + µ2 /4 − q 2 .
p
For a proof, see Ref. [218] under the additional assumption that the magnetic field vanishes (i.e.
that ξ · ⋆F = 0) and Ref. [328] for relaxing this assumption.
Example 10: For n = 5, Ωn−2 = Ω3 = 2π 2 [Eq. (5.20)] and Eqs. (5.111)-(5.112) become
−1
q2 q2
µ 2 µ
dr2 + r2 dθ12 + sin2 θ1 (dθ22 + sin2 θ2 dφ2 )
g = − 1 − 2 + 4 dt + 1 − 2 + 4
r r r r
µ0 Q
F = − 2 3 dt ∧ dr
2π r
8M µ0 Q2
µ= and q2 = .
3π 3π 3
Remark 5: A static and asymptotically flat black hole spacetime cannot carry a nonzero magnetic
monopole P for n ≥ 5 [185]. Hence Property 5.35 is not a direct generalization of Property 5.34 to
n > 4, even if H is assumed to be made of non-degenerate components.
Historical note (electrovacuum static solutions): The Reissner-Nordström solution (5.107) of
the Einstein-Maxwell system with a vanishing magnetic monopole charge (P = 0) has been found
independently by Hans Reissner in 1916 [429], Hermann Weyl in 1917 [517] and Gunnar Nordström in
176 Stationary black holes
1918 [385]. The generalization to P ̸= 0 can be found in Brandon Carter’s lecture at Les Houches (1972)
[95]. The generalization to dimensions n > 4 with P = 0, i.e. the solution (5.111), has been obtained by
Frank Tangherlini in 1963 [473]. The Majumdar-Papapetrou solution (5.108) has been found in 1947
independently by Sudhansu Datta Majumdar [353] and Achilles Papapetrou [399], as a solution to the
electrovacuum Einstein equation describing K charged point masses in gravito-electrostatic equilibrium.
The extremal Reissner-Nordström solution seems to have been discussed in details first by Papapetrou
in his 1947 article [399]. It was actually the starting point that lead him to the Majumdar-Papapetrou
solution (cf. Remark 4 above). The black hole character of the Majumdar-Papapetrou solution has been
truly recognized and analyzed by James Hartle and Stephen Hawking in 1972 [258]. The generalization
of the Majumdar-Papapetrou solution to spacetime dimensions n > 4 is due to Robert Myers in 1987
[375].
Historical note (uniqueness theorems for static electrovacuum black holes): The Reissner-
Nordström part (with P = 0) of the 4-dimensional uniqueness theorem 5.34 has been first proven by
Werner Israel in 1968 [290], under the same additional hypotheses as the ones used in his proof of
the vacuum theorem (cf. historical note on p. 172), notably that the event horizon H is connected,
has the topology R × S2 and is non-degenerate. The connectedness hypothesis has been removed by
Abul Masood-ul-Alam in 1992 [358]. The allowance for a magnetic charge (P ̸= 0) has been achieved
by Markus Heusler in 1994 [274] (see also Sec. 9.4 of Ref. [275]). Three years later, Heusler obtained
the Majumdar-Papapetrou part of the theorem by assuming that all the connected components of
H are degenerate [276]. This assumption has been relaxed by Piotr Chruściel and Paul Tod in 2007
[128], who could exclude configurations with both degenerate and non-degenerate components of H .
The higher-dimensional uniqueness theorem 5.35 has been established by Gary Gibbons, Daisuke Ida
and Tetsuya Shiromizu in 2002 [218] with the extra assumption of a vanishing magnetic field. This
assumption has been relaxed by Hari Kunduri and James Lucietti in 2018 [328].
5.6.2 Uniqueness theorems for stationary and axisymmetric black
holes
In this section, we restrict ourselves to the standard spacetime dimension n = 4. The strong
rigidity theorem discussed in Sec. 5.4 (Properties 5.25 and 5.26) basically states that any station-
ary black hole spacetime obeying the (electro)vacuum Einstein equation has to be axisymmetric.
A crucial step towards the no-hair theorem is then the following uniqueness theorem for ax-
isymmetric stationary black holes:
Property 5.36: Carter-Robinson theorem (Carter 1971 [93], Robinson 1975 [437])
Let (M , g) be a 4-dimensional asymptotically flat spacetime containing a black hole of
event horizon H . If
• (M , g) is stationary and axisymmetric,
• g fulfills the vacuum Einstein equation (1.44),
• H is connected,
5.6 The no-hair theorem 177
• there is no closed causal curve in the domain of outer communications ⟨⟨M ⟩⟩,
then ⟨⟨M ⟩⟩ is isometric to the domain of outer communications of the Kerr spacetimea .
a
We shall not define the Kerr spacetime here; this will be done in Chap. 10.
Remark 6: In their original works, Carter and Robinson assumed that H is a non-degenerate Killing
horizon (i.e. has non-zero surface gravity). However, this hypothesis can be relaxed [125] (see [123] for
an extended discussion).
The proof of Property 5.36 can be found in Chap. 10 of Heusler’s textbook [275] (see also
Sec. 5 of Carter’s review article [101]) with the additional assumption that the horizon is
non-degenerate; see Ref. [125] for the degenerate case.
Remark 7: The causality condition (absence of closed causal (i.e. null of timelike) curves in the black
hole exterior), which is one of the assumptions of Carter-Robinson’s theorem (cf. [103] for a discussion),
does not appear in Israel’s theorem (Property 5.33) because a static spacetime, which by definition has
hypersurface-orthogonal timelike curves, cannot contain any closed causal curve.
The electrovacuum version of the Carter-Robinson theorem is
Property 5.37: Bunting-Mazur theorem (Bunting 1983 [76], Mazur 1982 [361])
Let (M , g) be a 4-dimensional asymptotically flat spacetime containing a black hole of
event horizon H . If
• (M , g) is stationary and axisymmetric,
• g fulfills the electrovacuum Einstein equation (1.54),
• H is connected,
• there is no closed causal curve in the domain of outer communications ⟨⟨M ⟩⟩,
then ⟨⟨M ⟩⟩ is isometric to the domain of outer communications of the Kerr-Newman
spacetimea .
a
The Kerr-Newman solution extends the Reissner-Nordström solution (5.107) to the rotating case; it
depends on four parameters: M , Q, P (as for Reissner-Nordström) and J, the latter being the total angular
momentum. One may say as well that the Kerr-Newman solution extends the Kerr solution, which depends
only on (M, J), to the electromagnetic case. See e.g. Eq. (5.54) of Ref. [95] for the explicit form of the
Kerr-Newman metric.
The proof of Property 5.37 with the additional assumption that the horizon is non-degenerate
can be found in Chap. 10 of Heusler’s textbook [275] as well as in Carter’s articles [100, 101]
and Mazur’s review article [364]; the degenerate case is treated in Ref. [125].
Remark 8: Some of the Majumdar-Papapetrou solutions (5.108) are axisymmetric (those for which
all the points (xk , yk , zk ) are aligned). However, they are excluded from the conclusion of the above
theorem because they do not fulfill the hypothesis of H being connected.
178 Stationary black holes
Historical note : The first stationary-axisymmetric uniqueness result has been obtained by Brandon
Carter in 1971 [93]. Under the assumptions of a non-degenerate horizon with spherical cross-sections,
Carter reduced the 4-dimensional stationary-axisymmetric vacuum Einstein equation to a 2-dimensional
non-linear elliptic system of two partial differential equations, with boundary conditions depending
on only two parameters: the mass M and c := κA/(4π), where κ is the surface gravity and A is the
horizon area (κ ̸= 0 since the horizon was assumed non-degenerate). Then Carter could show that
the solutions of this system form disjoint 2-parameter families and that within a given family, the
members are fully specified by the pair (M, c). The Kerr family is such a family and is the only one
containing the Schwarzschild solution, but Carter could not exclude that there exist other families. This
step has been achieved by David Robinson in 1975 [437], who, via a tour de force [103], showed that
the solution has to belong to the Kerr family. The electrovacuum case turned out to be too complicated
to be dealt with the same techniques; only a result similar to Carter’s one (i.e. reduction to disjoint
4-parameter families of solutions) could be achieved by Robinson in 1974 [436]. New techniques have
been introduced independently by Paweł Mazur in 1982 [361] and Gary Bunting in 1983 [76], which
enabled them to get the uniqueness theorem 5.37 under the non-degeneracy assumption. The case of
a degenerate horizon has been dealt with only in 2010, when Piotr Chruściel and Luc Nguyen [125]
showed that electrovacuum rotating black holes with a degenerate horizon necessarily belong to the
Kerr-Newman family (they are the so-called extremal members of that family). For further details, see
the historical accounts [103, 364, 439] by the actors themselves. As for the Kerr-Newman solution itself,
it has been discovered by Ezra Newman and his collaborators in 1965 [380], two years after Roy Kerr
found his famous solution [312]. A historical note about the discovery of the Kerr metric is to be found
in Chap. 10, p. 326.
5.6.3 The 4-dimensional no-hair theorem
We are now is position to state the famous theorem:
Property 5.38: no-hair theorem
Let (M , g) be a spacetime that
• is 4-dimensional,
• is analytic,
• is asymptotically flat,
• is stationary,
• fulfills the electrovacuum Einstein equation (1.54) (with the vacuum Einstein equation
(1.44) considered as a subcase),
• contains a black hole with a connected event horizon that is either (i) rotatinga or (ii)
non-rotating and non-degenerate,
• does not contain any closed causal curve in the domain of outer communications
⟨⟨M ⟩⟩.
5.6 The no-hair theorem 179
Then ⟨⟨M ⟩⟩ is isometric to the domain of outer communications of a Kerr-Newman
spacetime. The latter depends on four parameters: the total mass M , the total angular
momentum J, the electric charge Q and the magnetic monopole charge P , with the
following subcases:
• J = 0: Reissner-Nordström spacetime;
• Q = 0, P = 0: Kerr spacetime;
• Q = 0, P = 0, J = 0: Schwarzschild spacetime.
In the last two cases, the electromagnetic field vanishes and g fulfills the vacuum Einstein
equation (1.44).
a
Let us recall that the concepts of rotating and non-rotating horizons have been defined in Property 5.2.
Proof. Let H be the black hole event horizon. In case (i) (H rotating), we may invoke
the strong rigidity theorem 5.25 to get that (M , g) is axisymmetric, in addition to being
stationary. Then the Bunting-Mazur theorem 5.37 leads to ⟨⟨M ⟩⟩ being isometric to the
domain of outer communications of a Kerr-Newman spacetime. In case (ii) (H is non-rotating
and non-degenerate), we may invoke the staticity theorem 5.24 to assert that (M , g) is static.
Then the generalized Israel theorem 5.34 leads to ⟨⟨M ⟩⟩ being isometric to the domain of
outer communications of a Reissner-Nordström spacetime (the Majumdar-Papapetrou case in
theorem 5.34 is excluded since H is assumed non-degenerate). Since the Reissner-Nordström
family is a subfamily of the Kerr-Newman one, this completes the proof.
Remark 9: A stationary black hole spacetime with a non-rotating degenerate event horizon and not
belonging to the Kerr-Newman family is not excluded by the above no-hair theorem. Such a spacetime
cannot be static, otherwise the generalized Israel theorem 5.34 would imply that it belongs to the
Reissner-Nordström subfamily. The existence of such a black hole is not excluded by the staticity
theorem 5.24 due to the degeneracy hypothesis and is an open question.
Remark 10: The no-hair theorem does not hold for dimensions n > 4, even if one restricts oneself to
the vacuum Einstein equation. Indeed, for n > 4, the Kerr solution is generalized to the Myers-Perry
solution discussed in Sec. 5.2.3, but there exist other families of black holes solutions. For n = 5, some
of these families are the black rings (cf. Sec. 5.2.3) and the black saturns (cf. Remark 9 on p. 161).
From an astrophysical perspective, the weaknesses of the no-hair theorem are the hy-
potheses of analyticity and of absence of any closed causal curve in the domain of outer
communications. The analyticity hypothesis is inherited from the strong rigidity theorem
(Property 5.25). In physics, one usually assumes that the fields are smooth, not necessarily ana-
lytic. As discussed in Sec. 5.4.2, the only successful attempt to date to get rid of the analyticity
requirement regards slowly rotating black holes [11]. As for the hypothesis of non-existence
of closed causal curves in ⟨⟨M ⟩⟩, which holds if ⟨⟨M ⟩⟩ is assumed globally hyperbolic (cf.
p. 127), it would be much more satisfactory if this would be a consequence20 of the theorem
and not one of its premises. In the current state, we cannot exclude that there is somewhere
20
The Kerr-Newmann spacetime has a globally hyperbolic domain of outer communications.
180 Stationary black holes
in our Universe a stationary black hole that is distinct from a Kerr-Newman one and around
which one can travel backward in time...
Historical note : The first hints towards the “hairlessness” of black holes arised in 1964-65 from studies
by Vitaly Ginzburg [222] and by Andrei Doroshkevich, Yakov Zeldovich and Igor Novikov [172] (cf.
Chap. 7 of Thorne’s book [479] for some historical details). Ginzburg showed that the gravitational
collapse of a (electrically neutral) magnetized star gives birth to an unmagnetized black hole, while
Doroshkevich, Zeldovich and Novikov showed that the collapse of a non-rotating body slightly departing
from spherical symmetry leads to a perfectly spherical (i.e. Schwarzschild) black hole; the deformations
away from spherical symmetry (the “hairs”) in the external metric are decaying to zero as the black hole
forms. These authors have also shown in the same article [172] that any static quadrupolar deformation
of the Schwarzschild metric leads to a curvature singularity at the event horizon and hence cannot
provide a regular black hole solution to the vacuum Einstein equation. This result is quoted as a
motivation for proving the uniqueness of the Schwarzschild solution by Werner Israel in the famous
1967 article [289] presenting the theorem that bears his name (Property 5.33). In the same article,
Israel raised the question about the uniqueness of the Kerr solution. In 1968, Brandon Carter [90]
conjectured that the Kerr-Newman family — which contains the Schwarzschild one — may represent all
stationary black holes. This conjecture became known as the Carter-Israel conjecture [439, 266]. It has
been popularized by the phrase “a black hole has no hair” by John Wheeler in a review article published
in 1971 [443], as well as in the famous 1973 MTW textbook [371] (Box 33.1). In a 1996 interview,
Wheeler [519] stated that Jacob Bekenstein (then his PhD student) coined that phrase. The conjecture
made most of its way in becoming a theorem in 1972 when Stephen Hawking [260] established the
first version of the strong rigidity theorem (Property 5.25). Indeed the latter basically states that all
stationary black holes are necessarily axisymmetric and a year before, Brandon Carter [93] had shown
that all axisymmetric stationary and 4-dimensional black holes belong either to the Kerr family or to a
disconnected 2-parameter family, the latter possibility being eventually excluded by David Robinson in
1975 [437] (cf. the historical note on p. 178). The no-hair theorem has been strengthened in the following
years by making explicit and/or relaxing certain hypotheses (cf. the historical notes on p. 172, 176 and
178). A critical review of the theorem as of 1994, distinguishing the folklore around it from what has
been mathematically established, has been performed by Piotr Chruściel [115]. Further details about the
theorem history can be found in the accounts by Carter [103] and Robinson [439].
5.6.4 To go further
See Heusler’s textbook (1996) [275] and the review articles by Chruściel, Lopes Costa & Heusler
(2012) [123], Hollands & Ishibashi (2012) [280] and Ionescu & Klainerman (2015) [288]. The
no-hair theorem has been extended from general relativity to scalar-tensor theories, under
some assumptions (among which the scalar field obeys the same symmetries as the spacetime
ones), see the article by Capuano, Santoni & Barausse (2023) [82] for details.
Part II
Schwarzschild black hole
Chapter 6
Schwarzschild black hole
Contents
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.2 The Schwarzschild-(anti-)de Sitter solution . . . . . . . . . . . . . . . 183
6.3 Radial null geodesics and Eddington-Finkelstein coordinates . . . . 189
6.4 Black hole character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.1 Introduction
After having discussed stationary black holes in Chap. 5, we examine here the simplest of them:
the Schwarzschild black hole. Let us recall that the prime importance of this object in general
relativity stems from Israel uniqueness theorem (Property 5.33, Sec. 5.6), which states that any
static black hole in an asymptotically flat 4-dimensional vacuum spacetime ruled by general
relativity must be a Schwarzschild black hole.
In this chapter, we derive the Schwarzschild metric as a solution to the Einstein equation,
possibly with a non-vanishing cosmological constant Λ (Sec. 6.2); we then focus on the case
Λ = 0 (the true Schwarzschild solution) and explore it by means of the Eddington-Finkelstein
coordinates, which have the advantage to be regular on the horizon (Sec. 6.3). Finally, in
Sec. 6.4, we check formally that the Schwarzschild spacetime has a region that obeys the general
definition of a black hole given in Sec. 4.4.2. The maximal extension of the Schwarzschild
spacetime and its bifurcate Killing horizon is discussed Chap. 9, after two chapters (Chap. 7
and 8) devoted to the timelike and null geodesics in Schwarzschild spacetime.
6.2 The Schwarzschild-(anti-)de Sitter solution
6.2.1 Vacuum Einstein equation with a cosmological constant
Let us search for a static and spherically symmetric solution to the Einstein equation (1.43) in a
vacuum 4-dimensional spacetime (M , g) with some arbitrary cosmological constant Λ. For
184 Schwarzschild black hole
n = 4, Eq. (1.43) becomes
R = Λg . (6.1)
An equivalent form is obtained by setting T = 0 in the original Einstein equation (1.40):
1
R+ Λ− R g =0. (6.2)
2
6.2.2 Static and spherically symmetric metric
Let us assume that the spacetime (M , g) is static, in the sense defined in Sec. 5.2.1: the
translation group (R, +) is a isometry group of (M , g) (cf. Sec. 3.3.1), with orbits that are
timelike, at least near some conformal boundary (stationarity property) and hypersurface-
orthogonal (staticity property). Let us denote by ξ the associated Killing vector field (unique
up to some constant rescaling), i.e. the generator of the isometry group (R, +) (cf. Sec. 3.3.1).
We may foliate M by a 1-parameter family of hypersurfaces (Σt )t∈R , such that ξ is normal
to all Σt ’s and t is a parameter associated to ξ:
ξ(t) = 1 (6.3)
or equivalently, ⟨dt, ξ⟩ = 1 [cf. Eq. (5.4)].
In addition to being static, we assume that (M , g) is spherically symmetric, i.e. that it is
invariant under the action of the rotation group SO(3), whose orbits are spacelike 2-spheres
(cf. Sec. 3.3.1). Let S be some generic 2-sphere orbit. The static Killing vector field ξ must
be orthogonal to S , otherwise the orthogonal projection of ξ onto S would define some
privileged direction on S , which is incompatible with spherical symmetry. The orthogonality
of ξ and S implies that S ⊂ Σt . Let (xa ) = (θ, φ) be spherical coordinates on S . The
(Riemannian) metric q induced by g on S is given by
q = r2 dθ2 + sin2 θ dφ2 . (6.4)
The positive coefficient r2 in front of the standard spherical element must be constant over
S , by virtue of spherical symmetry. The area of S is then A = 4πr2 . For this reason, r is
called the areal radius of S . Letting S vary, r can be considered as a scalar field on M . If
dr ̸= 0, we may use it as a coordinate1 . Since S ⊂ Σt , (r, θ, φ) is a coordinate system on each
hypersurface Σt . The set (t, r, θ, φ), where t is adapted to ξ thanks to (6.3), is then a spacetime
coordinate system and, by construction, the expression of the metric tensor with respect to
this system is
g = −A(r) dt2 + B(r) dr2 + r2 dθ2 + sin2 θ dφ2 . (6.5)
Note that this is a special case of the general static metric element (5.5) and that Eq. (5.4) holds:
ξ = ∂t . (6.6)
1
An example of spherically symmetric spacetime with dr = 0 somewhere is the maximally extended
Schwarzschild spacetime, to be studied in Chap. 9: dr = 0 on the so-called bifurcation sphere S (cf. Sec. 9.3.3).
Hence r cannot be used as a coordinate in the vicinity of S ; one shall use Kruskal-Szekeres coordinates instead.
6.2 The Schwarzschild-(anti-)de Sitter solution 185
In particular, gtt = −A(r) and grr = B(r) do not depend on t as a result of the spacetime
stationarity, while gtr = gtθ = gtφ = 0 expresses the orthogonality of ξ and Σt , i.e. the
spacetime staticity. The coordinates (t, r, θ, φ) are called areal coordinates, reflecting the fact
that r is the areal radius.
6.2.3 Solving the Einstein equation
The Christoffel symbols of the metric (6.5) with respect to the areal coordinates are (cf. Sec. D.4.2
for the computation):
1 dA 1 dA 1 dB r
Γt tr = Γt rt = Γr tt = Γr rr = Γr θθ = −
2A dr 2B dr 2B dr B
r sin2 θ 1 (6.7)
Γr φφ = − Γθ rθ = Γθ θr = Γθ φφ = − sin θ cos θ
B r
1 1
Γφrφ = Γφφr = Γφθφ = Γφφθ = ,
r tan θ
the Christoffel symbols not listed above being zero.
The tt component of the Einstein equation (6.2) leads to (cf. Sec. D.4.2 for the computation)
dB
r − B + (1 − Λr2 )B 2 = 0, (6.8)
dr
while the rr component leads to
dA
r + A − (1 − Λr2 )AB = 0. (6.9)
dr
Finally, the θθ and φφ components lead to the same equation:
2
d2 A 2 dA
1 dA 2A dB 1 dA
2 2 + − + − + 4ΛAB = 0. (6.10)
dr r dr B dr r dr A dr
All the other components of the Einstein equation (6.2) are identically zero.
Adding Eq. (6.8) multiplied by A to Eq. (6.9) multiplied by B yields
dA dB d
B +A = (AB) = 0.
dr dr dr
The solution to this equation is obviously A(r)B(r) = C, where C is a constant. Without any
loss of generality, we may choose C = 1. Indeed, substituting C/B(r) for A(r) in Eq. (6.5)
results in
C
dt2 + B(r) dr2 + r2 dθ2 + sin2 θ dφ2 .
g=−
B(r)
√
Assuming C > 0, the change of variable√t′ = Ct, which is equivalent to changing the
stationary Killing vector from ξ to ξ ′ = 1/ C ξ, yields
1
dt′2 + B(r) dr2 + r2 dθ2 + sin2 θ dφ2 ,
g=−
B(r)
186 Schwarzschild black hole
which is exactly the solution corresponding to C = 1. Hence from now on, we set C = 1, i.e.
1
B(r) = . (6.11)
A(r)
Substituting this expression in Eq. (6.9) yields an ordinary differential equation for A(r):
dA
r + A − 1 + Λr2 = 0,
dr
the solution to which is
2m Λ 2
A(r) = 1 − − r , (6.12)
r 3
where m is a constant. The general static and spherically symmetric solution to the vacuum
Einstein equation (6.1) is therefore
−1
2m Λ 2 2 2m Λ 2
dr2 + r2 dθ2 + sin2 θ dφ2 .
g =− 1− − r dt + 1 − − r
r 3 r 3
(6.13)
It is called the Kottler metric (cf. the historical note below). The Schwarzschild metric is
the particular case Λ = 0. If Λ > 0, (6.13) is called the Schwarzschild-de Sitter metric, often
abridged as Schwarzschild-dS metric, while if Λ < 0, it is called the Schwarzschild-anti-de
Sitter metric, often abridged as Schwarzschild-AdS metric.
In the rest of this chapter, we will focus on the Schwarzschild metric, i.e. on the version
Λ = 0 of Eq. (6.13):
−1
2m 2m
2
dr2 + r2 dθ2 + sin2 θ dφ2 . (6.14)
g =− 1− dt + 1 −
r r
The areal coordinates (t, r, θ, φ) are then called the Schwarzschild-Droste coordinates2 .
Since A(r) = 1 − 2m/r and B(r) = (1 − 2m/r)−1 for the Schwarzschild metric, the
non-vanishing Christoffel symbols (6.7) become3
m m(r − 2m) m
Γt tr = Γt rt = Γr tt = Γr rr = −
r(r − 2m) r3 r(r − 2m)
1
Γr θθ = 2m − r Γr φφ = (2m − r) sin2 θ Γθ rθ = Γθ θr = (6.15)
r
1 1
Γθ φφ = − sin θ cos θ Γφrφ = Γφφr = Γφθφ = Γφφθ = .
r tan θ
2
In the literature they are often referred to as simply Schwarzschild coordinates; we follow here Deruelle &
Uzan [162, 163].
3
See also the notebook D.4.3 for a check.
6.2 The Schwarzschild-(anti-)de Sitter solution 187
6.2.4 The mass parameter
The Schwarzschild metric (6.14) depends on a single parameter: m. This parameter has a direct
physical interpretation: it is the gravitational mass (or simply mass) that is felt by an observer
located at large values of r. Indeed, we will see in Chap. 7 that an observer on a circular orbit at
a large value of r has an orbital period T obeying Kepler’s third law: T 2 = 4π 2 r3 /m. Without
waiting for Chap. 7, we may notice that for r ≫ |m|, the metric (6.14) takes the standard
weak-field form (see e.g. [87, 371]):
g ≃ − (1 + 2Φ(r)) dt2 + (1 − 2Φ(r)) dr2 + r2 dθ2 + sin2 θ dφ2 , (6.16)
where Φ(r) := −m/r is the Newtonian gravitational potential outside a spherically symmetric
body of mass m.
Another argument for identifying m with a mass is provided by Example 3 of Sec. 5.3.3,
where it has been shown that m is nothing but the Komar mass associated with the stationarity
of Schwarzschild spacetime.
Historical note : The Schwarzschild metric (6.14) is actually the first non-trivial (i.e. different from
Minkowski metric) exact solution to the Einstein equation ever found. It has been obtained by the
astrophysicist Karl Schwarzschild in the end of 1915 [449], only a few weeks after the publication of
the articles funding general relativity by Albert Einstein. It is also quite remarkable that Schwarzschild
found the solution while serving in the German army at the Russian front. Unfortunately, he died from
a rare skin disease a few months later. The way Schwarzschild proceeded was quite different from that
exposed above: instead of the coordinates (t, r, θ, φ) named today after him, he used the coordinates
(t, x1 , x2 , φ) where x1 = r∗3 /3, with r∗3 = r3 − 8m3 , and x2 = − cos θ. Such a choice was made to
enforce det(gαβ ) = −1, a condition prescribed by Einstein in an early version of general relativity,
which had been presented on 18 November 1915 and on which Schwarzschild was working. Only in
the final version, published on 25 November 1915, did Einstein relax the condition det(gαβ ) = −1,
allowing for full covariance. Schwarzschild however exhibited the famous metric (6.14), via what he
called the “auxiliary quantity” r = (r∗3 + 8m3 )1/3 . For him, the “center”, namely the location of the
“point mass” generating the field, was at r∗ = 0, i.e. at r = 2m. Independently of Schwarzschild,
Johannes Droste, then PhD student of Hendrik Lorentz, arrived at the solution (6.14) in May 1916 [175].
Contrary to Schwarzschild, Droste performed the computation with a spherical coordinate system,
(t, r̄, θ, φ), yet distinct from the standard “Schwarzschild-Droste” coordinates (t, r, θ, φ) by the fact
that the radial coordinate r̄ was not chosen to be the areal radius, but instead a coordinate for which
gr̄r̄ = 1. At the end, by a change of variable, Droste exhibited the metric (6.14). The generalization to
a non-vanishing cosmological constant, i.e. Eq. (6.13), has been obtained by Friedrich Kottler in 1918
[322] and, independently, by Hermann Weyl in 1919 [518]. We refer to Eisenstaedt’s article [180] for a
detailed account of the early history of the Schwarzschild solution.
6.2.5 The Schwarzschild-Droste domain
We immediately notice on (6.14) that the metric components are singular at r = 0 and r = 2m.
Accordingly, the Schwarzschild-Droste coordinates (t, r, θ, φ) cover the following subset of M ,
188 Schwarzschild black hole
which we call the Schwarzschild-Droste domain:
MSD := MI ∪ MII , (6.17a)
MI := R × (2m, +∞) × S2 , (6.17b)
MII := R × (0, 2m) × S2 , (6.17c)
with the coordinate t spanning R, the coordinate r spanning (2m, +∞) on MI and (0, 2m) on
MII , and the coordinates (θ, φ) constituting a standard spherical chart of S2 . Note that MSD
is a disconnected open subset of the full spacetime manifold M (to be specified later), whose
connected components are MI and MII .
Remark 1: To cover entirely S2 in a regular way, one needs a second chart, in addition to (θ, φ); this is
related to the standard singularities of spherical coordinates at θ = 0 and θ = π. It is fully understood
that the metric g, as expressed by (6.14), is fully regular on S2 . The fact that det(gαβ ) = −r2 sin2 θ is
zero at θ = 0 and θ = π reflects merely the coordinate singularity of the (θ, φ) chart there. We shall
not discuss this coordinate singularity any further.
The boundary value rS := 2m of r between MI and MII is conventionaly called the
Schwarzschild radius. A more appropriate name would have been the Schwarzschild areal
radius, for r does not describe a radius (in the sense of a distance from some “origin”) but rather
an area, as discussed in Sec. 6.2.2.
Immediate properties of the Schwarzschild-Droste domain are:
Property 6.1: the Schwarzschild exterior MI
Region (MI , g) is strictly static, in the sense defined in Sec. 5.2.1: the Killing vector ξ = ∂t
is timelike on all MI and is orthogonal to the hypersurfaces t = const. Moreover, (MI , g)
is asymptotically flat: the metric g tends to Minkowski metric when r → +∞.
Proof. From expression (6.14), we see that gtt < 0 for r > 2m. Since gtt = g(∂t , ∂t ), this
implies that ∂t is timelike. Moreover ∂t is orthogonal to any hypersurface Σt defined by a
constant value of t, since we read on (6.14) that g(∂t , ∂i ) = gti = 0 for i ∈ {1, 2, 3} and by
definition, the vectors (∂i ) = (∂r , ∂θ , ∂φ ) form a basis of the tangent planes to Σt . Finally, for
r → +∞, all the metric components given by (6.14) tend to those of the Minkowksi metric
(4.3) [see also Eq. (6.16)].
Property 6.2: the Schwarzschild interior MII
In region MII , the Killing vector ξ = ∂t is spacelike. It follows that (MII , g) is not static: the
translation group (R, +) is still an isometry group of (MII , g), but its orbits are spacelike
curves (cf. Sec. 5.2.1). Besides, in MII , t is a spacelike coordinate (i.e. the hypersurfaces
t = const are timelike, cf. Sec. A.3.2) and r is a timelike one (i.e. the hypersurfaces
r = const are spacelike).
6.3 Radial null geodesics and Eddington-Finkelstein coordinates 189
Proof. We see from (6.14) that gtt > 0 for r < 2m, which implies that ∂t is spacelike there.
Moreover, we have g tt > 0 and g rr < 0 in MII , which, according to the criterion (A.56), implies
that t is a spacelike coordinate and r a timelike one.
Remark 2: In the region MII , grr < 0, so that the metric g keeps a Lorentzian signature, as it should!
Remark 3: As a consequence of Property 6.2 and of the diagonal character of the metric components
(6.14), the axes of the light cones are horizontal lines for r < 2m in Fig. 6.1 below.
6.3 Radial null geodesics and Eddington-Finkelstein coor-
dinates
6.3.1 Radial null geodesics
Let us search for the null geodesics of the Schwarzschild metric (6.14) that are radial, i.e. along
which θ = const and φ = const. They are found by setting dθ = 0 and dφ = 0 in the line
element (1.3) and searching for ds2 = gµν dxµ dxν = 0, with the gµν ’s read on (6.14):
dr2
ds2 = 0 ⇐⇒ dt2 = . (6.18)
2m 2
1− r
Hence the radial null geodesics are governed by
dr
dt = ± . (6.19)
1 − 2m
r
This equation is easily integrated:
r
t = ± r + 2m ln − 1 + const. (6.20)
2m
We have thus two families of curves, one for each choice of the sign ±:
• the outgoing radial null geodesics L(u,θ,φ)
out
, whose equation is
r
L(u,θ,φ)
out
: t = r + 2m ln −1 +u, (6.21)
2m
where u ∈ R is a constant along L(u,θ,φ)
out
;
• the ingoing radial null geodesics L(v,θ,φ)
in
, whose equation is
r
L(v,θ,φ)
in
: t = −r − 2m ln −1 +v , (6.22)
2m
where v ∈ R is a constant along L(v,θ,φ)
in
.
190 Schwarzschild black hole
t/m
4
1 2 3 4 5 6 7 8
r/m
-1
-2
-3
-4
Figure 6.1: Radial null geodesics of Schwarzschild spacetime, plotted in terms of Schwarzschild-Droste
coordinates (t, r): the solid (resp. dashed) lines correspond to outgoing (resp. ingoing) geodesics L(u,θ,φ)
out
(resp.
L(v,θ,φ) ), as given by Eq. (6.21) (resp. Eq. (6.22)). The interiors of some future light cones are depicted in yellow.
in
Note that along a given outgoing radial null geodesic, (u, θ, φ) are constant and that two
outgoing radial null geodesics that have distinct (u, θ, φ) are distinct. We can thus use the
triplet (u, θ, φ) to label the outgoing radial null geodesics, leading to the notation L(u,θ,φ)
out
.
Moreover, the family (L(u,θ,φ) ), with u ∈ R, θ ∈ [0, π] and φ ∈ [0, 2π), forms a congruence:
out
there is one, and only one, such curve through every point of MSD . Similar considerations
apply to the ingoing radial null geodesics L(v,θ,φ)
in
.
By introducing the tortoise coordinate
r
r∗ := r + 2m ln −1 , (6.23)
2m
one may rewrite Eqs. (6.21)-(6.22) as respectively
L(u,θ,φ)
out
: t = r∗ + u (6.24)
L(v,θ,φ)
in
: t = −r∗ + v. (6.25)
The parameter u appears then as a retarded time: u = t − r∗ and v as an advanced time:
v = t + r∗ .
Strictly speaking, we have found radial null curves only, i.e. solutions of Eq. (6.18). Since
not all null curves are null geodesics4 , there remains to prove that the curves defined by (6.21)
4
A famous counterexample is the null helix in Minkowski spacetime, cf. Remark 2 on p. 35.
6.3 Radial null geodesics and Eddington-Finkelstein coordinates 191
and (6.22) obey the geodesic equation [Eq. (B.10) in Appendix B]:
d2 xα µ
α dx dx
ν
+ Γ µν = 0, (6.26)
dλ2 dλ dλ
where λ is an affine parameter (cf. Sec. B.2.1). Let us check that (6.26) is satisfied by choosing
λ = r. For the curves defined by (6.21), we have
r
(xα (r)) = r + 2m ln − 1 + u, r, θ, φ .
2m
Hence
dxα d2 xα
r 2m
= , 1, 0, 0 and = − , 0, 0, 0 .
dr r − 2m dr2 (r − 2m)2
Given the Christoffel symbols (6.15), it is then a simple exercise to show that Eq. (6.26) is
satisfied. The same property holds for the family (6.22). Hence we conclude
Property 6.3: radial null geodesics
The radial null geodesics in the Schwarzschild-Droste domain form two congruences,
out
(L(u,θ,φ) ) and (L(v,θ,φ)
in
), obeying Eqs. (6.21)-(6.22). Moreover, the areal radius r is an affine
parameter along them.
The two congruences of radial null geodesics are depicted in Fig. 6.1. The singularity of
Schwarzschild-Droste coordinates at the Schwarzschild radius r = 2m appears clearly on this
figure.
Remark 1: Despite their name, geodesics of the outgoing family L(u,θ,φ)out are actually ingoing in the
region r < 2m, in the sense that r is decreasing along them when moving towards the future. Indeed,
as noticed in Sec. 6.2.5, for r < 2m, ∂r is a timelike vector and we shall see in Sec. 6.3.6 that −∂r is
oriented towards the future (cf. the “tilted” light cone in Fig. 6.1).
6.3.2 Eddington-Finkelstein coordinates
The parameter v is one of the three parameters labelling the ingoing radial null geodesics
L(v,θ,φ)
in
. Let us promote it to a spacetime coordinate, instead of t, i.e. let us consider the
coordinate system (v, r, θ, φ) that is related to the Schwarzschild-Droste coordinates (t, r, θ, φ)
by Eq. (6.22):
r
v = t + r + 2m ln −1 . (6.27)
2m
By differentiation, it follows immediately that
dr
dt = dv − , (6.28)
1 − 2m/r
192 Schwarzschild black hole
the tensor square of which is [cf. Eq. (A.39)]
2 1
dt2 = dv 2 − dv dr + dr2 .
1 − 2m/r (1 − 2m/r)2
Substituting this expression for dt2 in Eq. (6.14) yields the metric components with respect to
the coordinates (xα̂ ) := (v, r, θ, φ):
2m
dv 2 + 2 dv dr + r2 dθ2 + sin2 θ dφ2 . (6.29)
g =− 1−
r
Remark 2: Since dv dr := 1/2 (dv ⊗ dr + dr ⊗ dv) [cf. Eq. (A.39)], the metric component gvr is one
half of the coefficient of the dv dr term in Eq. (6.29), i.e. gvr = 1.
The coordinates (xα̂ ) = (v, r, θ, φ) are called the null ingoing Eddington-Finkelstein
(NIEF) coordinates. The qualifier null stems from the fact that v is a null coordinate, i.e. the
level sets v = const are null hypersurfaces (cf. Sec. A.3.2). This can be seen from g vv = 0 [cf.
Eq. (A.56)].
To deal with a timelike coordinate instead of a null one, let us set
t̃ := v − r ⇐⇒ v = t̃ + r (6.30)
and define the ingoing Eddington-Finkelstein (IEF) coordinates to be
(xα̃ ) := (t̃, r, θ, φ). (6.31)
Remark 3: From (6.30), v appears as the “time” t̃ “advanced” by r, while from (6.25), v is the “time” t
“advanced” by r∗ .
The relation between the ingoing Eddington-Finkelstein coordinates (t̃, r, θ, φ) and the
Schwarzschild-Droste ones (t, r, θ, φ) is obtained by combining Eqs. (6.27) and (6.30):
r
t̃ = t + 2m ln −1 . (6.32)
2m
The hypersurfaces t = const are plotted in Fig. 6.2, in terms of the IEF coordinates.
From (6.30), we have dv = dt̃ + dr. Substituting into (6.29) yields
2m 4m 2m
2
dr2 + r2 dθ2 + sin2 θ dφ2 . (6.33)
g =− 1− dt̃ + dt̃ dr + 1 +
r r r
To avoid any ambiguity, we shall denote by ∂r̃ the coordinate vector of the IEF frame and
by ∂r the coordinate vector of the Schwarzschild-Droste frame:
∂ ∂
∂r̃ := and ∂r := . (6.34)
∂r t̃,θ,φ ∂r t,θ,φ
6.3 Radial null geodesics and Eddington-Finkelstein coordinates 193
t̃/m
4
1 2 3 4 5 6 7 8
r/m
-1
-2
-3
-4
Figure 6.2: Hypersurfaces of constant Schwarzschild-Droste coordinate t, drawn in terms of the ingoing
Eddington-Finkelstein coordinates (t̃, r). Since the dimensions along θ and φ are not represented, these 3-
dimensional surfaces appear as curves.
The relation between the two vectors is given by the chain rule:
∂ ∂ ∂t ∂ ∂r ∂ ∂θ ∂ ∂φ
= + + + ,
∂r t̃,θ,φ ∂t r,θ,φ ∂r t̃,θ,φ ∂r t,θ,φ ∂r t̃,θ,φ ∂θ t,r,φ ∂r t̃,θ,φ ∂φ t,r,θ ∂r t̃,θ,φ
| {z } | {z } | {z } | {z }
r −1 1 0 0
(1− 2m )
where (6.32) has been used to evaluate ∂t/∂r|t̃,θ,φ . Hence
r −1
∂r̃ = ∂r + 1 − ∂t . (6.35)
2m
On the other hand, we deduce from (6.32) that
∂ ∂
= ,
∂ t̃ r,θ,φ ∂t r,θ,φ
which implies:
∂t̃ = ∂t . (6.36)
In particular, the vector ∂t̃ of the IEF frame coincides with the Killing vector ξ:
∂t̃ = ξ . (6.37)
194 Schwarzschild black hole
Remark 4: The result (6.37) is not surprising since the metric components (6.33) are independent from
t̃. This implies that ∂t̃ is a Killing vector. t̃ being a timelike coordinate in MI , it follows that ∂t̃ = αξ,
where α is a constant. Since t̃ ∼ t when r → +∞, we get α = 1.
Remark 5: In region MII , the four vectors (∂t̃ , ∂r̃ , ∂θ , ∂φ ) are spacelike5 . There is nothing wrong
about that; in particular this does not contradict the signature (−, +, +, +) of the metric. The latter is
related to the type of the basis vectors only when the metric components take a diagonal form, which
is not the case here, since gt̃r ̸= 0 [Eq. (6.33)]. All that is demanded to (t̃, r, θ, φ) for being (locally) a
regular coordinate system is that (∂t̃ , ∂r̃ , ∂θ , ∂φ ) form a basis of the tangent space Tp M at each point
p. This can be achieved with any causal type of the vectors (∂t̃ , ∂r̃ , ∂θ , ∂φ ).
Remark 6: The IEF expression (6.33) for the metric can be recast in the following remarkable form:
2m 2
g = −dt̃2 + dr2 + r2 dθ2 + sin2 θ dφ2 + dt̃ + dr , (6.38)
| {z } r | {z }
f k⊗k
where the f is the (flat) Minkowski metric (expressed above in terms of the spherical coordinates
(t̃, r, θ, φ)) and k = −d(t̃ + r) = −dv. The 1-form k is dual to a vector k, which is null, as it can be
seen from g µ̃ν̃ kµ̃ kν̃ = 0. The latter property is easily deduced from kµ̃ = (−1, −1, 0, 0) and expression
(6.40) for g µ̃ν̃ below. A metric of the type (6.38) is said to be a Kerr-Schild metric; these peculiar metrics
are discussed in Appendix C.
Historical note : Eddington-Finkelstein coordinates have been introduced by Arthur Eddington in
1924 [177]. More precisely, Eddington introduced the outgoing version of these coordinates, while
we have focused above on the ingoing version. Indeed Eddington’s Eq. (2) is t̃ = t − 2m ln(r − m),
which mainly differs from our Eq. (6.32) by the minus sign in front of the logarithm6 , which means that
Eddington’s time coordinate is actually t̃ = u + r, instead of t̃ = v − r (our Eq. (6.30)). Eddington used
his transformation to get the Kerr-Schild form (6.38) of Schwarzschild metric, with (dt̃ + dr)2 replaced
by (dt̃ − dr)2 due to the change ingoing ↔ outgoing. For a modern reader, it is quite surprising that
Eddington did not point out that the metric components w.r.t. (t̃, r, θ, φ) are regular at r = 2m. Actually
the main purpose of Eddington’s article [177] was elsewhere, in the comparison of general relativity
to an alternative theory proposed in 1922 by the mathematician Alfred N. Whitehead (see e.g. [221]).
Only in 1958 did David Finkelstein reintroduce the Eddington transformation and conclude that the
Schwarzschild metric is analytic over the whole domain r ∈ (0, +∞) [195]. Meanwhile the regularity
of Schwarzschild metric at r = 2m had been proven by Georges Lemaître in 1932 [346], via another
coordinate system, which we shall introduce in Sec. 14.2.6 (see also [182] for a detailed discussion), as
well as by John Synge in 1950 [469], by means of yet another coordinate system (cf. the historical note
on p. 294).
Remark 7: In the literature, the terminology Eddington-Finkelstein coordinates is often used for the
coordinates (v, r, θ, φ) (or (u, r, θ, φ)), i.e. for what we have called the null Eddington-Finkelstein
coordinates, and the regularity of the metric tensor at r = 2m is demonstrated by considering the
5
This follows from the diagonal components gαα read on (6.33) being positive for r < 2m, but this can also be
seen graphically on Fig. 6.3 below: for r < 2m, both the t̃ and r coordinate lines, i.e. the vertical and horizontal
lines, are outside the light cones.
6
The other differences with (6.32) are a constant additive term and a misprint in Eddington’s formula: the
term ln(r − m) should be replaced by ln(r − 2m).
6.3 Radial null geodesics and Eddington-Finkelstein coordinates 195
components (6.29). However, neither Eddington [177] nor Finkelstein [195] considered this null version:
they used coordinates (t̃, r, θ, φ) and they exhibited (the outgoing version of) the metric components
(6.33). Hence our terminology is more faithful to history.
6.3.3 The Schwarzschild horizon
Contrary to the Schwarzschild-Droste components (6.14), the metric components (6.33) are
regular as r → 2m. Hence (6.33) defines a regular non-degenerate metric on the whole ingoing
Eddington-Finkelstein domain
MIEF := R × (0, +∞) × S2 , (6.39)
with the coordinate t̃ spanning R, the coordinate r spanning (0, +∞) and the coordinates
(θ, φ) forming the standard spherical chart of S2 . The components of the inverse metric with
respect to the IEF coordinates are
− 1 + 2m 2m
r r
0 0
2m 2m
r
1 − r
0 0
g α̃β̃ = . (6.40)
1
0 0 r2
0
1
0 0 0 r2 sin2 θ
In particular, the components g α̃β̃ are regular at r = 2m. Moreover we notice that g t̃t̃ < 0 for
all r ∈ (0, +∞). In view of the criterion (A.56), we may assert
Property 6.4: Timelike character of the IEF coordinate t̃
The coordinate t̃ is timelike in all MIEF .
This is in contrast with the Schwarzschild-Droste coordinate t, which is timelike in MI but
spacelike in MII (Property 6.2).
The IEF domain is an extension of the Schwarzschild-Droste domain introduced in Sec. 6.2.5:
MIEF = MSD ∪ H = MI ∪ MII ∪ H , (6.41)
where H is the subset of MIEF defined by r = 2m. Note that H has the topology
H ≃ R × S2 (6.42)
and that (t̃, θ, φ) is a coordinate system on H . Actually H is nothing but what has been
called the Schwarzschild horizon in the examples of Chaps. 2 and 3. Indeed, the metric (6.33)
is nothing but the metric (2.5) introduced in Example 3 of Chap. 2 (p. 29), up to the change of
notation t̃ ↔ t (compare (2.6) and (6.40) as well). We have thus the fundamental result, the
proof of which is given in Example 10 of Chap. 3 (p. 68):
196 Schwarzschild black hole
Property 6.5
The hypersurface H defined by r = 2m is a Killing horizon, the null normal of which is ξ.
In particular, H is a null hypersurface, whose null geodesic generators admit ξ = ∂t̃ as tangent
vector. It is a non-expanding horizon, whose area, as defined in Sec. 3.2.2, is (cf. Example 3 of
Chap. 3, p. 59)
A = 16πm2 . (6.43)
H is depicted in Fig. 2.7. We shall see in Sec. 6.4 that H is actually a black hole event horizon
in Schwarzschild spacetime.
Historical note : The first author to recognize that the hypersurface r = 2m in Schwarzschild spacetime
is a one-way membrane, i.e. a horizon, is David Finkelstein in 1958 [195]. Amazingly, Finkelstein stressed
rather r = 2m as a white hole boundary in a time-reversed version of (MIEF , g). In particular, he wrote
“causal influences propagating into the "future" can cross the Schwarzschild surface only in an outward
direction” (see also his Fig. 1). The reason is that Finkelstein considered the extension of the MI region
to r ≤ 2m constructed with the outgoing coordinate t̃˜ = t − 2m ln(r/(2m) − 1) [his Eq. (2.3) in our
notations] instead of t̃ as given by Eq. (6.32). However, he noticed that another extension of MI can be
obtained by time inversion, corresponding to a “surface that is permeable inwards”. As we shall see in
Chap. 9, both black hole and white hole regions actually exist in the maximal extension of Schwarzschild
spacetime.
6.3.4 Coordinate singularity vs. curvature singularity
The above considerations show that the divergence of the metric component grr in (6.14)
when r → 2m reflects a pathology of Schwarzschild-Droste coordinates and not a singularity
in the metric tensor g by itself: (MIEF , g) is perfectly regular spacetime, including at the
Schwarzschild radius r = 2m. The bad behavior of Schwarzschild-Droste coordinates is
obvious in Fig. 6.2: the hypersurfaces t = const fail to provide a regular slicing of spacetime.
This pathology is called a coordinate singularity, since it is intrinsic a given coordinate
system (here the Schwarzschild-Droste one).
Another pathology appears in the metric components in both the Schwarzschild-Droste
coordinates and the ingoing Eddington-Finkelstein ones: gtt and g̃t̃t̃ diverge when r → 0.
This type of singularity cannot be removed by a coordinate transformation. Indeed, the
Kretschmann scalar, defined as the following “square” of the Riemann curvature tensor
K := Rµνρσ Rµνρσ , (6.44)
is (cf. Sec. D.4.3 for the computation)
48m2
K= . (6.45)
r6
Hence K → +∞ when r → 0. Since K is a scalar field, its value is independent of any
coordinate system used to express it. Hence the divergence of K reflects a pathology of the
6.3 Radial null geodesics and Eddington-Finkelstein coordinates 197
t̃/m
4
1 2 3 4 5 6 7 8
r/m
-1
-2
-3
-4
Figure 6.3: Radial null geodesics of Schwarzschild spacetime, plottved in terms of ingoing Eddington-Finkelstein
coordinates (t̃, r): the solid (resp. dashed) lines correspond to outgoing (resp. ingoing) geodesics L(u,θ,φ)
out
(resp.
L(v,θ,φ) ), as given by Eq. (6.51) (resp. Eq. (6.47)). The interiors of some future light cones are depicted in yellow.
in
Note that the hypersurfaces of constant t̃ (horizontal lines) always lie outside the light cones, i.e. are spacelike, in
agreement with t̃ being everywhere a timelike coordinate (Property 6.4). Similarly the hypersurfaces of constant
r (vertical lines) lie outside the light cones for r < 2m, in agreement with r being a timelike coordinate in MII
(Property 6.2).
Riemann tensor per se: it is called a curvature singularity. Physically, this means that
unbounded tidal forces are felt by any system approaching r = 0. This interpretation holds
because tidal forces correspond to the acceleration of the separation vector between neighboring
timelike geodesics and that acceleration is governed by the Riemann tensor, via the geodesic
deviation equation, cf. Property B.22 (see Chap. 11 of MTW [371] for more details).
6.3.5 Radial null geodesics in terms of the Eddington-Finkelstein co-
ordinates
Let us search directly for the radial null geodesics on the IEF domain MIEF by looking for the
radial null curves of the metric (6.33). The vanishing of the line element ds2 = g(dx, dx) for
the radial displacement dx = dt̃ ∂t̃ + dr ∂r̃ leads to
2
dr 4m dr r − 2m
− + − = 0.
dt̃ r + 2m dt̃ r + 2m
The two solutions of this quadratic equation in dr/dt̃ are
dr ±r − 2m
= . (6.46)
dt̃ r + 2m
198 Schwarzschild black hole
Ingoing radial null geodesics
Choosing − for ± in Eq. (6.46), we get dr/dt̃ = −1, so that the integration is immediate and
leads to the ingoing radial null geodesics:
L(v,θ,φ)
in
: t̃ = −r + v , (θ, φ) = const, (6.47)
where the constant v ∈ R labels the null curve, along with (θ, φ). The simplicity of Eq. (6.47)
reflects the construction of the IEF coordinates on the ingoing radial null geodesics. Note that
in MSD , Eq. (6.47) is equivalent to Eq. (6.22), given relation (6.32) between t̃ and t.
We have seen in Sec. 6.3.1 that r is an affine parameter along L(v,θ,φ)
in
. It follows that λ = −r
is an affine parameter as well, in terms of which the equation of L(v,θ,φ) deduced from Eq. (6.47)
in
is
t̃(λ) = λ + v, r(λ) = −λ, θ(λ) = θ = const, φ(λ) = φ = const.
The tangent vector k associated with this parametrization of L(v,θ,φ)
in
has components k α =
dxα /dλ = (1, −1, 0, 0) with respect to the IEF coordinates. We have therefore
k = ∂t̃ − ∂r̃ . (6.48)
The IEF components of the 1-form k metric-dual to k are kα = gαµ k µ with gαµ given by
Eq. (6.33). We get kα = (−1, −1, 0, 0), hence k = −dt̃ − dr, i.e.
k = −dv. (6.49)
Outgoing radial null geodesics
For ± equal to + in Eq. (6.46), we get
dr r − 2m
= . (6.50)
dt̃ r + 2m
To proceed, we have to distinguish two cases. First, if r ̸= 2m, Eq. (6.50) can be rewritten as
r + 2m 4m
dt̃ = dr = 1 + dr,
r − 2m r − 2m
the integration of which leads to the outgoing radial null geodesics:
r
L(u,θ,φ)
out
: t̃ = r + 4m ln −1 +u, (θ, φ) = const, (6.51)
2m
where the integration constant u ∈ R labels the null curve, along with (θ, φ). Since r ̸= 2m,
Eq. (6.51) regards MSD and we actually recover Eq. (6.21), given relation (6.32) between t̃ and t.
On can easily show that a curve defined by (6.51) never crosses H , i.e. it either lies entirely
in MI or lies entirely in MII (cf. Fig. 6.3). We have thus two families of outgoing radial null
geodesics, each labelled by (u, θ, φ): L(u,θ,φ)
out,I
in MI and L(u,θ,φ)
out,II
in MII .
6.3 Radial null geodesics and Eddington-Finkelstein coordinates 199
Let us now consider the case r = 2m in Eq. (6.50); the equation reduces then to dr/dt̃ = 0,
so that the equation of the radial null curve is r = const. The constant being necessarily 2m,
we get the family of curves L(θ,φ)
out,H
defined by
L(θ,φ)
out,H
: r = 2m , (θ, φ) = const. (6.52)
These null curves, which form a family labelled by (θ, φ), are actually the null geodesic
generators of the Killing horizon H (cf. Secs. 6.3.3 and 2.3.3). Indeed, it follows from Eq. (6.52)
that a tangent vector to them is ∂t̃ = ξ, which, for r = 2m, is the null normal of H . The
null geodesics L(θ,φ)
out,H
had not been found in Sec. 6.3.1 for they don’t belong to MSD . They
extend the outgoing family L(u,θ,φ)
out
since they obey the same differential equation (6.50) as the
geodesics L(u,θ,φ) . Moreover, they correspond to the limiting case r = const separating the
out
two subfamilies of outgoing geodesics: L(u,θ,φ)
out,I
, which have r increasing towards the future
(and therefore are truly “outgoing”) and L(u,θ,φ) , which have r decreasing towards the future
out,II
(cf. Fig. 6.3).
Given that both L(θ,φ)
out,H
and L(θ,φ)
out,H
obey the differential equation (6.50), the vector field
tangent to them when parametrized by t̃ is ℓ̂ := ∂t̃ + (r − 2m)/(r + 2m)∂r̃ . For future
convenience, we shall actually consider the vector field ℓ := (1/2 + m/r) ℓ̂; we have then by
construction:
Property 6.6: outgoing radial null vector field
The vector field
1 2m 2m
ℓ= 1+ ∂t̃ + 1 − ∂r̃ (6.53)
2 r r
is a null regular vector field in all MIEF , which is tangent to the outgoing radial null
geodesics L(u,θ,φ)
out,I
in MI , to L(u,θ,φ)
out,II
in MII and to L(θ,φ)
out,H
on H .
H
Note that, on H , ℓ = ∂t̃ = ξ.
6.3.6 Time orientation of the spacetime manifold
From now on, we consider as Schwarzschild spacetime (M , g) the spacetime whose manifold
is the largest one considered so far, i.e. the ingoing Eddington-Finkelstein domain:
M := MIEF = MI ∪ H ∪ MII . (6.54)
We have then M = R × (0, +∞) × S2 [Eq. (6.39)]. Note that we shall extend this spacetime in
Chap. 9.
The vector k defined by Eq. (6.48) is a nonzero null vector field defined on the whole manifold
M . It may therefore be used to set the time orientation of the Schwarzschild spacetime (M , g)
(cf. Sec. 1.2.2). Since for r → +∞, k clearly points towards increasing t̃, we declare that k
defines the future direction:
200 Schwarzschild black hole
Property 6.7: time orientation of Schwarzschild spacetime
The time orientation of the Schwarzschild spacetime (M , g) is such that the null vector k
defined by Eq. (6.48) is everywhere future-directed.
The above choice induces a time orientation of the subdomains MI and MII of M . We
read on the metric components (6.14) that grr < 0 for r < 2m, so that the coordinate vector ∂r
of Schwarzschild-Droste coordinates is timelike in MII . According to Lemma 1.2 (Sec. 1.2.2)
with u = k, its time orientation is given by the scalar product k · ∂r . Given Eqs. (6.35) and
(6.36), we have −1
2m
∂r = − 1 − ∂t̃ + ∂r̃ .
r
Via (6.48) and (6.33), we deduce then that
r r −1
k · ∂r = 1− > 0 in MII .
2m 2m
In view of Eq. (1.6c) in Lemma 1.2, we conclude:
Property 6.8: ∂r past-directed timelike in the black hole region
In region MII , the vector ∂r of Schwarzschild-Droste coordinates is a past-directed timelike
vector.
This explains why −∂r lies within the future null cones in Fig. 6.1 for r < 2m.
A corollary is:
Property 6.9: decreasing of r in the black hole region
In region MII , r must decrease towards the future along any null or timelike worldline.
Proof. Let L be a causal curve in region MII and λ a parameter along L increasing towards
the future. The associated tangent vector v = dx/dλ is then future-directed. According to
Property 6.8, −∂r is a future-directed timelike vector in MII , so that we can apply Lemma 1.1
(Sec. 1.2.2) with u = −∂r and get g(−∂r , v) < 0. Now, using the Schwarzschild-Droste
components (6.14), we have
−1
µ r dr 2m dr
g(−∂r , v) = −grµ v = −grr v = −grr = −1 .
dλ r dλ
Since 2m/r − 1 > 0 in MII , g(−∂r , v) < 0 is thus equivalent to dr/dλ < 0, which proves
that r is decreasing along L as λ increases.
Thus not only an observer in MII cannot cross MII ’s outer boundary H to visit MI , H being
a null hypersurface, but he is forced to move to decreasing r until he reaches the curvature
singularity at r → 0. We shall study this motion in detail in Sec. 7.3.2.
6.4 Black hole character 201
Figure 6.4: Manifold with boundary M ′ = MII ∪ H ∪ MI ∪ I ′ , drawn in terms of the coordinates x and
(a compactified version of) v. The dashed lines are the ingoing radial null geodesics (as in Fig. 6.3), the arrows
marking the future orientation.
6.4 Black hole character
We have already seen in Sec. 6.3.3 that H is a Killing horizon. In particular, it is a null
hypersurface, and thereby a one-way membrane (cf. Sec. 2.2.2). Since H is the boundary of
MII , we conclude that no particle nor electromagnetic signal may emerge from MII (this is
pretty clear by looking to null geodesics on Fig. 6.3). Hence, with respect to the “outside” world,
represented by the asymptotically flat region MI , MII is a black hole.
It would be satisfactory though to check that MII fulfills the formal definition of a black hole
region that we have given in Sec. 4.4.2. The first step is to define a conformal completion at null
infinity (M˜, g̃) of the Schwarzschild spacetime (M , g), as defined by Eq. (6.54). To this aim, let
us start from the null ingoing Eddington-Finkelstein coordinates (xα̂ ) = (v, r, θ, φ) introduced
in Sec. 6.3.2; they cover entirely M and the metric tensor g is expressed in terms of them by
′
Eq. (6.29). Performing the change of coordinates (xα̂ ) = (v, r, θ, φ) 7→ (xα ) = (v, x, θ, φ) with
2m 2m
x=1− ⇐⇒ r = , x ∈ (−∞, 1), (6.55)
r 1−x
we deduce from (6.29) that
4m 4m2
g = −x dv 2 + 2 2 2
(6.56)
dv dx + dθ + sin θ dφ .
(1 − x)2 (1 − x)2
Defining
2m
Ω := 1 − x = , (6.57)
r
202 Schwarzschild black hole
we may rewrite the metric tensor as
g = Ω−2 g̃, (6.58)
with
g̃ = −x(1 − x)2 dv 2 + 4m dv dx + 4m2 dθ2 + sin2 θ dφ2 . (6.59)
Since (v, x, θ, φ) is a global coordinate system on M (up to the trivial coordinate singularities
of (θ, φ)), we can identify M with the following open subset of R2 × S2 :
M = R × (−∞, 1) × S2 , (6.60)
with v spanning R, x spanning (−∞, 1) and (θ, φ) spanning S2 . We can then extend M to the
manifold with boundary7
M ′ := R × (−∞, 1] × S2 . (6.61)
Notice the change (−∞, 1) → (−∞, 1] with respect to (6.60), which means that x = 1 is an
allowed value on M ′ ; it actually defines the boundary of M ′ , I ′ say. According to (6.55), I ′
corresponds to r → +∞. A view of the manifold M ′ is provided in Fig. 6.4. We note that the
conformal metric (6.59) can be extended to the boundary I ′ , yielding a regular metric. Indeed,
the determinant of the metric components (6.59) is
det (g̃α′ β ′ ) = −64m6 sin2 θ,
which does not vanish at x = 1 (except at the trivial coordinate singularity θ = 0 or θ = π),
showing that g̃ is a non-degenerate symmetric bilinear form at I ′ and hence a well-defined
metric on all M ′ . Furthermore we have Ω > 0 on M and Ω = 0 at I ′ [set x = 1 in Eq. (6.57)],
as well as
dΩ = −dx ̸= 0. (6.62)
Hence (M ′ , g̃) obeys all the conditions listed in Sec. 4.3 to be a conformal completion at infinity
of (M , g). However, it is not adapted to the black hole definition given in Sec. 4.4.2. Indeed
I ′ does not include any future infinity (I + ). Actually, I ′ is entirely a past infinity: a generic
point of I ′ has coordinates (v, x, θ, φ) = (v0 , 1, θ0 , φ0 ) and is the past end point of the ingoing
radial null geodesic defined by (v, θ, φ) = (v0 , θ0 , φ0 ). Therefore, we shall extend M ′ to include
some I + part. To achieve this, we shall construct I + as the set of endpoints of the outgoing
radial null geodesics in MI . In terms of the null ingoing Eddington-Finkelstein coordinates
(v, r, θ, φ), the equation of these geodesics is obtained by combining (6.51) and (6.47):
r
v = 2r + 4m ln − 1 + u, (6.63)
2m
where u ∈ R is a constant parameter along a given geodesic. We notice that on MI , we may
use (xα̌ ) = (u, r, θ, φ) as a coordinate system, naturally called the null outgoing Eddington-
Finkelstein coordinates. Since (6.63) implies
2
dv = du + dr,
1 − 2m/r
7
Cf. Sec. A.2.2 for the definition of a manifold with boundary.
6.4 Black hole character 203
Figure 6.5: Manifold with boundary MI′′ = MI ∪ I ′′ , drawn in terms of the coordinates x and (a compactified
version of) u. The green solid lines are the outgoing radial null geodesics (as in Fig. 6.3), the arrows marking the
future orientation. Note that H , which is drawn on this figure, is not part of MI′′ .
we easily deduce from (6.29) the metric components in these coordinates:
2m
du2 − 2 du dr + r2 dθ2 + sin2 θ dφ2 . (6.64)
g =− 1−
r
Remark 1: Contrary to (v, r, θ, φ), the coordinates (u, r, θ, φ) do not cover all M = MIEF , but only MI .
out,I
This is graphically evident from Fig. 6.3, where the outgoing radial null geodesics L(u,θ,φ) accumulate
on H as u → +∞ from the MI side.
′′
On MI , let us perform the change of coordinates (xα̌ ) = (u, r, θ, φ) → (xα ) = (u, x, θ, φ),
where x is related to r by the same formula as (6.55), except that on MI , x’s range is (0, 1) only.
We deduce from (6.64) and (6.55) the expression of g in terms of the coordinates (u, x, θ, φ):
4m 4m2
g = −x du2 − 2 2 2
(6.65)
du dx + dθ + sin θ dφ .
(1 − x)2 (1 − x)2
Let us identify MI with the following open subset of R2 × S2 :
MI = R × (0, 1) × S2 , (6.66)
with u spanning R, x spanning (0, 1) and (θ, φ) spanning S2 . Similarly to what we did above
for M , we may then extend MI to the manifold with boundary
MI′′ := R × (0, 1] × S2 . (6.67)
The boundary of MI′′ , I ′′ say, lies at x = 1 (cf. Fig. 6.5). It shall not be confused with the
boundary of MI as a submanifold of M ′ , which is I ′ . The difference arises from the fact that
u diverges (to −∞) when one approaches I ′ in M ′ , so that u cannot be used as a coordinate
on M ′ . This is clear on the relation (6.63) between u, v and r, which, once re-expressed in
terms of x, becomes
1 x
u = v − 4m + ln . (6.68)
1−x 1−x
For a fixed value of v in M ′ , this relation yields indeed diverging values of u at two places:
204 Schwarzschild black hole
• x → 0+ (the horizon H ): u → +∞;
• x → 1− (the boundary I ′ ): u → −∞.
Reciprocally, for a fixed value of u, relation (6.68) implies that v diverges (to +∞) when x → 1− ,
which shows that I ′′ is not included in M ′ .
The conformal metric g̃ on MI′′ is given by
g̃ = −x(1 − x)2 du2 − 4m du dx + 4m2 dθ2 + sin2 θ dφ2 . (6.69)
We notice that it is regular and non-degenerate in all MI′′ , including on I ′′ (x = 1), and that
on the submanifold MI , it is related to the physical metric g by g̃ = Ω2 g, with the scalar field
Ω taking the same expression in terms of x as that introduced in Eq. (6.57): Ω = 1 − x.
The conformal completion of (M , g) including both I ′ (as I − ) and I ′′ (as I + ) is con-
structed as follows. Let
M˜ = M ′ ∪ MI′′ . (6.70)
We endow M˜ with two coordinate charts:
Φ1 : M ′ −→ R × (−∞, 1] × S2 Φ2 : MI′′ −→ R × (0, 1] × S2
and
p 7−→ (v, x, θ, φ) p 7−→ (u, x, θ, φ)
(6.71)
and define the intersection of the two chart codomains:
M ′ ∩ MI′′ = {p ∈ M ′ , x(p) ∈ (0, 1)} = {p ∈ MI′′ , x(p) ∈ (0, 1)}, (6.72)
along with the transition map implementing (6.68):
Φ2 ◦ Φ−1
1 : R × (0, 1) × S
2
−→ R × (0, 1) × S2
1 (6.73)
x
(v, x, θ, φ) 7−→ u = v − 4m 1−x + ln 1−x
, x, θ, φ ,
The above construction makes M˜ a manifold with boundary (cf. Fig. 6.6), the boundary being
I = I + ∪ I −, (6.74)
with
I + := {p ∈ MI′′ , x(p) = 1} and I − := {p ∈ M ′ , x(p) = 1}. (6.75)
We then endow M˜ with a Lorentzian metric g̃, whose expression is given by (6.59) on M ′
and by (6.69) on MI′′ . By construction, (M˜, g̃) is then a conformal completion at null infinity
of the Schwarzschild spacetime (M , g), the conformal factor Ω being given by (6.57) in both
charts (M ′ , Φ1 ) and (MI′′ , Φ2 ): Ω = 1 − x. In particular, it is clear that no past-directed causal
curve originating in M intersects I + and that no future-directed causal curve originating in
M intersects I − . We also check immediately that I + and I − are null hypersurfaces with
respect to the metric g̃: both hypersurfaces are defined by x = 1, so that the induced metric on
them, as deduced from (6.59) and (6.69), is
g̃|I ± = 4m2 dθ2 + sin2 θ dφ2 , (6.76)
6.4 Black hole character 205
Figure 6.6: Schematic view of the manifold with boundary M˜, which defines a conformal completion at null
infinity of Schwarzschild spacetime (M , g). NB: contrary to Figs. 6.4 and 6.5, this figure is not drawn on some
specific coordinate system. As in Figs. 6.3, 6.4 and 6.5, the green solid (resp. dashed) lines are the outgoing (resp.
ingoing) radial null geodesics, the arrows marking the future orientation.
which is clearly degenerate (along the u direction for I + and along the v direction for I − ).
As it is clear from Fig. 6.6, MI is the interior of the causal past of I + within M :
MI = int J − (I + ) ∩ M . (6.77)
In view of the formal definition (4.37), we conclude:
Property 6.10: black hole region in Schwarzschild spacetime
The Schwarzschild spacetime (M = MIEF , g) has a black hole region B, the interior of
which is MII ; the event horizon is nothing but the Schwarzschild horizon H discussed in
Sec. 6.3.3.
Remark 2: As stated at the beginning of this section, the null character of the boundary H between
MI and MII and the fact that MII never intersect the asymptotically flat region r → +∞, was sufficient
to claim that MII represents what by any means should be called a black hole region. Therefore, we can
view the above demonstration more as a “sanity check” of the formal definition of a black hole given in
Sec. 4.4.2: this definition would not have been acceptable if it would not apply to the Schwarzschild
spacetime.
Remark 3: The above construction of the conformal completion at null infinity (M˜, g̃) involves two
coordinate charts, (v, x, θ, φ) and (u, x, θ, φ), with two different domains, M ′ and MI′′ . As will be
discussed in Chap. 9, one may construct a conformal completion with a single chart, as in the Minkowski
case, but its relation with the coordinates introduced so far is quite involved. In particular the standard
206 Schwarzschild black hole
compactification of Kruskal-Szekeres coordinates, which is used in many textbooks to construct the
Carter-Penrose diagram of Schwarzschild spacetime, does not provide any conformal completion, as it
will be discussed in Sec. 9.4.2.
Chapter 7
Geodesics in Schwarzschild spacetime:
generic and timelike cases
Contents
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.2 Geodesic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.3 Timelike geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
7.1 Introduction
We have already investigated some geodesics of Schwarzschild spacetime in Chap. 6, namely
the radial null geodesics (Secs. 6.3.1 and 6.3.5). Here, we perform an extensive study. After
having established the main properties of generic causal (timelike or null) geodesics in Sec. 7.2,
we investigate timelike geodesics in Sec. 7.3. They are of great physical importance, since they
represent orbits of planets or stars around the black hole, as well as worldlines of intrepid
observers freely falling into the black hole. The study of null geodesics, which govern images
received by observers, is deferred to Chap. 8.
Before studying this chapter, the reader might want to have a look at Appendix B, which
recaps the properties of geodesics in manifolds equipped with a metric. It could also be worth
to read again Sec. 1.3 about worldlines of particles.
7.2 Geodesic motion
Let L be a geodesic1 of Schwarzschild spacetime (M , g). We shall assume that L is causal,
i.e. either timelike or null2 . It therefore can be considered as the worldline of some particle P,
1
The definition and basic properties of geodesics are recalled in Appendix B; see also Sec. 1.3.2.
2
As shown in Sec. B.2.1, a geodesic cannot be partly timelike and partly null.
208 Geodesics in Schwarzschild spacetime: generic and timelike cases
either massive (L timelike) or massless (L null). As recalled in Sec. 1.3, the worldline of a
particle P is a geodesic if, and only if, P is submitted only to gravitation, i.e. P is in free fall.
7.2.1 First integrals of motion
The Schwarzschild spacetime (M , g) is static and spherically symmetric; the Killing vector ξ
associated with the staticity (cf. Sec. 6.2.2) and the Killing vector η associated with the rotation
symmetry along some axis, give birth to two conserved quantities along L :
Property 7.1: conserved quantities along causal geodesics
Denoting by p the 4-momentum of particle P that follows the geodesic L (cf. Sec. 1.3),
the scalar products
E := −ξ · p = −g(ξ, p) (7.1a)
L := η · p = g(η, p) , (7.1b)
are constant along L . The scalar E is called P’s conserved energy or energy at infin-
ity, while L is called P’s conserved angular momentum or angular momentum at
infinity.
Proof. The 4-momentum p is a tangent vector associated with an affine parameter of L , i.e. it
obeys the geodesic equation (1.11). The constancy of E and L follow then from Property B.20
(Sec. B.5).
In coordinates (t, r, θ, φ) adapted to the spacetime symmetries, i.e. coordinates such that ξ = ∂t
and η = ∂φ , for instance the Schwarzschild-Droste coordinates or the Eddington-Finkelstein
ones, one can rewrite (7.1) in terms of the components pt = gtµ pµ and pφ = gφµ pµ of the
1-form p associated to p by metric duality:
E = −pt (7.2a)
L = pφ (7.2b)
Indeed, in such a coordinate system, ξ µ = δ µt and η µ = δ µφ , so that E = −gµν ξ µ pν =
−gtν pν = −pt and L = gµν η µ pν = gφν pν = pφ .
It is worth stressing that E is not a genuine energy, i.e. it is not an energy measured by some
observer. Indeed, the latter is defined by Eq. (1.23), which resembles Eq. (7.1a) but differs from
it by ξ not being a unit vector in general: ξ · ξ ̸= −1. In other words, ξ cannot be interpreted as
the 4-velocity of some observer, so that the quantity E defined by (7.1a) cannot be a physically
measured particle energy. It is only in the asymptotic region, where ξ · ξ = gtt → −1, that ξ is
eligible as a 4-velocity, hence the name energy at infinity. Note that this name is commonly used,
even in the particle P never visits the asymptotic region. Similarly, L is not some (component
of a) genuine angular momentum. Only in the asymptotic region do we have
L ≃ gφφ pφ ≃ r2 sin2 θ pφ ≃ r2 sin2 θ P φ ≃ r sin θ P (φ) , (7.3)
7.2 Geodesic motion 209
where P (φ) is the azimuthal component of the momentum P of particle P as measured by
an asymptotic inertial observer O (cf. Sec. 1.4), i.e. the component of P along e(φ) in the
orthonormal basis (e(r) , e(θ) , e(φ) ), with e(φ) = (r sin θ)−1 ∂φ . In view of (7.3), we may say that
L is the angular momentum about the symmetry axis θ = 0 that O would attribute to particle
P if the latter would move close to him. Equivalently, in a Cartesian coordinate system defined
by (x, y, z) = (r sin θ cos φ, r sin θ sin φ, r cos θ), L is the component Lztot of the total angular
momentum of P as measured by O:
Ltot := r × P = Lxtot ∂x + Lytot ∂y + L ∂z . (7.4)
Property 7.2: positivity of the conserved energy
The conserved energy E is a positive quantity as soon as the geodesic L has some part in
MI , i.e. some part with r > 2m:
L ∩ MI ̸= ∅ =⇒ E > 0. (7.5)
Proof. In MI , the Killing vector ξ is timelike and future-directed. The 4-momentum p is either
timelike or null and always future-directed. By Eq. (1.5a) in Lemma 1.1 (Sec. 1.2.2), one has
then necessarily ξ · p < 0; hence Eq. (7.1a) implies E > 0 in MI . Since E is constant along L ,
it follows that E > 0 everywhere.
Remark 1: If the geodesic L is confined to MII , i.e. to the black hole region (cf. Sec. 6.4), where ξ
is spacelike (cf. Sec. 6.2.5), it is possible to have E ≤ 0, since the scalar product of p with a spacelike
vector can take any value.
Remark 2: The Killing vector η being always spacelike, the scalar product g(η, p) can a priori take
any real value, and thus there is no constraint on the sign of L.
To be specific, let us describe Schwarzschild spacetime in terms of the Schwarzschild-Droste
coordinates (t, r, θ, φ) introduced in Sec. 6.2.3. Without any loss of generality, we may choose
these coordinates so that at t = 0, the particle P is located in the equatorial plane θ = π/2 and
the spatial projection of the worldline L lies in that plane, i.e. p has no component along ∂θ :
t=0
p = pt ∂t + pr ∂r + pφ ∂φ . (7.6)
Now, for t > 0, if the geodesic L were departing from θ = π/2, this would constitute some
breaking of spherical symmetry, making a difference between the “Northern” hemisphere and
210 Geodesics in Schwarzschild spacetime: generic and timelike cases
the “Southern” one. Hence3 L must stay at θ = π/2, which implies
pθ = 0 . (7.7)
We conclude:
Property 7.3: planar character of geodesics
A geodesic L of Schwarzschild spacetime is necessarily confined to a timelike hypersurface.
Without any loss of generality, we can choose Schwarzschild-Droste coordinates (t, r, θ, φ)
such that this hypersurface is the “equatorial hyperplane” θ = π/2. Then the component pθ
of the 4-momentum of the particle having L as worldline vanishes identically [Eq. (7.7)].
Let us denote by µ the mass of particle P, with possibly µ = 0 if P is a photon. The scalar
square of the 4-momentum p is then [cf. Eq. (1.10)]
g(p, p) = −µ2 . (7.8)
So µ2 can be seen as an integral of motion.
7.2.2 Equations of motion and generic properties
Contemplating Eqs. (7.1a), (7.1b), (7.7) and (7.8), we realize that we have four first integral
of motions. The problem is then completely integrable. More specifically, let λ be the affine
parameter along the geodesic L associated with the 4-momentum p [cf. Eq. (B.2)]:
dx
p= , (7.9)
dλ
where dx is the infinitesimal displacement along L corresponding to the parameter change
dλ. Note that λ is dimensionless and necessarily increases towards the future4 , since p is
by definition future-directed (cf. Sec. 1.3.1). In terms of the components with respect to
Schwarzschild-Droste coordinates, this yields
dt dr dθ dφ
ṫ := = pt , ṙ := = pr , θ̇ := = pθ , φ̇ := = pφ . (7.10)
dλ dλ dλ dλ
3
More rigorously, Eq. (7.7) can be derived from the geodesic equation (1.11): given the expression of the
Christoffel symbols of g in Schwarzschild-Droste coordinates (cf. Sec. D.4.2), Eq. (1.11) yields
dpθ 2 2
+ pr pθ − sin θ cos θ (pφ ) = 0,
dλ r
where λ is the affine parameter of L associated with p, so that pθ = dθ/dλ. Whatever the values of r(λ), pr (λ)
and pφ (λ), the solution to this ordinary differential equation with the initial conditions pθ = 0 and cos θ = 0 is
pθ = 0 for all values of λ.
4
Let us recall that Schwarzschild spacetime is time-oriented, cf. Sec. 6.3.6.
7.2 Geodesic motion 211
In the present case, where θ(λ) = π/2, we have of course θ̇ = 0, in agreement with Eq. (7.7).
Given the components (6.14) of Schwarzschild metric with respect to the Schwarzschild-Droste
coordinates, Eq. (7.1a) can be written as
µ t 2m
E = −gtµ p = −gtt p = −gtt ṫ = 1 − ṫ,
r
hence
−1
dt 2m
=E 1− . (7.11)
dλ r
Similarly, Eq. (7.1b) becomes
L = gφµ pµ = gφφ pφ = gφφ φ̇ = r2 sin2 θ φ̇.
Since θ = π/2, we get
dφ L
= 2 . (7.12)
dλ r
We have already noticed that the sign of L is unconstrained (Remark 2 on p. 209). The above
equation shows that it corresponds to the increase (L > 0) or decrease (L < 0) of φ along the
geodesic L . In other words, we deduce from Eq. (7.12) that
Property 7.4: monotonic behavior of φ along geodesics
Along any timelike or null geodesic of Schwarzschild spacetime, the azimuthal coordinate
φ is either constant (L = 0) or increases (resp. decreases) monotonically (L > 0) (resp.
L < 0).
The last unexploited first integral of motion is Eq. (7.8); it yields
−1
2m 2 2m
− 1− (ṫ) + 1 − (ṙ)2 + r2 (θ̇)2 + r2 sin2 θ(φ̇)2 = −µ2 .
r r
Using (7.11), (7.12), as well as θ̇ = 0 and θ = π/2, we get
−1 −1
L2
2 2m 2m
−E 1 − + 1− (ṙ)2 + 2 = −µ2 ,
r r r
which can be recast as
2
2µ2 m L2
dr 2m
− + 2 1− = E 2 − µ2 . (7.13)
dλ r r r
To summarize, the geodesic motion in Schwarzschild spacetime is governed by Eqs. (7.11),
(7.12) and (7.13), where r = r(λ) and µ, E and L are constants. This constitutes a system of
3 differential equations for the 3 unknown functions t(λ), r(λ) and φ(λ). We observe that
212 Geodesics in Schwarzschild spacetime: generic and timelike cases
Eq. (7.13) is decoupled from the other two equations. The task is then to first solve this equation
for r(λ) and to inject the solution into Eqs. (7.11) and (7.12), which can then be integrated
separately.
A constraint to keep in mind is that the 4-momentum vector p, whose components are
related to the solution (t(λ), r(λ), φ(λ)) by Eq. (7.10), has to be a future-directed causal vector.
In MI , as we have seen above, this is guaranteed by choosing E > 0 [cf. Eq. (7.5)]. In MII ,
a future-directed timelike vector is −∂r (cf. Sec. 6.3.6). According to Eq. (1.5a) in Lemma 1.1
(Sec. 1.2.2), we have then p future-directed iff −∂r · p < 0, i.e. iff
−1
2m
−1 pr < 0.
r
Since 2m/r − 1 > 0 in MII , this is equivalent to pr < 0, i.e. to dr/dλ < 0. Hence
Property 7.5: decreasing of r in the black hole region
In the black hole region MII , i.e. for r < 2m, the solution r(λ) of Eq. (7.13) must be a
strictly decreasing function of λ.
Actually, we recover Property 6.9, which has been derived for any causal worldline (not
necessarily a geodesic) in Sec. 6.3.6.
Remark 3: We have derived the system of Eqs. (7.11), (7.12) and (7.13) without invoking explicitly the
famous geodesic equation, i.e. Eq. (B.10) in Appendix B. This is because we had enough first integrals of
the second-order differential equation (B.10) to completely reduce it to a system of first order equations.
7.2.3 Trajectories in the orbital plane
If the conserved angular momentum vanishes, L = 0, the equation of motion (7.12) implies
that φ = const = φ0 . The geodesic L is then confined to the 2-dimensional timelike surface
(θ, φ) = (π/2, φ0 ), which is spanned by the coordinates (t, r). One says that L is a radial
geodesic.
In the remainder of this section, we discuss the opposite case, namely we assume
L ̸= 0. (7.14)
We have stressed above that φ is then a strictly monotonic function of λ, increasing (resp.
decreasing) continuously along L for L > 0 (resp. L < 0). Consequently,
Property 7.6: φ as a parameter along geodesics
Along any timelike or null geodesic with L ̸= 0, φ can be chosen as a parameter, provided
one does not restrict its range to (0, 2π).
7.2 Geodesic motion 213
Contrary to λ, φ is not in general an affine parameter of L . Indeed, the dependency r = r(λ)
in Eq. (7.12) does not correspond to an affine relation between φ and λ, except for r(λ) = const
(case of circular orbits).
Let us use Eq. (7.12) to write dr/dλ = dr/dφ × dφ/dλ = L/r2 dr/dφ and substitute this
expression into the equation of motion (7.13). We get
2 µ 2 m E 2 µ 2
1 dr 2m 1
= 3 − 2 +2 + − . (7.15)
r4 dφ r r L r L L
To simplify this equation, it is natural to introduce the dimensionless variable
m
u := (7.16)
r
instead of r. We then get
2 2
du mµ 2 mE mµ 2
3 2
= 2u − u + 2 u+ − . (7.17)
dφ L L L
This differential equation determines entirely the (r, φ)-part of the geodesic L , which we shall
call the trajectory of L in the orbital plane. The term “orbital plane” is a slight abuse of
language for the 2-dimensional surface (t, θ) = (t0 , π/2), where t0 is a constant.
In general Eq. (7.17) is not solvable in terms of elementary functions. The exceptions are
circular orbits (u = const, the constant being one of the roots of the cubic polynomial in u in
the right-hand side) and the critical null geodesics, which we shall discuss in Sec. 8.3.2. For the
generic case, exact solutions are expressible in terms of some non-elementary special functions;
there are basically two strategies:
• The first one is to invoke the Weierstrass elliptic function5 ℘(z; ω1 , ω2 ), which is a doubly-
periodic meromorphic function of the complex variable z, of periods ω1 ∈ C and ω2 ∈ C.
Among the many properties of this function, the one relevant here is that ℘ is a solution
to the differential equation
2
(℘′ (z)) = 4℘(z)3 − g2 ℘(z) − g3 , (7.18)
where g2 and g3 are two constants entirely determined by the √ periods ω1 and ω2 of ℘.
Indeed, via the change of variables v := u − 1/6 and φ̃ := φ/ 2, Eq. (7.17) is equivalent
to
2 mµ 2 2
dv 1 mE 4 mµ 2 1
3
= 4v − −4 v+2 − − , (7.19)
dφ̃ 3 L L 3 L 27
which is obviously of type (7.18) (no square in the cubic polynomial). The solution is thus
m φ 1
u= = ℘ √ + C; ω1 , ω2 + , (7.20)
r 2 6
where C ∈ C is a constant and ω1 and ω2 are determined by m, µ, E and L (see e.g.
Ref. [220] for more details regarding this method applied to null geodesics).
5
The character ℘ is a kind of calligraphic lowercase p, which is standard to denote this function.
214 Geodesics in Schwarzschild spacetime: generic and timelike cases
• The second approach consists in noticing that the method of separation of variables can
easily be applied to Eq. (7.17), leading to
Z u
dū
φ=± q + φ0 , (7.21)
mµ 2 mE 2 mµ 2
u0 3 2
2ū − ū + 2 L ū + L − L
where u0 and φ0 are two constants, and the ± sign can be + on some parts of the geodesic
L and − on some other parts of L . The integral in right-hand side is expressible in
terms of the so-called incomplete elliptic integrals of the first kind. We shall detail such
a technique for null geodesics in Sec. 8.3. Note that this approach leads to φ = φ(r),
whereas the method involving the Weierstrass function leads to the “polar equation”
form: r = r(φ). It is possible though to get the polar form by invoking the inverses of
elliptic integrals, namely Jacobi elliptic functions.
Once the solution r = r(φ) of Eq. (7.17) has been obtained, it can be injected into the
equation for t = t(φ) that can be deduced from the equations of motions (7.11)-(7.12):
dt E r(φ)3
= . (7.22)
dφ L r(φ) − 2m
This is an ordinary differential equation for t = t(φ), the solution to which amounts to finding
a primitive with respect to φ of the right-hand side. Unfortunately, this is not an easy task in
general, the function r(φ) being quite involved, except for circular orbits (r(φ) = const).
In what follows, we discuss separately the resolution of the system (7.11)-(7.13) or of
Eq. (7.17) for timelike geodesics (Sec. 7.3) and for null geodesics (Chap. 8).
7.3 Timelike geodesics
7.3.1 Effective potential
When the geodesic L is timelike, it is natural to use the proper time τ as an affine parameter
along it, instead of the parameter λ associated with the 4-momentum p. Since the tangent
vector associated with τ is the 4-velocity u (cf. Sec. 1.3.3) and p and u are related by Eq. (1.18):
p = µ u, we get dx/dλ = µ dx/dτ , from which we infer the relation between τ and λ:
τ = µλ, (7.23)
up to some additive constant. This is of course a special case of the generic relation (B.3)
between two affine parameters of the same geodesic. Equation (7.13) becomes then
2
ε2 − 1
1 dr
+ Vℓ (r) = , (7.24)
2 dτ 2
where
ℓ2
m 2m
Vℓ (r) := − + 2 1− (7.25)
r 2r r
7.3 Timelike geodesics 215
and ε and ℓ are respectively the specific conserved energy and specific conserved angular
momentum of particle P:
E L
ε := = −ξ · u and ℓ := =η·u, (7.26)
µ µ
where u is the 4-velocity of P and the second equalities result from definitions (7.1) and the
relation p = µ u [Eq. (1.18)]. Note that ε is dimensionless (in units c = 1) and that it shares
the same positiveness property (7.5) as E:
Property 7.7: positivity of the specific conserved energy
The specific conserved energy ε is positive as soon as the timelike geodesic L has some
part in MI , i.e. some part with r > 2m:
L ∩ MI ̸= ∅ =⇒ ε > 0. (7.27)
On the contrary, ℓ can be either positive, zero or negative, depending on the variation of φ
along L , as was already noticed above for L.
We note that Eq. (7.24) has the shape of the first integral of the 1-dimensional motion
of a non-relativist particle in the potential Vℓ (called hereafter the effective potential), the
term 1/2 (dr/dτ )2 being interpreted as the kinetic energy per unit mass, Vℓ (r) as the potential
energy per unit mass and the constant right-hand side (ε2 − 1)/2 as the total mechanical energy
per unit mass.
Remark 1: The effective potential (7.25) differs from its non-relativistic (Newtonian) counterpart only
by the factor 1 − 2m/r instead of 1. This difference plays an important role for small values of r, leading
to some orbital instability, as we shall see in Sec. 7.3.3.
In MI , where the Killing vector ξ is timelike, we may introduce the static observer O,
whose 4-velocity uO is collinear to ξ:
−1/2
2m
uO = 1 − ξ, (7.28)
r
the proportionality coefficient ensuring that uO ·uO = −1 given that ξ·ξ = gtt = −(1−2m/r).
We have then, from (7.26),
1/2 1/2
2m 2m
ε=− 1− uO · u = Γ 1 − , (7.29)
r r
where Γ = −uO · u is the Lorentz factor of P with respect to O (cf. Sec. 1.4; in particular
Eq. (1.33)). We may express Γ in terms of the norm v of the velocity of P with respect to O,
according to Eq. (1.35): Γ = (1 − v 2 )−1/2 and get
1/2
2 −1/2 2m
(7.30)
ε= 1−v 1− .
r
216 Geodesics in Schwarzschild spacetime: generic and timelike cases
V` ( r )
2 4 6 8 10
r/m
-0.5
-1
`/m = 0. 0000
`/m = 2. 0000
`/m = 3. 0000
`/m = 3. 4641
`/m = 3. 8000
`/m = 4. 2000
`/m = 4. 6000
`/m = 5. 0000
-1.5
Figure 7.1: Effective potential Vℓ (r) governing the r-part of the motion along a timelike geodesic in
Schwarzschild spacetime via Eq. (7.24). The vertical dashed line marks r = 2m, i.e. the location of the event
horizon. The numerical value ℓ/m = 3.4641 is that of the critical specific angular momentum (7.32).
In the region r ≫ m, we may perform a first order expansion, assuming that P moves at
nonrelativistic velocity with respect to O (v ≪ 1), thereby obtaining:
1 m
ε − 1 ≃ v2 − (r ≫ m and v ≪ 1) . (7.31)
2 r
We recognize in the right-hand side the Newtonian mechanical energy per unit mass of particle
P with respect to observer O, who can then be considered as an inertial observer, v 2 /2 being
the kinetic energy per unit mass and −m/r the gravitational potential energy per unit mass.
The profile of Vℓ (r) for selected values of ℓ is plotted in Figs. 7.1 and 7.2 . Its extrema are
given by dVℓ /dr = 0, which is equivalent to
mr2 − ℓ2 r + 3ℓ2 m = 0.
This quadratic equation admits real roots iff |ℓ| ≥ ℓcrit , with
√
ℓcrit = 2 3 m ≃ 3.464102 m . (7.32)
For |ℓ| ≥ ℓcrit , the two roots are
ℓ ℓ
q q
rmax = 2
ℓ − ℓ2 − ℓcrit and rmin = 2
ℓ + ℓ2 − ℓcrit , (7.33)
2m 2m
7.3 Timelike geodesics 217
V` ( r )
0.15 `/m = 0. 0000
`/m = 2. 0000
`/m = 3. 0000
`/m = 3. 4641
0.1 `/m = 3. 8000
`/m = 4. 2000
`/m = 4. 6000
`/m = 5. 0000
0.05
5 10 15 20 25
r/m
-0.05
-0.1
-0.15
Figure 7.2: Same as Fig. 7.1, but with a zoom in along the y-axis and a zoom out along the x-axis. The dots
mark the mimima of Vℓ , locating stable circular orbits.
corresponding respectively to a maximum of Vℓ and a minimum of Vℓ , hence the indices “max”
and “min”. Note that rmax ≤ rmin . In the marginal case |ℓ| = ℓcrit , the two roots coincide and
correspond to an inflection point of Vℓ (the circled dot in Fig. 7.2).
For |ℓ| < ℓcrit , there is no extremum and Vℓ is a strictly increasing function of r.
To get a full solution in terms of the Schwarzschild-Droste coordinates, once Eq. (7.24) is
solved for r(τ ), one has still to solve Eqs. (7.11) and (7.12), which can be rewritten in terms of
the proper time τ as
−1
dt 2m
=ε 1− , (7.34)
dτ r(τ )
dφ ℓ
= . (7.35)
dτ r(τ )2
7.3.2 Radial free fall
Generic case
The radial geodesics correspond to a vanishing conserved angular momentum: ℓ = 0. Indeed,
setting ℓ = 0 in Eq. (7.12) yields φ = const, which defines a purely radial trajectory in the plane
θ = π/2. The effective potential (7.25) reduces then to Vℓ (r) = −m/r, so that the equation of
radial motion (7.24) becomes
2
1 dr m ε2 − 1
− = . (7.36)
2 dτ r 2
218 Geodesics in Schwarzschild spacetime: generic and timelike cases
This equation is identical to that governing radial free fall in the gravitational field generated
by a mass m in Newtonian gravity. The solution is well known and depends on the sign of the
“mechanical energy” in the right-hand side, i.e. of the position of ε with respect to 1:
• if ε > 1, the solution is given in parameterized form (parameter η) by
m
τ= 2
(sinh η − η) + τ0
(ε − 1)3/2
(7.37)
r = m (cosh η − 1) ,
ε2 − 1
• if ε = 1, the solution is 1/3
9m
r(τ ) = (τ − τ0 )2 , (7.38)
2
• if ε < 1, the solution is given in parameterized form (parameter η) by
m
τ=
(η + sin η) + τ0
(1 − ε2 )3/2
(7.39)
r = m (1 + cos η) ,
1 − ε2
In the above formulas, τ0 is a constant; for ε ≥ µ, τ0 is the value of τ for which r → 0, while
for ε < 1, it is the value of τ at which r takes its maximal value.
Radial free fall from rest
Let us focus on the radial free fall from rest, starting at some position r = r0 at τ = 0. Starting
from rest means dr/dτ = 0 at τ = 0. The equation of radial motion (7.36) leads then to
−m/r0 = (ε2 − 1)/2, or equivalently
2m
ε2 = 1 − . (7.40)
r0
The right-hand side of this equation must be non-negative. This implies r0 ≥ 2m. We recover
the fact that one cannot be momentarily at rest (in terms of r) if r0 < 2m, for r has to decrease
along any causal geodesic in the black hole region MII (cf. Sec. 7.2.2).
Equation (7.40) implies ε < 1, i.e. E < µ. The solution is thus given by Eq. (7.39); expressing
1 − ε2 in it via (7.40), we get
r
r03
τ = (η + sin η)
8m
0 ≤ η ≤ π, (7.41)
r
r = 0 (1 + cos η)
2
where the range of η is such that r = r0 for τ = 0 (η = 0) and r decays to 0 when η → π. The
function r(τ ) resulting from (7.41) is depicted in Fig. 7.3.
7.3 Timelike geodesics 219
r/m
6
r0 = 2. 10 m
5 r0 = 3. 00 m
r0 = 4. 00 m
4 r0 = 5. 00 m
r0 = 6. 00 m
3
2
1
5 10 15
τ/m
Figure 7.3: Coordinate r as a function of the proper time τ for the radial free fall from rest, for various initial
values r0 of r.
The solution for t = t(τ ) is obtained by combining dt/dτ as expressed by (7.34) and dτ /dη
deduced from (7.41): r r
dτ r03 r0
= (1 + cos η) = r.
dη 8m 2m
We get
r −1 r
r2
dt dt dτ r0 2m r0
= =ε r 1− = −1 ,
dη dτ dη 2m r 2m r − 2m
where we have used (7.40) and ε > 0 [Eq. (7.27)] to write ε = 1 − 2m/r0 . Substituting r
p
from Eq. (7.41), we get
r
dt r0 r0 (1 + cos η)2
= −1 .
dη 2 2m 1 + cos η − 4m/r0
This equation can be integrated to (cf. the SageMath computation in Sec. D.4.5)
(r p r0 η
)
r0 hr0 i − 1 + tan
t = 2m −1 η+ (η + sin η) + ln p 2m
r0
2
η , (7.42)
2m 4m 2m
− 1 − tan 2
where we have assumed t = 0 at τ = 0 (η = 0).
The solution for the radial free fall starting from rest at r = r0 is thus given in parametric
form by Eqs. (7.41) and (7.42) and is represented in the left panel of Fig. 7.4. It has been obtained
in the Schwarzschild-Droste coordinates (t, r, θ, φ), which are singular at the event horizon H .
So, one might wonder if such a solution can describe the full infall, with the crossing of H . In
particular, we notice that the differential equation for t(τ ), Eq. (7.34), is singular at r(τ ) = 2m,
i.e. on H . The solution t(η), as given by Eq. (7.42), is singular at η = ηh , where
r
r0
ηh := 2 arctan −1 (7.43)
2m
220 Geodesics in Schwarzschild spacetime: generic and timelike cases
t/m t̃/m
25 r0 = 2. 10 m 25 r0 = 2. 10 m
r0 = 3. 00 m r0 = 3. 00 m
r0 = 4. 00 m r0 = 4. 00 m
r0 = 5. 00 m r0 = 5. 00 m
r0 = 6. 00 m r0 = 6. 00 m
20 20
15 15
10 10
5 5
1 2 3 4 5 6 7 8
r/m 1 2 3 4 5 6 7 8
r/m
Figure 7.4: Radial free fall from rest, viewed in Schwarzschild-Droste coordinates (t, r) (left) and in the ingoing
Eddington-Finkelstein coordinates (t̃, r) (right), for various values r0 of the coordinate r at τ = 0. The grey area
is the black hole region MII .
is precisely the value of η yielding r = 2m in Eq. (7.41) [to see it, rewrite the second part of
Eq. (7.41) as r = r0 cos2 (η/2) = r0 /(1 + tan2 (η/2))]. This singularity of t(η) appears also
clearly on Fig. 7.4 (left panel). On the other hand, the equation for r, Eq. (7.36), does not exhibit
any pathology at r = 2m, nor its solution (7.42). Actually, had we started from the ingoing
Eddington-Finkelstein (IEF) coordinates (t̃, r, θ, φ), instead of the Schwarzschild-Droste ones,
we would have found6 exactly the same solution for r(τ ) (which is not surprising since r,
considered as a scalar field on M , is perfectly regular at H ). The solution for t̃(τ ) can be
deduced from that for t(τ ) by the coordinate transformation law (6.32). Noticing that and
r/(2m) = cos2 (η/2)/ cos2 (ηh /2), we get
cos2 (η/2)
r 2 1 1
−1 = − 1 = cos (η/2) −
2m cos2 (ηh /2) cos2 (ηh /2) cos2 (η/2)
= cos2 (η/2) tan2 (ηh /2) − tan2 (η/2) . (7.44)
Using this identity, as well as (7.43) to express r0 /(2m) − 1 in the logarithm term of Eq. (7.42),
p
6
Exercise: do it!
7.3 Timelike geodesics 221
τ/m
τf
15 τh
10
0
2 3 4 5 6
r0 /m
Figure 7.5: Elapsed proper time to reach the event horizon (τh , dashed curve) and the central singularity (τf ,
solid curve), as a function of the initial value of r for a radial free fall from rest.
the transformation law (6.32) yields
tan η2h + tan η2
r
r0 h r0 i
2 η
2 ηh 2 η
t̃ = 2m −1 η+ (η + sin η) + ln cos tan − tan
2m 4m tan η2h − tan η2 2 2 2
r
r0 h r0 i η ηh η 2
= 2m −1 η+ (η + sin η) + ln cos2 tan + tan
2m 4m 2 2 2
r
r0 h r0 i η ηh η
= 2m −1 η+ (η + sin η) + 2 ln cos tan + sin .
2m 4m 2 2 2
From this expression, we have t̃ = 4m ln tan(ηh /2) for η = 0. Now, we can change the origin
of the IEF coordinate t̃ to ensure t̃ = 0 for η = 0, i.e. τ = 0. We get then
r −1/2
r0 h r0 i η r0 η
t̃ = 2m −1 η+ (η + sin η) + 2 ln cos + −1 sin . (7.45)
2m 4m 2 2m 2
This expression is perfectly regular for all values of η in [0, π], reflecting the fact that the
ingoing Eddington-Finkelstein coordinates cover all M in a regular way. The radial free fall
solution in terms of (t̃, r) is represented in the right panel of Fig. 7.4. We note the smooth
crossing of the event horizon H .
In view of Eq. (7.41), we may say that the radial infall starts at η = 0, for which τ = 0
and r = r0 , and terminates at η = π, for which r = 0, which means that the particle hits the
curvature singularity (cf. Sec. 6.3.4). The final value of the particle’s proper time is obtained by
222 Geodesics in Schwarzschild spacetime: generic and timelike cases
∆τin /m
2.5
1.5
5 10 15 20
r0 /m
Figure 7.6: Proper time spent inside the black hole region as a function of the initial value of r for a radial free
fall from rest. Note that r0 = 2m does not correspond to any asymptote but to the finite value ∆τin = π m with
a vertical tangent. On the other side, there is an horizontal asymptote ∆τin → 4m/3 for r0 → +∞.
setting η = π in Eq. (7.41):
r
π r03
τf = . (7.46)
2 2m
Similarly, the final value of t̃ is obtained by setting η = π in (7.45):
r
r0 r
0
r
0
t̃f = 2m π −1 + 1 − ln −1 . (7.47)
2m 4m 2m
As noticed above, the event horizon H is crossed at η = ηh ; via (7.41) and (7.43), this corre-
sponds to the following value of the proper time:
r " r s #
r03
r0 2m 2m
τh = arctan −1+ 1− , (7.48)
2m 2m r0 r0
while (7.45) leads to the following value of the IEF coordinate t̃:
r r
r0 r0 r0 r0 r0
t̃h = 2m 2 1 + − 1 arctan −1+ − 1 − ln . (7.49)
4m 2m 2m 2m 2m
The variation of τh and τf with r0 are depicted in Fig. 7.5 and numerical values for r0 = 6m
and standard astrophysical black holes are provided in Table 7.1.
7.3 Timelike geodesics 223
m rS = 2m τh τf ∆τin
15 M⊙ (Cyg X-1) 44.3 km 1.10 ms 1.21 ms 0.11 ms
4.3 106 M⊙ (Sgr A*) 12.7 106 km = 0.085 au 5 min 14 s 5 min 46 s 32 s
6 109 M⊙ (M87*) 118 au 5.07 days 5.58 days 12 h 17 min
Table 7.1: Proper time to reach the event horizon (τh ) and the central curvature singularity (τf ), as well as
elapsed proper time inside the black hole region (∆τin ), when freely falling from rest at r0 = 6m. The numerical
values are given for various black hole masses m, corresponding to astrophysical objects: the stellar black hole
Cygnus X-1 [393, 225], the supermassive black hole at the center of our galaxy (Sagittarius A*) [215, 301] and the
supermassive black hole M87* in the nucleus of the galaxy Messier 87 [214, 6].
The proper time spent inside the black hole is
r " r s #
r03
π r0 2m 2m
∆τin = τf − τh = − arctan −1− 1− . (7.50)
2m 2 2m r0 r0
It varies between πm (r0 → 2m) and 4m/3 (r0 → +∞) (cf. Fig. 7.6 and Sec. D.4.5 for the
computation of limr0 →+∞ ∆τin ). Numerical values for astrophysical black holes are provided
in Table 7.1.
7.3.3 Circular orbits
Circular orbits are defined as timelike geodesics with r = const. We have then dr/dτ = 0 and
d2 r/dτ 2 = 0, so that Eq. (7.24) implies
ε2 − 1
Vℓ (r) = (7.51a)
2
dVℓ
= 0. (7.51b)
dr
Given the expression (7.25) of Vℓ , Eq. (7.51b) is equivalent to
mr2 − ℓ2 r + 3ℓ2 m = 0. (7.52)
As already noticed
√ in Sec. 7.3.1, this quadratic equation in r admits two real roots iff |ℓ| ≥ ℓcrit ,
with ℓcrit = 2 3 m [Eq. (7.32)], which are
ℓ
q
±
rcirc (ℓ) = 2 2
ℓ ± ℓ − ℓcrit . (7.53)
2m
+
rcirc (ℓ) corresponds to a minimum of the effective potential Vℓ and thus to a stable orbit (see the
dots in Fig. 7.2), while rcirc
−
(ℓ) corresponds to a maximum of Vℓ and thus to an unstable orbit.
When ℓ varies from ℓcrit to +∞, rcirc +
(ℓ) increases from 6m to +∞, while rcirc−
(ℓ) decreases
from 6m to 3m (cf. Fig. 7.7). We conclude that
224 Geodesics in Schwarzschild spacetime: generic and timelike cases
`/m
6
5.5
4.5
3.5
2 4 6 8 10 12
r/m
Figure 7.7: Specific conserved angular momentum ℓ = L/µ on circular orbits as a function of the orbit
−
circumferential radius r. The dashed part of the curve corresponds to unstable orbits (r = rcirc (ℓ), as given
by
√ Eq. (7.53)), while the solid part corresponds to stable orbits (r = r +
circ (ℓ)). The minimal value of ℓ is ℓcrit =
2 3 m ≃ 3.46 m.
Property 7.8: circular orbits
Circular orbits in Schwarzschild spacetime exist for all values of r > 3m. Those with
r < 6m are unstable and those with r > 6m are stable. The marginal case r = 6m is called
the innermost stable circular orbit, often abridged as ISCO.
Remark 2: In the Newtonian spherical gravitational field generated by a point mass m, there is
no unstable orbit, and thus no ISCO. The existence of unstable orbits in the relativistic case can be
understood by the extra term in the effective potential Vℓ (r) (cf. Remark 1 on p. 215), which adds the
attractive part −ℓ2 m/r3 to the two terms constituting the Newtonian potential: −m/r (attractive)
and ℓ2 /(2r2 ) (repulsive). The latter is responsible for the infinite “centrifugal barrier” at small r in the
Newtonian problem, leading always to a minimum of Vℓ (r) and thus to a stable circular orbit. In the
relativistic case, for r small enough, the attractive term, which is O(r−3 ), dominates over the centrifugal
one, which is only O(r−2 ). Equivalently, we may say that the “centrifugal barrier” is weakened by
the factor 1 − 2m/r (cf. the expression (7.25) of Vℓ (r)) and ceases to exist for small values of |ℓ| (i.e.
|ℓ| < ℓcrit ).
From Eq. (7.52), we can easily express ℓ as a function of r on a circular orbit:
r
m
|ℓ| = r . (7.54)
r − 3m
7.3 Timelike geodesics 225
This function is represented in Fig. 7.7 (for ℓ > 0).
If we substitute (7.54) for ℓ in the expression (7.25) of Vℓ and use Eq. (7.51a), we obtain the
value of the specific conserved energy along a circular orbit, in terms of r:
r − 2m
ε= p . (7.55)
r(r − 3m)
This function is represented in Fig. 7.8. The minimal value of ε is achieved for r = 6m, i.e. at
the ISCO: √
2 2
min ε = εISCO = ≃ 0.9428 . (7.56)
3
From Fig. 7.8, we notice that
r > 4m ⇐⇒ ε < 1 ⇐⇒ E < µ. (7.57)
This corresponds to bound orbits, i.e. to geodesics that, if slightly perturbed, cannot reach
the asymptotically flat region r ≫ 2m, since E ≥ µ there. Indeed, when r → +∞, the Killing
vector ξ can be interpreted as the 4-velocity of some asymptotically inertial observer (at rest
with respect to the black hole) and E is the particle energy measured by that observer; the
famous Einstein relation (1.32) is then E = Γµ, where Γ is the Lorentz factor of the particle
with respect to the observer. Since Γ ≥ 1 [Eq. (1.35)], we have obviously7 E ≥ µ. For this
reason, the circular orbit at r = 4m is called the marginally bound circular orbit. Note that
the marginally bound circular orbit is unstable, since it has r < 6m.
The track of circular orbits in the (ℓ, ε) plane is depicted in Fig. 7.9. The ISCO, which is a
minimum for both ε and ℓ, appears as a cusp point.
The angular velocity of a circular orbit L is defined by
dφ uφ
Ω := = , (7.58)
dt L ut
where uφ = dφ/dτ and ut = dt/dτ are the only nonzero components w.r.t. Schwarzschild-
Droste coordinates of the 4-velocity u along the worldline L . It follows from (7.58) that Ω
enters into the linear combination of the two Killing vectors ξ and η expressing the 4-velocity
on a circular orbit according to
u = ut (ξ + Ωη) . (7.59)
We have the following nice physical interpretation:
Property 7.9: angular velocity measured at infinity
The quantity Ω defined by Eq. (7.58) is nothing but the angular velocity of the orbiting
particle P monitored by a infinitely distant static observer O.
7
Similarly, the radial-motion solutions (7.37)-(7.38), which allow for r → +∞, have E ≥ µ, while the solution
(7.39), which is relevant for a free fall from rest, has E < µ.
226 Geodesics in Schwarzschild spacetime: generic and timelike cases
ε
1.2
1.15
1.1
1.05
0.95
2 4 6 8 10 12
r/m
Figure 7.8: Specific conserved energy ε = E/µ on circular orbits as a function of the orbit circumferential
radius r. The dashed part of the curve corresponds to unstable orbits, while the solid part corresponds to stable
ones. The horizontal red line ε = 1 marks the limit of bound orbits.
1.04
1.02
0.98
0.96
0.94
0.92
3 4 5 6 7 8
`/m
Figure 7.9: Circular orbits in the (ℓ, ε) plane. The solid (resp. dashed) curve corresponds to stable (resp.
unstable) orbits. The ISCO is located at the cusp point.
7.3 Timelike geodesics 227
ΩISCO
m rISCO = 6m TISCO TP,ISCO
2π
15 M⊙ (Cyg X-1) 133 km 147 Hz 6.80 ms 4.81 ms
4.3 106 M⊙ (Sgr A*) 38.1 106 km 5.11 10−4 Hz 32 min 37 s 23 min 4 s
(0.255 au)
6 109 M⊙ (M87*) 355 au 3.66 10−7 Hz 31 d 15 h 22 d 9 h
Table 7.2: Values of various quantities at the ISCO for masses m of some astrophysical black holes (see Table 7.1
for details): areal radius r, orbital frequency ΩISCO /(2π), orbital period seen from infinity TISCO and orbital
period measured by the orbiting observer/particle TP,ISCO (proper time).
Proof. Suppose that O is located at fixed coordinates (r, θ, φ) = (rO , π/2, 0) with rO ≫ m and
that P emits a photon at the event (t1 , r, π/2, 0) along a radial null geodesic. This photon is
received by O at t = t′1 . After one orbit, at the event (t2 , r, π/2, 2π), P emits a second photon
in the radial direction, which is received at t = t′2 by O. According to the definition (7.58) of Ω,
we have
2π = Ω(t2 − t1 ).
On the other hand, since rO ≫ m, the proper time of O is t, so that the angular velocity
measured by O is
2π
ΩO = ′ .
t2 − t′1
Now, since t is the coordinate associated to the spacetime invariance by time translation
(stationarity), we have necessarily t′2 − t2 = t′1 − t1 (the increment in coordinate t for the
second signal is the same as for the first one), so that t′2 − t′1 = t2 − t1 . Accordingly, the above
two equations combine to ΩO = Ω.
By combining Eqs. (7.34) and (7.35), we get
1 2m ℓ
Ω= 2 1− .
r r ε
Substituting expression (7.54) for ℓ and expression (7.55) for ε, we obtain
r
m
Ω=± , (7.60)
r3
with the + (resp. −) sign for ℓ > 0 (resp. ℓ < 0).
Remark 3: This formula is identical to that of Newtonian gravity (Kepler’s third law for circular orbits)
for all values of r. This is a mere coincidence, valid only for Schwarzschild-Droste coordinates. Only
for r ≫ m, i.e. in the weak-field limit, this agreement is physically meaningful; it can be then used
to interpret the parameter m as the gravitational mass of Schwarzschild spacetime, as mentioned in
Sec. 6.2.4.
228 Geodesics in Schwarzschild spacetime: generic and timelike cases
Remark 4: Ω is not the orbital angular frequency experienced by the particle/observer P on the
circular orbit L , because the proper time of P is τ and not t. The actual orbital frequency measured by
P is
dt
ΩP = Ω = ut Ω,
dτ
with ut = dt/dτ obtained from (7.34) and (7.55): ut = r/(r − 3m). Hence
p
r r
r 1 m
ΩP = Ω=± . (7.61)
r − 3m r r − 3m
√
Note that |ΩP | > |Ω|; in particular, at the ISCO (r = 6m), ΩP = 2Ω. The orbital period measured by
P is TP = 2π/|ΩP |. Some ISCO values of TP for astrophysical black holes are provided in Table 7.2.
At the ISCO, r = 6m and formula (7.60) yields (for orbits with positive ℓ)
1
ΩISCO = √ . (7.62)
6 6m
Numerical values of ΩISCO (actually the frequency ΩISCO /(2π), which is more relevant from
an observational point of view) are provided in Table 7.2.
7.3.4 Other orbits
Let us relax the assumption r = const and consider generic orbits L obeying
|ℓ| > ℓcrit and 0 < ε < 1. (7.63)
The first condition ensures that the effective potential Vℓ (r) takes the shape of a well in the
region r > 2m (cf. Fig. 7.10) and the second one that the particle P is trapped in this well.
Indeed, 0 < ε < 1 makes the right-hand of Eq. (7.24) negative, so that the region r → +∞,
where Vℓ (r) → 0, cannot be reached. We have also argued in Sec. 7.3.3 that ε < 1 is forbidden
in the region r → +∞ on physical grounds [cf. the discussion below Eq. (7.57)].
In the potential well, the r-coordinate along L varies between two extrema: a minimum
rper , for periastron (or pericenter or periapsis), and a maximum rapo , for apoastron (or
apocenter or apoapsis) (cf. Figs. 7.10 and 7.11). Being extrema of r(τ ), the values of rper and
rapo are obtained by setting dr/dτ = 0 in Eq. (7.24), which leads to
ℓ2
2m
1− 1 + 2 = ε2 . (7.64)
r r
This is a cubic equation in r−1 , which has three real positive roots, corresponding to the three
intersections of the curve Vℓ (r) with the horizontal line at (ε2 − 1)/2 in Fig. 7.10. However,
the smaller root has to be disregarded as a periastron since it would lead to a motion with
Vℓ (r) > (ε2 − 1)/2, which is forbidden by Eq. (7.24).
We get, from Eqs. (7.24)-(7.25),
s
dr 2m ℓ2
2
=± ε − 1− 1+ 2 . (7.65)
dτ r r
7.3 Timelike geodesics 229
0.03
0.02
0.01
0
V` ( r )
-0.01
-0.02
1 (ε 2 − 1)
2
-0.03
-0.04
0 5
rper10 15
r/m
20 25
rapo
30
Figure 7.10: Effective potential Vℓ (r) for ℓ = 4.2m (one of the values displayed in Figs. 7.1 and 7.2). The
horizontal red line marks Vℓ (r) = (ε2 − 1)/2 with ε = 0.973, leading to rper = 9.058 m and rapo = 25.634 m.
The corresponding orbit is shown in Fig. 7.11.
y/m
20
10
-20 -10 10 20
x/m
-10
-20
Figure 7.11: Timelike geodesic with ε = 0.973 and ℓ = 4.2m (same values as in Fig. 7.10), plotted in terms of
the coordinates (x, y) := (r cos φ, r sin φ). The dotted circles correspond to r = rper (periastron) and r = rapo
(apoastron). The grey disk indicates the black hole region r < 2m. [Figure produced with the notebook D.4.6]
230 Geodesics in Schwarzschild spacetime: generic and timelike cases
y/m y/m
15
10
50
-15 -10 -5 5 10 15
x/m -50 50
x/m
-5
-50
-10
-15
Figure 7.12: Timelike geodesics with the same value of ℓ as in Fig. 7.11 (ℓ = 4.2m), but for different values of ε:
ε = 0.967 (left) and ε = 0.990 (right). Note that the left and right figures have different scales. [Figure produced
with the notebook D.4.6]
The equation governing the trajectory of L in the orbital plane is Eq. (7.17), which can be
recast as r
du m 2 mε 2 m 2
= ± 2u3 − u2 + 2 u+ − . (7.66)
dφ ℓ ℓ ℓ
Let us recall that u := m/r and that methods for solving this differential equation have been
briefly discussed in Sec. 7.2.3.
Remark 5: Far from the black hole, i.e. in the region r ≫ m, one can easily recover the Newtonian
orbits from Eq. (7.66). Indeed, according to Eq. (7.31), ε = 1 + ε0 , where ε0 = v 2 /2 − m/r is the
Newtonian mechanical energy per unit mass. It obeys |ε0 | ≪ 1, so that ε2 ≃ 1 + 2ε0 . Moreover, for
r ≫ m, u ≪ 1 and we can neglect the u3 term in front of the u2 one in Eq. (7.66). Hence Eq. (7.66)
reduces to r
du m 2 m 2
≃± 2 ε0 − u2 + 2 u. (7.67)
dφ ℓ ℓ
Let us introduce the constants
r
ℓ2 ε0 ℓ2
p := and e := 1 + 2 2 . (7.68)
m m
Then Eq. (7.67) can be rewritten as
p
dφ me
= ±r ,
du p
u−1 2
1− m
e
which is readily integrated into
p
−1
mu
φ = ± arccos + φ0 ,
e
7.3 Timelike geodesics 231
p
where φ0 is a constant. We have then mu = 1 + e cos(φ − φ0 ), or equivalently,
p
r= (7.69)
1 + e cos(φ − φ0 )
Assuming a bound orbit, we have ε < 1, which implies ε0 < 0 and, via Eq. (7.68), e < 1. We recognize
then in (7.69) the equation of an ellipse of eccentricity e and semi-latus rectum p. Hence Keplerian orbits
are recovered for r ≫ m, as they should.
Generic bound orbits differ from the Keplerian ellipses by the fact that the variation of φ
between two successive periastron passages, i.e. two events along the worldline of P for
which r = rper , is strictly larger than 2π. This phenomenon is called periastron advance and
causes the orbits to be not closed, as illustrated in Figs. 7.11 and 7.12.
Historical note : The equations of geodesic motion (7.11)-(7.13), as well as Eq. (7.17) for the trajectories
in the orbital plane, have been first given by Karl Schwarzschild himself in January 1916 in the very
same article [449] in which he presented his famous solution8 . Schwarzschild discussed only the weak
field limit of these equations, to recover the Mercury’s perihelion advance computed by Einstein.
It is quite remarkable that the general solution to the geodesic motion in Schwarzschild spacetime9
has been given as early as May 1916 by Johannes Droste, in the same article [175] in which he derived
the Schwarzschild solution, independently of Karl Schwarzschild (cf. historical note on p. 187). Droste
derived the equations of geodesic motion in Schwarzschild-Droste coordinates (t, r, θ, φ), using t as the
parameter along the geodesics, as well as the equation governing the trajectories in the orbital plane10 .
He gave the solutions for the trajectories in terms of the Weierstrass elliptic function ℘ (cf. Sec. 7.2.3). He
classified the solutions in terms of the roots of the cubic polynomial in v that appears in the right-hand
side of Eq. (7.19). Droste noticed that it takes an infinite amount of coordinate time t for a particle to
reach r = 2m (cf. left panel of Fig. 7.4) and he concluded incorrectly that “a moving particle outside the
sphere r = 2m can never pass that sphere”. He missed that this is only a coordinate effect, reflecting
the pathology of Schwarzschild-Droste coordinates at the horizon. One shall keep in mind that in 1916,
general relativity was just in its infancy and disentangling coordinate artefacts from physical effects
was not so obvious, especially regarding time. One can be amused by the fact that the first analysis of
the static black hole of general relativity made the black hole appear, not as an object from which no
particle may escape, but as an object into which no particle may penetrate...
Finally, it is worth mentioning two early detailed studies of geodesic motion of massive particles in
Schwarzschild spacetime: one by by Carlo De Jans in 1923 [160] and other one by Yusuke Hagihara in
1931 [249]. For a detailed account about the history of geodesic motion in Schwarzschild spacetime see
Ref. [181].
8
Equation (7.17) for the trajectories is Eq. (18) in the article [449], the link between Schwarzschild’s notations
and ours being R = r, x = u/m, c = L, 1 = E, h = µ2 .
9
More precisely: in the MI region of Schwarzschild spacetime.
10
The link between Droste’s notations and ours is α = 2m, x = 2u, z = 2u − 1/3, A = µ2 /E 2 and B = L/E.
232 Geodesics in Schwarzschild spacetime: generic and timelike cases
Chapter 8
Null geodesics and images in
Schwarzschild spacetime
Contents
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
8.2 Main properties of null geodesics . . . . . . . . . . . . . . . . . . . . . 234
8.3 Trajectories of null geodesics in the equatorial plane . . . . . . . . . 241
8.4 Asymptotic direction from some emission point . . . . . . . . . . . . 260
8.5 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
8.1 Introduction
Having investigated the properties of generic causal geodesics in Schwarzschild spacetime in
Chap. 7, we focus here on null geodesics. The main interest of null geodesics is of course that
they are the carriers of information from the surroundings of the black hole to some observer,
in particular in the form of images. We start by studying the general property of null geodesics
in Sec. 8.2, distinguishing the radial geodesics, from the non-radial ones and focussing on
the r-motion. Then in Sec. 8.3, we study of the φ-motion and show that the trajectory of a
given null geodesic in the equatorial plane is fully integrable, via elliptic integrals. In Sec. 8.4
we compute the asymptotic direction taken by a null geodesic emitted at a given point; this
prepares the discussion of images in Sec. 8.5. The study of black hole images has received a
tremendous boost after the release of the first observed image by the Event Horizon Telescope
collaboration in 2019, that of the black hole M87* [6]. However, we differ the discussion of this
image to the chapter regarding null geodesics around a rotating black hole (Chap. 12), since
the black hole spin plays some role in the images and M87* is expected to be a fast rotator.
234 Null geodesics and images in Schwarzschild spacetime
8.2 Main properties of null geodesics
We use the same notations as in Sec. 7.2 of the preceding chapter: (M , g) is Schwarzschild
spacetime, (t, r, θ, φ) are Schwarzschild-Droste coordinates and P is a particle, the worldline
of which is a geodesic L . The particle P is characterized by its 4-momentum p and E and L
stand for the conserved energy and conserved angular momentum along L [cf. Eq. (7.1)].
In all this chapter, we consider that L is a null geodesic, or equivalently that P is a massless
particle, typically a photon. As in Chap. 7, we assume that the coordinates (θ, φ) are chosen so
that L lies in the hyperplane θ = π/2. Let us recall that a geodesic of Schwarzschild spacetime
lies necessarily in some hypersurface, which can be chosen to be the hyperplane θ = π/2
without any loss of generality (cf. Sec. 7.2.1).
8.2.1 Equations to be solved
In terms of Schwarzschild-Droste coordinates, the geodesic motion of P is governed by
Eqs. (7.11), (7.12) and (7.13) with µ = 0 (massless particle):
−1
dt 2m
=E 1− , (8.1)
dλ r
dφ L
= 2 , (8.2)
dλ r
2
L2
dr 2m
+ 2 1− = E2 . (8.3)
dλ r r
Let us recall that the variable λ with respect to which these differential equations hold is the
affine parameter associated with the 4-momentum p of P: p = dx/dλ [Eq. (7.9)] and that λ
increases towards the future, p being always future-directed (cf. Sec. 7.2.2).
8.2.2 Radial null geodesics
Let us first discuss the case of radial geodesics, which are characterized by L = 0. Equation
(8.2) yields immediately φ = const, while Eq. (8.3) simplifies drastically:
dr
= ±E, (8.4)
dλ
the solution to which is immediate:
r = ±Eλ + r0 , (8.5)
where r0 is some constant. Moreover, writing dt/dλ = dt/dr × dr/dλ and combining Eqs. (8.1)
and (8.4), we get
−1
dt 2m
=± 1− . (8.6)
dr r
8.2 Main properties of null geodesics 235
We recognize the equation governing the radial null geodesics L(u,θ,φ)
out
and L(v,θ,φ)
in
obtained in
Sec. 6.3.1 [Eq. (6.19)], the solution to which is given by Eq. (6.20):
r
t = ±r ± 2m ln − 1 + const . (8.7)
2m
Remark 1: Since λ is an affine parameter and E is constant, Eq. (8.5) shows explicitly that r is another
affine parameter along radial null geodesics of Schwarzschild spacetime — a feature that we had already
obtained in Sec. 6.3.1.
out,H
Remark 2: The radial null geodesics L(θ,φ) discussed in Sec. 6.3.5 have L = 0, but they are not
recovered from Eq. (8.6), since the latter is based on Schwarzschild-Droste coordinates (t, r, θ, φ) and
out,H
the L(θ,φ) ’s are the null geodesic generators of the event horizon H , which does not lie in the
Schwarzschild-Droste domain.
Some radial null geodesics of Schwarzschild spacetime are plotted in Fig. 6.1 in terms of the
Schwarzschild-Droste coordinates and in Fig. 6.3 in terms of the ingoing Eddington-Finkelstein
coordinates.
8.2.3 Generic null geodesics: effective potential
Having discussed the case L = 0 in the preceding section, we focus now on the case L ̸= 0
(“generic” case). We may then consider along the geodesic L the affine parameter
λ̃ := |L|λ , (8.8)
instead of λ, the latter being the affine parameter associated with the 4-momentum p via
Eq. (7.9). Since L is a non-vanishing constant along a geodesic, the above formula does define a
new affine parameter. The absolute value ensures that λ̃ is increasing towards the future, as λ,
whatever the sign of L. Note that λ̃ has the same dimension as L, i.e. a squared length, given
that λ is dimensionless (cf. Sec. 7.2.2). In terms of λ̃, the differential system (8.1)-(8.3) becomes
−1
dt 2m
=b −1
1− , (8.9)
dλ̃ r
dφ ϵL
= 2 , (8.10)
dλ̃ r
2
dr 1
+ U (r) = 2 , (8.11)
dλ̃ b
where
|L|
b := , (8.12)
E
L
ϵL := = sgn L (8.13)
|L|
236 Null geodesics and images in Schwarzschild spacetime
and
1 2m
U (r) := 2 1− . (8.14)
r r
As for the timelike case (cf. Eq. (7.24)), we note that Eq. (8.11) has the shape of the first integral
of the 1-dimensional motion of a non-relativist particle in the effective potential U (r). The
main difference is that U (r) does not depend on L, contrary to the effective potential Vℓ (r) for
timelike geodesics. Actually, U (r) is a function of r and the black hole mass m only. We thus
arrive at
Property 8.1: characterization of null geodesics by the impact parameter
A null geodesic L is entirely characterized by the constants b and ϵL = ±1. As soon as L
has some part in region MI , then E > 0 [cf. Eq. (7.5)], so that b, as defined by Eq. (8.12), is
finite and positive. It has the dimension of a length and can be interpreted as the impact
parameter in the case of a particle arising from infinity.
Proof. Let us consider that the massless particle P arises from a point of coordinates θ = π/2,
r = r0 ≫ m and φ = φ0 . Let us introduce Cartesian coordinates (x, y) in the plane θ = π/2:
x := r cos φ and y := r sin φ. Without any loss of generality, we may consider |φ0 | ≪ 1.
For large values of r, the motion of P is then essentially along the x-axis, with φ remaining
small: y ≃ y0 , where y0 = r0 sin φ0 ≃ r0 φ0 . The quantity |y0 | is the impact parameter. Since
y = r sin φ ≃ rφ, we get from y ≃ y0 the following relation
y0
φ≃ (r ≫ m).
r
Deriving with respect to t leads to
dφ y0 dr y0
≃− 2 ≃ 2 (r ≫ m), (8.15)
dt r dt r
where we have used dr/dt = −1 for r ≫ m. This last property is easy to admit since r ≫ m
corresponds to the flat part of spacetime; however, it can be derived rigorously by combining
Eqs. (8.9) and (8.11) to evaluate dr/dt. Now by combining Eqs. (8.9) and (8.10), we have
dφ b 2m
= ϵL 2 1 − . (8.16)
dt r r
For r ≫ m, the right-hand side reduces to ϵL b/r2 , so that the comparison with (8.15) yields
ϵL b = y0 , i.e. b = |y0 |, which proves that b is the impact parameter of the massless particle P
with respect to the central black hole.
The effective potential U (r) is plotted in Fig. 8.1. It has no minimum and a maximum at
r = 3m, which is
1
Umax = . (8.17)
27m2
8.2 Main properties of null geodesics 237
0.05 3
2
0.04
4
0.03
m 2 U( r )
1
0.02
0.01
-0.01
0 5 rper 10 15 20
r/m
Figure 8.1: Effective potential U (r) (rescaled by m2 to make it dimensionless) governing the r-part of the
motion along a null geodesic in Schwarzschild spacetime [Eq. (8.14)]. The vertical dashed line marks r = 2m,
i.e. the location of the event horizon. The horizontal lines marked “1” to “4” correspond to the r-motion of four
null geodesics (cf. Sec. 8.2.4 for details); their trajectories in the equatorial plane are depicted in Fig. 8.2. [Figure
produced with the notebook D.4.7]
y/m
6
1
3 4 2
2
x/m
-10 -5 5 10
-2
4
-4
Figure 8.2: Selected null geodesics in the equatorial plane of Schwarzschild spacetime, plotted in terms of the
coordinates (x, y) := (r cos φ, r sin φ), with the black hole region r < 2m depicted as a grey disk. The green
geodesic is the photon circular orbit at r = 3m. The red geodesic (label “1”) starts at r0 = 10m, φ0 = 0, with
the impact parameter b = 7.071 m (b−2 = 0.02 m−2 ). The brown geodesic (label “2”) starts at r0 = 10m, φ0 = 0
with b = 5m (b−2 = 0.04 m−2 ). The orange geodesic (label “3”) starts outward at r0 = 2.1m, φ0 = 0 with
b = 4.663m (b−2 = 0.046 m−2 ). The grey geodesic (label “4”) starts outward at r0 = 2.4m, φ0 = −π/2 with
b = 5.345m (b−2 = 0.035 m−2 ). The r-motion of these four gedeosics is depicted in Fig. 8.1. [Figure produced
with the notebook D.4.8]
238 Null geodesics and images in Schwarzschild spacetime
This extremum is a stationary position in r: it corresponds thus to a circular orbit, usually
called the circular photon orbit The set of all photon orbits (one per choice of equatorial
plane) is often called the photon sphere. However, since it corresponds to a maximum of the
effective potential, the circular orbit at r = 3m is an unstable orbit. So one should not imagine
that a Schwarzschild black hole is surrounded by any spherical shell of photons...
8.2.4 Radial behavior of null geodesics
For the sake of concreteness, in this section and the remaining ones, we refer to the massless
particle P as a “photon”. In most applications, in particular the astrophysical ones, P will
indeed be a photon. But one shall keep in mind that all results are valid for any other massless
particle.
Since the effective potential U (r) has no minimum, it does not offer any potential well, as
it is clear from Fig. 8.1. This is in sharp contrast with the effective potential Vℓ (r) for massive
particle (compare Fig. 7.10). Hence there are no bound orbits for photons1 .
We can infer various types of photon worldlines from Fig. 8.1. In view of the “first integral”
(8.11), each photon worldline can be represented by a horizontal line of ordinate b−2 in this
figure, which must lie above the curve U (r) by the positive quantity (dr/dλ̃)2 . The region
under the curve U (r) is thus excluded.
For an initially inward photon, i.e. a photon emitted with dr/dλ̃ < 0 from a position
r = rem , there are two possibilities, depending on the values of rem and of the impact parameter
b:
• if rem > 3m and b is large enough to fulfill b−2 < Umax (e.g. trajectory no. 1 in Fig. 8.1),
the photon “bounces” on the potential barrier constituted by U (r) at some minimal value
rp of r — the periastron, which is given by U (rp ) = b−2 , or equivalently by
rp3 − b2 rp + 2mb2 = 0, rp > 3m. (8.18)
Equation (8.11) implies then
dr
= 0. (8.19)
dλ̃ r=rp
Actually, dr/dλ̃ changes sign at r = rp and the photon subsequently moves away from
the black hole for ever (cf. geodesic no. 1 in Fig. 8.2). We may call such a worldline a
scattering trajectory.
• if rem < 3m (e.g. trajectory no. 4 in Fig. 8.1) or b is small enough to fulfill b−2 > Umax (e.g.
trajectory no. 2 in Fig. 8.1) the photon is not halted by the potential barrier constituted
by U (r). It then reaches arbitrarily small values of r and is eventually absorbed by the
black hole (r < 2m) (cf. geodesic no. 2 in Fig. 8.2).
For an initially outward photon, i.e. a photon emitted with dr/dλ̃ > 0, one has necessarily
rem > 2m according to the result obtained in Sec. 7.2.2 and there are then two possible
outcomes:
1
unless one counts as “bound” a worldline that terminates in the black hole region
8.2 Main properties of null geodesics 239
• if rem > 3m (e.g. trajectory no. 1 in Fig. 8.1) or b is small enough to fulfill b−2 > Umax (e.g.
trajectory no. 3 in Figs. 8.1), the photon escapes to infinity (cf. geodesic no. 3 in Fig. 8.2);
• if 2m < rem < 3m and b−2 < Umax (e.g. trajectory no. 4 in Fig. 8.1), the photon “bounces”
on the left side of the potential barrier, reaching a maximal value ra of r — the apoastron,
which is given by U (ra ) = b−2 , or equivalently by
ra3 − b2 ra + 2mb2 = 0, ra < 3m. (8.20)
The photon moves subsequently towards the black hole and is absorbed by it (cf. geodesic
no. 4 in Fig. 8.2).
The critical value of the impact parameter b separating the cases discussed above is deter-
mined by b−2
c = Umax . Given the value (8.17) of Umax , we get
√
bc = 3 3 m ≃ 5.196152 m . (8.21)
The above discussion leads us to
Property 8.2: Behaviour of r along a null geodesics
Along any null geodesic of Schwarzschild spacetime, the areal coordinate r either is a
monotonic function or has a single turning point. In the latter case, if the turning point
corresponds to a minimum of r (a periastron, which can occur only for r > 3m), the null
geodesic escapes to infinity, while if the turning point corresponds to a maximum of r (an
apoastron, which can occur only for 2m < r < 3m), the null geodesic terminates at the
central singularity (r = 0).
Note that for r < 2m, i.e. in the black hole region, r is always a monotonic function, since
it has been demonstrated in Sec. 7.2.2 that r(λ) is strictly decreasing. The impossibility of a
turning point for r < 2m can also be graphically inferred from Fig. 8.1, which shows that
the effective potential U (r) is negative for r < 2m, preventing the turning point condition
U (r) = b−2 to hold.
For b > bc , the explicit expression of the periastron radius rp (or the apoastron radius ra )
is obtained by solving the cubic equation (8.18) (or (8.20), which is the same cubic equation,
except for the range of the solution). Fortunately Eq. (8.18) is a depressed2 cubic equation,
which makes it simpler to solve. For b > bc , its discriminant −(4p3 + 27q 2 ) is positive, which
implies that it admits three distinct real roots. Two of them are positive and are precisely rp
and ra . The third root is negative, since the product of the roots is −2mb2 < 0; it has therefore
no physical significance. The roots of the generic depressed cubic equation x3 + px + q = 0
can be expressed via Viète’s formulas:
r r
p 1 3q 3 2kπ
xk = 2 − cos arccos − + , k ∈ {0, 1, 2}. (8.22)
3 3 2p p 3
2
A depressed cubic equation is a polynomial equation of the type x3 + px + q = 0.
240 Null geodesics and images in Schwarzschild spacetime
rper
8 rapo
7
6
r/m
5
4
3
2
5 6 7 8 9 10
b/m
Figure 8.3: Radial coordinate of the periastron, rp , and of the apoastron, ra , along null geodesics with
b > bc ≃ 5.196 m. Note that a given null geodesic has either a periastron or an apoastron, but not both. [Figure
produced with the notebook D.4.9]
In the present case, p = −b2 , q = 2mb2 and rp (resp. ra ) corresponds to k = 0 (resp. k = 2), so
that we obtain
2b π 1 bc
rp = √ cos − arccos . (8.23)
3 3 3 b
2b 5π 1 bc
ra = √ cos − arccos . (8.24)
3 3 3 b
√
As a√check, we note that for b ≫ bc , Eq. (8.23) yields rp ≃ 2b/ 3 cos (π/3 − 1/3 × π/2) ≃
2b/ 3 cos(π/6) ≃ b, as expected. Indeed, b ≫ bc implies b ≫ m, so that the photon stays far
from the black hole, a regime in which the impact parameter coincides with the distance of
closest approach: b ≃ rp .
The variation of rp and ra with b is plotted in Fig. 8.3. Note that rp = ra = 3m (the photon
orbit) at the limit b = bc .
Remark 3: It is clear on Fig. 8.3 that, for the same value of b, ra ≤ rp , which may seem contradictory
with the apoastron corresponding to the largest distance from the center and the periastron to the
distance of closest approach. However, one shall keep in mind that the apoastron and the periastron
always refer to different null geodesics: geodesics with an apoastron lie below the photon sphere
(r = 3m), while those with a periastron are always outside it. A null geodesic can of course cross the
photon sphere (examples are geodesics 2 and 3 on Figs. 8.1 and 8.2), but then it has neither an apoastron
nor a periastron.
8.3 Trajectories of null geodesics in the equatorial plane 241
8.3 Trajectories of null geodesics in the equatorial plane
In all this section, we assume L ̸= 0, i.e. we consider non-radial null geodesics, the radial case
having been discussed in Sec. 8.2.2.
Let us first recall the generic property of causal geodesics in Schwarzschild spacetime
established in Sec. 7.2.2 : for L ̸= 0, the azimuthal angle coordinate φ has no turning point: it
is either always increasing towards the future (L > 0) or always decreasing towards the future
(L < 0).
Remark 1: We recover the above property from Eq. (8.10): dφ/dλ̃ = ϵL /r2 , given that ϵL = sgn L and
λ̃ increases towards the future.
8.3.1 Differential equation and fundamental cubic polynomial
The equation governing the (r, φ)-part of a null geodesic, named trajectory in the orbital plane
in Sec. 7.2.3, is the generic differential equation (7.17) with µ (the particle’s mass) set to zero
and the ratio E 2 /L2 replaced by 1/b2 [cf. Eq. (8.12)]:
2
du
= Pb (u) (8.25)
dφ
where u := m/r [Eq. (7.16)] and Pb (u) stands for the cubic polynomial
m2
Pb (u) := 2u3 − u2 + . (8.26)
b2
Remark 2: Of course, Eq. (8.25) can be recovered by combining Eqs. (8.10), (8.11) and (8.14).
The graph of the polynomial Pb is shown in Fig. 8.4 for selected values of b. Note that along
a null geodesic, one must have u > 0 and Pb (u) ≥ 0. The first condition follows from the very
definition of u as m/r, while the second one is a direct consequence of Eq. (8.25).
By Eq. (8.25), the zeros of the polynomial Pb correspond to points at which du/dφ = 0.
They are thus stationary points for u, and hence for r. This can only occur at the circular
photon orbit r = const = 3m (then b = bc ) or at the periastron or apoastron discussed in
Sec. 8.2.4 (then b > bc ). The zeros of Pb are governed by its discriminant, which is
m4 b2
∆=4 4 − 27 . (8.27)
b m2
One may distinguish three cases:
√
• ∆ > 0 ⇐⇒ b > 27m = bc : Pb has three distinct real zeros (cf. the green and
cyan curves in Fig. 8.4); one of them, un say, is negative, and hence unphysical (since
u := m/r > 0), while the two other ones are positive, being nothing but
m m
up = and ua = , (8.28)
rp ra
where rp and ra are the periastron and apoastron radii given by Eqs. (8.23)-(8.24).
242 Null geodesics and images in Schwarzschild spacetime
Pb (u)
0.2
b=3m
b = 4pm
b=3 3 m 0.15
b=7m
b = 20 m 0.1
0.05
u
-0.4 -0.2 0.2 0.4 0.6 0.8
-0.05
-0.1
-0.15
-0.2
Figure 8.4: Graph of the cubic polynomial Pb (u) = 2u3 − u2 + m2 /b2 [Eq. (8.26)]. [Figure produced with the
notebook D.4.9]
• ∆ = 0 ⇐⇒ b = bc : Pb has a double zero: ua = up = 1/3 and a negative zero:
un = −1/6 (cf. red curve in Fig. 8.4); only the first one is physical and correspond to
r = 3m.
• ∆ < 0 ⇐⇒ b < bc : Pb has one real negative zero, un , and two complex zeros (cf. blue
and magenta curves in Fig. 8.4), so that Pb (u) never vanishes for physical values of u,
which are real positive.
Since they will be required in what follows, let us find explicit expressions for the real zeros
of the polynomial Pb . The equation Pb (u) = 0 can be brought to a depressed form (no square
term) by the change of variable u = x + 1/6. We get
1 m2 1
x3 − x+ 2 − = 0. (8.29)
12 2b 108
We shall discuss separately the cases b > bc and b < bc .
Zeros of Pb for b ≥ bc
For b > bc , we can express the solutions of Eq. (8.29) via Viète’s formulas (8.22) with p = −1/12
and q = m2 /(2b2 ) − 1/108. We then get the zeros of Pb as
b2c
1 1 2kπ 1
uk = cos arccos 1 − 2 2 + + , k ∈ {0, 1, 2}.
3 3 b 3 6
8.3 Trajectories of null geodesics in the equatorial plane 243
ua
0.4 up
un
0.2
u 0
-0.2
-0.4
0 2 4 6 8 10 12 14
b/m
Figure 8.5: Real zeros un , up and ua of the cubic polynomial Pb (u) = 2u3 − u2 + m2 /b2 as functions of b. The
vertical dashed line marks the critical value b = bc ≃ 5.196 m. [Figure produced with the notebook D.4.9]
Using the identity arccos(1 − 2x2 ) = 2 arcsin x and noticing that u1 < 0, which implies
u1 = un , and u0 ≥ u2 , which implies u0 = ua and u2 = up , we arrive at
1 2 bc 2π 1
un = cos arcsin + + (b ≥ bc ) (8.30a)
3 3 b 3 6
1 2 bc 4π 1
up = cos arcsin + + (b ≥ bc ) (8.30b)
3 3 b 3 6
1 2 bc 1
ua = cos arcsin + (b ≥ bc ). (8.30c)
3 3 b 6
These zeros are plotted in terms of b on Fig. 8.5. Note the ordering
1 1 1
− ≤ un < 0 < up ≤ ≤ ua < , (8.31)
6 3 2
with the inequalities ≤ being saturated for b = bc . Note also that
1
lim un = 0, lim up = 0, and lim ua = . (8.32)
b→+∞ b→+∞ b→+∞ 2
It is easy to express two of the zeros in terms of the third one. For instance, let us pick up .
From Pb (up ) = 0, we have immediately m2 /b2 = −2u3p + u2p . If we substitute this expression
for m2 /b2 into Pb (u), we get
Pb (u) = 0 ⇐⇒ 2u3 − u2 − 2u3p + u2p = 0
(u − up ) 2(u2 + up u + u2p ) − (u + up ) = 0
⇐⇒
(u − up ) 2u2 + (2up − 1)u + up (2up − 1) = 0.
⇐⇒
244 Null geodesics and images in Schwarzschild spacetime
The two zeros different from up of this equation, namely un and ua , must then obey
2u2 + (2up − 1)u + up (2up − 1) = 0.
Solving this quadratic equation leads to the sought expressions:
1
q
un = 1 − 2up − (1 − 2up )(1 + 6up ) (8.33a)
4
1
q
ua = 1 − 2up + (1 − 2up )(1 + 6up ) . (8.33b)
4
Real zero of Pb for b < bc
As mentioned above, for b < bc , Pb has only one real zero, un , which is negative. Its value
is obtained by means of Viète’s substitution, which consists in setting x = w + 1/(36w) in
Eq. (8.29), thereby turning it into a quadratic equation for w3 . Solving the latter yields
r !2/3 r !−2/3
1 bc b2c bc b2c
un = 1 − − − 1 − − −1 (b < bc ). (8.34)
6 b b2 b b2
un is plotted as a function of b in the left part of Fig. 8.5. Note that
1 1
un < − , lim− un = − and lim un = −∞. (8.35)
6 b→bc 6 b→0
The last limit results from the expansion of Eq. (8.34), which yields un ∼ −(2bc /b)2/3 /6 when
b → 0. Observe also from Fig. 8.5 the smooth transition with the value of un for b ≥ bc , which
is given by Eq. (8.30a).
Historical note : As mentioned in the historical note on p. 231, the equations of geodesic motion in
Schwarzschild metric have been first derived by Karl Schwarzschild in January 1916 [449] and Johannes
Droste in May 1916 [175], but these two authors focussed on the timelike case (orbit of a massive particle
or a “planet”). It seems that the first explicit writing of the equations governing null geodesics is due to
Ludwig Flamm in September 1916 [197]. In particular, Flamm derived3 Eqs. (8.1), (8.2) and a combination
of Eqs. (8.1) and (8.3). He also derived Eq. (8.25) governing the trajectories in the equatorial plane, as well
as Eq. (8.18) giving the periastron. Regarding the solutions of these equations, he got only approximate
ones, to the second order in m/b and recover Einstein’s famous result for the deflection of light by the
Sun. For details about early studies of null geodesics in Schwarzschild spacetime, see Ref. [181].
8.3.2 Critical null geodesics
√
For b = bc = 3 3 m, the differential equation (8.25) can be integrated by means of elementary
functions. Indeed, one has then Pb (u) = (u − 1/3)2 (2u + 1/3) and Eq. (8.25) is equivalent to
dφ 1
=± q . (8.36)
du u− 1 1
2u + 3
3
3
The link between Flamm’s notations and ours is α = 2m, R = r, x = u/m and ∆ = b.
8.3 Trajectories of null geodesics in the equatorial plane 245
r/m
10
ϕ − ϕ0
-8 -6 -4 -2−ϕ∗ ϕ∗ 2 4 6 8
Figure 8.6:
√ r as a function of φ along a null geodesic with an impact parameter equal to the critical one:
b = bc = 3 3 m; the blue curve is for a critical null geodesic with r > 3m [Eq. (8.40)], while the red one regards
r < 3m [Eq. (8.42)]. [Figure produced with the notebook D.4.10]
This equation can be easily integrated by noticing that
√
d
2 artanh x for x ∈ (0, 1)
1
dx
√ = (8.37)
(1 − x) x d √
2
arcoth x for x ∈ (1, +∞).
dx
We thus perform the change of variable x = 2u + 1/3 and treat separately two cases : u <
1/3 ⇐⇒ x ∈ (1/3, 1) and u > 1/3 ⇐⇒ x ∈ (1, +∞). We call geodesics in the first case
external critical null geodesics, since u < 1/3 ⇐⇒ r > 3m, and those in the second
case internal critical null geodesics, since u > 1/3 ⇐⇒ r < 3m. Note that the qualifiers
external and internal refer to the photon sphere r = 3m discussed in Sec. 8.2.3, and not to the
black hole region (r < 2m).
External critical null geodesics
For u < 1/3, x ∈ (1/3, 1), so that the first line of Eq.(8.37) is relevant and Eq. (8.36) is integrated
to r
1
φ = ±2 artanh 2u + + φ0 , (8.38)
3
where φ0 is some integration constant. This relation is easily inverted to
1 φ − φ0 1
u = tanh2 − . (8.39)
2 2 6
246 Null geodesics and images in Schwarzschild spacetime
y/m
5
4
3
2
1
x/m
5 10
-1
-2
-3
Figure 8.7: Trace in the equatorial
√ plane spanned by the coordinates x := r cos φ, y := r sin φ of an external
critical null geodesic (b = 3 3 m ≃ 5.196m and r > 3m) with φ∞ = 0. It obeys Eq. (8.40) with φ0 = −φ∗ and
φ ∈ (0, +∞) (right branch of the blue curve in Fig. 8.6). [Figure produced with the notebook D.4.10]
Moving back to r = m/u, we obtain the equation of the external critical null geodesic in polar
form:
2m
r= 2 φ−φ0
1 . (8.40)
tanh 2
−3
The constant φ0 can be related to the asymptotic value φ∞ of φ when r → +∞ by setting
u = 0 in Eq. (8.39); we get, using the identity artanh x = 1/2 ln[(1 + x)/(1 − x)],
√ !
3+1
φ0 = φ∞ ± φ∗ , with φ∗ := ln √ ≃ 1.316958. (8.41)
3−1
The function r(φ), as given by Eq. (8.40), is depicted on Fig. 8.6 (blue curve). The region
φ0 − φ∗ < φ < φ0 + φ∗ is excluded, since Eq. (8.40) would yield r < 0. For φ > φ0 + φ∗ , r(φ)
is a decaying function, which corresponds to the plus sign in Eq. (8.38) and to the minus sign in
Eq. (8.41): φ0 = φ∞ − φ∗ . Figure 8.7 shows such a null geodesic with φ∞ = 0, which implies
φ0 = −φ∗ and φ > 0. When φ → +∞, the geodesic rolls up indefinitely onto the photon orbit
discussed in Sec. 8.2.3; this behavior corresponds to the horizontal asymptote at r = 3m in
the right part of Fig. 8.6. Note that the geodesic approaches very fast the photon orbit, only
a single path round to it being graphically visible in Fig. 8.7. This is because the asymptotic
expansion of relation (8.40) is r ∼ 3m(1 + 6e−φ ) when φ → +∞.
It is worth stressing that Fig. 8.7 describes both (i) the trace in the (x, y)-plane of a geodesic
with L > 0 (so that φ increases towards the future, cf. Eq. (8.2)) arising from r → +∞ and
spiralling inwards to the photon orbit and (ii) the trace of a geodesic with L < 0 (so that φ
decays towards the future) arising from r = rem > 3m, φ = φem > 0, spiralling outwards
and escaping to r → +∞ as φ → 0. Had we restored the t dimension perpendicular to the
(x, y)-plane in a 3d plot, these two geodesics would have clearly appeared distinct.
On the contrary, for φ < φ0 − φ∗ , r(φ) is an increasing function, as it is clear on the left
part of the blue curve in Fig. 8.6. This corresponds to the minus sign in Eq. (8.38) and to the
plus sign in Eq. (8.41): φ0 = φ∞ + φ∗ . A null geodesic of this type is depicted in Fig. 8.8. It
8.3 Trajectories of null geodesics in the equatorial plane 247
y/m
3
2
1
x/m
5 10
-1
-2
-3
-4
-5
Figure 8.8: Same as Fig. 8.7, but for φ0 = φ∗ and φ ∈ (−∞, 0) (left branch of the blue curve in Fig. 8.6). [Figure
produced with the notebook D.4.10]
y/m y/m
3 3
2 2
1 1
x/m x/m
-3 -2 -1 1 2 3 -3 -2 -1 1 2 3
-1 -1
-2 -2
-3 -3
√
Figure 8.9: Trace in the equatorial plane of two internal critical null geodesics (b = 3 3 m ≃ 5.196m and
r < 3m). They both obey Eq. (8.42) with φ0 = 0. The left panel corresponds to φ ∈ (−∞, 0) (left part of the red
curve in Fig. 8.6), while the right one is for φ ∈ (0, +∞) (right part of the red curve in Fig. 8.6). [Figure produced
with the notebook D.4.10]
has φ∞ = 0, so that φ0 = φ∗ and φ < 0. As for Fig. 8.7, the curve depicted in Fig. 8.8 can be
interpreted as the trace in the (x, y)-plane of two distinct geodesics: one with L < 0 arising
from r → +∞ and spiralling inwards to the photon orbit when φ → −∞ and another one
with L > 0 arising from r = rem > 3m, φ = φem < 0, spiralling outwards and escaping to
r → +∞ as φ → 0.
Internal critical null geodesics
Let us now consider the “internal” case: r < 3m ⇐⇒ u > 1/3. This implies x ∈ (1, +∞) in
Eq. (8.37), so that Eq. (8.36) is integrated to
2m
r= 2 φ−φ0
1
. (8.42)
coth 2
− 3
248 Null geodesics and images in Schwarzschild spacetime
Again, φ0 is an integration constant. But this time, it cannot be determined by the value of φ
when r → +∞ since we are in the case r < 3m. The function r(φ) given by Eq. (8.42) is plotted
as the red curve in Fig. 8.6. Contrary to external critical null geodesics, there is no exclusion
interval for φ. However, φ cannot range from −∞ to +∞ along a given geodesic. Indeed, for
φ = φ0 , one gets r = 0, which corresponds to the curvature singularity of the Schwarzschild
black hole. This is necessarily a termination point for any geodesic. The maximum range of φ
along an internal critical null geodesic is therefore either (−∞, φ0 ) or (φ0 , +∞).
An internal critical null geodesic with φ0 = 0 and φ ∈ (−∞, 0) is depicted in the left
panel of Fig. 8.9. For φ → −∞, it rolls up indefinitely onto the photon orbit (r = 3m) from
below. Again the approach to the photon orbit is exponentially fast, the asymptotic expansion
of relation (8.42) being r ∼ 3m(1 − 6eφ ) when φ → −∞. The curve plotted in Fig. 8.9 actually
represents the trace in the equatorial plane of two distinct geodesics: (i) a geodesic with L > 0
arising from r = rem < 3m, φ = φem < 0 and inspiralling to the central singularity as φ → 0
and (ii) a geodesic with L < 0 arising from r = rem ∈ (2m, 3m), φ = φem < 0 and spiralling
outward to the photon orbit as φ → −∞. Note that for (ii), the emission point must fulfill
rem > 2m, i.e. must lie outside the black hole region. Indeed, as shown in Sec. 7.2.2, along any
geodesic, r must decrease towards the future in the black hole region. Hence no null geodesic
can be spiralling outwards there.
Remark 3: Actually, an outward spiralling critical null geodesic can emerge from the region r < 2m
when the latter corresponds to the white hole region in the extended Schwarzschild spacetime that
will be discussed in Chap. 9. This explains why nothing in Eq. (8.42) and Fig. 8.9 seems to prevent this
behavior.
The right panel of Fig. 8.9 corresponds to an internal critical null geodesic with φ0 = 0
and φ ∈ (0, ∞). For φ → +∞, it rolls up indefinitely onto the photon orbit (r = 3m) from
below. Again, the plotted curve describes two cases: (i) a geodesic with L < 0 arising from
r = rem < 3m, φ = φem > 0 and inspiralling to the central singularity as φ → 0 and (ii) a
geodesic with L > 0 arising from r = rem ∈ (2m, 3m), φ = φem > 0 and spiralling outward
to the photon orbit as φ → +∞.
Historical note : The photon circular orbit at r = 3m has been exhibited by David Hilbert in December
1916 [277, 278]. Hilbert also discussed the critical null geodesics and their spirals around the photon orbit,
interpreting the latter as a Poincaré limit cycle of the dynamical system governed by Eq. (8.25). Hilbert
pointed out that ingoing null geodesics with an impact parameter b > bc are deflected (possibly looping
around the photon orbit if b is close to bc ) and escape to infinity, while those with b < bc cross the photon
orbit and terminate on “the circle r = 2m”. Hilbert claimed that null geodesics stop there, because
their (coordinate!) velocity vanishes4 . We recover here the interpretation of the r = 2m (coordinate)
singularity as an impenetrable sphere advanced by Droste while discussing timelike geodesics (cf. the
historical note on p. 231). The distinction between coordinate effects and physical ones was definitely
not clear in the early days of general relativity, even for great minds like Hilbert! The same analysis and
conclusions are found in the general relativity treatise by Max von Laue published in 1921 [340].
4
The coordinate velocity for ingoing radial null geodesics is given by Eq. (8.6): dr/dt = −1 + 2m/r, which
clearly vanishes at r = 2m.
8.3 Trajectories of null geodesics in the equatorial plane 249
8.3.3 Null geodesics with b > bc and r > 3m
For b ̸= bc , the differential equation (8.25) cannot be integrated in terms of elementary functions.
As we are going to see, it is integrable though in terms of standard functions of mathematical
physics, namely elliptic integrals of the first kind. As a first step, we rewrite Eq. (8.25) as
dφ 1
= ϵL ϵin p , (8.43)
du Pb (u)
where ϵL = ±1 is the sign of the conserved angular momentum L [cf. Eq. (8.13)] and ϵin = ±1
is defined by
du
ϵin := sgn , (8.44)
dλ̃
i.e. ϵin = +1 if u increases towards the future along L , or equivalently if r decreases towards
the future along L (inward motion, hence the index “in”), and ϵin = −1 otherwise. Let us
recall that by vertue of the equation of motion (8.10), ϵL gives the sign of dφ/dλ̃, so that the
sign of dφ/du = dφ/d p λ̃ × (du/dλ̃) is ϵL ϵin , which justifies the factor ϵL ϵin in front of the
−1
positive quantity 1/ Pb (u) in the right-hand side of Eq. (8.43).
In this section, we consider null geodesics with b > bc and outside the photon sphere, i.e.
geodesics similar to that labelled 1 in Figs. 8.1 and 8.2. Each of these geodesics has a periastron,
at u = up , and obeys u → 0 (r → +∞) for both λ̃ → −∞ and λ̃ → +∞. The general solution
to the differential equation (8.43) can be written as
Z u
dū
φ = φp + ϵL ϵin p , (8.45)
up Pb (ū)
where φp is the value of φ at the periastron. We shall rewrite it as
φ = φp − ϵL ϵin Φb (u) , (8.46)
where we have introduced the function
Z up Z up
dū dū
Φb (u) := p = p (b > bc ). (8.47)
u Pb (ū) u 2ū − ū2 + (m/b)2
3
Since up is the function of b given by Eq. (8.30b), the above relation defines uniquely a function
of u and b, which we consider as a function of u parameterized by b. Note that, since u ≤ up
(by definition of the periastron), one has Φb (u) ≥ 0. To evaluate Φb (u), we rewrite it in terms
of the three zeros (un , up , ua ) of Pb :
Z up
dū
Φb (u) = p . (8.48)
u 2(ū − un )(up − ū)(ua − ū)
Let us perform the change of variable5
ū − un
t := ⇐⇒ ū = (up − un )t + un .
up − un
5
It should be clear that the variable t introduced here has nothing to do with the Schwarzschild-Droste
coordinate t.
250 Null geodesics and images in Schwarzschild spacetime
0.9
0.8
0.7
k
0.6
0.5
6 8 10 12 14 16 18 20
b/m
Figure 8.10: Modulus k of the elliptic integrals F and K that are involved in expression (8.53) for Φb (u). [Figure
produced with the notebook D.4.9]
We get
Z 1
1 dt
Φb (u) = p p , (8.49)
2(ua − un ) u−un
up −un
t(1 − t)(1 − k 2 t)
where k is the following constant:
r √
up − un 2
k := = q√ (b > bc ), (8.50)
ua − un 3 cot 2
arcsin bc
+1
3 b
the second equality following from the expressions (8.30) of un , up and ua in terms of b.
We can simplify further the integral via a second change of variable:
√
t = sin2 ϑ ⇐⇒ ϑ = arcsin t.
Note
p that 0 < u ≤ up implies 0 < t ≤ 1, so that 0 < ϑ ≤ π/2. Since dt = 2 sin ϑ cos ϑ dϑ =
2 t(1 − t) dϑ, we arrive immediately at
√ Z π/2
2 dϑ
Φb (u) = √ p , (8.51)
ua − un ϕb (u) 1 − k 2 sin2 ϑ
where6
s
u − un
ϕb (u) := arcsin (b > bc ). (8.52)
up − un
6
Do no confuse the letter ϕ with the symbol used for the coordinate φ.
8.3 Trajectories of null geodesics in the equatorial plane 251
R π/2 R π/2 Rϕ
By splitting the integral according to ϕ
= 0
− 0
, we rewrite (8.51) as
√
2
Φb (u) = √ [K(k) − F (ϕb (u), k)] (b > bc ), (8.53)
ua − un
where F (ϕ, k) is the incomplete elliptic integral of the first kind [79, 232, 3]:
Z ϕ
dϑ
F (ϕ, k) := p (8.54)
0 1 − k 2 sin2 ϑ
and K(k) is the complete elliptic integral of the first kind:
Z π
2 dϑ π
K(k) := p =F ,k . (8.55)
0 1 − k 2 sin2 ϑ 2
The notation F (ϕ, k) is the most common one in the literature [79, 232], but one may encounter
as well F (ϕ|m) for F (ϕ, k) with m = k 2 [3]. The parameter k is called the modulus of the
elliptic integral. From its expression (8.50), we see that k is a function of b. It is plotted in
Fig. 8.10. Given the ordering (8.31) and the limits (8.32), we deduce from expression (8.50) that
0 < k < 1, (8.56)
with
lim k = 1 and lim k = 0. (8.57)
b→b+
c
b→+∞
Given the range (0, up ) of u, we deduce from expression (8.52) that
un π
r
0 < arcsin < ϕb (u) ≤ , (8.58)
un − up 2
with ϕb (up ) = π/2.
The function Φb (u) is plotted in Fig. 8.11 for various values of b. Note that, by construction
[cf. Eq. (8.47)], Φb (u) ≥ 0 and Φb (u) = 0 ⇐⇒ u = up . Note also that the closer b is from
bc ≃ 5.1961452, the larger the amplitude of Φb (u). This point will be discussed further in
Secs. 8.3.6 and 8.4.
Remark 4: One can express Φb (u) in terms of a single elliptic integral thanks to the following property
of incomplete elliptic integrals of the first kind7 :
1
tan ψ tan ϕ = √ =⇒ F (ψ, k) = K(k) − F (ϕ, k). (8.59)
1 − k2
Then by setting r
1 up − u
ψb (u) := arcsin , (8.60)
k ua − u
7
√
See e.g. Eq. (117.01) of Ref. [79] with k ′ = 1 − k 2 or Eq. (17.4.13) of Ref. [3] with sin α = k.
252 Null geodesics and images in Schwarzschild spacetime
b = 5.196153 m
b = 5.19616 m
8 b = 5.1962 m
b = 5.197 m
b = 5.2 m
b = 5.3 m
b = 6.0 m
6 b = 10.0 m
b = 20.0 m
Φ b (u) b = 100.0 m
4
0
0 0.05 0.1 0.15 0.2 0.25 0.3
u
Figure 8.11: Function Φb (u) defined by Eq. (8.47) and evaluated via the elliptic integral expression (8.52)-(8.53),
for selected values of b > bc . For each value of b, the range of u is (0, up ], with the inverse periastron radius
up = up (b) given by Eq. (8.30b). [Figure produced with the notebook D.4.11]
we can rewrite Eq. (8.53) as √
2
Φb (u) = √ F (ψb (u), k) . (8.61)
ua − un
We prefer however the form (8.53) of Φb (u) because (i) expression (8.52) for ϕb (u) is simpler than
expression (8.60) for ψb (u) and (ii) the form (8.53) is better adapted to the study of the limit b → bc , to
be discussed in Sec. 8.4.1.
Remark 5: The trajectory of a null geodesic in the plane θ = π/2 as given by Eqs. (8.46) and (8.53)
is of the form φ = φ(r) on each of the two arcs with respect to the periastron. One can invert
this relation to express the trajectory in the polar form r = r(φ). This is performed thanks to the
inverse of the incomplete elliptic integral F (ϕ, k), which is the Jacobi elliptic sine sn(x, k), defined by
sin ϕ = sn(F (ϕ, k), k). We deduce then from Eqs. (8.60)-(8.61) that
√
up − u ua − un
2
= k sn √ Φb (u), k . (8.62)
ua − u 2
We refer the reader to Refs. [80, 374] for more details.
8.3.4 Null geodesics with b > bc and r < 3m
Let us consider now null geodesics still with b > bc but located below the photon sphere, i.e.
geodesics similar to that labelled 4 in Figs. 8.1 and 8.2. Each of these geodesics has an apoastron,
at u = ua , where ua is the function of b given by Eq. (8.30c). In other words, i.e. one has u ≥ ua
along the geodesic.
The treatment is similar to that of Sec. 8.3.3, but with ua playing the role of up .
To be detailed later...
8.3 Trajectories of null geodesics in the equatorial plane 253
8.3.5 Null geodesics with b < bc
A null geodesic L with an impact parameter b < bc has neither a periastron nor an apoastron,
since Pb has no zero in the physical range (0, +∞) of u in that case (cf. Sec. 8.3.1 and Fig. 8.4).
The only real zero of Pb is un < 0, which is the function of b given by Eq. (8.34). Examples of
such geodesics are those labelled 2 and 3 in Figs. 8.1 and 8.2. Having no periastron or apoastron
implies that r, and hence u, is a monotonic function along L . We shall then distinguish two
cases:
• L is ingoing: r decreases all along L , from +∞ to 0 as λ̃ varies from −∞ to some
value λ̃0 where L hits the curvature singularity at r = 0:
lim r = +∞ and lim r = 0 ⇐⇒ lim u = 0 and lim u = +∞. (8.63)
λ̃→−∞ λ̃→λ̃0 λ̃→−∞ λ̃→λ̃0
• L is outgoing: r increases all along L ; as shown in Sec. 7.2.2, this cannot happen in
the region r < 2m (the black hole interior). Hence, we must have r varying from 2m to
+∞, with λ̃ ranging from some finite value, λ̃1 say, to +∞:
1
lim r = 2m and lim r = +∞ ⇐⇒ lim u = and lim u = 0. (8.64)
λ̃→λ̃1 λ̃→+∞ λ̃→λ̃1 2 λ̃→+∞
That the range of λ̃ for outgoing geodesics is (λ1 , +∞) and not the whole real line can be
understood by considering the limit b → 0. We are then in the case of radial null geodesics,
for which it has been proved that r is an affine parameter (cf. Secs. 6.3.1 and 8.2.2). Since this
particular affine parameter obviously takes a finite value at r = 2m, any other affine parameter
must take a finite value as well. Actually, we shall see in Chap. 9 (cf. Sec. 9.3.1) that this r = 2m
limit in the past of outgoing null geodesics does not correspond to the black hole horizon but
to the event horizon of a white hole, which is a located in a part of the spacetime that is not
covered by the Schwarzschild-Droste coordinates considered here.
Since Pb (un ) = 0, we have m2 /b2 = −2u3n + u2n , so that we may write
Pb (u) = 2(u3 − u3n ) − (u2 − u2n ) = (u − un ) 2(u2 + uun + u2n ) − (u + un ) .
After a slight rearrangement of the term in square brackets, we arrive at
Pb (u) = 2(u − un ) (u − u0 )2 + (u∗ − un )2 − (u0 − un )2 , (8.65)
with
1 un
and (8.66)
p
u0 := − u∗ := un (3un − 1) + un .
4 2
Note that u0 and u∗ are functions of b, via the expression (8.34) of un . They are plotted in
Fig. 8.12. It is clear from this figure that u∗ > u0 ; consequently the term inside the square
brackets in Eq. (8.65) is always positive, in agreement with un being the only real zero of Pb for
b < bc . Furthermore, according to the limits (8.35), we have
1
lim u∗ = +∞ and lim− u∗ = . (8.67)
b→0 b→bc 3
254 Null geodesics and images in Schwarzschild spacetime
2.5
u∗
u0
2
1.5
u0 , u ∗
1
0.5
0
0 1 2 3 4 5
b/m
Figure 8.12: Parameters u0 and u∗ , defined by Eq. (8.66), as functions of b. The horizontal dashed line marks
u = 1/3. [Figure produced with the notebook D.4.12]
The general solution to the differential equation (8.43) can be written as
φ = φ∗ − ϵL ϵin Φb (u) , (8.68)
where Z u∗ Z u∗
dū dū
Φb (u) := p = p (b < bc ) (8.69)
u Pb (ū) u 2ū3 − ū2 + (m/b)2
and φ∗ is the value of φ at u = u∗ . Since
1
lim− u∗ = = lim up ,
b→bc 3 b→b+c
there is a kind of continuity of (8.69) with the definition (8.47) of Φb (u) for b > bc , despite the fact
that Φb (u) is not defined for b = bc (more precisely, as we shall see later, limb→b−c Φb (u) = +∞
for u < u∗ and limb→b+c Φb (u) = +∞ for u < up ).
To evaluate Φb (u), we shall use expression (8.65) for Pb (u):
Z u∗
1 dū
Φb (u) = √ p .
2 u (ū − un ) [(ū − u0 ) + (u∗ − un )2 − (u0 − un )2 ]
2
Performing the change of variable
u∗ − ū 1−t
t := ⇐⇒ ū = (u∗ − un ) + un (8.70)
u∗ + ū − 2un 1+t
yields
Z u∗ −u
u∗ +u−2un dt
Φb (u) = √ p . (8.71)
0 1 − t2 (u∗ − un )(1 + t2 ) − (u0 − un )(1 − t2 )
8.3 Trajectories of null geodesics in the equatorial plane 255
Given the range (0, +∞) for u, the range of t is
u∗
−1 < t ≤ < 1.
u∗ − 2un
To proceed, we shall distinguish the cases u ≤ u∗ and u > u∗ . The first case corresponds to
u∗
0 ≤ t < u∗ −2u n
< 1 and we perform the change of variable t = cos ϑ with ϑ ∈ (0, π/2) in the
integral (8.71), yielding
Z π/2
1 dϑ
Φb (u) = p p ,
2(u∗ − un ) ϕb (u)) 1 − k 2 sin2 ϑ
with
|u∗ − u|
ϕb (u) := arccos (b < bc ). (8.72)
u∗ + u − 2un
and8 s
u∗ − 5un /2 + 1/4
k := (b < bc ). (8.73)
2(u∗ − un )
The absolute value in the right-hand side of Eq. (8.72) is not necessary in the present case
since u ≥ u∗ . We keep it because the same expression will be used below for the case u > u∗ .
To let appear the incomplete elliptic integral of the first kind (8.54), let us rewrite the above
expression as
"Z #
π/2 Z ϕb (u)
1 dϑ dϑ
Φb (u) = p p − p .
2(u∗ − un ) 0 1 − k 2 sin2 ϑ 0 1 − k 2 sin2 ϑ
In view of respectively (8.55) and (8.54), the first integral is K(k), while the second one is
F (ϕb (u), k). We have thus
1
Φb (u) = p [K(k) − F (ϕb (u), k)] (u ≤ u∗ , b < bc ). (8.74)
2(u∗ − un )
As for the case b > bc , the modulus k of the elliptic integrals K and F is a function of b only,
which is given by combining Eqs. (8.73), (8.66) and (8.34). It is plotted in Fig. 8.13. Note that its
range is pretty limited:
√
q
1
2 + 3 < k < 1. (8.75)
|2 {z }
≃0.96592
Let us now turn to the case u > u∗ . It implies −1 < t < 0, so that we perform the change
of variable t = − cos ϑ in order to keep ϑ ∈ (0, π/2). We obtain then
Z ϕb (u)
1 dϑ
Φb (u) = p p ,
2(u∗ − un ) π/2 1 − k 2 sin2 ϑ
8
To get expression (8.73), we have used Eq. (8.66) to get rid of u0 .
256 Null geodesics and images in Schwarzschild spacetime
1
0.995
0.99
0.985
k
0.98
0.975
0.97
0 1 2 3 4 5
b/m
Figure 8.13: Modulus k of the elliptic integrals F and K that are involved in expressions (8.74) and (8.76) for
Φb (u). [Figure produced with the notebook D.4.12]
6 b = 5.196 m
b = 5.192 m
b = 5.19 m
4 b = 5.15 m
b = 5.0 m
b = 3.0 m
2 b = 1.0 m
b = 0.5 m
Φ b (u)
-2
-4
-6
0 0.2 0.4 0.6 0.8 1 1.2 1.4
u
Figure 8.14: Function Φb (u) defined by Eq. (8.69) and evaluated via the elliptic integral expressions (8.74) and
(8.76), for selected values of b < bc . [Figure produced with the notebook D.4.12]
8.3 Trajectories of null geodesics in the equatorial plane 257
where ϕb (u) and k are given by Eqs. (8.72) and (8.73). We conclude that
1
Φb (u) = p [F (ϕb (u), k) − K(k)] (u > u∗ , b < bc ). (8.76)
2(u∗ − un )
The function Φb evaluated via Eqs. (8.74) and (8.76) is plotted in Fig. 8.14 for various values
of b. Note that, by construction [cf. Eq. (8.69)], Φb (u) = 0 ⇐⇒ u = u∗ . Note also that, as in
the case b > bc (Sec. 8.3.3), the closer b is from bc ≃ 5.1961452, the larger the amplitude of
Φb (u).
8.3.6 Deflection angle and winding number
Let us consider a null geodesic L with b > bc arising from r → +∞, i.e. belonging to the
family studied in Sec. 8.3.3. L reaches some periastron and departs to r → +∞. The total
change of φ along the geodesic history is
∆φ = φ∞ − φ−∞ , (8.77)
with φ∞ := limλ̃→+∞ φ and φ−∞ := limλ̃→−∞ φ. Given that limλ̃→±∞ u = 0, Eq. (8.46) with
respectively ϵin = +1 and ϵin = −1 leads to
φ−∞ = φp − ϵL Φb (0) and φ∞ = φp + ϵL Φb (0). (8.78)
Hence
∆φ = 2ϵL Φb (0). (8.79)
Note that one can have |∆φ| > 2π; in such a case, the null geodesic is winding around the
black hole before leaving to infinity. We therefore introduce the winding number n ∈ Z by
(
∆φ ∈ [0, 2π) if L > 0
∆φ =: ∆φ + 2πn with (8.80)
∆φ ∈ (−2π, 0] if L < 0.
Note that n ≥ 0 for L > 0 and n ≤ 0 for L < 0. We then define the deflection angle Θ by
Θ := ∆φ − ϵL π. (8.81)
Let us recall that ϵL = ±1 is the sign of the conserved angular momentum L [cf. Eq. (8.13)].
The −ϵL π term in the above equation is chosen so that Θ = 0 in flat spacetime (no deflection
of light). By construction, the range of Θ is
−π ≤ Θ ≤ π. (8.82)
The above concepts are illustrated in Figs. 8.15-8.16, which show the trajectories of null
geodesics arising from infinity along the direction φ = 0 (i.e. having φ−∞ = 0), for various
values of the impact parameter b decaying from b = 12m to bc . They have been computed by
means of the geodesic integrator of SageMath (cf. Appendix D), instead of making use of the
elliptic-integral expression (8.53). All these geodesics have L > 0 and hence ϵL = +1. For
258 Null geodesics and images in Schwarzschild spacetime
y/m y/m
b = 12.000000 m b = 8.000000 m
10 10
5 5
x/m x/m
-10 -5 5 10 -10 -5 5 10
-5 -5
-10 -10
y/m y/m
b = 6.000000 m b = 5.355000 m
10 10
5 5
x/m x/m
-10 -5 5 10 -10 -5 5 10
-5 -5
-10 -10
Figure 8.15: Null geodesics in the plane θ = π/2 of Schwarzschild spacetime, plotted in terms of the coordinates
(x, y) := (r cos φ, r sin φ). All geodesics arise from x ≫ m with trajectories initially parallel to the x-axis; they
differ by the value of the impact parameter b. The grey disk marks the black hole region r < 2m, while the dashed
green circle indicates the photon orbit at r = 3m. [Figure produced with the notebook D.4.8]
8.3 Trajectories of null geodesics in the equatorial plane 259
y/m y/m
b = 5.230000 m b = 5.202500 m
10 10
5 5
x/m x/m
-10 -5 5 10 -10 -5 5 10
-5 -5
-10 -10
y/m y/m
b = 5.196430 m b = 5.196155 m
10 10
5 5
x/m x/m
-10 -5 5 10 -10 -5 5 10
-5 -5
-10 -10
Figure 8.16: Same as Fig. 8.15 but for values of b closer to bc ≃ 5.196152 m. [Figure produced with the notebook
D.4.8]
260 Null geodesics and images in Schwarzschild spacetime
b = 12m (upper left panel of Fig. 8.15), the geodesic suffers only some moderate bending: the
deflection angle is Θ ≃ π/6 and the winding number is n = 0. We recover here the standard
deflection of light by massive bodies in general relativity. For b = 8m and 6m, the bending is
more pronounced, exceeding Θ = π/2 for b = 6m, still with n = 0. For b = 5.355 m (lower
right panel of Fig. 8.15), the deflection angle is Θ ≃ π, i.e. the photon goes back in the direction
from which it was coming.
When the impact parameter becomes even closer to the critical value bc ≃ 5.196152 m
[Eq. (8.21)], the null geodesic starts to wind around the black hole before escaping to infinity
(Fig. 8.16). For b = 5.230 m (upper left panel of Fig. 8.16), the winding number is n = 1 and the
deflection angle is Θ ≃ −π/2. For b = 5.2025 m (upper right panel of Fig. 8.16), one has n = 1
and Θ = 0. For b = 5.19643 m (lower left panel of Fig. 8.16), one has n = 1 and Θ ≃ π and
for b = 5.196155 m (lower right panel of Fig. 8.16), one has n = 2 and Θ ≃ π/2. Note that the
winding is taking place almost at the photon circular orbit (r = 3m). That after a few turns
the null geodesic departs to infinity corroborates the fact that the photon orbit is unstable (cf.
Sec. 8.2.3).
We can understand the winding phenomenon around the photon circular orbit for b close
to bc without investigating the properties of elliptic integrals. Indeed, by combining Eqs. (8.79)
and (8.48), we get
√ Z up du
∆φ = ϵL 2 p . (8.83)
0 (up − u)(ua − u)(u − un )
Given the ordering (8.31), each of the three factors under the square root is positive on the
integration range (0, up ). For b ̸= bc , one has ua ̸= up (cf. Fig. 8.5) and the only diverging
√
term in the integrand of (8.83) is 1/ up − u, which diverges at the integral boundary u = up .
However, the integral Z up
du
√
0 up − u
√
is finite, being equal to 2 up , so that ∆φ remains finite. When b → bc , ua → up and the
integral (8.83) has a behavior similar to
Z up Z up
du du
p = .
0 (up − u) 2
0 up − u
Since the latter is a diverging integral, we conclude that
∆φ → ±∞ when b → bc . (8.84)
We recover the behavior observed for b = bc in Sec. 8.3.2: for an external critical null geodesic,
∆φ is infinite, the geodesic spiralling indefinitely around the photon orbit (cf. Fig. 8.7). We
shall refine (8.84) in Sec. 8.4.1 [Eq. (8.102)].
8.4 Asymptotic direction from some emission point
To discuss images in Sec. 8.5, we shall need the change in φ between some arbitrary point on a
null geodesic (the “emission” point) and a point far away from the black hole (the “reception”
point). The total change ∆φ along all the geodesic history considered in Sec. 8.3.6 is then a
special case: that for which the emission point is infinitely far from the black hole.
8.4 Asymptotic direction from some emission point 261
8.4.1 Asymptotic direction for b > bc
Let us consider a null geodesic L with b > bc and an event of (finite) affine parameter λ̃ = λ̃em
on L , which we shall call the emission point. We are interested in null geodesics which reach
the asymptotic flat region r → +∞ for λ̃ → +∞. This implies that the emission point is
located outside the photon sphere, i.e. obeys
rem := r(λ̃em ) > 3m. (8.85)
Indeed, we have seen in Sec. 8.3.4 that null geodesics with b > bc and emitted under the photon
sphere never cross the later (this is also obvious from the effective potential profile, as plotted
in Fig. 8.1, cf. trajectory no. 4). Our aim is to relate the value φ∞ of φ for λ̃ → +∞ (the
asymptotic direction) to its value φem at λ̃ = λ̃em .
If the emission point is past L ’s periastron, i.e. if λ̃em > λ̃p , φem is related to φp by Eq. (8.46)
with ϵin = −1:
φem = φp + ϵL Φb (uem ), (8.86)
where uem := m/rem . On the other side, the asymptotic value φ∞ is related to φp by Eq. (8.78).
By combining these two relations to eliminate φp , we get
φ∞ = φem + ϵL [Φb (0) − Φb (uem )] . (8.87)
Substituting expression (8.53) for Φb yields
√
2 (b > bc )
φ∞ = φem + ϵL √ [F (ϕb (uem ), k) − F (ϕb (0), k)] , (8.88)
ua − un (λ̃em > λ̃p )
where, according to Eq. (8.52),
s s
uem − un |un |
ϕb (uem ) = arcsin and ϕb (0) = arcsin . (8.89)
up − un up − un
If the emission point is located prior to the periastron, i.e. if λ̃em < λ̃p , φem is related to φp
by Eq. (8.46) with ϵin = +1:
φem = φp − ϵL Φb (uem ) (8.90)
and we get, by combining with Eq. (8.78),
φ∞ = φem + ϵL [Φb (0) + Φb (uem )] . (8.91)
Equation (8.53) then yields
√
2 (b > bc )
φ∞ = φem + ϵL √ [2K(k) − F (ϕb (uem ), k) − F (ϕb (0), k)] .
ua − un (λ̃em < λ̃p )
(8.92)
φ∞ − φem is plotted as a function of b in Fig. 8.17 for various values of rem = m/uem , with the
dashed curves corresponding to the case λ̃em > λ̃p . We notice that for b ≫ m (cf. the red curve
near b = 12m), one has either φ∞ − φem ≃ 0 (dashed red curve: λ̃em > λ̃p ) or φ∞ − φem ≃ π
(solid red curve: λ̃em < λ̃p ): null geodesics with large impact parameters suffer almost no
deflection, as expected. Another striking feature of Fig. 8.17 is the divergence of φ∞ − φem
when b tends to bc . Let us examine this in details.
262 Null geodesics and images in Schwarzschild spacetime
7π rem → 2m
2 rem = 3 m
rem = 4 m
3π rem = 6 m
rem = 10 m
rem = 20 m
5π rem = 100 m
ϕ∞ − ϕem
2 rem → + ∞
2π
3π
2
π
1π
2
0
0.0 2.0 4.0 6.0 8.0 10.0 12.0
b/m
Figure 8.17: Total change φ∞ − φem in φ from the emission point at (r, φ) = (rem , φem ) as a function of
the impact parameter b of the null geodesic, assuming ϵL = +1. The dashed curves correspond to trajectories
with b > bc that do not pass through the periastron. For each value of rem > 3m, the maximum of b is given by
rp (b) = rem , i.e. by b2 = rem
3
/(rem − 2m) [cf. Eq. (8.18) and Fig. 8.3]. [Figure produced with the notebook D.4.11]
Limit b → b+
c
From Eqs. (8.30), we have the following limits:
1 1
lim+ un = − and lim+ up = lim+ ua = . (8.93)
b→bc 6 b→bc b→bc 3
It follows then from expressions (8.89) that
r
1 1
lim+ ϕb (uem ) = arcsin 2uem + and lim ϕb (0) = arcsin √ . (8.94)
b→bc 3 b→b+
c 3
Besides, limb→b+c k = 1 [cf. Eq. (8.57)] and, for any ϕ ∈ [0, π/2), the definition (8.54) of F leads
to Z ϕ Z ϕ
dϑ dϑ 1 + sin ϕ
F (ϕ, 1) = p = = ln . (8.95)
0 1 − sin2 ϑ 0 cos ϑ cos ϕ
The limits (8.94) result then in
√ √ ! √ !
3 + 6uem + 1 3+1
lim F (ϕb (uem ), k) = ln p and lim F (ϕb (0), k) = ln √ .
b→b+c 2(1 − 3uem ) b→b+
c 2
(8.96)
For λ̃em > λ̃p , inserting these formulas into (8.88) leads to
p !
2 + 3uem + 3(6uem + 1)
φ∞ ∼+ φem + ϵL ln √ (λ̃em > λ̃p ). (8.97)
b→bc (2 + 3)(1 − 3uem )
8.4 Asymptotic direction from some emission point 263
Note that the constraint (8.85) is equivalent to 1 − 3uem > 0, so that the above formula is well
posed.
For the case λ̃em < λ̃p , φ∞ is given by Eq. (8.92) and we need to determine the behavior of
K(k) to conclude. Actually, when k → 1, K(k) is diverging with the following behavior9 :
4
lim K(k) − ln √ = 0. (8.98)
k→1 1 − k2
√
To relate 1 − k 2 to b − bc , let us introduce the small parameter ε > 0 such that
1
up =: − ε. (8.99)
3
The relation Pb (up ) = 0, once expanded to second order in ε2 , leads to the following relation
between b − bc and ε: √
b − bc 81 3 2
= ε + O(ε3 ). (8.100)
m 2
Besides, from the expressions (8.33) of un and ua in terms of up , we get, still to the second order
in ε,
1 1
un = − + 2ε2 + O(ε3 ) and ua = + ε − 2ε2 + O(ε3 ).
6 3
Substituting these values in the expression (8.50) of the modulus k and expanding to the first
order in ε leads to
k = 1 − 2ε + O(ε2 ),
from which √ p √
1 − k2 ∼ 1 − (1 − 4ε) ∼ 2 ε.
ε→0 ε→0
The property (8.98) then leads to
2 4
2K(k) ∼ 2 ln √ ∼ ln .
ε→0 ε ε→0 ε
Using this result, as well as (8.96), in Eq. (8.92) yields
" √ √ ! √ !#
4 3 + 6uem + 1 3+1
φ∞ ∼+ φem + ϵL × 2 ln − ln p − ln √
b→bc ε 2(1 − 3uem ) 2
16 2(1 − 3uem ) 2
∼ φem + ϵL ln √ √ √ .
b→b+
c ε2 ( 3 + 6uem + 1)2 ( 3 + 1)2
Note that we have used the limits (8.93) to evaluate the prefactor in Eq. (8.92) as
p
2/(ua − un ) ∼
2/(1/3 − (−1/6)) = 2. Expressing ε in terms of b − bc via Eq. (8.100) leads to the final
p 2
formula:
√ !
648(2 3 − 3)(1 − 3uem ) m
φ∞ ∼+ φem + ϵL ln p × (λ̃em < λ̃p ). (8.101)
b→bc 2 + 3uem + 3(1 + 6uem ) b − bc
9
√
See e.g. Eq. (112.1) of Ref. [79] with k ′ = 1 − k 2 or Eq. (17.3.26) of Ref. [3] with m = k 2 and m1 = 1 − k 2 .
264 Null geodesics and images in Schwarzschild spacetime
The prefactor of m/(b − bc ) in the logarithm is plotted as a function of rem = m/uem in Fig. 8.22
below.
The value of the total change ∆φ along the complete geodesic history [cf. Eq. (8.77)] is
obtained by taking the limit uem → 0 (i.e. rem → +∞) in this formula. We get
√ !
648(7 3 − 12)m
∆φ ∼+ ϵL ln . (8.102)
b→bc b − bc
We recover the result (8.84): ∆φ is diverging when b → b+ c . Moreover, Eq. (8.102) specifies this
divergence as logarithmic in b − bc . More generally, Eq. (8.101) shows that φ∞ − φem diverges
logarithmically in b − bc when b → b+ c , whatever the position of the emission point.
Let us express ∆φ is terms of the deflection angle Θ and the winding number n through
Eqs. (8.80) and (8.81):
∆φ = Θ + ϵL π + 2πn. (8.103)
We deduce then from (8.102) that
√
b − bc ∼+ 648(7 3 − 12)e−π m e−ϵL Θ−2π|n| . (8.104)
b→bc | {z }
≃3.482284
In the above writing, we have used the fact that the sign of n is the same as that of L, so that
ϵL n = |n|.
Remark 1: A check of Eq. (8.102) is obtained by comparing it with Eq. (268) in Chap. 3 of Ref. [107],
where δD = b−bc and Θ is the same as ours, or with Eq. (7.4.54) of Ref. [204], where ∆ℓ = (b−bc )/(2m).
Historical note : Equation (8.104) has been first derived by Charles Galton Darwin – the grandson of
the famous naturalist Charles Robert Darwin – in 1959 [155] (cf. Eqs. (31) and (32) of Ref. [155], where
µ = Θ).
8.4.2 Asymptotic direction for b < bc
When the impact parameter b is lower than the critical value, the null geodesic L has no
periastrion and φem and φ∞ are given by Eq. (8.68) with ϵin = −1 (outgoing motion):
φem = φ∗ + ϵL Φb (uem ) and φ∞ = φ∗ + ϵL Φb (0). (8.105)
We have then
φ∞ = φem + ϵL [Φb (0) − Φb (uem )] . (8.106)
Φb (0) is given by Eq. (8.74):
1
Φb (0) = p [K(k) − F (ϕb (0), k)] . (8.107)
2(u∗ − un )
If uem < u∗ , Φb (uem ) is given by Eq. (8.74) as well, so that Eq. (8.106) becomes
ϵL (b < bc )
φ∞ = φem + p [F (ϕb (uem ), k) − F (ϕb (0), k)] , (8.108)
2(u∗ − un ) (uem < u∗ )
8.4 Asymptotic direction from some emission point 265
with ϕb (uem ) and ϕb (0) given by Eq. (8.72):
|u∗ − uem | u∗
ϕb (uem ) = arccos and ϕb (0) = arccos . (8.109)
u∗ + uem − 2un u∗ − 2un
If uem > u∗ , then Φb (uem ) is given by Eq. (8.76). Combining with Eq. (8.107), we can then
write Eq. (8.106) as
ϵL (b < bc )
φ∞ = φem + p [2K(k) − F (ϕb (uem ), k) − F (ϕb (0), k)] ,
2(u∗ − un ) (uem > u∗ )
(8.110)
where ϕb (uem ) is still given by Eq. (8.109). φ∞ − φem is plotted as a function of b in Fig. 8.17
for various values of rem = m/uem . For rem ≫ m (cf. the red curve rem = 100 m for b < bc ),
we note that φ∞ − φem ≃ 0, as expected. As for the case b > bc , we also note that φ∞ − φem is
diverging when b tends to bc . Let us quantify this diverging behavior:
Limit b → b−
c
When b → b−
c , we have the following limits [cf. Eqs. (8.35) and (8.67)]:
1 1
lim− un = − and lim− u∗ = .
b→bc 6 b→bc 3
Consequently, Eq. (8.109) yields
|1 − 3uem | 1 π
lim ϕb (uem ) = arccos and lim− ϕb (0) = arccos = . (8.111)
b→b−c 2 + 3uem b→bc 2 3
Moreover, from expression (8.73) for k and the above limits for u∗ and un , we have (see also
Fig. 8.13)
lim− k = 1.
b→bc
Given the values (8.111) and expression (8.95) for the elliptic integral F when k = 1, we get
p !
2 + 3uem + 3(1 + 6uem )
lim F (ϕb (uem ), k) = ln (8.112a)
b→b−c |1 − 3uem |
√
lim− F (ϕb (0), k) = ln(2 + 3). (8.112b)
b→bc
When uem < u∗ , inserting these formulas into Eq. (8.108) leads to
p !
2 + 3uem + 3(6uem + 1) 1
φ∞ ∼− φem + ϵL ln √ uem < . (8.113)
b→bc (2 + 3)(1 − 3uem ) 3
We have written uem < 1/3 because u∗ → 1/3 when b → b− c . For this reason, we also get
rid of the absolute value around 1 − 3uem . Note that the value of φ∞ given by Eq. (8.113) is
266 Null geodesics and images in Schwarzschild spacetime
identical to that given by Eq. (8.97), which was obtained for b → b+ c and an emission point
beyond the periastron. This is not surprising if one invokes the continuity around b = bc of
null geodesics as regards their part beyond the periastron for b > bc and outside r = 3m for
b < bc . Indeed, the geodesics with b → b+c differ significantly from those with b → bc only on
−
parts including the periastron. This continuity appears clearly on Fig. 8.17: the curves with
b < bc and rem > 3m have a continuous prolongation with the dashed curves, which are the
curves with b > bc without any periastrion on the path from the emission point to infinity.
For uem > u∗ , we shall use formula (8.110), with K(k) having the diverging behavior (8.98)
since limb→b−c k = 1. To express K(k) in terms of |b − bc |, let us introduce the small parameter
ε > 0 such that
1
un = − − ε. (8.114)
6
The relation Pb (un ) = 0, once expanded to first order in ε, leads then to the following relation
between bc − b and ε: √
bc − b 81 3
= ε. (8.115)
m 4
Besides, by expanding formula (8.66) at first order in ε, we get
1
u∗ = + ε + O(ε2 ).
3
Substituting this value, as well as (8.114), into expression (8.73) for k yields
ε
k = 1 − + O(ε2 ),
4
so that
√
r r
ε ε
1 − k2 ∼ 1− 1− ∼
ε→0 2 ε→0 2
The property (8.98) then leads to
√ !
4 2 32
2K(k) ∼ 2 ln √ ∼ ln .
ε→0 ε ε→0 ε
Substituting this expression for K(k), as well as Eq. (8.112) for F (ϕb (uem ), k) and F (ϕb (0), k),
into Eq. (8.110) results in
" ! #
√
p
ϵL 32 2 + 3uem + 3(1 + 6uem )
φ∞ ∼− φem + q ln ε − ln − ln(2 + 3)
b→bc
2 31 + 16 3uem − 1
√ !
32 (2 − 3)(3uem − 1)
∼ φem + ϵL ln p .
b→b−c ε 2 + 3uem + 3(1 + 6uem )
Finally, using (8.115) to let appear bc − b instead of ε, we get
√ !
648(2 3 − 3)(3uem − 1) m 1
φ∞ ∼− φem + ϵL ln p × uem > . (8.116)
b→bc 2 + 3uem + 3(1 + 6uem ) bc − b 3
8.5 Images 267
Figure 8.18: Link between the observation angle b̂ and the impact parameter b for an asymptotic observer O
located at rO ≫ m.
Hence we recover a logarithmic divergence of φ∞ − φem when b tends to bc by lower values.
Remark 2: The prefactor of m/(bc − b) in formula (8.116) is exactly the opposite of the prefactor of
m/(b − bc ) in formula (8.101). It is plotted as a function of rem = m/uem in Fig. 8.22 below.
The upper bound on uem is 1/2, for this corresponds to a source just outside the black hole
event horizon at r = 2m. Let us evaluate φ∞ in this limit; setting uem → 1/2 in Eq. (8.116)
results in, after simplification,
√ !
648(26 3 − 45)m 1
φ∞ ∼− φem + ϵL ln uem → . (8.117)
b→bc bc − b 2
Remark 3: As a check, Eq. (8.117) agrees with Eq. (4) in Ref. [234].
8.5 Images
Being the worldlines of photons, null geodesics are the key ingredient in determining images
as seen by some observer of emitting material around a black hole. Computing such images is
of great interest, especially after the first image of a black hole vicinity obtained by the Event
Horizon Telescope team in 2019 [6, 83]. As stated in Sec. 8.1, we shall differ the discussion of
that image to the chapter dealing with rotating black holes (Sec. 12.5.3).
8.5.1 The asymptotic observer
Let us consider some “far-away” static observer, i.e. an observer O located at r = rO ≫ m.
Without any loss of generality, we may assume that O is located at θ = π/2 and φ = 0
(cf. Fig. 8.18). Furthermore, we suppose that O is equipped with an optical device (telescope)
pointing in the direction from O to the black hole, which is the x-axis in terms of the coordinates
(x, y) := (r cos φ, r sin φ). Images are formed by null geodesics reaching O’s screen with a
angle b̂ with respect to the telescope axis (in O’s frame) within the telescope aperture. The
angle b̂ is actually related to the geodesic impact parameter b by sin b̂ = b/rO (cf. Fig. 8.18),
which, for large rO , can be rewritten as
b
b̂ = . (8.118)
rO
268 Null geodesics and images in Schwarzschild spacetime
10
1
5
3
S
y/m
-5
2
-10
-10 -5 0 5 10 15 20
x/m
Figure 8.19: Null geodesics in Schwarzschild spacetime with φ∞ = 0 and various values of the impact
parameter b. The green ones have b ranging from 0 to 12m, by steps of 0.8m, while the olive ones have b close to
bc , namely b/m ∈ {5.0, 5.2, 5.4}. S is a luminous point source and three of its images for an observer at y = 0
and x → +∞ are obtained by following the drawn null geodesics through S. [Figure produced with the notebook
D.4.13]
A second parameter characterizing any incoming null geodesic is a polar angle α ∈ [0, 2π) so
that (b̂, α) are spherical coordinates on O’s celestial sphere, the North pole (b̂ = 0) of which
coinciding with the direction to the black hole. The image on O’s screen at the telescope output
may be then parameterized by polar coordinates (b̂, α), with b̂ representing the distance to
the screen’s center. Other projections from the celestial sphere to the planar screen may be
considered but this is rather unimportant here, given the small field of view of the telescope. In
what follows, we shall use directly b, instead of b̂, as the radial coordinate in the screen plane.
According to Eq. (8.118), this amounts to drop the constant scale factor 1/rO .
Moreover, we shall consider the asymptotic observer limit: rO → ∞. In that limit, null
geodesics that reach O must have
φ∞ = 0 mod 2π. (8.119)
8.5 Images 269
8.5.2 Images of a point source
Qualitative analysis from a concrete example
To understand the formation of images on O’s screen, a bunch of null geodesics with φ∞ = 0
is drawn in Fig. 8.19. O is located in the far right of this figure and all the drawn geodesics
eventually hit O’s screen. Let us consider a luminous point source S. From the drawing of
Fig. 8.19, three null geodesics of the φ∞ = 0 family goes through S. They give rise to three
distinct images of S on O’s screen, which are represented by the open circles labelled 1 to 3 in
the right part of the figure. These three geodesics can be distinguished by their deflection angle
Θ and their winding number n (cf. Sec. 8.3.6). The geodesic giving image 1 is the less deflected
one: it has L < 0, φ−∞ ≃ 5π/4, ∆φ ≃ −5π/4, n = 0 and hence, from Eq. (8.81), a deflection
angle Θ ≃ −π/4. The geodesic giving image 2 has L > 0, φ−∞ ≃ −4π/3, ∆φ ≃ 4π/3,
n = 0 and Θ ≃ π/3. Finally, the geodesic giving image 3 has L < 0, φ−∞ ≃ 2π + 7π/6,
∆φ ≃ −7π/6, n = −1 and Θ ≃ −π/6. It it the most deflected of the three, having a winding
number of n = −1, i.e. it makes a full turn around the black hole before reaching O. We note
that the more deflected the geodesic, the closer the image with respect to the circle b = bc on
O’s screen.
For the sake of clarity, only three images of the source S have been depicted in Fig. 8.19,
but we are going to see that there is actually an infinite number of them: two per value of |n|
(the number of turns around the black hole) — one with L > 0 and one with L < 0.
Remark 1: In Fig. 8.19, the source S is rather close to the black hole but it is worth to stress that the
multiple character of the images of a given source is not due to the proximity between the source and
the black hole. Multiple images are formed for any source, even very far ones. For instance, the source
can lie behind the observer, i.e. one can have xS > xO . This is clear if we consider the lower-right panel
of Fig. 8.15 (case b = 5.355 m): a source with xS > xO ≫ m and y = 5.355 m gives an image on O’s
screen at b = 5.355 m.
Link between the emission angle and the impact parameter
In the above discussion, we have parameterized the null geodesics arriving on the observer’s
screen by the impact parameter b. Let us relate the latter to the emission angle in the source
frame. To this aim, we consider that the point source S, as depicted in Fig. 8.19, is actually the
trace in the plane of the figure of the worldline of a static observer equipped with a source of
light, who we shall call the emitter and denote by Oem . Furthermore, we suppose that the rest
frame of Oem is an orthonormal tetrad uem , e(r) , e(θ) , e(φ) such that the first spacelike vector,
e(r) , is always pointing towards the black hole (cf. Fig. 8.20). Note that the first vector of the
270 Null geodesics and images in Schwarzschild spacetime
Figure 8.20: Part (e(r) , e(φ) ) of the vector frame of the emitter Oem and emission angle η. The coordinates
(x, y) are defined in terms of the Schwarzschild-Droste coordinates by x = r cos φ and y = r sin φ. P is the
photon momentum as measured by Oem .
tetrad is necessarily the 4-velocity uem of Oem . More precisely, we set
−1/2
2m
uem = 1 − ∂t (8.120a)
rem
1/2
2m
e(r) = − 1 − ∂r (8.120b)
rem
−1
e(θ) = −rem ∂θ (8.120c)
−1
e(φ) = −rem ∂φ , (8.120d)
where (∂t , ∂r , ∂θ , ∂φ ) is the natural basis associated with Schwarzschild-Droste coordinates
and rem is the (constant) r-coordinate of Oem . It is immediate that uem , e(r) , e(θ) , e(φ) is
an orthonormal tetrad with respect to the Schwarzschild metric (6.14). Moreover, having a
4-velocity uem collinear to the Killing vector ∂t makes the observer Oem static [cf. Eq. (7.28)].
The generic 4-momentum p of a photon evolving in the equatorial plane θ = π/2 is given
by Eqs. (7.10) and (8.1)-(8.3):
" −1 #
2m b
(8.121)
p
p=E 1− ∂t ± 1 − b2 U (r) ∂r + ϵL 2 ∂φ ,
r r
where we have used Eq. (8.14) to let appear U (r) and Eqs. (8.12)-(8.13) to express10 L in terms
of E and b. Taking the value of p at the emission point S and using (8.120) to express ∂t , ∂r
and ∂φ in terms of Oem ’s orthonormal tetrad, we get
p = εem (uem + n) , (8.122)
10
Note that formula (8.121) assumes E ̸= 0, otherwise b := |L|/E would be ill-defined. As discussed in
Sec. 8.2.3, the property E ̸= 0, and even E > 0, holds as soon as the photon travels in the black hole’s exterior.
8.5 Images 271
10.0 rem = 2.1 m
rem = 2.5 m
rem = 3.0 m
rem = 3.5 m
5.0 rem = 6.0 m
rem = 20.0 m
²L b/m 0.0
−5.0
−10.0
0 1π
2
π 3π
2
2π
η
Figure 8.21: Impact parameter b of null geodesics (multiplied by ϵL = ±1) as a function of the emission
angle η in the emitter’s rest frame, for various values of the r-coordinate rem of the static emitter. Dashed curves
correspond to values of rem lower than 3m. The two horizontal grey lines marks b = bc ≃ 5.196 m. [Figure
produced with the notebook D.4.14]
where −1/2
2m
εem := −uem · p = E 1 − (8.123)
rem
is the energy of the photon as measured by Oem (cf. Sec. 1.4) and
(8.124)
p p
n := ± 1 − b2 U (rem ) e(r) − ϵL b U (rem ) e(φ)
is a unit spacelike vector orthogonal to uem : n · n = 1 and uem · n = 0.
In view of (8.122) and uem · n = 0, the linear momentum of the photon as measured by
Oem is P = εem n [cf. Eq. (1.22) and Fig. 8.20]. We conclude that the unit vector n gives the
direction of emission of the photon in Oem ’s rest frame. Let us then define the emission angle
η by
n = cos η e(r) + sin η e(φ) . (8.125)
Equation (8.124) yields immediately
and (8.126)
p p
cos η = ± 1 − b2 U (rem ) sin η = −ϵL b U (rem ),
from which we can express the impact parameter b and in terms of the emission angle η:
sin η
b = −ϵL p . (8.127)
U (rem )
Remark 2: Equation (8.126) is well posed because U (r) ≥ 0 for r > 2m [cf. Eq. (8.14)] and the equation
of motion (8.11) implies b−2 − U (rem ) ≥ 0, from which we get 0 ≤ b2 U (rem ) ≤ 1.
272 Null geodesics and images in Schwarzschild spacetime
80
60
40
A(rem )
20
-20
0 5 10 15 20 25 30 35 40
rem /m
Figure 8.22:
√ Function A(rem ) defined by Eq. (8.130). The dashed line marks the horizontal asymptote
A = 648(7 3 − 12) ≃ 80.58246. [Figure produced with the notebook D.4.14]
The variation of b in terms of η and rem , as given by formula (8.127), is depicted in Fig. 8.21.
We notice that the smallest range of b is [0, bc ) and is reached for rem = 3m, which is not
surprising in view of (8.127) since the maximum of U precisely occurs at r = 3m (cf. Fig. 8.1).
Infinite sequence of images
It appears clearly from Fig. 8.21 that, whatever the location rem of the emitter, including
locations inside the photon sphere (rem < 3m), there always exist four emission angles η giving
birth to impact parameters b in the vicinity of the critical value bc . These four angles can be
gathered in two pairs of angles of opposite directions: (η1 , η1 + π) and (η2 , η2 + π). In each pair,
only a single emission direction is prior to the periastron for null geodesics with b > bc and
only a single emission direction is outgoing and reaches infinity for b < bc . In other words, on
the four values of η, only two of them define geodesics that may reach the asymptotic observer
O.
Let φem ∈ [0, 2π) be the φ coordinate of the emitter Oem . A null geodesic emitted by Oem
reaches the asymptotic observer O, who is located at φ = 0, iff φ∞ = 0 mod 2π [Eq. (8.119)],
i.e. iff
φ∞ = 2πϵL n, n ∈ N. (8.128)
The above formula takes into account the fact that the sign of φ∞ is that of L, hence the factor
ϵL and n ≥ 0. From Fig. 8.17, we see that values n ≥ 1 can be reached iff either (i) rem > 3m, b
is close to bc from above and λ̃em < λ̃p or (ii) rem < 3m and b is close to bc from below. In the
first case, we may use the approximate formula (8.101) with uem = m/rem , while in the second
case formula (8.116) is relevant. Inserting (8.128) in either of these formulas, we get
m
2πϵL n = φem + ϵL ln A(rem ) , (8.129)
b − bc
8.5 Images 273
with (cf. Remark 2, page 267)
√
648(2 3 − 3)(rem − 3m)
A(rem ) := p . (8.130)
2rem + 3m + 3rem (rem + 6m)
Note that A(rem ) > 0 for rem > 3m and A(rem ) < 0 for rem < 3m. Taking into account that
b − bc > 0 for case (i) and b − bc < 0 for case (ii), we check that the argument of the logarithm
in Eq. (8.129) is always positive. Equation (8.129) can be solved for b:
b = bc + mA(rem )eϵL φem −2πn ,
which we may write b = b+
n or bn , denoting the solution with ϵL = +1 by bn and that with
− +
ϵL = −1 by b−
n:
b+
n = bc + mA(rem ) e
φem −2πn
, n ∈ N∗ (8.131a)
b−
n := bc + mA(rem ) e
−φem −2πn
, n ∈ N∗ . (8.131b)
The function A(rem ) is plotted in Fig. 8.22. It varies monotonically between
√ √
lim A(rem ) = −648(26 3 − 45) and lim A(rem ) = 648(7 3 − 12) .
rem →2m | {z } rem →+∞ | {z }
≃−21.59201 ≃80.58246
One shall recall that formulas (8.131) have been derived under the assumption |b − bc | ≪ m.
Given the amplitude of A(rem ) shown in Fig. 8.22 and the value e−2π ≃ 1.9 × 10−3 , this is valid
for any n ≥ 1, except maybe for b+ n with n = 1 and φem large (i.e. close to 2π).
Via Eq. (8.127), the two sequences (8.131) of the impact parameter b correspond to two
sequences of the emission angle η:
(8.132a)
p
sin ηn+ := − U (rem ) bc + mA(rem )eφem −2πn , n ∈ N∗
(8.132b)
p
sin ηn− := U (rem ) bc + mA(rem )e−φem −2πn , n ∈ N∗ .
Note that 0 ≤ bc U (rem ) ≤ 1 and bc U (rem ) = 1 ⇐⇒ rem = 3m. As discussed above (cf.
p p
Fig. 8.21), for rem ̸= 3m, there are always some solutions for ηn+ and ηn− for n sufficiently large
(in practice, n ≥ 1). We therefore conclude:
Property 8.3: images of a point source
Any static point source S give birth to two infinite sequences of images on the screen of
the asymptotic observer O, at the effective distances from the screen’s center (b+ n )n∈N∗
and (b− )
n n∈N ∗ , as given by Eq. (8.131). If S, O and the black hole are not aligned, i.e. if
φem ̸∈ {0, π}, all the images of S are aligned on O’s screen: they all lie in the plane defined
by S, O and the black hole (plane of Fig. 8.19). If S is located outside the photon sphere
(rem > 3m), all images lie outside the circle b = bc on O’s screen, while if S is located inside
the photon sphere (rem < 3m), all images lie inside that circle. In both cases, the images of
the sequence (b+ n )n∈N∗ (positive L) are located on the opposite side of those of the sequence
(bn )n∈N∗ (negative L). The higher n, the closer the images b+
−
n and bn to the circle b = bc .
−
Actually, according to the laws (8.131), each sequence converge exponentially fast to that
274 Null geodesics and images in Schwarzschild spacetime
10
1
5
3
A
y/m
-5 4
2
-10
-10 -5 0 5 10 15 20
x/m
Figure 8.23: Same as Fig. 8.19 but underlining four null geodesics from a point source A located on the x-axis,
i.e. aligned with the black hole and the observer on the far right. [Figure produced with the notebook D.4.13]
circle. From n to n + 1, the distance to the circle is reduced by a factor e−2π ≃ 1.9 × 10−3 .
Remark 3: The alignment of the images is a direct consequence of the spherical symmetry of Schwarzschild
spacetime.
Remark 4: The reader may be puzzled by the compatibility between an infinite number of images from
a single source and the conservation of energy. There is actually no issue here because one can show
that the images are fainter as n increases. So in practice, only a few images would be visible.
8.5.3 Aligned source and Einstein rings
In the special case where the source is located on the x-axis, the images form a series of
concentric circles accumulating from above on the circle b = bc . This is illustrated by the
source A in Fig. 8.23. Four geodesics through the source are drawn, given birth to four images
that are symmetric with respect to the x-axis, labelled 1, 3 and 2, 4 respectively. Since in this
case the plane of the figure is not privileged (one cannot speak about the plane defined by
A, O and the black hole, since they are aligned); we deduce by rotation about the x-axis that
A generates images all along circles centered on the origin in O’s screen. Images 1 and 2 in
Fig. 8.23 lie actually on the same circle, as well as images 3 and 4. As for the non-aligned case,
8.5 Images 275
Fig. 8.23 shows only a limited number of images, but there is actually an infinite number of
such images: the full image of A on O’s screen is composed by a infinite sequence (Cn )n∈N
of concentric circles that accumulate exponentially fast onto the circle of radius b = bc . Each
circle Cn has a radius bn that is given by Eq. (8.131) with φem = 0 or π (case of Fig. 8.23), so that
n = bn =: bn (case φem = 0) or bn = bn−1 =: bn (case φem = π). The outermost circle, C0 , is
− −
b+ +
called the Einstein ring of A. It is the only significant ring in astronomical images involving
gravitational lensing by a massive foreground source (not necessarily a black hole), which are
in the context |xA | ≫ m and b ≫ m. We may call Cn the nth Einstein ring of A. For n ≥ 1,
Cn is also called a relativistic Einstein ring of A. Indeed, these rings exist only in the case of
a very relativistic central object, so that photons can wind around it. On the contrary C0 exists
even for non-relativistic objects. In particular, in astronomical observations of Einstein rings
performed up to now, the central deflecting object is a galaxy or a galaxy cluster, both being
highly non-relativistic (very low compactness).
Historical note : The infinite sequence of images of a point source located at r > 3m, with accumulation
just outside the circle b = bc , has been predicted in 1959 by Charles Galton Darwin [155] (cf. historical
note on p. 264). Darwin has also derived the sequence of relativistic Einstein rings for aligned sources.
8.5.4 Black hole shadow
Let us determine the image on observer O’s screen when all the sources of light are very far,
both from the black hole and from O. We may think of many stars shining on the celestial
sphere. To simplify the problem, we shall assume that the black hole and the observer are
surrounded by a distant sphere S that is uniformly bright. In such a setting, all values of
b > bc on O’s screen, whatever the polar angle α, correspond to a null geodesic that originates
from the shining sphere S , while this is not possible for b ≤ bc . This is clear on Fig. 8.19: when
traced backward from the right (O position), (i) null geodesics with b > bc eventually end up
far away from the black hole, i.e. necessarily on S , (ii) null geodesics with b < bc end up
infinitely close to the black hole horizon and (iii) null geodesics with b = bc roll up indefinitely
around the photon sphere (cf. Figs. 8.7 and 8.8).
The same conclusion can be reached by considering the effective potential diagram in
Fig. 8.1. For concreteness, imagine that O is located at r = 15m and that the shining sphere S
has a coordinate radius r = 20m. These values allow us to put O and S in Fig. 8.1 according
to our settings (S is encompassing both O and the black hole), but one shall keep in mind
that both O and S are assumed to lie at much larger values of r (fulfilling r ≫ m). Only
two kinds of geodesic trajectories in Fig. 8.1 diagram can reach r = 15m (O’s screen) with
increasing values of r (the direction of observation): those of type 1 and those of type 3. The
latter ones originate necessarily from a region between the black hole horizon and O’s location,
i.e. a region that does not intersect S . On the contrary, all the geodesics of type 1 must have
had r = 20m in their past history, i.e. we may consider that they all have been emitted by S
inwards and have bounced on the potential barrier before reaching O’s screen in their outward
motion.
Both reasonings, that based on Fig. 8.19 and that based on Fig. 8.1, led us to conclude that
276 Null geodesics and images in Schwarzschild spacetime
Figure 8.24: Shadow of a Schwarzschild black hole. The dashed grey circle delimitates the dark disk that would
appear on the screen of an observer looking at a black√ sphere of radius r = 2m in Minkowski spacetime. The
ratio between the radii of the two disks is bc /2m = 3 3/2 ≃ 2.60.
Property 8.4: shadow of a Schwarzschild black hole
For a distant observer, the image of a uniformly bright sphere S surrounding the black
hole and the observer is a bright area fulfilling the observer’s screen, except for a central
black disk of radius b = bc (cf. Fig. 8.24). This disk is called the black hole shadow.
Historical note : In his 1921 treatise [340], Max von Laue noted√that an emitting sphere of radius
r0 such that 2m < r0 < 3m will appear as having a radius of 3 3m (= bc ) to a distant observer,
independently of the value of r0 . Maybe the first mention of a shadow lies in the article by Charles
Galton Darwin in 1959 [155]. Actually, Darwin imagined that the source of the Schwarzschild metric
is a massive point particle, called by him the “sun”, and wrote “suppose that there is a roughly uniform
star-field all round the sky, and consider what will be seen and mapped from the telescope... To the area
inside this circle (the circle b = bc ) only the sun itself can contribute light. The most obvious assumption
is that its rays would emerge in straight lines, so that there would be a brilliant point of light surrounded
by blackness”. In the full black hole context, the first computation of the shadow has been presented
by James M. Bardeen at the famous 1972 Les Houches Summer School [38]; the shadow is called the
apparent shape of the black hole by Bardeen and the computation has been performed for the Kerr black
hole, which generalizes the Schwarzschild black hole to the rotating case (cf. Chap. 10); this Kerr black
hole shadow will be discussed in details in Sec. 12.4.
To go further, see the review article [144].
8.5 Images 277
Figure 8.25: Image of an accretion disk around a Schwarzschild black hole, as seen by a distant observer, with
various inclination angles: ι = 0 (upper left), ι = π/6 (upper right), ι = π/3 (lower left) and ι = 3/2 (lower
right). [Figure produced by Gyoto with the input files given in Sec. E.2.1]
8.5.5 Image of an accretion disk
A realistic “source of light” in the vicinity of a black hole is an accretion disk [2, 316]. Actually
most of astronomical observations of black holes in the electromagnetic domain are measure-
ments about an accretion disk, either a spectrum or an image, like the image of M87* released
in 2019 by the Event Horizon Telescope collaboration [6]. The matter (mostly hydrogen)
constituting the accretion disk arises either from a companion star (stellar-mass black hole in a
binary system, like Cyg X-1, cf. Table 7.1) or from gas clouds in a galactic center (supermassive
black holes, like Sgr A* and M87*, cf. Table 7.1).
Figure 8.25 shows various views of an accretion disk around a Schwarzschild black hole
as seen by a distant observer. The accretion disk is a model developed by Novikov & Thorne
278 Null geodesics and images in Schwarzschild spacetime
[388] and Page & Thorne [396], lying in some plane around the black hole. It is geometrically
thin but optically thick. The inner radius of the disk is the innermost stable circular orbit
(ISCO) at r = 6m (cf. Sec. 7.3.3). The images shown in Fig. 8.25 have been computed by
the open-source ray-tracing code Gyoto [493] (cf. Appendix E). This code integrates the null
geodesic equations backward, starting from the observer’s screen. For Fig. 8.25, the observer is
located at r = 1000 m and at various inclination angles ι with respect to the accretion disk,
ranging from ι = 0 (disk seen face-on) to ι = 3/2 (disk seen almost edge-on). The colors in
Fig. 8.25 encode the flux at a fixed wavelength.
Let us first discuss the face-on view (upper-left panel of Fig. 8.25). To interpret it, some null
geodesics generating it have been depicted in Fig. 8.26. One may distinguish three images of
the accretion disk:
• the primary image, which is formed by the outermost bright part; it is generated by null
geodesics that suffers a small deflection from the disk to the observer; more precisely,
they are arising from the side of the disk facing the observer and have ∆φ = ±π/2 in
their trajectory plane (cf. orange curves in Fig. 8.26); the inner boundary of the primary
image is the image of the ISCO under these geodesics; it is located at b = 6.932 m on the
observer’s screen;
• the secondary image, which is the bright yellow ring separated from the inner boundary
of the primary image by the thick black annulus; this image is generated by null geodesics
that perform half a turn around the photon sphere before reaching the observer; more
precisely, they are arising from the side of the disk opposite to the observer and have
∆φ = ±3π/2 in their trajectory plane (cf. brown curves in Fig. 8.26); the inner boundary
of the secondary image is located at b = 5.479 m on the observer’s screen;
• the tertiary image, which is the thin faint innermost ring; this image is generated by
null geodesics that perform a full turn around the photon sphere before reaching the
observer; more precisely, they are arising from the side of the disk facing the observer
and have ∆φ = ±5π/2 in their trajectory plane (cf. red curves in Fig. 8.26); the inner
boundary of the tertiary image is located at b = 5.208 m on the observer’s screen.
As discussed in Sec. 8.5.2, there are actually an infinite number of images of the accretion disk
but they are more and more faint and only three of them are visible in Fig. 8.25. Moreover
these images are formed by exponentially thinner rings and accumulate near the circle b = bc .
In Figs. 8.25 and 8.26, the tertiary image corresponds to n = 2 for ϵL = 1 and n = 1 for
ϵL = −1 in Eq. (8.128) 11 , so that Eqs. (8.131) √ with φem = π/2 (b = b− 1 ) and φem = 3π/2
(b = b+
2 ) yield (b − b c )/bc = A(r em )e−5π/2
/(3 3). With r em = 6m, we have A(rem ) ≃ 30.3, so
that (b − bc )/bc ≃ 0.0013 for the tertiary image. This we may conclude that, up to a relative
accuracy of 10−3 , the interior of the tertiary image corresponds to the black hole shadow as
defined in Sec. 8.5.4,
For a nonzero inclination angle, the images shown in Fig. 8.25 exhibit some asymmetry
between the left and the right. This is due to the rotation of the disk at relativistic speed, which
11
This asymmetry of n with respect to ϵL is due to the convention φem ∈ [0, 2π); had we chosen φem ∈ (−π, π],
it would not be present.
8.5 Images 279
y/m
10
8
6
4
2
x/m
5 10 15
-2
-4
Figure 8.26: Formation of the primary (orange), secondary (brown) and tertiary (red) images of an accretion
disk when viewed face-on. The figure plane is orthogonal to the disk plane, the intersection between the two
planes being the y-axis, so that disk is depicted by the thick segments located at φ = π/2 and φ = 3π/2. As in
Fig. 8.25, the inner boundary of the disk is rin = 6 m (the ISCO), but the outer boundary has been truncated at
rout = 9 m for pedagogical purposes. The observer is located at x ≫ m and y = 0. The dashed green circle is the
photon orbit at r = 3 m. The lower panel shows the geodesics emerging from the inner and outer edges of the
upper part of the disk. [Figure produced with the notebook D.4.15]
280 Null geodesics and images in Schwarzschild spacetime
generates a strong Doppler boost, making the left part of the disk, which moves towards the
observer, much brighter than the right part, which recedes from the observer. Note that for the
three views with ι ̸= 0, the primary, secondary and tertiary images are still present but are
no longer circular rings. Note that the almost edge-on view is drastically different from what
would get in flat (Minkowski) spacetime, since for the latter the accretion disk would appear as
a very thin ellipsoidal shape elongated along the horizontal axis.
Historical note : The first computation of the image of an accretion disk around a Schwarzschild black
hole has been performed by Jean-Pierre Luminet in 1979 [348], (see Ref. [349] for an historical account).
The accretion disk was the same Page-Thorne model as that considered here and Luminet obtained an
image pretty close to that of the last quadrant of Fig 8.25, the difference lying in the inclination angle:
ι = 80◦ ≃ 1.40 rad versus ι = 1.50 rad in Fig 8.25. In 1991, Jean-Alain Marck computed the first movie
of an observer plunging into a Schwarzschild black endowed with an accretion disk [355, 356]. In 2007,
Alain Riazuelo computed very precise images of a Schwarzschild black hole in front of a realistic stellar
field [432, 433], which illustrate magnificently the multiple character of images of stars (Sec. 8.5.2) and
the concept of black hole shadow (Sec. 8.5.4) (cf. Appendix F).
Chapter 9
Maximal extension of Schwarzschild
spacetime
Contents
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
9.2 Kruskal-Szekeres coordinates . . . . . . . . . . . . . . . . . . . . . . . 282
9.3 Maximal extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
9.4 Carter-Penrose diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
9.5 Einstein-Rosen bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
9.6 Physical relevance of the maximal extension . . . . . . . . . . . . . . 318
9.1 Introduction
In the preceding chapters, the Schwarzschild spacetime was considered to be (MIEF , g), where
the manifold MIEF is covered by the ingoing Eddington-Finkelstein coordinates. It turns out
that this spacetime can be smoothly (actually analytically) extended to a larger spacetime,
(M , g) say, which on its turn cannot be extended, i.e. it is maximal. To construct (M , g), we
first introduce the so-called Kruskal-Szekeres coordinates on MIEF in Sec. 9.2. These coordinates
restore some symmetry between the ingoing radial null geodesics and the outgoing ones: both
types of geodesics appear as straight lines in a spacetime diagram built on Kruskal-Szekeres
coordinates, with a slope +1 for the outgoing family and −1 for the ingoing one. Then it
appears clearly that the outgoing radial null geodesics are artificially halted1 at some past
boundary of MIEF , while no curvature singularity is located there. This calls for an extension
of the spacetime, which is performed in Sec. 9.3, thanks to Kruskal-Szekeres coordinates. This
extension is maximal; in particular, all incomplete geodesics are so because they either terminate
to or emanate from a curvature singularity. In Sec. 9.4, we construct the Carter-Penrose diagram
1
Technically, one says that they are incomplete geodesics, cf. Sec. B.3.2.
282 Maximal extension of Schwarzschild spacetime
of the maximal Schwarzschild spacetime (M , g), via two compactified versions of the Kruskal-
Szekeres coordinates. We show that the simplest one, which is standard in the literature, does
not lead to a regular conformal completion of (M , g), as defined in Sec. 4.3. The second version,
built on Penrose-Frolov-Novikov compactified coordinates, achieves this goal. We use this
completion to show explicitly that the maximal spacetime contains a white hole, in addition
to the black one. In Sec. 9.5, we investigate the hypersurfaces of constant Kruskal-Szekeres
time, which connect two asymptotically flat regions of (M , g) through the co-called Einstein-
Rosen bridge. Finally, Sec. 9.6 discusses the physical relevance of the maximally extended
Schwarzschild spacetime.
9.2 Kruskal-Szekeres coordinates
9.2.1 Definition
ˆ
On the open set MI , let us consider the “double-null” coordinate system xα̂ = (u, v, θ, φ). It is
related to Schwarzschild-Droste coordinates (t, r, θ, φ) by Eqs. (6.21)-(6.22):
u = t − r − 2m ln r − 1 t = 1 (u + v)
2m 2
⇐⇒ (9.1)
v = t + r + 2m ln r − 1 r + 2m ln r − 1 = 1 (v − u).
2m 2m 2
Although r cannot be explicitly expressed in terms of (u, v), the function r 7→ r+2m ln 2m
r
−1
is invertible on (2m, +∞) (cf. Fig. 9.1), so that (9.1) does define a coordinate system on MI .
The range of (u, v) is R2 .
The above relations imply
dr dr dr2
du = dt − , dv = dt + and du dv = dt2 − .
1 − 2m
r
1 − 2m
r 1− 2m 2
r
The Schwarzschild metric (6.14) can be then rewritten as
2m
du dv + r2 dθ2 + sin2 θ dφ2 . (9.2)
g =− 1−
r
In this formula, r is to be considered as a function of (u, v), given by (9.1).
The metric components (9.2) are regular on MI . Having a look at Fig. 9.1, we realize that we
cannot extend this coordinate system to include the Schwarzschild horizon H , since r → 2m
is equivalent to v − u → −∞: if u (resp. v) were taking a finite value on H , we would have
v → −∞ (resp. u → +∞). This impossibility of extending to H is also reflected by the fact
that 2
1 2m
det gα̂ˆ β̂ˆ = − 1− r4 sin2 θ
4 r
vanishes for r → 2m, which would make g a degenerate bilinear form at r = 2m, while it is
not of course.
9.2 Kruskal-Szekeres coordinates 283
r ∗ /m
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
r/m
-1
-2
-3
-4
-5
-6
-7
-8
Figure 9.1: Function r∗ (r) = r + 2m ln r
2m − 1 (the tortoise coordinate, cf. Eq. (6.23)). It relates r to (u, v)
via r∗ (r) = (u − v)/2 [Eq. (9.1)].
On MI , r > 2m and (9.1) yields
r 1 r
r + 2m ln − 1 = (v − u) =⇒ er/2m − 1 = e(v−u)/4m , (9.3)
2m 2 2m
This motivates to introduce the coordinates (U, V ) defined on MI by
U := −e−u/4m
(9.4)
V := ev/4m .
Since the range of (u, v) is R2 , the range of U is (−∞, 0) and that of V is (0, +∞). We have
1 −u/4m 1 v/4m
dU = e du, dV = e dv and du dv = 16m2 e(u−v)/4m dU dV.
4m 4m
In view of Eq. (9.3), we get
−1
32m3 −r/2m
r −1
2 −r/2m 2m
du dv = 16m e −1 dU dV = e 1− dU dV.
2m r r
Substituting this expression in (9.2) yields the expression of the metric with respect to the
coordinates X α̂ := (U, V, θ, φ):
32m3 −r/2m
dU dV + r2 dθ2 + sin2 θ dφ2 . (9.5)
g=− e
r
284 Maximal extension of Schwarzschild spacetime
In this formula, r has to be considered as a function of (U, V ), whose implicit expression is
found by combining (9.3) and (9.4):
r
er/2m − 1 = −U V . (9.6)
2m
This relation takes a very simple form in terms of the tortoise coordinate (cf. Eq. (6.23)):
er∗ /2m = −U V. (9.7)
We notice that the factor (1 − 2m/r) has disappeared in the metric components (9.5), which
are thus perfectly regular as r → 2m.
The coordinates U and V are null coordinates, since u and v are (cf. Sec. 6.3.2) and Eq. (9.4)
shows that the hypersurfaces of constant U (resp. V ) obviously coincide with those of constant
u (resp. v). To cope with a timelike-spacelike coordinate system instead, let us introduce on
MI the pair (T, X) such that U is T retarded by X and V is T advanced by X:
U =T −X T = 1 (U + V )
2
⇐⇒ (9.8)
V =T +X X = 1 (V − U )
2
Since the range of U on MI is (−∞, 0) and that of V is (0, +∞), the range of (T, X) is ruled
by T < X, T > −X and X > 0. In other words, the coordinates (T, X) span the following
quarter of R2 (cf. Fig. 9.2):
MI : X > 0 and − X < T < X. (9.9)
The coordinates X α := (T, X, θ, φ) are called the Kruskal-Szekeres coordinates.
We have dU dV = (dT − dX)(dT + dX) = dT 2 − dX 2 , so that the metric components
with respect to the Kruskal-Szekeres coordinates are easily deduced from Eq. (9.5):
32m3 −r/2m
−dT 2 + dX 2 + r2 dθ2 + sin2 θ dφ2 . (9.10)
g= e
r
Here r is to be considered as a function of (T, X), which is implicitly defined by
r
e r/2m
− 1 = X2 − T 2 . (9.11)
2m
This relation is a direct consequence of (9.6) and (9.8). We may rewrite it as F (r/2m) = X 2 −T 2 ,
where F is the function defined by
F : (0, +∞) −→ (−1, +∞)
(9.12)
x 7−→ ex (x − 1).
The graph of F is shown in Fig. 9.3. We see clearly that F is a bijective map. In particular, F
induces a bijection between (1, +∞) (the range of r/2m on MI ) and (0, +∞) (the range of
9.2 Kruskal-Szekeres coordinates 285
T
3
I
X
3 2 1 1 2 3
Figure 9.2: Submanifold MI in the Kruskal-Szekeres coordinates (T, X): MI is covered by the Schwarzschild-
Droste grid (in blue): the solid lines have t = const (spaced apart by δt = m), while the dashed curves have
r = const (spaced apart by δr = m/2). [Figure generated by the notebook D.4.16]
X 2 − T 2 on MI , according to (9.9)). The inverse of F can be expressed in terms of the Lambert
function W0 [3, 142], which is defined as the inverse of x 7→ xex :
W0 : (−1/e, +∞) −→ (−1, +∞)
(9.13)
x 7−→ y such that yey = x.
Noticing that F (x) = ex (x − 1) = e × (x − 1)ex−1 , we may write F −1 = W , where W is the
rescaled Lambert function defined by
x
W (x) := W0 +1. (9.14)
e
By construction, W is a bijection (−1, +∞) → (0, +∞), which obeys
eW (x) (W (x) − 1) = x. (9.15)
Its graph is shown in Fig. 9.4.
Since F −1 = W , we may invert relation (9.11) to r = 2m W (X 2 − T 2 ). This allows
us to eliminate r from the metric expression (9.10); using Eq. (9.15) to substitute W (X 2 −
T 2 )eW (X −T ) by X 2 −T 2 +eW (X −T ) , we arrive at a fully explicit expression of Schwarzschild
2 2 2 2
286 Maximal extension of Schwarzschild spacetime
F(x)
7
6
5
4
3
2
1
x
0.5 1.0 1.5 2.0
1
Figure 9.3: Function F : x 7→ ex (x − 1), yielding X 2 − T 2 = F (r/2m), cf. Eq. (9.11).
W(x)
1.5
1.0
0.5
x
1 1 2 3 4 5
Figure 9.4: Rescaled Lambert function W = F −1 , defined by (9.14) and obeying eW (x) (W (x) − 1) = x.
metric in terms of the Kruskal-Szekeres coordinates:
2 4 2 2
2 2 2 2 2 2
g = 4m −dT + dX + W (X − T ) dθ + sin θ dφ .
X 2 − T 2 + eW (X 2 −T 2 )
(9.16)
The relation between the Kruskal-Szekeres coordinates and the Schwarzschild-Droste ones
is obtained by combining (9.8), (9.4) and (9.1):
1 1 v/4m −u/4m
1 (t+r∗ )/4m (r∗ −t)/4m
r∗ /4m t
T = (U + V ) = e −e = e −e =e sinh ,
2 2 2 4m
where r∗ is related to r by (6.23). Similarly X = er∗ /4m cosh(t/4m). In particular, we have
T t
= tanh . (9.17)
X 4m
Since Eq. (6.23) yields er∗ /4m = er/4m r/(2m) − 1, the above relations can be written as
p
p r
t X+T
T = er/4m 2m
− 1 sinh 4m t = 2m ln X−T
MI : ⇐⇒ (9.18)
X = er/4m p r − 1 cosh t
r = 2m W (X 2 − T 2 ).
2m 4m
9.2 Kruskal-Szekeres coordinates 287
Note that we have used the identity artanh x = 1/2 ln [(1 + x)/(1 − x)]. The curves of constant
t and constant r in the (T, X) plane are drawn in Fig. 9.2. The fact that the curves of constant
t are straight lines from the origin follow immediately from Eq. (9.17).
Remark 1: Given the properties of the cosh and sinh functions, it is clear on these expressions that the
constraints (9.9) are satisfied.
Remark 2: In expression (9.16) the metric components gT T and gXX depend on both X and T ; this
shows that neither ∂T nor ∂X coincide with a Killing vector. In other words, the coordinates (T, X)
are not adapted to the spacetime symmetries, contrary to the Schwarzschild-Droste coordinates or to
the Eddington-Finkelstein ones.
9.2.2 Extension to the IEF domain
We notice that the metric components (9.10) are perfectly regular at r = 2m. Therefore,
the Kruskal-Szekeres coordinates can be extended to cover the Schwarzschild horizon H .
Actually they can be extended to all values of r ∈ (0, 2m], i.e. to the whole domain of
the ingoing Eddington-Finkelstein coordinates: the manifold MIEF introduced in Sec. 6.3.3:
MIEF = MI ∪ H ∪ MII . Let us show this in detail. Back on MI , we can express the IEF
coordinate t̃ in terms of (T, X) by combining t̃ = v − r [Eq. (6.30)], v = 4m ln V [Eq. (9.4)]
and V = T + X [Eq. (9.8)]:
t̃ = 4m ln(T + X) − r. (9.19)
The above relation is a valid expression as long as T + X > 0. Besides, we already noticed that
the function F defined by (9.12) is a bijection from the range of r/2m on MIEF , i.e. (0, +∞),
to (−1, +∞), with the (0, +∞) part of the latter interval representing the range of X 2 − T 2
on MI . We may use these properties to extend the Kruskal-Szekeres coordinates to all MIEF
by requiring
t̃ = 4m ln(T + X) − r (9.20a)
r
er/2m − 1 = X 2 − T 2. (9.20b)
| 2m
{z }
F (r/2m)
The range of the coordinates (T, X) on MIEF is then ruled by
MIEF : T + X > 0 and X 2 − T 2 > −1,
which can be rewritten as
√
MIEF : −X < T < X 2 + 1. (9.21)
We deduce from (9.20) that
X + T = e(t̃+r)/4m
(9.22)
r
X − T = e(r−t̃)/4m −1 .
2m
288 Maximal extension of Schwarzschild spacetime
T
3
0
r=
2
t̃ = 2m
1
t˜= 0
X
3 2 1 1 2 3
t˜=
−
1 2m
Figure 9.5: Domain of ingoing Eddington-Finkelstein coordinates, MIEF = MI ∪ H ∪ MII , depicted in terms
of the Kruskal-Szekeres coordinates (T, X): the solid red curves have t̃ = const (spaced apart by δ t̃ = m), while
the dashed red curves have r = const (spaced apart by δr = m/2). [Figure generated by the notebook D.4.16]
Hence the relation between the ingoing Eddington-Finkelstein coordinates and the Kruskal-
Szekeres ones on MIEF :
h i
T = er/4m cosh 4m − 4m e−t̃/4m
t̃ r
t̃ = 2m [2 ln(T + X) − W (X 2 − T 2 )]
h i ⇐⇒
X = er/4m sinh t̃ + r e−t̃/4m
4m 4m
r = 2m W (X 2 − T 2 )
(9.23)
The various subsets of MIEF correspond then to the following coordinate ranges (cf. Fig. 9.5):
MI : X > 0 and − X < T < X (9.24a)
H : X > 0 and T = X (9.24b)
√
MII : |X| < T < X 2 + 1. (9.24c)
Since the relation between IEF coordinates and Kruskal-Szekeres ones is the same in MII
as in MI (being given by (9.23) in both cases), we conclude that expressions (9.10) and (9.16)
for the metric components with respect to Kruskal-Szekeres coordinates are valid in all MIEF .
Let us determine the relation between the Kruskal-Szekeres coordinates and the Schwarz-
schild-Droste ones in MII . Since r < 2m in MII , Eq. (6.32) gives
r
r
MII : e t̃/4m
=e t/4m
1− ,
2m
9.2 Kruskal-Szekeres coordinates 289
T
3
0
r=
2
1
II I
X
3 2 1 1 2 3
Figure 9.6: Schwarzschild-Droste coordinates in MSD = MI ∪ MII depicted in terms of the Kruskal-Szekeres
coordinates (T, X): the solid blue curves have t = const (spaced apart by δt = m), while the dashed blue curves
have r = const (spaced apart by δr = m/2). [Figure generated by the notebook D.4.16]
so that (9.22) can be rewritten as
r
(t+r)/4m r
1−
X +T = e
2m
MII : r
r
X − T = −e(r−t)/4m 1 − .
2m
We obtain then
r t T +X
T = er/4m 1 −
p
2m
cosh 4m t = 2m ln
T −X
MII : ⇐⇒ (9.25)
r t
X = er/4m p1 −
r = 2m W (X 2 − T 2 ).
2m
sinh 4m
This is to be compared with (9.18). The curves of constant t and constant r in the (T, X) plane
are drawn in Fig. 9.6, which extends Fig. 9.2 to MII .
As discussed in Sec. 6.3.4, one approaches a curvature singularity as r → 0. According to
(9.23) or (9.25), this corresponds to X 2 − T 2 → −1 (see also√Fig. 9.3), with T > 0. Hence, in
the (T, X) plane, the curvature singularity is located at T = X 2 + 1, i.e. at the upper branch
of the hyperbola T 2 − X 2 = 1.
290 Maximal extension of Schwarzschild spacetime
9.2.3 Radial null geodesics in Kruskal-Szekeres coordinates
By construction, the Kruskal-Szekeres coordinates (T, X, θ, φ) are adapted to the radial null
geodesics. This is clear on the expression (9.10) of the metric tensor, where the (T, X) part is
conformal to the flat metric −dT 2 + dX 2 . Consequently the radial null geodesics are straight
lines of slope ±45◦ in the (T, X) plane (cf. Fig. 9.7):
• the ingoing radial null geodesics obey
T = −X + V, (9.26)
where V is a positive constant (the constraint V > 0 following from (9.21)), so that each
geodesic of this family can be labelled by (V, θ, φ);
• the outgoing radial null geodesics obey
T = X + U, (9.27)
where U is an arbitrary real constant, so that each geodesic of this family can be labelled
by (U, θ, φ).
In particular, the Schwarzschild horizon H is generated by the outgoing radial null geodesics
having U = 0: Eqs. (9.27) and (9.11) clearly imply r = 2m for U = 0, i.e. X = T . The outgoing
radial null geodesics not lying on H , which are denoted L(u,θ,φ)
out
in Sec. 6.3.5, have an equation
in terms of the IEF coordinates given by Eq. (6.51): t̃ = r + 4m ln |r/2m − 1| + u, where the
constant u is related to U by
U = −e−u/4m on MI (9.28a)
U = 0 on H (9.28b)
U = e−u/4m on MII . (9.28c)
These relations are easily established by combining (6.51) and (9.23).
Remark 3: The relation U = −e−u/4m introduced in Sec. 9.2.1 by Eq. (9.4) is thus valid only in MI . On
the contrary the relation V = ev/4m is valid in all MIEF .
9.3 Maximal extension
9.3.1 Construction
The spacetime (MIEF , g) is not geodesically complete (cf. Sec. B.3.2 in Appendix B). Indeed, let
us consider the radial null geodesics discussed above. We have seen in Sec. 6.3.1 that r is an
affine parameter along them, except for those that are null generators of H (the outgoing ones
with U = 0). Now, for the ingoing radial null geodesics, r is decreasing towards the future
and all of them terminate at r = 0 (the left end-point of the dashed lines in Fig. 9.7). They
are thus incomplete geodesics. However, they cannot be extended to negative values of the
9.3 Maximal extension 291
T
3
0
r=
2
X
3 2 1 1 2 3
Figure 9.7: Radial null geodesics in MIEF = MI ∪ H ∪ MII depicted in terms of the Kruskal-Szekeres
coordinates (T, X): the solid lines correspond to the outgoing family, with u spanning [−6m, 8m] (with steps
δu = 2m), from the left to the right in MII and from the right to the left in MI ; the dashed lines correspond to
the ingoing family, with v spanning [−8m, 6m] (with steps δv = 2m) from the left to the right. [Figure generated
by the notebook D.4.16]
affine parameter r by extending the spacetime since r = 0 marks a spacetime singularity (cf.
Sec. 6.3.4).
On the other hand, the outgoing radial null geodesics are limited by the constraint T +X > 0,
which corresponds to r > 2m in MI , with r increasing towards the future, and to r < 2m in
MII , with r decreasing towards the future. Thus all outgoing radial null geodesics terminate
towards the past at the finite value 2m of the affine parameter r (the left end point of the solid
lines in Fig. 9.7) and are therefore incomplete geodesics. However, contrary to ingoing radial
null geodesics, they can be extended since r = 2m does not mark any spacetime singularity.
More precisely, the limit at which outgoing radial null geodesics terminate is T = −X, which
by virtue of (9.11) yields r = 2m. This does not correspond to the Schwarzschild horizon H ,
since for the latter T = X, but rather to t̃ → −∞, as it is clear when comparing Fig. 9.7 with
Fig. 9.5.
Another hint regarding the extendability of (MIEF , g) is the fact that the Killing horizon
H is non-degenerate, having a non-zero surface gravity (cf. Sec. 3.3.6); the latter has been
computed in Example 10 of Chap. 2: κ = 1/4m. Now, we have seen in Sec. 3.4 that non-
degenerate Killing horizons have incomplete null generators and, if they can be extended,
they must be part of a bifurcate Killing horizon (Property 3.15). In the present case, the null
generators of H are nothing but outgoing radial null geodesics. They are thus as incomplete
as those discussed above, i.e. those that admit r as an affine parameter.
292 Maximal extension of Schwarzschild spacetime
The possibility of spacetime extension beyond MIEF is clear on the metric element (9.16):
it is invariant by the transformation
Φ: R2 −→ R2
(9.29)
(T, X) 7−→ (−T, −X).
Thus we may include the part T + X < 0 by adding a copy of MIEF , symmetric to the original
one with respect to the “origin” (T, X) = (0, 0). The whole spacetime manifold is then the
following open subset of R2 × S2 :
M := {p ∈ R2 × S2 , T 2 (p) − X 2 (p) < 1} , (9.30)
where (T, X, θ, φ) is the canonical coordinate system on R2 ×S2 , called in this context Kruskal-
Szekeres coordinates. The metric g on the whole M is then defined by (9.16):
2 4 2 2
2 2 2 2 2 2
g = 4m −dT + dX + W (X − T ) dθ + sin θ dφ ,
X 2 − T 2 + eW (X 2 −T 2 )
(9.31)
where W is the rescaled Lambert function defined by (9.14) (cf. Fig. 9.4); it is the inverse of
the function x 7→ ex (x − 1), which establishes a bijection from (0, +∞) to (−1, +∞). Note
that the metric components gαβ appearing in Eq. (9.31) are regular functions of (T, X) in all
M . In particular, the denominator X 2 − T 2 + eW (X −T ) of gT T and gXX never vanishes for
2 2
X 2 − T 2 > −1, which is the range of variation of (T, X) on M according to Eq. (9.30). For
|T | = |X|, and in particular at (T, X) = (0, 0), one has gXX = −gT T = 16m2 /e, given that
W (0) = 1 (cf. Fig. 9.4),
Let us define the following open subsets of M , which are respectively the images of MI
and MII by the reflection through the origin (9.29):
MIII : X < 0 and X < T < −X (9.32a)
√
MIV : − X 2 + 1 < T < −|X|. (9.32b)
On MIII ∪ MIV , one may introduce coordinates (t′ , r′ , θ, φ) of Schwarzschild-Droste type; they
are related to the Kruskal-Szekeres coordinates by formulas analogous to (9.18) and (9.25),
simply changing T to −T and X to −X:
q
r′ /4m r′ t′
t′ = 2m ln X+T
T = −e
2m
− 1 sinh 4m
X−T
MIII : q ⇐⇒ (9.33)
X = −er′ /4m r′ − 1 cosh t′ r′ = 2m W (X 2 − T 2 ).
2m 4m
q
r′ /4m r′ t′
t′ = 2m ln T +X
T = −e
1 − 2m cosh 4m
T −X
MIV : q ⇐⇒ (9.34)
X = −er′ /4m 1 − r′ sinh t′ r′ = 2m W (X 2 − T 2 ).
2m 4m
9.3 Maximal extension 293
T
3
0
r=
2
III II I
X
3 2 1 1 2 3
IV
1
r0 =
0
Figure 9.8: Kruskal diagram: Schwarzschild spacetime M depicted in terms of Kruskal-Szekeres coordinates
(T, X). Each point in this diagram, including the one at (T, X) = (0, 0), is actually a sphere S2 , spanned by
the coordinates (θ, φ). Solid lines denote the hypersurfaces t = const in MI and MII and the hypersurfaces
t′ = const in MIII and MIV , whiles dashed curves denote the hypersurfaces r = const in MI and MII and the
hypersurfaces r′ = const in MIII and MIV . The bifurcate Killing horizon is marked by thick black lines, while
the singularities at r = 0 and r′ = 0 are depicted by the heavy dashed brown curve. [Figure generated by the
notebook D.4.16]
The extended Schwarzschild spacetime (M , g) is depicted in Fig. 9.8, which is usually
called a Kruskal diagram. There are two curvature singularities, which formally are not
part of M : the hypersurfaces r = 0 and r′ = 0, which are the two branches of the hyperbola
T 2 − X 2 = 1. As discussed in Sec. 9.2.3, the radial null geodesics appear as straight lines of
slope ±45◦ (+ for the outgoing family, and − for the ingoing one). As in (MIEF , g), they are
still not complete but the only locations where they terminate are the curvature singularities
at r = 0 (future end point) and r′ = 0 (past end point). Therefore, they cannot be extended
further. For this reason, (M , g) is called the maximal extension of Schwarzschild spacetime.
Remark 1: The extended manifold M is not just the union MI ∪ MII ∪ MIII ∪ MIV , since the latter
does not contain the hypersurfaces T = ±X (cf. the strict inequalities in Eqs. (9.24) and (9.32)), which
are parts of M according to the definition (9.30). Actually, we have
M = MI ∪ MII ∪ MIII ∪ MIV ∪ Hˆ , (9.35)
where Hˆ is the bifurcate Killing horizon, to be discussed in Sec. 9.3.3.
294 Maximal extension of Schwarzschild spacetime
9.3.2 Global null coordinates
In Secs. 9.2.1 and 9.2.3, we have introduced on MIEF the null coordinates (U, V ); they are
related to the coordinates (T, X) by Eq. (9.8) (or equivalently Eqs. (9.26)-(9.27)), which we can
use to define (U, V ) in all the maximal extension M :
U =T −X T = 1 (U + V )
2
⇐⇒ (9.36)
V =T +X X = 1 (V − U )
2
The range of (U, V ) of M is deduced from the constraint T 2 − X 2 < 1 [cf. Eq. (9.30)]: since
T 2 − X 2 = U V , we get:
M : (U, V ) ∈ R2 and U V < 1. (9.37)
The expression of the metric with respect to the null coordinates X α̂ = (U, V, θ, φ) is
deduced from (9.31):
4
2 2 2 2 2
(9.38)
g = 4m dU dV + W (−U V ) dθ + sin θ dφ .
U V − eW (−U V )
We can also rewrite it as (9.5):
32m3 −r/2m
dU dV + r2 dθ2 + sin2 θ dφ2 , (9.39)
g=− e
r
where r is the function of (U, V ) given by
r = 2m W (−U V ). (9.40)
Note that relation (9.6) between r and (U, V ) holds in all M :
r
r/2m
e − 1 = −U V . (9.41)
2m
Remark 2: In Sec. 9.3.1, we have distinguished the coordinate r in MI ∪ MII from the coordinate r′ in
MIII ∪ MIV . Here, r is the function (9.40) of (U, V ), which has the same expression in MI ∪ MII and
MIII ∪ MIV . There is no need to make any distinction. Hence there is no mention of r′ in (9.39).
Historical note : The Kruskal-Szekeres coordinates have been introduced in 1960 independently by
Martin Kruskal [326] and George Szekeres [472]. Actually the coordinates introduced by Szekeres
√ √
were2 (2T / e, 2X/ e). Both Kruskal and Szekeres have used these coordinates to construct the
maximal extension of Schwarzschild spacetime. Its graphical representation in the (X, T ) plane (the
Kruskal diagram, cf. Fig. 9.8) has been presented by Kruskal (Fig. 2 of Ref. [326]). However, the first
author to obtain the maximal extension of Schwarzschild spacetime is John Synge in 1950 [469], i.e. 10
2
They are denoted by (v, u) in Szekeres’ article [472].
9.3 Maximal extension 295
years before Kruskal and Szekeres. Synge used coordinates (T ′ , X ′ ) whose relation to Schwarzschild-
Droste coordinates is more complicated than the Kruskal-Szekeres one: T ′ = R(r) sinh 4m t
and
h q i
X = R(r) cosh 4m , with R(r) := 2m arcosh 2m + 2m 2m − 1 ; compare with (9.18). Albeit
′ t
p r r r
−1/2
looking complicated, R(r) is nothing but the primitive vanishing at r = 2m of r 7→ 2m r
−1 .
Interestingly, in his article [472], Szekeres says that the transformations (9.18) “are essentially due to
Synge”,
pprobably because they differ only in the choice of the function R(r), the latter being RKS (r) =
er/4m r/2m − 1 for Kruskal-Szekeres coordinates. For this reason, both coordinate systems share
some similarities: in Synge diagram (Figs. 8 and 9 in Ref. [469]), the bifurcate horizon appears as the
two bisector lines T ′ = ±X ′ and the singularity r = 0 as the hyperbola T ′2 − X ′2 = π 2 m2 (compare
with T 2 − X 2 = 1 for Kruskal-Szekeres coordinates). A major difference is that Synge diagram is not
“conformal”: the radial null geodesics are generally not lines with ±45◦ slope. Even, in some regions,
the coordinate X ′ ceases to be spacelike3 . The maximal extension of Schwarzschild spacetime has also
been found by Christian Fronsdal [205] in 1959, not via any explicit change of coordinates but rather via
an isometric embedding of the spacetime in the 6-dimensional Minkowski spacetime.
9.3.3 Bifurcate Killing horizon
As discussed in Sec. 9.3.1, the Schwarzschild horizon H is a non-degenerate Killing horizon
and therefore shall be part of a bifurcate Killing horizon (cf. Sec. 3.4) in the extended spacetime.
The bifurcate Killing horizon, Hˆ say, is easily found by considering the Killing vector field
ξ in the maximal extension of Schwarzschild spacetime. The components of ξ w.r.t. to the
Kruskal-Szekeres coordinates are obtained from the property ξ = ∂t :
∂T ∂X ∂θ ∂φ
ξT = , ξX = , ξθ = = 0, ξφ = = 0.
∂t ∂t ∂t ∂t
Given the coordinate transformation laws (9.18) and (9.25), we get in MI and MII :
1 1
ξT = X, ξX = T, ξ θ = ξ φ = 0.
4m 4m
Hence in MI ∪ MII ,
1
ξ= (X ∂T + T ∂X ) . (9.42)
4m
Now, this formula defines a smooth vector field in all M . Moreover, in MIII ∪ MIV , this vector
coincides with ∂t′ since ξ T = ∂T /∂t′ and ξ X = ∂X/∂t′ , with the partial derivatives with
respect to t′ evaluated from (9.33)-(9.34). Hence the vector field ξ defined by (9.42) is a Killing
vector field of maximal extension (M , g). This vector field is depicted in Fig. 9.9.
The bifurcate Killing horizon with respect to ξ that extends H is Hˆ = H1 ∪ H2 , where
• H1 is the null hypersurface T = X (or equivalently U = 0);
• H2 is the null hypersurface T = −X (or equivalently V = 0).
3
We refer the reader to Fig. 2 of Ref. [485] for a plot of Synge coordinates in terms of Kruskal-Szekeres ones
296 Maximal extension of Schwarzschild spacetime
T
3
0
r=
2
X
3 2 1 1 2 3
r0 =
0
Figure 9.9: Killing vector field ξ on the extended Schwarzschild manifold. [Figure generated by the notebook
D.4.16]
The bifurcate Killing horizon Hˆ is depicted in black in Fig. 9.9. The Schwarzschild horizon
H is the part of H1 defined by X > 0. In terms of the null coordinates (U, V ) introduced in
Sec. 9.3.2, we have, given (9.36),
Hˆ : U = 0 or V = 0 (9.43a)
H : U = 0 and V > 0. (9.43b)
The bifurcation surface is S = H1 ∩ H2 , which is the 2-surface defined by T = 0 and
X = 0, or equivalently by U = 0 and V = 0. It is a 2-sphere, since any fixed value of the pair
(T, X) defines a 2-sphere, according to the definition of M as a part of R2 × S2 [cf. Eq. (9.30)].
Accordingly, S is called the bifurcation sphere. It is located at the center of Fig. 9.9. The areal
radius of S is found by setting dT = 0, dX = 0 and (T, X) = (0, 0) in the metric expression
(9.31): rS
2
= 4m2 W (0)2 . Since W (0) = 1 (cf. Fig. 9.4), we get
rS = 2m . (9.44)
Moreover, setting (T, X) = (0, 0) in Eq. (9.42), we recover Property 3.12 (Sec. 3.4.1): the Killing
vector field vanishes at the bifurcation sphere:
ξ|S = 0. (9.45)
9.4 Carter-Penrose diagram 297
9.4 Carter-Penrose diagram
9.4.1 First construction
To have a compact representation of the maximal extension of Schwarzschild spacetime,
one can use the same trick as for Minkowski spacetime (cf. Sec. 4.2.1), namely employ the
arctangent function to map the range (−∞, +∞) of the null coordinates U and V to the
interval (−π/2, π/2), thereby defining the finite-range coordinates (Û , V̂ ):
Û = arctan U U = tan Û
⇐⇒ (9.46)
V̂ = arctan V V = tan V̂ .
The range of (Û , V̂ ) is deduced from (9.37):
U V < 1 ⇐⇒ tan Û tan V̂ < 1.
Since for Û , V̂ ∈ (−π/2, π/2), we have cos Û > 0 and cos V̂ > 0, we may write
π π
U V < 1 ⇐⇒ sin Û sin V̂ < cos Û cos V̂ ⇐⇒ cos(Û + V̂ ) > 0 ⇐⇒ − < Û + V̂ < .
2 2
Hence the range of (Û , V̂ ) on the maximal extension of Schwarzschild spacetime:
π π π π π π
M : − < Û < , − < V̂ < and − < Û + V̂ < . (9.47)
2 2 2 2 2 2
Since (9.46) yields dU = dÛ / cos2 Û and dV = dV̂ / cos2 V̂ , we deduce immediately from
(9.39) the expression of the metric in terms of the coordinates xα = (Û , V̂ , θ, φ):
32m3 −r/2m dÛ dV̂
+ r2 dθ2 + sin2 θ dφ2 , (9.48)
g=− e
r cos2 Û cos2 V̂
where [cf. Eq. (9.40)]
r = 2m W (− tan Û tan V̂ ). (9.49)
To depict M , let us introduce “time+space” coordinates (T̂ , X̂), which are related to (Û , V̂ )
in exactly the same way as the coordinates (τ, χ) were related to the finite-range null coordinates
(U, V ) for Minkowski spacetime [cf. Eq. (4.14)]:
T̂ = Û + V̂ Û = 1 (T̂ − X̂)
2
⇐⇒ (9.50)
X̂ = V̂ − Û V̂ = 1 (T̂ + X̂).
2
The range of (T̂ , X̂) is deduced from (9.47):
π π
M : − < T̂ < , T̂ − π < X̂ < T̂ + π and − T̂ − π < X̂ < −T̂ + π. (9.51)
2 2
298 Maximal extension of Schwarzschild spacetime
T̂
r=0
1.5
0 + 1.0 II +
0.5
III I
X̂
3 2 1 1 2 3
0.5
0− −
1.0 IV
1.5
r0 = 0
Figure 9.10: Carter-Penrose diagram of the Schwarzschild spacetime constructed with the compactified
coordinates (T̂ , X̂). Solid curves denote hypersurfaces of constant Schwarzschild-Droste coordinate t: in region
MI , from the X̂-axis to the top: t = 0, 2m, 5m, 10m, 20m and 50m, the last two being barely visible; in region
MII , from the T̂ -axis to the right: t = 0, 2m, 5m, 10m, 20m and 50m, Dashed curves denote hypersurfaces of
constant Schwarzschild-Droste coordinate r: in region MI , from the left to the right: r = 2.01m, 2.1m, 2.5m
(almost vertical), 4m, 8m, 12m, 20m and 100m, the last three being barely visible; in region MII , from the bottom
to the top: r = 1.98m, 1.9m, 1.7m, 1.5m, 1.25m, m, 0.5m and 0.1m. The color code is the same as in Fig. 9.8.
[Figure generated by the notebook D.4.17]
Via (9.46) and (9.36), the relation between (T̂ , X̂) and the Kruskal-Szekeres coordinates
(T, X) is then the same as that between (τ, χ) and (t, r) for Minkowksi spacetime [Eq. (4.16)]:
sin T̂
T =
T̂ = arctan(T + X) + arctan(T − X)
cos T̂ + cos X̂
⇐⇒ (9.52)
X̂ = arctan(T + X) − arctan(T − X) sin X̂
X= .
cos T̂ + cos X̂
The maximal extension of Schwarzschild spacetime is depicted with respect to the coordi-
nates (T̂ , X̂) in Fig. 9.10. Such a plot is called a Carter-Penrose diagram (see the historical
note p. 300). As the Kruskal diagram (Fig. 9.8), it has the feature of displaying radial null
geodesics as straight lines with slope ±45◦ . This holds since Û (resp. V̂ ) is a function of U
only (resp. V only), cf. Eq. (9.46), so that Û (resp. V̂ ) is constant on outgoing (resp. ingoing)
radial null geodesics. In particular, the bifurcate Killing horizon and the Schwarzschild horizon
are obtained for specific values of Û and V̂ :
Hˆ : Û = 0 or V̂ = 0 (9.53a)
H : Û = 0 and V̂ > 0. (9.53b)
These relations follow immediately from (9.43) and (9.46).
9.4 Carter-Penrose diagram 299
We have seen in Sec. 6.4 that the future null infinity I + corresponds to v → +∞ and that
the past null infinity I − to u → −∞ (cf. Fig. 6.6). Since on MI , U = −e−u/4m and V = ev/4m
[cf. Eq. (9.4)], we may write equivalently:
I + : V → +∞ and U ∈ (−∞, 0) (9.54a)
I − : U → −∞ and V ∈ (0, +∞). (9.54b)
In view of (9.46), we get then:
π π
I + : V̂ → and Û ∈ − , 0 (9.55a)
2 2 π
π
I− : Û → − and V̂ ∈ 0, . (9.55b)
2 2
By symmetry, the extension MIII ∪ MIV of Schwarzschild spacetime has the following null
infinity:
′+ π π
I : Û → and V̂ ∈ − , 0 (9.56a)
2 2
− π π
I ′ : V̂ → − and Û ∈ 0, . (9.56b)
2 2
9.4.2 Discussion: Carter-Penrose diagram and conformal completion
The Carter-Penrose diagram in Fig. 9.10 can be compared with the conformal diagram of
Minkowski spacetime in Fig. 4.3. The right asymptotics of the Carter-Penrose diagram (i.e. the
part X̂ > π/2) looks similar to that of Minkowski conformal diagram. However, there is a
difference: the coordinates (T̂ , X̂) employed in the construction of the diagram of Fig. 9.10 are
not related to any (regular) conformal completion — as defined in Sec. 4.3 — contrary to the
coordinates (τ, χ) used for Minkowski spacetime.
To see this, let us rewrite the metric components (9.48) in a form that makes clear their
behavior near null infinity. Given (9.49) and (9.15), we have
r
e r/2m
− 1 = − tan Û tan V̂ , (9.57)
2m
from which we get
2m −r/2m 1 − 2m/r
e =− .
r tan Û tan V̂
Hence
2m e−r/2m 1 − 2m/r 4(1 − 2m/r)
=− =− .
r cos2 Û cos2 V̂ sin Û cos Û sin V̂ cos V̂ sin 2Û sin 2V̂
Therefore, we may rewrite expression (9.48) for the metric tensor as
2m dÛ dV̂
2
+ r2 dθ2 + sin2 θ dφ2 , (9.58)
g = 64m 1 −
r sin 2Û sin 2V̂
with r given by (9.49). To get a conformal completion, we should write (cf. Sec. 4.3)
g = Ω−2 g̃, (9.59)
300 Maximal extension of Schwarzschild spacetime
where Ω = 0 and dΩ ̸= 0 on the spacetime boundary I and g̃ is a regular metric on the
completion M ∪ I . Since in Eq. (9.58), the term sin 2Û sin 2V̂ vanishes at I = I + ∪ I − ∪
I ′ + ∪ I ′ − [cf. Eqs. (9.55)-(9.56)], we would have, up to some constant factor,
q
Ω = − sin 2Û sin 2V̂ , (9.60)
the minus sign taking into account that sin 2Û sin 2V̂ approaches zero via negative values near
I . A first issue is that the square root in (9.60) makes Ω not differentiable on I , where either
sin 2Û = 0 or sin 2V̂ = 0. In other words, dΩ is diverging on I . Suppose we accept this and
are ready to introduce a slight deviation (given that Ω2 , which is involved in (9.59), is smooth)
from the definition given in Sec. 4.3. Then the conformal metric should be
2m
2
dÛ dV̂ − r2 sin 2Û sin 2V̂ dθ2 + sin2 θ dφ2 . (9.61)
g̃ = −64m 1 −
r
Near I , r → +∞ and we have g̃Û V̂ → −32m2 . On the contrary, g̃θθ is of the type “∞ × 0”; in
order to determine its behavior, let us rewrite it as follows:
g̃θθ = −r2 sin 2Û sin 2V̂ = −4r2 sin Û sin V̂ × cos Û cos V̂ ,
with cos Û cos V̂ expressed via (9.57):
e−r/2m
cos Û cos V̂ = − sin Û sin V̂ .
r/2m − 1
Hence
re−r/2m
g̃θθ = 8m sin2 Û sin2 V̂
1 − 2m/r
and (9.61) becomes
re−r/2m
2m
2
dÛ dV̂ + 8m sin2 Û sin2 V̂ dθ2 + sin2 θ dφ2 . (9.62)
g̃ = −64m 1 −
r 1 − 2m/r
On this expression, we can read directly the value of the conformal metric at I , where r → +∞,
2m/r → 0, re−r/2m → 0 and sin2 Û → 1 or sin2 V̂ → 1:
I
g̃ = −64m2 dÛ dV̂ . (9.63)
This bilinear form is clearly degenerate (cf. Sec. A.3.1). Therefore, g̃ is not a regular metric
on the whole manifold M ∪ I . We conclude that (9.60)-(9.61) does not define a conformal
completion of (M , g).
Historical note : The first compactified conformal diagram of the (maximal extension of) Schwarzschild
spacetime has been constructed by Brandon Carter in 1966 [88], using the same coordinates (T̂ , X̂) as
here4 : compare Fig. 9.10 with Fig. 1c of Ref. [88]. In his article, Carter notes that “the manner in which
4
(T̂ , X̂) are denoted (ψ, ξ) by Carter [88].
9.4 Carter-Penrose diagram 301
the distant flat-space parts (...) are compressed into finite parts of the (ξ, ψ) plane by the coordinate
transformations recalls the conformal diagrams used by R. Penrose” in 1964 [404], the diagrams in
Penrose’s article [404] regarding Minkowski and de Sitter spacetimes only. This justifies the name
Carter-Penrose diagram used in the literature (e.g. [204]) for the graphical representation shown in
Fig. 9.10, while the mere Penrose diagram or conformal diagram is quite common; one encounters as
well the name Penrose-Carter diagram (e.g. [416]).
9.4.3 A regular conformal completion based on Penrose-Frolov-Novikov
coordinates
In order to get a regular conformal completion of the maximally extended Schwarzschild
spacetime (M , g), let us introduce the finite-range coordinates (Ũ , Ṽ ) that are related to the
null Kruskal-Szekeres coordinates (U, V ) by
Ũ = arctan(arsinh U ) U = sinh(tan Ũ )
⇐⇒ (9.64)
Ṽ = arctan(arsinh V ) V = sinh(tan Ṽ ).
The range of (Ũ , Ṽ ) is deduced from (9.37):
π π π π
M : − < Ũ < , − < Ṽ < and sinh(tan Ũ ) sinh(tan Ṽ ) < 1. (9.65)
2 2 2 2
Note that contrary to what happened for (Û , V̂ ), these conditions do not yield to a simple
polygonal region in the (Ũ , Ṽ ) plane. The presence of the sinh function in the expression (9.64)
of (U, V ) in terms of (Ũ , Ṽ ) does not alter the values of the finite-range coordinates at null
infinity, as compared to (Û , V̂ ) [cf. (9.55)-(9.56)]:
π π
I + : Ṽ → and Ũ ∈ − , 0 (9.66a)
2 2
π π
I − : Ũ → − and Ṽ ∈ 0, (9.66b)
2 2
+ π π
I ′ : Ũ → and Ṽ ∈ − , 0 (9.66c)
2 2
− π π
I ′ : Ṽ → − and Ũ ∈ 0, . (9.66d)
2 2
We shall call (Ũ , Ṽ , θ, φ) the Penrose-Frolov-Novikov coordinates (cf. the historical note on
p. 306). From (9.64), we get
cosh(tan Ũ ) cosh(tan Ṽ )
dU = dŨ and dV = dṼ ,
cos2 Ũ cos2 Ṽ
so that the metric expression in terms of the coordinates xα = (Ũ , Ṽ , θ, φ) is easily deduced
from (9.39)-(9.40):
32m3 −r/2m cosh(tan Ũ ) cosh(tan Ṽ )
dŨ dṼ + r2 dθ2 + sin2 θ dφ2 , (9.67)
g=− e
r cos2 Ũ cos2 Ṽ
302 Maximal extension of Schwarzschild spacetime
where r is the function of (Ũ , Ṽ ) given by
r = 2m W − sinh(tan Ũ ) sinh(tan Ṽ ) . (9.68)
As we did for (Û , V̂ ), let us rewrite (9.67) in a form that is better adapted to the null asymptotics.
Given (9.68) and (9.15), we have
r
er/2m − 1 = − sinh(tan Ũ ) sinh(tan Ṽ ), (9.69)
2m
from which we get
2m −r/2m 2m 1
e =− 1− .
r r sinh(tan Ũ ) sinh(tan Ṽ )
Hence (9.67) becomes
2m2 dŨ dṼ
g = 16m 1 −
r tanh(tan Ũ ) tanh(tan Ṽ ) cos2 Ũ cos2 Ṽ
+r2 dθ2 + sin2 θ dφ2 . (9.70)
Given the values (9.66) of Ũ and Ṽ near I , tanh(tan Ũ ) tanh(tan Ṽ ) does not vanish there. A
natural choice of conformal factor is then
Ω := cos Ũ cos Ṽ . (9.71)
The corresponding conformal metric is
2 2m dŨ dṼ
g̃ = 16m 1 −
r tanh(tan Ũ ) tanh(tan Ṽ ) . (9.72)
+r2 cos2 Ũ cos2 Ṽ dθ2 + sin2 θ dφ2
Considering (Ũ , Ṽ , θ, φ) as a canonical coordinate system on R2 × S2 , we define the conformal
completion manifold as
π π 2
˜
M := 2 2
p ∈ R × S , (Ũ (p), Ṽ (p)) ∈ − , and sinh(tan Ũ (p)) sinh(tan Ṽ (p)) < 1
2 2
+ −
∪I + ∪ I − ∪ I ′ ∪ I ′ , (9.73)
with
n π π o
I + := p ∈ R2 × S2 , Ṽ (p) = and Ũ (p) ∈ − , 0 (9.74a)
2 2 π o
n π
I− := p ∈ R2 × S2 , Ũ (p) = − and Ṽ (p) ∈ 0, (9.74b)
2 π 2 o
+
n π
I′ := p ∈ R2 × S2 , Ũ (p) = and Ṽ (p) ∈ − , 0 (9.74c)
2 2 π o
−
n π
I′ := p ∈ R2 × S2 , Ṽ (p) = − and Ũ (p) ∈ 0, . (9.74d)
2 2
9.4 Carter-Penrose diagram 303
Note that the first line in (9.73) corresponds to M , identified as a subset of R2 ×S2 [cf. Eq. (9.65)]
and that the definitions of I + , I − , I ′ + and I ′ − are in agreement with (9.66). It is clear that
M˜ is a manifold with boundary and that
+ −
∂ M˜ = I := I + ∪ I − ∪ I ′ ∪ I ′ . (9.75)
Moreover, the scalar field Ω defined by (9.71) satisfies Ω ≥ 0 on M˜, along with Ω = 0 on I
and dΩ ̸= 0 on I . The last property follows from
dΩ = − sin Ũ cos Ṽ dŨ − cos Ũ sin Ṽ dṼ ,
which implies dΩ|I + = − cos Ũ dṼ ̸= 0, dΩ|I − = cos Ṽ dŨ ̸= 0, dΩ|I ′ + = − cos Ṽ dŨ ̸=
0 and dΩ|I ′ − = cos Ũ dṼ ̸= 0. Hence the conditions 1, 3 and 4 of the definition of a conformal
completion given in Sec. 4.3 are fulfilled. There remains to check condition 2, namely that
the tensor g̃ defined by (9.72) is a regular metric on the whole M˜. This was the main failing
point in the attempt of Sec. 9.4.2. Since Ω2 > 0 on M , g̃ is well-behaved on M . Let us thus
examine its behavior on I . We shall focus on I + , the behavior on the other parts of I
being obtained by some trivial symmetry. As one approaches I + , r → +∞, Ṽ → π/2 and
tanh(tan(Ṽ )) → 1; accordingly we read from (9.72) that
I+ 8m2
g̃Ũ Ṽ = .
tanh(tan Ũ )
Besides, we have g̃θθ = r2 cos2 Ũ cos2 Ṽ , which is of the type “+∞ × 0” near I + . Noticing
that tan Ṽ ∼ 1/ cos Ṽ when Ṽ → π/2, we get from (9.69)
rer/2m
1 π
sinh ∼− when Ṽ → .
cos Ṽ 2m sinh(tan Ũ ) 2
√
Since arsinh x = ln(x + x2 + 1) ∼ ln(2x) when x → +∞, we obtain
rer/2m
1 r r
∼ ln − = + ln − ln − sinh(tan Ũ )
cos Ṽ m sinh(tan Ũ ) 2m m
r π
∼ when Ṽ → .
2m 2
Hence cos2 Ṽ ∼ 4m2 /r2 and g̃θθ ∼ 4m2 cos2 Ũ . Gathering the above results, we have
I+ 4
2 2 2 2 2
(9.76)
g̃ = 4m dŨ dṼ + cos Ũ dθ + sin θ dφ .
tanh(tan Ũ )
Since cos2 Ũ ̸= 0 on I + [cf. Eq. (9.74a)], this bilinear form is non-degenerate. Moreover, since
tanh(tan Ũ ) < 0 on I + [again by (9.74a)], its signature is (−, +, +, +). We conclude that g̃ is
a well-behaved metric on the whole manifold M˜. This completes the demonstration of the
following result:
304 Maximal extension of Schwarzschild spacetime
Property 9.1: conformal completion of Schwarzschild spacetime
The pair (M˜, g̃), with M˜ defined by (9.73)-(9.74) and g̃ defined by (9.72) is a conformal
completion of the maximally extented Schwarzschild spacetime (M , g), the conformal fac-
tor being given by (9.71). The Penrose-Frolov-Novikov coordinates (Ũ , Ṽ , θ, φ) employed
in this construction are related to the null Kruskal-Szekeres coordinates (U, V, θ, φ) by
(9.64).
Remark 1: At first sight, the metric g̃ given by (9.72) looks degenerate at the bifurcate Killing horizon
H , since 1 − 2m/r = 0 there. But one shall not forget that on H , which is defined by (Ũ = 0 or
Ṽ = 0), one has tanh(tan Ũ ) tanh(tan Ṽ ) = 0, which compensate the vanishing of 1 − 2m/r in the
term g̃Ũ Ṽ . Actually, to deal with g̃ near H , it is more appropriate to use the form that is deduced from
(9.67) and (9.71):
32m3 −r/2m
cosh(tan Ũ ) cosh(tan Ṽ ) dŨ dṼ + r2 cos2 Ũ cos2 Ṽ dθ2 + sin2 θ dφ2 . (9.77)
g̃ = − e
r
Remark 2: The conformal completion constructed above cannot be analytically extended “beyond” I ,
because the function Ṽ 7→ 1/ tanh(tan Ṽ ), which appears in (9.72), is C ∞ but not analytic at Ṽ = π/2.
It is possible to construct an analytic conformal completion, but it involves more complicated coordinate
transformations. The latter start, not from the Kruskal-Szekeres coordinates (U, V ), but from the null
coordinates (u, v) defined by (9.1). We refer to the article [254] for details.
To depict M˜, let us introduce “time+space” coordinates (T̃ , X̃), which are related to (Û , V̂ )
in exactly the same way as (T̂ , X̂) are related to (Û , V̂ ) [cf. Eq. (9.50)]:
T̃ = Ũ + Ṽ Ũ = 1 (T̃ − X̃)
2
⇐⇒ (9.78)
X̃ = Ṽ − Ũ Ṽ = 1 (T̃ + X̃).
2
The range of (T̃ , X̃) is deduced from (9.65):
M : −π < T̃ − X̃ < π, −π < T̃ + X̃ < π
sinh[tan((T̃ − X̃)/2)] sinh[tan((T̃ + X̃)/2)] < 1. (9.79)
The picture of (M , g) in the (T̃ , X̃) plane is shown in Fig. 9.11. We shall call it a regular
Carter-Penrose diagram of Schwarzschild spacetime. As the singular Carter-Penrose diagram
of Fig. 9.10, it has the feature of displaying radial null geodesics as straight lines with slope
±45◦ , since Ũ (resp. Ṽ ) is a function of U only (resp. V only) [cf. Eq. (9.64)]. In particular, the
bifurcate Killing horizon and the Schwarzschild horizon are defined by:
Hˆ : Ũ = 0 or Ṽ = 0 ⇐⇒ T̃ = X̃ or T̃ = −X̃ (9.80a)
H : Ũ = 0 and Ṽ > 0 ⇐⇒ T̃ = X̃ and T̃ > 0. (9.80b)
These relations follow immediately from (9.43), (9.64) and (9.78).
9.4 Carter-Penrose diagram 305
T̃
1.5 r=0
0 + 1.0 II +
0.5
III I
X̃
3 2 1 1 2 3
0.5
0− −
1.0 IV
1.5 r0 = 0
Figure 9.11: Carter-Penrose diagram of the Schwarzschild spacetime based on the Penrose-Frolov-Novikov
coordinates. Solid curves denote the same hypersurfaces of constant Schwarzschild-Droste coordinate t as in
Fig. 9.10: in region MI , from the X̂-axis to the top: t = 0, 2m, 5m, 10m, 20m and 50m; in region MII , from the
T̂ -axis to the right: t = 0, 2m, 5m, 10m, 20m and 50m, Dashed curves denote the same hypersurfaces of constant
Schwarzschild-Droste coordinate r as in Fig. 9.10: in region MI , from the left to the right: r = 2.01m, 2.1m,
2.5m, 4m, 8m, 12m, 20m and 100m; in region MII , from the bottom to the top: r = 1.98m, 1.9m, 1.7m, 1.5m,
1.25m, m, 0.5m and 0.1m. The color code is the same as in Figs. 9.8 and 9.10. Contrary to the Carter-Penrose
of Fig. 9.10, this one is associated to a regular conformal completion at null infinity of Schwarzschild spacetime.
[Figure generated by the notebook D.4.18]
At first sight, the main difference with the “standard” Carter-Penrose diagram of Fig. 9.10
is the more complicated shape of the boundary around the T̃ -axis. This follows from the
third condition in (9.79), which is more involved than the third condition in (9.51). Actually
this boundary corresponds to the curvature singularity limit r → 0 or r′ → 0. Indeed, from
Eq. (9.41), we have
r = 0 ⇐⇒ U V = 1. (9.81)
In terms of the coordinates (Û , V̂ ), we have then [cf. Eq. (9.46)]
r=0 ⇐⇒ tan Û tan V̂ = 1 ⇐⇒ sin Û sin V̂ = cos Û cos V̂
π
⇐⇒ cos Û cos V̂ − sin Û sin V̂ = 0 ⇐⇒ cos(Û + V̂ ) = 0 ⇐⇒ Û + V̂ = ± .
2
Since T̂ = Û + V̂ , we get the simple relation
π
r = 0 ⇐⇒ T̂ = ± . (9.82)
2
On the contrary, in terms of the coordinates (Ũ , Ṽ ), Eq. (9.81) becomes [cf. Eq. (9.64)]
r = 0 ⇐⇒ sinh(tan Ũ ) sinh(tan Ṽ ) = 1,
306 Maximal extension of Schwarzschild spacetime
which yields to the complicated formula
r = 0 ⇐⇒ sinh[tan((T̃ − X̃)/2)] sinh[tan((T̃ + X̃)/2)] = 1. (9.83)
This explains the more complex boundary of Fig. 9.11 diagram with respect to Fig. 9.10 diagram.
Remark 3: The shape of the Carter-Penrose diagram in Frolov & Novikov’s book (Fig. 5.2 of Ref. [202];
see also Fig. 10.6 of Ref. [204]) differs slightly from the diagram obtained here (Fig. 9.11). This is because
the coordinates used by Frolov & Novikov are constructed from the Szekeres’ version (up to a factor
√ √
2) of Kruskal-Szekeres coordinates: T ′ = T / e and X ′ = X/ e (cf. the historical note on p. 294).
Accordingly, in Frolov & Novikov’s version, one shall replace the 1 in the right-hand side of Eq. (9.83)
by 1/e, yielding to a different shape of the boundary r = 0.
Remark 4: As noticed by Frolov and Novikov [202] (see their Sec. 5.1.3), one can perform some
coordinate transformation from (Ũ , Ṽ ) to get a Carter-Penrose diagram with a straight line for the
boundary r = 0.
Besides the shape of the boundary r = 0, another difference between the Carter-Penrose
diagram based on Penrose-Frolov-Novikov coordinates (Fig. 9.11) and the “standard” diagram
of Fig. 9.10 is that the t = const hypersurfaces of the former (solid curves in Fig. 9.11) are all
tangent to the horizontal axis when X̃ → ±π. On the contrary, the same hypersurfaces in
Fig. 9.10 reach the point (T̂ , X̂) = (0, ±π) with a finite slope. We note that in this respect, the
Carter-Penrose diagram of Fig. 9.11 is similar to the conformal diagram of Minkowski spacetime,
as shown in Fig. 4.3, and therefore display correctly the asymptotic flatness structure. The
failure of diagram of Fig. 9.10 to reproduce this behavior reflects the fact that the coordinates
(T̂ , X̂) are singular on the boundary, as discussed in Sec. 9.4.2.
Historical note : The transformation (9.64) from (U, V ) to (Ũ , Ṽ ) has been suggested by Roger Penrose
in 1967 (p. 209 of Ref. [407]) to get a “version of the Kruskal diagram in which conformal infinity is
represented”. The figure in Penrose’s article (Fig. 37 in Ref. [407]) ressembles Fig. 9.11, except that the
curve r = 0 does not look tangent to the lines T̃ = ±X̃, as it is in Fig. 9.11. The same coordinate
change (9.64) has been discussed in details by Valeri P. Frolov and Igor D. Novikov in their 1998 textbook
[202] (Sec. 5.1.3), stressing that it leads to a regular conformal completion, contrary to the “standard”
coordinate change (9.46) introduced by Brandon Carter [88] and used in Hawking & Ellis [266] and
MTW [371] textbooks.
9.4.4 Black hole and white hole regions
(M˜, g̃) is not only a conformal completion of the maximally extended Schwarzschild spacetime
(M , g), as established above, but it is a conformal completion at null infinity, in the sense
defined in Sec. 4.3.1. Indeed, we can rewrite the boundary I of M˜ expressed by Eq. (9.75) as
I = I∗+ ∪ I∗− , (9.84)
with
+ −
I∗+ := I + ∪ I ′ and I∗− := I − ∪ I ′ . (9.85)
It is clear that any point of I∗+ is the furure end point of a future-directed null curve from M ,
while any point of I∗− is the past end point of a future-directed null curve to M (cf. Fig. 9.11).
9.5 Einstein-Rosen bridge 307
Hence the writing (9.84) matches the definition of a conformal completion at null infinity given
in Sec. 4.3.1. I∗+ is then the future null infinity of (M , g) and I∗− its past null infinity.
We are thus in position to apply the definitions of a black hole and a white hole given in
Sec. 4.4.2. The causal past of I∗+ is J − (I∗+ ) = MI ∪ MIII ∪ MIV (cf. Fig. 9.11). In view of the
definition (4.37), we get
Property 9.2: black hole in the maximal Schwarzschild spacetime
The maximal extension of Schwarzschild spacetime admits a black hole region, the interior
of which is MII . The black hole event horizon is the part of the bifurcate Killing horizon
Hˆ (cf. Sec. 9.3.3) that has T̃ > 0.
The black hole region has thus the same interior MII as the black hole region of the Schwarzschild
spacetime (MIEF , g) considered in Chap. 6. It differs only a larger boundary: it is not reduced
to the Schwarzschild horizon H , but contains the part X̃ < 0 and T̃ > 0 of Hˆ .
The novelty with respect to the original Schwarzschild spacetime (MIEF , g) is the existence
of a white hole region. Indeed, the causal future of I∗− is J + (I∗− ) = MI ∪ MII ∪ MIII (cf.
Fig. 9.11). In view of the definition (4.41), we get
Property 9.3: white hole in the maximal Schwarzschild spacetime
The maximal extension of Schwarzschild spacetime admits a white hole region, the interior
of which is MIV . The corresponding past event horizon is the part of the bifurcate Killing
horizon Hˆ (cf. Sec. 9.3.3) that has T̃ < 0.
9.5 Einstein-Rosen bridge
To get some insight on the maximally extended Schwarzschild spacetime (M , g), let us examine
the geometry of a slice of constant Kruskal-Szekeres time T .
9.5.1 Hypersurfaces of constant Kruskal-Szekeres time
Let ΣT0 be a hypersurface of M defined in terms of the global Kruskal-Szekeres coordinates
(T, X, θ, φ) by T = T0 , where T0 ∈ R is a constant (cf. Fig. 9.12). The 3-tuple (xi ) = (X, θ, φ)
is then a coordinate system on ΣT0 subject to the constraint expressed in (9.30):
X 2 > T02 − 1. (9.86)
Consequently
• if |T0 | < 1, the hypersurface ΣT0 is connected and diffeomorphic to R×S2 , the coordinate
X spanning R and (θ, φ) spanning S2 .
308 Maximal extension of Schwarzschild spacetime
T
3
0
r=
2 T0 = 2.0
T0 = 1.5
1 T0 = 1.0
T0 = 0.5
T0X= 0.0
3 2 1 1 2 3
T0 = − 0.5
1 T0 = − 1.0
T0 = − 1.5
2 T0 = − 2.0
r0 =
0
Figure 9.12: Kruskal diagram with the hypersurfaces ΣT0 (defined by T = T0 = const) as blue horizontal
lines. For |T0 | > 1, the dotted part of ΣT0 corresponds to a region that cannot be embedded isometrically in the
Euclidean space. When T0 varies, the limit of these regions form the grey dotted curve. [Figure generated by the
notebook D.4.19]
• if |T0 p
| ≥ 1, ΣT0 has two connected components, defined by X < − T02 − 1 and
p
X > T02 − 1 respectively (cf. Fig. 9.12). Each of them is diffeomorphic to R × S2 .
For future convenience, we split ΣT0 in two disjoint parts, according to the sign of X:
Σ+
T0 = {p ∈ ΣT0 , X(p) ≥ 0} and Σ−
T0 = {p ∈ ΣT0 , X(p) < 0} . (9.87)
For |T0 | < 1, there is a slight asymmetry between the two parts: Σ+ T0 is a manifold with
boundary (cf. Sec. A.2.2), the boundary corresponding to X = 0, while Σ−
T0 is not. For |T0 | ≥ 1,
ΣT0 and ΣT0 are nothing but the two connected components of ΣT0 .
+ −
The geometry of ΣT0 is defined by the metric γ induced on it by g:
32m3 −r/2m
dX 2 + r2 dθ2 + sin2 θ dφ2 , (9.88)
γ= e
r
where r is the function of X defined by
r = r(X) = 2mW (X 2 − T02 ). (9.89)
The metric (9.88) is obtained by setting T = T0 and dT = 0 in (9.31). Since r > 0, the
metric (9.88) is clearly positive definite, i.e. γ is a Riemannian metric and ΣT0 is a spacelike
hypersurface.
9.5 Einstein-Rosen bridge 309
r/m
T0 = 0.0
T0 = 0.5 1
T0 = 1.0
T0 = 1.5
T0 = 2.0
X
4 3 2 1 1 2 3 4
Figure 9.13: Function r = r(X) on the hypersurface ΣT0 , for the same values of T0 as in Fig. 9.12. [Figure
generated by the notebook D.4.19]
The graph of the function r(X) is shown in Fig. 9.13. Once restricted to positive (resp.
negative) values of X, this function is a bijection (X0 , +∞) → (r0 , +∞) (resp. (−∞, −X0 ) →
(r0 , +∞)), where
0 if |T0 | < 1 2m W (−T 2 ) if |T | < 1
0 0
X0 = and r0 = (9.90)
T − 1 if |T0 | ≥ 1. if |T0 | ≥ 1.
p
2 0
0
The inverse of this bijection is5
r r
X = X(r) = ± er/2m − 1 + T02 , (9.91)
2m
with the + sign on Σ+T0 and the − sign on ΣT0 .
−
We may use the above bijection to introduce coordinates (r, θ, φ) instead of (X, θ, φ) on
each of the two regions Σ+T0 and ΣT0 . Differentiating (9.91) leads to
−
rer/2m
dX = ± q dr.
r
8m2 er/2m 2m
− 1 + T02
Substituting in (9.88) we get the expression of the metric on ΣT0 in terms of the coordinates
(xi ) = (r, θ, φ):
−1 2
2m 2 −r/2m
dr + r2 dθ2 + sin2 θ dφ2 . (9.92)
γ = 1− 1 − T0 e
r
5
Let us recall that W −1 (x) = ex (x − 1).
310 Maximal extension of Schwarzschild spacetime
Remark 1: As a check of the above formula, we notice that for T0 = 0 it reduces to the metric of a
slice t = const in Schwarzschild-Droste coordinates [set dt = 0 in Eq. (6.14)]. This is correct since the
positive-X half of the hypersurface T = 0 in Kruskal-Szekeres coordinates, i.e. Σ+0 , coincides with the
hypersurface t = 0 in Schwarzschild-Droste coordinates, as it can be seen by setting T = 0 in Eq. (9.18)
(see also Fig. 9.12).
9.5.2 Isometric embedding in 3-dimensional Euclidean space
We may visualize the geometry of the spacelike hypersurface ΣT0 via some isometric embedding
of some 2-dimensional slice of it in the 3-dimensional Euclidean space (R3 , f ), f being the
standard flat (Euclidean) metric. By isometric embedding of a 2-dimensional Riemannian
manifold (S , g) in (R3 , f ), it is meant a smooth embedding Φ : S → R3 , as defined in
Sec. A.2.7, such that the metric induced on Φ(S ) by the Euclidean metric of R3 coincides with
the original metric g on S :
∀p ∈ S , ∀(u, v) ∈ (Tp S )2 , f (Φ∗ u, Φ∗ v) = g(u, v), (9.93)
where Φ∗ u is the vector of R3 that is the the pushforward of u by Φ, as defined in Sec. A.2.8.
Another phrasing of the isometry property (9.93) is: the pullback of f on S by Φ coincides
with g: Φ∗ f = g [cf. Eq. (A.33)].
Taking into account the spherical symmetry of ΣT0 , there is no loss of generality in choosing
the equatorial plane θ = π/2 as the 2-dimensional slice. We shall denote it by Σeq T0 . Coordinates
on ΣT0 are (x ) = (X, φ), or on each of the two parts ΣT0 (X ≥ 0) and Σ−,eq
eq a +,eq
T0 (X < 0),
(x ) = (r, φ). If |T0 | < 1, the topology of ΣT0 is R × S , i.e. that of a cylinder, while for
a eq 1
|T0 | ≥ 1, it has two connected components, Σ+,eq T0 and Σ−,eq
T0 , each of them having the topology
of a cylinder.
The metric induced by g on Σeq T0 , q say, is obtained by setting θ = π/2 and dθ = 0 in
Eq. (9.92):
−1 2
2m
q = 1− 2 −r/2m
1 − T0 e dr + r2 dφ2 . (9.94)
r
Given the invariance in φ, it is quite natural to embed (Σeq
T0 , q) as a surface of revolution in
the Euclidean space (R3 , f ). Describing R3 with cylindrical coordinates (xi ) = (r, z, φ), the
Euclidean metric f is
f = dr2 + dz 2 + r2 dφ2 . (9.95)
A surface of revolution S in R3 is described by an equation of the type z = Z(r). On such a
surface, one has therefore dz = Z ′ (r) dr, so that the metric h induced by f on it is
h = 1 + Z ′ (r)2 dr2 + r2 dφ2 . (9.96)
Comparing (9.96) with (9.94), we see that a possible isometric embedding of (Σeq
T0 , q) into
(R3 , f ) is
Φ : Σeq −→ R3
T0
(9.97)
(X, φ) 7−→ (r, z, φ) = (r(X), ±Z(r(X)), φ) ,
9.5 Einstein-Rosen bridge 311
with the function r(X) is given by Eq. (9.89), the sign ± is + on Σ+,eq
T0 and − on Σ−,eq
T0 and the
function Z(r) obeys
−1
′ 2 2m 2 −r/2m
1 + Z (r) = 1 − 1 − T0 e .
r
Thanks to Eq. (9.91), this expression can be recast as
1 − T02 e−r/2m er/2m − T02
Z ′ (r)2 = r = . (9.98)
T02 e−r/2m + 2m −1 X(r)2
For |T0 | < 1, i.e. when Σeq
T0 is connected, the map (9.97) defines a smooth embedding if, and
only if, at the boundary X = 0 between Σ+,eqT0 and ΣT−,eq
0
, the following holds:
Z(r(0)) = 0 and Z ′ (r(0)) = ∞. (9.99)
The condition Z(r(0)) = 0 insures the continuity of the embedded surface Φ(Σeq T0 ), while
Z (r(0)) = +∞ insures that it has a vertical tangent at the junction between Φ(Σ+,eq
′
T0 ) and
−,eq
Φ(ΣT0 ), so that it is a smooth surface. Fortunately, the condition Z (r(0)) = ∞ is automati-
′
cally fulfilled from the second expression of Z ′ (r)2 in (9.98): Z ′ (r)2 clearly diverges at X = 0.
Moreover, in order for the isometric embedding Φ to be well-defined, the right-hand side of
(9.98) must be non-negative. Since the denominator of the last term is manifestly non-negative,
the sign is determined by the numerator. Hence the condition er/2m ≥ T02 , or equivalently,
r ≥ 4m ln |T0 |. (9.100)
For |T0 | ≤ 1, this condition is always fulfilled, since ln |T0 | ≤ 0 and r ≥ 0. For |T0 | > 1, it
implies the existence of a minimal value of r,
remb (T0 ) := 4m ln |T0 |, (9.101)
such that the part of Σeq
T0 with r < remb (T0 ) cannot be embedded isometrically in the Euclidean
3-space.
Remark 2: The above result should not be surprising since there is no guarantee that a 2-dimensional
Riemannian manifold can be isometrically embedded in the 3-dimensional Euclidean space. The relevant
theorem here is Nash embedding theorem [379], which states that any smooth Riemannian manifold of
dimension n can be isometrically embedded in the Euclidean space (Rm , f ), with m ≤ n(n + 1)(3n +
11)/2. For n = 2, we get m ≤ 51, so there is really no guarantee that m = 3 is sufficient...
Via (9.89) and the fact that the rescaled Lambert function W is an increasing function (cf.
Fig. 9.4), of inverse F (x) = ex (x − 1), the condition (9.100) can be turned into a condition on
X:
(9.102)
p
|X| ≥ Xemb (T0 ) := |T0 | 2 ln |T0 |.
This limit is shown as the grey dotted curve in Fig. 9.12.
312 Maximal extension of Schwarzschild spacetime
Figure 9.14: Flamm paraboloid: isometric embedding in the Euclidean R3 of the spacelike slice T = 0 and
θ = π/2 of Schwarzschild spacetime. The (x, y) coordinates are the standard Cartesian coordinates of R3 related
to (r, φ) via x = r cos φ and y = r sin φ. The labels are in units of m. [Figure generated by the notebook D.4.19]
Summarizing, the minimal value of r on the embedded surface is r0 [cf. Eq. (9.90)] for
|T0 | ≤ 1 or remb (T0 ) for |T0 | > 1:
2m W (−T 2 ) if |T | ≤ 1
0 0
rmin (T0 ) = (9.103)
4m ln |T0 | if |T0 | > 1.
Note that rmin (T0 ) is a continuous function, with the peculiar values rmin (0) = 2m and
rmin (1) = 0. The embedding function Z(r) is found by integration of Z ′ (r), as given by (9.98),
from rmin (T0 ) to r: s
Z r
2m 1 − T02 e−x
Z(r) = 2m dx. (9.104)
rmin (T0 ) T02 e−x + x − 1
2m
The integral cannot be computed exactly in terms of elementary functions, except for T0 = 0,
where it reduces to Z r
2m dx
Z(r) = 2m √ (T0 = 0).
1 x−1
Hence r
r
Z(r) = 4m −1 (T0 = 0). (9.105)
2m
According to (9.97), the whole surface Φ(Σeq
T0 ) is obtained by considering z = −Z(r) as
well. The surface equation in terms of the cylindrical coordinates (r, z, φ) of R3 is then
9.5 Einstein-Rosen bridge 313
T0 = 0 T0 = 0.5 T0 = 0.9
T0 = 1 T0 = 1.5 T0 = 2
Figure 9.15: Sequence of isometric embeddings in the Euclidean space of spacelike slices of Schwarzschild
spacetime defined by T = T0 and θ = π/2. The slices are those shown in the Kruskal diagram of Fig. 9.12 (except
for T0 = 0.9). The first embedding (T0 = 0) is the Flamm paraboloid depicted in Fig. 9.14. In the disconnected
case (T0 = 1.5 and T0 = 2.0), the distance between the upper and lower parts is arbitrary (chosen here to be
∆z = 1). [Figure generated by the notebook D.4.19]
z 2 = 16m2 (r/2m − 1), or
z 2 = 8m(r − 2m) . (9.106)
We recognize the equation of a paraboloid of revolution around the z-axis. It is known as Flamm
paraboloid [197] and is depicted in Fig. 9.14. Its topology is clearly that of a cylinder (R × S1 ).
The geometry is different though: from top to bottom, the radius of the “cylinder” decreases to
a minimal value, rmin = 2m, and then increases. The “neck” around r = rmin , or equivalently
X = 0, is called the Einstein-Rosen bridge [179]. Contemplating the slice T = 0 in the
Kruskal diagram of Fig. 9.12, we realize that this “bridge” connects the two asymptotically flat
regions MI and MIII . The Einstein-Rosen bridge is also called the Schwarzschild wormhole.
However, it is not a traversable wormhole: it is clear from the Kruskal diagram (Figs. 9.8 and
9.12) or the Carter-Penrose diagram (Fig. 9.11) that no timelike or null worldline can go from
MI to MIII .
314 Maximal extension of Schwarzschild spacetime
Figure 9.16: Same Flamm paraboloid as in Fig. 9.14 but seen from farther away. Despite being more and more
flat, none of the two sheets is asymptotic to a plane. [Figure generated by the notebook D.4.19]
When T0 ̸= 0, the integral in (9.104) has to be computed numerically (see Sec. D.4.19 for the
computation with SageMath). The resulting embedded surfaces Φ(Σeq T0 ) are shown in Fig. 9.15
for the values of T0 involved in the Kruskal diagram of Fig. 9.12. When T0 increases from 0, the
“neck” becomes thinner and thinner. At T0 = 1, it ceases to be connected. As mentioned above,
for T0 > 1, the surface Σeq
T0 can no longer be entirely isometrically embedded in the Euclidean
3-space. Hence the holes in the central parts of the surfaces for T0 = 1.5 and T0 = 2. These
holes correspond to the dotted segments in Fig. 9.12 and their radii are given by Eq. (9.101).
Note that the tangents to the embedded surfaces at their inner boundaries are horizontal.
The evolution of ΣT0 as T0 increases is not surprising if one remembers that the Kruskal-
Szekeres time coordinate T is not associated with any timelike Killing vector of Schwarzschild
spacetime. The sequence shown in Fig. 9.15 can be thought of as representing the dynamics
of the Schwarzschild wormhole, in particular its “pinching-off” at T0 = 1, which forbids any
traveler to go through it.
Remark 3: We have restricted ourselves to slices T = const of Schwarzschild spacetime, with the
isometric embedding limitation for |T | > 1. We refer the reader to Ref. [134] for more general slices
and the corresponding embedding diagrams.
Remark 4: There are many inexact plots of embeddings of spatial sections of Schwarzschild spacetime
in the literature, including renown textbooks. A first common error is to draw the two ends of the
embedded surface as asymptotic to flat planes, which a paraboloid is not (the vertical distance between
√
the two ends grows unbounded, as r, cf. Eq. (9.105) and Fig. 9.16). This is correct from a topological
point of view, but not from the geometrical one, i.e. the embedding depicted in this way is not an
isometry. Probably this results from some confusion with asymptotic flatness: it is true that the metric
(9.94) tends to a flat metric when r → +∞, reflecting the asymptotic flatness of Schwarzschild spacetime,
but the associated curvature does not decay fast enough to allow the embedded surface to be tangent
to a plane. A second error regards the embeddings for |T0 | > 1, which are depicted as variants of that
T0 = 1 (cf. Fig. 9.15), with two spikes at r = 0, simply pushed apart. However, as discussed above, the
isometric embeddings with T = const cannot reach the region near r = 0 for |T0 | > 1.
9.5 Einstein-Rosen bridge 315
Historical note : In 1916, very soon after the publication of Schwarzschild solution [449], the Austrian
physicist Ludwig Flamm (1885-1964) showed that the slice t = const and θ = π/2 in Schwarzschild-
Droste coordinates (t, r, θ, φ) can be isometrically embedded in the Euclidean space as a paraboloid of
revolution obeying Eq. (9.106) [197]. Let us recall that the positive-X part of the hypersurface T = 0
considered here coincides with the hypersurface t = 0 (cf. Remark 1 on p. 310). Although he draw
the whole paraboloid (actually a parabola in a 2-dimensional plot — Fig. 2 of Ref. [197]), Flamm did
not seem to have considered the negative-z part as physically relevant. In other words, he limited his
considerations to MI and did not contemplate any bridge to the extension MIII .
9.5.3 Isotropic coordinates
Let us consider a hypersurface of constant Schwarzschild-Droste time t in MI . According to
Eq. (9.17), this hypersurface obeys
t
T = tanh X, (9.107)
4m
with X > 0, which implies that it is represented by a straight half-line from the origin in the
Kruskal diagram (cf. Fig. 9.8). Similarly, a hypersurface of constant t′ in MIII obeys an equation
identical to (9.107), except for t replaced by t′ and X < 0 [cf. Eq. (9.33)]. Accordingly, for t′ = t,
the union of these two hypersurfaces forms a hypersurface of M ruled by Eq. (9.107), with
X < 0 or X > 0. If we add the points (T, X) = (0, 0) to it (i.e. the bifurcation sphere S
(cf. Sec. 9.3.3), we obtain a connected hypersurface in which X takes all values in the range
(−∞, +∞). Let us call St this hypersurface. In other words, St is the hypersurface of M
defined by Eq. (9.107) with X ∈ R. Note that for t = 0, this hypersurface coincides with the
hypersurface T = 0 introduced in Sec. 9.5.1: S0 = Σ0 . But for t ̸= 0, St ̸= ΣT .
There are two Schwarzschild-Droste coordinate systems on St : (r, θ, φ) on St ∩ MI and
(r′ , θ, φ) on St ∩ MIII , with both r and r′ ranging (2m, +∞). Let us introduce on St a third
coordinate system (xī ) = (r̄, θ, φ) as follows:
m m 2
on St ∩ MI : r̄ ∈ , +∞ , r = r̄ 1 + (9.108a)
2 2r̄
1
(9.108b)
p
⇐⇒ r̄ = r − m + r(r − 2m)
2
m m 2
on St ∩ MIII : r̄ ∈ 0, , r′ = r̄ 1 + (9.108c)
2 2r̄
1 ′
(9.108d)
p
⇐⇒ r̄ = ′ ′
r − m − r (r − 2m)
2
m
on St ∩ S : r̄ = . (9.108e)
2
The range of r̄ is thus (0, +∞). The graph of the function r̄ 7→ r̄(1 + m/(2r̄))2 is depicted
in Fig. 9.17. We can separate this graph in two parts: r̄ ∈ (0, m/2) (the MIII part) and
r̄ ∈ (m/2, +∞) (the MI part). In each of these part, there is a one-to-one correspondence
316 Maximal extension of Schwarzschild spacetime
r/m
5
1 2 3 4
r̄/m
Figure 9.17: Areal radius r as a function of the isotropic coordinate r̄.
between r̄ and r (or r′ ). Note that
when r → +∞, r̄ ∼ r (9.109a)
m2
when r′ → +∞, r̄ ∼ ′ . (9.109b)
4r
When t varies, St constitute a foliation of
Miso := MI ∪ S ∪ MIII . (9.110)
This foliation is regular in both MI and MIII , but is singular at the bifurcation sphere S , since
all the hypersurfaces St intersect there (cf. Fig. 9.8). We may then consider (xᾱ ) = (t, r̄, θ, φ)
as a coordinate system on Miso , which is regular on MI and MIII , but is singular at S , i.e. at
r̄ = m/2. This system is called isotropic coordinates.
From (9.108a), we get
m 2
1−
m m 2m
dr = 1 + 1− dr̄ and 1− = 2r̄
m .
2r̄ 2r̄ r 1+ 2r̄
It is then immediate to deduce from (6.14) the expression of the metric tensor in terms of the
isotropic coordinates (xᾱ ) = (t, r̄, θ, φ):
m 2
1−
m 4 2
2r̄
dt2 + 1 + dr̄ + r̄2 dθ2 + sin2 θ dφ2 . (9.111)
g=− m
1+ 2r̄
2r̄
9.5 Einstein-Rosen bridge 317
Since the relation between r′ and r̄ is identical to that between r and r̄ [cf. Eqs. (9.108a) and
(9.108c)], the above expression of g is valid on MI and MIII . Note that all metric coefficients
are regular on MI ∪ MIII (except for the standard coordinate singularity of the spherical
coordinates (θ, φ) for θ ∈ {0, π}). On the contrary Eq. (9.111) yields det(gᾱβ̄ ) = 0 for r̄ = m/2,
which reflects the fact that the isotropic coordinates are singular on the bifurcation sphere S .
A remarkable feature of expresssion (9.111) is that the spatial part is proportional to the flat
metric f of the Euclidean 3-space:
f = dr̄2 + r̄2 dθ2 + sin2 θ dφ2 . (9.112)
In other words, the metric γ induced by g on St is conformal to the flat metric f (cf. Sec. 4.2.2):
γ = Ψ4 f , (9.113)
with the conformal factor6
m
Ψ=1+ . (9.114)
2r̄
The conformally-flat feature explains the name isotropic given to the coordinates (t, r̄, θ, φ).
Remark 5: Sometimes, isotropic coordinates are simply presented as coordinates deduced from the
standard Schwarzschild-Droste ones by formula (9.108a). But much more than a mere change of
coordinate r ↔ r̄ is involved: the two coordinate systems do not cover the same part of the extended
Schwarzschild spacetime: Schwarzschild-Droste coordinates cover the region MI ∪ MII , while isotropic
coordinates cover the region MI ∪ MIII . In particular, isotropic coordinates cannot be used to describe
the black hole interior (i.e. MII ). This last feature can be inferred directly from the square in the gtt
component read on (9.111), which implies an everywhere timelike Killing vector ∂t , while ∂t is spacelike
in MII (cf. Sec. 6.2.5).
9.5.4 Recapitulation: coordinates on Schwarzschild spacetime
At this point, we have introduced many coordinate systems on Schwarzschild spacetime:
• Schwarzschild-Droste (t, r, θ, φ) (Sec. 6.2.3);
• ingoing Eddington-Finkelstein (t̃, r, θ, φ) (Sec. 6.3.2);
• null ingoing Eddington-Finkelstein (v, r, θ, φ) (Sec. 6.3.2);
• null outgoing Eddington-Finkelstein (u, r, θ, φ) (Sec. 6.4);
• Kruskal-Szekeres (T, X, θ, φ) (Sec. 9.2.1);
• global null (U, V, θ, φ) (Sec. 9.3.2);
• Penrose-Frolov-Novikov (Ũ , Ṽ , θ, φ) (Sec. 9.4.3);
6
Note a different convention with respect to Sec. 4.2.2: with respect to the latter, what would be called the
conformal factor is the square of Ψ: Ω = Ψ2 .
318 Maximal extension of Schwarzschild spacetime
• isotropic (t, r̄, θ, φ) (Sec. 9.5.3).
Two other coordinate systems will be introduced in Chap. 14:
• Lemaître synchronous (τ, χ, θ, φ) (Sec. 14.2.6);
• Painlevé-Gullstrand (τ, r, θ, φ) (Sec. 14.2.7).
9.6 Physical relevance of the maximal extension
9.6.1 Naked singularity
Beside harboring a white hole (cf. Sec. 9.4.4), the maximal extension of Schwarzschild spacetime
contains a naked singularity, i.e. a curvature singularity that can be seen by arbitrarily far
observers in the asymptotically flat regions MI and MIII . Indeed, it is clear from the Carter-
Penrose diagram of Fig. 9.11 that the past null cone of any event in MI or MIII encounters
the curvature singularity r′ = 0 at the past boundary of MIV . In other words, any observer in
MI or MIII can receive signals from this singularity. Contrary to the singularity r = 0 at the
future boundary of MII , it is not “clothed” by a black hole horizon, but by a white hole horizon,
which does not prevent null geodesics from moving from the singularity to the observer.
In a given spacetime, a naked singularity constitutes a limitation to predictability, since
one cannot compute what may come out of the singularity at any instant. That no “reasonable”
physical process can generate a naked singularity is known as the (weak) cosmic censorship
conjecture, first formulated by Penrose in 1969 [408] (see also Sec. 10.1 of Wald’s textbook
[499]).
9.6.2 Astrophysical relevance
We shall see in Chap. 14 that, in spherical symmetry, the formation of a black hole by the
astrophysical process of gravitational collapse of a star or a cloud of matter yields a spacetime
that contains some parts of regions MI and MII of Schwarzschild spacetime, but that do not
contain any part of MIII nor MIV (cf. the Carter-Penrose diagram of Fig. 14.4). In particular,
it does not contain any white hole nor any naked singularity. In other words, the maximal
extension of Schwarzschild spacetime cannot be formed by gravitational collapse of a star or
a cloud of matter. This, of course, does not exclude by itself the existence of (approximate)
maximally extended Schwarzschild regions in our universe. Such regions could exist because
the universe is born with them built in. For instance, we shall see in Chap. 14 that there exist
solutions of the Einstein equation describing a spherically symmetric ball of matter that is
expanding from an initial singularity, reaches some maximal extension and then collapses and
whose exterior contains parts of regions MI , MII and MIV , and even MIII (in which case, it is
called a semiclosed world), cf. Figs. 14.5 and 14.6, as well as Remark 3 on p. 559. For the above
reasons, the maximal extension of the Schwarzschild spacetime is sometimes called the eternal
Schwarzschild black hole.
9.6 Physical relevance of the maximal extension 319
9.6.3 Use in theoretical physics
Even if the astrophysical motivation for the Schwarzschild maximal extension is pretty weak,
this spacetime plays some role in theoretical physics. For instance, it has been advanced that
quantum entanglement between two black holes can be realized by an Einstein-Rosen bridge
[354], a conjecture that is known as ER = EPR, where ER stands for Einstein-Rosen and EPR for
Einstein-Podolsky-Rosen, from the famous EPR paradox involving entangled particles (see e.g.
Ref. [341]).
320 Maximal extension of Schwarzschild spacetime
Part III
Kerr black hole
Chapter 10
Kerr black hole
Contents
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
10.2 The Kerr solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
10.3 Extension of the spacetime manifold through ∆ = 0 . . . . . . . . . . 334
10.4 Principal null geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
10.5 Event horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
10.6 Global quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
10.7 Families of observers in Kerr spacetime . . . . . . . . . . . . . . . . . 354
10.8 Maximal analytic extension . . . . . . . . . . . . . . . . . . . . . . . . 364
10.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
10.1 Introduction
Having studied the Schwarzschild black hole in the preceding chapters, we turn now to its
rotating generalization: the Kerr black hole. The Kerr metric is arguably the most important
solution of general relativity, largely because of the no-hair theorem, according to which all
stationary black holes in the Universe are Kerr black holes (cf. Sec. 5.6).
In this chapter, the Kerr solution is first presented in terms of the standard Boyer-Lindquist
coordinates and its basic properties are discussed (Sec. 10.2). Then Kerr coordinates are
introduced in Sec. 10.3; contrary to Boyer-Lindquist coordinates, they are regular on the two
Killing horizons of Kerr spacetime. Kerr coordinates are also tied to one of the two congruences
of null geodesics related to the spacetime conformal structure: the principal null geodesics,
which are introduced in Sec. 10.4. The second congruence, that of the so-called outgoing
principal null geodesics, provides the generators of the black hole event horizon, which is
studied in Sec. 10.5. Then Sec. 10.6 focuses on global quantities characterizing Kerr spacetime:
mass, angular momentum and horizon area. Section 10.7 presents various standard families of
324 Kerr black hole
observers in Kerr spacetime. Finally, Sec. 10.8 discusses the maximal analytical extension of
Kerr spacetime and the concept of Cauchy horizon.
10.2 The Kerr solution
10.2.1 Expression in Boyer-Lindquist coordinates
The Kerr solution depends on two constant non-negative real parameters:
• the mass parameter m > 0, to be interpreted in Sec. 10.6.1 as the spacetime total mass;
• the spin parameter a ≥ 0, to be interpreted in Sec. 10.6.2 as the specific angular
momentum a = J/m, J being the spacetime total angular momentum.
In this chapter, we focus on Kerr solutions for which
0 < a < m, (10.1)
postponing the case a = m to Chap. 13. The Kerr solution is usually presented in the so-called
Boyer-Lindquist coordinates (t, r, θ, φ). Except for the standard singularities of the spherical
coordinates (θ, φ) on S2 at θ ∈ {0, π}, we may consider that Boyer-Lindquist coordinates cover
the manifold R2 × S2 , with t spanning R, r spanning1 R, θ spanning (0, π) and φ spanning
(0, 2π). Hence (t, r) is a Cartesian chart covering R2 and (θ, φ) is the standard spherical chart
of S2 .
In this section, we choose the spacetime manifold to be the open subset MBL of R2 × S2
formed by the disjoint union of the following three connected components (cf. Fig. 10.1):
MBL := MI ∪ MII ∪ MIII , (10.2a)
MI := R × (r+ , +∞) × S2 (10.2b)
MII := R × (r− , r+ ) × S2 (10.2c)
MIII := R × (−∞, r− ) × S2 \ R, (10.2d)
where √ √
r+ := m + m 2 − a2 and r− := m − m2 − a2 (10.3)
and R is the subset of R2 × S2 defined in terms of Boyer-Lindquist coordinates (t, r, θ, φ) by
n πo
R := p ∈ R × S , r(p) = 0 and θ(p) =
2 2
. (10.4)
2
Remark 1: By construction, the points of R2 × S2 obeying r = r− and r = r+ are excluded from the
spacetime manifold MBL . In a latter stage (Sec. 10.3), we shall extend the spacetime manifold to include
these points, so that the spacetime manifold will be M = R2 × S2 \ R. Even latter on, after having
1
This contrasts with r spanning only (0, +∞) for the standard spherical coordinates (r, θ, φ) on R3 .
10.2 The Kerr solution 325
I
II
III
plane
rotation axis
Figure 10.1: View of a section t = const of R2 × S2 in terms of the coordinates (R, θ, φ), with R := er , so
that the region r → −∞ is reduced to a single point at the centre of all the pictured spheres. Such coordinates
have been introduced for pictorial purposes by O’Neill [391].
noticed that (M , g) is not geodesically complete, we shall extend the spacetime manifold further and
discuss the maximal analytic extension (Sec. 10.8).
Remark 2: One shall stress that the points having r = 0 are not special points in R2 × S2 spanned
by the coordinates (t, r, θ, φ). In particular, (t, r) = (t0 , 0), where t0 is a constant, defines a regular
sphere S0 diffeomorphic to S2 . On the spacetime manifold MBL , the points r = 0 are part of MIII and
because of the exclusion of R in the definition (10.2d), (t, r) = (t0 , 0) defines a sphere minus its equator
(cf. Fig. 10.1, where the equator is the tick orange circle).
Note that thanks to the constraint (10.1), r+ and r− are well-defined and obey
0 < r− < m < r+ < 2m. (10.5)
Note also that R is spanned by the coordinates (t, φ) and is diffeomorphic to the 2-dimensional
cylinder R × S1 :
R ≃ R × S1 . (10.6)
This is so because r = 0 is not a peculiar value of r in R2 × S2 (cf. Remark 2 above). In view of
Eqs. (10.2b)-(10.2d) and (10.4), it is clear that the various connected components of MBL are
defined in terms of the Boyer-Lindquist coordinates (t, r, θ, φ) by
∀p ∈ MBL , p ∈ MI ⇐⇒ r(p) > r+ (10.7a)
p ∈ MII ⇐⇒ r− < r(p) < r+ (10.7b)
π
p ∈ MIII ⇐⇒ r(p) < r− and r(p) ̸= 0 or θ(p) ̸= . (10.7c)
2
326 Kerr black hole
The Kerr metric is defined in terms of the Boyer-Lindquist coordinates (t, r, θ, φ) by
4amr sin2 θ ρ2 2
2mr
g = − 1− 2 dt2 − dt dφ + dr
ρ ρ2 ∆
(10.8)
2a2 mr sin2 θ
2 2 2 2
+ρ dθ + r + a + sin2 θ dφ2 ,
ρ2
with
ρ2 := r2 + a2 cos2 θ (10.9)
and
∆ := r2 − 2mr + a2 = (r − r− )(r − r+ ) . (10.10)
Remark 3: Since dt dφ := 1/2 (dt ⊗ dφ + dφ ⊗ dt) [cf. Eq. (A.39)], the metric component gtφ is one
half of the coefficient of the dt dφ term in Eq. (10.8), i.e. gtφ = −2amr sin2 θ/ρ2 .
Remark 4: On MBL , ρ ̸= 0 and ∆ ̸= 0 (by construction of MBL !), so that the metric components (10.8)
are regular in MBL , except for the standard singularities of the spherical coordinates (θ, φ).
By means of a computer algebra system (cf. the SageMath notebook D.5.1), one easily
checks:
Property 10.1: Kerr metric as a solution of the vacuum Einstein equation
The Kerr metric g [Eq. (10.8)] is a solution of the vacuum Einstein equation (1.44) on MBL .
Historical note : The Kerr solution has been found by the New Zealand mathematician Roy P. Kerr
(then at the University of Texas at Austin) in the spring of 1963 [312]. Kerr was searching for algebraically
special metrics, i.e. metrics whose Weyl conformal curvature tensor admits a doubly degenerate principal
null direction (to be defined in Sec. 10.4 below), in the case where the principal null congruence has a
non-vanishing twist (or “rotation”). The special case of vanishing twist (i.e. hypersurface-orthogonal
congruence) had been treated by Ivor Robinson and Andrzej Trautman in 1962 [440]. Kerr used Cartan’s
structure equations in a null tetrad to manipulate the Einstein equation; he obtained the solution in
coordinates different from the Boyer-Lindquist ones, known today as Kerr coordinates and to be discussed
in Sec. 10.3.1 (cf. the historical note page 336). For more details about this fantastic discovery, see the
account by Kerr himself in Refs. [313, 324]. Boyer-Lindquist coordinates have been introduced in 1966
by Robert H. Boyer (see the historical note on p. 87) and Richard W. Lindquist [72] (compare Eq. (2.13) of
Ref. [72] with Eq. (10.8) above, keeping in mind that Boyer and Lindquist used −a instead of a, following
Kerr’s convention in the discovery article [312], cf. the historical note on p. 336).
10.2.2 Basic properties
Various properties of the Kerr metric are immediate:
10.2 The Kerr solution 327
Property 10.2: asymptotic flatness of Kerr spacetime
The spacetime (MBL , g) has two asymptotically flat ends: one in MI for r → +∞, which
is equivalent to the asymptotics of a Schwarzschild spacetime of mass m and the other one
in MIII for r → −∞, which is equivalent to the asymptotics of a Schwarzschild spacetime
of (negative!) mass −m.
Proof. For r → +∞ or r → −∞, one has ρ2 ∼ r2 and ρ2 /∆ ∼ (1 − 2m/r)−1 , and
4amr/ρ2 dt dφ ∼ 4am/r2 dt rdφ, so that the metric (10.8) becomes
−1
2m 2m 1
2 2 2 2 2 2
(10.11)
g ≃− 1− dt + 1 − dr + r dθ + sin θ dφ + O 2
r r r
For r > 0, we recognize the Schwarzschild metric expressed in Schwarzschild-Droste coor-
dinates [Eq. (6.14)]. For r < 0, the change of coordinate r′ = −r leads to the Schwarzschild
metric as well, but with the negative mass parameter m′ = −m.
Property 10.3: stationarity and axisymmetry of Kerr spacetime
The spacetime (MBL , g) admits two isometries: stationarity and axisymmetry, generated
respectively by the Killing vectors
ξ := ∂t and η := ∂φ . (10.12)
Proof. All the metric components gαβ in Eq. (10.8) are independent from t and φ. This implies
that the vector fields (10.12) are Killing vectors (cf. Sec. 3.3.1). Since t spans R, the isometry
group generated by ξ is clearly the translation group (R, +). Moreover, in view of (10.11), we
have ξ · ξ = gtt < 0 as r → +∞, which means that the Killing vector ξ is asymptotically
timelike. Given the definition of stationarity stated in Sec. 5.2.1, we conclude that the Kerr
spacetime is stationary. On the other side, since φ is an azimuthal coordinate on S2 , the
isometry group generated by η is the rotation group SO(2) = U(1). Moreover, ∂φ is spacelike
for r → +∞ (even for r > 0). Hence, the Kerr spacetime is axisymmetric (cf. the definition
given in Sec. 5.3.6).
Property 10.4: non-staticity for a ̸= 0
For a ̸= 0, as we have assumed in (10.1), the Kerr spacetime is not static (cf. the definition
in Sec. 5.2.1): the stationary Killing vector ξ is not hypersurface-orthogonal.
Proof. From Eq. (10.8), we have ξ · η = gtφ ̸= 0 for a ̸= 0. Since η = ∂φ is tangent to the
hypersurfaces t = const, this implies that ξ is not normal to these hypersurfaces. That there
does not exist any hypersurface family orthogonal to ξ can be seen from the non-vanishing of
328 Kerr black hole
the twist 3-form defined by Eq. (3.63): ω := ξ ∧ dξ. We have indeed (see the notebook D.5.1
for the computation)
2am sin θ h 2 2 2
i
ω= sin θ(a cos θ − r ) dt ∧ dr ∧ dφ + 2∆r cos θ dt ∧ dθ ∧ dφ .
ρ4
Clearly ω ̸= 0 as soon as a ̸= 0. According to the Frobenius theorem (see e.g. Eq. (B.3.6) in
Wald’s textbook [499]), it follows that ξ is not hypersurface-orthogonal.
Property 10.5: Schwarzschild limit
For a → 0, the Kerr metric reduces to the Schwarzschild metric.
Proof. When a → 0, we have r+ → 2m, r− → 0, ρ2 ∼ r2 , and ρ2 /∆ ∼ (1 − 2m/r)−1 ,
and we see on (10.8) that the Kerr metric reduces to the Schwarzschild metric expressed in
Schwarzschild-Droste coordinates [Eq. (6.14)].
Property 10.6: the double flat disk at r = 0
The metric induced by the Kerr metric on the hypersurface r = 0 of MBL is
h = −dt2 + a2 cos2 θ dθ2 + sin2 θ dφ2 . (10.13)
For a ̸= 0, h is a flat Lorentzian metric, which can be brought to a manifestly Minkowskian
form via the change coordinates x := a sin θ cos φ, y := a sin θ sin φ: h = −dt2 +dx2 +dy 2 .
In particular, for a fixed value of t, the r = 0 subset
S0,t := {p ∈ MIII , r(p) = 0, t(p) = t} (10.14)
is made of two connected components, which are two flat open disks of radius a, corre-
sponding respectively to θ < π/2 and θ > π/2, since the equator θ = π/2 is excluded by
the very definition of MIII (cf. Remark 2 above).
Proof. On the hypersurface r = 0, we have ρ2 = a2 cos2 θ, ∆ = a2 and dr = 0, so that the
Kerr metric (10.8) induces the metric (10.13). That S0,t is made of two disks is obvious from
the range of variation of the coordinates (x, y), which by construction obey x2 + y 2 < a2 (cf.
Fig. C.3 in Appendix C, where the Kerr-Schild coordinates (x, y) coincide with the current
coordinates (x, y), up to some rotation).
The set S0,t is depicted by the dotted red line in Fig. 10.1. It is also depicted in terms of the
so-called Kerr-Schild coordinates in Figs. C.1 - C.4 of Appendix C.
10.2 The Kerr solution 329
10.2.3 Determinant and inverse metric
The determinant of the metric g with respect to Boyer-Lindquist coordinates is deduced from
(10.8); it takes a relatively simple form (see the notebook D.5.1 for the computation):
det(gαβ ) = −ρ4 sin2 θ. (10.15)
The inverse metric is (see the notebook D.5.1 for the computation)
2 2
− ∆1 r2 + a2 + 2a mrρ2sin θ 0 0 − 2amr
ρ2 ∆
∆
0 ρ2
0 0
αβ
g = . (10.16)
0 0 ρ12 0
− 2amr
2
ρ ∆
0 0 1
2
∆ sin θ
1 − 2mr
ρ 2
10.2.4 Ergoregion
Let us investigate the causal character of the stationary Killing vector ξ. We have, according to
(10.8) and (10.9),
2mr
ξ · ξ = gtt = −1 + 2 .
r + a2 cos2 θ
Thus
ξ timelike ⇐⇒ r2 − 2mr + a2 cos2 θ > 0 ⇐⇒ r < rE − (θ) or r > rE + (θ),
with √
rE ± (θ) := m ± m2 − a2 cos2 θ . (10.17)
Comparing with (10.3), we note that
0 ≤ rE − (θ) ≤ r− ≤ m ≤ r+ ≤ rE + (θ) ≤ 2m, (10.18)
with
rE − (0) = rE − (π) = r− and rE − (π/2) = 0 (10.19a)
rE + (0) = rE + (π) = r+ and rE + (π/2) = 2m. (10.19b)
Given the definition of MI , MII and MIII , we conclude that:
Property 10.7: ergoregion of Kerr spacetime
The stationary Killing vector ξ of Kerr spacetime obeys:
• ξ is timelike in the region of MI defined by r > rE + (θ) and in the region of MIII
defined by r < rE − (θ);
• ξ is null on the hypersurface E + of MI defined by r = rE + (θ) and on the hypersurface
E − of MIII defined by r = rE − (θ);
330 Kerr black hole
e r/m cosθ
I 4
II 2
in +
III −
e r/m sinθ
-5 5
r=0
-2
-4
Figure 10.2: Meridional view of a section t = const of Kerr spacetime with a/m = 0.90 in O’Neill exponential
coordinates x = er/m sin θ and z = er/m cos θ (cf. Fig. 10.1). The right (resp. left) half of the figure corresponds
to φ = 0 (resp. φ = π). The Roman numbers I, II, III denote the components MI , MII and MIII of the manifold
MBL . The dotted orange circle marks the location of r = 0, while the small black circle at the center of the figure
corresponds to r → −∞. The two red dots marks the curvature singularity R. The ergoregion (cf. Sec. 10.2.4) is
shown in grey, while the yellow part is Carter time machine (cf. Sec. 10.2.5).
e r/m cosθ
6
I
4
II 2
+
in
III e r/m sinθ
-5 5
r=0
-2
-4
-6
Figure 10.3: Same as Fig. 10.2 but for a/m = 0.50.
10.2 The Kerr solution 331
e r/m cosθ
4
I
2 in
II III +
−
e r/m sinθ
-5 5
r=0
-2
-4
Figure 10.4: Same as Fig. 10.2 but for a/m = 0.99.
• ξ is spacelike in all MII and in the region G + of MI defined by r < rE + (θ), as well
as in the region G − of MIII defined by r > rE − (θ).
According to the nomenclature introduced in Sec. 5.4.2, one calls E + (resp. E − ) the outer
ergosphere (resp. inner ergosphere) and G + (resp. G − ) the outer ergoregion (resp. inner
ergoregion). The part of MBL where ξ is spacelike, i.e. G = G + ∪ MII ∪ G − , is called the
ergoregion.
Following the standard notation in topology, we shall denote by G the closure of G , i.e. the
union of the ergoregion and the two ergospheres:
G := G ∪ E − ∪ E + . (10.20)
The locus where the Killing vector ξ is timelike is then MBL \ G .
The ergoregion is depicted in Figs. 10.2–10.4. It is worth noticing that the locations of
the inner and outer ergospheres in the equatorial plane (θ = π/2) do not depend on the spin
parameter a: rE − (π/2) = 0 [Eq. (10.19a)] and rE + (π/2) = 2m [Eq. (10.19b)], a feature that
appears clearly in Figs. 10.2–10.4, given that e2 ≃ 7.39.
Remark 5: By taking the limit a → 0 in Eq. (10.17), we get rE + (θ) → 2m. Since r+ → 2m in the same
limit, we conclude that, for a Schwarzschild black hole, the outer ergosphere coincides with the event
horizon. Consequently, there is no outer ergoregion for such a black hole: G + = ∅.
Remark 6: Sometimes the word ergosurface is used instead of ergosphere. One may encounter as
well the names infinite redshift surface (e.g. [495]) and static limit (cf. Sec. 10.7.2 below) for ergosphere,
especially in old texts.
Remark 7: By construction, the Killing vector ξ is null on the ergospheres E + and E − . It is moreover
tangent to these hypersurfaces (see below). However, for a ̸= 0, E + and E − are not Killing horizons,
332 Kerr black hole
as defined in Sec. 3.3.2, since ξ fails to be normal to them, or equivalently, E + and E − are not null
hypersurfaces. They are actually timelike hypersurfaces, except on the rotation axis. This can be seen by
considering E ± as the level set f = 0 of the scalar field f := r − rE ± (θ) = r − m ∓ (m2 − a2 cos2 θ)1/2
[cf. Eq. (10.17)]. The differential of f is df = dr ± a2 sin θ cos θ(m2 − a2 cos2 θ)−1/2 dθ, so that
→
−
⟨df, ξ⟩ = 0, which shows that ξ is tangent to E ± . The scalar square of the gradient vector field ∇f
associated to df by metric duality is easily evaluated via the inverse metric (10.16):
a4 sin2 θ cos2 θ E ± a2 m2 sin2 θ
→
− →
− µν 1
∇f · ∇f = g ∂µ f ∂ν f = 2 ∆ + 2 = .
ρ m − a2 cos2 θ ρ2 (m2 − a2 cos2 θ)
→
− →
−
Away from the rotation axis, i.e. for sin θ ̸= 0, we have clearly ∇f · ∇f > 0 for a ̸= 0, which shows
→
−
that the normal ∇f to E ± is spacelike, making E ± a timelike hypersurface. Further details about the
outer ergosphere E + , especially the 2-dimensional geometry of its time slices, are discussed in Ref. [295].
We shall see in Sec. 11.3.2 that the outer ergoregion plays a key role in an energy extraction
mechanism known as the Penrose process.
10.2.5 Carter time machine
Let us now focus on the second Killing vector, η. From (10.8) and (10.9), we have
2a2 mr sin2 θ
2
η · η = gφφ = r + a + 22
sin2 θ. (10.21)
r + a2 cos2 θ
Hence
η spacelike ⇐⇒ (r2 + a2 )(r2 + a2 cos2 θ) + 2a2 mr sin2 θ > 0.
For θ → 0 or θ → π, the left-hand side of the above inequality is always positive, but for
θ = π/2 and r negative with |r| small enough so that 2a2 m|r| > r2 (r2 + a2 ), it is negative. This
feature is apparent on Fig. 10.5: for θ close to π/2, there is a region T defined by rT (θ) < r < 0
for some negative function rT (θ), such that gφφ < 0. Since T corresponds to negative values
of r, we have T ⊂ MIII . Hence we conclude:
Property 10.8: Carter time machine
The axisymmetric Killing vector η of Kerr spacetime obeys:
• η is spacelike in all MI and MII , as well as outside the region T in MIII ;
• η is timelike in the subset T of MIII ;
• η is null at the boundary of T .
The region T is called Carter time machine. This name stems from the fact that thanks
to T , there is a future-directed timelike curve connecting any two points of MIII .
See e.g. Proposition 2.4.7 of O’Neill’s textbook [391] for a demonstration, or Carter’s original
article [90]) for the proof of the last sentence. The Carter time machine T is depicted in yellow
in the meridional diagrams of Figs. 10.2-10.4.
10.2 The Kerr solution 333
ρ 2 (r 2 + a 2 ) + 2a 2 mr sin 2 θ
θ=0
θ = π/4
θ = π/3
8 θ = π/2
-1.5 -1 -0.5 0.5 1 1.5
r/m
Figure 10.5: Graph of the function giving the sign of gφφ for a = 0.9m and various values of θ.
10.2.6 Singularities
The components gαβ of the Kerr metric as given by (10.8) are diverging at various locations:
• when ρ2 → 0, which, given (10.9) and assuming a ̸= 0, is equivalent to approaching the
cylinder R defined by (10.4);
• when ∆ → 0, which, given (10.10), is equivalent to either r → r− or r → r+ ; the first
case corresponds to the boundary (within R2 × S2 ) between MII and MIII and the second
case to the boundary between MI and MII .
The divergence when ρ2 → 0 corresponds to a curvature singularity:
Property 10.9: ring singularity
The curvature of Kerr spacetime, as measured by the Kretschmann scalar K := Rµνρσ Rµνρσ
[cf Eq. (6.44)], diverges for ρ2 → 0, i.e. for r → 0 and θ → π/2. Accordingly the subset R
of R2 × S2 defined by ρ2 = 0 [Eq. (10.4)] is called the ring singularity of Kerr spacetime;
the word ring reflects that t = const sections of R are circles [cf. Eq. (10.6)].
Proof. The Kretschmann scalar expressed in terms of Boyer-Lindquist coordinates is (cf. the
notebook D.5.1 for the computation)
m2 6
r − 15r4 a2 cos2 θ + 15r2 a4 cos4 θ − a6 cos6 θ . (10.22)
K = 48 12
ρ
The value for θ = π/2 is thus K = 48m2 /r6 , which clearly diverges for r → 0 (i.e. ρ2 → 0).
334 Kerr black hole
See Ref. [124] for an extended discussion of the ring singularity.
On the contrary, the divergence of the metric components (10.8) when ∆ → 0 corresponds
to a mere coordinate singularity, i.e. to a pathology of Boyer-Lindquist coordinates. The latter
can be cured by switching to other coordinates, as we shall see in the next section.
10.3 Extension of the spacetime manifold through ∆ = 0
10.3.1 Advanced Kerr coordinates
The advanced Kerr coordinates are coordinates (xα̂ ) = (v, r, θ, φ̃) defined on R2 × S2 and
related to the Boyer-Lindquist coordinates (xα ) = (t, r, θ, φ) introduced in Sec. 10.2.1 by
r 2 + a2
dv = dt + dr (10.23a)
∆
a
dφ̃ = dφ + dr . (10.23b)
∆
If a = 0, we note that the advanced Kerr coordinates are nothing but the null ingoing Eddington-
Finkelstein coordinates on Schwarzschild spacetime (cf. Sec. 6.3.2 and compare (10.23a) with
(6.28)).
Remark 1: In the literature, the advanced Kerr coordinates are often simply called Kerr coordinates (e.g.
[371, 464]). However, we prefer to keep this short name for closely related coordinates to be introduced
in Sec. 10.3.3. Both systems originate in Kerr’s seminal article [312] (cf. the historical notes on p. 336
and p. 340). The advanced Kerr coordinates are also called Kerr-star coordinates by O’Neill [391].
Given that ∆ = (r − r− )(r − r+ ) = r2 + a2 − 2mr [Eq. (10.10)], we have the identities
r 2 + a2
2m r+ r− a a 1 1
=1+ − and = − ,
∆ r+ − r− r − r+ r − r− ∆ r+ − r− r − r+ r − r−
√
with r+ − r− = 2 m2 − a2 , so that Eqs. (10.23) can be readily integrated to
m r − r+ r − r−
v =t+r+ √ r+ ln − r− ln (10.24a)
m2 − a2 2m 2m
a r − r+
φ̃ = φ + √ ln , (10.24b)
2
2 m −a 2 r − r−
up to some additive constants. When a → 0, we have r+ → 2m and r− → 0 and by comparing
Eq. (10.24a) with Eq. (6.27), we recover that the advanced Kerr coordinates reduces to the null
ingoing Eddington-Finkelstein ones in this limit.
The expression of the metric tensor g in terms of the advanced Kerr coordinates (v, r, θ, φ̃)
is computed from that in terms the Boyer-Lindquist ones, as given by Eq. (10.8), via (10.23).
10.3 Extension of the spacetime manifold through ∆ = 0 335
One gets (cf. Appendix D or Eq. (5.31) of Ref. [266], or Lemma 2.5.2 of [391]):
4amr sin2 θ
2mr
g = − 1− 2 dv 2 + 2dv dr − dv dφ̃
ρ ρ2
(10.25)
2a2 mr sin2 θ
2 2 2 2
2
−2a sin θ dr dφ̃ + ρ dθ + r + a + sin2 θ dφ̃2 .
ρ2
The inverse metric is pretty simple (cf. the notebook D.5.2-1):
a2 sin2 θ r2 + a2 0 a
2 2
r +a
1 ∆ 0 a
g α̂β̂ = 2 . (10.26)
ρ 0 0 1 0
a a 0 sin12 θ
We note that the metric components (10.25) do not show any divergence when ∆ → 0,
contrary to the Boyer-Lindquist ones. Hence, we may extend the Kerr metric to the points of
R2 × S2 where ∆ = 0, i.e. to the hypersurfaces (cf. Fig. 10.1)
H := p ∈ R2 × S2 , r(p) = r+ (10.27)
and
Hin := p ∈ R2 × S2 , (10.28)
r(p) = r− .
The hypersurface H is actually the boundary between the regions MI and MII , while Hin is
the boundary between MII and MIII (cf. Eq. (10.7) and Fig. 10.2). We thus consider
M := MBL ∪ H ∪ Hin = R2 × S2 \ R (10.29)
as the spacetime manifold. In order for g defined by (10.25) to be a well-defined metric on M ,
it does not suffice that the components gα̂β̂ do not diverge at H and Hin : one shall check as
well that the bilinear form g is non-degenerate there. This is easily proven by considering the
determinant of the metric components (10.25), which turns out to have a simple form:
det(gα̂β̂ ) = −ρ4 sin2 θ. (10.30)
Except at θ = 0 and θ = π (the usual singularities of spherical coordinates), we have det(gα̂β̂ ) ̸=
0 everywhere on M , since ρ vanishes only on R, which is excluded from M . Hence we
conclude
Property 10.10: Kerr spacetime
The symmetric bilinear form g given by Eq. (10.25) is regular and non-degenerate on the
manifold M defined by (10.29) and thus (M , g) is a well-behaved spacetime — our Kerr
spacetime from now on. We note that, contrary to MBL , M is a connected manifold.
336 Kerr black hole
We deduce from (10.23) that
∂v ∂v r 2 + a2 ∂ φ̃ a ∂ φ̃
= 1, = , = , = 1.
∂t r,θ,φ ∂r t,θ,φ ∆ ∂r t,θ,φ ∆ ∂φ t,r,θ
It follows from the chain rule that the advanced Kerr coordinate frame is related to the Boyer-
Lindquist coordinate frame by
∂v = ∂t (10.31a)
2 2
a +r a
∂r̂ = ∂r − ∂t − ∂φ (10.31b)
∆ ∆
∂θ = ∂θ (10.31c)
∂φ̃ = ∂φ . (10.31d)
Note that we are using the notation ∂r̂ for the ∂/∂r vector of the advanced Kerr coordinates
(xα̂ ) = (v, r, θ, φ̃), to distinguish it from the ∂/∂r vector of Boyer-Lindquist coordinates.
Remark 2: Contrary to the Schwarzschild case (cf. Sec. 6.3.2), v is not a null coordinate for a ̸= 0,
but rather a spacelike one, i.e. the hypersurfaces of constant v are timelike (cf. the definitions given in
Sec. A.3.2). Indeed, we read on (10.26) that g vv = a2 sin2 θ/ρ2 . For a ̸= 0, this implies g vv > 0 outside
the rotation axis. The criterion (A.56c) allows us to conclude that v is a spacelike coordinate.
Historical note : The advanced Kerr coordinates are those in which Roy P. Kerr originally presented
his solution in 1963 [312]. As noted by Kerr himself later on [313], he used −a instead of a, because
he was “rather hurried in performing this calculation (angular momentum) and got the sign wrong”
(footnote in Sec. 2.5 of Ref. [313]). Actually, it was shown in 1964 by Robert H. Boyer and T.G. Price [73]
that the angular momentum about the rotation axis of the Kerr solution is J = −am, where a is Kerr’s
one. Taking this into account, the correspondence between our notations and those of Kerr’s article
[312] is v ↔ u and a ↔ −a. Then one can check that the metric (10.25) coincides with that given by an
unnumbered (!) equation in Kerr’s article [312].
10.3.2 Time orientation of Kerr spacetime
We read on (10.25) that g(∂r̂ , ∂r̂ ) = grr = 0, which implies that ∂r̂ is a global null vector field
on M . We may then use it to set the time orientation of (M , g) (cf. Sec. 1.2.2):
Property 10.11: time orientation of Kerr spacetime
The Kerr spacetime (M , g) is time-orientable, and we choose its time orientation such that
k := −∂r̂ (10.32)
is a future-directed null vector field in all M .
Remark 3: The minus sign in the above definition, along with Eq. (10.31b), ensures that
k ∼ ∂t − ∂r when r → +∞,
10.3 Extension of the spacetime manifold through ∆ = 0 337
which shows that the time orientation set by k agrees asymptotically with that of ξ = ∂t . The latter
vector field could not have been chosen to set the time orientation of (M , g) since it is not causal
everywhere, being spacelike in the ergoregion.
The field lines of k are future-directed null curves, which may be qualified of ingoing since,
by definition, −∂r̂ points towards decreasing values of r. Note that, by the very definition of
the coordinate vector ∂r̂ , the values of the coordinates (v, θ, φ̃) are fixed along each of these
null curves. We may therefore denote them by L(v,θ, in
φ̃) . We shall see in Sec. 10.4 that each
L(v,θ,φ̃) is actually a null geodesic.
in
Decaying of r towards the future in MII
The scalar square of the vector ∂r of Boyer-Lindquist coordinates is read from the metric
components (10.8): ∂r · ∂r = g(∂r , ∂r ) = grr = ρ2 /∆. Since ∆ is positive in MI and MIII and
negative in MII , we conclude that ∂r is spacelike in MI ∪ MIII and timelike in MII . Moreover,
the above choice of time orientation leads to
Property 10.12
In region MII , the vector ∂r of Boyer-Lindquist coordinates is a past-directed timelike
vector.
Proof. Applying Lemma 1.2 (Sec. 1.2.2) with u = k and v = ∂r , we get that ∂r is past-directed
iff g(k, ∂r ) > 0. Now, in terms of the Boyer-Lindquist components (10.8), g(k, ∂r ) = gµr k µ =
grr k r = (ρ2 /∆)k r . The Boyer-Lindquist component k r is given by Eq. (10.31b) where ∂r̂ = −k;
we get k r = −1. Hence g(k, ∂r ) = −ρ2 /∆ > 0 in MII , for ∆ < 0 there.
An important consequence of the above property is
Property 10.13: decreasing of r in MII
In region MII , the coordinate r must decrease towards the future along any causal (i.e.
timelike or null) worldline.
Proof. Let L be a causal curve in region MII and λ a parameter along L increasing towards
the future. The associated tangent vector v = dx/dλ is then future-directed. According to
Property 10.12, −∂r is a future-directed timelike vector in MII , so that we can apply Lemma 1.1
(Sec. 1.2.2) with u = −∂r and get g(−∂r , v) < 0. Now, using Boyer-Lindquist components,
we have
dr ρ2 dr
g(−∂r , v) = −grµ v µ = −grr v r = −grr =− .
dλ ∆ dλ
Since −ρ2 /∆ > 0 in MII , g(−∂r , v) < 0 is thus equivalent to dr/dλ < 0, which proves that r
is decreasing along L as λ increases.
338 Kerr black hole
10.3.3 Kerr coordinates
As in Sec. 6.3.2, we shall move from the coordinate v to a (asymptotically) timelike one by
setting
t̃ = v − r ⇐⇒ v = t̃ + r (10.33)
so that v appears as the advanced time t̃ + r (compare with Eq. (6.30)). We thus consider the
coordinates (xα̃ ) = (t̃, r, θ, φ̃), which we shall call Kerr coordinates (cf. the historical note
below). It is worth to relate them to Boyer-Lindquist coordinates (t, r, θ, φ). This is easily
achieved by combining (10.23) with dt̃ = dv − dr:
2mr
dt̃ = dt + dr (10.34a)
∆
a
dφ̃ = dφ + dr . (10.34b)
∆
The integrated version is obtained by substituting (10.33) in Eq. (10.24):
m r − r+ r − r−
t̃ = t + √ r+ ln − r− ln (10.35a)
m2 − a2 2m 2m
a r − r+
φ̃ = φ + √ ln . (10.35b)
2
2 m −a 2 r − r−
Since the transformation (10.33) leads to dv = dt̃ + dr, the expression of the metric with
respect to the Kerr coordinates (t̃, r, θ, φ̃) is easily deduced from (10.25):
4amr sin2 θ
2mr 4mr
g = − 1− 2 dt̃2 + 2 dt̃ dr − dt̃ dφ̃
ρ ρ ρ2
2mr 2mr
+ 1+ 2 2
dr − 2a 1 + 2 sin2 θ dr dφ̃ (10.36)
ρ ρ
2a2 mr sin2 θ
2 2 2 2
+ρ dθ + r + a + sin2 θ dφ̃2 .
ρ2
Since we kept r, θ and φ̃ and simply changed v to t̃ via (10.33) when moving from the
advanced Kerr coordinates to the Kerr ones, we easily get the link between the two coordinate
frames:
∂t̃ = ∂v (10.37a)
∂r̃ = ∂v + ∂r̂ (10.37b)
∂θ = ∂θ (10.37c)
∂φ̃ = ∂φ̃ . (10.37d)
Note that we have denoted by ∂r̃ the second vector of the coordinate frame associated to the
Kerr coordinates (xα̃ ) = (t̃, r, θ, φ̃), in order to distinguish it from the coordinate vector ∂r̂ of
10.3 Extension of the spacetime manifold through ∆ = 0 339
the advanced Kerr coordinates (xα̂ ) = (v, r, θ, φ̃), as well as from the coordinate vector ∂r of
the Boyer-Lindquist coordinates (xα ) = (t, r, θ, φ).
By combining (10.31) and (10.37), we get the relation between the Kerr coordinate frame
and the Boyer-Lindquist coordinate frame:
∂t̃ = ∂t (10.38a)
2mr a
∂r̃ = ∂r − ∂t − ∂φ (10.38b)
∆ ∆
∂θ = ∂θ (10.38c)
∂φ̃ = ∂φ . (10.38d)
We notice on (10.38a) and (10.38d) that the coordinate frame vectors ∂t̃ and ∂φ̃ coincide with
the Killing vectors ξ and η:
∂t̃ = ξ and ∂φ̃ = η . (10.39)
That ∂t̃ and ∂φ̃ are Killing vectors is not surprising since the metric components (10.36) do not
depend on t̃ nor on φ̃.
The determinant of the metric components (10.36) takes a very simple form:
det gα̃β̃ = −ρ4 sin2 θ. (10.40)
The inverse metric also takes a rather simple form in terms of Kerr coordinates (see the
notebook D.5.2-2 for the computation):
−1 − 2mr ρ2
2mr
ρ2
0 0
2mr ∆ a
ρ 2 ρ2 0 ρ 2
α̃β̃
g = . (10.41)
0 0 ρ12 0
a 1
0 ρ2
0 ρ2 sin2 θ
Comparing (10.36) with (10.8), we note that the metric components in Kerr coordinates are
slightly more complicated than those in Boyer-Lindquist coordinates, for they contain extra
off-diagonal terms: gt̃r and grφ̃ . However the determinant (10.40) and the inverse metric (10.41)
are pretty simple. Morever Kerr coordinates are as well adapted to the spacetime symmetries
as the Boyer-Lindquist ones, as (10.39) shows, and they have the great advantage to be regular
on the boundary hypersurfaces H and Hin , contrary to Boyer-Lindquist coordinates. The last
feature is all the more important that H is the future event horizon of Kerr spacetime, as we
are going to see. Therefore, we shall continue our study of Kerr spacetime, and especially of
the black hole feature, by means of Kerr coordinates.
Remark 4: The coordinate t̃ is not everywhere timelike, i.e. the hypersurfaces of constant t̃ are
not everywhere spacelike. Indeed, according to the criterion (A.56), t̃ is timelike iff g t̃t̃ < 0 and we
read on Eq. (10.41) that g t̃t̃ = −1 − 2mr/ρ2 . Hence t̃ is timelike iff ρ2 + 2mr > 0, or equivalently
r2 + 2mr + a2 cos2 θ > 0. This quadratic polynomial in r is positive everywhere except in the region
of MIII defined by
p p
−m − m2 − a2 cos2 θ ≤ r ≤ −m + m2 − a2 cos2 θ. (10.42)
340 Kerr black hole
Note that this region is contained in the negative-r part of MIII . We conclude that the coordinate t̃ is
timelike in MI , MII and in the part of MIII outside the region defined by (10.42). Consequently, the
Kerr coordinates (t̃, r, θ, φ̃) are related to a 3+1 slicing of spacetime only outside the region (10.42). By
3+1 slicing, it is meant a foliation of M by spacelike hypersurfaces (see e.g. [227]).
Historical note : The coordinate t̃ has been introduced by Roy Kerr in the 1963 discovery article [312],
by exactly the same transformation as (10.33) (t̃ is denoted by t and v by u in Ref. [312]). Kerr considered
t̃ along with Cartesian-type coordinates (x, y, z) deduced from (r, θ, φ̃) by spheroidal transformations,
to form the coordinate system (t̃, x, y, z), which is known today as Kerr-Schild coordinates (cf. Ap-
pendix C), despite they have been introduced first in Kerr’s article [312] and not in the subsequent
article by Kerr and Schild [314] (1965). Accordingly, the Kerr coordinates (t̃, r, θ, φ̃) defined above are a
mix of the coordinates (v, r, θ, φ̃) in which Kerr exhibited his solution (cf. the historical note on p. 336),
called here advanced Kerr coordinates, and the Kerr-Schild coordinates (t̃, x, y, z). The Kerr coordinates
(t̃, r, θ, φ̃) have been first explicitely considered by Robert Boyer and Richard Lindquist in 1966 [72],
being called by them the “(E) frame” — “(E)” standing for Eddington, since these coordinates generalize
Eddington-Finkelstein coordinates to the rotating case. We can check that the metric (10.36) coincides
with Eq. (2.7) of Boyer and Lindquist’s article [72], keeping in mind that these authors were using −a
for a, as Kerr in 1963 (cf. the discussion about the sign of a in the historical note on p. 336).
10.4 Principal null geodesics
10.4.1 Ingoing principal null geodesics
We have seen that the advanced Kerr coordinates (v, r, θ, φ̃) introduced in Sec. 10.3.1 are such
that the curves L(v,θ,
in
φ̃) defined by (v, θ, φ̃) = const are null curves (cf. Sec. 10.3.2). Their
future-directed tangent vector field is k = −∂r̂ [Eq. (10.32)], which can expressed in terms of
the Kerr basis via (10.37):
k = ∂t̃ − ∂r̃ . (10.43)
The 1-form k associated to k by g-duality is easily computed from kα̃ = gα̃µ̃ k µ̃ , with gα̃µ̃ given
by Eq. (10.36). We get kα̃ = (−1, −1, 0, a sin2 θ), i.e.
k = −dt̃ − dr + a sin2 θ dφ̃ . (10.44)
A direct computation (cf. the notebook D.5.2-2) shows that
∇k k = 0. (10.45)
It follows that each curve L(v,θ,
in
φ̃) is a geodesic and that the parameter λ associated with k is
an affine parameter of this geodesic (cf. Sec. B.2.1 in Appendix B). Since Eq. (10.43) implies
k r = dr/dλ = −1, we have, up to some additive constant,
λ = −r. (10.46)
The geodesics L(v,θ,
in
φ̃) are called the ingoing principal null geodesics. The qualifier ingoing
stems from the fact that r is decreasing towards the future along L(v,θ,
in
φ̃) , which is an immediate
10.4 Principal null geodesics 341
consequence of λ = −r being a future-directed parameter along L(v,θ,
in
φ̃) . The qualifier principal
arises from a specific connection between k and the Weyl conformal curvature tensor C of
Kerr spacetime (cf. Sec. A.5.4), namely:
C αµν[β kγ] k µ k ν = 0 and ∗
C αµν[β kγ] k µ k ν = 0, (10.47)
where ∗ C stands for the dual of the Weyl tensor:
1 α
∗
C αβγδ := C ϵµν , (10.48)
2 βµν γδ
ϵ being the Levi-Civita tensor (cf. Sec. A.3.4). In view of (10.47), one says that the vector field k
constitutes a doubly degenerate, principal null direction of C (see e.g. Chap. 5 of O’Neill
textbook [391] for details). We note that the ingoing principal null geodesics form a congruence:
through each point of M , there is one, and only one, curve L(v,θ, in
φ̃) .
Remark 1: For the Kerr spacetime, as for any solution of the vacuum Einstein equation (R = 0), the
Weyl conformal curvature tensor C is equal to the Riemann curvature tensor Riem: setting R = 0 in
Eq. (A.114) yields Riem = C.
Remark 2: At the Schwarzschild limit, a = 0, φ̃ = φ and the ingoing principal null geodesics L(v,θ, in
φ̃)
reduce to the ingoing radial null geodesics L(v,θ,φ)
in discussed in Secs. 6.3.1 and 6.3.5 [compare Eqs. (6.48)
and (10.43)].
The components of k with respect to the Boyer-Lindquist coordinate frame are immediately
deduced from Eq. (10.31b) and k = −∂r̂ :
r 2 + a2 a
k= ∂t − ∂r + ∂φ . (10.49)
∆ ∆
Substituting Eqs. (10.34) for dt̃ and dφ̃ in Eq. (10.44), we get the expression of the associated
1-form k with respect to the Boyer-Lindquist coordinate coframe:
ρ2
k = −dt − dr + a sin2 θ dφ. (10.50)
∆
Remark 3: Expressions (10.49) and (10.50) are singular on the two horizons H and Hin , where ∆ = 0.
This reflects the singularity of Boyer-Lindquist coordinates on H and Hin , not any pathology of the
vector field k. Indeed, the expressions of k and k in terms of Kerr coordinates [Eqs. (10.43) and (10.44)]
are perfectly regular in all M .
10.4.2 Outgoing principal null geodesics
One can construct a second congruence of principal null geodesics by considering the retarded
Kerr coordinates instead of the advanced ones discussed in Sec. 10.3.1. The retarded Kerr
342 Kerr black hole
t̃/m
in
4
k
2 `
r/m
8 6 4 2 2 4 6 8
Figure 10.6: Principal null geodesics of Kerr spacetime viewed in terms of the Kerr coordinates (t̃, r) for
a/m = 0.9. The solid (resp. dashed) curves correspond to outgoing geodesics L(u,θ,
out
˜ (resp. ingoing geodesics
φ̃)
L(v,θ,
in
φ̃) ), as given by Eq. (10.53) with u = const (resp. Eq. (10.33) with v = const). The increment in u between
two depicted outgoing geodesics is δu = 2m; similarly, two depicted ingoing geodesics differ in their values of v
by δv = 2m. Note that the outgoing principal null geodesics tend to become tangent to the horizon H (resp.
Hin ) when r → r+ (resp. r → r− ). Actually, we shall see in Sec. 10.5.3 and 10.8.3 that the generators of these
two horizons belong to the outgoing principal null congruence. [Figure generated by the notebook D.5.3]
˜ are defined by relations to Boyer-Lindquist coordinates that are similar
coordinates (u, r, θ, φ̃)
to (10.23), up to a change of sign:
r 2 + a2
du = dt − dr (10.51a)
∆
a
dφ̃˜ = dφ − dr. (10.51b)
∆
The coordinates (u, r, θ, φ̃)
˜ generalize the null outgoing Eddington-Finkelstein coordinates
introduced in Sec. 6.4 to the case a ̸= 0. Thanks to the symmetry (t, φ) 7→ (−t, −φ) of the
Kerr metric (10.8), which turns (u, φ̃)
˜ into (−v, −φ̃), it is clear that the curves L out ˜ defined
(u,θ,φ̃)
by (u, θ, φ̃) = const constitute a second congruence of principal null geodesics, called the
˜
outgoing principal null geodesics. A priori, this congruence is defined only in MBL , i.e. where
∆ ̸= 0, but we shall see below that we can extend it to all M . As −r was a affine parameter
along the ingoing principal null geodesics, r is an affine parameter along the outgoing principal
null geodesics in MBL . A difference with respect to the ingoing family is that, while −r was
always increasing towards the future along all ingoing geodesics, r is increasing towards the
future along outgoing principal null geodesics only in regions MI and MIII ; in region MII , r is
decreasing towards the future, in agreement with the general Property 10.13 (Sec. 10.3.2).
10.4 Principal null geodesics 343
t̃/m
in
4
k
2 `
r/m
8 6 4 2 2 4 6 8
Figure 10.7: Same as Fig. 10.6, but for a/m = 0.5.
The fact that the Weyl tensor C admits two, and exactly two, congruences of principal
null geodesics means that the Kerr metric is an algebraically special solution of the Einstein
equation: it belongs to the so-called Petrov type D [391].
Let us find the expression of the outgoing principal null geodesics in terms of Kerr coor-
dinates (which have been constructed on the ingoing principal null congruence). Combining
(10.51) with (10.34), we get
r2 + 2mr + a2
du = dt̃ − dr (10.52a)
∆
2a
dφ̃˜ = dφ̃ − dr. (10.52b)
∆
These equations can be integrated (cf. the computation leading to Eq. (10.24)), yielding
2m r − r+ r − r−
u = t̃ − r − √ r+ ln − r− ln (10.53a)
m2 − a2 2m 2m
a r − r+
φ̃˜ = φ̃ − √ ln , (10.53b)
m2 − a2 r − r−
The quantities u − t̃ and φ̃˜ − φ̃ are plotted in terms of r in Fig. 10.8. We see there that u takes
all values in the range (−∞, +∞) in each of the regions MI , MII and MIII . Accordingly, the
3-tuple (u, θ, φ̃)
˜ defines uniquely an outgoing principal null geodesic only in each of these
three regions. In other words, the congruence of outgoing principal null geodesics can be split
in three families L(u,θ,
out,I
˜ , L(u,θ,φ̃)
φ̃) ˜ and L(u,θ,φ̃)
out,II
˜ , each family being labelled by (u, θ, φ̃). In what
out,III ˜
follows, we shall denote by L(u,θ, out
˜ any member of one of these three families.
φ̃)
344 Kerr black hole
8 8
6 6
4 4
2 2
(u − t̃)/m
ϕ̃˜ − ϕ̃
0 0
2 2
4 4
6 6
8 8
8 6 4 2 0 2 4 6 8 8 6 4 2 0 2 4 6 8
r/m r/m
Figure 10.8: u − t̃ (left panel) and φ̃˜ − φ̃ (right panel) as functions of r given by Eqs. (10.53) for a/m = 0.9.
The dashed vertical lines correspond to r = r− (Hin ) and r = r+ (H ) and delimitate the regions MIII , MII and
MI (from the left to the right). [Figure generated by the notebook D.5.3]
The outgoing principal null geodesics are depicted, along with the ingoing ones, in Figs. 10.6
and 10.7. Note that the φ̃-motion of the outgoing geodesics, as expressed by Eq. (10.53b) with
φ̃˜ held fixed, is not shown in these figures, which represent only traces in the (t̃, r) plane.
Actually the geodesics L(u,θ,
out
˜ are winding with respect the Kerr coordinates (t̃, r, θ, φ̃) at the
φ̃)
coordinate speed
dφ̃ 2a
= 2 . (10.54)
dt̃ L out r + 2mr + a2
˜
(u,θ,φ̃)
Proof. Along a null geodesic L(u,θ,
out
˜ , we have du = 0 and dφ̃ = 0, so that (10.52) yields
φ̃)
˜
dt̃ r2 + 2mr + a2 dφ̃ 2a
= and = . (10.55)
dr L out ∆ dr L out ∆
˜
(u,θ,φ̃) ˜
(u,θ,φ̃)
Dividing the second expression by the first one yields (10.54).
Note that in both asymptotically flat regions, when r → ±∞, the winding speed (10.54)
goes to zero and the two congruences of geodesics tend to ±45◦ lines in Figs. 10.6 and 10.7, as
expected. Note also that, despite their name, the outgoing principal null geodesics are actually
ingoing in MII (between Hin and H ), i.e. have r decreasing towards the future, in agreement
with Property 10.13 (Sec. 10.3.2).
Remark 4: In Fig. 10.7, the outgoing geodesics seem to go “backward in time” in the region −2m ≲ r ≲ 0.
This is an artefact due to the hypersurfaces t̃ = const being not spacelike there, as discussed in Remark 4
p. 339. Consequently it is possible to move to the future with decaying values of t̃ in this region. The
same effect exists, but is less pronounced, for a/m = 0.9 (Fig. 10.6).
Another view of ingoing principal null geodesics is provided by Figs. C.1, C.2 and C.4 of
Appendix C, which depict them in terms of Kerr-Schild coordinates.
10.4 Principal null geodesics 345
10.4.3 Regular null tangent vector to the outgoing congruence
The presence of ∆ in the denominators of expressions (10.55) shows that r can no longer be
considered as a parameter along L(u,θ,
out
˜ when ∆ = 0, i.e. on H and Hin . Actually, the family
φ̃)
L(u,θ,φ̃)
out
˜ is not even defined there, since we read on (10.53) and see on Fig. 10.8 that u → ±∞
and φ̃˜ → ±∞ when r → r± (see also Fig. 6.5 for a pictorial view of u → +∞ when r → r+
in the Schwarzschild limit). In order to extend the outgoing principal null family to H and
Hin , let us introduce along any geodesic L(u,θ,
out
˜ in MBL the parameter λ that is related to the
φ̃)
affine parameter r by
dr ∆
= . (10.56)
dλ 2(r + a2 )
2
The tangent vector to L(u,θ,
out
˜ associated to λ, ℓ say, has the following components w.r.t. Kerr
φ̃)
coordinates, obtained by combining Eqs. (10.55) and (10.56):
dt̃ dt̃ dr r2 + 2mr + a2 1 mr
ℓt̃ = = × = 2 2
= + 2
dλ dr dλ 2(r + a ) 2 r + a2
dr ∆ 1 mr
ℓr = = 2 2
= − 2
dλ 2(r + a ) 2 r + a2
dθ
ℓθ = =0
dλ
dφ̃ dφ̃ dr a
ℓφ̃ = = × = 2 .
dλ dr dλ r + a2
In other words, we have
1 mr 1 mr a
ℓ= + 2 2
∂t̃ + − 2 2
∂r̃ + 2 ∂φ̃ . (10.57)
2 r +a 2 r +a r + a2
It is clear that this vector field is regular everywhere in M . Given the metric components
(10.36), it is easy to check that g(ℓ, ℓ) = 0, i.e. that ℓ is a null vector. Moreover, an explicit
computation (cf. the notebook D.5.2-2) reveals that
m(r2 − a2 )
∇ℓ ℓ = κℓ ℓ, with κℓ := . (10.58)
(r2 + a2 )2
Equation (10.58) shows that the integral curves of the vector field ℓ are geodesics (cf. Secs. 2.3.3
and B.2.2). In MBL , they are nothing but the null geodesics L(u,θ,out
˜ . On H or Hin , they are
φ̃)
null geodesics evolving at constant r (= r+ or r− ) (since ℓ = 0 there), constant θ (since ℓθ = 0)
r
and constant ψ, where
a
ψ := φ̃ − t̃. (10.59)
2mr±
Indeed, in view of the components (10.57), we have along these geodesics
dφ̃ dφ̃ dλ ℓφ̃ 2a a
= = t̃ = 2 = .
dt̃ dλ dt̃ ℓ r± + 2mr± + a2 2mr±
Since a/(2mr± ) is constant, we get φ̃ = a/(2mr± ) t̃ + ψ, where ψ is some integration constant.
346 Kerr black hole
Property 10.14: outgoing principal null geodesics
Given that ℓ is a smooth vector field, the congruence of outgoing principal null geodesics
can be smoothly extended to include the integral curves of ℓ on H and Hin , which we shall
denote by L(θ,ψ)
out,H
and L(θ,ψ)
out,Hin
, given that θ and ψ [Eq. (10.59)] are constant along them.
Consequently, the congruence of outgoing principal null geodesics on M is made of 5
families: L(u,θ,
out,I
˜ , L(u,θ,φ̃)
φ̃) ˜ , L(u,θ,φ̃)
out,II
˜ , L(θ,ψ) and L(θ,ψ)
out,III out,H out,Hin
, each family covering a different
subset of M .
Remark 5: At the Schwarzschild limit, a = 0, φ̃˜ = φ, ψ = φ and the outgoing principal null geodesics
out,H out,H
L(u,θ,
out
˜ and L(θ,ψ) reduce respectively to the outgoing radial null geodesics L(u,θ,φ) and L(θ,φ)
φ̃)
out
discussed in Secs. 6.3.1 and 6.3.5. This can be seen from the fact that, for a = 0, the tangent vector ℓ
given by Eq. (10.57) reduces to the tangent vector ℓ given by Eq. (6.53).
The 1-form ℓ associated to ℓ by g-duality is obtained from ℓα̃ = gα̃µ̃ ℓµ̃ , with gα̃µ̃ given by
Eq. (10.36):
∆ 2ρ2 − ∆ a∆ sin2 θ
ℓ=− 2 d t̃ + dr + dφ̃ . (10.60)
2(r + a2 ) 2(r2 + a2 ) 2(r2 + a2 )
The components of ℓ and ℓ with respect to the Boyer-Lindquist coordinate frame/coframe
are obtained by combining Eq. (10.57) with Eq. (10.38) and Eq. (10.60) with Eq. (10.34):
1 ∆ a
ℓ = ∂t + 2 2
∂r + ∂φ . (10.61)
2 2(r + a ) 2(r + a2 )
2
∆ ρ2 a∆ sin2 θ
ℓ=− 2 dt + dr + dφ. (10.62)
2(r + a2 ) 2(r2 + a2 ) 2(r2 + a2 )
Remark 6: The reader may have noticed a certain dissymmetry between the chosen tangent vector
k of ingoing principal null geodesics, which obeys ∇k k = 0 [Eq. (10.45)] and the tangent vector ℓ of
the outgoing ones, which obeys ∇ℓ ℓ ̸= 0 [Eq. (10.58)]. The last property implies that the parameter λ
associated to ℓ is not affine, while the parameter −r associated to k is (cf. Sec. B.2.2). The non-affine
choice is the price to pay to have a parametrization of the outgoing congruence well-defined everywhere
in M , even where ∆ = 0. We shall see in Sec. 10.8 that in the maximal extension of the Kerr spacetime,
there are other regions where these features are reversed, thereby restoring the symmetry between
ingoing and outgoing principal null geodesics on the extended spacetime.
Using either the Kerr components (10.43) and (10.57) or the Boyer-Lindquist components
(10.49) and (10.61), one can easily evaluate the scalar product of k and ℓ:
ρ2
g(k, ℓ) = − . (10.63)
r 2 + a2
This implies that g(k, ℓ) < 0 everywhere (note that ρ2 = 0 would correspond to the ring
singularity R, which has been excluded from the spacetime manifold M ). Using Lemma 1.2
(Sec. 1.2.2) with u = k and v = ℓ, we conclude
10.5 Event horizon 347
Property 10.15: ℓ future-directed
The tangent vector ℓ to the outgoing principal null geodesics is future-directed in all the
Kerr spacetime M .
Remark 7: Instead of ℓ, another natural choice for the tangent vector to the outgoing principal null
geodesics would have been the tangent vector ℓ̂ such that g(k, ℓ̂) = −1. In view of (10.63), the two
tangent vectors are related by ℓ̂ = (r2 + a2 )/ρ2 ℓ As for ℓ, the tangent vector ℓ̂ does not correspond to
an affine parameter of the outgoing principal null geodesics, i.e. its non-affinity coefficient κℓ̂ is nonzero.
While ℓ̂ is used in many studies, we prefer ℓ here because it coincides with the Killing vector ξ + ΩH η
on the event horizon, as we shall see in Sec. 10.5 [Eq. (10.72)]. In particular, its non-affinity coefficient
κℓ yields the so-called black hole’s surface gravity (Sec. 10.5.4 below).
10.5 Event horizon
10.5.1 The two Killing horizons of Kerr spacetime
Let us consider the hypersurfaces of M defined by a fixed value of the coordinate r. H and
Hin are two particular cases, corresponding to r = r+ and r = r− respectively. The normal
→
−
1-form to a hypersurface r = const is dr; the corresponding gradient vector field is ∇r, the
components of which with respect to Kerr coordinates are ∇α̃ r = g α̃µ̃ ∂µ̃ r = g α̃r , with g α̃r read
on the second raw of the matrix (10.41). Hence
→
− 2mr ∆ a
∇r = 2 ∂t̃ + 2 ∂r̃ + 2 ∂φ̃ .
ρ ρ ρ
→
− →
−
Instead of ∇r, it is quite natural to consider the rescaled vector field n := ρ2 ∇r as the normal
to the hypersurfaces r = const, for it has simpler components in the Kerr frame:
n = 2mr ∂t̃ + ∆ ∂r̃ + a ∂φ̃ . (10.64)
The scalar square of n is n · n = nµ nµ = ρ2 (∇µ r)nµ = ρ2 nr . Hence, in view of (10.64),
n · n = ρ2 ∆. (10.65)
Since ρ2 > 0 everywhere on M and ∆ = (r − r+ )(r − r− ) [Eq. (10.10)], we conclude:
Property 10.16: causal type of the hypersurfaces of constant r
• The hypersurfaces r = const are timelike in regions MI and MIII ;
• The hypersurfaces r = const are spacelike in region MII ;
• H (where r = r+ ) and Hin (where r = r− ) are null hypersurfaces.
348 Kerr black hole
On H and Hin , ∆ = 0, so that Eq. (10.64) yields
n = 2mr± ∂t̃ + a ∂φ̃ = 2mr± ξ + a η, (10.66)
where we have used (10.39) and r± stands for r+ on H and r− on Hin . On H , we may rewrite
this expression as
n = 2mr+ χ, (10.67)
with
χ := ξ + ΩH η (10.68)
and
a a a
ΩH := = 2 2
= √ . (10.69)
2mr+ r+ + a 2m m + m2 − a2
ΩH being a constant, the vector field χ defined by (10.68) is a Killing vector field of (M , g).
Moreover, Eq. (10.67) shows that this Killing vector is normal to the null hypersurface H . In
view of the definition given in Sec. 3.3.2, we conclude:
Property 10.17: H as a Killing horizon
The hypersurface H defined by r = r+ is a Killing horizon with respect to the Killing
vector field χ [Eq. (10.68)].
Similarly, on Hin , we may rewrite (10.66) as n = 2mr− χin , with
χin := ξ + Ωin η (10.70)
and
a a a
Ωin := = 2 2
= √ , (10.71)
2mr− r− + a 2m m − m2 − a2
thereby arriving at:
Property 10.18: Hin as a Killing horizon
The hypersurface Hin defined by r = r− is a Killing horizon with respect to the Killing
vector field χin [Eq. (10.70)].
We shall call Hin the inner horizon. We shall see in Sec. 10.8.3 that Hin is actually (part of) a
so-called Cauchy horizon.
Historical note : The identification of the hypersurfaces H and Hin as the only two null hypersurfaces
of Kerr spacetime that are stationary (i.e. that are Killing horizons in the modern language) has been
first performed in 1964 by Robert H. Boyer and T.G. Price [73], who claimed: “These are ‘horizons’ in
the sense that there can be a flow of matter or radiation across them in only one direction. They are the
analogues of the Schwarzschild null sphere or ‘singularity’.”
10.5 Event horizon 349
10.5.2 Black hole character
As a null hypersurface, H is a one-way membrane (cf. Sec. 2.2.2), therefore any (massive or
null) particle that crossed it from MI to MII can never be back in MI . Let us show that H is
actually a black hole event horizon, as defined in Sec. 4.4.2.
We have seen in Sec. 10.2.2 that the asymptotics of region MI is the same as that of
Schwarzschild spacetime. Hence one can perform a conformal completion of (MI , g) endowed
with a future null infinity I + and a past null infinity I − (an explicit construction of I +
and I − for Schwarzschild spacetime has been performed in Sec. 9.4.3). A conformal diagram
representing MI along with I + and I − is given in Fig. 10.9 below.
Let us show that the causal past of the future null infinity coincide with MI : J − (I + ) = MI .
Since, as stressed above, no future-directed causal curve can move from MII to MI and I + is
a boundary of MI , we have MII ∩ J − (I + ) = ∅. A fortiori MIII ∩ J − (I + ) = ∅. We have
thus J − (I + ) ⊂ MI . To show the equality between the two sets there remains to show that
any point p ∈ MI can emit a signal reaching I + . Let L be the null geodesic through p of
the outgoing principal null congruence L(u,θ, out
˜ introduced in Sec. 10.4, i.e. L is the unique
φ̃)
geodesic departing from p with the tangent vector ℓ given by (10.57). Along L , one has
dr 1 mr
= ℓr = − 2 ,
dλ 2 r + a2
where λ is the parameter associated with ℓ. In particular, at p, if we denote by r0 the r-coordinate
of p in the Kerr system,
dr 1 mr0
= − 2 > 0.
dλ p 2 r0 + a2
The above inequality simply translates the fact that r0 > r+ wherever p lies in MI . Hence,
initially r is increasing along L and we get, since −mr/(r2 + a2 ) is an increasing function of
r,
dr 1 mr0
≥ − 2 =: α > 0.
dλ 2 r0 + a2
Since α is a constant, we deduce that
r ≥ r0 + α(λ − λ0 ),
where λ0 is the value of L ’s parameter at p. When λ → +∞, we get r → +∞, which proves
that the null curve L reaches I + . Hence we conclude:
Property 10.19: black hole region in Kerr spacetime
B = M \ MI is the black hole region, the event horizon of which is H .
Incidentally, since we have already shown that H is a Killing horizon (Property 10.17), this
illustrates the strong rigidity theorem (Property 5.25 in Sec. 5.4.2): the black hole event horizon
of Kerr spacetime is a Killing horizon.
According to the discussion in Sec. 5.4.2, we may then call the quantity ΩH introduced in
Eqs. (10.68)-(10.69) the black hole rotation velocity.
350 Kerr black hole
10.5.3 Null generators of the event horizon
The null vector field ℓ defined by Eq. (10.57) coincides with the Killing vector χ on H , since
H H
mr/(r2 + a2 ) = 1/2 and a/(r2 + a2 ) = ΩH :
H
ℓ = χ. (10.72)
Since (i) χ is tangent to the null geodesic generators of H , being a null normal to it (cf.
Sec. 2.3.3) and (ii) ℓ is the tangent vector to the outgoing principal null geodesics L(θ,ψ)
out,H
(Sec. 10.4), we get
Property 10.20: null generators of the event horizon
The null generators of the event horizon H are the geodesics L(θ,ψ)
out,H
of the outgoing
principal null congruence.
Similarly, we have
Property 10.21: null generators of the inner horizon
The null generators of the inner horizon Hin are the geodesics L(θ,ψ)
out,Hin
of the outgoing
principal null congruence.
The reader is referred to Fig. 5.1 for a pictorial view of the event horizon H spanned by
the rotating null generators, as well as to Figs. 10.6 and 10.7, where it appears clearly that at
r = r+ (resp. r = r− ) the outgoing principal null geodesics are tangent to H (resp. Hin ).
Remark 1: Since ∆ = 0 on H , we read immediately from Eq. (10.60) that
H ρ2
ℓ= dr,
r 2 + a2
3 which shows that, at any point of H , the vector ℓ is normal to the hypersurface r = const through
this point. This hypersurface being nothing but H itself, we recover the fact that ℓ is normal to H .
In view of expression (10.69) for ΩH , Eq. (10.59) for the parameter ψ labelling the outgoing
principal null geodesics L(θ,ψ)
out,H
can be rewritten as
ψ = φ̃ − ΩH t̃. (10.73)
Hence ψ appears as a corotating azimuthal coordinate on H . Moreover, we verify that the
winding speed of the outgoing principal null geodesics given by Eq. (10.54) tends toward ΩH
when approaching the event horizon:
dφ̃
lim = ΩH . (10.74)
r→r+ dt̃ L out ˜
(u,θ,φ̃)
H
This follows immediately from ℓ = χ = ξ + ΩH η = ∂t̃ + ΩH ∂φ̃ and the identity r+
2
+ a2 =
2mr+ (compare Eq. (10.54) with r → r+ and Eq. (10.69)).
10.6 Global quantities 351
10.5.4 Surface gravity
In view of the pre-geodesic equation (10.58) satisfied by ℓ and the identity (10.72), we deduce
that
H
∇χ χ = κ χ , (10.75)
with the non-affinity coefficient given by Eq. (10.58): κ = κℓ |r=r+ = m(r+2
− a2 )/(r+
2
+ a2 )2 .
Since r+ is a zero of ∆, we have r+2
+ a2 = 2mr+ , so that κ can be rewritten as
2
r+ − a2 2
r+ − a2
κ= 2
= 2
. (10.76)
2r+ (r+ + a2 ) 4mr+
Substituting (10.3) for r+ , we get an expression involving the two basic Kerr parameters:
√
m 2 − a2
κ= √ . (10.77)
2m(m + m2 − a2 )
Given the strict inequality a < m assumed in this chapter [Eq. (10.1)], we have κ ̸= 0. According
to the classification introduced in Sec. 3.3.6, we may state:
Property 10.22: non-degeneracy of the event horizon
As long as a < m, the event horizon H is a non-degenerate Killing horizon.
In Sec. 3.3.7, we have seen that the non-affinity coefficient κ can be interpreted as a “rescaled”
surface gravity. Hence κ is called the black hole surface gravity. The fact that κ is a constant
(i.e. does not depend on θ) is an illustration of the zeroth law of black hole dynamics established
in Sec. 3.3.5 (cf. in particular Example 14 in that section).
Remark 2: As a check, if we let a → 0 in Eq. (10.77), we get κ = 1/(4m), i.e. we recover the
Schwarzschild horizon value computed in Example 10 of Chap. 2 [cf. Eq. (2.29)].
10.6 Global quantities
10.6.1 Mass
We have seen in Sec. 10.2.2 that when r → +∞, the Kerr metric tends towards the Schwarzschild
metric of parameter m (cf. Eq. (10.11)); we conclude that m is nothing but the gravitational mass
M (cf. Sec. 6.2.4). However, for any asymptotically flat spacetime endowed with a stationary
Killing vector ξ, as the Kerr spacetime, there is a well-defined concept of mass, namely the
Komar mass introduced in Sec. 5.3.3. Let us show explicitly that for the Kerr spacetime, the
Komar mass coincides with m. To this aim, we shall use expression (5.46) with n = 4 for the
Komar mass. We shall consider the Boyer-Lindquist coordinates (xα ) = (t, r, θ, φ) and define
S to be a sphere (t, r) = const. Coordinates on S are then (x2 , x3 ) = (θ, φ). Moreover, since
the value of MS does not depend on the choice of S (cf. Property 5.15), we may set r → +∞
352 Kerr black hole
and use the asymptotic flatness of Kerr metric to get simple expressions. The unit normals to
√
S are then n = ∂t and s = ∂r . Moreover, when r → +∞, q = r2 sin θ. Hence Eq. (5.46)
with n = 4 yields Z
1
M =− lim (∂r ξt − ∂t ξr )r2 sin θ dθ dφ,
8π r→+∞ S
|{z}
0
with
2m
ξt = gtµ ξ µ = gtµ δ µt = gtt ≃ −1 +
,
r
the last expression resulting from the expansion (10.11). We have then ∂r ξt = −2m/r2 , so that
the above integral yields
M =m. (10.78)
10.6.2 Angular momentum
Since the Kerr spacetime is axisymmetric, a well-defined notion of angular momentum is
provided by the Komar angular momentum J introduced in Sec. 5.3.6. Let us compute J via
Eq. (5.66) by means of Boyer-Lindquist coordinates, choosing for S a 2-sphere (t, r) = const.
In evaluating the terms sµ ∂µ ην nν and nµ ∂µ ην sν as r → +∞, we have to be a little more
cautious than in Sec. 10.6.1, since one of the components ηα is diverging when r → +∞:
ηα = gαµ η µ = gαµ δ µφ = gαφ = (gtφ , 0, 0, gφφ )
with, according to (10.11), gφφ ∼ r2 sin2 θ as r → +∞. Moreover, given the value of gtφ read
on (10.8), we may write
2am sin2 θ
ηα ∼ − 2 2
, 0, 0, r sin θ when r → +∞.
r
Let us choose for the timelike normal n to S the future-directed unit normal to the hypersur-
faces t = const:
n = −N dt, (10.79)
where N is a normalization factor ensuring n · n = −1. We do not need the precise value2 of
N , but simply the property N → 1 as r → +∞. We have then nα = (−N, 0, 0, 0), so that
nα = g αµ nµ = g αt (−N ) = (−N g tt , 0, 0, −N g tφ ),
where the last equality follows from the expression (10.16) of g αβ in Boyer-Lindquist coordinates,
with g tt ∼ −1 and g tφ ∼ −2am/r3 when r → +∞; hence
2am
α
n ∼ 1, 0, 0, 3 when r → +∞.
r
The choice of n completely determines that of s:
√ !
∆
sα = 0, , 0, 0 ∼ (0, 1, 0, 0) when r → +∞.
ρ
2
It is given by Eq. (10.93) below.
10.6 Global quantities 353
Indeed, given
√ the metric components (10.8), we immediately check that n · s = 0, s · s = 1
and s = (ρ/ ∆) dr, which does imply that s is normal to S .
Given the above expressions for ηα , nα and sα , we get, for r → +∞,
2am sin2 θ 6am sin2 θ
µ ν
2am
s ∂µ ην n ∼ ∂r − × 1 + ∂r r2 sin2 θ × 3 ∼
r r r2
and
2am
nµ ∂µ ην sν ∼ ∂t ηr + 3 ∂φ ηr = 0.
|{z} r |{z}
0 0
Hence Eq. (5.66) leads to
π
6am sin2 θ
Z Z Z
1 3am 3am
J= lim 2
× r2 sin θ dθ dφ = 3
sin θ dθ dφ = sin3 θ dθ,
16π r→+∞ S r 8π S 4 0
| {z }
4/3
i.e.
J = am . (10.80)
We conclude that the parameter a is nothing but the total angular momentum divided by the
total mass.
10.6.3 Black hole area
Since the event horizon H is a Killing horizon (cf. Sec. 10.5.1), it is a non-expanding horizon.
As such, it has a well-defined area A, which is the area of any of its cross-sections, as we have
seen in Sec. 3.2.2. To compute A, we shall not use Boyer-Lindquist coordinates as for M and J,
because they are singular on H ; instead, we shall use the Kerr coordinates (t̃, r, θ, φ̃), which
are regular on H . H is defined by r = r+ and it is natural to consider a cross-section of it, S
say, defined by {t̃ = const, r = r+ }. Then S is spanned by the coordinates (xa ) = (θ, φ̃) and
the metric q induced on S by g is obtained by setting r = r+ , dt̃ = 0 and dr = 0 in (10.36):
2a2 mr+ sin2 θ
2 2 2 22 2
q= (r+ + a cos θ) dθ + r+ + a + 2 sin2 θ dφ̃2 .
r+ + a2 cos2 θ
Now, since r+ is a zero of ∆ [cf. Eq. (10.10)], we have 2mr+ = r+
2
+ a2 . This brings us to
2
(r+ + a2 ) 2
q= 2
(r+ 2 2
+ a cos θ) dθ + 2 2
2 2
sin2 θ dφ̃2 . (10.81)
r+ + a cos θ
The area of the cross-section S is
√
Z Z
2 2
A= q dθ dφ̃ = (r+ + a ) sin θ dθ dφ̃,
S
| S {z }
4π
354 Kerr black hole
where we have used (10.81) to write q = det(qab ) = (r+
2
+ a2 )2 sin2 θ. We have thus
2
A = 4π(r+ + a2 ) = 8πmr+ . (10.82)
Via (10.3), one may recast this result to let appear only m and a:
√
A = 8πm(m + m2 − a2 ) . (10.83)
10.6.4 Smarr formula
By combining the relations ΩH = a/(2mr+ ) [Eq. (10.69)], κ = (r+
2
− a2 )/(4mr+
2
) [Eq. (10.76)]
and A = 8πmr+ [Eq. (10.82)], we get an interesting identity:
κA
+ 2ΩH am = m.
4π
If we let appear the total angular momentum J = am [Eq. (10.80)] and the Komar mass M = m
[Eq. (10.78)], we can turn this identity into
κA
M= + 2ΩH J . (10.84)
4π
This is actually the particular case T = 0 (vacuum) of the generic Smarr formula (5.91)
established in Chap. 5, or the particular case QH = 0 (vanishing electric charge) of the
electrovacuum Smarr formula (5.104).
10.7 Families of observers in Kerr spacetime
The concept of observer in a relativistic spacetime has been recalled in Sec. 1.4. We discuss
here some families of observers well adapted to Kerr spacetime: the ZAMOs (Sec. 10.7.3) and
the Carter observers (Sec. 10.7.4). These two families are actually particular cases of a more
general concept, that of a stationary observer, which we introduce first.
10.7.1 Stationary observers
A stationary observer is an observer O in Kerr spacetime whose 4-velocity u is a linear
combination of the Killing vectors ξ and η with constant coefficients3 :
u = α ξ + β η, α = const, β = const. (10.85)
It follows that the worldline L of O is an orbit of the isometry group R × SO(2) of Kerr
spacetime. In physical terms, this means that the spacetime geometry as perceived by observer
O does not evolve, hence the name stationary observer.
3
In the definition (10.85), the coefficients α and β are required to be constant along a given observer’s worldline.
When considering a family of observers, they may vary from one worldline to the other.
10.7 Families of observers in Kerr spacetime 355
A stationary observer moves necessarily at fixed values of the non-ignorable coordinates
in a coordinate system adapted to the spacetime symmetries, like Boyer-Lindquist coordinates
or Kerr ones. For instance, considering the Boyer-Lindquist coordinates (t, r, θ, φ), we have
ξ = ∂t and η = ∂φ and we deduce from (10.85) and the definition (1.16) of the 4-velocity that
along the worldline L :
dt dr dθ dφ
= α, = 0, = 0, = β,
dτ dτ dτ dτ
where τ is the observer’s proper time. It follows that the stationary observer evolves at fixed
values of the coordinates r and θ.
Outside the Carter time machine T (cf. Sec. 10.2.5), we have necessarily α ̸= 0, because
η is spacelike in M \ T and the 4-velocity u is necessarily timelike. Hence, by introducing
ω := β/α, we may rewrite (10.85) as
u = α (ξ + ω η) , α = const, ω = const. (10.86)
The coefficient ω gives the rate of variation of the azimuthal coordinate along the observer’s
worldline L in any coordinate system adapted to the spacetime symmetries, like the Boyer-
Lindquist ones (t, r, θ, φ), the advanced Kerr ones (v, r, θ, φ̃) or the Kerr ones (t̃, r, θ, φ̃), ac-
cording to
dφ dφ̃ dφ̃
ω= = = . (10.87)
dt L dv L dt̃ L
Proof. Let us consider Boyer-Lindquist coordinates (t, r, θ, φ). Denoting by τ the proper time
of O, we have dφ/dt|L = dφ/dτ × dτ /dt. Now by the definition (1.16) of the 4-velocity,
dφ/dτ = uφ and dτ /dt = (ut )−1 . We have thus dφ/dt|L = uφ /ut . From Eq. (10.86) along
with ξ = ∂t and η = ∂φ , we read ut = α and uφ = αω. Hence dφ/dt|L = ω. The same
demonstration applies to Kerr and advanced Kerr coordinates because ξ = ∂v = ∂t̃ and
η = ∂φ̃ .
Remark 1: That the ratios dφ/dt, dφ̃/dv and dφ̃/dt̃ along L are all equal, as expressed in (10.87), is
not surprising if one considers the links between the various coordinates expressed by Eqs. (10.23a),
(10.23b) and (10.34). Setting dr = 0 in these equations, since r is constant along L , we get
dt|L = dv|L = dt̃|L and dφ|L = dφ̃|L .
Equation (10.87) allows for a nice physical interpretation of ω. Indeed, we have seen in
Sec. 7.3.3 [cf. Eq. (7.58)] that dφ/dt|L is nothing but the angular velocity around the symmetry
axis as measured by an asymptotically distant inertial observer. The demonstration was
performed in the Schwarzschild case and for a circular orbit in the equatorial plane, but it used
only the stationarity of Schwarzschild spacetime and the fact that dφ/dt was constant along
L , so it applies to the present case as well.
356 Kerr black hole
10.7.2 Static observers
A static observer is a stationary observer O having ω = 0. The denomination stems from the
fact that the three coordinates (r, θ, φ) (or (r, θ, φ̃)) remain constant along O’s worldline, since
ω = 0 in Eq. (10.87) implies φ = const and φ̃ = const. Moreover, such an observer appears
not moving to an asymptotic inertial observer.
According to (10.86) with ω = 0, the 4-velocity of O is collinear to the Killing vector ξ:
u = αξ. We conclude that a static observer can exist only where ξ is timelike, i.e. outside
the ergoregion (cf. Sec. 10.2.4). This explains why the outer boundary of the ergoregion (the
ergosphere) is sometimes called the static limit (cf. Remark 6 on p. 331).
Since static observers are not very useful for describing physical processes in the vicinity of
a Kerr black hole (in particular, they cannot exist close to the event horizon, which is located
in the ergoregion), we shall not discuss them further.
10.7.3 Zero-angular-momentum observers (ZAMO)
Let us consider an observer O whose worldline is normal to the hypersurfaces of constant
Boyer-Lindquist coordinate t, Σt say. Of course, such an observer exists only where the
hypersurface Σt is spacelike, so that the latter has a timelike normal (cf. Sec. 2.2.2). A normal
1-form to Σt is of course dt. The associated normal vector, N say, is obtained by metric duality
[cf. Eq. (A.46)-(A.47)]:
→
− →
−
N = −dt = −∇t. (10.88)
In components:
N α = −g αµ (dt)µ = −g αµ δ t µ = −g tα .
The minus sign has been chosen to have N future-directed, as we shall see below. In view of
the Boyer-Lindquist components (10.16) of the inverse metric, we get
2a2 mr sin2 θ
1 2amr
N= 2
r +a +2
2
∂t + 2 ∂φ . (10.89)
∆ ρ ρ∆
The scalar square of N is
2a2 mr sin2 θ
1
µ t
N · N = g(N , N ) = Nµ N = δ µ N = N = − µ t 2 2
r +a + . (10.90)
∆ ρ2
Now, the quantity inside the parentheses is positive everywhere except in the Carter time
machine T ⊂ MIII discussed in Sec. 10.2.5. Indeed, up to the factor sin2 θ ≥ 0, it coincides
with expression (10.21) of η · η. Since ∆ = (r − r+ )(r − r− ) is positive on MI and MIII , and
negative on MII , we conclude that the locus where N is timelike is
MZAMO := MI ∪ (MIII \ T ). (10.91)
This is thus the region where the observer O is defined. Note that it does not contain the
horizons H and Hin , which is not surprising since Boyer-Lindquist coordinates are singular
there, notably in terms of the spacetime slicing by the hypersurfaces Σt , as illustrated in Fig. 6.2
for the case a = 0.
10.7 Families of observers in Kerr spacetime 357
In all MZAMO , the timelike vector N is future-directed with respect to the time orientation
chosen in Sec. 10.3.2. Indeed, the latter is set by the global null vector field k and we have,
using the Boyer-Lindquist components (10.49) of k:
r 2 + a2
k · N = Nµ k µ = −k t = − < 0 on MZAMO ,
∆
so that Lemma 1.2 [cf. Eq. (1.6a)] let us conclude that N is future-directed.
Choosing the observer O as having his worldline orthogonal to Σt means that O’s 4-velocity
n is the unit timelike vector introduced by Eq (10.79). Equivalently, n is N rescaled to form a
unit vector:
→
−
n = N N = −N ∇t, (10.92)
with4 N := (−N · N )−1/2 . In view of Eq. (10.90), we get
−1/2
√ 2a2 mr sin2 θ
N = ∆ r +a + 2 2
. (10.93)
ρ2
N is called the lapse function, for it relates the increment dτ in the proper time of O to the
change dt of the coordinate t when moving from Σt to Σt+dt via
dτ = N dt . (10.94)
Proof. The infinitesimal vector that connects the point of proper time τ to that of proper
time τ + dτ along O’s worldline is dx = dτ n (by the very definition of the 4-velocity n,
compare Eq. (1.16)). The corresponding increment in the coordinate t is given by formula
→
−
(A.21): dt = ⟨dt, dx⟩; we have then, using Eq. (10.92) to express ∇t,
→
− →
− dτ dτ
dt = ⟨dt, dx⟩ = ∇t · dx = dτ ∇t · n = − n · n} = ,
N | {z N
−1
hence formula (10.94).
A zero-angular-momentum observer (ZAMO) is an observer O of the above type, i.e.
whose worldline L is normal to the hypersurfaces Σt of constant Boyer-Lindquist time t, and
whose orthonormal frame (e(α) ) is related to the Boyer-Lindquist coordinate frame (∂α ) by
p
ρ2 (r2 + a2 ) + 2a2 mr sin2 θ 2amr
e(0) = n = √ ∂t + p ∂φ
ρ ∆ ρ ∆[ρ2 (r2 + a2 ) + 2a2 mr sin2 θ]
(10.95a)
√
∆
e(r) = ∂r (10.95b)
ρ
1
e(θ) = ∂θ (10.95c)
ρ
ρ
e(φ) = p ∂φ , (10.95d)
sin θ ρ2 (r2 + a2 ) + 2a2 mr sin2 θ
4
Note that N is not defined as the (pseudo)norm of N , as the notation might suggest, but rather as the inverse
of it.
358 Kerr black hole
√
where ρ := r2 + a2 cos2 θ (the positive square root of ρ2 , which is defined by Eq. (10.9)).
Expression (10.95a) for n has been obtained by combining Eqs. (10.92), (10.93) and (10.89).
Given the Boyer-Lindquist components (10.8) of g, one readily check that (e(r) , e(θ) , e(φ) ) is an
orthonormal basis of spacelike vectors (see the notebook D.5.5 for a SageMath computation).
Moreover, these vectors are all tangent to Σt , since (∂r , ∂θ , ∂φ ) are. They are thus orthogonal to
n = e(0) , which is a unit timelike vector. This completes the proof that (e(α) ) is an orthonormal
frame.
The ZAMO coframe is the 4-tuple of 1-forms (e(α) )0≤α≤3 that constitutes, at each point
p ∈ L , a dual basis of (e(α) p ), namely (e(α) ) obeys ⟨e(α) , e(β) ⟩ = δ αβ (cf. Eq. (A.24) in
Appendix A). Its expression in terms of the Boyer-Lindquist coordinate coframe (dxα ) is (cf.
the notebook D.5.5)
√
ρ ∆
(0)
e =p dt (10.96a)
ρ2 (r2 + a2 ) + 2a2 mr sin2 θ
ρ
e(r) = √ dr (10.96b)
∆
e(θ) = ρ dθ (10.96c)
2amr sin θ sin θ
q
e(φ) = − p dt + ρ2 (r2 + a2 ) + 2a2 mr sin2 θ dφ.
2 2 2 2
ρ ρ (r + a ) + 2a mr sin θ 2 ρ
(10.96d)
Each ZAMO can be characterized by its coordinates (r0 , θ0 , φ0 ) at some fixed value t0
of the Boyer-Lindquist time t. The set of ZAMOs is thus a 3-parameter family of observers
filling MZAMO . The coordinates (r, θ, φ) span each hypersurface Σt . Contrary to the ZAMO
worldlines, the curves of fixed (r, θ, φ), the tangent vector of which is ∂t = ξ, are not orthogonal
to Σt , except for a = 0. The orthogonal decomposition of ξ into a part along the normal n and
a part tangent to Σt defines the shift vector β:
ξ =Nn+β , n · β = 0. (10.97)
From Eq. (10.95a), we get
2amr
β=− ∂φ . (10.98)
ρ2 (r2 + a2 )+ 2a2 mr sin2 θ
Remark 2: The terms lapse and shift vector are those used in the so-called 3+1 formalism of general
relativity (see e.g. Refs. [10, 44, 227, 456]). In this context, the ZAMO is called the Eulerian observer,
which is the generic denomination of the observer whose worldline is normal to the hypersurfaces Σt
that constitute the 3+1 foliation of spacetime.
The ZAMO rotation velocity seen from infinity ω is obtained by comparing Eqs. (10.86) and
Eq. (10.97) rewritten as n = N −1 (ξ − β φ η). We thus get immediately ω = −β φ . Hence
dφ 2amr
ω := = . (10.99)
dt L ρ2 (r2 + a2 ) + 2a2 mr sin2 θ
10.7 Families of observers in Kerr spacetime 359
Note that ω = 0 for a = 0 (Schwarzschild black hole) and that ω decays quite rapidly with r:
2am 2J
ω ∼ 3
= 3, (10.100)
r→±∞ r r
where Eq. (10.80) has been used to let appear the black hole angular momentum J.
A ZAMO is not an inertial observer: it is not in free-fall, since its 4-acceleration a := ∇n n
is nonzero. Indeed, for any family of observers orthogonal to a spacelike foliation (Σt )t∈R , it
can be shown that a is the orthogonal projection onto Σt of the gradient of the logarithm of the
→
−
lapse function (see e.g. Eq. (4.19) in Ref. [227]): a = ∇ ln N + (∇n ln N ) n. In the present case,
this expression simplifies since ∇n ln N = nµ ∂µ ln N = 0, as a result of n = nt ∂t + nφ ∂φ . We
→
−
then get a = ∇ ln N , i.e. aα = g αµ ∂µ ln N , so that
∆ ∂N 1 ∂N
a= ∂ r + ∂θ .
ρ2 N ∂r ρ2 N ∂θ
Using expression (10.93) for N , we get5
m[ρ2 (r4 − a4 ) + 2∆(ra sin θ)2 ] 2a2 mr(r2 + a2 ) sin θ cos θ
a= √ e (r) − e(θ) .
ρ3 ∆[ρ2 (r2 + a2 ) + 2a2 mr sin2 θ] ρ3 [ρ2 (r2 + a2 ) + 2a2 mr sin2 θ]
(10.101)
We have thus a ̸= 0 as soon as m ̸= 0. In other words, the ZAMO’s worldline is not a geodesic
(cf. Eq. (B.1) in Appendix B) and the ZAMO feels√some acceleration, the stronger, √ the closer to
the black hole. Far from the black hole, we have ∆ ∼ |r|, ρ ∼ |r| so that ρ3 ∆ ∼ r4 and we
get
m
a ∼ e(r) . (10.102)
r→±∞ r 2
In the asymptotic regions, the non-relativistic gravitational field felt by the observer is g = −a,
so that we recover the standard Newtonian√expression for r → +∞ (MI side). In the asymptotic
region of MIII , i.e. for r → −∞, e(r) = ( ∆/ρ) ∂r is oriented towards the black hole, so that
g points outward, i.e. is a repelling field. This is of course in agreement with the negative mass
aspect of the Kerr metric in region MIII discussed in Sec. 10.2.2.
Despite ω as given by Eq. (10.99) is nonzero for a ̸= 0, a ZAMO has a vanishing angular
momentum about the rotation axis, hence the name zero-angular momentum observer. Indeed,
the observer’s specific “angular momentum”, loosely defined by ℓ := η · n [cf. Eq. (7.26)], is
identically zero, the observer’s 4-velocity n being orthogonal to η = ∂φ . By losely defined, we
mean that the above definition of ℓ fully makes sense for a geodesic, for which it leads to a
conserved quantity along the worldline (cf. Sec. 7.2.1), and we are going to see that a ZAMO’s
worldline is not a geodesic. However, a ZAMO shares with a ℓ = 0 geodesic crossing his
worldline the same value ω of the angular velocity as seen from infinity, as we shall see in
Sec. 11.3.4 [cf. Eq. (11.72)].
The name locally non-rotating observer, initially given to a ZAMO (cf. historical note below),
conveys other specificities of such a observer O:
5
See the notebook D.5.5; see also Eqs. (70)-(71) of Ref. [450] or Eqs. (A.9) and (A.10) of Ref. [61].
360 Kerr black hole
• O does not measure any component along e(φ) for the 3-momentum P of any null or
timelike geodesic that has a zero conserved angular momentum (L = 0):
P (φ) = e(φ) · P = e(φ) · [p + (n · p) n)] = e(φ) · p = eφ(φ) η · p = eφ(φ) L = 0,
where p stands for the particle’s 4-momentum and we have used expression (1.24) for
the 3-momentum, along with e(φ) · n = 0, formula (10.95d) for e(φ) and the definition
L := η · p of the conserved angular momentum [cf. Eq. (7.1b), as well as Eq. (11.2b)
below].
• for O, the directions e(φ) and −e(φ) are equivalent, insofar as the proper time measured
by O for a light signal to perform a full circuit on a circle at r = const, θ = const is the
same for a forward motion (increasing φ) of the signal as for a backward one (decreasing
φ) (see Ref. [37] for details).
However, a ZAMO is not totally “non rotating”, for he has a nonzero 4-rotation as soon
as a ̸= 0. Let us recall that the 4-rotation vector of an observer O of 4-velocity n and 4-
acceleration a is defined as the spacelike vector ωrot orthogonal to n such that the evolution
of O’s orthonormal frame (e(α) ) along O’s worldline L takes the form (see e.g. Sec. 13.6 of
Ref. [371] or Sec. 4.5 of Ref. [228])
∇n e(α) = (a · e(α) )n − (n · e(α) )a + ωrot ×n e(α) , (10.103)
| {z }
ΩFW (e(α) )
where the cross product in the hyperplane orthogonal to n, ωrot ×n e(α) , is the unique vector
orthogonal to both n and ωrot such that (ωrot ×n e(α) ) · v = ϵ(n, ωrot , e(α) , v) for any vector v,
ϵ being the Levi-Civita tensor (cf. Sec. A.3.4). The ΩFW operator that appears in the right-hand
side of Eq. (10.103) is called the Fermi-Walker operator. It appears as soon as the observer
is accelerating, even if he is non-rotating. It corrects the parallel transport of (e(α) ) along L ,
which would be realized by ∇n e(α) = 0, to ensure that (e(α) ) remains an orthonormal frame.
A vector field v that obeys ∇n e = ΩFW (v) is said to be Fermi-Walker transported along L .
Physically, Fermi-Walker transport is realized by a free gyroscope: its spin vector with respect
to O is Fermi-Walker transported along L . Hence the 4-rotation ωrot of an observer, which
is an absolute quantity (like the 4-acceleration a, it does not depend on any other observer),
measures the motion of his spacelike triad (e(i) ) with respect to an orthonormal triad whose
vectors are aligned along gyroscopes axes. For the ZAMO, the 4-rotation turns out to be6
√ ρ2 2
ω
ωrot =− 3 2 2 2 2 2
a ∆ cos θ sin θ e(r) + r(r + a ) + (r − a ) sin θ e(θ) , (10.104)
ρ 2r
where ω is given by Eq. (10.99). Hence, as soon as a ̸= 0, ωrot ̸= 0. Far from the black hole,
ρ ∼ |r| and we get
3ω r 3J
ωrot ∼ − sin θ e(θ) = − 3 sin θ e(θ) . (10.105)
r→±∞ 2 |r| |r|
6
See the notebook D.5.5 for the computation; see also Eqs. (73)-(74) of Ref. [450].
10.7 Families of observers in Kerr spacetime 361
In particular, in the equatorial plane (θ = π/2), ωrot ∼ 3J/|r|3 ez , where ez = −e(θ) is parallel
the symmetry axis, with the same direction as the black hole spin.
Remark 3: As a family of observers, the ZAMOs form a zero-vorticity congruence. The vorticity
2-form of any congruence of timelike worldlines of 4-velocity u is defined as the “magnetic” part Ω in
the “electric/magnetic” decomposition7 of the 2-form du with respect to u (cf. Sec. A.4.3):
du = a ∧ u + Ω with Ω(u, ·) = 0, (10.106)
where a is the 1-form associated by metric duality to the 4-acceleration a := ∇u u of the worldlines.
That Ω = 0 for the ZAMO congruence follows from its orthogonality to a family of hypersurfaces,
namely (Σt )t∈R . Indeed, we deduce from (10.92) that dn = −d(N dt) = −dN ∧ dt, since ddt = 0 [cf.
Eq. (A.92)]. Using (10.92) again, we get dn = d ln N ∧ n. Comparing with (10.106) with u = n, we
conclude that Ω = 0 for the ZAMO congruence8 . Physically, this means that if each ZAMO sets up a
orthonormal spatial frame (e′i ) with axes aligned along the spin of free gyroscopes, instead of the frame
(e(i) ) defined by (10.95), then this frame will not rotate with respect to the frame defined similarly by a
nearby ZAMO.
Historical note : The concept of ZAMO has been introduced by James M. Bardeen in 1970 [37] under
the name of locally non-rotating observers and for generic stationary and axisymmetric spacetimes.
The application to the exterior part of Kerr spacetime has been performed by Bardeen, William H. Press
and Saul A. Teukolsky in 1972 [42]. The name ZAMO has been coined by Kip S. Thorne and Douglas
MacDonald in 1982 [480].
10.7.4 Carter observers
A Carter observer is a stationary observer O defined in the region MI ∪MIII of Kerr spacetime,
whose worldline L has the following equation in Boyer-Lindquist coordinates:
r 2 + a2 a
t(τ ) = √ τ + const, r(τ ) = const, φ(τ ) = √ τ + const,
θ(τ ) = const,
ρ ∆ ρ ∆
(10.107)
where τ is O’s proper time, and which is equipped with the following orthonormal frame:
r 2 + a2 a
ε(0) = √ ∂t + √ ∂φ (10.108a)
ρ ∆ ρ ∆
√
∆
ε(r) = ∂r (10.108b)
ρ
1
ε(θ) = ∂θ (10.108c)
ρ
a 1
ε(φ) = sin θ ∂t + ∂φ . (10.108d)
ρ ρ sin θ
7
This name stems from the decomposition of the electromagnetic field 2-form F (cf. Sec. 1.5.2) with respect to
an observer of 4-velocity u into the electric field E and the magnetic field B, both measured by that observer,
according to F = u ∧ E + ⋆(u ∧ B), where ⋆ stands for the Hodge dual, cf. Eq. (5.38).
8
Incidentally, we also get a = d ln N + α n and Eq. (10.101) shows that α is actually zero.
362 Kerr black hole
We note that the frame (ε(α) ) is well-defined because ∆ > 0 in MI ∪ MIII . It is an easy exercise
to check that (ε(α) ) is an orthonormal frame (see the notebook D.5.6). In particular, we have
ε(0) · ε(0) = −1 and, using the Boyer-Lindquist components (10.50) of k:
r 2 + a2 a ρ
k · ε(0) = kµ εµ(0) =− √ + a sin2 θ √ = − √ < 0,
ρ ∆ ρ ∆ ∆
which, by virtue of Lemma 1.2 (Sec. 1.2.2), shows that ε(0) is future-directed. Moreover, we
notice from (10.107) and (10.108a) that εα(0) = dxα /dτ , which proves that the worldline L
is timelike and ε(0) is the 4-velocity of observer O. Since r and θ are constant along L , we
note also that expression (10.108a) for ε(0) agrees with the general form (10.85) of a stationary
observer’s 4-velocity.
The algebraic dual of the Carter frame, called the Carter coframe, is the 4-tuple of 1-forms
(ε )0≤α≤3 such that ⟨ε(α) , ε(β) ⟩ = δ αβ (cf. Sec. A.2.4 in Appendix A). It is related to the
(α)
Boyer-Lindquist coordinate coframe (dxα ) by9
√
∆
(0)
dt − a sin2 θ dφ (10.109a)
ε =
ρ
ρ
ε(r) = √ dr (10.109b)
∆
ε(θ) = ρ dθ (10.109c)
sin θ
ε(φ) = −adt + (r2 + a2 )dφ . (10.109d)
ρ
Remark 4: By definition of an orthonormal frame, the spacetime metric can be written
g = −ε(0) ⊗ ε(0) + ε(r) ⊗ ε(r) + ε(θ) ⊗ ε(θ) + ε(φ) ⊗ ε(φ) . (10.110)
Substituting Eqs. (10.109) for the 1-forms ε(α) and using the notation ω 2 := ω ⊗ ω for the tensor-
product square of a 1-form [cf. Eq. (A.39)], we get the following expression of the metric with respect to
Boyer-Lindquist coordinates:
∆ 2 ρ2 2 sin2 θ 2
g=− 2
dt − a sin2
θ dφ + dr + ρ2
dθ 2
+ 2
(r2 + a2 )dφ − adt . (10.111)
ρ ∆ ρ
This expression, which is equivalent to (10.8), is often found in textbooks (cf. e.g. Eq. (33.2) in Ref. [371]).
It is called the canonical form by Carter [95].
The specificity of the Carter observer is to be linked to the principal null geodesic con-
gruences of Kerr spacetime, which have been introduced in Sec. 10.4. Indeed, by adding and
subtracting the first two vectors of the Carter frame, as given by Eqs. (10.108a) and (10.108b)
and comparing the result with Eqs. (10.49) and (10.61), we can express the tangent vectors k
and ℓ of respectively the ingoing and outgoing principal null geodesics as
√
ρ ρ ∆
and (10.112)
k=√ ε(0) − ε(r) ℓ= ε (0) + ε (r) .
∆ 2(r2 + a2 )
9
cf. the notebook D.5.6.
10.7 Families of observers in Kerr spacetime 363
In other words, the principal null vectors k and ℓ lie in Span(ε(0) , ε(r) ). Physically, this means
that the Carter observer see the principal null geodesics having a pure radial motion (no
component along ε(θ) or ε(φ) ). Since the principal null geodesics are related to the Weyl tensor
and to the algebraically special character of the Kerr metric (cf. Sec. 10.4), this makes the Carter
frame well tailored to the computation of the Riemann’s curvature tensor via Cartan’s formula
(see Refs. [95, 391]).
By comparing expression (10.108a) for a Carter observer’s 4-velocity ε(0) with Eq. (10.86),
we get immediately the angular velocity around the symmetry axis of the Carter observer as
measured by an asymptotically distant static observer:
dφ a
ωC := = . (10.113)
dt L r2 + a2
Remark 5: Expression (10.113) is much simpler than that of the angular velocity ω of a ZAMO, as given
by Eq. (10.99). In particular, ωC does not depend on θ, contrary to ω.
The 4-acceleration aC := ∇ε(0) ε(0) of the Carter observer is10
m(r2 − a2 ) + a2 (m − r) sin2 θ a2
aC = √ ε(r) − 3 sin θ cos θ ε(θ) . (10.114)
ρ3 ∆ ρ
√ √
Far from the black hole, we have ∆ ∼ |r|, ρ ∼ |r| so that ρ3 ∆ ∼ r4 and we get
m
aC ∼ ε(r) , (10.115)
r→±∞ r2
i.e. the same Newtonian behavior as for the ZAMO 4-acceleration [Eq. (10.102)], with the
attractive character in MI and the repelling one in MIII .
The 4-rotation vector ωrot
C
of the Carter observer is computed in a way similar to that of the
ZAMO [Eq. (10.104)], i.e. by subtracting the Fermi-Walker part from ∇ε(0) ε(i) for i ∈ {1, 2, 3}
and writing it as ωrot
C
×ε(0) ε(i) . We get11
a h√ i
C
ωrot = 3 ∆ cos θ ε(r) − r sin θ ε(θ) . (10.116)
ρ
√
Due to ∆ ∼ |r| and ρ ∼ |r| when r → ±∞, the asymptotic behavior is
a
C
(10.117)
ωrot ∼ 2
cos θ ε(r) − (sgn r) sin θ ε(θ) ,
r→±∞ r
with sgn r = +1 in MI and sgn r = −1 in the asymptotic region of MIII . Introducing in both
the region MI and in the asymptotic region of MIII a “Cartesian” coordinate system (x, y, z)
defined x = |r| sin θ cos φ, y = |r| sin θ sin φ and z = |r| cos θ, the above formula turns out to
be equivalent to
a
C
ωrot ∼ (sgn r) 2 ∂z . (10.118)
r→±∞ r
10
See the notebook D.5.6 for the computation; see as well Eqs. (90)-(91) of Ref. [450].
11
See the notebook D.5.6 for the computation; see as well Eq. (93) of Ref. [450].
364 Kerr black hole
We have thus ωrot
C
always orthogonal to the equatorial plane in the asymptotic regions.
Remark 6: This is not the case of the ZAMO’s 4-rotation vector ωrot : Eq. (10.105) shows that ωrot is
collinear to ∂z only at θ = π/2. Note also that ωrot decays faster than ωrot C : |r|−3 versus r −2 . Besides
note that the full expression of ωrot , Eq. (10.116), is much simpler than that of ωrot , Eq. (10.104). The same
C
remark holds for the 4-acceleration aC [Eq. (10.114)] versus the ZAMO 4-acceleration a [Eq. (10.101)].
This reflects the fact that the Carter observer is tightly bound to the spacetime structure, being closely
related to the principal null geodesics, as shown by Eq. (10.112).
Historical note : The Carter frame has been introduced by Brandon Carter in 1968 [91], in the form of its
dual, namely the Carter coframe, for a quite general class of spacetimes including the Kerr one (Eqs. (81)-
(82) in Ref. [91]). The explicit form for Kerr spacetime, i.e. the system (10.109), can be found in Carter’s
lecture at the 1972 Les Houches Summer School [95] (Eqs. (5.19)-(5.24) and (5.31)-(5.34); cf. footnote 41
in the reprinted version [95]). The Carter frame has been popularized by Roman Znajek in 1977 [533] for
studying electromagnetic processes around a black hole and has been heavily used in Barrett O’Neill’s
monograph about Kerr black holes [391], where it is called√the Boyer-Lindquist frame field. In particular
O’Neill’s canonical Kerr vector fields V and W are V = ρ ∆ ε(0) and W = ρ sin θ ε(φ) .
10.7.5 Asymptotic inertial observers
For r → ±∞, we see on Eqs. (10.102), (10.105), (10.115) and (10.117) that the 4-acceleration and
4-rotation of both the ZAMOs and the Carter observers tend to zero. Hence, these observers
become inertial observers. Actually, at the same value of (r, θ, φ), the ZAMO and Carter
observer reduce to the same inertial observer. The latter has ω = 0 (take the limit r → ±∞ in
Eqs. (10.100) and (10.113)). It is therefore a static observer (cf. Sec. 10.7.2). We shall call it the
asymptotic inertial observer. His 4-velocity is nothing but the Killing vector ξ.
10.8 Maximal analytic extension
10.8.1 Carter-Penrose diagrams
It is useful to have a “compactified view” of the whole Kerr spacetime (M , g), as we had
for Schwarzschild spacetime via the Carter-Penrose diagrams discussed in Sec. 9.4. However,
because the Kerr spacetime with a ̸= 0 is not spherically symmetric, such a 2-dimensional
diagram can only offer a truncated view. With this in mind, we shall call a Carter-Penrose
diagram12 of Kerr spacetime (M , g) the image Π(M ) of a differentiable map Π : M → R2 ,
(t̃, r, θ, φ̃) 7→ (T, X) such that
1. T = T (t̃, r), X = X(t̃, r);
2. Π(M ) is a bounded region of R2 ;
3. the ingoing principal null geodesics are mapped to lines T = −X + const and the
outgoing ones are mapped to lines T = X + const, with T increasing towards the future
of (M , g) in both cases.
12
See the historical notes on p. 300 and p. 368.
10.8 Maximal analytic extension 365
Property 1 implies that Π acts as a projection, leaving out any information in the non-ignorable
coordinate θ. Property 2 provides the “compactified” view13 , and Property 3 makes Π(M )
look “conformal” to the 2-dimensional Minkowski spacetime generated by the metric f =
−dT ⊗ dT + dX ⊗ dX on R2 . However, let us stress that Π is not a priori connected to a
proper conformal completion of (M , g), as defined in Sec. 4.3. Note that to fulfill Property 3, it
suffices to choose the functions T (t̃, r) and X(t̃, r) defining Π such that in each of the regions
MI , MII and MIII ,
T (t̃, r) = U (u) + V (v) and X(t̃, r) = V (v) − U (u), (10.119)
where (i) U (u) is a function of the retarded Kerr time u = u(t̃, r) given by Eq. (10.53a) that is
monotonic increasing in MI and MIII and decreasing in MII (cf. Fig. 10.8) and (ii) V (v) is a
monotonic increasing function of the advanced Kerr time v = t̃ + r [Eq. (10.33)].
This follows from u (resp. v) being constant along the outgoing (resp. ingoing) principal
null geodesics. Note also that Property 3 ensures that the two horizons H and Hin , which
are generated by outgoing principal null geodesics, appear as lines inclined by +45◦ with
respect to the X-axis, as for the black hole event horizon in the Carter-Penrose diagrams of
Schwarzschild spacetime (Figs. 9.10 and 9.11).
Remark 1: A related but distinct concept of projection diagram has been introduced in Ref. [126] (see
also Chap. 7 of Ref. [119]); it differs from that considered here by demanding, instead of Properties 1
to 3, that Π maps any timelike curve of14 (M \ T , g) to a timelike curve of the Minkowski spacetime
(R2 , f ) and that any timelike curve in Π(M \ T ) is the image of a timelike curve of (M \ T , g).
A schematic Carter-Penrose diagram of the Kerr spacetime (M , g) is shown in Fig. 10.9.
By schematic it is meant that we do not provide any explicit construction via a map Π. Let
us mention however that defining Π by choosing U (u) = arctan(u/m) − kπ and V (v) =
arctan(v/m) in Eq. (10.119), with k = 0, 1 and 2 on respectively MI , MII and MIII , would
lead to such a diagram. By virtue of Property 1, one may think of each point in the diagram
of Fig. 10.9 as being a 2-sphere, spanned by (θ, φ̃), except along the curve r = 0 (thick dotted
line), where each point is the union S0,t of two flat open disks (cf. Sec. 10.2.2, Eq. (10.14)).
Each of the regions MI , MII and MIII of the Kerr spacetime M is mapped to the interior of
a square tilted by 45◦ (a “diamond”) in the diagram of Fig. 10.9. In each of these diamond blocks,
v increases from −∞ to +∞ in the South-West to North-East direction, while u increases from
−∞ to +∞ in the South-East to North-West direction in MI and MIII and in the opposite
direction in MII . These ranges and directions follow directly from v = t̃ + r and the graph of
u − t̃ shown in Fig. 10.8.
The dotted curves in Fig. 10.9 represent some hypersurfaces r = const. According to the
results of Sec. 10.5.1, such hypersurfaces are timelike in MI and MIII , spacelike in MII and null
for r = r− or r = r+ . Since the Killing vector field ξ is tangent to the hypersurfaces r = const,
the dotted curves in Fig. 10.9 can also be seen as the projections of the field lines of ξ, i.e. of
the orbits of the stationary group action.
13
Property 2 refers to a “bounded region” instead of “compact region”, because Π(M ) is in general an open
subset of R2 and therefore not a compact subset.
14
T is the Carter time machine discussed in Sec. 10.2.5.
366 Kerr black hole
Figure 10.9: Carter-Penrose diagram of the Kerr spacetime (M , g), with M = R2 × S2 \ R (cf. Eq. (10.29)
in Sec. 10.3.1), spanned by the ingoing principal null geodesics (dashed green lines). The solid green lines are
outgoing principal null geodesics, while the dotted curves mark some hypersurfaces r = const.
As pointed out in Sec. 10.5.2, the asymptotic structure of region MI is identical to that of
MI in Schwarzschild spacetime. It can thus be endowed with a conformal completion at null
infinity, as constructed in Secs. 9.4.3 and 9.4.4. This allows us to add the future (resp. past)
null infinity I + (resp. I − ) to the diagram of Fig. 10.9. Note that I + corresponds to the limit
v → +∞ in MI , while I − corresponds to the limit u → −∞ in MI .
10.8.2 Constructing the maximal extension
The ingoing principal null geodesics L(v,θ, in
φ̃) (dashed green lines in Fig. 10.9) are complete (cf.
Sec. B.3.2), except for those lying in the equatorial plane, which hit the curvature singularity
at r = 0. Indeed, the affine parameter λ = −r of any ingoing principal null geodesic with
θ ̸= π/2 ranges from −∞ (lower right of the diagram) to +∞ (upper left). On the contrary, the
outgoing principal null geodesics L(u,θ,out
˜ (solid green lines in Fig. 10.9) are not complete: in
φ̃)
MI , their affine parameter λ = r is bounded from below by r+ ; in MII , their affine parameter
λ = −r ranges in (−r+ , −r− ) only and in MIII , their affine parameter λ = r is bounded from
above by r− . Since these geodesics are not ending at any spacetime singularity (it can be shown
that all curvature scalar invariants remain bounded along them), except those with θ = π/2
in MIII , this indicates that the spacetime (M , g) can be extended. Moreover, for 0 ≤ a < m,
the event horizon H is a non-degenerate Killing horizon (cf. Sec. 10.5.4) and we have seen in
Sec. 3.4.2 that, generically, such horizons are part of a bifurcate Killing horizon in an extended
10.8 Maximal analytic extension 367
Figure 10.10: Carter-Penrose diagram of the minimal extension of MI to ensure complete outgoing principal
null geodesics (one of them is drawn as a solid green line).
spacetime.
A spacetime extending (MI , g) “to the past”, so that the outgoing principal null geodesics
are complete, is shown in Fig. 10.10. It is made by attaching to (MI , g) a time-reversed copy15
of (MII , g), (MII∗ , g) say, and then attaching to the latter a copy of (MIII , g), (MIII
′′
, g) say16 .
By construction, the outgoing principal null geodesics are complete in this extension, but the
ingoing ones are not: the affine parameter λ = −r of that denoted by v = v0 in Fig. 10.10 is
bounded from above by −r+ . Actually the situation is completely symmetric to that of the
original Kerr spacetime (M , g) (Fig. 10.9). In particular, the region MII∗ ∪ Hin′′ ∪ MIII ′′
, where
Hin is the (null) hypersurface r = r− , is a white hole, since it is the complement of J (I − ) (cf.
′′ +
the definition (4.41) of a white hole). The hypersurface r = r+ (black thick line in Fig. 10.10) is
the corresponding past event horizon.
By combining the diagrams of Figs. 10.9 and 10.10, one obtains a spacetime which still
contains incomplete null geodesics: the outgoing ones in regions MII and MIII , and the ingoing
ones in regions MII∗ and MIII ′′
. To go further, one should add new regions isometric to one
of the three blocks (MI , g), (MII , g) and (MIII , g) and iterate indefinitely, leading to the
Carter-Penrose diagram of Fig. 10.11. In this process, one shall make sure to have some analytic
continuation of the metric between the various blocks. This is done by introducing Kruskal-type
coordinates in the vicinity of the boundaries between the various blocks. We shall not do it
15
By copy, it is meant that (MII∗ , g) is a spacetime isometric to (MII , g) and by time-reversed that r is increasing
towards the future in MII∗ , while it is decreasing towards the future in MII .
16
The notation MIII ′′
instead of MIII′
is for later convenience.
368 Kerr black hole
here and refer the reader to the seminal articles by Boyer & Lindquist [72] and Carter [90], the
famous Les Houches lectures by Carter [95] or the textbook by O’Neill [391].
In the diagram of Fig. 10.11, it is clear that the event horizon H and the inner horizon Hin
have been extended to bifurcate Killing horizons (cf. Sec. 3.4), the bifurcation surface of which
being a 2-sphere depicted by a circular dot.
Historical note : In 1966, Brandon Carter [88] obtained the maximal analytic extension of the 2-
dimensional manifold constituted by the rotation axis17 A of the Kerr spacetime and drew a diagram
similar to that of Fig. 10.11 (Fig. 1a of Ref. [88]). More precisely, Carter introduced on A a coordinate
system (T, X) (denoted (ψ, ξ) by him) in which the metric induced by g on A is explicitely conformal
the 2-dimensional Minkowski metric f = −dT ⊗ dT + dX ⊗ dX. Thus Fig. 1a of Carter’s article [88],
which is reproduced as Fig. 28 of Hawking & Ellis’ textbook [266], is a true conformal representation of
A , and not the mere “compactified projection” that we used to define a generic Carter-Penrose diagrams
in Sec. 10.8.1. This was of course made possible because the rotation axis A is a 2-dimensional manifold.
For the whole Kerr spacetime, the “projection” aspect is inevitable. In the same article [88], Carter
suggested that the maximal analytic extension of the whole 4-dimensional manifold would be similar to
that of A . This was proven rigorously a year later by Robert H. Boyer and Richard W. Lindquist [72]
and generalized to Kerr-Newmann spacetimes by Carter himself in 1968 [90].
10.8.3 Cauchy horizon
Let us consider a spacelike hypersurface Σ running from the asymptotically flat end of MI
to the asymptotically flat end of M ′ I , possibly through18 MII or MII ∗ (cf. Fig. 10.12) that is
acausal, in the sense that no causal curve intersects it more than once. One says that Σ is a
partial Cauchy surface, the definition of the latter being an acausal hypersurface without
edge [266]. The future Cauchy development (resp. past Cauchy development) of Σ is the
set D+ (Σ) (resp. D− (Σ)) of all spacetime points p such that each past-directed (resp. future
directed) inextendible causal curve through p intersects Σ. The future Cauchy development of
Σ is the hatched region in Fig. 10.12. The spacetime metric at every point in D+ (Σ) is entirely
determined by initial data on Σ through the Einstein equation, in its 3+1 form (see e.g. Chap. 5
of Ref. [227]); this reflects the well-posedness of general relativity as a Cauchy problem.
One says that Σ is a Cauchy surface iff D+ (Σ) ∪ D− (Σ) is the whole spacetime, i.e. iff
every inextendible causal curve intersects19 Σ. It is clear on Fig. 10.12 that Σ is a Cauchy
surface for (M1 , g), with M1 := MI ∪ M ′ I ∪ MII ∪ M ∗ II , but not for the whole extended
Kerr spacetime. The future boundary of D+ (Σ) is called the future Cauchy horizon of Σ
and denoted by H + (Σ). In the present case, the future Cauchy horizon does not depend upon
the choice of Σ, being the same for any partial Cauchy surface through MI ∪ M ′ I . We shall
therefore denote it by HC . It is depicted as the blue thick line in Fig. 10.12. HC is the union
of what we called the inner horizon in Sec. 10.5.1, i.e. the Killing horizon Hin , and the null
17
Let us recall that the rotation axis is the 2-dimensional Lorentzian submanifold A where the Killing vector η
vanishes (cf. the definition given in Sec. 5.3.6); in terms of the Kerr coordinates (t̃, r, θ, φ̃), A is defined by by
θ = 0 or π and is naturally spanned by the coordinates (t̃, r).
18
In Fig. 10.12, this is not the case: Σ goes from MI to M ′ I via the bifurcation sphere.
19
There is a single intersection since Σ is assumed acausal; the Cauchy surface definition given here agrees
thus with that given on p. 127.
10.8 Maximal analytic extension 369
Figure 10.11: Carter-Penrose diagram of the maximal analytic extension of the Kerr spacetime. As in Figs. 10.9
and 10.10, the dotted curves mark some hypersurfaces r = const. The central black or light brown dots mark the
bifurcation spheres of bifurcate Killing horizons.
370 Kerr black hole
Figure 10.12: The partial Cauchy surface Σ, its future Cauchy development D+ (Σ) (hatched) and the Cauchy
horizon HC . As in Figs. 10.9-10.11, the dotted curves marks some hypersurfaces r = const.
boundary between MII and M ′ III , H ′ in say, which is a Killing horizon as well:
HC = Hin ∪ H ′ in . (10.120)
Note that HC corresponds to a fixed value of the r coordinate of Kerr-type coordinate systems,
namely r = r− , in agreement with the primary definition of Hin given in Eq. (10.28).
Remark 2: There is no such Cauchy horizon in the Schwarzschild spacetime, even in its maximally
extended version. For instance the hypersurface defined in terms of the Kruskal-Szekeres coordinates
by T = 0 and whose equatorial section is depicted in Fig. 9.14 (Flamm paraboloid), is a Cauchy surface
for the whole maximally extended Schwarzschild spacetime: by looking at Fig. 9.12, you may convince
yourself that any inextendible causal curve in Schwarzschild spacetime must go through T = 0.
Let us evaluate the non-affinity coefficient κin of the Killing generator χin of the part Hin of
the Cauchy horizon [cf. Eq. (10.70)]. Since mr− /(r− 2
+ a2 ) = 1/2, we notice, from Eqs. (10.57),
(10.70) and (10.71), that the Killing vector χin coincides with the vector ℓ tangent to the outgoing
principal null geodesics on Hin :
H
χin =in ℓ. (10.121)
Hence, as the event horizon H , the inner horizon Hin is generated by outgoing principal null
geodesics [compare Eq. (10.72) and see Figs. 10.6 and 10.7]. Equation (10.58) implies then
H
∇χin χin =in κin χin , (10.122)
10.9 Further reading 371
with the non-affinity coefficient κin obtained by specializing Eq. (10.58) to r = r− : κin =
2
κℓ |r=r− = m(r− − a2 )/(r−
2
+ a2 )2 . Using the expression of r− in terms of m and a [Eq. (10.3)],
we get
√
m2 − a2
κin = − √ . (10.123)
2m(m − m2 − a2 )
Given the assumption 0 < a < m, we have κin ̸= 0, which implies (cf. Sec. 3.3.6)
Property 10.23: non-degeneracy of the inner horizon
As long as a < m, the part Hin of the Cauchy horizon HC is a non-degenerate Killing
horizon.
We note that κin < 0. According to the results of Sec. 3.4.2, this implies
Property 10.24: Cauchy horizon as part of a bifurcate Killing horizon
Hin is contained in a bifurcate Killing horizon, the bifurcation surface of which being the
future boundary of Hin (light brown small disk in Fig. 10.12). Actually, the whole Cauchy
horizon HC is the past part of the bifurcate Killing horizon, the future part being formed
by the future boundaries r = r− of MIII and M ′ III (light brown thick lines in Fig. 10.12).
10.8.4 Physical relevance of the maximal extension
As for the maximal extension of Schwarzschild spacetime (Chap. 9), the maximal extension
of the Kerr spacetime discussed above corresponds to an “eternal” black hole, not to any
“astrophysical” black hole formed by gravitational collapse. Moreover, contrary to the non-
rotating case, where the Schwarzschild geometry is exact outside the collapsing star (by virtue
of the Jebsen-Birkhoff theorem, cf. Sec. 14.2.5), the spacetime outside a collapsing rotating
star is not a part of Kerr spacetime. In particular, it contains gravitational waves and is not
stationary. Only at “late times”, when all the “hairs” have been radiated away, does it settle to
the Kerr spacetime.
Another physical issue regards the Cauchy horizon: it has been shown to be unstable,
suffering from the so-called mass inflation instability discovered by Éric Poisson and Werner
Israel [417] (see Ref. [78] for a more recent study of this instability).
10.9 Further reading
For more material about the Kerr black hole, we refer the reader to O’Neill’s very nice mono-
graph [391], as well to the review articles by Heinicke & Hehl [270], Teukolsky [476] and Visser
[497].
372 Kerr black hole
Chapter 11
Geodesics in Kerr spacetime: generic and
timelike cases
Contents
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
11.2 Equations of geodesic motion . . . . . . . . . . . . . . . . . . . . . . . 374
11.3 Main properties of geodesics . . . . . . . . . . . . . . . . . . . . . . . . 388
11.4 Timelike geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
11.5 Circular timelike orbits in the equatorial plane . . . . . . . . . . . . . 418
11.6 Going further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
11.1 Introduction
In various occasions during our study of Kerr spacetime in Chap. 10, we have already encoun-
tered some peculiar geodesics, namely the principal null geodesics. In this chapter, we begin
the systematic study of causal (timelike or null) geodesics in Kerr spacetime. First, taking
advantage of an algebraic particularity of this spacetime, giving birth to a non-trivial Killing
tensor and the associated Carter constant, we shall see in Sec. 11.2 that Kerr geodesic motion is
fully governed by a system of first order equations, the solution of which can be obtained by
quadrature. The basic properties of geodesics that can be inferred from this system are derived
in Sec. 11.3. Then we focus on timelike geodesics (Sec. 11.4) and in particular on those that
are bound; they represent orbits of massive particles or bodies around a Kerr black hole. In
Sec. 11.5, we shall investigate the particular case of circular orbits in the equatorial plane and
discuss their stability. The detailed study of null geodesics is deferred to the next chapter.
Of course, when the Kerr spin parameter a tends to zero, all results in this chapter reduce to
those obtained for Schwarzschild geodesics in Chap. 7. As already suggested in the introduction
of Chap. 7, before moving on, the reader might want to have a look at Appendix B, which
recaps the properties of geodesics in pseudo-Riemannian manifolds.
374 Geodesics in Kerr spacetime: generic and timelike cases
11.2 Equations of geodesic motion
11.2.1 Introduction
In all this chapter, we are concerned with the motion of a particle P in the Kerr spacetime
(M , g), under the hypothesis that P feels only gravity, as described by g (freely falling particle).
The worldline L of P is then necessarily a geodesic1 of (M , g). It is a timelike geodesic if P
is a massive particle and a null geodesic if P is massless (e.g. a photon).
In a given coordinate system (xα ), the geodesic L is described by a system of the form2
xα = xα (λ), where λ is an affine parameter3 along L . We choose λ to be the affine parameter
associated with P’s 4-momentum p, i.e. such that dxα /dλ = pα [Eq. (1.21)]. In particular, λ is
dimensionless and is increasing towards the future4 along L . The curve L is a geodesic iff
the functions xα (λ) obey the geodesic equation [Eq. (B.10) in Appendix B]:
d 2 xα µ
α dx dx
ν
+ Γ µν = 0, 0 ≤ α ≤ 3, (11.1)
dλ2 dλ dλ
where the Γαµν ’s are the Christoffel symbols of the metric g with respect to the coordinates (xα ),
as given by Eq. (A.71). Equation (11.1) is the component expression of ∇p p = 0 [Eq. (1.11)]. It
is a system of four coupled second-order differential equations, which are non-linear. We are
going to see that it is actually not necessary to solve this system to compute the geodesics of
Kerr spacetime. Indeed, as in the Schwarzschild case studied in Chap. 7, there exists enough
first integrals of Eq. (11.1) to reduce the problem to four first-order equations of the type
dxα /dλ = F α (x0 , x1 , x2 , x3 ).
Three integrals of motion are similar to those of Schwarzschild geodesics: one is the
particle’s mass µ (Sec. 11.2.3 below) and the two other ones are the conserved energy E and the
conserved angular momentum L (Sec. 11.2.2), which arise from the common symmetries of Kerr
and Schwarzschild spacetimes: stationarity (outside the ergoregion) and axisymmetry. In the
Schwarzschild case, the fourth integral of motion was provided by spherical symmetry, which
constrained all geodesics to be planar, so that a suitable choice of coordinates (t, r, θ, φ) makes a
given geodesic confined to the hyperplane θ = π/2, yielding the first integral pθ = 0 [Eq. (7.7)].
The Kerr spacetime with a ̸= 0 being not spherically symmetric, we loose this property here.
Fortunately there exists another integral of motion, known as the Carter constant; it arises
from a remarkable property of Kerr spacetime: the existence of a non-trivial Killing tensor
(Sec. 11.2.4). This makes a total of four integral of motions, which makes the problem integrable
(Secs. 11.2.5 to 11.2.8).
11.2.2 Integrals of motion from spacetime symmetries
As for Schwarzschild spacetime (cf. Sec. 7.2.1), we have the following property:
1
The definition and basic properties of geodesics are recalled in Appendix B; see also Sec. 1.3.2.
2
In Appendices A and B, a different symbol is used for the function X α (λ) defining L and the coordinate xα
(cf. Secs. A.2.3 and B.3.1). Following standard usage in the physics literature, we shall not do this in this chapter.
3
See Sec. B.2.1 for the definition of an affine parameter along a geodesic.
4
Let us recall that the Kerr spacetime is time-oriented, cf. Sec. 10.3.2.
11.2 Equations of geodesic motion 375
Property 11.1: conserved quantities along causal geodesics
The Killing vectors ξ and η of Kerr spacetime, associated respectively with stationarity
and axisymmetry [cf. Eq. (10.12)], give birth to two conserved quantities along any (causal)
geodesic L :
E := −ξ · p = −g(ξ, p) (11.2a)
L := η · p = g(η, p) , (11.2b)
where p is the 4-momentum of the particle P having L as worldline (cf. Sec. 1.3). For
the same reasons as in Sec. 7.2.1, E is called the conserved energy or energy at infinity
of P, while L is called the conserved angular momentum or angular momentum at
infinity of P.
In particular, if L reaches the asymptotic region |r| → +∞, E is nothing but P’s energy as
measured by the asymptotic inertial observer introduced in Sec. 10.7.5, given that the 4-velocity
of the latter coincides with the Killing vector ξ [compare Eq. (1.23) with Eq. (11.2a)]. Similarly,
L is the component along the rotation axis of P’s (total) angular momentum vector Ltot as
measured by the asymptotic inertial observer [cf. Eq. (7.4)].
Remark 1: Because it represents only a component of the total angular momentum, L is sometimes
denoted by Lz .
In coordinates (t, r, θ, φ) adapted to spacetime symmetries, i.e. coordinates such that ξ = ∂t
and η = ∂φ , for instance Boyer-Lindquist coordinates (Sec. 10.2.1), advanced Kerr coordinates
(Sec. 10.3.1) or Kerr coordinates (Sec. 10.3.3), one can rewrite (11.2) in terms of the components
pt = gtµ pµ and pφ = gφµ pµ of the 1-form p associated to p by metric duality:
E = −pt (11.3a)
L = pφ (11.3b)
Indeed, in such a coordinate system, ξ µ = δ µt and η µ = δ µφ , so that E = −gµν ξ µ pν =
−gtν pν = −pt and L = gµν η µ pν = gφν pν = pφ .
Example 1 (Generators of the event and inner horizons): Let us choose for L a null geodesic
generator of the event horizon H (cf. Sec. 10.5.3). L is then an outgoing principal null geodesic
out,H H
L(θ,ψ) and the 4-momentum vector p is proportional to the null vector ℓ = χ = ξ + ΩH η [Eq. (10.72)
and (10.68)]. By definition of a generator of a null hypersurface (cf. Sec. 2.3.3), p is normal to H .
Since the Killing vector fields ξ and η are tangent to H (for H is globally preserved by the spacetime
symmetries), we get immediately from the definitions (11.2) E = 0 and L = 0. Similarly, if L is a null
out,Hin
geodesic generator of the inner horizon Hin , L is an outgoing principal null geodesic L(θ,ψ) , with p
H
proportional to the null normal ℓ =in χin = ξ + Ωin η [Eq. (10.121) and (10.70)], so that we have E = 0
and L = 0 as well. Hence we conclude:
L null geodesic generator of H or Hin =⇒ E = 0 and L = 0. (11.4)
376 Geodesics in Kerr spacetime: generic and timelike cases
Example 2 (Ingoing principal null geodesics): For the ingoing principal null geodesics L(v,θ,
in
φ̃)
introduced in Sec. 10.4, the 4-momentum vector is
p = αk, (11.5)
where α is a (positive) constant, since k is a geodesic vector [Eq. (10.45)], as p [Eq. (1.11)]. Equations (11.3)
with the components relative to Kerr coordinates (t̃, r, θ, φ̃) lead then to
E = −αkt̃ = −α(−1) = α and L = αkφ̃ = αa sin2 θ,
where the components kt̃ and kφ̃ have been read on Eq. (10.44). Hence for any ingoing principal null
geodesic L(v,θ,
in
φ̃) ,
E>0 and L = aE sin2 θ. (11.6)
Recall that θ is constant along L(v,θ,
in
φ̃) , so that the above formula does yield a constant value for L.
Moreover, it fulfills L ≥ 0 with L = 0 only for a = 0 or θ ∈ {0, π} (rotation axis).
Example 3 (Outgoing principal null geodesics): For the outgoing principal null geodesics L(u,θ,
out
˜ ,
φ̃)
out,H out,Hin
L(θ,ψ) and L(θ,ψ) introduced in Sec. 10.4, the 4-momentum vector is
p = β(λ)ℓ. (11.7)
where β(λ) is a function of the affine parameter λ associated to p that obeys β(λ) > 0, since both p and ℓ
are future-directed (cf. Sec. 10.4). Contrary to the coefficient α in Eq. (11.5), β(λ) is not constant because
ℓ is not a geodesic vector, but only a pregeodesic one: it fulfills ∇ℓ ℓ = κℓ ℓ with κℓ ̸= 0 [Eq. (10.58)]
(cf. Remark 6 on p. 346). The geodesic equation ∇p p = 0 implies that β obeys the differential equation
∇p β + κℓ β 2 = 0, or equivalently
dβ
+ κℓ β 2 = 0. (11.8)
dλ
The outgoing principal null geodesics either (i) are confined to one of the horizons H and Hin , generating
out,H out,Hin
them (case of L(θ,ψ) and L(θ,ψ) ), or (ii) never intersect them (case of L(u,θ,
out
˜ , cf. the solid curves
φ̃)
in Figs. 10.6 – 10.7). In the first case, κℓ is a constant on H and Hin , given by Eq. (10.58) with r = r+
for H and r = r− for Hin . The solution of Eq. (11.8) is then
1
β(λ) = ,
κℓ (λ − λ0 )
where λ0 is a constant. Outside the horizons, κℓ is the function of r given by Eq. (10.58). We then search
for a solution of Eq. (11.8) in the form β(λ) = B(r(λ)), where r(λ) is the function giving the coordinate
r along the geodesic. Since the latter obeys dr/dλ = pr = βℓr with ℓr read on Eq. (10.57), we get the
following linear differential equation for B:
dB r2 − a2
(r2 − 2mr + a2 ) + 2m 2 B = 0.
dr r + a2
The solution is
r2 + a2
B(r) = 2 B0 = β(λ),
r2 − 2mr + a2
where B0 is a positive constant in MI and MIII and a negative constant in MII , to ensure that β(λ) > 0.
11.2 Equations of geodesic motion 377
Plugging (11.7) into Eqs. (11.3) with the components relative to Kerr coordinates yields
∆ a∆ sin2 θ
E = −β(λ)ℓt̃ = β(λ) and L = β(λ)ℓφ̃ = β(λ) ,
2(r + a2 )
2 2(r2 + a2 )
where the components ℓt̃ and ℓφ̃ have been read on Eq. (10.60). Given the above results for β(λ), we get
E = 0 and L = 0 on H and Hin , since ∆ = 0 there, and E = B0 and L = aB0 sin2 θ elsewhere. We
conclude that for the outgoing principal null geodesics,
in H ∪ Hin , E=0 and L=0 (11.9a)
in MI ∪ MIII , E>0 and L = aE sin θ2
(11.9b)
in MII , E<0 and 2
L = aE sin θ. (11.9c)
In particular, (11.9a) is nothing but the result (11.4) already obtained in Example 1. We note also that
the relation between L and E is identical to that obtained in Example 2 for the ingoing principal null
geodesics [Eq. (11.6)].
In what follows, we will use Boyer-Lindquist coordinates (xα ) = (t, r, θ, φ) as introduced in
Sec. 10.2.1. Given the components (10.8) of the metric tensor g in these coordinates, evaluating
E and L via E = −gtµ pµ and L = gφµ pµ yields
2amr sin2 θ φ
2mr
E = 1− 2 pt + p . (11.10)
ρ ρ2
2amr sin2 θ t 2a2 mr sin2 θ
L=− 2
p + r +a + 2
sin2 θ pφ , (11.11)
ρ2 ρ2
where ρ2 := r2 + a2 cos2 θ [Eq. (10.9)].
Let us recall that the components (pα ) of the 4-momentum are related to the parametric
equation xα = xα (λ) of the geodesic L in terms of the affine parameter λ by pα = dxα /dλ
[cf. Eq. (1.21)], i.e.
dt dr dθ dφ
pt = , pr = , pθ = , pφ = . (11.12)
dλ dλ dλ dλ
11.2.3 Mass as an integral of motion
The mass µ of particle P is related to the scalar square of the 4-momentum vector p via
Eq. (1.10):
µ2 = −g(p, p). (11.13)
The fact that L is a geodesic implies that µ is constant along L (cf. Eq. (B.6) in Appendix B).
It therefore provides a third integral of motion, after E and L.
It is convenient to express (11.13) in terms of the inverse metric in order to let appear
pt = −E and pφ = L:
µ2 = −g µν pµ pν .
Given the components (10.16) of the inverse metric in Boyer-Lindquist coordinates, we get
2a2 mr sin2 θ L2
2 1 2 2 2 4amr 1 2mr
µ = r +a + E − EL − 1 −
∆ ρ2 ρ2 ∆ ∆ ρ2 sin2 θ
ρ2 2
− (pr )2 − ρ2 pθ , (11.14)
∆
378 Geodesics in Kerr spacetime: generic and timelike cases
where ∆ := r2 − 2mr + a2 [Eq. (10.10)]. Note that we have expressed pr and pθ in terms of
pr and pθ thanks to the relations pr = grµ pµ and pθ = gθµ pµ , which are very simple for the
Boyer-Lindquist components (10.8) of g:
ρ2 r
pr = p and pθ = ρ 2 pθ . (11.15)
∆
11.2.4 The fourth integral of motion: Carter constant
It turns out that the Kerr spacetime is endowed with a non-trivial Killing tensor of valence 2:
the Walker-Penrose Killing tensor K, which is the symmetric tensor of type (0, 2) defined
by
K := (r2 + a2 ) (k ⊗ ℓ + ℓ ⊗ k) + r2 g , (11.16)
where ℓ and k are the 1-forms associated by metric duality to the null vector fields k and ℓ
tangent to the principal null geodesics introduced in Sec. 10.4. In index notation, Eq. (11.16)
writes
Kαβ = (r2 + a2 ) (kα ℓβ + ℓα kβ ) + r2 gαβ . (11.17)
K is called a Killing tensor because its symmetrized covariant derivative vanishes identically:
∇(α Kβγ) = 0 . (11.18)
This property can be seen as a generalization of the Killing equation (3.19) to tensors of valence
2. That the tensor K defined by (11.16) obeys the Killing identity (11.18) is established in the
SageMath notebook D.5.7.
Killing tensors are discussed in Sec. B.5.2 of Appendix B. It is shown there that the Killing
identity (11.18) implies that the following quantity is constant along any geodesic L (Prop-
erty B.21):
K := K(p, p) = Kµν pµ pν . (11.19)
K is named Carter constant (cf. the historical note on p. 387). From the definition (11.16) of
K, we have
K = 2(r2 + a2 )⟨k, p⟩⟨ℓ, p⟩ + r2 g(p, p). (11.20)
Now by Eq. (11.13), g(p, p) = −µ2 . Besides, we have ⟨k, p⟩ = kµ pµ = k µ pµ . The last form lets
appear the constants of motion pt = −E and pφ = L [Eq. (11.3)]. Using it with the components
of k as given by Eq. (10.49), we get
r2 + a2 a
⟨k, p⟩ = − E − pr + L.
∆ ∆
Similarly, from the components (10.61) of ℓ, we obtain
1 ∆ a
⟨ℓ, p⟩ = − E + 2 2
pr + L.
2 2(r + a ) 2(r + a2 )
2
Accordingly, Eq. (11.20) becomes
1 2
K = (r + a2 )E + ∆pr − aL (r2 + a2 )E − ∆pr − aL − r2 µ2 ,
∆
11.2 Equations of geodesic motion 379
which can be rewritten as
1 h 2 2 i
K = (r + a2 )E − aL − ρ4 (pr )2 − r2 µ2 . (11.21)
∆
Note that we have expressed pr in terms of pr via Eq. (11.15).
Some physical interpretation of the Carter constant can be inferred from the above expres-
sion, as soon as the particle P of worldline L visits the asymptotic region r → +∞. Indeed,
given that ∆ := r2 − 2mr + a2 ∼ r2 and ρ4 := (r2 + a2 cos2 θ)2 ∼ r4 as r → +∞, we deduce
from Eq. (11.21) that
K ∼ r2 E 2 − µ2 − (pr )2 . (11.22)
r→+∞
As discussed above, E is P’s energy as measured by the asymptotic inertial observer O. Then,
according to Einstein’s formula (1.26), E 2 − µ2 = P · P , where P is P’s linear momentum as
measured by O. Given that asymptotically, pr ∼ P r [cf. Eq. (1.22)], Eq. (11.22) becomes
K ∼ r2 P · P − (P r )2 = r2 (P (θ) )2 + (P (φ) )2 ,
r→+∞
where P (θ) = rP θ ∼ rpθ and P (φ) = r sin θ P φ ∼ r sin θ pφ are the angular components of P
in the orthonormal basis (e(r) , e(θ) , e(φ) ) := (∂r , r−1 ∂θ , (r sin θ)−1 ∂φ ). Now the total angular
momentum of P measured by O is
Ltot := r × P = −rP (φ) e(θ) + rP (θ) e(φ) .
Hence we may conclude that asymptotically, the Carter constant coincides with the square of
P’s angular momentum as measured by the inertial observer O:
K ∼ Ltot · Ltot . (11.23)
r→+∞
We shall see later that one has always K ≥ 0. For now, let us establish the following
characterization of null geodesics with K = 0:
Property 11.2: null geodesics with vanishing Carter constant
A null geodesic L has a vanishing Carter constant K if, and only if, L is a principal null
geodesic:
K = 0 ⇐⇒ L = L(v,θ, in
φ̃) or L = L out , (11.24)
where L(v,θ,
in
φ̃) (resp. L
out
) is one of the ingoing (resp. outgoing) principal null geodesic
introduced in Sec. 10.4, with L out standing for L(u,θ,out
˜ , L(θ,ψ) or L(θ,ψ)
φ̃)
out,H out,Hin
.
Proof. Consider expression (11.20) for K . If L is a null geodesic, the term g(p, p) vanishes
identically, so that one is left with
K = 2(r2 + a2 )(k · p)(ℓ · p).
380 Geodesics in Kerr spacetime: generic and timelike cases
Given that r2 + a2 ̸= 0 (this is clear for a ̸= 0, while if a = 0 (Schwarzschild case), r = 0 is
excluded from the spacetime manifold), we have then
K = 0 ⇐⇒ k · p = 0 or ℓ · p = 0.
The vectors k, ℓ and p are all null. Now, according to Property 1.4 in Sec. 1.2.2, two null vectors
are orthogonal iff they are collinear. Hence K = 0 is equivalent to p = αk or p = αℓ with
α > 0. Since k (resp. ℓ) is tangent to L(v,θ,
in
φ̃) (resp. L
out
), this completes the proof.
Remark 2: Property 11.2 is consistent with the interpretation (11.23) of K , since asymptotically the
principal null geodesics are purely radial5 , and hence have Ltot = 0.
Example 4 (Generators of the event and inner horizons): We have seen in Example 1 in Sec. 11.2.2
that the null geodesic generators of the event horizon H and the inner horizon Hin have E = 0 and
L = 0. Since these generators are principal null geodesics (cf. Secs. 10.5.3 and 10.8.3), the above results
shows that in addition K = 0. As µ = 0 by definition of a null geodesic, we see that the four integrals
of motion µ, E, L and K all vanish for these geodesics.
The converse is true:
Property 11.3: characterization of the null generators of the two horizons
The null geodesic generators of the event horizon H and of the inner horizon Hin are the
only geodesics of Kerr spacetime having their four integrals of motion vanishing:
L null geodesic generator of H or Hin ⇐⇒ (µ, E, L, K ) = (0, 0, 0, 0). (11.25)
Proof. If a geodesic L has (µ, K ) = (0, 0), L is necessary null and (11.24) shows that L is a
principal null geodesic. Moreover Eq. (11.21) with (µ, E, L, K ) = (0, 0, 0, 0) implies pr = 0,
i.e. L lies at a constant value of r. If L is ingoing, then p ∝ k, with the Boyer-Lindquist
components of k given by Eq. (10.49). Since k r = −1, this precludes pr = 0. Hence L is an
outgoing principal null geodesic and one has p ∝ ℓ, with the Boyer-Lindquist components of ℓ
given by Eq. (10.61). We read ℓr = ∆/(2(r2 + a2 )) with ∆ ̸= 0, except precisely on H and
Hin 6 . Hences pr = 0 is possible only on H and Hin .
Historical note : That K = 0 for principal null geodesics has been pointed out by Jiří Bičák and
Zdeněk Stuchlík in 1976 [60].
11.2.5 First order equations of motion
We have thus four first integrals of the geodesic equation (11.1) at disposal: E [Eq. (11.10)], L
[Eq. (11.11)], µ2 [Eq. (11.14)] and K [Eq. (11.21)]. In the expressions of each of these integrals,
5
Indeed, we deduce from Eqs. (10.49) and (10.61) that, for r → +∞, k ∼ ∂t − ∂r and ℓ ∼ (∂t + ∂r )/2.
6
This is graphically confirmed by Figs. 10.6 and 10.7, which show that H and Hin are the only locations
where a principal null geodesic can have r = const.
11.2 Equations of geodesic motion 381
pα has to be thought of as the first order derivative dxα /dλ [Eq. (11.12)]. Two first integrals,
namely E and L, are linear in the pα ’s, while the two others, namely µ2 and K , are quadratic.
Furthermore, Eqs. (11.10) and (11.11) constitute a decoupled subsystem for (pt , pφ ), which can
easily be solved7 , yielding
1 2
ρ 2 pt = [(r + a2 )2 E − 2amrL] − a2 E sin2 θ (11.26)
∆
L a
ρ 2 pφ =
2 + (2mrE − aL). (11.27)
sin θ ∆
Besides, Eq. (11.21) involves only p and can be recast as
r
ρ4 (pr )2 = R(r), (11.28)
with
2
R(r) := (r2 + a2 )E − aL − ∆(r2 µ2 + K ) . (11.29)
Recall that ∆ is the function of r given by Eq. (10.10): ∆ := r2 − 2mr + a2 . All other
quantities appearing in Eq. (11.29) are constant. Accordingly, R(r) is a 4th order polynomial in
r. Equation (11.28) implies that this polynomial must be non-negative along the geodesic L :
R(r) ≥ 0 . (11.30)
Finally, if we substitute pr by the value given by Eqs. (11.28)-(11.29) in the mass first integral
(11.14), we get, after simplification,
ρ4 (pθ )2 = Θ(θ), (11.31)
with
2
L
Θ(θ) := K − − aE sin θ − µ2 a2 cos2 θ . (11.32)
sin θ
Equation. (11.31) imposes that Θ(θ) is non-negative along the geodesic L :
Θ(θ) ≥ 0 . (11.33)
The following constant is often used instead of K :
Q := K − (L − aE)2 (11.34)
Thanks to it, we may rewrite (11.32) as
L2
2
Θ(θ) = Q + cos θ a (E − µ ) −2 2 2
. (11.35)
sin2 θ
7
An intermediate step is combining Eqs. (11.10) and (11.11) to get aE − L/ sin2 θ = apt − (r2 + a2 )pφ and
(r + a2 )E − aL = ∆(pt − a sin2 θpφ ).
2
382 Geodesics in Kerr spacetime: generic and timelike cases
Following the standard usage, we call Q the Carter constant as well. To distinguish between
the two Carter constants, we shall specify Carter constant Q or Carter constant K . As we shall
see in Sec. 11.3.7, Q is well adapted to the description of the θ-motion of geodesics. On the
other hand, a nice property of K , which is not shared by Q, is to be always non-negative, as
Eqs. (11.32) and (11.33) show:
K ≥0. (11.36)
If the particle P reaches the asymptotic region r ≫ m, we deduce from Eqs. (11.34) and
(11.23) the following behavior of Q:
Q ∼ Ltot · Ltot − L2 + aE(2L − aE). (11.37)
r→+∞
Hence, if a = 0, Q can be interpreted as the square of the part of P’s angular momentum
(measured by the asymptotic inertial observer) that is not in L.
Example 5 (Carter constant Q of the principal null geodesics): As (11.24) shows, for a principal
null geodesic, be it ingoing or outgoing, the Carter constant K vanishes identically. According to
Eq. (11.34), the Carter constant Q is then Q = −(L − aE)2 . In view of the relation L = aE sin2 θ for
these geodesics [Eqs. (11.6) and (11.9)], we get
Q = −a2 E 2 cos4 θ, (11.38)
where θ is the constant value of the θ-coordinate along the principal null geodesic. Note that the above
relation holds in all Kerr spacetime, including on the horizons H and Hin , where it reduces to Q = 0,
for E = 0 there [Eq. (11.9a)]. Equation (11.38) implies
Q ≤ 0, (11.39)
with Q = 0 only for principal null geodesics lying in the equatorial plane or for the outgoing principal
out,H out,Hin
null geodesics L(θ,ψ) and L(θ,ψ) generating the horizons H and Hin .
Remark 3: As for K , one may derive the Carter constant Q from a Killing tensor. Indeed, from the
Walker-Penrose Killing tensor K [Eq. (11.16)], let us form the tensor field
K̃ := K − η̃ ⊗ η̃, where η̃ := η + aξ. (11.40)
Being a linear combination with constant coefficients of the Killing vectors η and ξ, η̃ is itself a Killing
vector. It follows that η̃ ⊗ η̃ is a Killing tensor (cf. Example 3 in Sec. B.5.2), so that K̃ a Killing tensor,
the Killing equation (11.18) being linear. Applying K̃ to the 4-momentum p, we get
K̃(p, p) = K(p, p) −(⟨η, p⟩ +a ⟨ξ, p⟩)2 .
| {z } | {z } | {z }
K L −E
Comparing with the definition (11.34), we conclude that
Q = K̃(p, p). (11.41)
The Boyer-Lindquist components of the contravariant tensor associated to K̃ by metric duality have an
expression particularly simple in terms of those of the inverse metric (cf. the notebook D.5.7):
K̃ αβ = −a2 cos2 θ g αβ + diag(−a2 cos2 θ, 0, 1, tan−2 θ)αβ . (11.42)
11.2 Equations of geodesic motion 383
In view of the relation (11.12) between the pα ’s and the derivatives of the functions xα (λ),
we may collect Eqs. (11.26), (11.27), (11.28) and (11.32) as the first-order system
dt 1
ρ2 = [(r2 + a2 )2 E − 2amrL] − a2 E sin2 θ (11.43a)
dλ ∆
dr
(11.43b)
p
ρ2 = ϵr R(r)
dλ
dθ
(11.43c)
p
ρ2 = ϵθ Θ(θ)
dλ
dφ L a
ρ2 = 2 + (2mrE − aL) , (11.43d)
dλ sin θ ∆
where ϵr := sgn pr = ±1, ϵθ := sgn pθ = ±1 and the functions R(r) and Θ(θ) are defined by
Eq. (11.29) and Eq. (11.32) or (11.35). Since pr = dr/dλ, ϵr is +1 (resp. −1) in the parts of the
geodesic L where r increases (resp. decreases) with λ. Similarly, ϵθ is +1 (resp. −1) in the
parts of L where θ increases (resp. decreases) with λ.
We may rewrite Eq. (11.29) for R(r) in terms of Q instead of K , via Eq. (11.34):
2
R(r) := (r2 + a2 )E − aL − ∆ r2 µ2 + Q + (L − aE)2 ) . (11.44)
Note that, beside the constants of motion E, L, µ and Q, R(r) depends on both Kerr parameters
a and m (via ∆ = r2 − 2mr + a2 ), while Θ(θ) depends on a only [cf. Eq. (11.35)]. Along L ,
these functions must obey R(r) ≥ 0 [Eq. (11.30)] and Θ(θ) ≥ 0 [Eq. (11.33)].
11.2.6 Turning points
Let L be a geodesic that is not stuck at some constant value of the coordinate r. We define a
r-turning point of L as a point p0 ∈ L , the r-coordinate r0 of which obeys
R(r0 ) = 0 and R′ (r0 ) ̸= 0, (11.45)
i.e. r0 is a simple root of the polynomial R.
We have then
dr d2 r R′ (r0 )
= 0 and = ̸= 0, (11.46)
dλ λ0 dλ2 λ0 2ρ40
where λ0 is the value of the affine parameter λ at p0 and ρ0 := ρ(p0 ).
Proof. The vanishing of dr/dλ at λ0 follows immediately from Eq. (11.43b) with R(r0 ) = 0,
since ρ2 never vanishes on M . Besides, by taking the derivative of Eq. (11.43b) with respect to
λ, we get
d2 r R′ (r) dr R′ (r)
dr 2 dθ dr
2 r + a cos θ sin θ + ρ2 2 = ϵr p = .
dλ dλ dλ dλ 2 R(r) dλ 2ρ2
At λ = λ0 , the first term in left-hand side vanishes identically, due to the dr/dλ factor, and we
get the second part of (11.46).
384 Geodesics in Kerr spacetime: generic and timelike cases
We deduce from the result (11.46) that at λ = λ0 , dr/dλ moves from positive to negative
values or vice-versa (depending on the sign of R′ (r0 )), which means that the function r(λ)
switches from increasing to decreasing or vice-versa, hence the name r-turning point. The
factor ϵr = ±1 in Eq. (11.43b) necessarily changes sign at λ = λ0 .
Remark 4: For a generic smooth function r(λ) with dr/dλ = 0 at λ0 , the condition d2 r/dλ2 ̸= 0 at λ0
is sufficient but not necessary for r to change its direction of variation there. Indeed, the same property
holds with d2 r/dλ2 = 0 and higher order derivatives vanishing up to some even order k for which
dk r/dλk ̸= 0. However, in the present case, d2 r/dλ2 = 0 would imply R′ (r0 ) = 0 and we shall see
Sec. 11.3.6 that a geodesic can reach such a point only asymptotically, i.e. for λ → +∞. Hence it cannot
be a turning point.
Similarly, if L is a geodesic that is not stuck at some constant value of the coordinate θ,
we define a θ-turning point of L as a point p0 ∈ L , the θ-coordinate θ0 of which obeys
Θ(θ0 ) = 0 and Θ′ (θ0 ) ̸= 0. (11.47)
We deduce then from the equation of motion (11.43c):
dθ d2 θ Θ′ (θ0 )
= 0 and = ̸= 0. (11.48)
dλ λ0 dλ2 λ0 2ρ40
This implies that at a θ-turning point, the function θ(λ) switches from increasing to decreasing
or vice-versa. The factor ϵθ = ±1 in Eq. (11.43c) necessarily changes sign at such a point.
Remark 5: A comment similar to Remark 4 can be made: a geodesic with varying θ that has dθ/dλ = 0
for some finite value of λ cannot have d2 θ/dλ2 = 0 at the same point, since we shall see in Sec. 11.3.6
that a value of θ with both Θ(θ) = 0 and Θ′ (θ) = 0 can only be reached asymptotically along a geodesic.
Hence (11.47) is a necessary and sufficient condition for a θ-turning point.
11.2.7 Equations of motion in terms of Mino parameter
In view of the right-hand sides of the system (11.43), it is quite natural to introduce a new
parameter λ′ along the geodesic L such that
dλ dλ
dλ′ = 2
= . (11.49)
ρ r(λ) + a2 cos2 θ(λ)
2
Since ρ2 never vanishes on the spacetime manifold M (by construction of the latter, cf.
Eq. (10.29)), the above relation leads to a well-defined parameter8 along L . Moreover, since
ρ2 > 0, λ′ increases towards the future, as λ. A difference between the two parametrizations is
that λ′ is not an affine parameter of L in general9 , contrary to λ. The parameter λ′ is called
Mino parameter [368].
8
Note however that λ′ may blow up if L comes arbitrarily close to the ring singularity, i.e. if ρ → 0.
9
The only exception is for a circurlar orbit at θ = π/2, since then ρ2 is constant and Eq. (11.49) reduces to an
affine relation between λ and λ′ .
11.2 Equations of geodesic motion 385
In terms of Mino parameter, the system (11.43) becomes
dt 1 2
′
= [(r + a2 )2 E − 2amrL] − a2 E sin2 θ (11.50a)
dλ ∆
dr
(11.50b)
p
= ϵr R(r)
dλ′
dθ
(11.50c)
p
= ϵθ Θ(θ)
dλ′
dφ L a
′
= 2 + (2mrE − aL) , (11.50d)
dλ sin θ ∆
where R is the quartic polynomial defined by Eq. (11.44) and Θ is the function defined by
Eq. (11.35). It is remarkable that Eqs. (11.50b) and (11.50c) are two fully decoupled equations:
Eq. (11.50b) is an ordinary differential equation (ODE) for the function r(λ′ ), while Eq. (11.50c)
is an ordinary differential equation for the function θ(λ′ ). This was not the case for Eqs. (11.43b)
and (11.43c) since ρ2 involves both r and θ.
11.2.8 Integration of the geodesic equations
The ODE (11.50b) can be integrated by the method of separation of variables. On a part of L
where R(r) ̸= 0, this yields Z r
ϵ dr̄
′ ′
λ − λ0 = pr , (11.51)
r0 R(r̄)
with r0 := r(λ′0 ). The hypothesis R(r) ̸= 0 excludes any r-turning point between λ′0 and λ′ ,
so that ϵr = ±1 is constant along the considered part of L .
Actually, the solution (11.51) can be extended to include a turning point at one or two of
its boundaries, despite R(r) = 0 there. Indeed, let us assume that r = r1 corresponds to a
r-turning point of L . Due to R′ (r1 ) ̸= 0 [Eq. (11.45)], the integral in the right-hand side of
(11.51) with r = r1 is finite. Indeed, the Taylor expansion R(r̄) = R′ (r1 )(r̄ − r1 ) + O((r̄ − r1 )2 )
makes the integral behave near r1 as10
Z r1
1 dr̄
p √ ,
−R′ (r1 ) r0 r1 − r̄
which is a convergent improper integral.
Remark 6: This is the same reasoning as in Sec. 8.3.6.
Let us assume that there are M ≥ 1 r-turning points p1 , . . . , pM between λ′0 and λ′ . Their
radial coordinates take at most two distinct values, r1 and r2 , such that r(p1 ) = r1 , r(p2 ) = r2 ,
r(p3 ) = r1 , r(p4 ) = r2 , etc. From the above convergence property, the solution of Eq. (11.50b)
is then Z r1 Z r2 Z r
ϵr dr̄ ϵr dr̄ ϵ dr̄
′ ′
λ − λ0 = p + (M − 1) p + pr , (11.52)
r0 R(r̄) r1 R(r̄) r1,2 R(r̄)
10
We assume here r0 < r1 , so that the constraint R(r̄) ≥ 0 on the interval [r0 , r1 ] implies R′ (r1 ) < 0.
386 Geodesics in Kerr spacetime: generic and timelike cases
where r1,2 = r1 for M odd and r1,2 = r2 for M even. Note that each term in the above sum is
positive, the sign of ϵr compensating the order of the integral boundaries.
The right-hand side of Eq. (11.52) is actually a path integral and is often abriged by means
of a slash notation: Z r
ϵr dr̄
′ ′
λ − λ0 = − p . (11.53)
r0 R(r̄)
Similarly, if Θ(θ) ̸= 0 along L , except possibly at some θ-turning points, Eq. (11.50c) can
be integrated as
Z θ
′ ϵθ dθ̄
λ − λ′0 = − p
θ Θ(θ̄)
0Z θ
ϵ dθ̄
pθ if N = 0
θ0 Θ(θ̄)
= Z θ1 Z θ2 Z θ
ϵθ dθ̄ ϵθ dθ̄ ϵ dθ̄
+ (N − 1) pθ if N ≥ 1,
p p +
Θ(θ̄) Θ(θ̄) Θ(θ̄)
θ0 θ1 θ1,2
(11.54)
where θ0 := θ(λ′0 ), N is the number of θ-turning points between λ′0 and λ′ , θ1 (resp. θ2 ) is the
value of θ at the first (resp. second) turning point, if any, θ1,2 = θ1 for N odd and θ1,2 = θ2 for
N even.
We are now in position to state the full expression of the general solution for geodesic
motion:
Property 11.4: solution for geodesic motion
Let L be a null or timelike geodesic in Kerr spacetime, with conserved energy E, conserved
angular momentum L, mass µ and Carter constant Q. We assume that the associated quartic
polynomial R(r), as defined by Eq. (11.44) (see also Eq. (11.95) below), and the associated
function Θ(θ), as defined by Eq. (11.35), do not vanish along L except possibly at some
r-turning points or θ-turning points. Let λ be the affine parameter of L associated with
the 4-momentum p and λ′ the Mino parameter. If for λ = λ0 L lies at the point of
Boyer-Lindquist coordinates (t0 , r0 , θ0 , φ0 ), then at any value of λ, the Boyer-Lindquist
11.2 Equations of geodesic motion 387
coordinates (t, r, θ, φ) along L obey
Z r Z θ
ϵr dr̄ ϵθ dθ̄
′
λ′0
λ − =− p =− p (11.55a)
r0 R(r̄) θ0 Θ(θ̄)
Z r 2 Z θ
ϵr r̄ dr̄ ϵθ cos2 θ̄ dθ̄
λ − λ0 = − p 2
+a − p (11.55b)
r0 R(r̄) θ0 Θ(θ̄)
Z r 2 2 2 Z θ
(r̄ + a ) E − 2amr̄L ϵr dr̄ ϵθ dθ̄
t − t0 = − 2 2
p − a E− sin2 θ̄ p
2
(11.55c)
r0 r̄ − 2mr̄ + a R(r̄) θ0 Θ(θ̄)
Z r Z θ
2mr̄E − aL ϵr dr̄ 1 ϵ dθ̄
φ − φ0 = a− 2 2
p + L− 2
pθ . (11.55d)
r0 r̄ − 2mr̄ + a R(r̄) θ0 sin θ̄ Θ(θ̄)
Proof. Equation (11.55a) is nothing but the gathering of Eqs. (11.53) and (11.54). For Eq. (11.55b),
it suffices to rewrite Eq. (11.49) as
dλ = r2 dλ′ + a2 cos2 θ dλ′
and to substitute dλ′ by ϵr dr/ R(r) in the first term [cf. Eq. (11.50b)] and by ϵθ dθ/ Θ(θ) in
p p
the second term [cf. Eq. (11.50c)]. Similarly, by rewriting Eq. (11.50a) as
(r2 + a2 )2 E − 2amrL ′
dt = dλ − a2 E sin2 θ dλ′
r2 − 2mr + a2
and performing the same substitutions for dλ′ as above, we get Eq. (11.55c). Finally, Eq. (11.55d)
is deduced in the same fashion from Eq. (11.50d).
The system (11.55) shows that the geodesic motion can be fully solved in terms of the Mino
parameter λ′ . Indeed, the integral on r in Eq. (11.55a) can be evaluated by means of elliptic
integrals since R is a polynomial of degree 4. This provides λ′ = λ′ (r). Inverting this relation
via Jacobi elliptic functions (cf. Remark 5 on p. 252) yields r = r(λ′ ). The integral on θ in
Eq. (11.55a) can be evaluated by means of elliptic integrals as well since the change of variable
ζ := cos θ along with expression (11.35) for Θ(θ) results in
Z θ Z ζ
ϵθ dθ̄ ϵθ dζ̄
− p = −− p ,
θ0 Θ(θ̄) ζ0 Q(1 − ζ̄ 2 ) + ζ̄ 2 [a2 (E 2 − µ2 )(1 − ζ̄ 2 ) − L2 ]
with the term under the square root in the right-hand side being a polynomial of degree 4 in
ζ̄. The use of Jacobi elliptic functions leads then to θ = θ(λ′ ). From r(λ′ ) and θ(λ′ ) one can
get the functions λ(λ′ ), t(λ′ ) and φ(λ′ ) by evaluating the integrals in the right-hand sides of
Eqs. (11.55b)–(11.55d). Again, these integrals are reducible to elliptic integrals. We shall not
give the details of all the elliptic integrals computations, referring the reader to Ref. [206] for
bound timelike geodesics and to Ref. [236] for null geodesics.
Historical note : The constant of motion K has been discovered by Brandon Carter in 1968 [90], in a
study about Kerr-Newman spacetimes, which generalize the Kerr ones to nonzero global electric charge.
Carter actually did not get K from the Killing tensor K, which was discovered only two years later
388 Geodesics in Kerr spacetime: generic and timelike cases
by Martin Walker and Roger Penrose [508]; he started instead from the Lagrangian (B.39) governing
both timelike and null geodesics, derived the corresponding Hamiltonian as H = g µν pµ pν (for the
uncharged case, i.e. Kerr) and discovered that the Hamilton-Jacobi equation is separable, i.e. can be
solved by separation of variables. K appeared then as a separation constant. Note that Carter used
the Kerr coordinates11 (v, r, θ, φ̃) described in Sec. 10.3.1 and not the Boyer-Lindquist ones. Carter also
introduced the constant Q via Eq. (11.34). He obtained the equivalent of the first-order system (11.43)
for the Kerr coordinates. Actually two of his equations, those for dr/dλ and dθ/dλ, are identical to
Eqs. (11.43b) and (11.43c) of the Boyer-Lindquist system. This is not surprising since the coordinates r
and θ are the same in both systems. Carter’s equations for the Kerr coordinates v and φ̃ are slightly
more complicated than Eqs. (11.43a) and (11.43d) for the Boyer-Lindquist coordinates t and φ. It seems
that the Boyer-Lindquist first-order system (11.43) has been first derived by Daniel Wilkins in 1972 [522],
starting from Carter’s system and performing the transformation to Boyer-Lindquist coordinates. The
integrated geodesic equations (11.55) have been obtained by Carter in 1968 [90]: Eq. (11.55a) (without
the λ′ − λ′0 part) is Eq. (58) of Ref. [90] and Eq. (11.55b) is Eq. (59) of Ref. [90]. Regarding Eqs. (11.55c)
and (11.55d), Carter obtained equivalent ones for the Kerr coordinates v and φ̃, which he was using [his
Eqs. (60) and (61)].
11.3 Main properties of geodesics
11.3.1 Sign of E
We have:
Property 11.5: positivity of the conserved energy outside the ergoregion
If a null or timelike geodesic L has some part lying in the exterior of the ergoregion G (cf.
Sec. 10.2.4), then the conserved energy E defined by Eq. (11.2a) is necessarily positive:
L ̸⊂ G =⇒ E > 0. (11.56)
Proof. By the very definition of the ergoregion G (cf. Sec. 10.2.4), the Killing vector ξ is
timelike in the exterior of G , i.e. in M \ G . Moreover, it is future-directed there, given the
time orientation defined in Sec. 10.3.2. The 4-momentum p is either timelike or null and
always future-directed. By Eq. (1.5a) in Lemma 1.1, one has then necessarily ξ · p < 0; hence
E := −ξ · p > 0 in M \ G . Since E is constant along L , it follows that E > 0 everywhere.
In particular, any timelike or null geodesic that reaches one of the asymptotic regions r → ±∞
has E > 0.
Remark 1: Property 11.5 generalizes Property 7.2 obtained for Schwarzschild spacetime to the case
a ̸= 0. Indeed, for Schwarzschild spacetime, the exterior of the ergoregion is nothing but the exterior of
the black hole region.
Inside the ergoregion, the Killing vector ξ is spacelike and E can be either positive, zero or
negative. A particle with E < 0 (resp. E = 0) is called a negative-energy particle (resp. a
11
Carter’s u is our v.
11.3 Main properties of geodesics 389
Figure 11.1: Projection in the equatorial plane t = const and θ = π/2 of the worldline of a particle and its
4-momentum p, which decay at event A in two particles: one with 4-momentum p′ , which leaves to infinity, and
one with 4-momentum p∗ , which falls into the black hole (black region). The grey zone is the outer ergoregion.
zero-energy particle). Note that according to Property 11.5, neither a negative-energy particle
nor a zero-energy one can exist outside the ergoregion.
Example 6 (Generators of the event and inner horizons): We have already encountered zero-energy
particles in Example 1 of Sec. 11.2.2: the photons whose worldlines are the null generators of the horizons
H and Hin [cf. Eq. (11.4)].
Example 7 (Outgoing principal null geodesics in MII ): We have seen in Example 3 (p. 376) that
in the region MII , the outgoing principal null geodesics L(u,θ,
out
˜ have E < 0 [Eq. (11.9c)]. This is
φ̃)
consistent with Property 11.5 because MII is entirely contained in the ergoregion G (cf. Fig. 10.2).
Remark 2: It should be stressed that, for a particle lying in the ergoregion, the quantity E is not the
particle’s energy measured by some physical observer, as given by formula (1.23). Indeed, since ξ is a
spacelike vector in the ergoregion, it cannot be identified with any observer 4-velocity. In other words,
any local observer measuring the energy of a negative-energy particle or a zero-energy one, according
to the process described in Sec. 1.4, will find a positive value. Hence there is nothing mysterious about
such particles.
11.3.2 The Penrose process
The possibility of having negative-energy particles in the outer ergoregion opens the path to
a mechanism of energy extraction from a rotating black hole. To see this, consider a particle
P in free fall from infinity into the outer ergoregion G + . At some point A ∈ G + , suppose
that P splits (or decays) into two particles: P ′ and P∗ , such that P ′ subsequently exits the
ergoregion and leaves to infinity, while P∗ falls into the black hole (cf. Fig. 11.1). The particles
P, P ′ , and P∗ can be either massive or massless and we assume that, once created, they are
subject only to gravitation, so that their worldlines are timelike or null geodesics. Let p, p′ and
390 Geodesics in Kerr spacetime: generic and timelike cases
p∗ be the 4-momentum vectors of respectively P, P ′ and P∗ . We define the energy gain in
the above process by
∆E := Eout − Ein , (11.57)
where Ein (resp. Eout ) is the energy of P (resp. P ′ ) as measured by an asymptotic inertial
observer at rest with respect to the black hole (cf. Sec. 10.7.5). Ein and Eout are actually nothing
but the conserved energies of particles P and P ′ as defined in Sec. 11.2.2. They are thus given
by Eq. (11.2a):
Ein = − ξ · p|∞ and Eout = − ξ · p′ |∞ , (11.58)
where the subscript ∞ refers to the location of the asymptotic inertial observer. Since Ein
and Eout are constant along the geodesic worldlines of P and P ′ (Property 11.1), we have
Ein = − ξ · p|A and Eout = − ξ · p′ |A . Accordingly, Eqs. (11.57) and (11.58) lead to
∆E = ξ · p|A − ξ · p′ |A = ξ|A · (p|A − p′ |A ).
Now, the conservation of energy-momentum at event A implies p|A = p′ |A + p∗ |A , hence
∆E = ξ · p∗ |A = −E∗ , (11.59)
where E∗ is the conserved energy of particle P∗ . If P∗ happens to be a negative-energy
particle, which is allowed since P∗ is confined to the ergoregion, then E∗ < 0 and Eq. (11.59)
implies ∆E > 0, i.e. the outgoing particle has more energy than the ingoing one. The above
process is then called a Penrose process. It is particulary relevant to astrophysics, via some
electromagnetic variants of it, notably the so-called Blandford-Znajek mechanism (see e.g.
Ref. [339] for an extended discussion and Box. 15.1 of Ref. [257] for an elementary presentation
of the Blandford-Znajek mechanism).
Remark 3: No peculiar property of the Kerr spacetime, except for the existence of an outer ergoregion,
has been invoked in the above reasoning. Consequently, the Penrose process can occur for any stationary
rotating body (not even necessarily a black hole) that is endowed with an ergoregion in which a round
trip from infinity is possible.
Since the particle P∗ is assumed to fall into the black hole, it must cross the event horizon
H at some point, B say. At B, the Killing vector χ := ξ + ΩH η of Kerr spacetime [Eq. (10.68)]
is null (being a null normal to H , cf. Eq. (10.72)) and future-directed (Property 10.15). On
the other side, the 4-momentum p∗ is necessarily a future-directed timelike or null vector.
Moreover, p∗ is transverse to H at B (for P∗ is crossing H ) and therefore cannot be collinear
to χ, which is tangent to H . We may then invoke Lemma 1.2 (case (1.6a)) to assert that
χ · p∗ |B < 0, or equivalently [thanks to Eq. (10.68)],
ξ · p∗ | +ΩH η · p∗ |B < 0,
| {z B} | {z }
−E∗ L∗
where L∗ is the conserved angular momentum of P∗ [Eq. (11.2b)]. We thus get
E∗ > ΩH L∗ . (11.60)
11.3 Main properties of geodesics 391
Given that ΩH > 0 [cf. Eqs. (10.69) and (10.1)], we deduce that if a Penrose process occurs,
i.e. if E∗ < 0, then necessarily L∗ < 0. Similarly to the energy gain (11.57), we may define the
angular momentum gain of the process P → P ′ + P∗ by
∆L := Lout − Lin , (11.61)
where Lin (resp. Lout ) is the angular momentum of P (resp. P ′ ) as measured by an asymptotic
inertial observer, i.e. the conserved angular momenta of particles P and P ′ as defined in
Sec. 11.2.2: Lin = η · p|∞ and Lout = η · p′ |∞ [Eq. (11.2b)]. The same computation as above
for ∆E, except for ξ replaced by η, shows that the energy-momentum conservation law
p|A = p′ |A + p∗ |A leads to a result similar to (11.59):
∆L = − η · p∗ |A = −L∗ . (11.62)
Along with Eq. (11.59), this relation turns the inequality (11.60) into ΩH ∆L > ∆E. This shows
that any Penrose process yields ∆L > 0 in addition to ∆E > 0. Let us summarize:
Property 11.6: Penrose process
The Penrose process as described above and illustrated in Fig. 11.1 extracts both some
energy ∆E > 0 and some angular momentum ∆L > 0 from a rotating black hole, in such
a way that
ΩH ∆L > ∆E > 0, (11.63)
where ΩH is the black hole rotation velocity [cf. Eq. (10.69)]. The secondary particle P∗
falling into the black hole during the process has both a negative conserved energy E∗ and
a negative conserved angular momentum L∗ , which obey Eq. (11.60): ΩH L∗ < E∗ < 0.
Remark 4: Setting ΩH = 0 in Eq. (11.63) shows that the Penrose process is not possible for a non-
rotating (Schwarzschild) black hole. This is not surprising since there is no outer ergoregion for such a
black hole, the ergosphere coinciding with the event horizon (cf. Remark 5 on p. 331).
Ultimately, the energy and angular momentum extracted via the Penrose process are taken
from the mass m and the angular momentum J = am of the black hole, so that these two
quantities must decrease slightly (i.e. in a infinitely small manner, given that P, P ′ and P∗
are assumed to be “test particles” (geodesic worldlines), which implies ∆E ≪ m and ∆L ≪ J).
This will be discussed in details in Sec. 16.3.1.
The Penrose process can be extended from particles to fields; it is then known as the
phenomenon of superradiance, also called superradiant scattering. This occurs when the
scattering of a (scalar, vector or tensor) field by a rotating black hole results in a outgoing
field energy larger than the ingoing one. The energy of the field is defined from its energy-
momentum tensor T by constructing the “energy density” vector P as P α := −T αRµ ξ µ (compare
with Eq. (11.2a)). The energy of the field through a hypersuface Σ is then EΣ := Σ ⋆P , where
⋆P is the the Hodge dual of the 1-form P , i.e. the 3-form defined by (⋆P )αβγ := P µ εµαβγ [cf.
Eq. (5.38) with n = 4 and p = 1]. Now, by combining the energy-momentum conservation
law ∇µ T µα = 0 [Eq. (1.45)] and the Killing equation for ξ, it is easy to show that P is a
392 Geodesics in Kerr spacetime: generic and timelike cases
conserved current: ∇µ P µ = 0. This is equivalent to the 3-form ⋆P being closed: d⋆P = 0,
since d ⋆ P = (∇µ P µ ) ε. The Stokes theorem (A.94) leads then to EΣ = 0 as soon as Σ
is a closed oriented hypersurface. Let us choose Σ = Σ1 ∪ ∆H ∪ Σ2 ∪ Σext , where (i) Σ1
and Σ2 are two spacelike hypersufaces intersecting the event horizon H and such that Σ2
lies in the future of Σ1 (12 ), (ii) ∆H is the portion of H between H ∩ Σ1 and H ∩ Σ2
and (iii) Σext is a timelike hypersurface connecting Σ1 and Σ2 far from H . We have then
EΣ = EΣ1 + E∆H + EΣ2 + EΣext = 0. Assuming that T decays sufficiently fast so that
EΣext = 0, we arrive at
E2 = E1 − E∗ , (11.64)
where E2 := EΣ2 , E1 := −EΣ1 (the change of sign occurs because the orientation of Σ1
inherited from that of Σ isRthe opposite of that induced by the future-directed unit normal n
to Σ1 ) and E∗ := E∆H = ∆H ⋆P . Viewing the spacelike hypersurfaces Σ1 and Σ2 as “time
slices”, we may say that E1 (resp. E2 ) is the field energy at “time Σ1 ” (resp. “time Σ2 ”) 13 , while
E∗ is the field energy captured by the black hole. Superradiance occurs iff E2 > E1 , i.e. iff
E∗ < 0, as in the Penrose process. For a scalar field of the form Φ = Φ0 (r, θ)ei(ωt−mφ) obeying
the Klein-Gordon equation, it can be shown (by a direct computation of E∗ = ∆H ⋆P ) that
R
E∗ < 0 is equivalent to [99, 499, 339]
0 < ω < mΩH . (11.65)
Actually the same superradiance condition holds for the Fourier modes of electromagnetic
waves [477, 99], as well as gravitational waves [477].
Historical note : The Penrose process has first been suggested by Roger Penrose in 1969, in the review
article [408] (cf. the footnote 7 in this article); the detailed calculation has been presented subsequently
in an article with Roger Floyd published in 1971 [409]. In particular, Penrose and Floyd have derived the
inequality (11.60) [Eq. (4) in Ref. [409], taking into account that a/(2mr+ ) = ΩH , cf. Eq. (10.69)]. The
extension to fields (superradiance) was introduced by Yakov Zeldovich in 1971 [527, 528] and explicit
computations for a scalar field in the Kerr metric have been performed by his doctoral student Alexei
Starobinsky in 1972 (published: 1973) [460], as well as by William Press and Saul Teukolsky at the same
time [420]. It seems that the superradiance of a scalar field scattered by a Kerr black hole has also been
discovered around 1971 by Charles Misner: mention of unpublished results by him can be found in his
article [369], as well as in a 1973 article by Jacob Bekenstein [51].
11.3.3 Future-directed condition
Not all values of the constant of motions (µ, E, L, Q) lead to a solution xα (λ) of the first-order
system (11.43) that is future-directed, i.e. such that p = dx/dλ is oriented everywhere towards
the future. The latter condition is equivalent to have the affine parameter λ increase towards
the future along the geodesic L , as we are demanding in all our study (cf. Sec. 11.2.1). In
particular, this allows one to identify λ with the proper time when L is timelike.
12
For instance, Σ1 is a hypersurface of constant Kerr coordinate t̃ = t̃1 and Σ2 is a hypersurface t̃ = t̃2 , with
t̃2 > t̃1 .
13
OneR can easily express E1 and E2 in terms of the normal volume element vector dV introduced by Eq. (5.48):
E1,2 = Σ1,2 Pµ dV µ .
11.3 Main properties of geodesics 393
As discussed in Sec. 10.3.2, the time orientation of the Kerr spacetime (M , g) is set by the
global null vector field k generating the ingoing principal null geodesics, cf. Eq. (10.32). By
virtue of Lemma 1.2 [cf. Eq. (1.6a)], we have then, for p not collinear with k (in particular for
L timelike),
r 2 + a2 a
p future-directed ⇐⇒ g(k, p) < 0 ⇐⇒ k µ pµ < 0 ⇐⇒ pt − pr + pφ < 0,
∆ ∆
where the last inequality follows from the Boyer-Lindquist components (10.49) of k. Given the
identities pt = −E, pφ = L [Eq. (11.3)] and pr = ϵr R(r)/∆ [Eqs. (11.15) and (11.43b)], we
p
conclude:
Property 11.7
L timelike or 1 h 2 i
(11.66)
p
=⇒ (r + a2 )E − aL + ϵr R(r) > 0 .
k not tangent to L ∆
Given expression (11.44) for R(r) and ∆ := r2 − 2mr + a2 [Eq. (10.10)], we note that this
constraint involves (µ, E, L, Q) and r, but not θ.
Another global future-directed null vector field on the Kerr spacetime (M , g) is the null
tangent ℓ to the outgoing principal null geodesics, introduced in Sec. 10.4 [Eq. (10.57)]. If we
apply Lemma 1.2 with ℓ instead of k, we get, using the Boyer-Lindquist components (10.61) of
ℓ,
Property 11.8
L timelike or
(11.67)
p
=⇒ (r2 + a2 )E − aL − ϵr R(r) > 0 .
ℓ not tangent to L
In MI ∪ M
pIII , ∆ > 0 and we can combine (11.66) and (11.67) to (r + a )E − aL >
2 2
|ϵr R(r)| = R(r), thereby getting a constraint that is independent from ϵr :
p
Property 11.9
L not principal null geod. =⇒ (r2 + a2 )E − aL > in MI ∪ MIII . (11.68)
p
R(r)
A related constraint in MI ∪ MIII is obtained from the vector ε(0) of the Carter frame
introduced in Sec. 10.7.4. As shown there, ε(0) is a linear combination of k and ℓ [cf. Eq. (10.112)]
and is a future-directed timelike vector in all MI ∪ MIII . We can then apply Lemma 1.1, using
expression (10.108a) of ε(0) in terms of ξ = ∂t and η = ∂φ :
p future-directed in MI ∪ MIII ⇐⇒ g(ε(0) , p) < 0 ⇐⇒ (r2 + a2 ) g(ξ, p) +a g(η, p) < 0.
| {z } | {z }
−E L
Hence the constraint:
394 Geodesics in Kerr spacetime: generic and timelike cases
Property 11.10
(r2 + a2 )E − aL > 0 in MI ∪ MIII . (11.69)
Notep
that if L is not a principal null geodesic, (11.69) is a mere consequence of (11.68), given
that R(r) ≥ 0.
An immediate corollary of Property 11.10 is:
Property 11.11
In the outer ergoregion G + = G ∩ MI , a geodesic that has E ≤ 0 must have L < 0.
Another constraint is obtained by considering the vector field N normal to the hyper-
surfaces of constant Boyer-Lindquist coordinate t, which has been introduced in Sec. 10.7.3:
→
−
N = −dt [Eq. (10.88)]. As shown in Sec. 10.7.3, N is future-directed timelike in the subpart
MZAMO := MI ∪ (MIII \ T ) of Kerr spacetime. Lemma 1.1 [cf. Eq. (1.5a)] then yields
p future-directed in MZAMO ⇐⇒ g(N , p) < 0 ⇐⇒ − ⟨dt, p⟩ < 0 ⇐⇒ pt > 0.
| {z }
pt
Hence the Boyer-Lindquist component pt of the 4-momentum p must be positive in MZAMO :
Property 11.12
pt > 0 in MI ∪ (MIII \ T ). (11.70)
Let us recall that T is the Carter time machine (cf. Sec. 10.2.5), which occupies a very lim-
ited portion of MIII . Since pt = dt/dλ and λ increases towards the future (cf. Sec. 11.2.1),
Property 11.12 implies
Property 11.13
In the region MI ∪(MIII \T ) of Kerr spacetime, the Boyer-Lindquist coordinate t increases
towards the future along any geodesic L .
Using Eq. (11.43a) with dt/dλ = pt and the fact that ∆ > 0 in MI ∪ MIII , we can rewrite
Property 11.12 as
Property 11.14
11.3 Main properties of geodesics 395
y/m
3
2
1
x/m
2 4 6 8 10 12 14
-1
-2
Figure 11.2: Trajectory in the equatorial plane of an incoming timelike geodesic with E = µ, L = 0 and Q = 0,
plunging into a Kerr black hole with a = 0.998 m. The figure is drawn in terms of the Cartesian Boyer-Lindquist
coordinates (x, y) defined by Eq. (11.73) with θ = π/2 (equatorial plane) and the grey disk marks the black hole
region. [Figure generated by the notebook D.5.8]
ρ2 (r2 + a2 )E − 2amr(L − aE sin2 θ) > 0 in MI ∪ (MIII \ T ). (11.71)
Remark 5: In the outer ergoregion G + = G ∩ MI (i.e. the part of the ergoregion outside of the black
hole), E ≤ 0 is allowed, but Property 11.12 shows that pt ≤ 0 is not allowed.
Historical note : That E ≤ 0 in the outer ergoregion implies L < 0 (Property 11.11) has been shown
for equatorial geodesics by George Contopoulos in 1984 [141].
11.3.4 Lense-Thirring effect
Let us consider a null or timelike geodesic L with a vanishing angular momentum: L = 0.
The equations of motion (11.43a) and (11.43d) reduce to
2
(r + a2 )2
2 dt 2 2 E 2 2
ρ (r + a2 ) + 2a2 mr sin2 θ
ρ =E − a sin θ =
dλ ∆ ∆
dφ 2amrE
ρ2 = ,
dλ ∆
so that we get
dφ 2amr
= . (11.72)
dt L ρ2 (r2 + a2 )+ 2a2 mr sin2 θ
L=0
Hence, for a ̸= 0, a geodesic with zero angular momentum about the symmetry axis has
nevertheless a rotational motion around that axis. Moreover, this motion is independent of the
geodesic’s characteristics, i.e. (µ, E, Q). This is a manifestation of the Lense-Thirring effect,
also called dragging of inertial frames, or, in short, frame dragging.
The Lense-Thirring effect is illustrated in Fig. 11.2 which shows the trajectory a timelike
particle with L = 0 which is asymptotically at rest (marginally bound particle, having E = µ, to
be discussed in Sec. 11.3.8). The trajectory is initialy radial, but as r decreases, dφ/dt increases
according to formula (11.72). In Fig. 11.2 and in all figures of this chapter, we are using the
396 Geodesics in Kerr spacetime: generic and timelike cases
Cartesian Boyer-Lindquist coordinates (t, x, y, z), which are defined in the r > 0 region of
Kerr spacetime and are related to the Boyer-Lindquist coordinates (t, r, θ, φ) by the standard
transformation from spherical to Cartesian coordinates:
x := r sin θ cos φ, y := r sin θ sin φ, z := r cos θ. (11.73)
We note that, at a given point (r, θ), the angular velocity (11.72) coincides with that of the
zero-angular momentum observer (ZAMO) at that point [compare Eq. (10.99)].
11.3.5 Winding near the event horizon and the inner horizon
Let us consider a null or timelike geodesic L in the vicinity of the black hole event horizon H .
On H , r = r+ and ∆ = 0. Then, the term ∆−1 in Eq. (11.43a) makes dt/dλ diverge as r → r+ ,
except in the very special where (r+ 2
+ a2 )E − aL = 0, which is equivalent to E = ΩH L
according to Eq. (10.69). Similarly, dφ/dλ, as given by Eq. (11.43d), diverges as r → r+ , except
for E = ΩH L. These two divergences are not a pathology of L per se; they reflect merely
the singularity of Boyer-Lindquist coordinates (t, r, θ, φ) at H (cf. Sec. 10.2.6). However, we
read on Eqs. (11.43a) and (11.43d) that the ratio dφ/dt|L := dφ/dλ × (dt/dλ)−1 converges to
a finite value:
dφ a
lim = 2 = ΩH , (11.74)
r→r+ dt
L r+ + a2
where the second equality follows from Eq. (10.69), making the black hole rotation velocity
ΩH appear. Hence we conclude:
Property 11.15: behavior of φ near the event horizon
Any null or timelike geodesic that approaches the event horizon H is winding around it
in terms of the Boyer-Lindquist coordinates at exactly the black hole rotation velocity ΩH .
Remark 6: For a timelike geodesic on a circular orbit, we have argued in Sec. 7.3.3 that dφ/dt|L is
the angular velocity of as seen by an asymptotic inertial observer. The reasoning was given in the
Schwarzschild spacetime context but it actually involved only the spacetime symmetry by translation in
t, so it is applicable here.
Remark 7: When a ̸= 0, a geodesic that starts far from the black hole with dφ/dt|L < 0 [according to
Eq. (11.43d) with r ≫ m, this occurs for L < 0] must necessarily have a turning point in φ if it reaches
the event horizon, in order to fulfill (11.74), where ΩH is positive. This is illustrated in Fig. 11.3 and is
in sharp contrast with the Schwarzschild case, where φ is always a monotonic function of λ, as shown
in Sec. 7.2.2.
Remark 8: The winding property does not hold for the Kerr or advanced Kerr coordinates. Indeed,
we have seen in Sec. 10.4 that the ingoing principal null geodesics L(v,θ,
in
φ̃) are geodesics along which
φ̃ is constant. They are therefore not winding in terms of neither the Kerr coordinates (t̃, r, θ, φ̃) nor
the advanced Kerr ones (v, r, θ, φ̃). This difference of (coordinate) behavior is understandable if one
11.3 Main properties of geodesics 397
y/m
1
x/m
2 4 6 8 10 12
-1
-2
-3
Figure 11.3: Trajectory in the equatorial plane of an incoming null geodesic with L = −6E < 0 and Q = 0,
plunging into a Kerr black hole with a = 0.998 m. Note the turning point in φ and the final winding in the
direction of the black hole rotation (counterclockwise in the figure). The figure is drawn in terms of the Cartesian
Boyer-Lindquist coordinates (x, y) defined by Eq. (11.73) with θ = π/2 (equatorial plane) and the grey disk marks
the black hole region. [Figure generated by the notebook D.5.8]
considers the diverging behavior in ∆−1 of the relation dφ̃ = dφ + a/∆ dr [Eq. (10.23b)] between the
angular coordinates φ and φ̃.
Regarding the Boyer-Lindquist coordinate behavior in the vicinity of the inner horizon Hin
(corresponding to the second root r− of ∆), we deduce from Eqs. (11.43a) and (11.43d) that
dφ a
lim = 2
= Ωin , (11.75)
r→r− dt L r− + a2
where the second equality follows from Eq. (10.71); it involves the rotation velocity Ωin of the
inner horizon Hin . Hence
Property 11.16: behavior of φ near the inner horizon
Any null or timelike geodesic that approaches the inner horizon Hin is winding around it
in terms of the Boyer-Lindquist coordinates at exactly the rotation velocity Ωin of Hin .
11.3.6 Asymptotic r-values and θ-values
Let L be a geodesic and p0 a point of L at which r varies, i.e. such that R(r0 ) ̸= 0, where
r0 = r(p0 ). Let us assume that L comes close to r = r1 , where r1 is a double root of R, i.e.
fulfills R(r1 ) = 0, R′ (r1 ) = 0 and R′′ (r1 ) ̸= 0, such that R(r) > 0 on the interval [r0 , r1 ). Let
us reconsider the argument in Sec. 11.2.7 that lead to extend the integral (11.51) to r = r1 ,
where r1 corresponded to a simple root of R (R(r1 ) = 0 and R′ (r1 ) ̸= 0). For a double root,
the integral in the right-hand side of Eq. (11.51) with r = r1 behaves near r1 as
√ Z r1
2 dr̄
p ,
R (r1 ) r0 r1 − r̄
′′
which is a divergent improper integral. If r1 is a higher order root of R, the divergence is even
worse, being triggered by a higher power of 1/(r1 − r̄). Consequently, Eq. (11.51) with r = r1
398 Geodesics in Kerr spacetime: generic and timelike cases
implies that the Mino parameter diverges: λ′ → +∞. Since ρ2 > 0, provided that ρ2 does not
tend to 0 (the ring singularity) as λ′ → +∞, this implies that the affine parameter λ tends to
+∞ as well, given the relation (11.49) between the two parameters. Hence we conclude:
Property 11.17: asymptotic r-value
Let L be a geodesic of Kerr spacetime that does not lie at a fixed value of r and let R(r)
be the associated polynomial (11.44). A point away from the ring singularity R and of r-
coordinate r1 such that both R(r1 ) = 0 and R′ (r1 ) = 0 can only be reached asymptotically
by L , i.e. in the limit λ → ±∞ of the affine parameter λ. We call r1 an asymptotic
r-value of the geodesic L .
This property explains why R(r0 ) = 0 and R′ (r0 ) ̸= 0 is a necessary condition to have a
r-turning point (cf. Remark 4 on p. 384).
By the same reasoning, we have
Property 11.18: asymptotic θ-value
Let L be a geodesic of Kerr spacetime that does not lie at a fixed value of θ and let Θ(θ)
be the associated function (11.32). A point away from the ring singularity R and of θ-
coordinate θ1 such that both Θ(θ1 ) = 0 and Θ′ (θ1 ) = 0 can only be reached asymptotically
by L , i.e. in the limit λ → ±∞ of the affine parameter λ. We call θ1 an asymptotic
θ-value of the geodesic L .
11.3.7 Latitudinal motion
We start by analysing the variation of the θ coordinate along a geodesic, as governed by the
decoupled equation (11.50c), since this provides some constraints to discuss later the r-motion.
Given expression (11.35) for Θ(θ), the first-order equation of motion (11.50c) can be rewrit-
ten as
2
dθ
+ V (θ) = Q , (11.76)
dλ′
with
L2
2 2 2 2
V (θ) := cos θ a (µ − E ) + . (11.77)
sin2 θ
Note that V (θ) is related to Θ(θ) by
Θ(θ) = Q − V (θ). (11.78)
In particular, the θ-turning points (cf. Sec. 11.2.6) are characterized by V (θ) = Q and V ′ (θ) ̸= 0,
while the asymptotic θ-values (cf. Sec. 11.3.6) correspond to V (θ) = Q and V ′ (θ) = 0.
We recognize in Eq. (11.76) the first integral of a (non-relativistic) 1-dimensional motion
in the potential V (θ), which we shall call the effective θ-potential. The discussion of the
11.3 Main properties of geodesics 399
geodesic θ-motion is then based on the properties of that potential, Q in the right-hand side
of Eq. (11.76) playing the role of the constant “total energy”. Since (dθ/dλ′ )2 ≥ 0, Eq. (11.76)
implies
V (θ) ≤ Q. (11.79)
Accordingly, given a plot of V (θ), as in Figs. 11.4 and 11.5, the allowed range of θ is determined
by the part of the graph of V (θ) that lies below the horizontal line of ordinate equal to Q.
We shall distinguish the case L = 0 from the case L ̸= 0, since they lead to different shapes
of the potential V (θ).
Geodesics with L = 0
If L = 0, the effective θ-potential reduces to V (θ) = a2 (µ2 − E 2 ) cos2 θ. We have then three
subcases:
• Case a2 (E 2 − µ2 ) < 0 ⇐⇒ a ̸= 0 and |E| < µ: the corresponding graph of V (θ) is
shown in Fig. 11.4 (left part); we deduce immediately from it that Q ≥ 0, with
◦ Q = 0: the motion is confined to the equatorial plane θ = π/2; since it corresponds
to a minimum of the effective potential, this is a stable configuration.
◦ 0 < Q < a2 (µ2 − E 2 ): θ oscillates between two θ-turning points, which are
14
symmetric about the equatorial plane : θm := arccos Q/(a2 (µ2 − E 2 )) and
p
π − θm (cf. the trajectory Q = Q1 in Fig. 11.4, left).
◦ Q = a2 (µ2 − E 2 ): θ = 0 and θ = π are either unstable positions or asymptotic
θ-values (cf. Sec. 11.3.6), i.e. the geodesic reaches the rotation axis for λ → ±∞.
◦ Q > a2 (µ2 −E 2 ): the range of θ is not limited (cf. the trajectory Q = Q2 in Fig. 11.4,
left) and each time the geodesic reaches the rotation axis (θ = 0 or θ = π), it crosses
it, since the velocity dθ/dλ′ does not vanish there. This leads to θ < 0 or θ > π; to
keep θ within the interval [0, π], one shall use the identification of the points (θ, φ),
(−θ, φ + π) and (θ − π, φ + π), which holds on the sphere S2 .
• Case a2 (E 2 − µ2 ) = 0 ⇐⇒ a = 0 or |E| = µ: V (θ) = 0 and√ Eq.(11.76) reduces to
(dθ/dλ ) = Q. This implies Q ≥ 0 and the solution is θ(λ ) = ± Q λ′ + θ0 , so that
′ 2 ′
◦ for Q = 0, the geodesic lies at a constant value of θ, within the range [0, π];
◦ for Q > 0, θ varies monotonically along the geodesic, which therefore crosses the
rotation axis an infinite number of times.
• Case a2 (E 2 − µ2 ) > 0 ⇐⇒ a ̸= 0 and |E| > µ: the corresponding graph of V (θ) is
shown in Fig. 11.4 (right part); the Carter constant must obey Q ≥ −a2 (E 2 − µ2 ), with
◦ Q = −a2 (E 2 − µ2 ): only θ = 0 and θ = π are possible and they correspond to
minima of V (θ); the geodesic is then stably located on the rotation axis.
14
The index m in θm stands for minimal.
400 Geodesics in Kerr spacetime: generic and timelike cases
L = 0, a 0, |E| < µ L = 0, a 0, |E| > µ
0.4
1.2
Q = Q2 Q = Q2
0.2
1.0 Q = a 2 (µ 2 − E 2 ) Q=0
0.0
0.8 Q = Q1 0.2 Q = Q1 Q = Q1
V(θ) /V0
V(θ) /V0
0.6 0.4
0.4 0.6
0.2 0.8
Q=0 Q = Qmin Q = Qmin
0.0 1.0
0 θm 1π
4
1π
2
3π
4 π − θm π 0 1π
4 θv 1π
2 π − θv34 π π
θ θ
Figure 11.4: Effective θ-potential V (θ) in the case L = 0 and a ̸= 0. V (θ) is plotted in units of V0 := a2 |µ2 −E 2 |.
The left figure is for |E| < µ, with colored dots or horizontal lines corresponding to geodesic trajectories for four
values of the Carter constant Q: 0, Q1 = 0.75V0 , a2 (µ2 − E 2 ) and Q2 = 1.2V0 . The right figure is for |E| > µ,
with trajectories corresponding to four values of Q: Qmin = −a2 (E 2 − µ2 ), Q1 = −0.3V0 , 0 and Q2 = 0.2V0 .
Dashed lines indicate trajectories with asympotic θ-values.
L 0, L 2 ≥ a 2 (E 2 − µ 2 ) L 0, L 2 < a 2 (E 2 − µ 2 )
4
8 Q = Q2
2
6 Q=0
V(θ) /L 2
V(θ) /L 2
Q = Q1 0
4 Q = Q1 Q = Q1
2
2
Q=0 4 Q = Qmin Q = Qmin
0
0 θm 1π
4
1π
2
3π
4
π − θm π 0 θm 1π
4 θv 1π
2 π − θv 34 π π − θm π
θ θ
Figure 11.5: Effective θ-potential V (θ) in the case L ̸= 0. The left figure is for L2 ≥ a2 (E 2 − µ2 ), with colored
dots or horizontal lines corresponding to geodesic trajectories for two values of the Carter constant Q: 0 and
Q1 = 4L2 . The right figure is for L2 < a2 (E 2 − µ2 ), with trajectories corresponding to four values of Q: Qmin ,
Q1 = −2L2 , 0 and Q2 = 2L2 . The dashed line (Q = 0) indicates the trajectory with an asympotic θ-value, which
is π/2.
◦ −a2 (E 2 − µ2 ) < Q < 0: the geodesic oscillates about the rotation axis, without
reaching the equator; one has
p either θ ∈ [0, θv ] or θ ∈ [π − θv , π], with the turning
15
point value θv := arccos |Q|/(a2 (E 2 − µ2 )) < π/2 (cf. the trajectories Q = Q1
in Fig. 11.4, right).
◦ Q = 0: θ = π/2 is either an unstable position or an asymptotic θ-value, i.e. the
geodesic approaches the equatorial plane when λ → ±∞.
◦ Q > 0: θ varies in all the range [0, π]; when the geodesic reaches the rotation axis,
it crosses it (cf. the trajectory Q = Q2 in Fig. 11.4, right).
15
The index v in θv stands for vortical, the definition of which is given at the end of this section.
11.3 Main properties of geodesics 401
Geodesics with L ̸= 0
When L ̸= 0, we see from expression (11.77) that V (θ) → +∞ when θ → 0 or π. We conclude
immediately that the geodesic cannot reach the rotation axis for L ̸= 0. Moreover, we have
L2
dV 2 2 2
= −2 cos θ sin θ a (µ − E ) + .
dθ sin4 θ
The extrema of V in (0, π) are then given by
dV π L2
= 0 ⇐⇒ θ = or sin4 θ = . (11.80)
dθ 2 a2 (E 2 − µ2 )
We are thus led to distinguish two cases, depending whether or not the equation involving
sin4 θ has solutions distinct from π/2:
q
• Case L ≥ a (E − µ ) ⇐⇒ a = 0 or |E| ≤ µ2 + La2 : the only extremum solution
2 2 2 2 2
is θ = π/2. Its value is V (π/2) = 0 and it is necessarily a minimum, since V (θ) → +∞
at the boundaries of the interval [0, π] (cf. left part of Fig. 11.5). Hence one must have
Q ≥ 0, with
◦ Q = 0: the geodesic stays stably in the equatorial plane.
◦ Q > 0: the geodesic oscillates about the equatorial plane, between two turning
points at θ = θm and π − θm , θm being the solution of V (θm ) = Q in (0, π/2) (cf.
the trajectory Q = Q1 in Fig. 11.5, left).
q
• Case L2 < a2 (E 2 − µ2 ) ⇐⇒ a ̸= 0 and |E| > µ2 + La2 : V (θ) has three extrema (cf.
2
right part of Fig. 11.5), which are located at θ = θ∗ , π/2 and π − θ∗ with
s
|L| π
θ∗ := arcsin p and 0 < θ∗ < . (11.81)
a E 2 − µ2 2
θ∗ and π − θ∗ correspond to the minimum of V (θ), while π/2 corresponds to a lo-
cal maximum. The minimum is V (θ∗ ) = (1 − sin2 θ∗ )(a2 (µ2 − E 2 ) + L2 / sin2 θ∗ ) =
−(a E 2 − µ2 − |L|)2 . This is necessary the minimal value of Q [cf. Eq. (11.79)]:
p
p 2
Qmin 2 2
= − a E − µ − |L| . (11.82)
We have then (cf. right panel of Fig. 11.5)
◦ Q = Qmin : the geodesic stays stably at a fixed value of θ, either θ∗ or π − θ∗ .
◦ Qmin < Q < 0: the geodesic oscillates between two θ-turning points either in the
Northern hemisphere (θ < π/2) or the Southern one (θ > π/2), without reaching
the equator nor the rotation axis (cf. the trajectories Q = Q1 in Fig. 11.5, right).
402 Geodesics in Kerr spacetime: generic and timelike cases
◦ Q = 0: the geodesic lies unstably in the equatorial plane or moves asymptotically
towards it (possibly after a turning point at θm ̸= π/2), π/2 being an asymptotic
θ-value, since Θ′ (π/2) = −V ′ (π/2) = 0 (cf. Sec. 11.3.6).
◦ Q > 0: the geodesic oscillates about the equatorial plane, between two turning
points at θ = θm and π − θm , θm being the solution of V (θm ) = Q in (0, π/2) (cf.
the trajectory Q = Q2 in Fig. 11.5, right).
Expression of the θ-turning points
The θ-turning points θm and θv mentioned above are solutions of Θ(θ) = 0 and Θ′ (θ) ̸= 0
[Eq. (11.47)]. We search only for θ-turning points with 0 < θ < π/2, since 0 and π/2 cannot be
θ-turning points (for Θ′ (0) = 0 and Θ′ (π/2) = 0) and θ-turnings points with π/2 < θ < π are
deduced from those with 0 < θ < π/2 by θ 7→ π − θ (symmetry with respect to the equatorial
plane). Using expression (11.35) for Θ and introducing
x := cos2 θ, (11.83)
the equation Θ(θ) = 0 is equivalent to
a2 (E 2 − µ2 )x2 + L2 + Q − a2 (E 2 − µ2 ) x − Q = 0. (11.84)
If a2 (E 2 − µ2 ) = 0, the solution is x = Q/(L2 + Q). Moreover, in this case one has necessarily
Q ≥ 0, so that 0 ≤ x ≤ 1 and the unique solution in (0, π/2) is
s
Q
θm = arccos . (11.85)
L2 +Q
a2 (E 2 −µ2 )=0
In the rest of this section, we assume that a2 (E 2 − µ2 ) ̸= 0. Equation (11.84) is then a quadratic
equation in x, the discriminant of which is
2
∆ = L2 + Q − a2 (E 2 − µ2 ) + 4Qa2 (E 2 − µ2 )
2
= L2 + Q + a2 (E 2 − µ2 ) − 4L2 a2 (E 2 − µ2 ).
It turns out that ∆ is always non-negative. Indeed, from the second equality above, we have
clearly ∆ ≥ 0 if a2 (E 2 − µ2 ) ≤ 0. In the complementary case, i.e. when a2 (E 2 − µ2 ) > 0,
then ∆ ≥ 0 as soon as Q ≥ Q2 , where Q2 is the larger of the two roots of ∆ considered as
a quadratic polynomial in Q. Again, considering
p the second equalitypin the expression of ∆,
we have Q2 = −L2 − a2 (E 2 − µ2 ) + 2|L|a E 2 − µ2 = −(|L| − a E 2 − µ2 )2 . In view of
Eq. (11.82), we realize that Q2 = Qmin , so that Q ≥ Q2 always holds and ∆ ≥ 0. Accordingly
the two roots of Eq. (11.84) are real and are given by
s
2 2
2
1 L +Q L +Q 4Q
x± = 1 − 2 2 2
± 1− 2 2 2
+ 2 2 . (11.86)
2 a (E − µ ) a (E − µ ) a (E − µ2 )
11.3 Main properties of geodesics 403
Since x := cos2 θ and θ = 0 and θ = π/2 cannot correspond to a θ-turning point, acceptable
solutions must obey
0 < x± < 1. (11.87)
For L = 0, Eq. (11.86) simplifies to16
Q
x+ = 1 and x− = −
a2 (E 2 − µ2 )
Now, x+ = 1 is excluded by (11.87). There remains only x− , leading to the turning point value
found in the L = 0 cases discussed above, which we can combine into a single formula:
s
Q
θm = θv = arccos . (11.88)
a (µ − E 2 )
2 2
L=0
This formula assumes that 0 < Q/(a2 (µ2 − E 2 )) < 1 (as for the trajectories Q = Q1 in both
panels of Fig. 11.4), otherwise there is no θ-turning point (as for the trajectories Q = Q2 in
both panels of Fig. 11.4). More precisely, for Q > 0 and µ2 − E 2 > 0, it leads to θm , while for
Q < 0 and µ2 − E 2 < 0, it leads to θv .
For L ̸= 0, one shall distinguish between two cases, since the case |E| = µ has been covered
above [Eq. (11.85)]:
• if |E| < µ, then only x− fulfills the criterion (11.87), leading to the turning point value:
v
u s 2
u1
u 2
L +Q 2
L +Q 4Q
θm = arccos t 1 + 2 2 − 1 + −
2 a (µ − E 2 ) a2 (µ2 − E 2 ) a2 (µ2 − E 2 )
(11.89)
• if |E| > µ, then x+ always fulfills the criterion (11.87), leading to the turning point value:
v
u s 2
u1
u 2
L +Q 2
L +Q 4Q
θm = arccos t 1 − 2 2 2
+ 1− 2 2 2
+ 2 2 .
2 a (E − µ ) a (E − µ ) a (E − µ2 )
(11.90)
If, in addition Q < 0, then x− also fulfills (11.87) and we get a second turning point value
(cf. the case Q = Q1 in the right panel of Fig. 11.5):
v
u s 2
u1
u 2
L +Q 2
L +Q 4Q
θv = arccos t 1 − 2 2 2
− 1− 2 2 2
+ 2 2 .
2 a (E − µ ) a (E − µ ) a (E − µ2 )
(11.91)
Remark 9: At the limit L → 0, formulas (11.89) and (11.91) reduce both to (11.88).
16
A switch between x+ and x− is performed if 1 + Q/(a2 (E 2 − µ2 )) < 0.
404 Geodesics in Kerr spacetime: generic and timelike cases
Summary
We can summarize the above results by
Property 11.19: latitudinal motion of Kerr geodesics
• A geodesic L of Kerr spacetime cannot encounter the rotation axis unless it has
L = 0.
• If L2 ≥ a2 (E 2 − µ2 ), the Carter constant Q is necessarily non-negative:
Q ≥ 0. (11.92)
• The Carter constant Q can take negative values only if L2 < a2 (E 2 − µ2 ), which
implies a ̸= 0 and |E| > µ; the range of Q is then limited from below:
p 2
Q ≥ Qmin = − a E 2 − µ2 − |L| . (11.93)
A geodesic with Q < 0 is called vortical; it never encounters the equatorial plane.
• If Q > 0, L oscillates symmetrically about the equatorial plane, between two
θ-turning points, at θ = θm and θ = π − θm , where θm ∈ (0, π/2) is given by
Eq. (11.85) for a2 (E 2 − µ2 ) = 0, Eq. (11.88) for L = 0 and Q < a2 (µ2 − E 2 ),
Eq. (11.89) for |E| < µ and Eq. (11.90) for |E| > µ, except for two subcases with
L = 0: (i) Q = a2 (µ2 − E 2 ): L lies unstably along the rotation axis or approaches
it asymptotically and (ii) Q > a2 (µ2 − E 2 ): L crosses repeatedly the rotation axis,
with θ taking all values in the range [0, π].
• If Q = 0, L is stably confined to the equatorial plane for L2 > a2 (E 2 − µ2 ) or
L2 = a2 (E 2 −µ2 ) ̸= 0; for L2 < a2 (E 2 −µ2 ), L either lies unstably in the equatorial
plane or approaches it asymptotically from one side, while for L2 = a2 (E 2 − µ2 ) = 0,
L lies at a constant value θ = θ0 ∈ [0, π].
• If Qmin < Q < 0, L never encounters the equatorial plane, having a θ-motion
entirely confined either to the Northern hemisphere (0 < θ < π/2) or to the
Southern one (π/2 < θ < π); if L ̸= 0, L oscillates between two θ-turning points, at
θ = θm and θ = θv (Northern hemisphere) or at θ = π − θv and θ = π − θm (Southern
hemisphere), where θm and θv are given by Eqs. (11.90) and (11.91) respectively; if
L = 0, L oscillates about the rotation axis, with a θ-turning point at θ = θv or
θ = π − θv , where θv is given by Eq. (11.88).
11.3 Main properties of geodesics 405
• If Q = Qmin , L lies stably at a constant value θ = θ∗ or θ = π −θ∗ , with θ∗ ∈ [0, π/2)
given bya s
|L|
θ∗ := arcsin p . (11.94)
a E 2 − µ2
a
This is Eq. (11.81) generalized to encompass the case L = 0.
Remark 10: We could have deduced that L = 0 is a necessary condition for a geodesic to encounter
the rotation axis without studying the potential V (θ). Indeed, by definition, L = η · p [Eq. (11.2b)],
with the Killing vector η being zero on the rotation axis, since the latter is pointwise invariant under
the action of the rotation group SO(2). Hence L = 0 on the rotation axis. Since L is constant along the
geodesic, the result follows immediately.
Remark 11: That a vortical geodesic never intersects the equatorial plane has been obtained by
examining the various cases Q < 0. This result can be derived directly from the constraint Θ(θ) ≥ 0
[Eq. (11.33)], by noticing the identity Θ(π/2) = Q, which follows immediately from Eq. (11.35).
Example 8 (Schwarzschild geodesics): It is instructive to apply the above results to a = 0 and recover
the geodesics in Schwarzschild spacetime studied in Chaps. 7 and 8. There, spherical symmetry was
used to select the coordinates (t, r, θ, φ) so that the geodesic was confined to the hyperplane θ = π/2.
Consequently there was no θ-motion. Here, we keep the coordinates (t, r, θ, φ) fixed and do not assume
that they are adapted to the geodesic L under consideration. So θ may vary along L . In Schwarzschild
spacetime, the Carter constant Q is always non-negative, since the inequality L2 ≥ a2 (E 2 − µ2 ) is
trivially satisfied for a = 0 [cf. Eq. (11.92)].
If Q = 0, then for L = 0, L lies at a constant value of θ: this actually corresponds to a purely radial
geodesic. Indeed, the total angular momentum measured in the asymptotic region is Ltot = 0 in that
case [set Q = 0, L = 0 and a = 0 in Eq. (11.37)]. If L ̸= 0, still with Q = 0, L lies stably in the
equatorial plane (this is the only possibility for a = 0 among the Q = 0 cases listed above).
If Q > 0, then L = 0 is necessarily the subcase (ii) listed above: L crosses repeatedly the z-axis
(θ ∈ {0, π}); this corresponds to the case where the orbital plane contains the z-axis. All the angular
momentum measured asymptotically is then contained in Q [cf. Eq. (11.37)]. For Q > 0 and L ̸= 0, L
oscillates symmetrically about the equatorial plane: this is the case where the orbital plane is inclined by
an angle ι ∈ (0, π/2) with respect to the equatorial plane; the θ-turning point of L is then θm = π/2−ι.
11.3.8 Radial motion
The r-part of geodesic motion is constrained by Eq. (11.30): R(r) ≥ 0. Let us expand expression
(11.44) for the quartic polynomial R(r) in powers of r:
R(r) = (E 2 − µ2 )r4 + 2mµ2 r3 + a2 (E 2 − µ2 ) − Q − L2 r2 + 2m Q + (L − aE)2 r − a2 Q .
(11.95)
Geodesics with |E| < µ
For r → ±∞ and |E| = ̸ µ, we have R(r) ∼ (E 2 −µ2 )r4 and in particular limr→±∞ R(r) = +∞
for |E| > µ and limr→±∞ R(r) = −∞ for |E| < µ. In the latter case, the constraint R(r) ≥ 0
406 Geodesics in Kerr spacetime: generic and timelike cases
cannot be satisfied for large values of |r|. A geodesic with |E| < µ is therefore located within
a bounded region in terms of the radial coordinate r. Moreover, |E| < µ implies necessarily
L2 > a2 (E 2 − µ2 ), so that the results of Sec. 11.3.7 lead to Q ≥ 0 [cf. Eq. (11.92)]. Therefore,
−a2 Q ≤ 0. It follows then that r cannot be negative. Indeed, we see on expression (11.95) that
for |E| < µ (which implies µ > 0 and Q ≥ 0) and r < 0,
R(r) = (E 2 − µ2 )r4 + 2mµ2 r3 + a2 (E 2 − µ2 ) − Q − L2 r2 + 2m Q + (L − aE)2 r −a2 Q,
| {z } | {z } | {z } | {z } | {z }
<0 <0 ≤0 ≤0 ≤0
which contradicts R(r) ≥ 0. Hence we conclude:
Property 11.20: bound orbits
Any geodesic L with |E| < µ is necessarily timelike (µ > 0), has a non-negative Carter
constant Q, is confined to the region r ≥ 0 of Kerr spacetime and cannot reach arbitrarily
large values of r. L is called a geodesic with a bound orbit, or in short, a bound geodesic.
Remark 12: That L cannot reach the asymptotic region r → +∞ could have been found directly from
the definition of E as the “energy at infinity” of the particle P having L as worldline [Eq. (11.2a)].
Indeed, since L is timelike, the 4-momentum of P is p = µu [Eq. (1.18)], where u is the 4-velocity
of P, so that Eq. (11.2a) yields E = −µ ξ · u. Now, when r → +∞, ξ tends to the 4-velocity of an
inertial observer (cf. Sec. 10.7.5). If L could reach the asymptotic region, the scalar product of the
two 4-velocities ξ and u would be necessarily lower or equal to −1, or equivalently Γ := −ξ · u ≥ 1
(the proof lies in expression (1.35) of Γ with 0 ≤ V · V < 1), so that we would have E ≥ µ, which
contradicts |E| < µ.
Geodesics with |E| = µ
Contrary to the case |E| < µ, a geodesic with |E| = µ can be null, provided it has E = 0 (cf.
Example 1 on p. 375). We have
Property 11.21
Any geodesic with |E| = µ has Q ≥ 0 and is necessarily confined to the region r ≥ 0 of
Kerr spacetime.
Proof. Q ≥ 0 follows directly from the criteria |E| ≤ µ2 + L2 /a2 [cf. Eq. (11.92)], which is
p
evidently fulfilled for |E| = µ. Furthermore, for |E| = µ, expression (11.95) for R(r) simplifies
to
R(r) = 2mµ2 r3 − (Q + L2 )r2 + 2m Q + (L − aE)2 r − a2 Q.
For r < 0, all the four terms in the above sum are ≤ 0. If L is timelike, then µ ̸= 0 and the
first term is < 0, so that r < 0 =⇒ R(r) < 0, which violates the constraint R(r) ≥ 0. Let us
now assume that L is null. We have then µ = 0 and E = 0, so that
R(r) = −(Q + L2 )r2 + 2m(Q + L2 )r − a2 Q = (Q + L2 )r(2m − r) − a2 Q.
11.3 Main properties of geodesics 407
If Q + L2 ̸= 0, R(r) is a quadratic polynomial that is either negative everywhere or negative
outside the interval [r1 , r2 ], when the two roots r1 and r2 of R(r) are reals. In the latter case, we
deduce from the signs of the coefficients of R(r) that r1 + r2 > 0 and r1 r2 ≥ 0, so that r1 ≥ 0
and r2 ≥ 0. This implies that R(r) < 0 for r < 0, which is not permitted. If Q + L2 = 0, the
property Q ≥ 0 implies Q = 0 and L = 0. The Carter constant K is then zero as well, since
K = Q + (L − aE)2 [Eq. (11.34)]. Then by the result (11.25), L is a null geodesic generator
of either the event horizon H , which is located at r = r+ , or the inner horizon Hin , which is
located at r = r− . Since both r+ and r− are positive [Eq. (10.5)], we cannot have r < 0 in this
case either.
A timelike geodesic L with E = µ can reach the asymptotic region r → +∞. It has then
a Lorentz factor with respect to the asymptotic static observer of 4-velocity ξ equal to one (cf.
Remark 12 above), which implies that p is collinear to ξ. In that sense, L is asymptotically “at
rest”. Such a geodesic is called marginally bound.
On the opposite, a null geodesic with E = µ(= 0) cannot reach the asymptotic region
r → +∞. Actually, it cannot even exist outside the ergoregion, by virtue of Property 11.5.
Example 9: The null geodesics generating the horizons H and Hin considered in Example 1 on p. 375
have E = 0 [Eq. (11.4)] and are indeed fully located in the ergoregion G , since H ⊂ G and Hin ⊂ G
(cf. Fig. 10.2).
Geodesics with |E| > µ
An immediate corollary of the properties obtained for |E| < µ and |E| = µ is
Property 11.22: necessary condition to visit the negative-r region
Only geodesics with |E| > µ may have some part in the region r < 0 of Kerr spacetime,
M− :
L ∩ M− ̸= ∅ =⇒ |E| > µ. (11.96)
Decay of r towards the future in MII
As a particular case of the result established in Sec. 10.3.2 for any causal worldline (not neces-
sarily a geodesic), we have
Property 11.23: decreasing of r in MII
In region MII , the coordinate r must decrease towards the future along any timelike or
null geodesic:
dr
<0 . (11.97)
dλ MII
408 Geodesics in Kerr spacetime: generic and timelike cases
Example 10 (Principal null geodesics): For the ingoing principal null geodesics L(v,θ, in
φ̃) , a future-
directed tangent vector is k and we have k = dr/dλ = −1, since the affine parameter associated with
r
k is λ = −r [Eq. (10.46)]. Hence for these geodesics, r is decreasing towards the future everywhere
in M , and in particular in MII . For the outgoing principal null geodesics L(u,θ,
out
˜ , a future-directed
φ̃)
tangent vector is ℓ and the associated (non-affine) parameter obeys Eq. (10.56):
dr ∆
= .
dλ 2(r + a2 )
2
Thus dr/dλ < 0 in MII , since ∆ < 0 there.
11.3.9 Geodesics reaching or emanating from the ring singularity
In the Schwarzschild spacetime, any causal geodesic that enters into the black hole region
inevitably terminates at the curvature singularity r = 0, as it is clear on the Kruskal or Carter-
Penrose diagrams constructed in Chap. 9. For the Kerr spacetime with a ̸= 0, we are going to
see that, on the contrary, most causal geodesics in the black hole region avoid the curvature
singularity. In all this section, we assume a ̸= 0, so that the curvature singularity is the ring
singularity R discussed in Sec. 10.2.6.
Formally, R is not part of the Kerr spacetime M but of the larger manifold R2 × S2 [cf.
the construction (10.29)]. It is located at ρ2 = 0, i.e. at r = 0 and θ = π/2. We shall say
that a geodesic L of affine parameter λ (oriented towards the future) approaches the ring
singularity if L has both r(λ) → 0 and θ(λ) → π/2 as λ → λ∗ , with λ∗ being finite or
infinite. If λ∗ is finite, we shall say that L hits the ring singularity for λ → λ−
∗ and emanates
from the ring singularity for λ → λ+ ∗ . If λ∗ = ±∞, we shall say that L asymptotically
approaches the ring singularity.
A first key result is
Property 11.24: vanishing of Q required to approach the ring singularity
A null or timelike geodesic L that approaches the ring singularity has a vanishing Carter
constant Q.
Proof. From Eqs. (11.95) and (11.35), we have
lim R(r) = −a2 Q and lim Θ(θ) = Q.
r→0 θ→π/2
The constraints R(r) ≥ 0 [Eq. (11.30)] and Θ(θ) ≥ 0 [Eq. (11.33)] imply then respectively
Q ≤ 0 and Q ≥ 0, from which we get Q = 0.
The above property can be refined17 :
17
This is a refinement because not all geodesics with Q = 0 lie in the equatorial plane. For instance the null
geodesic generators of the event horizon H have Q = 0 [Eq. (11.25)] but those with θ ̸= π/2 lie outside the
equatorial plane.
11.3 Main properties of geodesics 409
Property 11.25: ring singularity reachable only from the equatorial plane
A null or timelike geodesic cannot approach the ring singularity unless it lies entirely in
the equatorial plane.
Proof. Let L be a causal geodesic that approaches R. From the previous result, L has Q = 0.
Reviewing the subcases Q = 0 amongp all the cases considered in Sec. 11.3.7, we see that L can
approach θ = p
π/2 iff (i) L has |E| ≤ µ2 + L2 /a2 and lies stably in the equatorial plane or (ii)
L has |E| > µ2 + L2 /a2 and approaches asymptotically θ = π/2 as the Mino parameter λ′
tends to ±∞. Let us show that (ii) is not compatible with r → 0. For Q = 0, expression (11.95)
for R(r) reduces to
R(r) = r (E 2 − µ2 )r3 + 2mµ2 r2 + a2 (E 2 − µ2 ) − L2 r + 2m(L − aE)2 ,
with the constant term inside the square brackets 2m(L − aE)2 ̸= 0, since L = aE is not
compatible with |E| > µ2 + L2 /a2 . It follows that r = 0 is a simple root of R(r). If L
p
would reach both r = 0 and θ = π/2 when λ tends to some value λ∗ , then Eq. (11.55a) would
yield
Z 0 Z π
ϵr dr 2 ϵθ dθ
′ ′
λ∗ − λ0 = − p =− p , (11.98)
r0 R(r) θ0 Θ(θ)
where λ′∗ is the Mino parameter corresponding to λ∗ . Note that λ′∗ may be infinite even if λ∗ is
finite, due to the relation (11.49) with ρ2 → 0 when λ → λ∗ . Since r = 0 is a simple root of
R(r), the integral on r has a finite value. Now, for Q = 0, expression (11.35) for Θ reduces to
L2
2 2 2 2
Θ(θ) = cos θ a (E − µ ) − .
sin2 θ
The cos2 θ term, which behaves as (θ − π/2)2 for θ near π/2, makes the integral on θ in
Eq. (11.98) divergent, which is incompatible with the finite value of the integral on r in the
left-hand side. Hence only (i) is possible, which completes the proof.
Property 11.25 has been obtained for a generic approach to the ring singularity, i.e. for
a geodesic L that hits R, emanates from R or asymptotically approach L in the future or
the past. If we apply it to null geodesics (light rays) emanating from R, we conclude that an
intrepid observer diving into the black hole would not see the ring singularity at all, except
when he crosses the equatorial plane. At this instant, the singularity would appear to him as
a 1-dimensional segment, and the image would disappear as soon as the observer leaves the
equatorial plane. In particular, the observer will never see a ring-like image.
11.3.10 Moving to the negative-r side
If it maintains θ ̸= π/2 in the vicinity of r = 0, a geodesic L can a priori move from the
region r > 0 of M to the region r < 0, or vice-versa, through one the two open disks r = 0
(either the disk θ < π/2 or the disk θ > π/2) delimited by the ring singularity (cf. Sec. 10.2.2).
410 Geodesics in Kerr spacetime: generic and timelike cases
However, such a motion is possible only if R(0) ≥ 0 (condition (11.30) at the boundary r = 0).
The case R(0) = 0 is excluded for it would correspond either to a r-turning point (Sec. 11.2.6)
or to an asymptotic r-value (Sec. 11.3.6). In both cases, L would remain on a single side of
the hypersurface r = 0. The necessary condition for r = 0 crossing is thus R(0) > 0. Now, in
view of expression (11.95) of R, we have R(0) = −a2 Q, so that R(0) > 0 ⇐⇒ Q < 0. We
thus conclude
Property 11.26: crossing r = 0 only possible for vortical geodesics
Only a vortical geodesic (Q < 0) can cross the hypersurface r = 0 and thus move from
the positive-r region of spacetime (M+ ) to the negative-r one (M− ), or vice-versa. In
particular, such a geodesic must have a high energy, i.e. it must obey |E| > µ2 + L2 /a2 .
p
The energy condition is simply the necessary condition for Q < 0 stated in Sec. 11.3.7. We note
that it implies |E| > µ, which is consistent with the property (11.96) required to travel in M− .
Example 11: The ingoing principal null geodesics with θ ̸= π/2 cross the hypersurface r = 0 (cf. the
dashed green lines in Figs. 10.6 and 10.9, as well as the green lines for θ = π/6 in Fig. C.2) and are
indeed vortical, since their Carter constants (11.38) obey Q < 0 for θ ̸= π/2. Moreover,p
they have µ = 0
and L = aE sin θ [Eq. (11.6)], with sin θ < 1 for θ ̸= π/2, so that they fulfill |E| > µ2 + L2 /a2 .
2 2
11.4 Timelike geodesics
11.4.1 Parametrization
Whenever the geodesic L is timelike, it is relevant to rescale everything by the mass µ of the
particle P whose worldline is L . In particular, as in the Schwarzschild case treated in Sec. 7.3,
let us parameterize L by the proper time τ , which is the affine parameter τ related to the
affine parameter λ associated to the 4-momentum p by
τ = µλ (11.99)
and let us introduce the specific conserved energy ε, specific conserved angular momentum
ℓ , and reduced Carter constant q:
E L Q
ε := , ℓ := and q := . (11.100)
µ µ µ2
We shall refer to ε, ℓ and q as the reduced integrals of motion of the geodesic L . Note that
in geometrized units (c = G = 1), ε is dimensionless, ℓ has the dimension of a length and q
that of a squared length.
The first-order equations of motion (11.50) can be rewritten in terms of the above quantities:
11.4 Timelike geodesics 411
dt
= T1 (r) + T2 (θ) (11.101a)
dτ ′
2
dr
− R(r) = 0 (11.101b)
dτ ′
2
dθ
− Θ̃(θ) = 0 (11.101c)
dτ ′
dφ
= Φ1 (r) + Φ2 (θ) (11.101d)
dτ ′
where τ ′ is related to the Mino parameter λ′ by
τ ′ = µλ′ (11.102)
and
ε(r2 + a2 )2 − 2amℓr
T1 (r) := , T2 (θ) := −a2 ε sin2 θ, (11.103)
r2 − 2mr + a2
a(2mεr − aℓ) ℓ
Φ1 (r) := , Φ2 (θ) := , (11.104)
2
r − 2mr + a 2 sin2 θ
R(r) := (ε2 − 1)r4 + 2mr3 + a2 (ε2 − 1) − q − ℓ2 r2 + 2m q + (ℓ − aε)2 r − a2 q ,
(11.105)
2
ℓ
Θ̃(θ) := q + cos2 θ a2 (ε2 − 1) − . (11.106)
sin2 θ
Note that R(r) = R(r)/µ2 , Θ̃(θ) := Θ(θ)/µ2 and that expressions (11.105) and (11.106) follow
respectively from Eqs. (11.95) and (11.35).
Using Eqs. (11.99) and (11.49), we can relate τ ′ to the proper time τ :
dτ dτ
dτ ′ = = . (11.107)
ρ2 r(τ )2 + a2 cos2 θ(τ )
We shall call τ ′ Mino time along the geodesic L .
Remark 1: Despite its name, the Mino time has not the dimension of a time, but rather that of a time
inverse or length inverse (in the units G = c = 1 that we are using).
11.4.2 Bound orbits
We consider here a timelike geodesic L with |E| < µ, or equivalently
|ε| < 1. (11.108)
412 Geodesics in Kerr spacetime: generic and timelike cases
−R(r)/m 4
−R(r)/m 4
1.5
30
1
0.5
rp r/m
20 -0.5 0.5 1 1.5 2 2.5
10
r/m
2 rp 4 6 ra
-10
-20
Figure 11.6: Effective potential −R(r) corresponding to a = 0.998 m, ε = 0.9, ℓ = 2m and q = 1.3 m2 . The
black vertical line marks the black hole horizon at r = r+ ≃ 1.063 m. Note that the polynomial −R(r) has
four real roots, one of them being very close to, but distinct from, r+ . The two largest roots give the periastron
and apoastron of the bound orbit with the above values of (ε, ℓ, q); they are respectively rp ≃ 2.175 m and
ra ≃ 6.853 m. [Figure generated by the notebook D.5.8]
As shown in Sec. 11.3.8, such a geodesic has necessarily a non-negative Carter constant:
q≥0 (11.109)
and is located in a radially bounded part of the r ≥ 0 region of Kerr spacetime.
Equation (11.101b) can be viewed as the first-integral of a 1-dimensional motion in the
effective potential U (r) := −R(r), with a vanishing “total energy” — the right-hand side
of Eq. (11.101b). The motion is thus possible wherever U (r) ≤ 0 or, equivalently, wherever
R(r) ≥ 0, which is nothing but the constraint (11.30). Property (11.108) implies that the
coefficient of r4 in formula (11.105) for R(r) is negative. It follows then that limr→±∞ U (r) =
+∞ and the segments with U (r) ≤ 0 are located between two roots of R(r) (cf. Fig. 11.6).
Let us denote the lower of these two roots by rp , for periastron, and the larger one by ra , for
apoastron. The periastron and apoastron are of course r-turning points of the geodesic, as
defined in Sec. 11.2.6.
It is clear that the motion in the potential well U (r) = −R(r) as governed by Eq. (11.101b)
(cf. Fig. 11.6) is periodic:
∀n ∈ Z, r(τ ′ + nΛr ) = r(τ ′ ), (11.110)
the period Λr being the Mino time ∆τp
′
spent to perform a round-trip between rp and ra . By
rewriting Eq. (11.101b) as dτ = ±dr/ R(r), we get
′
Z ra
dr
Λr = 2 p . (11.111)
rp R(r)
11.4 Timelike geodesics 413
Since R(r) is a polynomial of degree 4 [cf. Eq. (11.105)], the integral in the right-hand side of
Eq. (11.111) can be evaluated by means of elliptic integrals. We shall not give the detail here,
referring the interested reader to the article [206].
Regarding the θ-motion, we are in the case L2 > a2 (E 2 − µ2 ) considered in Sec. 11.3.7,
since for bound geodesics E 2 − µ2 < 0. The effective θ-potential V (θ) = Q − Θ(θ) has then
the shape shown in the left panel of Fig. 11.4 for ℓ = 0 and in the left panel of Fig. 11.5 for
ℓ ̸= 0. We can then distinguish three types of bound orbits:
• polar orbit: L crosses the rotational axis an infinite number of times; this occurs iff
ℓ = 0 and q > a2 (1 − ε2 );
• equatorial orbit: L is entirely contained in the equatorial plane θ = π/2; for a bound
orbit, this occurs iff q = 0;
• non-polar and non-equatorial orbit: L never crosses the rotational axis and oscillates
symmetrically about the equatorial plane between two θ-turning points at θ = θm ∈
(0, π/2) and θ = π − θm ; for a bound orbit, this occurs iff q > 0 and (ℓ ̸= 0 or q <
a2 (1 − ε2 )).
Strictly speaking, there is also the exceptional case ℓ = 0 and q = a2 (1 − ε2 ), for which the
rotation axis is reached asymptotically (cf. Sec. 11.3.6 and the grey dashed curve in the left
panel of Fig. 11.4).
Circular equatorial orbits will be discussed in Sec. 11.5. In the following, we focus on
non-polar and non-equatorial orbits, which constitute the generic category of bound timelike
orbits. One has then θm ≤ θ ≤ π − θm , with θm given by Eq. (11.89):
v
u s 2
u1
u 2
ℓ +q 2
ℓ +q 4q
θm = arccos t 1 + 2 2
− 1+ 2 2
− 2 , (11.112)
2 a (1 − ε ) a (1 − ε ) a (1 − ε2 )
which reduces to Eq. (11.88) for ℓ = 0:
r
q
θm = arccos . (11.113)
a2 (1 − ε2 )
ℓ=0
From the viewpoint of Eq. (11.101c) above, non-polar orbits have a periodic θ-motion in the
potential well −Θ̃(θ):
∀n ∈ Z, θ(τ ′ + nΛθ ) = θ(τ ′ ). (11.114)
The period Λθ is the Mino time
q ∆τ spent in a round-trip between θm and π − θm . By rewriting
′
Eq. (11.101c) as dτ ′ = ±dθ/ Θ̃(θ), we get
Z π−θm Z π/2
dθ dθ
Λθ = 2 q =4 q , (11.115)
θm Θ̃(θ) θm Θ̃(θ)
414 Geodesics in Kerr spacetime: generic and timelike cases
Figure 11.7: Meridional section of the torus occupied by a bound timelike geodesic. The dot-dashed line is the
rotation axis and the solid line marks the equatorial plane.
where the last equality results from the symmetry of Θ̃(θ) with respect to π/2. One naturally
associates to the periods Λr and Λθ the Mino angular frequencies:
2π 2π
Υr := and Υθ := . (11.116)
Λr Λθ
For non-polar orbits, the motion is restricted by
rp ≤ r ≤ ra and θm ≤ θ ≤ π − θm . (11.117)
This means that in terms of the Cartesian Boyer-Lindquist coordinates (11.73), the geodesic is
confined inside a torus (cf. Fig. 11.7), which is symmetric about the equatorial plane. Moreover,
if the two frequencies Υr and Υθ are not commensurable, i.e. if Υθ /Υr ̸∈ Q, the geodesic fills
the torus. Examples of orbits with Υθ /Υr ∈ Q, i.e. orbits with a closed trajectory in the (r, θ)
plane, can be found in Ref. [244].
As an illustration, Figs. 11.8–11.10 show a bound timelike geodesic L of reduced integrals
of motion ε = 0.9, ℓ = 2m and q = 1.3 m2 orbiting around a Kerr black hole with a = 0.998 m.
It has rp ≃ 2.175 m, ra ≃ 6.853 m and θm ≃ 1.060 rad ≃ 60.75◦ . The effective radial potential
−R(r) of this geodesic is the one depicted in Fig. 11.6. Figures 11.8–11.10 show actually
the segment 0 ≤ τ ≤ 600 m of L , starting at the point of Boyer-Lindquist coordinates
(0, (rp + ra )/2, π/2, 0). That θ(τ ) lies in the interval [θm , π − θm ] ∼ [60◦ , 120◦ ] appears clearly
in the right panel of Fig. 11.9. On Figs. 11.9 (left panel) and 11.10, we notice a generic feature of
eccentric orbits in the Kerr metric, known as zoom-whirl18 : the particle falls from its apoastron
to the central region (“zoom in”) and performs some quasi-circular revolutions at close distance
to the black hole (it “whirls”), reaching the periastron, and finally goes back to the apoastron.
The zoom-whirl behavior exists as well for eccentric orbits in Schwarzschild spacetime (cf.
right panel of Fig. 7.12), but is less pronounced, i.e. the number of whirls is smaller (only one
in Fig. 7.12).
Let us now consider the evolution of the coordinates t and φ along the geodesic, as governed
by Eqs. (11.101a) and (11.101d). Since r and θ are periodic functions of τ ′ , so are the terms
18
According to Ref. [223], the name “zoom-whirl” has been forged by Curt Cutler and Eric Poisson and may
have been suggested by Kip Thorne.
11.4 Timelike geodesics 415
Figure 11.8: Trajectory with respect to the Cartesian Boyer-Lindquist coordinates (x, y, z) [Eq. (11.73)] of a
bound timelike geodesic of reduced integrals of motion ε = 0.9, ℓ = 2m and q = 1.3 m2 , orbiting around a Kerr
black hole with a = 0.998 m. These parameters are the same as for the −R(r) plot in Fig. 11.6. [Figure generated
by the notebook D.5.8]
y/m
6
4 z/m
2 3
2
x/m
-6 -4 -2 2 4 6 1
-2 x/m
-6 -4 -2 2 4 6
-1
-4
-2
-6 -3
Figure 11.9: Projection in the (x, y) plane (equatorial plane; left panel) and the (x, z) plane (meridional plane;
right panel) of the bound timelike geodesic considered in Fig. 11.8. The grey disk depicts the black hole region.
On the left panel, there appears clearly that the minimal value of r is rp ≃ 2.2 m, while its maximal value is
ra ≃ 6.9 m. On the right panel, we check that θm ≤ θ ≤ π − θm with θm ∼ 60◦ , in agreement with the generic
behavior illustrated in Fig. 11.7. [Figure generated by the notebook D.5.8]
416 Geodesics in Kerr spacetime: generic and timelike cases
Figure 11.10: Spacetime diagram based on the Cartesian Boyer-Lindquist coordinates (t, x, y) [Eq. (11.73)]
depicting the bound timelike geodesic considered in Figs. 11.8 and 11.9. The black cylinder is the black hole event
horizon. [Figure generated by the notebook D.5.8]
T1 (r), T2 (θ), Φ1 (r) and Φ2 (θ) that appear in the right-hand side of Eqs. (11.101a) and (11.101d).
We can thus expand them in Fourier series:
+∞ +∞
′ ′
X X
T1 (r) = T̂1k eikΥr τ = ⟨T1 (r)⟩ + T̂1k eikΥr τ + c.c. (11.118a)
k=−∞ k=1
+∞ +∞
ikΥθ τ ′ ikΥθ τ ′
X X
T2 (θ) = T̂2k e = ⟨T2 (θ)⟩ + T̂2k e + c.c. (11.118b)
k=−∞ k=1
+∞ +∞
ikΥr τ ′ ′
X X
Φ1 (r) = Φ̂1k e = ⟨Φ1 (r)⟩ + Φ̂1k eikΥr τ + c.c. (11.118c)
k=−∞ k=1
+∞ +∞
ikΥθ τ ′ ′
X X
Φ2 (θ) = Φ̂2k e = ⟨Φ2 (θ)⟩ + Φ̂2k eikΥθ τ + c.c. , (11.118d)
k=−∞ k=1
where “c.c.“ stands for complex conjugate and ⟨T1 (r)⟩ = T̂10 , ⟨T2 (θ)⟩ = T̂20 , ⟨Φ1 (r)⟩ = Φ̂10
and ⟨Φ2 (θ)⟩ = Φ̂20 are the mean values of the functions T1 (r(τ ′ )), T2 (θ(τ ′ )), Φ1 (r(τ ′ )) and
11.4 Timelike geodesics 417
Φ2 (θ(τ ′ )) over one period:
Z Λr Z ra
1 2 T (r)
⟨T1 (r)⟩ = ′ ′
T1 (r(τ )) dτ = p1 dr (11.119a)
Λr 0 Λr rp R(r)
Z Λθ Z π/2
1 4 T (θ)
⟨T2 (θ)⟩ = ′ ′
T2 (θ(τ )) dτ = q2 dθ, (11.119b)
Λθ 0 Λθ θm Θ̃(θ)
with similar formulas for ⟨Φ1 (r)⟩ and ⟨Φ2 (θ)⟩.
We may then rewrite Eqs. (11.101a) and (11.101d) as
+∞ +∞
dt X
ikΥr τ ′
X
ikΥθ τ ′
= Γ + T̂1k e + c.c. + T̂2k e + c.c. (11.120a)
dτ ′ k=1 k=1
+∞ +∞
dφ X
ikΥr τ ′
X
ikΥθ τ ′
= Υφ + Φ̂1k e + c.c. + Φ̂2k e + c.c. , (11.120b)
dτ ′ k=1 k=1
with
Γ := ⟨T1 (r)⟩ + ⟨T2 (θ)⟩, (11.121)
and
Υφ := ⟨Φ1 (r)⟩ + ⟨Φ2 (θ)⟩. (11.122)
Γ and Υφ represent the constant (or average) parts of respectively dt/dτ ′ and dφ/dτ ′ , while
the oscillatory parts are the terms involved in the sums over k ≥ 1. Since Γ is the average value
of dt/dτ ′ , we can define the average angular frequencies with respect to Boyer-Lindquist
time by [173, 206]
Υr Υθ Υφ
Ωr := , Ωθ := , Ωφ := . (11.123)
Γ Γ Γ
We may summarize the above results as follows:
Property 11.27: periodicities of bound timelike orbits
The motion of a generic bound timelike geodesic in Kerr spacetime is periodic in r and θ
only in terms of the Mino time τ ′ , the corresponding angular frequencies being Υr and Υθ .
In terms of the Boyer-Lindquist time t, the motion in r or θ is not periodic. In this respect,
the “average frequencies” Ωr and Ωθ defined above are not true frequencies like Υr or Υθ .
Regarding the motion in φ, it is not periodic, neither in terms of the Mino time, nor in
terms of the Boyer-Lindquist time, so that both Υφ and Ωφ are average frequencies.
Quasi-Keplerian parametrization
Given a bound timelike geodesic L of Kerr spacetime, one may define its eccentricity e and
(dimensionless) semilatus rectum p by means of Keplerian-like formulas [447, 174, 461]:
pm pm
rp =: and ra =: , (11.124)
1+e 1−e
418 Geodesics in Kerr spacetime: generic and timelike cases
or equivalently
2ra rp ra − rp
p= and e= . (11.125)
m(ra + rp ) ra + rp
One may also introduce the inclination angle θinc by
π
θinc := − (sgn ℓ)θm . (11.126)
2
The sign of ℓ appears in this formula to enforce θinc ∈ [0, π/2] for prograde orbits (ℓ > 0) and
θinc ∈ [π/2, π] for retrograde orbits (ℓ < 0).
Instead of (ε, ℓ, q), a bound timelike geodesic can be parameterized19 by (p, e, θinc ). Actually,
there is a one-to-one correspondence between (ε, ℓ, q) and (p, e, θinc ): it is clear from formulas
(11.125) and (11.125) that (p, e, θinc ) are functions of (ε, ℓ, q), because (i) ra and rp are some
roots of the quartic polynomial R(r), whose coefficients are functions of (ε, ℓ, q) only (at fixed
Kerr parameters (m, a)) [cf. Eq. (11.105)] and (ii) θm in the right-hand side of expression (11.126)
for θinc is the function of (ε, ℓ, q) given by Eq. (11.112). Conversely, to express (ε, ℓ, q) in terms
of (p, e, θinc ), one shall write
R(rp ) = 0
R(ra ) = 0
Θ̃(θm ) = 0,
where rp and ra are the functions of (e, p) given by Eq. (11.124) and θm is the function of θinc
given by Eq. (11.126). The above system is then a system of 3 equations for the 3 unknowns
(ε, ℓ, q); see Appendix B of Ref. [447] or Appendix A of Ref. [174] for details.
Example 12: The geodesic L depicted in Figs. 11.8–11.10 has ε = 0.9, ℓ = 2m and q = 1.3 m2 , from
which we get rp = 2.175 m, ra = 6.853 m and θm = 1.060 rad = 60.75◦ . Then Eqs. (11.125) and
(11.126) yield p = 3.302, e = 0.518 and θinc = 29.26◦ .
The parametrization (p, e, θinc ) allows one to introduce along the geodesic L the radial
phase angle ψ(τ ) and the meridional phase angle χ(τ ) as monotonically increasing functions
of τ such that
pm
r(τ ) =: and θ(τ ) =: arccos (cos θm cos χ(τ )) . (11.127)
1 + e cos ψ(τ )
For numerical integration, ψ(τ ) and χ(τ ) are preferred to r(τ ) and θ(τ ) since the latter are
tricky at the turning points, where dr/dτ and dθ/dτ vanish and change sign [173].
11.5 Circular timelike orbits in the equatorial plane
11.5.1 Equations of motion in the equatorial plane
Let us focus on timelike geodesics confined to the equatorial plane: θ = π/2. They thus have
dθ/dτ ′ = 0 and Eq. (11.101c) leads to Θ̃(θ) = 0. Given expression (11.106) for Θ̃(θ), this implies
q=0. (11.128)
19
Some authors use x := cos θinc instead of θinc [461].
11.5 Circular timelike orbits in the equatorial plane 419
Remark 1: The above result follows as well from the analysis of the θ-motion performed in Sec. 11.3.7
and according to which the only possible value of the Cartan constant Q for a motion confined to the
equatorial plane is Q = 0, be the geodesic timelike or null. The converse is not true: there exist (timelike
or null) geodesics with Q = 0 outside the equatorial plane; they are asymptotically approaching the
equatorial from one side, except for the exceptional case L = 0 and a2 E 2 = a2 µ2 , for which they are
moving with a constant value of θ (cf. Sec. 11.3.7).
When θ = π/2, we have sin2 θ = 1, ρ2 = r2 and dτ ′ = r−2 dτ [cf. Eq. (11.107)], so that the
system (11.101) reduces to
dt 1 2am
= 2 2 2
ε(r + a ) + (aε − ℓ) (11.129a)
dτ r − 2mr + a2 r
2
dr
+ V(r) = 0 (11.129b)
dτ
dφ 1 2m 2amε
= 2 ℓ 1− + , (11.129c)
dτ r − 2mr + a2 r r
with
2m ℓ2 + a2 (1 − ε2 ) 2m(ℓ − aε)2
V(r) := 1 − ε2 − + − . (11.130)
r r2 r3
Note that V(r) = −R(r)/r4 = −R(r)/(µ2 r4 ) [cf. Eqs. (11.105) and (11.95)].
Remark 2: At the Schwarzschild limit (a = 0) we recover the differential equations of Chap. 7, namely
Eq. (11.129a) reduces to Eq. (7.34), Eq. (11.129b) to Eq. (7.24) and Eq. (11.129c) to Eq. (7.35). In particular
we have V(r) = 1 − ε2 + 2Vℓ (r), where Vℓ (r) is the effective potential (7.25). Note however that when
a ̸= 0, V(r) cannot be considered as an effective potential for the radial motion, because it depends
on ε, in addition to ℓ, even if we subtract from it the constant term 1 − ε2 . This is why we did not
bother to add a 1/2 factor in Eq. (11.129b), as we did for Eq. (7.24) to make it match the first integral of a
1-dimensional motion in a potential well.
Historical note : The potential (11.130), which is ruling the radial motion of timelike geodesics confined
to the equatorial plane, has been exhibited first by Robert Boyer and Richard Lindquist in 1966 (published
1967) [72], in the form r3 V(r), cf. their Eq. (5.6), taking into account that they were using −a instead of
a (cf. the historical note on p. 340).
11.5.2 Equatorial circular orbits: definition and existence
The simplest equatorial geodesics are of course the circular ones: a circular orbit in the
equatorial plane is a geodesic that obeys
∀τ ∈ R, θ(τ ) = π/2 and r(τ ) = r0 = const. (11.131)
The constant r0 is called the radius of the circular orbit. One has then dr/dτ = 0, so that
Eq. (11.129b) implies
V(r0 ) = 0 . (11.132)
420 Geodesics in Kerr spacetime: generic and timelike cases
The above condition is not sufficient to single out a geodesic worldline: there exist worldlines
with constant r = r0 that obey the whole system (11.129) (which includes (11.132)) but that are
not solution of the geodesic equation (11.1) (see Ref. [489] for concrete examples). One must
add a second condition, which is obtained as follows. For any timelike equatorial geodesic (not
necessarily circular), the following relation must hold:
dr d2 r
′
2 + V (r) = 0.
dτ dτ 2
It is obtained by differentiating Eq. (11.129b) with respect to τ . If the geodesic is not circular, at
any point where r is not stationary (i.e. excluding r-turning points), we have dr/dτ ̸= 0 and
the above relation implies
′ d2 r
V (r) = −2 2 .
dτ
Since in the vicinity of any circular geodesic there exist non-circular ones (those with small
eccentricity e, as defined by Eq. (11.125)), by continuity, we shall ask that this relation holds for
circular geodesics as well. For the latter ones, r = r0 and d2 r/dτ 2 = 0, so that it simplifies to
V ′ (r0 ) = 0 . (11.133)
One can verify20 that this relation, in conjunction with Eq. (11.132), is sufficient to eliminate
the non-geodesic circular worldlines.
To summarize, circular timelike geodesics in the equatorial plane are obtained by solving
the system (11.132)-(11.133). Given expression (11.130) for V, this system takes the form
(1 − ε2 )r03 − 2mr02 + ℓ2 + a2 (1 − ε2 ) r0 − 2m(ℓ − aε)2 = 0 (11.134a)
2
(11.134b)
2 2 2
2
mr0 − ℓ + a (1 − ε ) r0 + 3m(ℓ − aε) = 0.
This is a system of 2 nonlinear equations for 3 unknowns: r0 , ε and ℓ. We thus expect a
1-parameter family of solutions. It is convenient to choose r0 as the parameter. We have thus
to solve (11.134) for (ε, ℓ). The change of variable
ℓ̃ := ℓ − aε (11.135)
turns (11.134) into the system
(11.136a)
3 2
r0 ε = mℓ̃2 + r02 (r0 − m)
2ar0 εℓ̃ = (3m − r0 )ℓ̃2 + r0 (mr0 − a2 ). (11.136b)
Let us consider the square of Eq. (11.136b):
h i2
2 2 2 2 2 2
4a r0 ε ℓ̃ = (3m − r0 )ℓ̃ + r0 (mr0 − a ) , (11.137)
and substitute (11.136a) for ε2 in it; we get
r0 (r0 − 3m)2 − 4a2 m ℓ̃4 + 2r02 (3m2 − a2 )r0 − m(r02 + a2 ) ℓ̃2 + r03 (mr0 − a2 )2 = 0.
This is a quadratic equation for X := ℓ̃2 . Its discriminant turns out to have a simple form:
∆ = 16a2 mr03 (r02 − 2mr0 + a2 )2 .
It follows immediately that ∆ ≥ 0 ⇐⇒ r0 ≥ 0. Hence, there is no solution for r0 < 0:
20
See Ref. [489] for details.
11.5 Circular timelike orbits in the equatorial plane 421
Property 11.28
There does not exist any equatorial circular timelike orbit in the region r < 0 of Kerr
spacetime.
Remark 3: For r0 < 0 with |r0 | ≫ m, the above result is not surprising since we have seen in Sec. 10.2.2
that under these conditions, the Kerr metric appears as a Schwarzschild metric with a negative mass.
The asymptotic gravitational field is then repulsive and certainly does not admit any circular orbit.
In the region r < 0 and |r| small, the above argument does not apply. However, the results obtained
in Sec. 11.3.8 show that if a circular orbit would exist there, it would necessarily have |E| > µ [cf.
Eq. (11.96)], i.e. would be unbound.
The case r0 = 0 is excluded as well, since in the equatorial plane, it would correspond to the
curvature singularity. In what follows, we therefore assume
r0 > 0 . (11.138)
√ √
We have then ∆ = 4ar0 mr0 |r02 − 2mr0 + a2 | and the solutions of the quadratic equation
are21
√
(a2 − 3m2 )r02 + mr0 (r02 + a2 ) ∓ 2a mr0 (r02 − 2mr0 + a2 )
ℓ̃2± = r0 . (11.139)
r0 (r0 − 3m)2 − 4a2 m
The denominator of the right-hand side can be written as
2 1 h
2 2 √ 2
i 1
r0 (r0 − 3m) − 4a m = (r0 (r0 − 3m)) − (2a mr0 ) = A+ A− ,
r0 r0
with
√
A± := r0 (r0 − 3m) ± 2a mr0 .
√
In the numerator of (11.139), we may use the identity ∓2a mr0 = A∓ − r0 (r0 − 3m) and get,
after simplification,
r02 r2
ℓ̃2± = A∓ (r02 − 2mr0 + a2 ) − A+ A− = 0 r02 − 2mr0 + a2 − A± .
A+ A− A±
√ √
Using r02 − 2mr0 + a2 − A± = a2 ∓ 2a mr0 + mr0 = (a ∓ mr0 )2 , we obtain
√
r02 (a ∓ mr0 )2
ℓ̃2± = 2 √ . (11.140)
r0 − 3mr0 ± 2a mr0
√
Since obviously ℓ̃2± ≥ 0, this expression gives birth to the constraint r02 − 3mr0 ± 2a mr0 > 0,
which is equivalent to
3/2 1/2 √
r0 − 3mr0 ± 2a m > 0. (11.141)
21
For future convenience, we have chosen the sign ∓ in the numerator to be the opposite of the label ± in ℓ̃2± .
422 Geodesics in Kerr spacetime: generic and timelike cases
1/2
The left hand-side of this inequality is a cubic polynomial in x := r0 and determining
its sign amounts to compute its roots. Fortunately, this corresponds
√ to a depressed cubic
equation x + px + q = 0, with p := −3m and q := ±2a m, the discriminant of which is
3
−(4p3 + 27q 2 ) = 108m(m2 − a2 ) ≥ 0. The roots (xk )k∈{0,1,2} are then all real√ and are given by
Viète’s formula (8.22). They obey x0 + x1 + x2 = 0 and x0 x1 x2 = −q = ∓2a m, from which
we deduce that for ± = + in Eq. (11.141), i.e. for ℓ̃2+ , there are two positive roots, which are x0
and x2 as given by Eq. (8.22), while for ± = −, i.e. for ℓ̃2− , there is a single positive root, which
is x0 as given by Eq. (8.22). Going back to r0 = x2 , with the roots x0 and x2 given by Viète’s
formula (8.22), we get the following constraints for circular orbits in the equatorial plane:
for ℓ̃2+ : ∗
0 < r0 < rph or +
r0 > rph (11.142a)
for ℓ̃2− : −
r0 > rph , (11.142b)
with
1 a
±
rph := 4m cos 2
arccos ∓ (11.143)
3 m
and
1 a 4π
∗
rph := 4m cos 2
arccos − + . (11.144)
3 m 3
The index “ph” stands for photon, because we shall see in Sec. 11.5.5 and in Sec. 12.3 that these
radii actually correspond to photon orbits (circular null geodesics). rph
±
and rph
∗
are plotted as
functions of a in Fig. 11.11 (green curves). Note that m ≤ rph ≤ 3m and 3m ≤ rph ≤ 4m, with
+ −
lim r± = 3m, lim r+ = m, lim r− = 4m. (11.145)
a→0 ph a→m ph a→m ph
Besides, 0 ≤ rph
∗
≤ m, with
∗
lim rph = 0, ∗
lim rph = m. (11.146)
a→0 a→m
Furthermore, one can check that
±
rph ≥ r+ (11.147)
and
∗
0 ≤ rph ≤ r− , (11.148)
√ √
where r+ := m2 + m2 − a2 and r− := m2 − m2 − a2 [Eq. (10.3)] are the radii of respectively
the event horizon H and the inner horizon Hin . Regarding the upper bound in Eq. (11.148),
one can see from Fig. 11.11 that rph ∗
is always very close to r− . The maximum discrepancy is
r− − rph ≃ 0.032 m and is achieved for a ≃ 0.9 m (cf. the notebook D.5.9). We conclude that
∗
the first permitted range for circular orbits in (11.142a) lies entirely in the region MIII of Kerr
spacetime, while the second range in (11.142a) lies in entirely in MI . Regarding the orbits for
ℓ̃2− , the range (11.142b) lies in MI as well.
Substituting expression (11.140) for ℓ̃2 in Eq. (11.136a), we get, after simplification,
2 √ 2
r − 2mr 0 ± a mr 0
ε2± = 2 0 2 √ . (11.149)
r0 r0 − 3mr0 ± 2a mr0
11.5 Circular timelike orbits in the equatorial plane 423
4.0
3.5
3.0
2.5
rph+
r0 /m
2.0 rph−
rph∗
rergo
1.5 r+
r−
1.0
0.5
0.0
0.0 0.2 0.4 0.6 0.8 1.0
a/m
Figure 11.11: Domain of existence of circular equatorial timelike orbits in the (a, r0 ) plane: orbits of the
(ε+ , ℓ+ ) family exist in both the light and dark grey regions, while orbits of the (ε− , ℓ− ) family exist only in
the dark grey region. The orange line marks the location of the outer ergosphere in the equatorial plane, which
is rergo = rE + (π/2) = 2m [cf. Eq. (10.19b)], the black curve corresponds to the black hole event horizon H
(r = r+ ) and the maroon one to the inner horizon Hin (r = r− ). [Figure generated by the notebook D.5.9]
The ±’s in Eqs. (11.140) and (11.149) have to be consistent. In other words, we have two pairs
of solutions for (ε2 , ℓ̃2 ), namely (ε2+ , ℓ̃2+ ) and (ε2− , ℓ̃2− ). A priori,
q ε+ leads to two values for
2
ε, namely ± ε2+ , and ℓ̃2+ leads to two values for ℓ̃, namely ± ℓ̃2+ , so that the pair (ε2+ , ℓ̃2+ )
p
would generate four solutions for (ε, ℓ̃), and similarly the pair (ε2− , ℓ̃2− ) would generate four
extra solutions for (ε, ℓ̃), leading to a total of eight solutions. However, by construction these
solutions satisfy the “squared” equation (11.137) but not all of them satisfy the original equation
(11.136b). To see this, let us consider the following square roots of Eqs. (11.149) and (11.140):
√
r02 − 2mr0 ± a mr0
ε± = p 2 √ , (11.150)
r0 r0 − 3mr0 ± 2a mr0
√ √
r0 (a ∓
mr0 ) r0 (± mr0 − a)
ℓ̃± = − p 2 √ =p 2 √ (11.151)
r0 − 3mr0 ± 2a mr0 r0 − 3mr0 ± 2a mr0
and write the eight solutions of (11.137) as
(ε, ℓ̃) = ϵ1 ε± , ϵ2 ℓ̃± , with ϵ1 = ±1 and ϵ2 = ±1.
By a direct calculation, we get
√ √
2 2(1 − ϵ1 ϵ2 )ar0 (a ∓ mr0 )(r02 − 2mr0 ± a mr0 )
2
2ar0 εℓ̃ − (3m − r0 )ℓ̃ − r0 (mr0 − a ) = √ .
r02 − 3mr0 ± 2a mr0
424 Geodesics in Kerr spacetime: generic and timelike cases
Hence Eq. (11.136b) is fulfilled iff 1 − ϵ1 ϵ2 = 0, i.e. iff ϵ1 ϵ2 = 1. This reduces the number of
possible solutions from eight to four:
(ε, ℓ̃) = (ε+ , ℓ̃+ ) or (−ε+ , −ℓ̃+ ) or (ε− , ℓ̃− ) or (−ε− , −ℓ̃− ). (11.152)
A further reduction of the number of solutions is provided by the future-directed condition
(11.66). Since R(r0 ) = −µ2 r04 V(r0 ) = 0 by virtue of Eq. (11.132), for circular equatorial orbits,
Eq. (11.66) reduces to
1 a
ε− 2 ℓ > 0,
∆0 r 0 + a2
where ∆0 := r02 − 2mr0 + a2 = (r0 − r+ )(r0 − r− ). Now, we have observed above that
circular orbits lie either in MI or MIII , where ∆0 > 0. Therefore, we can further simplify the
future-directed condition to
a
ε− 2 ℓ > 0.
r0 + a2
Once reexpressed in terms of ℓ̃ = ℓ − aε, it is equivalent to
r02 ε − aℓ̃ > 0. (11.153)
Let us check each of the four solutions (11.152):
r0 ∆0
r02 ε+ − aℓ̃+ = p 2 √ >0
r0 − 3mr0 + 2a mr0
r02 (−ε+ ) − a(−ℓ̃+ ) = −(r02 ε+ − aℓ̃+ ) < 0
r0 ∆0
r02 ε− − aℓ̃− = p 2 √ >0
r0 − 3mr0 − 2a mr0
r02 (−ε− ) − a(−ℓ̃− ) = −(r02 ε− − aℓ̃− ) < 0.
We conclude that only two solutions remain:
(ε, ℓ̃) = (ε+ , ℓ̃+ ) or (ε− , ℓ̃− ). (11.154)
Given that ℓ = ℓ̃ + aε [Eq. (11.135)] and expressions (11.150) and (11.151) for respectively ε±
and ℓ̃± , these two solutions can be reexpressed in terms of (ε, ℓ) as
(ε, ℓ) = (ε+ , ℓ+ ) or (ε− , ℓ− ) , (11.155)
where
√
m r02 + a2 ∓ 2a mr0
r
ℓ± = ± p √ . (11.156)
r0 r02 − 3mr0 ± 2a mr0
The quantities ε± and ℓ± are plotted in terms of r0 in Figs. 11.12 and 11.13. Orbits with ℓ > 0
(resp. ℓ < 0) are called prograde (resp. retrograde). We have immediately ℓ− < 0, while
√
the sign of ℓ+ is that of of r02 + a2 − 2a mr0 . Outside the event horizon, i.e. in MI , we have
√
r0 > m, which implies r02 + a2 − 2a mr0 > r02 + a2 − 2ar0 = (r0 − a)2 > 0. Hence
+
r0 > rph =⇒ ℓ+ > 0. (11.157)
But for r0 < rph∗
, i.e. in region MIII , we may have ℓ+ < 0 (cf. Fig. 11.13). In view of this and
the result (11.142), we introduce the following nomenclature:
11.5 Circular timelike orbits in the equatorial plane 425
1.02 a = 0.0 m
2 a = 0.5 m
a = 0.9 m
a = 0.98 m
1.00 a = 1.0 m
1
0.98
ε±
ε±
0 0.96
1
a = 0.0 m
a = 0.5 m 0.94
a = 0.9 m
a = 0.98 m 0.92
a = 1.0 m
2
0 1 2 3 4 5 6 7 8 4 5 6 7 8 9 10
r0 /m r0 /m
Figure 11.12: Specific conserved energy ε = ε+ (solid curves) or ε = ε− (dashed curves) along circular timelike
orbits in the equatorial plane, as a function of the orbital radius r0 [Eq. (11.150)], for selected values of the Kerr
spin parameter a. Curves in the inner region (r0 < m) terminate by a vertical asymptote at r = rph ∗
(a) given by
±
Eq. (11.144), while the other curves start along a vertical asymptote at r = rph (a) given by Eq. (11.143). Dots on
ε+ curves and open circles on ε− ones mark the ISCO: all configurations on the left of these points are unstable.
The right panel is a zoom on the region 4 m ≤ r0 ≤ 10 m. Note that the red curve (a = 0) is the same as in
Fig. 7.8. [Figure generated by the notebook D.5.9]
10 a = 0.0 m
4
a = 0.5 m 3
a = 0.9 m
a = 0.98 m a = 0.0 m
5 a = 1.0 m 2 a = 0.5 m
a = 0.9 m
1 a = 0.98 m
a = 1.0 m
` ± /m
` ± /m
0 0
1
2
5
3
4
10
0 1 2 3 4 5 6 7 8 4 5 6 7 8 9 10
r0 /m r0 /m
Figure 11.13: Specific conserved angular momentum ℓ = ℓ+ (solid curves) or ℓ = ℓ− (dashed curves) along
circular timelike orbits in the equatorial plane, as a function of the orbital radius r0 [Eq. (11.156)], for selected
values of the Kerr spin parameter a. Curves in the inner region (r0 < m) terminate by a vertical asymptote at
±
∗
r = rph (a) given by Eq. (11.144), while the other curves start along a vertical asymptote at r = rph (a) given by
Eq. (11.143). Dots on ℓ+ curves and open circles on ℓ− ones mark the ISCO: all configurations on the left of these
points are unstable. The right panel is a zoom on the region 4 m ≤ r0 ≤ 10 m. [Figure generated by the notebook
D.5.9]
426 Geodesics in Kerr spacetime: generic and timelike cases
1.2
1.0
0.8
0.6
0.4
ε
0.2
0.0
a = 0.0 m
0.2 a = 0.5 m
a = 0.9 m
a = 0.98 m
a = 1.0 m
0.4
4 2 0 2 4
`/m
Figure 11.14: Circular timelike orbits in the (ℓ, ε) plane, ℓ being the specific conserved angular momentum
and ε the specific conserved energy. Solid (resp. dotted) curves correspond to stable (unstable) orbits. Retrograde
outer circular orbits are on the left side, inner circular orbits in the middle and prograde outer circular orbits on
the right side. The cusps in the left and right curves correspond to ISCOs. [Figure generated by the notebook D.5.9]
• orbits of the (ε+ , ℓ+ ) family with r0 > rph
+
are called prograde outer circular orbits;
• orbits of the (ε+ , ℓ+ ) family with 0 < r0 < rph
∗
are called inner circular orbits;
• all orbits of the (ε− , ℓ− ) family are called retrograde outer circular orbits.
We note from Figs. 11.12 and 11.13, or from formulas (11.150) and (11.156), that both ε and
ℓ are diverging at the boundaries of the various domains of existence of circular orbits. We
shall comment further on this behavior at the end of Sec. 11.5.5.
Figure 11.14 shows ε in terms of ℓ for the three families of circular orbits, with the indication
of the stability of the various branches, as determined in the next section.
11.5.3 Stability of circular timelike orbits
Latitudinal stability
A natural question regarding the stability of equatorial circular orbits is whether these orbits
are stably confined into the equatorial plane θ = π/2. The answer is very simple:
Property 11.29: latitudinal stability of equatorial circular timelike orbits
All equatorial circular timelike orbits are stable with respect to any perturbation away
from the equatorial plane.
11.5 Circular timelike orbits in the equatorial plane 427
r 0 = 2m
0.02 r 0 = 3m
0.01
0.00
V(r)
0.01
0.02
0.03
0.04
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
r/m
Figure 11.15: Function V(r), as defined by Eq. (11.130), with a = 0.9 m, ε = ε+ (r0 ) [Eq. (11.150)], ℓ = ℓ+ (r0 )
[Eq. (11.156)] for r0 = 2 m (red curve) and r0 = 3 m (blue curve). By definition of ε+ and ℓ+ , one has both
V(r0 ) = 0 and V ′ (r0 ) = 0. [Figure generated by the notebook D.5.9]
Proof. All equatorial orbits have a vanishing Carter constant Q [cf. Eq. (11.128)] and we have
seen in Sec. 11.3.7 that a Q = 0 geodesic stays stably at θ = π/2 iff a2 (E 2 − µ2 ) ≤ L2 , i.e. iff
a2 (ε2 − 1) ≤ ℓ2 (11.158)
for a timelike geodesic. Now, for a circular orbit of radius r0 , Eq. (11.134b) implies
ℓ2 − a2 (ε2 − 1) = mr0 + 3m(ℓ − aε)2 /r0 > 0,
where the inequality follows from r0 > 0 [Eq. (11.138)]. Hence the stability condition (11.158)
is always fulfilled.
Radial stability
Let us now investigate the stability with respect to a radial perturbation. A circular orbit at
r = r0 obeys both V(r0 ) = 0 and V ′ (r0 ) = 0 [Eqs. (11.132) and (11.133)]. The latter property
means that it corresponds to a local extremum of the function V. This extremum can be either
a local maximum (red curve in Fig. 11.15) or a local minimum (blue curve in Fig. 11.15). Now,
in view of Eq. (11.129b) rewritten as V(r) = −(dr/dτ )2 , any motion in the equatorial plane
must obey V(r) ≤ 0. If the circular orbit corresponds to a local minimum, then for r close to
r0 , but distinct from it, V(r) > 0, since V(r0 ) = 0 (cf. the blue curve in Fig. 11.15). This means
that no geodesic motion with the same values of the conserved quantities ε and ℓ is possible
in the vicinity of r0 except for precisely r = r0 . We conclude that the circular orbit at r0 is
stable in that case. On the contrary, when r0 is a local maximum of V, one has V(r) < 0 for r
close to r0 , but distinct from it (cf. the red curve in Fig. 11.15). Motion away from r0 is then
possible for the same values of ε and ℓ; we conclude that the circular orbit is unstable in that
case. Assuming that V ′′ (r0 ) ̸= 0, a local minimum (resp. maximum) is equivalent to V ′′ (r0 ) > 0
(resp. V ′′ (r0 ) < 0). We have then
428 Geodesics in Kerr spacetime: generic and timelike cases
Property 11.30: criterion for radial stability of circular orbits
The circular orbit of radius r0 is stable ⇐⇒ V ′′ (r0 ) > 0. (11.159)
From the definition (11.130) of V(r) and the relation ℓ = ℓ̃ + aε [Eq. (11.135)], we get
4m 6(ℓ̃2 + 2aεℓ̃ + a2 ) 24mℓ̃2
V ′′ (r0 ) = − + − .
r03 r04 r05
Substituting Eq. (11.136b) for 2aεℓ̃, we get a simple expression, involving only ℓ̃:
!
2
2m ℓ̃
V ′′ (r0 ) = 3 1 − 3 2 .
r0 r0
For a circular orbit, ℓ̃ is the function (11.151) of r0 . Using it, we get
√
′′ 2m(r02 − 6mr0 ± 8a mr0 − 3a2 )
V (r0 ) = √ .
r03 (r02 − 3mr0 ± 2a mr0 )
Since the denominator of the right-hand-side expression is always positive, by virtue of
Eqs. (11.138) and (11.141), the sign of V ′′ (r0 ) is entirely determined by the numerator, so
that we may rewrite (11.159) as
√
stable orbit (radius r0 ) ⇐⇒ r02 − 6mr0 ± 8a mr0 − 3a2 > 0
⇐⇒ P (x) := x4 − 6x2 ± 8āx − 3ā2 > 0, (11.160)
where ± is + (resp. −) for prograde (resp. retrograde) orbits and we have introduced the
dimensionless variables r
r0 a
x := and ā := . (11.161)
m m
The problem amounts to finding the range of x where the quartic polynomial P (x) is positive.
This requires computing the roots of P . We shall do it via Ferrari’s method. The first step is to
introduce a parameter Z1 and rewrite P (x) as
P (x) = (x2 − Z1 )2 − T (x), with T (x) := 2(3 − Z1 )x2 ∓ 8āx + Z12 + 3ā2 . (11.162)
The above expression is an identity, which holds for any value of Z1 ; the core of Ferrari’s
method it to find Z1 so that the quadratic
p polynomial T (x) has a double root, x0 say. We will
have then T (x) = S(x) , with S(x) := 2(3 − Z1 )(x−x0 ), and P (x) = (x −Z1 )2 −S(x)2 =
2 2
(x2 − Z1 − S(x))(x2 − Z1 + S(x)), so that
P (x) = 0 ⇐⇒ x2 − Z1 − S(x) = 0 or x2 − Z1 + S(x) = 0. (11.163)
In other words, the four (possibly complex) solutions of the quartic equation P (x) = 0 are
the two solutions of the quadratic equation x2 − Z1 − S(x) = 0 plus the two solutions of
11.5 Circular timelike orbits in the equatorial plane 429
3.0
2.5
2.0
Z1
1.5
1.0
0.0 0.2 0.4 0.6 0.8 1.0
a/m
Figure 11.16: Function Z1 , as defined by Eq. (11.165). [Figure generated by the notebook D.5.9]
x2 − Z1 + S(x) = 0. A necessary and sufficient condition for T (x) to have a double root is
that its discriminant vanishes, which is equivalent to
Z13 − 3Z12 + 3ā2 Z1 − ā2 = 0. (11.164)
We have thus to solve a cubic equation in Z1 . Let us reduce it to a depressed cubic equation (i.e.
an equation free of any square term) via the change of variable Z1 =: Z + 1:
Z 3 + 3(ā2 − 1)Z + 2(ā2 − 1) = 0.
The discriminant of this cubic equation is ∆ = −(4p3 + 27q 2 ), where p := 3(ā2 − 1) and
q := 2(ā2 − 1). We get ∆ = −108ā2 (1 − ā2 )2 . Hence ∆ < 0: there exist only one real solution.
It is given by Cardano’s formula:
v ! v !
r r
√ √
u u
u1
3 −∆ 3 1
u −∆ 3 3
√3
Z= t −q + + t −q − = 1 − ā 2 1 + ā + 1 − ā .
2 27 2 27
Hence √ √ √
(11.165)
3 3 3
Z1 = 1 + 1− ā2 1 + ā + 1 − ā .
Z1 is plotted in terms of ā = a/m in Fig. 11.16. In particular, we notice that 1 ≤ Z1 ≤ 3. When
Z1 takes the value (11.165), the double root of T (x) is x0 = ±2a/(3 − Z1 ), where the ± is the
opposite of the ∓ in the definition (11.162) of T (x), and therefore indicates which family of
circular orbits, among (ε+ , ℓ+ ) or (ε− , ℓ− ), is considered. We have then
√
p 2 2ā
S(x) = 2(3 − Z1 ) x ∓ √ .
3 − Z1
√
When ā → 0, the ratio ā/ 3 − Z1 is of the undetermined type “0/0”. We can rearrange it by
noticing the identity 8ā2 /(3 − Z1 ) = Z12 + 3ā2 , which is a direct consequence of Eq. (11.164).
We then write
q
S(x) = 2(3 − Z1 ) x ∓ Z2 , with Z2 := Z12 + 3ā2 .
p
430 Geodesics in Kerr spacetime: generic and timelike cases
The solutions of P (x) = 0 are obtained by solving the two quadratic equations [cf. Eq. (11.163)]
(11.166a)
p
x2 − 2(3 − Z1 ) x − Z1 ± Z2 = 0
(11.166b)
p
x2 + 2(3 − Z1 ) x − Z1 ∓ Z2 = 0.
Moreover, physically acceptable solutions must obey x > 0 (recall that x := r0 /m). The
p
discriminant of (11.166a) is ∆ = 2(3 + Z1 ∓ Z2 ). It is non-negative only when ∓ = +, i.e. for
orbits in the (ϵ− , ℓ− ) family. The positive solution of (11.166a) is then
1 p p
x− = √ 3 + Z1 + 2Z2 + 3 − Z1 .
2
On the other side, the discriminant of (11.166b) is ∆ = 2(3 + Z1 ± 2Z2 ). It is non-negative only
when ± = +, i.e. for orbits in the (ϵ+ , ℓ+ ) family. The positive solution of (11.166b) is then
1 p p
x+ = √ 3 + Z1 + 2Z2 − 3 − Z1 .
2
We conclude that the quartic polynomial P (x) has a single positive root, which is
1 p p
x± = √ 3 + Z1 + 2Z2 ∓ 3 − Z1 .
2
Going back to r0 = mx2 , we get the single solution of V ′′ (r0 ) = 0 on (0, +∞):
h i
(11.167)
±
p
rISCO = m 3 + Z2 ∓ (3 − Z1 )(3 + Z1 + 2Z2 ) ,
√
3
√ √ q
Z1 := 1 + 1 − ā2 3 1 + ā + 3 1 − ā ; Z2 := Z12 + 3ā2 ; ā := a/m.
where ± indicates which family among (ε+ , ℓ+ ) and (ε− , ℓ− ) is considered and ISCO stands
for innermost stable circular orbit. Indeed, rISCO
±
being the unique zero of r02 − 6mr0 ±
√
8a mr0 − 3a , (11.160) is equivalent to
2
Property 11.31: stability of circular orbits
A circular orbit of radius r0 is radially stable ⇐⇒ ±
r0 > rISCO . (11.168)
Note that rISCO
±
/m, as given by Eq. (11.167), is a function of ā := a/m only; this function is
depicted in Fig. 11.17. We notice immediately from it that
∗
rph +
< rISCO (11.169)
and conclude that all the inner circular orbits are unstable. We also see on Fig. 11.17 that in
the limit a = 0, rISCO
+ −
= rISCO = 6m, i.e. we recover the Schwarzschild ISCO discussed in
Sec. 7.3.3.
11.5 Circular timelike orbits in the equatorial plane 431
8
rph+
rph−
rph∗
6 rergo
r+
r −+
r0 /m
rISCO
rISCO
−
4 rmb+
rmb−
0
0.0 0.2 0.4 0.6 0.8 1.0
a/m
Figure 11.17: Various critical radii for circular orbits as functions of the Kerr spin parameter a. The red curves
±
correspond to the innermost stable circular orbit rISCO [Eq. (11.167)], the light brown ones to the marginally
±
bound circular orbit rmb [Eq. (11.186)], while the other curves are the same as in Fig. 11.11. [Figure generated by
the notebook D.5.9]
The function ±
rISCO (ā) is easily inverted by solving the quadratic equation (rISCO
±
)2 −
6mrISCO ± 8a mrISCO − 3a2 = 0 for a [cf. Eq. (11.160)]; one gets
±
p ±
r r !
± ±
m rISCO rISCO
a=± 4− 3 −2 , (11.170)
3 m m
where the overall factor ± must be the same as in22 rISCO
±
.
Remark 4: The ISCO is also called marginally stable circular orbit by some authors (e.g. [42]), rISCO
is then denoted by rms .
Example 13: For a = 0.9 m, Eq. (11.167) yields rISCO
+
= 2.32088 m. Accordingly, the prograde circular
orbit at r0 = 3 m is stable, while that at r0 = 2m is unstable, in agreement with the plot of V(r) in
Fig. 11.15.
An interesting property of the ISCO is that it corresponds to extrema of ε± (r0 ) and ℓ± (r0 ):
22 −
Note that for ± = −, formula (11.170) still yields a ≥ 0, since rISCO ≥ 6m.
432 Geodesics in Kerr spacetime: generic and timelike cases
Property 11.32: characterization of the ISCO
Among all prograde outer circular orbits, the ISCO is that for which the functions ε+ (r0 )
and ℓ+ (r0 ) are minimal. Similarly, among all retrograde outer circular orbits, the ISCO is
that for which the function ε− (r0 ) is minimal and the function ℓ− (r0 ) is maximal.
Proof. In what precedes, we have considered V, defined by Eq. (11.130), as a function of r only.
Let us consider it instead as a function of (r, ε, ℓ): V = V(r, ε, ℓ). What we have denoted by
V ′ (r) is then ∂V/∂r and Eqs. (11.132) and (11.133) can be rewritten as
V (r0 , ε± (r0 ), ℓ± (r0 )) = 0 (11.171a)
∂V
(r0 , ε± (r0 ), ℓ± (r0 )) = 0. (11.171b)
∂r
These equations are valid for any circular orbit, i.e. any value of r0 . Let us take the derivative
of Eq. (11.171a) with respect to r0 . Using the chain rule, we get
∂V ∂V ∂V
+ ε′± (r0 ) + ℓ′± (r0 ) = 0,
∂r ∂ϵ ∂ℓ
| {z }0 0 0
=0
where |0 means that the quantity is evaluated at (r, ε, ℓ) = (r0 , ε± (r0 ), ℓ± (r0 )) and the
vanishing of the first term results from Eq. (11.171b). Similarly, deriving Eq. (11.171b) with
respect to r0 yields
∂ 2V ∂ 2V ∂ 2V
+ ε′± (r0 ) + ℓ′± (r0 ) = 0.
∂r2 0 ∂ϵ∂r 0 ∂ℓ∂r 0
Now, at the ISCO, one has precisely ∂ 2 V/∂r2 |0 = 0 (V ′′ (r0 ) = 0 in the preceding notations).
Hence, at the ISCO, the following two equations must hold:
∂V ε′ (r0 ) + ∂V ℓ′ (r0 ) = 0
∂ϵ 0 ± ∂ℓ 0 ±
(11.172)
2
∂ V ε′± (r0 ) + ∂ 2 V ℓ′± (r0 ) = 0
∂ϵ∂r ∂ℓ∂r
0 0
This constitutes a linear homogeneous system for the two unknowns ε′± (r0 ), ℓ′± (r0 ) . Since
there are no obvious reason for its determinant to vanish, we deduce that the only solution is
ε′± (r0 ), ℓ′± (r0 ) = (0, 0). Hence the ISCO realizes an extremum of both ε± (r0 ) and ℓ± (r0 ). A
quick look at Figs. 11.12 and 11.13, especially their bottom panels, enables us to conclude that
the extremum is a minimum for ε± (r0 ) and ℓ+ (r0 ) and a maximum for ℓ− (r0 ).
Property 11.32 explains why the ISCO is located at the cusp in the (ℓ, ε) curves shown in
Fig. 11.14: this corresponds to an extremum of both ε(r0 ) and ℓ(r0 ). Property 11.32 has also a
particular astrophysical significance, to be discussed in Sec. 11.5.4.
Summary
11.5 Circular timelike orbits in the equatorial plane 433
Property 11.33: existence and stability of equatorial circular timelike orbits
Equatorial circular timelike orbits exist only in the region r > 0 of Kerr spacetime. They
are all stable with respect to perturbations away from the equatorial plane. They are of
three kinds:
1. The prograde outer circular orbits: they are located outside the black hole event
horizon (i.e. in region MI ), having a Boyer-Lindquist radius r0 ranging from rph +
=
4m cos2 [arccos(−a/m)/3] [Eq. (11.143)] to +∞; the minimal radius rph +
decreases
with a monotonically from 3m (a = 0) to m (a = m). These orbits are radially
unstable for rph+ +
< r0 ≤ rISCO and stable for r0 > rISCO
+
, where rISCO
+
is given by
Eq. (11.167), decreasing monotonically from 6m (a = 0) to m (a = m). Their specific
conserved energy and angular momentum are ε = ε+ (r0 ) > 0 and ℓ = ℓ+ (r0 ) > 0,
with ε+ (r0 ) given by Eq. (11.150) and ℓ+ (r0 ) by Eq. (11.156), both being plotted as
the solid curves in the region r0 ≥ m of Figs. 11.12 and 11.13. The functions ε+ (r0 )
and ℓ+ (r0 ) are minimal at the ISCO.
2. The retrograde outer circular orbits: they are located outside the black hole event
horizon (i.e. in region MI ), having a Boyer-Lindquist radius r0 ranging from rph −
=
4m cos [arccos(a/m)/3] [Eq. (11.143)] to +∞; the minimal radius rph increases with
2 −
a monotonically from 3m (a = 0) to 4m (a = m). These orbits are radially unstable
for rph
− −
< r0 ≤ rISCO and stable for r0 > rISCO
−
, where rISCO
−
is given by Eq. (11.167),
increasing monotonically from 6m (a = 0) to 9m (a = m). Their specific conserved
energy and angular momentum are ε = ε− (r0 ) > 0 and ℓ = ℓ− (r0 ) < 0, with ε− (r0 )
given by Eq. (11.150) and ℓ− (r0 ) by Eq. (11.156), both being plotted as the dashed
curves in Figs. 11.12 and 11.13. The functions ε− (r0 ) and |ℓ− (r0 )| are minimal at the
ISCO.
3. The inner circular orbits: they are located inside the inner horizon (in the part
r > 0 of region MIII ), having a Boyer-Lindquist radius r0 ranging from 0 (the ring
singularity) to rph
∗
= 4m cos2 [arccos(−a/m)/3 + 4π/3] [Eq. (11.144)]; the maximal
radius rph increases with a monotonically from 0 (a = 0) to m (a = m). These orbits
∗
are all radially unstable. Their specific conserved energy and angular momentum
are ε = ε+ (r0 ) and ℓ = ℓ+ (r0 ), with ε+ (r0 ) given by Eq. (11.150) and ℓ+ (r0 ) by
Eq. (11.156), both being plotted as the solid curves in the region r0 ≤ m of Figs. 11.12
and 11.13.
In particular, there are no equatorial circular orbits in region MII of Kerr spacetime.
Historical note : The solutions for timelike circular orbits in the equatorial plane (outer cases only, i.e.
excluding the case 3 above) have been first published by James M. Bardeen, William H. Press and Saul A.
Teukolsky in 1972 [42]. In particular, they have obtained Eq. (11.150) for ε± (r0 ), Eq. (11.156) for ℓ± (r0 )
±
and Eq. (11.167) for rISCO . In his lecture notes at the 1972 Les Houches School [38], Bardeen says that
these solutions have been derived by Teukolsky. In the article [42], it is mentioned that they have been
obtained by means of computer algebra techniques.
434 Geodesics in Kerr spacetime: generic and timelike cases
1.0
3
0.9 2
1
0.8
/m
0
εISCO
+
`ISCO
`ISCO
`ISCO
−
±
0.7 1
±
2
0.6 3
+
εISCO
εISCO
−
0.5 4
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
a/m a/m
Figure 11.18: Specific conserved energy ε (left) and specific conserved angular momentum ℓ (right) at the
prograde ISCO (solid curve) and retrograde ISCO (dashed curve), as functions of the Kerr spin parameter a. [Figure
generated by the notebook D.5.9]
11.5.4 Radiated energy from an accretion disk around a black hole
The prograde ISCO around a Kerr black hole plays an important role in astrophysics: infall
of matter onto black holes proceeds via an accretion disk (cf. Sec. 8.5.5) and the inner edge
of the disk is located at the ISCO, either exactly (low luminosity thin disk) or very close to it
[2, 316, 431]. Matter slowly inspirals in the accretion disk because it loses energy and angular
momentum by the friction resulting from the differential rotation in the disk. This friction
heats the disk at high temperatures and the energy is taken away by electromagnetic radiation.
At a given orbit in the disk, the conserved energy of a particle of mass µ is E = µε and the
difference Erad = |E − µ| = µ(1 − ε) can be considered as the energy radiated away to reach
this orbit starting from rest far from the black hole, where its energy was E∞ = µ (µ − E is
generally called the binding energy, cf. Property 11.20). According to Property 11.32, Erad is
maximal when the particle has reached the prograde ISCO. Let us then evaluate ε there.
The values of ε and ℓ at the ISCO depend on a (cf. Fig. 11.18); they take a simple form when
expressed in terms of rISCO
±
, instead of directly a:
Property 11.34: specific conserved energy and angular momentum at the ISCO
The values ε+
ISCO and ℓISCO (resp. εISCO and ℓISCO ) of the specific conserved energy and
+ − −
angular momentum of a particle at the prograde ISCO (resp. retrograde ISCO) are
s r !
2m 2m r±
ε±
ISCO = 1− ± and ℓ±
ISCO =± √ 2 3 ISCO − 2 + 1 . (11.173)
3rISCO 3 3 m
√ √
Proof. Let us set u := rISCO
±
/m, so that Eq. (11.170) becomes a = ±m u(4 − 3u − 2)/3.
11.5 Circular timelike orbits in the equatorial plane 435
Substituting this value for a in Eq. (11.150) yields
√
3u − 2 − 3u − 2
ε± =√ p
ISCO √ .
3u 3u − 1 − 2 3u − 2
√ √
Noticing that 3u − 1 − 2 3u − 2 = ( 3u − 2 − 1)2 leads to the first formula in Eq. (11.173).
Similarly, Eq. (11.156) becomes
√ √
± 2m 6u − 5 − 3u − 2 2m 6u − 5 − 3u − 2
ℓISCO = ± √ p √ =± √ √ .
3 3 3u − 1 − 2 3u − 2 3 3 3u − 2 − 1
√
Multiplying both the numerator and denominator by 3u − 2 + 1 and simplifying yields the
second formula in Eq. (11.173).
For rISCO
±
= 6m (i.e. a = 0), formula (11.173) yields
√
2 2 √
±
εISCO = ≃ 0.943 and ℓ±
ISCO = ±2 3 m ≃ 3.46 m (a = 0), (11.174)
3
while for rISCO
+
= m and rISCO
−
= 9m (i.e. a = m), formula (11.173) yields
1 2m
ε+
ISCO = √ ≃ 0.577 and ℓ+
ISCO = √ ≃ 1.15 m (a = m) (11.175a)
3 3
5 22m
ε−
ISCO = √ ≃ 0.962 and ℓ−ISCO = − √ ≃ −4.23 m (a = m). (11.175b)
3 3 3 3
For a = 0, we recover the value of εISCO obtained in Chap. 7 for the Schwarzschild metric
[Eq. (7.56)], while the value of |ℓ±
ISCO | coincides with the value ℓcrit given by Eq. (7.32), as it
should. For a = m, the values of ε± ISCO and ℓISCO are the extremal ones among all values of a,
±
as it appears clearly on Fig. 11.18.
From the numerical values (11.174)-(11.175a), the energy per unit mass radiated away to
reach the prograde ISCO, i.e. Erad /µ = 1 − ε+ ISCO , is 0.057 for a = 0 and 0.423 for a = m. This
leads us to the important conclusion:
Property 11.35: energy release from an accretion disk around a black hole
Accretion onto a black hole through an accretion disk releases a significant fraction of the
rest mass energy of matter: from 6% for a non-rotating black hole (Schwarzschild) up to
42% for a maximally rotating black hole (extremal Kerr):
Erad ≃ 0.057 µ (a = 0) (11.176a)
Erad ≃ 0.423 µ (a = m). (11.176b)
It must be stressed that the above numbers are much larger than the energy release achieved
through thermonuclear reactions: the most efficient of such reactions is hydrogen fusion —
the main energy source for stars — and it releases only 0.8% of the rest mass! This explains
436 Geodesics in Kerr spacetime: generic and timelike cases
why accretion onto a black hole is invoked to account for the most energetic sources in the
Universe, such as quasars and active galactic nuclei (AGNs) (cf. e.g. [135]).
Remark 5: The energy-release mechanism described above is distinct from the Penrose process discussed
in Sec. 11.3.2. In the former, the energy is extracted from the thermal energy of the disk, while in the
latter, it is extracted from the “rotational energy” of the black hole, as we shall discuss in Sec. 16.3.1, and
this requires the existence of an ergoregion, contrary to the disk accretion process.
Historical note : In 1969, Donald Lynden-Bell [351] suggested that the large luminosity of quasars is
due to an accretion disk surrounding a massive black hole and terminating at the ISCO. He focused on
Schwarzschild black holes, for which the radiated energy per unit mass is Erad /µ = 0.057 [Eq. (11.176a)].
Soon after, in 1970, James M. Bardeen [36] considered the rotating case, i.e. Kerr black holes. He obtained
expression (11.170) for a, as well as formulas (11.173) for ε± ±
ISCO and ℓISCO , and he pointed out that
Erad /µ can reach much higher values than in the non-rotating case, up to 0.423 for a = m [Eq. (11.176b)]
(the number 0.432 given in Ref. [36] is a typo and should be 0.423). In 1974, Kip Thorne [478] declared
that Bardeen’s article [36] is “a landmark paper — a paper that persuaded astrophysicists to stop using the
Schwarzschild metric to describe black holes and start using the Kerr metric”.
11.5.5 4-velocity and angular velocities
The orbiting angular velocity as seen by an asymptotic inertial observer is Ω := dφ/dt|L (cf.
Sec. 7.3.3 for the Schwarzschild case and Sec. 10.7.1 for the extension to Kerr spacetime). We
evaluate it by combining Eqs. (11.129a) and (11.129c):
2m
dφ dτ 1 − r0
ℓ + 2amε
r0
Ω= × = 2 2 2am .
dτ dt (r0 + a )ε + r0 (aε − ℓ)
Using ℓ̃ = ℓ − aε [Eq. (11.135)] instead of ℓ, we get
(r0 − 2m)ℓ̃ + ar0 ε
Ω= .
r0 (r02 + a2 )ε − 2amℓ̃
Substituting ε and ℓ̃ by the actual values ε± and ℓ̃± taken on a circular orbit [Eqs. (11.150) and
(11.151)], we obtain, after simplification,
√
m
Ω± = ± 3/2 √ , (11.177)
r0 ± a m
where Ω+ is the value of Ω for prograde outer circular orbits and inner circular orbits and Ω− is
the value for retrograde outer circular orbits. Ω+ and Ω− are drawn in terms of r0 in Fig. 11.19.
Note that for r0 ≳ 7 m, the effect of spin parameter a on Ω± is hardly perceptible.
Remark 6: For a → 0, Eq. (11.177) reduces to Eq. (7.60), as it should.
From the very definition (11.131) of a circular orbit in the equatorial plane, the 4-velocity
u along such an orbit obeys ur = dr/dτ = 0 and uθ = dθ/dτ = 0. Since moreover Ω =
dφ/dt = uφ /ut , ∂t = ξ and ∂φ = η, we can write the 4-velocity as
u = ut (ξ + Ω± η) . (11.178)
11.5 Circular timelike orbits in the equatorial plane 437
2.0 0.15
a = 0.0 m a = 0.0 m
a = 0.5 m a = 0.5 m
a = 0.9 m
a = 0.98 m 0.10 a = 0.9 m
a = 0.98 m
1.5 a = 1.0 m a = 1.0 m
0.05
1.0
mΩ±
mΩ±
0.00
0.5 0.05
0.10
0.0
0.15
0 1 2 3 4 5 6 7 8 4 5 6 7 8 9 10
r0 /m r0 /m
Figure 11.19: Angular velocity Ω := dφ/dt = Ω+ (solid curves) or Ω = Ω− (dashed curves) along circular
timelike orbits in the equatorial plane, as a function of the orbital radius r0 [Eq. (11.177)], for selected values of
the Kerr spin parameter a. Dots on Ω+ curves and open circles on Ω− ones mark the ISCO: all configurations on
the left of these points are unstable. The right panel is a zoom on the region 4 m ≤ r0 ≤ 10 m. [Figure generated
by the notebook D.5.9]
The component ut = dt/dτ is given by Eq. (11.129a). Substituting in it Eq. (11.150) for ε and
Eq. (11.151) for ℓ̃ := ℓ − aε, we get, after simplification:
p
r0 ± a m/r0
ut = p 2 √ . (11.179)
r0 − 3mr0 ± 2a mr0
Equation (11.178) shows that the 4-velocity is a linear combination of the two Killing vectors
ξ and η with coefficients that are constant along the worldline L . An observer on a circular
orbit in the equatorial plane is thus a stationary observer as defined in Sec. 10.7.1 [compare
Eq. (10.86)]. He does not notice any change in the spacetime geometry. We shall call him a
circular geodesic observer. Contrary to the families of stationary observers considered in
Sec. 10.7, a circular geodesic observer is in free fall: by construction, his worldline is a geodesic,
so that his 4-acceleration is zero.
The orbital velocity measured by the circular geodesic observer himself is ΩP = dφ/dτ =
dφ/dt × dt/dτ = ut Ω. Using the values (11.177) and (11.179), we get
√ −3/2
m ± amr0
Ω±
P =± √ √ p √ . (11.180)
( r0 ± a m/r0 ) r02 − 3mr0 ± 2a mr0
The orbital period measured by the circular geodesic observer is TP ±
= 2π/|Ω± P |.
It is instructive to evaluate the velocity V of a particle P on a circular orbit as measured by
a zero-angular momentum observer (ZAMO) (cf. Sec. 10.7.3). Let us first note that all circular
orbits are within the domain of ZAMOs, MZAMO = MI ∪ (MIII \ T ) [Eq. (10.91)]. Indeed,
the outer circular orbits are in MI and the inner ones are in the part r > 0 of MIII , while the
Carter time machine T is located in the part r < 0 (cf. Sec. 10.2.5). Let us then rewrite formula
(11.178) for P’s 4-velocity u by expressing ξ in terms of the ZAMO’s 4-velocity n, the lapse
438 Geodesics in Kerr spacetime: generic and timelike cases
1.0
a = 0.0 m
a = 0.5 m
a = 0.9 m
a = 0.98 m
a = 1.0 m
0.5
V (±ϕ)
0.0
0.5
1.0
0 1 2 3 4 5 6 7 8
r0 /m
Figure 11.20: Component V+(φ) (solid curves) or V−(φ) (dashed curves) of the velocity V = V±(φ) e(φ) of a
particle moving along a circular orbit as measured by the ZAMO [Eq. (11.183)] for selected values of the Kerr spin
(φ) (φ)
parameter a. Dots on V+ curves and open circles on V− ones mark the ISCO: all configurations on the left of
these points are unstable. [Figure generated by the notebook D.5.9]
N and the shift vector β via Eq. (10.97):
t t 1 t
u = u (N n + β + Ω± η) = u [N n + (Ω± − ω) η] = N u n + (Ω± − ω) η ,
N
where the second equality results from β = β φ ∂φ = β φ η = −ωη, ω being the rotation angular
velocity of the ZAMO seen from infinity [Eq. (10.99)]. Since n · η = 0, the above formula
constitute the orthogonal decomposition of u with respect to the ZAMO’s 4-velocity n, which
we can compare to the generic formula (1.36) and thereby conclude that the velocity of P with
respect to the ZAMO is
1
V = (Ω± − ω) η (11.181)
N
and the Lorentz factor of P with respect to the ZAMO is
Γ = (1 − V · V )−1/2 = N ut . (11.182)
Let us expand V on the ZAMO’s orthonormal fame (e(α) ) [Eq. (10.95)]. Given relation (10.95d)
between η = ∂φ and e(φ) we get, after substituting Eq. (10.93) (with θ = π/2) for N :
r02 + a2 + 2a2 m/r0
V = p (Ω± − ω) e(φ) .
r02 − 2mr0 + a2
Finally, let us substitute Eq. (11.177) for Ω± and Eq. (10.99) (with θ = π/2) for ω. We obtain,
11.5 Circular timelike orbits in the equatorial plane 439
after simplification:
√ √
(φ) (φ) m r02 ∓ 2a mr0 + a2
V = V± e(φ) with V± = ± 3/2 √ p . (11.183)
(r0 ± a m) r02 − 2mr0 + a2
As in all formulas of this section, the sign ± is + for a prograde outer circular orbit or an
(φ)
inner circular orbit and − for a retrograde outer circular orbit. The velocity V± is depicted in
Fig. 11.20. We note that, except for a = m,
(φ) (φ) (φ) (φ)
lim V+ = 1, lim V+ = −1,
∗
lim V+ = 1, lim V− = −1. (11.184)
r0 →0 r0 →rph +
r0 →rph −
r0 →rph
Since e(φ) is a unit vector, this means that
• the inner orbits rotate close to the speed of light with respect to the ZAMO near the ring
singularity (r0 → 0) and near the outer boundary of their domain of existence (r0 → rph∗
),
this motion being in the prograde (resp. retrograde) direction in the first (resp. second)
case;
• the outer orbits rotate at the speed of light with respect to the ZAMO near the minimal
radius of their domain of existence (r0 → rph ±
).
Given that the velocity of a massive particle with respect to any observer cannot be larger
than the speed of light, this provides a physical explanation for the boundaries of the domains
of existence of circular orbits obtained in Sec. 11.5.2. We shall see in Sec. 12.3 that these
boundaries correspond to photon circular orbits. This also provides a physical explanation why
the specific conserved energy ε and the specific conserved angular momentum ℓ diverge at
these boundaries, as observed in Figs. 11.12 and 11.13: ε := E/µ and ℓ := L/µ must tend to
±∞ when the particle’s mass µ tends to 0 (the photon limit).
The particular case a = m will be discussed in Sec. ??.
11.5.6 Marginally bound circular orbit
We see on Fig. 11.12 that for a ̸= m and r0 lower than a critical value, rmb +
say, the prograde
outer circular orbits have ε > 1 (cf. the grey horizontal line in Fig. 11.12), which is equivalent
to E > µ; they are thus unbound (cf. Property 11.20). Conversely, all orbits with r0 > rmb +
are bound. Similarly, but this time of all values of a ≤ m, the retrograde outer circular orbits
are bound iff r0 is larger than a critical value, rmb
−
say. The orbit at r0 = rmb ±
is called the
marginally bound circular orbit.
We note from Fig. 11.12 that
±
rph ±
< rmb ±
< rISCO . (11.185)
Accordingly, all unbound orbits are unstable.
To evaluate rmb±
, let us solve the equation ε± (r0 ) = 1 for r0 . Using expression (11.150) for
ε± (r0 ), taking the square and simplifying, we get
√
r02 − 4mr0 ± 4a mr0 − a2 = 0,
440 Geodesics in Kerr spacetime: generic and timelike cases
which we can write
√
r02 = (2 mr0 ∓ a)2 .
√ √
Since we are looking for solutions r0 ≥ r+ = m + m2 − a2 , we have 2 mr p0 ∓ a > 0, so that
√
we can reduce the above equation to r0 = 2 mr0 ∓ a. Introducing x := r0 /m, we end up
solving the quadratic equation x2 − 2x ± a/m = 0. The only solution x ≥ 1, which is implied
by r0 ≥ r+ , is x = 1 + 1 ∓ a/m. This yields
p
(11.186)
±
p
rmb = 2m ∓ a + 2 m(m ∓ a) .
±
rmb is plotted as a function of a in Fig. 11.17.
11.5.7 Circular orbits in the ergoregion
We see in Fig. 11.17 that for a sufficiently large, prograde outer circular orbits can exist in
the outer ergoregion G + (cf. Sec. 10.2.4), while no retrograde outer circular orbits can exist
there. In the equatorial plane, the coordinate r of the external boundary of G + (the outer
ergosphere) is simply rE + (π/2) = 2m [Eq. (10.19b)] — the orange horizontal line in Fig. 11.17.
Thus, prograde outer circular orbits exist in the ergoregion if rph
+
< 2m and they can be stable
if rISCO < 2m. The limiting value of a for the first case is obtained by setting r0 = 2m in the
+
equation governing√the existence
√ of prograde circular orbits, namely Eq. (11.141) with the +
sign: (2m) − 3m 2m + 2a m > 0, from which we get immediately
3/2
Property 11.36: timelike circular orbits in the ergoregion
timelike circular orbits
m
⇐⇒ a > √ ≃ 0.707 m. (11.187)
exist in the outer ergoregion 2
Similarly, the limiting value of a for the stability of circular orbits in the ergoregion
√ is obtained
by setting r0 = 2m in Eq. (11.160) with the + sign: (2m)2 − 12m2 + 8am 2 − 3a2 > 0. The
right-hand side being a quadratic polynomial in a, we get easily, given the constraint a ≤ m,
Property 11.37: stable timelike circular orbits in the ergoregion
√
timelike circular orbits
2 2
⇐⇒ a > m ≃ 0.943 m. (11.188)
exist stably in the ergoregion 3
Both in (11.187) and (11.188), the orbits referred to belong to the prograde outer family.
Remark 7: Regarding the limit (11.187), the existence is specified to be in the outer ergoregion, because
timelike circular orbits always exist, as soon as a > 0, in the inner ergoregion: they are the (unstable)
inner circular orbits found in Sec. 11.5.2. Indeed, these orbits exist in the range 0 < r0 < rph
∗ , which is
entirely contained in the inner ergoregion: the boundaries of the latter in the equatorial plane are r = 0
11.6 Going further 441
[Eq. (10.19a)] and r = r− , with r− > rph ∗ . On the contrary, in the limit (11.188), we have dropped the
qualifier outer for the ergoregion, since there is no other place where stable circular orbits can be found,
the inner circular orbits being all unstable.
Remark 8: As discussed in Sec. 11.3.1, negative-energy or zero-energy particles can exist in the outer
ergoregion. However, Fig. 11.12 shows that none of them can follow a circular orbit.
11.6 Going further
For an extended discussion of geodesics of Kerr spacetime, including those that cross the
various blocks of the maximal analytic extension presented in Sec. 10.8, see Chap. 4 of O’Neill
textbook [391]. For more details about bound orbits, in particular their description in terms
of action angles, see Sec. 6 of the recent review by Pound and Wardell [419] and references
therein.
442 Geodesics in Kerr spacetime: generic and timelike cases
Chapter 12
Null geodesics and images in Kerr
spacetime
Contents
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
12.2 Main properties of null geodesics . . . . . . . . . . . . . . . . . . . . . 443
12.3 Spherical photon orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
12.4 Black hole shadow and critical curve . . . . . . . . . . . . . . . . . . . 479
12.5 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
12.1 Introduction
Having investigated the properties of generic causal geodesics in Kerr spacetime in Chap. 11,
we focus here on null geodesics, with application to images of a Kerr black hole. First we
discuss the main properties of null geodesics in Sec. 12.2, in great part by taking the µ = 0
limit of results obtained for generic causal geodesics in Chap. 11. Then, in Sec. 12.3, we focus
on null geodesics evolving at a fixed value of the coordinate r — the so-called spherical photon
orbits. These geodesics play a crucial role in the formation of the images perceived by an
observer. In particular, they are related to the key concepts of critical curve and shadow in the
observer’s screen, which are investigated in Sec. 12.4. Finally, we discuss the images themselves
in Sec. 12.5, first by considering computed images from a simplified model of accretion disk
and then by analyzing the actual image of the surroundings of the black hole M87*, as obtained
recently by the Event Horizon Telescope [6].
12.2 Main properties of null geodesics
We shall distinguish the null geodesics with E = 0 (the so-called zero-energy geodesics, cf.
Sec. 11.3.1) from those having E ̸= 0. Indeed, in the latter case, we will rescale the angular
444 Null geodesics and images in Kerr spacetime
momentum L and the Carter constant Q by E, so that only two constants of motion become
pertinent for the study: L/E and Q/E 2 . We thus treat first the particular case E = 0.
12.2.1 Zero-energy null geodesics
First, we note that a geodesic L with E = 0 cannot exist outside the ergoregion G , by virtue
of the result (11.56). In particular, it cannot exist far from the black hole.
Another property of L is to have a non-negative Carter constant:
Q≥0 . (12.1)
E=0
This follows immediately Property 11.19 in Sec. 11.3.7, which,
p among other things, states that
a necessary condition for Q < 0 is a ̸= 0 and |E| > µ2 + L2 /a2 . Specializing this last
inequality to µ = 0 and E = 0, we get 0 > |L|, which is impossible.
Besides, if L has some part in MI (necessarily in the outer ergoregion) or in MIII (neces-
sarily in the inner ergoregion), the constraint (11.69) reduces to L < 0:
L ∩ (MI ∪ MIII ) ̸= ∅ =⇒ L < 0. (12.2)
We shall see below that actually L ≤ 0 for all zero-energy null geodesics, as soon as a ̸= 0.
The equations of geodesic motion expressed in terms of the Mino parameter λ′ [system
(11.50)] simplify considerably for a geodesic L with µ = 0 and E = 0:
dt 2amLr
=− (12.3a)
dλ′ ∆
dr
(12.3b)
p
= ϵr R(r)
dλ′
dθ
(12.3c)
p
= ϵθ Θ(θ)
dλ′
dφ L 2 2 2
(12.3d)
= r − 2mr + a cos θ ,
dλ′ ∆ sin2 θ
with [cf. Eqs. (11.95) and (11.35)]:
R(r) = −(Q + L2 )r2 + 2m(Q + L2 )r − a2 Q (12.4)
L2
Θ(θ) = Q − . (12.5)
tan2 θ
By combining (12.3a) and (12.3d), we get
dφ 2mr − r2 − a2 cos2 θ
= . (12.6)
dt L 2amr sin2 θ
E=0
It is remarkable that this expression does not depend on L or Q; it is therefore the same for
all zero-energy null geodesics. Moreover, we note that the numerator of the right-hand side
is always positive or zero in the closure G of the ergoregion, which is precisely defined by
12.2 Main properties of null geodesics 445
2mr − r2 − a2 cos2 θ ≥ 0 (cf. Sec. 10.2.4) and where L is necessarily confined. Since moreover
r > 0 in G , we conclude that
dφ
≥ 0. (12.7)
dt L
To proceed, we shall distinguish the subcases Q ̸= 0 and Q = 0.
Case Q ̸= 0
This case actually corresponds to Q > 0, since Q < 0 is forbidden by (12.1). We set
L
L̄ := √ (12.8)
Q
and rewrite expression (12.4) for R(r) as
R(r)/Q = −(1 + L̄2 )r2 + 2m(1 + L̄2 )r − a2 . (12.9)
Since 1 + L̄2 ̸= 0, this is a second-order polynomial in r, the two roots of which are
r r
a 2 a2
rmin = m − m2 − and rmax = m + m 2− . (12.10)
1 + L̄2 1 + L̄2
Since m2 ≥ a2 , the two roots are real. They are distinct except for a = m and L = 0. The
range of radial motion being determined by R(r) ≥ 0 [Eq. (11.30)], we get
rmin ≤ r ≤ rmax , (12.11)
√
with
√ a turning point at rmin and at rmax . Given that r− = m − m2 − a2 and r+ = m +
m2 − a2 [Eq. (10.3)], we note that
0 ≤ rmin ≤ r− ≤ m ≤ r+ ≤ rmax ≤ 2m, (12.12)
with rmin = 0 for a = 0 or L̄2 → +∞, rmin = r− for L = 0, rmax = 2m for a = 0 or L̄2 → +∞
and rmax = r+ for L = 0. If L ̸= 0 and a ̸= 0, then rmax > r+ , so that L has a part in the
outer ergoregion and (12.2) implies that L < 0. Hence
a ̸= 0 =⇒ L ≤ 0. (12.13)
Let us consider a zero-energy null geodesic L emitted outward (i.e. with ϵr = +1) from
a point A in the outer ergoregion G + . The coordinate r increases along L from rA to rmax ,
which corresponds to a r-turning point. Then r decreases to r+ , which means that L crosses
the black hole event horizon H and enters the region MII . In all MII , r keeps decreasing
and reaches r− . There L crosses the inner horizon Hin and enters the region MIII , where
r continues to decrease until it reaches rmin . The latter corresponding to a r-turning point,
r starts to increase and reaches r− again. There one might think that L crosses the inner
horizon Hin and enters into MII . But this is impossible since Hin is a 1-way membrane: it can
be crossed by a causal curve from MII to MIII but not in the reverse way. Moreover, r could
446 Null geodesics and images in Kerr spacetime
not continue to increase into MII since r must be decreasing towards the future in all this
region (this follows from the hypersurfaces r = const being spacelike in MII , cf. Sec. 10.5.1).
The solution to this apparent puzzle is immediate as soon as one realizes that the boundary
r = r− of MIII is not entirely constituted by Hin : it also comprises a null hypersurface that
separates MIII from a region distinct from MII in the maximally extended Kerr spacetime, cf.
Fig. 10.11. This region is a “time-reversed” copy of MII and is denoted by MII∗ ′ in Fig. 10.11
(cf. Sec. 10.8 for details). So actually, when it reaches r = r− , the null geodesic L enters MII∗ ′ .
There r necessarily increases towards the future, at the opposite of MII . It reaches then r = r+ ,
where L crosses a white hole horizon and emerges into the asymptotically flat region MI′′ , as
illustrated in Fig. 12.1. The region MI′′ is similar to MI . In particular, L is confined into the
outer ergoregion of MI′′ , having a r-turning point at r = rmax (same value (12.10) as in MI ).
Then a new cycle begins, with L entering the future event horizon of MI′′ .
The θ-motion of L is constrained by Θ(θ) ≥ 0 [Eq. (11.33)], which, given expression (12.5)
for Θ, is equivalent to
θm ≤ θ ≤ π − θm with θm := arctan(−L̄). (12.14)
Remark 1: The general formula for θm in the
p case a (E − µ ) = 0, Eq. (11.85), which holds here
2 2 2
since µ = 0 and E = 0, yields θm = arccos 1/(1 + L̄2 ) = arctan |L̄|. Hence we recover the above
formula.
For L = 0, one has θm = 0, so that θ takes all values in the range [0, π], which means that
L crosses repeatedly the rotation axis. For L < 0, one has 0 < θm < π/2 and L oscillates
symmetrically about the equatorial plane, having two θ-turning points, at θm and π − θm . Of
course, we recover the general results for Q > 0 of Sec. 11.3.7.
We can obtain r as a function of θ along L by evaluating the integrals in the identity
(11.55a):
Z r Z θ
ϵr dr̄ ϵθ dθ̄
− p =− p
r0 R(r̄) θ0 Θ(θ̄)
Using (12.9) and (12.5), we get on any portion of L where ϵr and ϵθ are constant,
Z r Z θ
dr̄ dθ̄
ϵr p = ϵθ p .
r0 −(1 + L̄2 )r̄2 + 2m(1 + L̄2 )r̄ − a2 θ0 1 − L̄2 / tan2 θ̄
The changes of variables
r/m − 1
x= q and µ = cos θ
a2
1 − m2 (1+ L̄2 )
lead to Z x Z cos θ
ϵr dx̄ dµ
√ √ = −ϵθ p .
1 + L̄2 x0 1 − x̄2 cos θ0 1 − (1 + L̄2 )µ2
12.2 Main properties of null geodesics 447
Figure 12.1: Trajectory in the extended Kerr spacetime of a null geodesic with E = 0, Q > 0 and L < 0,
emitted from a point A in the outer ergoregion.
√
The integration is then immediate: arcsin x = −ϵr ϵθ arcsin( 1 + L̄2 cos θ) + K, where K is a
constant, from which we get
s
a2 h p i
r =m+m 1− sin K − ϵ ϵ
r θ arcsin 1 + L̄2 cos θ . (12.15)
m2 (1 + L̄2 )
√
Since 1 + L̄2 cos θm = 1, we see that the constant K is related to the value of r at θ = θm by
r(θm )/m − 1 π
K = arcsin q + ϵr ϵθ . (12.16)
a2
1 − m2 (1+ 2
L̄2 )
448 Null geodesics and images in Kerr spacetime
e r/m cosθ
4 r=r +
2 r=r −
A e r/m sinθ
1 2 3 4 5 6 7 8
r=0
2
Figure 12.2: Trajectory
√ in a meridional plane, as given by Eq. (12.15), of a null geodesic (green curve) with
E = 0, Q > 0, L = − Q and r(θm ) = 1.5 m in the Kerr spacetime with a/m = 0.9. The meridional plane is
described in terms of O’Neill exponential coordinates x = er/m sin θ and z = er/m cos θ, as in Figs. 10.2 – 10.4.
The ergoregion is shown in grey. The black (resp. light brown) half-circle at r = r+ (resp. r = r− ) is the trace of
the outer (resp. inner) Killing horizon. The dotted orange half-circle marks the locus of r = 0, with the red dot
indicating the curvature singularity at r = 0 and θ = π/2. The area r > r+ corresponds to the regions MI and
MI′′ in Fig. 12.1, the area r− < r < r+ corresponds to the regions MII and MII∗ ′ in Fig. 12.1 and the area r < r−
corresponds to the region MIII in Fig. 12.1. [Figure generated by the notebook D.5.10]
Note that K is not a constant all along L , but only on portions where ϵr and ϵθ are constant.
Expression 12.15 gives the trace of the zero-energy null
√ geodesic L in a meridional plane. It
depends on Q and L only through the ratio L̄ := L/ Q. It depends as well on the value of
r at√θm via K, as it appears in Eq. (12.16). An example is shown in Fig. 12.2 for a/m = 0.9,
L/ Q = −1 and r(θm ) = 1.5 m. It has θm = π/4, rmin ≃ 0.229 m and rmax ≃ 1.771 m, while
for a/m = 0.9, one has r− ≃ 0.564 m and r+ ≃ 1.436 m. For concreteness, the arrows indicate
some direction of motion, but depending upon some initial conditions, the opposite direction
is possible. In particular, one may consider that the geodesic is the same as that shown in
Fig. 12.1, being emitted outward in the outer ergoregion from a point A in the equatorial plane
(θ = π/2).
Case Q = 0
If the zero-energy null geodesic L has a vanishing Carter constant Q, Eq. (12.5) reduces to
Θ(θ) = −L2 / tan2 θ, so that the constraint Θ(θ) ≥ 0 [Eq. (11.33)] implies L = 0 or θ = π/2 .
In the first case, the four constants of motion µ, E, L and Q are zero. By virtue of the result
(11.25), L is nothing but a null geodesic generator of the event horizon H or of the inner
12.2 Main properties of null geodesics 449
horizon Hin .
In the second case (θ = π/2), L is confined to the equatorial plane. If L = 0, we are back
to the first case: L is null geodesic generator of H or Hin lying in the equatorial plane. If
L ̸= 0, the radial motion of L is governed by Eq. (12.3b) with the expression (12.4) of R(r)
reduced to
R(r) = L2 r(2m − r). (12.17)
The constraint R(r) ≥ 0 [Eq. (11.30)] implies then that the motion is within the range 0 ≤ r ≤
2m, with r = 2m being a r-turning point, since it is a simple root of R(r) (cf. Sec. 11.2.6). It
corresponds to the outer edge of the ergoregion in the equatorial plane, cf. Eq. (10.19b). Hence
we have necessarily L ∩ MI ̸= ∅ and (12.2) applies: L < 0. The inner boundary of the radial
motion, r = 0, is the ring singularity. Accordingly, in the maximally extended Kerr spacetime,
L starts at the ring singularity in a MIII -type region (cf. Fig. 12.3), has r increasing, enters
a MII∗ -type region (time reversed copy of MII ), emerges in MI via the white hole horizon
at r = r+ and reaches a r-turning point at r = 2m, then r decreases continuously until L
terminates at the ring singularity of MIII , after having crossed the black hole horizon H and
the inner horizon Hin . This trajectory, depicted in Fig. 12.3, is similar to that for Q ̸= 0 shown
in Fig. 12.1, except that it is “blocked” by two ring singularities and cannot oscillate forever
between distinct MI -type regions.
Remark 2: For Q ̸= 0 and L ̸= 0, the limit Q → 0 corresponds to L̄2 → +∞ [cf. Eq. (12.8)], so that
Eq. (12.10) yields rmin → 0 and rmax → 2m. We recover then the range [0, 2m] for r obtained here for
Q = 0.
We conclude:
Property 12.1: null geodesic with E = 0 and Q = 0
Any null geodesic with E = 0 and Q = 0 is either a null generator of one of the two Killing
horizons H or Hin (in which case, it has L = 0) or it has L < 0, lies in the equatorial
plane, emanates from a ring singularity, reaches the outer ergosphere (r = 2m), where it
has a r-turning point, and terminates at a ring singularity.
Regarding the sign of L, we can combine the above result with that obtained for Q ̸= 0
[Eq. (12.13)] to get:
Property 12.2: negative angular momentum for zero-energy null geodesics
For a ̸= 0, any null geodesic with E = 0 has necessarily
L≤0 a̸=0 . (12.18)
E=0
Historical note : The zero-energy null geodesics in Kerr spacetime appear to have been first studied by
Zdeněk Stuchlík in 1981, in the appendix of the article [466]; some corrections and refinement of his
results have been performed by George Contopoulos in 1984 [141], who studied zero-energy timelike
geodesics as well.
450 Null geodesics and images in Kerr spacetime
Figure 12.3: Trajectory in the extended Kerr spacetime of a null geodesic with E = 0, Q = 0 and L < 0.
12.2.2 Equations of geodesic motion for E ̸= 0
For any null geodesic L with E ̸= 0, we introduce the reduced constants of motion
L Q
ℓ := and q := . (12.19)
E E2
Note that, in geometrized units (G = 1 and c = 1), ℓ has the dimension of a length and q that
of a squared length.
Remark 3: In the literature, ℓ is sometimes denoted by λ (e.g. Refs. [38, 236, 169]) or by ξ (e.g. Ref. [107]),
while q is sometimes denoted by η (e.g. Refs. [38, 107, 236]). Moreover, in studies restricted to q ≥ 0,
it may happen that the notation q stands for the square root of the quantity q defined by (12.19) (e.g.
Refs. [165, 238, 169]).
12.2 Main properties of null geodesics 451
Remark 4: Contrary to L and Q, the quantities ℓ and q are independent from the affine parametrization
of the geodesic L . Indeed, if instead of the affine parameter λ associated with the particle’s 4-momentum
p, one considers the affine parameter λ′ = αλ, where α is a positive constant, the tangent vector field
becomes p′ = α−1 p, so that the associated conserved quantities are E ′ = −ξ · p′ = α−1 E [cf.
Eq. (11.2a)], L′ = η · p′ = α−1 L [cf. Eq. (11.2b)] and Q′ = K̃(p′ , p′ ) = α−2 Q [cf. Eq. (11.41)], so that
ℓ′ := L′ /E ′ = ℓ and q ′ := Q′ /E ′ 2 = q.
The non-negative property of Carter constant K [Eq. (11.36)], along with the identity
(11.34) leads to the following constraint on the parameters ℓ and q:
q + (ℓ − a)2 ≥ 0 . (12.20)
Example 1 (Principal null geodesic): A principal null geodesic has E ̸= 0 if, and only if, it does
not belong to the outgoing family generating the horizon H or Hin . This follows immediately from
Eqs. (11.6) and (11.9). One has then L = aE sin2 θ0 [Eqs. (11.6) and (11.9)] and Q = −a2 E 2 cos2 θ0
[Eq. (11.38)], where θ0 is the constant value of θ along the geodesic. We have thus
ℓ = a sin2 θ0 and q = −a2 cos4 θ0 . (12.21)
This yields
q + (ℓ − a)2 = 0, (12.22)
so that the inequality (12.20) is saturated for principal null geodesics.
The equations of motion in terms of the Mino parameter λ′ , Eqs. (11.50) specialized to
µ = 0, can be rewritten as
1 dt 1 2
(r + a2 )2 − 2amℓr − a2 sin2 θ (12.23a)
′
=
E dλ ∆
1 dr
(12.23b)
p
= ϵr R(r)
|E| dλ′
1 dθ
q
= ϵθ Θ̃(θ) (12.23c)
|E| dλ′
1 dφ ℓ a
′
= 2 + (2mr − aℓ) , (12.23d)
E dλ sin θ ∆
with
R(r) Θ(θ)
R(r) := 2
and Θ̃(θ) := . (12.24)
E E2
From the general expressions (11.44), (11.95) and (11.35) specialized to µ = 0, we get
R(r) = (r2 + a2 − aℓ)2 − ∆ q + (ℓ − a)2 (12.25a)
R(r) = r4 + (a2 − ℓ2 − q)r2 + 2m q + (ℓ − a)2 r − a2 q (12.25b)
452 Null geodesics and images in Kerr spacetime
and
ℓ2
Θ̃(θ) = q + cos θ a2 −
2
. (12.26)
sin2 θ
It suffices to use the parameter λ′′ := |E|λ′ to make E disappear from the system (12.23).
We therefore conclude:
Property 12.3
In Kerr spacetime, a null geodesic with E ̸= 0 is entirely determined by the two constants
(ℓ, q), by the sign of E and by the values of the two signs ϵr = ±1 and ϵθ = ±1 at a given
point.
Example 2 (Principal null geodesic): Given the values (12.21) of ℓ and q for a principal null geodesic,
Eq. (12.25b) reduces to a simple expression for the quartic polynomial R:
2
R(r) = r2 + a2 cos2 θ0 . (12.27)
We note that R(r) = ρ4 , which makes sense because θ = θ0 is constant along such a geodesic. For any
principal null geodesic lying in the equatorial plane, the polynomial simplifies even further:
π
R(r) = r4 θ0 = . (12.28)
2
12.2.3 Position on a remote observer’s screen
The constants (ℓ, q) are closely related to the impact coordinates (α, β) on the screen of an
asymptotic inertial observer (cf. Sec. 10.7.5) in case the null geodesic L reaches the asymptotic
region r → +∞. Indeed, let us consider an asymptotic inertial observer O located at (fixed)
Boyer-Lindquist coordinates (rO , θO , φO ), with rO ≫ m (cf. Fig. 12.4). In order to reach O,
the geodesic L must be such that the constraints constraints R(rO ) ≥ 0 [Eq. (12.46)] and
Θ̃(θO ) ≥ 0 [Eq. (11.33)] are fulfilled. The first one is always satisfied due to the assumption
that O is an asymptotic observer, since R(r) ∼ r4 for r → +∞ [cf. Eq. (12.25b)]. In view of
expression (12.26) for Θ̃, the second constraint is equivalent to
q + a2 cos2 θO sin2 θO − ℓ2 cos2 θO ≥ 0. (12.29)
If sin θO is small, this constraint limits significantly the amplitude of ℓ.
Observer not located on the rotation axis
Here we treat the generic case of an observer O that is not located on the rotation axis, i.e.
we assume θO ̸∈ {0, π}. The orthonormal frame of O is e(0) = ξ, e(r) = ∂r , e(θ) = rO−1 ∂θ and
e(φ) = (rO sin θO )−1 ∂φ [Eq. (10.95) with r = rO → +∞]. Let us assume that the observer set
up a screen (via a telescope) centered on the direction to the black hole, i.e. such that e(r) is the
12.2 Main properties of null geodesics 453
Figure 12.4: Impact of a null geodesic L onto the screen of a remote observer. (x, y, z) are the Cartesian
Boyer-Lindquist coordinates defined by Eq. (11.73).
normal to the screen. The 4-momentum of the photon having L as worldline at the location
of O writes
p = pt ξ + pr e(r) + pθ rO e(θ) + pφ rO sin θO e(φ) = E(ξ + n),
where the second equality follows from the orthogonal decomposition (1.31) with respect to
O, ξ being the 4-velocity of O and the unit spacelike vector n being the photon’s velocity1
relative to O; it obeys ξ · n = 0 and n · n = 1 [Eq. (1.37)]. The appearance of the conserved
energy E in the above equation is a direct consequence of the definition (11.2a): E := −ξ · p,
with ξ · ξ = −1 for rO → +∞. The above identity implies pt = E and
pr pθ pφ
n= e(r) + rO e(θ) + rO sin θO e(φ) .
E E E
With respect to O, the incoming direction of the photon is given by the vector −n and the
trace on the screen is indicated by the component of −n that is tangent to the screen, namely
pθ pφ
m = −n∥ = − rO e(θ) − rO sin θO e(φ) (12.30)
E E
since the screen plane is spanned by e(θ) and e(φ) (cf. Fig. 12.4). Let us choose the screen
dimensionless Cartesian coordinates (α, β) so that the black hole’s rotation axis appears as the
β-axis (cf. Fig. 12.4), then
m = αe(α) + βe(β) , with e(α) = e(φ) and e(β) = −e(θ) . (12.31)
1
n is denoted by V in Eq. (1.31).
454 Null geodesics and images in Kerr spacetime
By comparing with (12.30), we get
pφ pθ
α=− rO sin θO and β= rO .
E E
Now, for rO → +∞,
pθ 1 dθ 1 dθ ϵθ
q
= = = 2 Θ̃(θO )
E E dλ ErO2 dλ′ rO
φ
p 1 dφ 1 dφ ℓ
= = = 2 ,
E E dλ 2
ErO dλ ′ rO sin2 θO
where we have used Eqs. (12.23c) and (12.23d), with the term involving a/∆ neglected since
∆ ∼ r2 for r → +∞. By inserting these formula into the above expressions of α and β, and
using Eq. (12.26) for Θ̃(θO ), we get the sought relation between the constants of motion (ℓ, q)
and the screen coordinates:
ℓ
α=− (12.32a)
rO sin θO
s
ϵθ ℓ2
β= q+ cos2 θO a2 − . (12.32b)
rO sin2 θO
We have defined (α, β) as dimensionless Cartesian coordinates on the screen, cf. Eq. (12.31),
where m is dimensionless and (e(α) , e(β) ) is the screen’s orthonormal basis. In practice, their
values are tiny, being exactly zero at the limit rO → +∞. (α, β) can thus be interpreted as
angular coordinates, measuring the departure from the direction of the black hole “center” on
the celestial sphere of observer O. We shall then call (α, β) the screen angular coordinates.
Remark 5: When studying null geodesics in Schwarzschild spacetime in Chap. 8, we introduced the
impact parameter b as b := |L|/E [Eq. (8.12)], hence b is related to ℓ by
b = |ℓ|. (12.33)
Moreover, thanks to spherical symmetry, we could reduce the study to the case where both the observer
and the geodesic lie in the equatorial plane, which imply θO = π/2 and q = 0. Equations (12.32) yield
then
b
|α| = = b̂ and β = 0, (12.34)
rO
where b̂ is the angle introduced by formula (8.118).
Remark 6: The angular impact parameters (α, β) depend on the geodesic L as a curve in spacetime
and not on any affine parametrization of L . Their direct connection with (ℓ, q) expressed by (12.32) is
thus in perfect agreement with the invariance of (ℓ, q) in any affine reparametrization of L , as noticed
in Remark 4 p. 451.
We deduce from Eqs. (12.32) a simple relation between the squared angular distance to the
screen center, α2 + β 2 , and the constants of motion (ℓ, q) of the incoming geodesic:
1 2
α2 + β 2 = ℓ + q + a2 cos2 θO . (12.35)
2
rO
12.2 Main properties of null geodesics 455
Observer on the rotation axis
If the asymptotic inertial observer O is located on the rotation axis, i.e. if θO = 0 or θO = π,
the only value of ℓ compatible with the constraint (12.29) is
ℓ=0 (θO = 0 or θO = π). (12.36)
Given that ℓ = L/E, we recover one of the properties listed in Sec. 11.3.7: a geodesic cannot
encounter the rotation axis unless it has L = 0.
Moreover, on the rotation axis, the vectors e(θ) and e(φ) are not defined, due to the singularity
of spherical coordinates there. Consequently, the screen coordinates (α, β) cannot be defined
by (12.31). In particular, the rotation axis appears as single point on the screen, which forbids
to use it to define the β-axis. One has then to pick an arbitrary orthonormal frame (e(α) , e(β) )
in the screen plane to define (α, β). Formulas (12.88) do no longer hold, but formula (12.35)
is still valid, since the distance from the screen’s center is a quantity independent from the
orientation of the frame (e(α) , e(β) ). Another way to see that (12.35) is still valid is to notice
that it admits a well-defined limit for θO → 0 or π. Taking into account ℓ = 0, we obtain
1
α2 + β 2 = q + a2 (θO = 0 or θO = π). (12.37)
2
rO
12.2.4 Latitudinal motion
Specializing the general results expressed in Property 11.19 to µ = 0 and E ̸= 0, we get
Property 12.4: latitudinal motion of null geodesics
• A null geodesic L of Kerr spacetime cannot encounter the rotation axis unless it
has ℓ = 0.
• If |ℓ| ≥ a, the reduced Carter constant q is necessarily non-negative:
q ≥ 0. (12.38)
• The reduced Carter constant q can take negative values only if |ℓ| < a (which implies
a ̸= 0); its range is then limited from below:
q ≥ qmin = − (a − |ℓ|)2 . (12.39)
If q < 0, L is called a vortical null geodesic; it never encounters the equatorial
plane.
• If q > 0 and ℓ ̸= 0, L oscillates symmetrically about the equatorial plane, between
two θ-turning points, at θ = θm and θ = π − θm , where θm ∈ (0, π/2) is given by
456 Null geodesics and images in Kerr spacetime
Eqs. (11.85) and (11.90):
r
q
θm = arccos 2
for a = 0 (12.40)
ℓ +q
v
u s 2
u
1 ℓ2 + q ℓ2 + q 4q
θm = arccos t 1 − + 1− + 2 for a ̸= 0. (12.41)
u
2 a2 a2 a
If q > 0 and ℓ = 0, L crosses repeatedly the rotation axis, with θ taking all values
in the range [0, π].
• If q = 0, L is stably confined to the equatorial plane for |ℓ| > a or |ℓ| = a ̸= 0; for
|ℓ| < a, L either lies unstably in the equatorial plane or approaches it asymptotically
from one side, while for ℓ = 0 and a = 0, L lies at a constant value θ = θ0 ∈ [0, π].
• If qmin < q < 0, L never encounters the equatorial plane, having a θ-motion entirely
confined either to the Northern hemisphere (0 < θ < π/2) or to the Southern one
(π/2 < θ < π); if ℓ ̸= 0, L oscillates between two θ-turning points, at θ = θm
and θ = θv (Northern hemisphere) or at θ = π − θv and θ = π − θm (Southern
hemisphere), where θm is given by Eq. (12.41) above and θv is given by Eq. (11.91):
v
u s 2
u1
u 2
ℓ +q 2
ℓ +q 4q
θv = arccos t 1 − 2
− 1− 2
+ 2 ; (12.42)
2 a a a
if ℓ = 0, L oscillates about the rotation axis, with a θ-turning point at θ = θv or
θ = π − θv , where θv is given by Eq. (11.88), or equivalently by the ℓ → 0 limit of
Eq. (12.42): √
−q
θv = arccos (ℓ = 0). (12.43)
a
• If q = qmin , L lies stably at a constant value θ = θ∗ or θ = π − θ∗ , with θ∗ ∈ [0, π/2)
given by r
|ℓ|
θ∗ := arcsin . (12.44)
a
Remark 7: For ℓ < 0, the constraint (12.39) is tighter than (12.20).
Example 3 (Principal null geodesic): A principal null geodesic moves at a constant angle θ = θ0 and
has ℓ = a sin2 θ0 [Eq. (12.21)]. For θ0 ̸= π/2, we have |ℓ| < a and Eq. (12.39) yields qmin = −a2 cos4 θ0 .
Comparing with the value of q given by Eq. (12.21), we note that
q = qmin , (12.45)
which agrees with the last caseplisted above, i.e. motion at constant θ, with θ0 = θ∗ or θ0 = π − θ∗
according to Eq. (12.44), since |ℓ|/a = sin θ0 .
12.2 Main properties of null geodesics 457
40 40 40 40
` = 0.5 m ` = 0.863 m `=m ` = 2mr + /a
R(r)/m 4 30 q = 16 m 2 30 q = 18.04 m 2 30 q = 20 m 2 30 q = 8 m2
20 20 20 20
10 10 10 10
0 0 0 0
r0 r1 r2 r0
10 10 10 10
1.5 2.0 2.5 3.0 3.5 1.5 2.0 2.5 3.0 3.5 1.5 2.0 2.5 3.0 3.5 1.5 2.0 2.5 3.0 3.5
r/m r/m r/m r/m
(a) (b) (c) (d)
Figure 12.5: Quartic polynomial R(r) in the region MI for four values of (ℓ, q) and for a = 0.95 m. The grey
area marks the black hole region, with the vertical black line at the event horizon, at r = r+ ≃ 1.312 m. The
green part of the r axis corresponds to R(r) ≥ 0, i.e. to regions where the geodesic motion is allowed. [Figure
generated by the notebook D.5.13]
12.2.5 Radial motion
As for any geodesic, the radial motion of a null geodesic L is ruled by R(r) ≥ 0 [Eq. (11.30)],
which, in terms of R(r) [Eq. (12.25)], writes
R(r) ≥ 0 . (12.46)
Example 4 (Principal null geodesic): Given the value (12.27) of R(r) for a principal null geodesic, we
note that R(r) > 0 in all Kerr spacetime. This is consistent with the fact that, for E ̸= 0 and θ0 ̸= π/2,
ingoing principal null geodesics travel from r = +∞ to r = −∞ (cf. the dashed green curve in Fig. 10.9)
and outgoing principal null geodesics travel from r = −∞ to r = +∞ (cf. the solid green curve in
Fig. 10.10).
Since R(r) is a polynomial of degree 4 in r, its behavior can be relatively complicated.
However, it turns out that in the black hole exterior, its behavior is quite simple, according to
the following lemma:
Lemma 12.5: behavior of R(r) in MI
In region MI of Kerr spacetime, i.e. for r > r+ , the quartic polynomial R(r) associated to
a given geodesic [Eq. (12.25)] has one of the following behaviors, depending on the value
of (ℓ, q):
1. R(r) has no root in (r+ , +∞) and R(r) > 0 there (Fig. 12.5a);
2. R(r) has a double root in (r+ , +∞), at r = r0 say, and R(r) > 0 iff r ∈ (r+ , r0 ) or
r ∈ (r0 , +∞) (Fig. 12.5b);
3. R(r) has two simple roots in (r+ , +∞), at r = r1 and r = r2 say, and R(r) > 0 iff
r ∈ (r+ , r1 ) or r ∈ (r2 , +∞) (Fig. 12.5c);
458 Null geodesics and images in Kerr spacetime
4. R(r) has a unique simple root in (r+ , +∞), at r = r0 say; then necessarily ℓ =
2mr+ /a, R(r+ ) = 0 and R(r) > 0 iff r ∈ (r0 , +∞) (Fig. 12.5d).
Proof. First, we observe that R(r) is positive or zero at both ends of MI . This is clear for the
asymptotic flat end since, according to expression (12.25b), R(r) ∼ r4 > 0 when r → +∞. At
the inner end, namely for r → r+ , we have ∆ → 0 and expression (12.25a) yields the limit
value R(r+ ) = (r+ 2
+ a2 − aℓ)2 ≥ 0. Now, as a polynomial of degree four, R(r) has at most
four real roots. Given the above boundary conditions, R(r) can have a unique simple root in
(r+ , +∞) only if R(r+ ) = 0 (cf. Fig. 12.5d), which occurs for a very specific value of ℓ, namely
ℓ = (r+2
+ a2 )/a = 2mr+ /a. This is case 4 above. Similarly, the case of three roots in (r+ , +∞),
either three simple roots or one double and one simple root, is compatible with the boundary
conditions only if R(r+ ) = 0, i.e. if r+ is a fourth root of R(r). But then the four roots of R(r)
would be positive (since r+ > 0). Now, since there is no r3 term in the expression (12.25b) of
R(r), the sum of the roots of R(r) is zero, which is impossible with all the roots being positive.
Hence there cannot be three roots in (r+ , +∞). The same argument about the zero sum of
the roots excludes as well the case of four roots of R(r) in (r+ , +∞). There remains then the
cases of no root at all in (r+ , +∞) (case 1 of the lemma) and that of two roots (cases 2 and 3).
These three cases are compatible with the boundary conditions and the vanishing of the sum
of the roots.
Remark 8: Case 4 of Lemma 12.5 can be seen as the limit r1 → r+ of case 3.
Remark 9: In case 4, R(r+ ) = 0 occurs for ΩH ℓ = 1, where ΩH is the black hole rotation velocity, as
given by Eq. (10.69). Similarly, R(r− ) = 0 for Ωin ℓ = 1, where Ωin is the rotation velocity of the inner
horizon, cf. Eq. (10.71).
Remark 10: The region MIII shares with MI that R(r) is non-negative at each of its ends: R(r) ∼
r4 > 0 when r → −∞ and R(r− ) = (r− 2 + a2 − aℓ)2 ≥ 0. However, the argument used to limit the
number of roots in MI cannot be applied to MIII because the latter can accommodate for both positive
and negative values of r and thus four roots of R(r) can be located in MIII .
According to the definition given in Sec. 11.2.6, a r-turning of a null geodesic L is a point
p0 ∈ L such that r0 := r(p) is a simple root of R(r). Lemma 12.5 leads then to:
Property 12.6: r-turning points of null geodesics
A null geodesic in Kerr spacetime has
• at most one r-turning point in region MI ;
• no r-turning point in region MII .
Proof. If a geodesic would have two turning points in MI , this would mean that there exist
two simple roots in MI and that R(r) is positive between them (so that the motion is possible
there). But this is excluded by Lemma 12.5. In region MII , we have ∆ < 0 and Eqs. (12.25a)
12.2 Main properties of null geodesics 459
and (12.20) show that R(r) is the sum of two non-negative terms. The only possibility to have
R(r) = 0 is then that each term vanishes separately: r2 + a2 − aℓ = 0 and q + (ℓ − a)2 = 0, i.e.
r2 = a(ℓ − a) and q = −(ℓ − a)2 .
The second equation implies q ≤ 0. The case q = 0 would lead to ℓ = a and r2 = 0, which is
impossible in MII . There remains q < 0, but according to the results of Sec. 12.2.4, this implies
|ℓ| < a, so that ℓ − a < 0 and the first equation above would yield r2 < 0, which is impossible.
There is thus no r-turning point in MII .
Remark 11: That no r-turning point can exist in MII has been established above by some considerations
on R(r). This property can also be deduced as an immediate consequence of the result (11.97), namely
that r must be a strictly decreasing function of λ at any point in MII .
Let us consider case 2 of Lemma 12.5 (double root of R(r) in MI ); it corresponds to two
distinct situations regarding the null geodesic L having R(r) as radial polynomial. First, L
can lie at a constant value of r, which is necessarily the double root r0 of R(r) by virtue of
Eq. (12.23b); L belongs then to the category of the spherical photon orbits, which will be studied
in Sec. 12.3. If r is not constant along L , then according to the definition in Sec. 11.3.6, L has
as an asymptotic r-value, which is the double root r0 . Given that r0 is the only root of R(r) in
(r+ , +∞), Eq. (12.23b) implies dr/dλ ̸= 0 all along L , so that r(λ) → r0 for λ → +∞ (future
asymptotic value) or λ → −∞ (past asymptotic value). Such a geodesic belong to the category
of the critical null geodesics, which will be studied in Sec. 12.4.1.
In view of the above results, we can state:
Property 12.7: radial behavior of null geodesics
In the region MI of Kerr spacetime, any null geodesic L has one of the following behaviors.
For generic values of the constant of motions (ℓ, q), the possibilities are:
1. L arises from the past null infinity of MI , I − , has a r-turning point, which we may
call the periastron, and terminates at the future null infinity of MI , I + ;
2. L arises from the past null infinity I − , has r decreasing monotonically and crosses
the black hole event horizon H ;
3. L arises from the past event horizon H − separating MI from the white hole region
MII∗ , has r increasing monotonically and terminates at the future null infinity I + ;
4. L arises from the past event horizon H − , has a r-turning point, which we may call
the apoastron, and crosses the black hole event horizon H ;
For some specific values of the constant of motions (ℓ, q), forming a 1-dimensional (hence
zero-measure) subset of the set of all possible valuesa , the possibilities are:
5. L evolves at a fixed value of r;
460 Null geodesics and images in Kerr spacetime
6. L arises from the past null infinity I − and has r decreasing monotonically to a
future asymptotic r-value at r0 > r+ ;
7. L arises from a past asymptotic r-value at r0 > r+ , has r increasing monotonically
and terminates at the future null infinity I + ;
8. L arises from a past asymptotic r-value at r0 > r+ , has r decreasing monotonically
and crosses the black hole event horizon H ;
9. L arises from the past event horizon H − , has r increasing monotonically to a future
asymptotic r-value at r0 > r+ .
a
This subset is given in parametric form by Eqs. (12.56)-(12.57) below.
Case 1 corresponds to a scattering trajectory, leading to the standard phenomenon of
deflection of light. The polynomial R(r) belongs then to case 3 or 4 of Lemma 12.5. Ingoing
principal null geodesics belong to case 2, while the outgoing ones with E ̸= 0 belong to case 3
(cf. Example 5 below). Both cases 2 and 3 correspond to case 1 of Lemma 12.5 (no root of R(r)
in MI ). Cases 5–9 correspond to case 2 of Lemma 12.5 (double root of R(r) in MI ). Case 5 is
that of spherical photon orbits and will be discussed in Sec. 12.3, while cases 6–9 are those of
critical null geodesics, to be discussed in Sec. 12.4.1.
Remark 12: The terminology (periastron, apoastron) employed here agrees with that introduced for the
Schwarzschild case in Sec. 8.2.4.
Example 5 (principal null geodesics): That the principal null geodesics with E ̸= 0 belong to cases 2
and 3 above is clear from their value (12.27) for R(r): R(r) = ρ4 > 0 in all M , which precludes the
existence of any r-turning point nor any r-asymptotic value along these geodesics.
Remark 13: As a sequel of Remark 10 above, a null geodesic can have two turnings points in region
MIII . There can thus exist null geodesics that are trapped between two distinct values of r in MIII .
12.3 Spherical photon orbits
In the Schwarzschild case studied in Chap. 8, circular photon orbits at r = 3m played a central
role in the computation of the black hole shadow and the image of an accretion structure. For
the Kerr black hole, a similar role is played by spherical photon orbits. They are also null
geodesics evolving at a fixed value of r but, contrary to circular photon orbits of Schwarzschild
spacetime, they are not planar in general.
12.3.1 Existence of spherical null geodesics
We shall say that a null geodesic L is spherical or is a spherical photon orbit iff L lies
at a constant value of the coordinate r, r0 say. We have already encountered such geodesics,
namely the null generators L(θ,ψ)out,H
and L(θ,ψ)
out,Hin
of the two Killing horizons H and Hin (cf.
Secs. 10.5.3 and 10.8.3). Indeed, they lie at r0 = r+ for H and r0 = r− for Hin . These peculiar
12.3 Spherical photon orbits 461
spherical orbits have E = 0 [cf. Eq. (11.25)]. From the results of Sec. 12.2.1, there is no other
null geodesic with E = 0 at constant r. Hence we conclude:
Property 12.8: zero-energy spherical photon orbits
In Kerr spacetime, the only spherical photon orbits with E = 0 are the null generators of
the two Killing horizons H and Hin .
Remark 1: A spherical photon orbit of Kerr spacetime is an example of what has been called a
fundamental photon orbit in a generic stationary and axisymmetric spacetime [144, 145], namely a
null geodesic L of affine parameter λ such that (i) L is restricted a spatially compact region and (ii)
there exists Λ ∈ R+ such that
∀λ ∈ R, ∃Φ ∈ R × SO(2), p(λ + Λ) = Φ(p(λ)),
where p(λ) stands for the point of affine parameter λ along L and R × SO(2) is the symmetry group
implementing stationarity and axisymmetry. Basically (i) says that L is a bound orbit and (ii) that L
is periodic, of period Λ in terms of λ, up to some isometry. Note that the definition of a fundamental
photon orbit is coordinate-independent. Introducing coordinates (t, x1 , x2 , φ) adapted to the spacetime
symmetries, the requirement (ii) is equivalent to demanding that x1 and x2 are periodic functions of λ.
In the present case x1 = r and x2 = θ. For a spherical photon orbit, r is obviously a periodic function
of λ, being a constant function, while we shall see in Sec. 12.3.2 that θ is indeed a periodic function of λ.
In the remainder of this section, we focus on spherical photon orbits with E ̸= 0. We shall
then describe these geodesics by the reduced constants of motion ℓ := L/E and q := Q/E 2
introduced in Sec. 12.2.2.
For a spherical photon orbit L , the quartic polynomial R(r) associated to (ℓ, q) by
Eq. (12.25) must obey
R(r0 ) = 0 and R′ (r0 ) = 0. (12.47)
Proof. Since r = r0 with r0 constant, we have dr/dλ′ = 0 where λ′ is the Mino parameter
along L , so that Eq. (12.23b) implies R(r0 ) = 0. If R′ (r0 ) ̸= 0, then r0 would correspond to a
r-turning point of L (cf. Eq. (11.46)), which would contradict r being constant.
In other words, r0 is a double root the polynomial R(r). A spherical photon orbit in MI
correspond then to case 2 of Lemma 12.5 and an example of R(r) is shown for a = 0.95 m and
r0 = 2.2 m in Fig. 12.5b.
In view of expression (12.25a) for R(r), the system (12.47) is equivalent to
(r02 + a2 − aℓ)2 − q̃∆0 = 0 (12.48a)
2r0 (r02 + a2 − aℓ) − q̃(r0 − m) = 0, (12.48b)
where ∆0 := r02 − 2mr0 + a2 and q̃ is the reduced Carter constant K :
K
q̃ := = q + (ℓ − a)2 , (12.49)
E2
462 Null geodesics and images in Kerr spacetime
where the second equality follows from relation (11.34) between K and Q. The system (12.48)
has 2 equations for 3 unknowns: (r0 , ℓ, q). We thus expect a one-parameter family of solutions.
It is convenient to consider r0 as the parameter and to solve the system (12.48) for (ℓ, q). We
shall distinguish the case r0 = m from r0 ̸= m. If r0 = m, the system (12.48) reduces to
(m2 + a2 − aℓ)2 − q̃(a2 − m2 ) = 0 ℓ = a + m2
a
⇐⇒
2 2
m + a − aℓ = 0 q̃ = 0 or a = m.
Now, q̃ = 0 is equivalent to q = −(ℓ−a)2 = −m4 /a2 , which implies q < 0, which is impossible
2
for ℓ = a + ma > a, due to the property |ℓ| > a =⇒ q ≥ 0 (cf. Sec. 12.2.4). There remains
a = m, which implies ℓ = 2m. Hence we conclude
a=m
r0 = m ⇐⇒ (12.50)
ℓ = 2m.
We shall discuss this case further in Sec. 12.4.4, as well as in Chap. 13, which is devoted to
the extreme Kerr black hole (a = m). In the remainder of this section, we assume r0 ̸= m.
Equation (12.48b) is then equivalent to
2r0 (r02 + a2 − aℓ)
q̃ = . (12.51)
r0 − m
Substituting this relation into Eq. (12.48a), we get an equation involving ℓ only:
2 2 2 2 2r0
(r0 + a − aℓ) r0 + a − aℓ − ∆0 = 0.
r0 − m
The two solutions are immediate:
r02
ℓ=a+ (12.52)
a
or
r0
ℓ=a+ [r0 (r0 − m) − 2∆0 ] . (12.53)
a(r0 − m)
The solution (12.52), once inserted in (12.51) leads to q̃ = 0, i.e. to q = −(ℓ − a)2 ≤ 0. Now,
q < 0 is excluded since Eq. (12.52) implies |ℓ| ≥ a (cf. Sec. 12.2.4). There remains q = 0, which
yields ℓ = a. Then Eq. (12.52) leads to r0 = 0. However, according to the results stated in
Sec. 12.2.4, q = 0 and ℓ = a imply that the geodesic is confined in the equatorial plane, where
r0 = 0 corresponds to the ring singularity. We thus conclude that the solution (12.52) is not
permitted. The remains then the solution (12.53). Substituting it into Eq. (12.51), we get
4r02 ∆0
q̃ = , (12.54)
(r0 − m)2
so that Eqs. (12.49) and (12.53) yield, after simplifications,
r03
(12.55)
3 2 2 2
q= 2 −r0 + 6mr0 − 9m r 0 + 4a m .
a (r0 − m)2
12.3 Spherical photon orbits 463
2.0
q q
25 ` `
|`| = a 1.5 |`| = a
20
1.0
15
`/m, q/m 2
`/m, q/m 2
0.5
10 rph∗ ∗ rph∗ rph+
0.0
5
rphms
rph∗ ∗ rph∗ rph+ rph− 0.5
0
1.0
5
1.5
1 0 1 2 3 4 1.0 0.5 0.0 0.5 1.0 1.5
r0 /m r0 /m
Figure 12.6: Functions ℓc (r0 ) (in red) and qc (r0 ) (in blue) giving the reduced angular momentum and reduced
Carter constant of a spherical photon orbit of radius r0 , according to Eqs. (12.56) and (12.57) and for a = 0.95 m.
The right figure is a zoom on the part −m ≤ r0 ≤ 3m/2. The thick vertical lines mark the two horizons: H
(black) and Hin (light brown). The horizontal dotted lines mark the boundary of the region |ℓ| < a, where q can
take negative values. Values of q in the grey zone are unphysical, i.e. do not fulfill the conditions (12.58), so that
only values of r0 for which the blue curve lies above the grey zone correspond to spherical photon orbits. This
−
occurs for rph
∗∗ ∗
≤ r0 ≤ rph and rph
+
≤ r0 ≤ rph . The thick segments of the ℓ and q curves correspond to stable
orbits, for which 0 ≤ r0 ≤ rph . [Figure generated by the notebook D.5.11]
ms
Recasting expressions (12.53) and (12.55), we conclude that a spherical photon orbit at r =
r0 ̸= m has a reduced angular momentum ℓ and a reduced Carter constant q given by
r02 (3m − r0 ) − a2 (r0 + m)
ℓ = ℓc (r0 ) := , (12.56)
a(r0 − m)
r03
(12.57)
2
4a m − r0 (r0 − 3m)2 .
q = qc (r0 ) := 2 2
a (r0 − m)
For a solution to exist, the constraint (12.20) must be obeyed; it writes q̃ ≥ 0. Given expression
(12.54) for q̃, we see that it is equivalent ∆0 ≥ 0. Hence spherical photon orbits do not exist in
region MII of Kerr spacetime. This is in agreement with the general result (11.97).
Equations (12.56) and (12.57) provide the general solution to the system R(r0 ) = 0 and
R (r0 ) = 0 [Eq. (12.47)], but not all of them correspond to spherical photon orbits. Indeed, we
′
do not expect spherical photon orbits to exist for any value of r0 , in particular for |r0 | ≫ m.
Actually, not all values of q given by Eq. (12.57) are permitted, but only those that fulfill the
constraints established in Sec. 12.2.4, namely
q≥0 if |ℓ| ≥ a (12.58a)
q ≥ − (a − |ℓ|) 2
if |ℓ| < a. (12.58b)
The solutions ℓ and q given by Eqs. (12.56)-(12.57) are plotted as functions of r0 for a = 0.95 m
in Fig. 12.6, where the region excluded by (12.58) is colored in grey. Consequently spherical
photon orbits exist only for values of r0 for which the q curve (in blue) lies above the grey
region. We see that this occurs in three intervals:
∗∗
r0 ∈ [rph , 0) , ∗
r0 ∈ (0, rph ] and +
r0 ∈ [rph −
, rph ], (12.59)
464 Null geodesics and images in Kerr spacetime
where rph∗
, rph
+
and rph
−
are the three (ordered) roots distinct from 0 of the equation qc (r0 ) =
0 (boundary for condition (12.58a)) and rph ∗∗
is the unique root of the equation qc (r0 ) =
− (a − |ℓc (r0 )|) when |ℓc (r0 )| < a (boundary for condition (12.58b)). The above reason-
2
ing is based on Fig. 12.6, which has been drawn for a = 0.95 m; however the conclusions (12.59)
hold for any value of a (see the notebook D.5.11 for figures with a = 0.5 m or a = 0.998 m). We
have excluded r0 = 0 in (12.59) because it would yield ℓ = a and q = 0 following Eqs. (12.56)-
(12.57). But according to the results of Sec. 12.2.4, such an orbit would be confined to the
equatorial plane, where r = 0 is not permitted (the ring singularity!).
Given expression (12.57) for q, we see that rph∗
, rph
+
and rph
−
are the three roots of the cubic
equation
r0 (r0 − 3m)2 − 4a2 m = 0. (12.60)
We can solve this equation by bringing it to a depressed form in order to use Viète’s formu-
las (8.22). However, we may rely on an already solved equation by noticing the following
equivalences:
√ √
r0 (r0 − 3m)2 − 4a2 m = 0 ⇐⇒ r0 ≥ 0 and r0 |r0 − 3m| = 2a m
√ √
⇐⇒ r0 ≥ 0 and r0 (r0 − 3m) ± 2a m = 0
3/2 1/2 √
⇐⇒ r0 ≥ 0 and r0 − 3mr0 ± 2a m = 0,
where ± is + for r0 ≤ 3m and − for r0 ≥ 3m. We recognize the function of r0 which appears
in the left-hand side of Eq. (11.141). As shown in Sec. 11.5.2, there are three real roots, rph
∗
, rph
+
and rph
−
, with rph
∗ +
, rph ≤ 3m (± = +) and rph −
≥ 3m (± = −). They are given by Eqs. (11.144)
and (11.143):
a 4π
2 1
∗
rph := 4m cos arccos − + . (12.61)
3 m 3
a
2 1
±
rph := 4m cos arccos ∓ (12.62)
3 m
As for rph
∗∗
, since qc (r0 ) = −(a − |ℓc (r0 )|)2 with |ℓc (r0 )| < a occurs in a region where ℓc (r0 ) < 0
(cf. Fig. 12.6), we get that rph∗∗
is a solution of the equation qc (r0 ) = −(a + ℓc (r0 ))2 . Given
expressions (12.56) and (12.57) for respectively ℓc (r0 ) and qc (r0 ), we get that rph ∗∗
is a root of the
cubic equation
2r03 − 3mr02 + a2 m = 0. (12.63)
By the change of variable r0 =: x + m/2, we turn this equation into a depressed one:
m2
3 3 2 m 2
x − m x+ a − = 0,
4 2 2
i.e. x3 + px + q = 0, with p := −3m2 /4 and q := (m/2)(a2 − m2 /2). The discriminant is
−(4p3 + 27q 2 ) = 27a2 m2 (m2 − a2 )/4 ≥ 0. The three roots (xk )k∈{0,1,2} are then all real and
are given by Viète’s formula (8.22). Only the root x1 leads to a negative value of r0 , which is
the value we are looking for (cf. Fig. 12.6). Viète’s formula (8.22) with k = 1 yields
a2
a 2π
1 2π 2
x1 = m cos arccos 1 − 2 2 + = m cos arcsin + ,
3 m 3 3 m 3
12.3 Spherical photon orbits 465
3
rph+
rph−
rph∗
2 rph∗ ∗
r0 /m
rphms
r+
r−
1
0.0 0.2 0.4 0.6 0.8 1.0
a/m
Figure 12.7: Domain of existence of spherical photon orbits in the (a, r0 ) plane (in green). The boundaries of
−
the domain are the radii rph , rph , rph
∗∗ ∗ +
and rph given by Eqs. (12.64), (12.61) and (12.62). The shaded area correspond
to stable spherical orbits; its upper boundary (blue curve) is the radius rph ms
given by Eq. (12.78). The black curve
indicates the black hole horizon H and the light brown one the inner horizon Hin . [Figure generated by the
notebook D.5.11]
from which we obtain
a 2π
m 2
∗∗
rph = + m cos arcsin + . (12.64)
2 3 m 3
The four critical radii rph
∗∗
, rph
∗
, rph
+
and rph
−
are plotted in terms of a in Fig. 12.7. We notice
that
m
− ≤ rph ∗∗
≤ 0 ≤ rph ∗
≤ r− ≤ m ≤ r+ ≤ rph +
≤ 3m ≤ rph −
≤ 4m , (12.65)
2
with the limits (11.145) and (11.146). In addition,
m
∗∗
lim rph =0 and ∗∗
lim rph =− . (12.66)
a→0 a→m 2
As already stressed in Sec. 11.5.2, rph
∗
is lower than, but very close to, the inner horizon radius
r− , with max(r− − rph ) ≃ 0.032 m, achieved for a ≃ 0.9 m. In view of the above inequalities
∗
and the ranges (12.59), we conclude:
Property 12.9: spherical photon orbits with E ̸= 0
Spherical photon orbits with E ̸= 0 exist in two regions of Kerr spacetime:
• orbits with r0 ∈ [rph
∗∗ ∗
, 0) ∪ (0, rph ] are located in MIII ; we shall call them the inner
spherical photon orbits;
466 Null geodesics and images in Kerr spacetime
• orbits with r0 ∈ [rph
+ −
, rph ] are located in MI ; we shall call them the outer spherical
photon orbits.
Sign of E
All E ̸= 0 spherical photon orbits lie in MI ∪ MIII . We may then apply the general result
(11.69) to them. Since L = Eℓ, we get E(r02 + a2 − aℓ) > 0. Hence, using formula (12.56) for ℓ,
E > 0 ⇐⇒ r02 + a2 − aℓ > 0
r0 ∆0
⇐⇒ >0
r0 − m
r0
⇐⇒ >0
r0 − m
⇐⇒ r0 > m or r0 < 0,
where the third line follows from ∆0 > 0 in MI ∪ MIII . In view of (12.65), we conclude:
Property 12.10: energy sign of spherical photon orbits
All outer spherical photon orbits have E > 0, as well as inner spherical photon orbits with
∗∗
r0 ∈ [rph , 0), while inner spherical photon orbits with r0 ∈ (0, rph
∗
] have E < 0.
12.3.2 Latitudinal motion
The latitudinal motion of spherical photon orbits is deduced from the general results of
Sec. 12.2.4, where geodesics with ℓ = 0 appear as a special case. We thus treat this case
first.
Polar spherical photon orbits
We see on Fig. 12.6 that the reduced conserved angular momentum ℓ (red curve) vanishes at
two places in the physically allowed range of r0 , i.e. where the blue curve lies above the grey
region: r0 = rph pol,in
and r0 = rph
pol
with rph
∗∗ pol,in
< rph < 0 and rph
+ pol
< rph −
< rph . The value rph
pol,in
(resp. rph
pol
) corresponds thus to inner (resp. outer) spherical orbits. The results of Sec. 12.2.4
show that these orbits are the only ones that encounter the rotation axis; moreover, they cross
it repeatedly. We therefore call them polar spherical photon orbits.
The values of rph pol,in
and rph
pol
are obtained by solving ℓc (r0 ) = 0, with ℓc (r0 ) given by
Eq. (12.56). We get the cubic equation r03 − 3mr02 + a2 r0 + a2 m = 0. Setting r0 =: x + m
reduces it to the depressed cubic x3 + px + q = 0, with p := a2 − 3m2 and q := 2m(a2 − m2 ).
The discriminant ∆ = −4p3 − 27q 2 being positive, the solutions are provided by Viète’s
formulas, Eq. (8.22), with k = 0 and k = 1 (k = 2 leads to the third root of ℓc (r0 ) seen on
12.3 Spherical photon orbits 467
Fig. 12.6, which lies in the unphysical range of r0 — blue curve in the grey region):
r " !#
2 2 2
a 1 m(m − a )
pol
rph = m + 2 m2 − cos arccos (12.67a)
3 3 (m2 − a2 /3)3/2
r " ! #
2 2 2
a 1 m(m − a ) 2π
pol,in
rph = m + 2 m2 − cos arccos + . (12.67b)
3 3 (m2 − a2 /3)3/2 3
As a function of a, rph
pol
decays monotonically from lima→0 rph
pol
= 3m to lima→m rphpol
=
√
(1 + 2)m ≃ 2.414214 m, while rph decays monotonically from lima→0 rph
pol,in pol,in
= 0 to
pol,in
√
lima→m rph = (1 − 2)m ≃ −0.414214 m.
Sign of ℓ and L
For spherical orbits with ℓ ̸= 0, the sign of ℓ is deduced from expression (12.56), whose
numerator has a sign governed by the position of r0 with respect to the roots rph
pol,in
and rph
pol
determined above and whose denominator is positive iff r0 > m, which occurs only for outer
spherical orbits. We get then
Property 12.11: sign of ℓ for spherical photon orbits
• Outer spherical orbits with r0 ∈ [rph
+ pol
, rph ) have ℓ > 0;
• outer spherical orbits with r0 ∈ (rph , rph ] have ℓ < 0;
pol −
• inner spherical orbits with r0 ∈ (rph , rph ] have ℓ > 0;
pol,in ∗
• inner spherical orbits with r0 ∈ [rph
∗∗ pol,in
, rph ) have ℓ < 0.
The sign of L is deduced from that of ℓ = L/E by combining with the sign of E obtained in
Sec. 12.3.1.
Latitudinal motion
By applying the results of Sec. 12.2.4, we get
Property 12.12: latitudinal behavior of spherical photon orbits
• All outer spherical photon orbits with r0 ̸∈ rph , rph and the inner ones with
+ −
r0 ∈ (0, rph ) have q > 0 (cf. Fig. 12.6); they therefore cross the equatorial plane.
∗
Moreover,
◦ those with r0 = rphpol
(outer polar spherical photon orbits) have ℓ = 0 and cross
repeatedly the rotation axis and the equatorial plane, with θ taking all values
in the range [0, π];
468 Null geodesics and images in Kerr spacetime
◦ those with r0 ̸= rph
pol
oscillate about the equatorial plane, having two θ-turning
points symmetrical about it, at θ = θm and θ = π − θm , with θm given by
Eq. (12.41), in which ℓ and q are to be considered as the functions (12.56)-(12.57)
of r0 .
• Inner spherical photon orbits with r0 ∈ (rph ∗∗
, 0) have q < 0 (cf. right panel of
Fig. 12.6); they are thus vortical and never encounter the equatorial plane. Moreover,
◦ those with r0 = rphpol,in
(inner polar spherical photon orbits) have ℓ = 0 and
oscillate about the rotation axis, with a θ-turning point at θ = θv (North-
ern hemisphere) or θ = π − θv (Southern hemisphere), where θv is given by
Eq. (12.43) in which q is to be considered as the function (12.57) of rph
pol,in
.
◦ those with r0 ̸= rph
pol,in
neither encounter the rotation axis nor the equatorial
plane, having either θ ∈ [θm , θv ] (Northern hemisphere) or θ ∈ [π − θv , π − θm ]
(Southern hemisphere), with θm and θv given by Eqs. (12.41) and (12.42), in
which ℓ and q are to be considered as the functions (12.56)-(12.57) of r0 .
Spherical photon orbits at r0 = rph ∗∗
, rph
∗
, rph
+
or rph
−
, which have been excluded from the
above list, will be discussed in details in Sec. 12.3.3.
In addition, we have
Property 12.13: periodicity of the θ-motion
For a spherical photon orbit, θ is either a constant or a periodic function of the affine
parameter λ, the period being
θmax
r02 + a2 cos2 θ
Z
2
Λθ = q dθ, (12.68)
|E| θmin Θ̃(θ)
with (θmin , θmax ) = (0, π) (outer polar orbit), (0, θv ) (Northern inner polar orbit), (π −θv , π)
(Southern inner polar orbit), (θm , π − θm ) (non-polar orbit with r0 ∈ (0, rph ∗ +
) ∪ (rph −
, rph )),
(θm , θv ) (Northern non-polar vortical inner orbit) or (π − θv , π − θm ) (Southern non-polar
vortical inner orbit).
Proof. Switching from the Mino parameter λ′ to the affine parameter λ via Eq. (11.49) with
r(λ) = r0 , we may rewrite the equation of motion (12.23c) as
2
E 2 Θ̃(θ)
dθ
+ V(θ) = 0, with V(θ) := − .
dλ (r02 + a2 cos2 θ)2
This is the 1-dimensional equation of motion in the potential well V. It is then clear that θ(λ)
q is a
periodic function. The period Λθ is evaluated by integrating dλ = ϵθ (r02 +a2 cos2 θ)2 /(|E| Θ̃(θ))
over a “round-trip” to the same values of θ and dθ/dλ.
12.3 Spherical photon orbits 469
Figure 12.8: Spherical photon orbit at r0 = 1.6 m around a Kerr black hole with a = 0.95 m, depicted in terms
of the Cartesian Boyer-Lindquist coordinates (x, y, z) defined by Eq. (11.73) and scaled in units of m. The geodesic
starts at λ = 0 and φ = 0 in the equatorial plane, in the direction of the Southern hemisphere (dθ/dλ > 0). The
left panel corresponds to 0 ≤ λ ≤ 7.7 m/E, while the right one extends the range to 0 ≤ λ ≤ 70 m/E. The grey
sphere is the black hole event horizon. [Figure generated by the notebook D.5.12]
Remark 2: The azimuthal coordinate φ is not a periodic function of the affine parameter λ in general,
nor of the Mino parameter λ′ . An exception regards polar spherical geodesics, since setting ℓ = 0 in the
equation of motion (12.23d) results in dφ/dλ′ = const, so that φ is a periodic function of λ′ (but still
not of λ).
Example 6 (Outer spherical photon orbits): Let us consider a Kerr black hole with a = 0.95 m (same
value as in Fig. 12.6). The event and inner horizon radii are r+ = 1.312 m and r− = 0.688 m and one
∗∗ = −0.478 m, r ∗ = 0.658 m, r + = 1.386 m and r − = 3.955 m. Some outer spherical photon
has rph ph ph ph
orbits are plotted in Figs. 12.8–12.11 for various values of r0 :
• r0 = 1.6 m (Fig. 12.8): this orbit has ℓ = 2.171 m, q = 5.976 m2 , θm = 0.706 rad and dφ/dλ > 0;
• r0 = 2.8 m (Fig. 12.9): this orbit has ℓ = −1.089 m, q = 26.260 m2 and θm = 0.206 rad; φ(λ) is
not a monotonic function (cf. the projection onto the xy-plane): dφ/dλ < 0 almost everywhere,
except near the equator, where the Lense-Thirring effect (cf. Sec. 11.3.4) is the strongest2 and
enforces dφ/dλ > 0;
• r0 = 3m (Fig. 12.10): this orbit has ℓ = −1.9 m, q = 27 m2 , θm = 0.346 rad and dφ/dλ ≤ 0,
with dφ/dλ = 0 in the equatorial plane, as it can be easily checked by plugging r = 3m, θ = π/2
as well as the above values of ℓ and q into Eq. (12.23d); this orbit maximizes the value of q among
all spherical photon orbits (cf. Fig. 12.6 and Eq. (12.83) below);
• r0 = 3.9 m (Fig. 12.11): this orbit has ℓ = −6.574 m, q = 3.525 m2 , θm = 1.290 rad and
dφ/dλ < 0.
The next example is devoted to inner spherical photon orbits. The Cartesian Boyer-Lindquist
coordinates (t, x, y, z) are not well suited to depict these orbits, in particular those that have
2
This can be seen by considering the limit θ → π/2 in Eq. (12.23d).
470 Null geodesics and images in Kerr spacetime
y/m
2
x/m
2 1 1 2
Figure 12.9: Spherical photon orbit at r0 = 2.8 m around a Kerr black hole with a = 0.95 m, depicted in terms
of the Cartesian Boyer-Lindquist coordinates (x, y, z) defined by Eq. (11.73) and scaled in units of m, with the
grey sphere representing the black hole event horizon. The right panel depicts the projection of the orbit onto the
xy-plane (the overlap with the grey area is a mere projection effect, since of course the orbit never crosses the
event horizon). The geodesic starts at λ = 0 and φ = 0 in the equatorial plane, in the direction of the Southern
hemisphere. The plotted range is 0 ≤ λ ≤ 38 m/E. [Figure generated by the notebook D.5.12]
y/m
2
x/m
1 1 2 3
Figure 12.10: Same as in Fig. 12.9 but for a spherical photon orbit at r0 = 3m, with 0 ≤ λ ≤ 32 m/E. [Figure
generated by the notebook D.5.12]
12.3 Spherical photon orbits 471
y/m
4
3
2
1
x/m
4 3 2 1 1 2 3 4
1
2
3
4
Figure 12.11: Same as in Fig. 12.9 but for a spherical photon orbit at r0 = 3.9 m, with 0 ≤ λ ≤ 55 m/E.
[Figure generated by the notebook D.5.12]
Figure 12.12: Marginally stable spherical photon orbit inside a Kerr black hole with a = 0.95 m, depicted
in terms of the coordinates (x̂, ŷ, ẑ) defined by Eq. (12.69) and scaled in units of m. The geodesic is located
at r0 = 0.540 m and starts at λ = 0 and φ = 0 in the equatorial plane, in the direction of the Southern
hemisphere (dθ/dλ > 0). The left panel corresponds to 0 ≤ λ ≤ 3 m/|E|, while the right one extends the range
to 0 ≤ λ ≤ 20, m/|E|. The orange red circle is the ring singularity and the vertical black line is the rotation axis,
with a black dot marking the asymptotic end r → −∞. [Figure generated by the notebook D.5.12]
r < 0. We could use O’Neill exponential coordinates, as in Fig. 12.2. However, we are going
to use instead the radial coordinate r̂ and the associated Cartesian-type coordinates (x̂, ŷ, ẑ)
defined by
1 √
r̂ := r + r2 + 4m2 (12.69a)
2
x̂ := r̂ sin θ cos φ, ŷ := r̂ sin θ sin φ, ẑ := r̂ cos θ. (12.69b)
As O’Neill coordinates, this brings the whole range (−∞, +∞) for r to (0, +∞) for r̂, with
r̂ → 0 corresponding to r → −∞. Contrary to O’Neill coordinate er/m , the coordinate r̂ does
not enlarge too much the region exterior to the black hole, which makes it better suited for
plots covering both the black hole interior and exterior, as in Fig. 12.17 below. Note that in MI ,
r̂ is asymptotically equivalent to r: r̂ ∼ r as r → +∞.
472 Null geodesics and images in Kerr spacetime
Figure 12.13: Same as in Fig. 12.12 but for a vortical spherical photon orbit at r0 = −0.46 m, with 0 ≤ λ ≤
20 m/E. Also shown is the Northern vortical circular photon orbit at r = rph
∗∗
(in light green). [Figure generated
by the notebook D.5.12]
6
4
2
0 `ph∗
`/m
`ph+
2 `ph−
4
6
8
0.0 0.2 0.4 0.6 0.8 1.0
a/m
Figure 12.14: Reduced angular momentum ℓ of the three circular photon orbits in the equatorial plane, as a
function of the Kerr spin parameter a. [Figure generated by the notebook D.5.11]
Example 7 (Inner spherical photon orbits): We consider the same Kerr spacetime as in Example 6,
i.e. a = 0.95 m, but this time, we focus on inner spherical photon orbits:
• r0 = 0.540 m (Fig. 12.12): this orbit has ℓ = 1.539 m, q = 0.282 m2 , θm = 1.173 rad dφ/dλ > 0,
E < 0 and L < 0; it is actually a marginally stable photon orbit (r0 = rph ms ), as we shall see in
Sec. 12.3.4;
• r0 = −0.46 m (Fig. 12.13): this orbit has ℓ = −0.176 m, q = −0.461 m2 (hence it is vortical),
θm = 0.282 rad, θv = 0.731 rad, dφ/dλ < 0, E > 0 and L < 0;
• r0 = −0.478 m = rph ∗∗ (Fig. 12.13, light green curve): this orbit has ℓ = −0.229 m, q = −0.519 m2
(hence it is vortical), θm = θv = θph ∗∗ = 0.514 rad, dφ/dλ < 0, E > 0 and L < 0.
12.3 Spherical photon orbits 473
12.3.3 Circular photon orbits
In any stationary and axisymmetric spacetime, such as the Kerr spacetime, one may define a
circular photon orbit as a null geodesic L whose tangent vector field p = dx/dλ is a linear
combination of the two Killing vectors ξ and η generating respectively the stationarity and the
axisymmetry, with a non-vanishing component along η:
p = αξ + βη, with β ̸= 0. (12.70)
Note that the above definition is independent from any coordinate system. It is worth to
compare Eq. (12.70) with expression (11.178) for the 4-velocity u = µ−1 p of a circular timelike
orbit in the equatorial plane.
Remark 3: Circular photon orbits are sometimes called light rings [144].
In Kerr spacetime, when using coordinates adapted to the spacetime symmetries, such as
Boyer-Lindquist coordinates (t, r, θ, φ), one has ξ = ∂t , η = ∂φ , α = pt , β = pφ and the
definition (12.70) is equivalent to pr = dr/dλ = 0 and pθ = dθ/dλ = 0. It follows immediately
that a circular photon orbit is any null geodesic along which both r and θ are constant. It is
thus a spherical photon orbit lying at a constant value of θ.
Remark 4: In Schwarzschild spacetime, the photon circular orbits forming the photon sphere at r = 3m
(cf. Sec. 8.2.3) do not have θ = const, except for the orbit in the equatorial plane. However, for a
given non-equatorial orbit on the photon sphere, one may use the spherical symmetry of Schwarzschild
spacetime to perform a change of coordinates (θ, φ) 7→ (θ′ , φ′ ) such that θ′ = const is constant for that
orbit. In Kerr spacetime with a ̸= 0, such “oblique” circular orbits cannot exist due to Lense-Thirring
precession.
Example 8: The spherical photon orbits with E = 0 discussed in Sec. 12.3.1, namely the null generators
of the horizons H and Hin are circular photon orbits, since they are the outgoing principal null
out,H out,Hin
geodesics L(θ,ψ) and L(θ,ψ) , which have θ = const.
The spherical photon orbits with r0 = rph∗
, rph
+
and rph
−
have q = 0 and |ℓ| > a. According
to the results of Sec. 12.2.4, they necessary lie in the equatorial plane θ = π/2. They are
thus circular. According to expression (12.57) for q, they are the only circular orbits in the
equatorial plane, because the only other root of qc (r0 ) = 0 is r0 = 0 (see also Fig. 12.6), which
would correspond to the ring singularity. Let us denote by ℓ∗ph , ℓ+ph and ℓph the reduced angular
−
momentum ℓ of respectively the circular photon orbit at rph ∗
, rph
+
and rph
−
. These values of ℓ are
deduced from Eq. (12.56) and are plotted in terms of a in Fig. 12.14. We have ℓ∗ph > 0, ℓ+ ph > 0
and ℓph < 0. Moreover,
−
√
lim ℓ+ −
ph = lim ℓph = 3 3m ≃ 5.196 m, (12.71)
a→0 a→0
in agreement with the Schwarzschild result (8.21), while
lim ℓ+ = lim ℓ∗ph = 2m, (12.72)
a→m ph a→m
in agreement with Eq. (12.50), since lima→m rph
+ ∗
= lima→m rph = m.
474 Null geodesics and images in Kerr spacetime
0.5
0.4
0.3
θph∗ ∗
0.2
0.1
0.0
0.0 0.2 0.4 0.6 0.8 1.0
a/m
Figure 12.15: Angle θ of the Northern vortical circular photon orbit at r = rph
∗∗
as a function of Kerr spin
parameter a. [Figure generated by the notebook D.5.11]
The spherical orbits with r0 = rph ∗∗
have q < 0, i.e. are vortical geodesics. Moreover, they
fulfill q = qmin = −(a − |ℓ|) . According to the results of Sec. 12.2.4, this corresponds to two
2
orbits at a fixed value of θ, which are symmetrical with respect to the equatorial plane. They lie
at θ = θph∗∗
(Northern hemisphere) and θ = π − θph ∗∗
(Southern hemisphere), where θph ∗∗
is given
by Eq. (12.44), using for ℓ the value (12.56) with r0 = rph . Since 3m(rph ) = 2(rph ) + a2 m by
∗∗ ∗∗ 2 ∗∗ 3
virtue of Eq. (12.63), some simplification occurs and we get
s
∗∗ ∗∗ 2
|rph | (rph )
∗∗
θph = arcsin ∗∗
1− 2
. (12.73)
m − rph a
∗∗
θph is an increasing function of a/m, plotted in Fig. 12.15. We have lima→0 θph
∗∗
= 0 (the rotation
axis) and lima→m θph = π/6. Since they occur at a fixed value of θ, the two vortical spherical
∗∗
photon orbits are actually circular. Such an orbit is shown in Fig. 12.13 (light green circle).
The reduced angular momentum and Carter constant of these orbits, denoted by ℓ∗∗ ph and qph
∗∗
respectively, are depicted in terms of a in Fig. 12.16. They tend to zero as a → 0 and obey
m 9 2
lim ℓ∗∗ and
ph = −
∗∗
lim qph =− m. (12.74)
4
a→m a→m 16
We may summarize the above results by
Property 12.14: circular photon orbits in Kerr spacetime
The geodesics at the boundaries of the domain of existence of spherical photon orbits
[Eq. (12.59)] are circular orbits:
• the orbit at (r, θ) = (rph
∗∗ ∗∗
, θph ) is called the Northern vortical circular photon orbit;
• the orbit at (r, θ) = (rph
∗∗ ∗∗
, π − θph ) is called the Southern vortical circular photon
orbit;
12.3 Spherical photon orbits 475
0.0
`ph∗ ∗
qph∗ ∗
0.1
`ph∗ ∗ /m, qph∗ ∗ /m 2
0.2
0.3
0.4
0.5
0.0 0.2 0.4 0.6 0.8 1.0
a/m
Figure 12.16: Reduced angular momentum ℓ∗∗
ph and reduced Carter constant qph of the two vortical circular
∗∗
photon orbits at r = rph
∗∗
, as functions of Kerr spin parameter a. [Figure generated by the notebook D.5.11]
• the orbit at (r, θ) = (rph
∗
, π/2) is called the equatorial inner circular photon orbit;
• the orbit at (r, θ) = (rph
+
, π/2) is called the prograde outer circular photon orbit;
• the orbit at (r, θ) = (rph
−
, π/2) is called the retrograde outer circular photon orbit;
Moreover, there is no other circular photon orbit with E ̸= 0. The circular photon orbits
with E = 0 are the null generators of the two Killing horizons H and Hin .
That the orbits listed above are the only E ̸= 0 circular orbits follows from the fact that all the
other E ̸= 0 spherical photon orbits have either q > 0 or qmin < q < 0 (cf. Fig. 12.6), which
imply that they have a varying θ (cf. Sec. 12.2.4).
12.3.4 Stability of spherical photon orbits
The radial stability of spherical photon orbits is derived by the same argument as that used in
Sec. 11.5.3 for timelike circular orbits: a spherical photon orbit at r = r0 is stable iff no geodesic
motion with the same values of the conserved quantities ℓ and q is possible in the vicinity of
r0 except for r = r0 . Given the same value of (ℓ, q) imply the same polynomial R and that a
geodesic motion is possible only if R(r) ≥ 0 [Eq. (12.46)], the above criteria is equivalent to
R(r) < 0 for r distinct from r0 but close to it. Since R(r0 ) = 0 and R′ (r0 ) = 0 [Eq. (12.47)],
this is equivalent to R having a maximum at r0 . In other words:
Property 12.15: stability criterion for spherical photon orbits
The spherical photon orbit of radius r0 is stable ⇐⇒ R′′ (r0 ) < 0. (12.75)
Remark 5: The criterion (11.159) for timelike circular orbits involves V ′′ (r0 ) > 0 simply because
476 Null geodesics and images in Kerr spacetime
of the minus sign in the definition of V from R: V(r) := −R(r)/(µ2 r4 ), whereas we have here
R(r) := R(r)/E 2 .
An immediate consequence of Lemma 12.5 is
Property 12.16: unstability of spherical photon orbits in the black hole exterior
All spherical photon orbits in the black hole exterior are unstable.
Proof. The quartic polynomial R(r) of spherical photon orbits in MI belong to case 2 of
Lemma 12.5, which states that R(r) > 0 for r ∈ (r+ , r0 ) ∪ (r0 , +∞), i.e. r0 corresponds to a
minimum of R (cf. Fig. 12.5b).
We are however going to see that there exist stable spherical photon orbits in region MIII .
From expression (12.25b) for R, we get R′′ (r0 ) = 12r02 + 2(a2 − ℓ2 − q). Substituting the values
(12.56) and (12.57) for respectively ℓ and q yields
a2
8r0
′′
R (r0 ) = 3 3
(r0 − m) + m 1 − 2 . (12.76)
(r0 − m)2 m
Hence the stability criterion (12.75) becomes
a2
0
r < 0 and (r0 − m) 3
+ m3
1− m2
>0
R′′ (r0 ) < 0 ⇐⇒ or (12.77)
r > 0 and (r − m)3 + m3 1 −
a2
0 0 m2
< 0.
Now
1/3
a2 a2
3 3 ms
(r0 − m) + m 1 − 2 > 0 ⇐⇒ r0 − m > −m 1 − 2 ⇐⇒ r0 > rph ,
m m
where " 1/3 #
a2
ms
rph := m 1 − 1 − 2 . (12.78)
m
Since obviously rph
ms
≥ 0, we conclude that the first case in (12.77) is excluded and that the
second case holds for 0 < r0 < rph
ms
:
Property 12.17: stability region for spherical photon orbits
A spherical photon orbit of radius r0 is stable ⇐⇒ 0 < r0 < rph
ms
. (12.79)
The index ‘ms’ in rph
ms
stands for marginally stable. rph
ms
is plotted as a function of a in Fig. 12.7.
By comparing the blue solid curve with the green dotted one in that figure, we note that
ms
rph ∗
≤ rph , (12.80)
with equality iff a = m. We conclude:
12.3 Spherical photon orbits 477
Property 12.18: stability of spherical photon orbits
All spherical photon orbits are unstable with respect to radial perturbations, except for a
subclass of inner orbits with negative energy (E < 0): those that have a radius r0 ∈ (0, rph
ms
),
where rph is given by Eq. (12.78). In particular, all spherical photon orbits in MI (the black
ms
hole exterior) are unstable, as well as all circular photon orbits discussed in Sec. 12.3.3.
That all stable orbits have E < 0 follows from Property 12.10, given the inequality (12.80).
The stable spherical photon orbits have q > 0 and ℓ > 0 (cf. right panel of Fig. 12.6).
According to the results of Sec. 12.2.4, they thus oscillate symmetrically about the equatorial
plane, between two θ-turning points, θm and π − θm , such that 0 < θm < π/2. In particular,
r0 = rms does not correspond to a unique orbit, but to a 1-parameter family of marginally stable
orbits; the parameter can be chosen to be the azimuthal coordinate φ at the first value θ = θm
(or θ = π/2) after t = 0. A marginally stable spherical photon orbit is shown in Fig. 12.12.
We have the following property, which appears clearly on Fig. 12.6:
Property 12.19: characterization of marginally stable spherical photon orbits
Among all inner spherical photon orbits, the marginally stable orbits at r0 = rph
ms
are those
for which the reduced angular momentum ℓ and reduced Carter constant q are maximal.
Proof. From Eqs. (12.56) and (12.57), we get
h i
a2
dqc 4r02 (r0 − 3m) (r0 − m) + m 3 3
1− m2
=− (12.81)
dr0 a2 (r0 − m)3
a2
dℓc (r0 − m)3 + m3 1 − m2
= −2 (12.82)
dr0 a(r0 − m)2
Since rph
ms
is the unique real root of (r0 − m)3 + m3 (1 − a2 /m2 ) = 0 [cf. Eq. (12.76)], it is clear
that the function ℓc (r0 ) has a unique extremum, which is achieved by the marginally stable
orbits. From the graph of ℓc (r0 ) shown in Fig. 12.6, we see that this extremum is a maximum.
Regarding the function qc (r0 ), the above expression of dqc /dr0 leads to two extrema: r0 = rph ms
and r0 = 3m. The former regards the inner spherical orbits, while the latter regards the outer
ones. Again, from the graph shown in Fig. 12.6, it is clear that r0 = rph
ms
realizes a maximum of
qc among all inner spherical orbits.
Remark 6: Another proof can be given by using the same general argument as that employed in
Sec. 11.5.3 for showing that the ISCO realizes an extremum of the specific energy and specific angular
momentum of timelike circular equatorial orbits (Property 11.32). Indeed, considering R as a function
of ℓ and q, in addition to r, i.e. writing R = R(r, ℓ, q), the marginal stable orbit obeys
∂R ∂2R
R(r0 , ℓc (r0 ), qc (r0 )) = 0, (r0 , ℓc (r0 ), qc (r0 )) = 0, (r0 , ℓc (r0 ), qc (r0 )) = 0.
∂r ∂r2
478 Null geodesics and images in Kerr spacetime
r̂
m cosθ
r=0
3 r = r−
r = r+
r̂
m cosθ
1.0
2
1 0.5
r̂ r̂
m sinθ m sinθ
1 2 3 4 0.5 1.0 1.5 2.0
1 0.5
2 1.0
Figure 12.17: Trace of the photon region (pale and dark green areas) in a meridional plane (t, φ) = const
of a Kerr spacetime with a = 0.95 m (as in Fig. 12.6). The right panel is a zoom on the inner √ part of the left
one. The meridional plane is described in terms of the coordinates (r̂, θ) where r̂ := (r + r2 + 4m2 )/2 [cf.
Eq. (12.69)]. Some spherical photon orbits are shown as green circular arcs, with the dashed ones representing the
polar spherical orbits, i.e. ℓ = 0 orbits. The black (resp. light brown) half-circle at r = r+ (resp. r = r− ) is the
trace of the outer (resp. inner) Killing horizon. The dotted orange half-circle marks the locus of r = 0, with the
red dot indicating the curvature singularity at r = 0 and θ = π/2. The ergoregion is shown in grey. The region
of stable spherical orbits is colored in dark green, with its boundary at r = rph ms
drawn in blue. Green dots mark
photon circular orbits: from the left to the right, they are the two vortical circular orbits at r = rph
∗∗
, the equatorial
inner circular orbit at r = rph , the prograde outer circular orbit at r = rph and the retrograde outer circular orbit
∗ +
−
at r = rph . The thin dotted grey half-circle marks r = rph ∗∗
. [Figure generated by the notebook D.5.11]
Deriving the first and second equations with respect to r0 and using the third equation, we get a
homogeneous linear system for (dℓc /dr0 , dqc /dr0 ), similar to the system (11.172). The unique solution
is then (dℓc /dr0 , dqc /dr0 ) = (0, 0), yielding the extremum in the functions ℓc (r0 ) and qc (r0 ) at
ms .
r0 = rph
Equation (12.81) shows that, in addition to that at r0 = rph
ms
, the function qc (r0 ) admits
a second extremum at r0 = 3m, i.e. for outer spherical orbits. This extremum is actually a
maximum and its value, obtained by setting r0 = 3m in Eq. (12.57), turns out to be independent
from a:
max qc (r0 ) = 27m2 . (12.83)
This is the maximum of the reduced Carter constant over all the spherical photon orbits (cf.
Fig. 12.6).
12.4 Black hole shadow and critical curve 479
12.3.5 Photon region
A corollary of the results obtained in Sec. 12.2.5 is that outside the black hole, i.e. in region
MI , a photon cannot be trapped (i.e. have a limited range of r) unless it moves on a (unstable)
spherical orbit. The part of Kerr spacetime made of points through which a spherical photon
orbit can pass is called the photon region or sometimes the photon shell [304]. In terms of the
Boyer-Lindquist r-coordinate, the photon region lies in the three intervals (12.59). The range of
the θ-coordinate has been discussed in Sec. 12.3.2: spherical photon orbits with r0 > 0 oscillate
about the equatorial plane with the limiting angle θm given by Eq. (12.41) (θm = π/2 for the
three circular orbits at r0 = rph
∗
, r0 = rph
+
and r0 = rph−
, which lie in the equatorial plane). On
the other side, the inner orbits with r0 < 0 are all vortical, with the limiting angles θm and θv
given by Eqs. (12.41) and (12.42). There is no constraint on the Boyer-Lindquist φ-coordinate:
the photon region occupies all the range [0, 2π).
The photon region is depicted in Fig. 12.17, which represents a meridional plane (t, φ) =
const of a Kerr spacetime with a = 0.95 m — the same value of a as in Fig. 12.6. Polar spherical
photon orbits, at r0 = rphpol
and r0 = rph
pol,in
(cf. Sec. 12.3.2), are plotted as dashed green curves.
It is graphically clear that they are the only orbits that encounter the rotation axis. We also
recover from this figure that, apart from those generating the two horizons Hin and H , the
only circular photon orbits of Kerr spacetime are the five ones considered in Sec. 12.3.3 (the five
green dots). We note also that a part of the outer spherical photon orbits lie in the ergoregion.
These orbits have however E > 0, according to the results of the end of Sec. 12.3.1.
Historical note : The prograde and retrograde outer circular photon orbits in the equatorial plane of a
Kerr black hole have been found by James M. Bardeen, William H. Press and Saul A. Teukolsky in 1972
[42]. The existence of stable spherical photon orbits under the inner horizon of a Kerr black hole has
been shown by Zdeněk Stuchlík in 1981 [466]. The systematic study of spherical photon orbits in the
black hole exterior has been performed by Edward Teo in 2003 [475].
12.4 Black hole shadow and critical curve
12.4.1 Critical null geodesics
As in the Schwarzschild case (Sec. 8.3.2), let us define a critical null geodesic as a null geodesic
with E ̸= 0 that has the same constants of motion (ℓ, q) as a spherical photon orbit, but that
does not stay at a fixed value of r. Equivalently, a critical null geodesic is a null geodesic
with varying r and for which the quartic polynomial R(r) [Eq. (12.25)] admits a double root
[Eq. (12.47)].
In Schwarzschild spacetime (Chap.√ 8), a critical null geodesic was simply any null geodesic
with varying r that has |ℓ| = bc = 3 3 m. In the Kerr case with 0 < a < m, we have instead a
1-parameter family of critical values: the family (ℓc (r0 ), qc (r0 )) given by Eqs. (12.56)–(12.57),
the parameter being r0 . By construction, the polynomial R(r) of a critical null geodesic L of
parameter3 r0 has a double root at r = r0 [cf. Eq. (12.47)]. More precisely, inserting formula
3
The word parameter is used here for the index among the family of all null critical geodesics and shall not be
confused with the affine parameter along L .
480 Null geodesics and images in Kerr spacetime
(12.56) for ℓ and formula (12.54) for q̃ := q + (ℓ − a)2 into expression (12.25a) for R(r) leads to
a2 qc (r0 )
2 2
R(r) = (r − r0 ) r + 2r0 r − , (12.84)
r02
where qc (r0 ) is the function (12.57). This writing of R(r) clearly exhibits the double root at
r = r0 . As discussed in Sec. 11.3.6 and 12.2.5, a consequence of the double-root behavior is that
any critical null geodesic L of parameter r0 has r0 as an asymptotic r-value, i.e. r(λ) → r0
when λ → +∞ or λ → −∞, where λ is the affine parameter along L .
Let us consider a critical null geodesic L of parameter r0 that goes through a point A of
Boyer-Lindquist coordinates (tA , rA , θA , φA ). By plugging (12.84) into the integrated equation
of motion (11.55a), we get, at any point of coordinates (t, r, θ, φ) along L ,
Z r Z θ
ϵr dr̄ ϵθ dθ̄
− p
2
=− q . (12.85)
2 2
rA |r̄ − r0 | r̄ + 2r0 r̄ − a qc (r0 )/r0 θA Θ̃(θ̄)
When r → r0 , it is clear that the integral in the left-hand side diverges logarithmically in terms
of |r − r0 |. The path integral in the right-hand side must therefore diverge as well, which
implies that L has an endless oscillatory θ-motion as r → r0 . Similarly the integrated equation
of motion (11.55d) with expression (12.84) for R(r) becomes
Z r Z θ
(2mr̄ − aℓ) ϵr dr̄ ϵθ dθ̄
φ − φem = a− p
2
+ ℓ− q .
2 2 2 2
rA |r̄ − r0 |(r̄ − 2mr̄ + a ) r̄ + 2r0 r̄ − a qc (r0 )/r0 θA sin2 θ̄ Θ̃(θ̄)
(12.86)
Again, the first integral diverges when r → r0 . The second one, which is a path integral on θ,
diverges as well because the path integral in the right-hand side of Eq. (12.85) diverges. From
expression (12.56) for ℓ it can be checked that 2mr0 − aℓ > 0 in the allowed range of r0 (i.e.
in the range (12.59), where spherical orbits exist). Given that ϵr dr̄ > 0 and ϵθ dθ̄ > 0, we
conclude that the integral on r and the path integral on θ both tend to +∞ when r → r0 . If
ℓ ≥ 0, we get then immediately φ → +∞ when r → r0 . If ℓ < 0, the two diverging terms in
the right-hand side of Eq. (12.86) have opposite signs. Disregarding some unexpected subtle
compensation, one term (in practice the second one) dominates over the other one, so that we
conclude that, whatever the sign of ℓ,
φ → ±∞ when r → r0 . (12.87)
We may summarize the above results as follows:
Property 12.20: radial motion of critical null geodesics
A critical null geodesic L of parameter r0 has r0 as an asymptotic r-value, either in the
future (r(λ) → r0 when λ → +∞) or in the past (r(λ) → r0 when λ → −∞), λ being the
(future-directed) affine parameter of L . In particular, L never crosses the sphere r = r0 .
Moreover L is winding endlessly on the sphere r = r0 when either λ → +∞ or λ → −∞,
12.4 Black hole shadow and critical curve 481
Figure 12.18: Critical null geodesic of parameter r0 = 2.2 m in a Kerr spacetime with a = 0.95 m, depicted
in terms of the Cartesian Boyer-Lindquist coordinates (x, y, z) [Eq. (11.73)]. This geodesic has ℓ = 0.863 m,
q = 18.042 m2 and is emitted at λ = 0 from the point of Boyer-Lindquist coordinates (r, θ, φ) = (40 m, π/2, 0)
⇐⇒ (x, y, z) = (40 m, 0, 0), towards the black hole (ϵr = −1). The drawing is interrupted at λ = 80 m/E. The
grey sphere is the black hole event horizon at r = r+ = 1.312 m. [Figure generated by the notebook D.5.13]
mimicking there the behavior of a spherical photon orbit.
Example 9: Figure 12.18 shows a critical null geodesic emitted from the equatorial plane at a large
distance from the black hole (the emission point is not shown on the figure that is truncated at r ∼ 10 m).
We note the winding around the sphere r = r0 = 2.2 m, in the same fashion as a spherical photon orbit
(compare with Figs. 12.8 and 12.9).
12.4.2 Critical curve and black hole shadow
In this section, we focus on null geodesics emitted in the black hole exterior, i.e. in region
MI , having in mind the formation of images on the screen of a distant observer. As in the
Schwarzschild case studied in Chap. 8, the family of critical null geodesics separates the null
geodesics emitted far from the black hole between those that can escape to infinity and those
that fall into the black hole. More precisely, according to Lemma 12.5, this family separates null
geodesics whose quartic polynomial R has no root in MI from those for which R has a double
root in MI . It follows that the family of critical null geodesics separates null geodesics that
have a r-turning point in MI from those that do not have any, given that a null geodesic has at
most one r-turning point in MI (cf. Sec. 12.2.5). This is illustrated by the following example.
Example 10: Figure 12.19 depicts two null geodesics initially very close to a critical one. They are
emitted at the same point as the critical geodesic considered in Example 9 and with parameters close
to critical: the same reduced Carter constant q and values of the reduced angular momentum ℓ almost
equal to the critical one, ℓc (r0 ), up to a relative difference of 10−4 . The geodesic L1 (green curve) has
ℓ = 1.0001 ℓc (r0 ); it performs a few turns onto (actually very close to) the sphere S0 of coordinate
radius r0 = 2.2 m and eventually depart to infinity; this means that L1 has a r-turning point, somewhere
482 Null geodesics and images in Kerr spacetime
Figure 12.19: Two null geodesics with constants of motion (ℓ, q) close to those of a critical null geodesic,
(ℓc (r0 ), qc (r0 )) [Eqs. (12.56)–(12.57)]. Here, as in Fig. 12.18, r0 = 2.2 m and a = 0.95 m. The two geodesics depart
from the same point at (x, y, z) = (40 m, 0, 0) (outside the scope of the figure) as the critical null geodesic shown
in Fig. 12.18. Both geodesics have q = qc (r0 ), but the green one has ℓ = 1.0001 ℓc (r0 ) while the orange one has
ℓ = 0.9999 ℓc (r0 ). [Figure generated by the notebook D.5.13]
close to S0 . The geodesic L2 (orange curve) has ℓ = 0.9999 ℓc (r0 ); it is graphically indistinguishable
from L1 until a full turn onto S0 (this is best seen in the interactive 3D view in the online notebook
D.5.13), it then clearly departs from it and eventually terminates into the black hole. Hence L2 has no
r-turning point.
Let us consider an asymptotic inertial observer O (cf. Sec. 10.7.5), i.e. a static observer
located at Boyer-Lindquist coordinates (t, rO , θO , φO ), where (rO , θO , φO ) are constant and
rO ≫ m. As in the Schwarzschild case (Sec. 8.5.4), the concept of black hole shadow is
defined by considering an emitting large sphere S of constant Boyer-Lindquist coordinate
rS that encompasses both the black hole and observer O; this means that rS > rO . In a more
astrophysical setting, one could image S as being made of many far-away light sources. We
assume that O is equipped with a screen in the direction of the black hole, in the same set up
as described in Sec. 12.2.3 (cf. Fig. 12.4). Given the relative position of O, the black hole and
the emitting sphere S , any photon emitted from S that reaches O’s screen has necessarily a
point of minimal approach to the black hole, i.e. the associated null geodesic has necessarily a
r-turning point between S and O. Reciprocally, any null geodesic that impacts O’s screen and
has no r-turning point in its past cannot have emerged from S and therefore results in a black
dot on the screen. According to the above discussion, the boundary of the black area on O’s
screen is made by the impact points of critical null geodesics and is called the critical curve
[234, 235]; we shall denote it by C . The black area itself is called the black hole shadow. It
is called so essentially because if the black hole were not present, it would not exist and O’s
screen would be uniformly bright.
Remark 1: The concept of black hole shadow is rather academic, since it requires the sources of light
12.4 Black hole shadow and critical curve 483
to be far from the black hole and to surround it, as well as the observer, from any direction. On the
contrary, astrophysical black holes are illuminated by close sources (e.g. an accretion disk) and we
shall see in Sec. 12.5 that the black area on astronomical images has little resemblance with the shadow
defined above. On the other hand, the critical curve has a true observational significance, as we shall
discuss in Sec. 12.5, and is definitely worth to study.
In what follows, we shall assume that the observer O in not located on the black hole rotation
axis, i.e. θO ̸∈ {0, π}, leaving the special case θO = 0 or π to Sec. 12.4.3. Then sin θO ̸= 0 and
we have seen in Sec. 12.2.3 that the impact point of a null geodesic L on O’s screen, measured
by the screen angular coordinates (α, β), is related to the reduced angular momentum ℓ and
reduced Carter constant q of L by formulas (12.32). Given the above definition, the critical
curve C is obtained by using for (ℓ, q) in (12.32) the critical values (ℓc (r0 ), qc (r0 )) given by
Eqs. (12.56)–(12.57):
ℓc (r0 )
α=− (12.88a)
rO sin θO
s
ϵθ ℓc (r0 )2
β= qc (r0 ) + cos2 2
θO a − . (12.88b)
rO sin2 θO
This provides the equation of C in parametric form, the parameter being r0 — the radius of the
spherical orbits that have the same value of (ℓ, q) as the critical null geodesic that impacts O’s
screen at the point (α, β). Let us recall that (α, β) are angular coordinates and are therefore
dimensionless (cf. Sec. 12.2.3). The range of r0 for spherical orbits in the black hole exterior is
+
rph ≤ r0 ≤ rph −
[Eq. (12.59)], with rph+
and rph
−
given by Eq. (12.62). The corresponding range of
ℓ = ℓc (r0 ) is then ℓc (rph ) ≤ ℓ ≤ ℓc (rph ) (cf. the red curve in Fig. 12.6) and the corresponding
− +
range of q = qc (r0 ) is 0 ≤ q ≤ 27 m2 [cf. Eq. (12.83)]. However, not all these values of (ℓ, q)
are allowed, since in order for the corresponding geodesic to reach O, they have to obey the
constraint (12.29):
qc (r0 ) + a2 cos2 θO sin2 θO − ℓc (r0 )2 cos2 θO ≥ 0. (12.89)
Except for θO = π/2 (observer in the equatorial plane), this constraint limits the range of
r0 to a subinterval [r0min , r0max ] of [rph
+ −
, rph ]. We note that the radius rphpol
of polar spherical
photon orbits, as given by Eq. (12.67a), lies necessarily in that subinterval; indeed by definition
pol
ℓc (rph ) = 0, which ensures that the constraint (12.89) is fulfilled for r0 = rph pol
, whatever the
value of θO .
The critical curve C in the observer’s screen is obtained by first solving Eq. (12.89) with
equality in the sign ≥ to determine r0min and r0max . Then, the upper half of C is computed
by means of formulas (12.88) with ϵθ = +1 and r0 ranging in [r0min , r0max ]. The lower half is
obtained similarly, but with ϵθ = −1 in Eq. (12.88b), so that it is the symmetric of the upper
half with respect to the α-axis.
The result of the computation is shown in Fig. 12.20 for a Kerr black hole with a = 0.95 m
and various inclination angles of observer O. For θO = π/2 the parameter r0 of the critical
null geodesics forming C spans the full interval [rph + −
, rph ] ≃ [1.386 m, 3.955 m]. For θO = π/6,
the range is restricted to [r0 , r0 ] ≃ [1.768 m, 3.237 m], while for θO = 0 (to be discussed
min max
484 Null geodesics and images in Kerr spacetime
2
(r /m) β
1.39 m r0 1.90 m
1.90 m r0 2.41 m
0 2.41 m r0 2.93 m
2.93 m r0 3.44 m
3.44 m r0 3.96 m
2
6
4 2 0 2 4 6 8
(r /m) α
6 6
4 4
2 2
(r /m) β
(r /m) β
0 0
2 2
4 4
6 6
4 2 0 2 4 6 6 4 2 0 2 4 6
(r /m) α (r /m) α
Figure 12.20: Shadow of a Kerr black hole of mass m and spin parameter a = 0.95 m on the screen of an
asymptotic inertial observer O located at r = rO and θ = θO , for three values of θO : θO = π/2 (O in the
equatorial plane; upper panel), θO = π/6 (lower left) and θO = 0 (O on the rotation axis; lower right). The screen
is spanned by the angular coordinates (α, β), rescaled by the inverse of the factor m/rO , whose values for some
astrophysical black holes can be found in Table 12.1. The shadow is bounded by the critical curve C , which is
depicted with colors indicating the range of the parameter r0 of the critical null geodesics forming it. [Figure
generated by the notebook D.5.14]
12.4 Black hole shadow and critical curve 485
Sgr A* M87* M31* Cyg X-1
m [M⊙ ] 4.1 106 6.2 109 1.5 108 15
rO [kpc] 8.12 1.67 104 7.6 102 1.86
m/rO 2.4 10−11 1.8 10−11 9.4 10−12 3.9 10−16
m/rO [µas] 5.0 3.7 1.9 8.0 10−5
Table 12.1: Scale factor m/rO for various astrophysical black holes: the supermassive black hole at the center
of our galaxy, Sagittarius A* (data taken from Table A.1 of Ref. [4]), the supermassive black hole M87* in the
nucleus of the galaxy Messier 87 (data from Refs. [214, 6]; see also Appendix I and Table 9 of Ref. [8]), the
supermassive black hole M31* in the nucleus of the Andromeda Galaxy (Messier 31) (data from Ref. [55]) and
the stellar black hole Cygnus X-1 (data from Refs. [393, 428, 225]). The last line gives m/rO in microarcseconds
(1 µas = 4.848 10−12 rad).
in Sec. 12.4.3), r0 can take only one value: r0 = rphpol
≃ 2.493 m. At a fixed value of a/m, the
black hole shadow depends on m and rO only through the dimensionless ratio m/rO , which
sets the global scale of the shadow. Accordingly, in Fig. 12.20, the screen angular coordinates
(α, β) have been rescaled by (m/rO )−1 . Values of m/rO for some astrophysical black holes are
provided in Table 12.1; the observer’s radial coordinate rO is then nothing but the distance of
the black hole to the Earth. These values of m/rO are tiny, being at most 2.4 10−11 = 5.0 µas
(microarcseconds; 1 µas = 4.848 10−12 rad) for the (known) black hole of largest apparent size
as seen from Earth, Sgr A*. Given that the diameter of the shadow is ∼ 10 in the scale m/rO
used in Fig. 12.20, this means that the angular size of Sgr A* shadow as seen from Earth is only
∼ 50 µas. Such a small value4 has entered recently in the realm of observational astronomy,
with the advent of the Event Horizon Telescope [6, 521], whose angular resolution is of order
20 µas. We also note from Table 12.1 that the size of the shadow of a stellar-mass black hole in
our galaxy, such as Cygnus X-1, is even more tiny, by a factor 10−5 , which makes the shadow
of this kind of black holes out of reach by the current technology.
The main feature that appears in Fig. 12.20 is that for θO = π/6 and even more for θO = π/2,
the shadow is shifted to the right and its left edge is flattened, as compared to the shadow for
θO = 0 (or π). This can be understood by noticing that the critical null geodesics forming the
left edge arise form regions of smaller r0 than those on the right edge (cf. the color code). For
instance, the left critical null geodesic at β = 0 has α < 0 and ℓ > 0 (cf. the minus sign in
Eq. (12.88a)) and arises from the prograde outer circular photon orbit lying at r0 = rph +
in the
equatorial plane (cf. Sec. 12.3.3), while the right one at β = 0 has α > 0 and ℓ < 0 and arises
from the retrograde outer circular photon orbit at r0 = rph −
. That prograde (resp. retrograde)
geodesics impact the screen on the left (resp. right) side is easily recovered by remembering
that the β-axis (α = 0) coincides with the orthogonal projection of the black hole’s spin onto
the observer’s screen, with the spin being oriented upward.
4
For comparison, the angular resolution of the Hubble Space Telescope is ∼ 0.1′′ = 105 µas!
486 Null geodesics and images in Kerr spacetime
5.20
5.15
5.10
(r /m) Rshad
5.05
5.00
4.95
4.90
4.85
0.0 0.2 0.4 0.6 0.8 1.0
a/m
Figure 12.21: Angular radius Rshad of the (circular) critical curve of a Kerr black hole as seen by an asymptotic
inertial observer located at r = rO on the black hole’s rotation axis, as a function of the Kerr spin parameter a
[Eq. (12.90)]. Rshad is given in units of m/rO (cf. Table 12.1). [Figure generated by the notebook D.5.14]
12.4.3 Shadow for an observer on the rotation axis
As stressed in Sec. 12.2.3, if θO ∈ {0, π}, the screen angular coordinates (α, β) can no longer be
defined from the orthonormal frame (e(θ) , e(φ) ) associated with the Boyer-Lindquist coordinates
(θ, φ). In this case, we pick an arbitrary orthonormal frame (e(α) , e(β) ) in the screen plane to
define (α, β). Due to the axisymmetry of spacetime, when O lies on the rotation axis, the black
hole shadow is symmetric by any rotation around the screen’s center. It is therefore necessarily
a disk. Its boundary, the critical curve C , is then a circle. A critical null geodesic impacting
the screen on C has necessarily ℓ = 0 [Eq. (12.36)]. It follows that the parameter r0 can take
only a single value: r0 = rph pol
, which is given by Eq. (12.67a). The radius of C is then given by
Eq. (12.37), with q = qc (rph ):
pol
2 1 2 pol 2
α + β = 2 qc (rph ) + a .
rO
Given expression (12.57) of the function qc and the fact that rph
pol
obeys (rph
pol 4 pol
) (rph − 3m)2 =
a4 (rph + m)2 , as a solution of ℓc (r0 ) = 0, we get
pol
s q
pol pol 2
p m rph (rph ) − a2
Rshad := α2 + β 2 = 2 pol
(θO = 0 or θO = π) (12.90)
rO m rph −m
where rph
pol
is the function (12.67a) of (m, a) and the scale factor m/rO is given for some
astrophysical black holes in Table 12.1.
Example 11: An example of shadow seen from the rotation axis is shown in the lower right panel of
pol
Fig. 12.20. The Kerr spin parameter is a = 0.95 m, for which formula (12.67a) yields rph ≃ 2.493 m,
so that Eq. (12.90) results in Rshad ≃ 4.875 m/rO . We note that the critical curve C is depicted in the
12.4 Black hole shadow and critical curve 487
same color (green) as the part of the critical curve in the other panels that crosses the β-axis, which is
expected since the critical null geodesics that reach the screen along that axis have ℓ = 0 and thus the
pol
same parameter r0 = rph as all the critical null geodesics forming C for θO = 0 or π.
The shadow radius Rshad is a decreasing function of a, which is depicted in Fig. 12.21. Note
that its variation range is pretty limited, since the limits lima→0 rph
pol
= 3m and lima→m rphpol
=
√
( 2 + 1)m (cf. Sec. 12.3.2) yield respectively
√ m m √ m m
lim Rshad = 3 3 ≃ 5.196 and lim Rshad = 2( 2 + 1) ≃ 4.828 . (12.91)
a→0 rO rO a→m rO rO
The value for a = m is thus only 7% lower than the value for a = 0.
Remark 2: The limit a → 0 is in agreement with the result for the shadow of a Schwarzschild black
hole obtained in Sec. 8.5.4.
12.4.4 Shadow of an extremal Kerr black hole
The case a = m corresponds to the extremal Kerr black hole, which will be studied in Chap. 13.
However, we shall discuss its shadow and critical curve here, by taking some appropriate limits.
For a = m, formulas (12.56)-(12.57) for ℓc (r0 ) and qc (r0 ) simplify significantly:
r02
ℓc (r0 ) = − + 2r0 + m (a = m) (12.92)
m
r03
(4m − r0 )
qc (r0 ) = (a = m). (12.93)
m2
Substituting these values into the system (12.88) leads to the screen angular coordinates
determining the critical curve C of an extremal Kerr black hole:
(r0 − m)2 − 2m2
α= (12.94a)
mrO sin θO
ϵθ
q
β= r03 (4m − r0 ) − m2 cos2 θO [2r0 (r0 + 2m) + m2 cos2 θO ]. (12.94b)
mrO sin θO
Besides, Eq. (12.62) yields
+
rph =m and −
rph = 4m (a = m). (12.95)
The range of r0 is determined by rph
+
≤ r0 ≤ rph −
and Θ̃(θO ) ≥ 0. The last condition is
equivalent to demanding that the quantity under the square root in expression (12.94b) for β
is non-negative. This leads5 to some interval [rmin , rmax ] ⊂ [rph
+ −
, rph ]. As in the case a < m
treated above, we have rmax ≤ rph with rmax = rph ⇐⇒ θO = π/2. However, rmin behaves
− −
5
The values of rmin and rmax can be computed exactly as rmin = max(r1 , m) and rmax = r2 , where r1 and r2
are two roots of the quartic polynomial in r0 that appears under the square root in Eq. (12.94b). However, doing
so would lead to complicated expressions, while a computation by numerical root finding, as in the notebook
D.5.14, is sufficient in practice.
488 Null geodesics and images in Kerr spacetime
differently. As we going to see, rmin = rph +
(= m) for a finite-width interval of values of θO
around π/2 and not only for π/2. To see this, let us start by noticing that expression (12.93)
results in qc (rph
−
) = 0, while qc (rph+
) = 3m2 ̸= 0. This last result seems to contradict the
fact that rph has been obtained in Sec. 12.3.1 by searching for the zeros of qc . However, the
+
generic expression (12.57) for qc has a factor (r0 − m)2 in its denominator, so that taking the
limits a → m and r0 → rph +
= m yields the indeterminate form “0/0”. As a consequence of
qc (rph ) ̸= 0, one has β ̸= 0 for r0 = rph
+ +
= m and |θO − π/2| sufficiently small. This implies
that rmin = m for a finite range of values of θO around π/2. More precisely, according to (12.94),
the screen coordinates for r0 = rph +
= m are
m
α|r0 =m = −2 (12.96a)
rO sin θO
ϵθ m p
β|r0 =m = 3 − 6 cos2 θO − cos4 θO . (12.96b)
rO sin θO
In particular, for θO = π/2 (observer O in the equatorial plane), we get
m √ m π
α|r0 =m = −2 and β|r0 =m = ϵθ 3 θO = . (12.97)
rO rO 2
Equation (12.96b) shows that, for ϵθ = +1, β|r0 =m > 0 ⇐⇒ θcrit < θO < π − θcrit , where
θcrit is such that cos2√θcrit is the positive root of the quadratic
√ polynomial
√ −x2 − 6x + 3 = 0.
We get cos θcrit = 2 3 − 3, from which sin θcrit = 4 − 2 3 = ( 3 − 1) , hence
2 2 2
√
θcrit = arcsin( 3 − 1) ≃ 0.8213 rad ≃ 47.06◦ . (12.98)
For θO = θcrit or π − θcrit , we have exactly β|r0 =m = 0, while for θO < θcrit or θO > π − θcrit ,
β|r0 =m is imaginary. This means that for θcrit ≤ θO ≤ π − θcrit , the range of the parameter r0
is [m, rmax ], while for for θO < θcrit or θO > π − θcrit , the range is [rmin , rmax ] with rmin > m,
as for the case θO ̸= π/2 of the shadows with a < m discussed in Sec. 12.4.2. In this last case,
the critical curve is a smooth curve with β|r0 =rmin = 0 (in addition to β|r0 =rmax = 0, which
always holds).
Let us focus on the first case, i.e. θcrit ≤ θO ≤ π − θcrit . The parametric curve defined by
the system (12.94) terminates at r0 = m on the two points given by Eq. (12.96), or Eq. (12.97)
in the particular case θO = π/2 (one point for ϵθ = +1, and the other one for ϵθ = −1). For
θcrit < θO < π − θcrit , one has β ̸= 0 at these two points, so that the curve is not closed, as one
can see on Fig. 12.22 (drawn for θO = π/2). One may be puzzled by this feature: the shadow
boundary has to be closed! We are thus missing some critical null geodesics to complete the
boundary. It is easy to find the missing ones as soon as we remember that in the special case
a = m, we had found an extra family of spherical photon orbits in Sec. 12.3.1: those given
by Eq. (12.50), namely the spherical photon orbits at r0 = m that have ℓ = 2m. Obviously,
this family cannot be parameterized by r0 ; on the other hand, the reduced Carter constant
q is a valid parameter. Since q is not constrained by Eq. (12.50), it can take all values in the
range [0, +∞). Note that q < 0 is not permitted here since ℓ = 2m > a [cf. property (12.38)].
However, not any value of q corresponds to a spherical photon orbit associated with a critical
null geodesic that reaches the asymptotic inertial observer O. Indeed, q must give birth to a
12.4 Black hole shadow and critical curve 489
(r /m) β
1.00 m r0 1.60 m
1.60 m r0 2.20 m
0 2.20 m r0 2.80 m
2.80 m r0 3.40 m
3.40 m r0 4.00 m
2
6
4 2 0 2 4 6 8
(r /m) α
Figure 12.22: Part of the critical curve of an extremal Kerr black hole (a = m) in the screen of an asymptotic
inertial observer O in the equatorial plane (θO = π/2), as defined by the parametric equations (12.94),
√ with
r0 ∈ [m, 4m]. The two endpoints are achieved at r0 = m and are given by Eq. (12.97): (ᾱ, β̄) = (−2, ± 3) ≃
(−2, ±1.732), where ᾱ := αrO /m and β̄ := βrO /m. [Figure generated by the notebook D.5.14]
radial polynomial R(r) positive for all r > m, so that the radial motion is possible between the
spherical orbit at r0 = m and the observer [condition (12.46)]. Specializing expression (12.25)
for R(r) to a = m and ℓ = 2m, we get
R(r) = (r − m)2 (r2 + 2mr − q). (12.99)
r = m appears as a double root of R, which confirms that we are dealing with spherical photon
orbits at r0 = m. There are two other roots, which depend on q:
p
rq± = ± m2 + q − m. (12.100)
Since q ≥ 0, we have rq+ ≥ 0 and rq− ≤ −2m. It is then clear (cf. Fig. 12.23) that
(∀r ∈ (m, +∞), R(r) > 0) ⇐⇒ rq+ ≤ m. (12.101)
The critical value is rq+ = m, which correspond to q = 3m2 (red curve in Fig. 12.23). r = m is
then a triple root of R: q = 3m2 =⇒ R(r) = (r − m)3 (r + 3m). Since rq+ is an increasing
function of q, we conclude:
Property 12.21
For a = m, a critical null geodesic of constants of motion (ℓ, q) can reach the asymptotic
region r ≫ m from the vicinity of a spherical photon orbit at r0 = m iff
ℓ = 2m and 0 ≤ q ≤ 3m2 . (12.102)
Furthermore, such a critical null geodesic can reach the asymptotic inertial observer O,
who is located at θ = θO , only if the Θ̃ function associated with (ℓ, q) obeys Θ̃(θO ) ≥ 0. Setting
490 Null geodesics and images in Kerr spacetime
1.5
q = m2
q = 3m 2
q = 5m 2
1.0
0.5
R(r)/m 4
0.0
0.5
1.0
0.0 0.5 1.0 1.5 2.0
r/m
Figure 12.23: Radial polynomial R(r) for a = m, ℓ = 2m and three values of q. All the polynomials have a
double root at r = m and the dots mark the root rq+ [Eq. (12.100)], the fourth root rq− ≤ −2m being out of the
figure’s scope. The dotted horizontal lines indicate the range of r where a null geodesic motion to the asymptotic
inertial observer is possible. [Figure generated by the notebook D.5.14]
a = m and ℓ = 2m into expression (12.26) for Θ̃, we get the criterion
3 + cos2 θO 2
q≥ m. (12.103)
tan2 θO
The screen angular coordinates corresponding to the critical null geodesic are obtained by
plugging the above values of ℓ and q, as well as a = m, into formulas (12.32):
2m
α=− (12.104a)
rO sin θO
r
ϵθ m q 3 + cos2 θO q
β= sin2 θO − cos2 θO (3 + cos2 θO ), 2
≤ 2 ≤ 3. (12.104b)
rO sin θO m2 tan θO m
The system (12.104) defines a curve parameterized by q, with the range of q obtained by
combining (12.102) and (12.103). For O in the equatorial plane (θO = π/2), this parametric
equation simplifies to
r
m m q q π
α = −2 and β = ϵθ , 0≤ 2 ≤3 θO = . (12.105)
rO rO m2 m 2
In all cases, the curve (12.104) is actually a segment of a vertical straight line on O’s screen,
since it has α = const. Following Ref. [238], we shall call this segment the NHEK line, NHEK
standing for Near-Horizon Extremal Kerr, given that spherical photon orbits with r0 close to
m are located in the near-horizon region, which we shall study in details in Sec. 13.4. The
key feature is that the end points of the NHEK line, which are obtained for q/m2 = 3, are
exactly the two points defined by Eq. (12.96), i.e. the end points of the curve parameterized by
r0 (cf. Fig. 12.22). By adding the NHEK line, we are thus closing the boundary of the black hole
shadow! The result is shown in Fig. 12.24, where the NHEK line is drawn in maroon on the left
edge of the plots for θO = π/2 and θO = π/3 > θcrit .
12.4 Black hole shadow and critical curve 491
2 r0 = m
(r /m) β
1.00 m < r0 1.60 m
1.60 m r0 2.20 m
0 2.20 m r0 2.80 m
2.80 m r0 3.40 m
3.40 m r0 4.00 m
2
6
4 2 0 2 4 6 8
(r /m) α
6 6
4 4
2 2
(r /m) β
(r /m) β
0 0
2 2
4 4
6 6
4 2 0 2 4 6 8 4 2 0 2 4 6 8
(r /m) α (r /m) α
Figure 12.24: Critical curve C (colored curve) and shadow (grey area) of an extremal Kerr black hole on the
screen of an asymptotic inertial observer O located at r = rO and θ = θO , for three values√ of θO : θO = π/2
(O in the equatorial plane; upper panel), θO = π/3 (lower left) and θO = θcrit = arcsin( 3 − 1) ≃ 0.821
[Eq. (12.98)]. The NHEK line (plotted in maroon) is present in the first two images, while it is vanishing in the
third one (marginal case), since there is no NHEK line in C for θ < θcrit or θ > π − θcrit . As in Fig. 12.20, the
color code corresponds to some selected ranges for the parameter r0 of the critical null geodesics forming C .
[Figure generated by the notebook D.5.14]
492 Null geodesics and images in Kerr spacetime
Cardioid shape
As seen from Fig. 12.24, the departure of the critical curve C from a perfect circle is maximal
when the observer lies in the equatorial plane, i.e. when θO = π/2. The critical curve is often
said to have a D-shape (from the letter D). It is interesting that this shape corresponds actually
to a simple mathematical curve: the convex hull of a cardioid [193]. Indeed, for θO = π/2,
equations (12.94), which govern the part of C parameterized by r0 , simplify to
(r0 − m)2 − 2m2
α= (12.106a)
mrO
ϵθ
(12.106b)
p
β= r0 r0 (4m − r0 ).
mrO
Let us then extract r0 from Eq. (12.106a):
r
rO
r0 = m 1 + 2 + α
m
and substitute it into the square of Eq. (12.106b); we get (using ϵ2θ = 1)
r 3 r
m2
rO rO
2
β = 2 1+ 2+ α 3− 2+ α . (12.107)
rO m m
At this stage, it is worth to introduce explicitly the rescaled angular coordinates used in
Figs. 12.22 and 12.24:
rO rO
ᾱ := α and β̄ := β. (12.108)
m m
Then, by expanding its right-hand side, we can rewrite Eq. (12.107) as
√
β̄ 2 = −ᾱ2 + 2ᾱ + 8 ᾱ + 2 + 11. (12.109)
Let us introduce new screen coordinates (α̂, β̂) by shifting the origin to (ᾱ, β̄) = (−1, 0):
α̂ := ᾱ + 1 and β̂ := β̄. (12.110)
We can then recast Eq. (12.109) as
√
α̂2 + β̂ 2 − 4α̂ = 8 α̂ + 1 + 1 . (12.111)
√
The square of this expression is (α̂2 + β̂ 2 − 4α̂)2 = 64(α̂ + 2 α̂ + 1 + 2). Using Eq. (12.111)
to get rid of the square root, we obtain
2
2 2 2
α̂ + β̂ − 4α̂ = 16 α̂ + β̂ . 2
(12.112)
The reader may have recognized the Cartesian equation of a cardioid. To put it in a more familiar
form, let us introduce the polar coordinates (ρ, ϕ) defined by ρ2 := α̂2 + β̂ 2 , cos ϕ := α̂/ρ
and sin ϕ := β̂/ρ. Then Eq. (12.112) becomes (ρ2 − 4ρ cos ϕ)2 = 16ρ2 , which is equivalent to
12.4 Black hole shadow and critical curve 493
β̄
2
6
2 0 2 4 6 8
ᾱ
Figure 12.25: Cardioid defined by the polar equation ρ = 4 (1 + cos ϕ) (dotted blue curve) and critical curve of
an extremal Kerr black hole seen from the equatorial plane (red curve). The screen coordinates (p ᾱ, β̄) are defined
by Eq. (12.108), while (ρ, ϕ) are polar coordinates around the point (ᾱ, β̄) = (−1, 0), i.e. ρ := (ᾱ + 1)2 + β̄ 2
and sin ϕ := β̄/ρ. [Figure generated by the notebook D.5.14]
(ρ − 4 cos ϕ)2 = 16, i.e. to ρ = ±4 + 4 cos ϕ. Given that ρ ≥ 0, the ± sign must be +, so that
we end up with
ρ = 4 (1 + cos ϕ) . (12.113)
We recognize the polar equation of a cardioid, generated by a circle of radius 2 rolling around a
circle of the same radius and centered at (α̂, β̂) = (2, 0) ⇐⇒ (ᾱ, β̄) = (1, 0).
As it is clear from the starting point of the above calculation, the cardioid corresponds only
to the part of the critical curve parameterized by r0 , i.e. the part depicted in Fig. 12.22. It does
not reproduce the NHEK line. Actually, adding the NHEK line results in the convex hull of the
cardioid, as shown in Fig. 12.25.
Remark 3: It is amusing to note that the cardioid is involved in two optical phenomena of very distinct
origin: in classical optics, it appears as the caustic generated by reflection of light on a circular material,
such as a bowl or a coffee cup, and at the same time, it delineates the shadow of some black holes of
general relativity.
Remark 4: For θO ̸= π/2, the critical curve C of the extremal Kerr black hole is no longer a cardioid.
It is however still a quartic algebraic curve, i.e. (ᾱ, β̄) obey an algebraic equation of degree 4, as in
Eq. (12.112). Moreover, C is a classical elementary curve, namely a Cartesian oval, also known as
oval of Descartes, as shown in Ref. [237]. More precisely, C is exactly such an oval for θO ≤ θcrit or
θO ≥ π − θcrit . For θcrit < θO < π − θcrit , i.e. when the NHEK line is present, C is the convex hull of a
Cartesian oval.
12.4.5 Comparing the critical curves at fixed inclination
From an observational point of view, it may happen that the inclination angle θO is known, as
we shall see for M87* in Sec. 12.5.3. Figure 12.26 compares then the critical curves at a fixed
494 Null geodesics and images in Kerr spacetime
θ = π/2
6
(r /m) β
0 a=0
a = 0.5 m
a=m
2
6
6 4 2 0 2 4 6
(r /m) α
θ = π/6 θ =0
6 6
4 4
2 2
(r /m) β
(r /m) β
0 0
2 2
4 4
6 6
6 4 2 0 2 4 6 6 4 2 0 2 4 6
(r /m) α (r /m) α
Figure 12.26: Critical curves for different values of a at fixed observer’s inclination angle: θO = π/2 (top
panel), θO = π/6 (lower left panel) and θO = 0 (lower right panel). [Figure generated by the notebook D.5.14]
value of θO for various values of the black hole spin parameter a. For θO = 0, we recover the
feature found in Sec. 12.4.3, namely that the critical curve depends very weakly on the spin.
Historical note : The critical curve of a Kerr black hole has been computed first by James M. Bardeen
in 1972 [38] (cf. the historical note on p. 276), in the form of the parametric system (12.88). He wrote:
The rim of the “black hole”6 corresponds to photon trajectories which are marginally trapped by the black
hole; they spiral around many times before they reach the observer. It is conceptually interesting, if not
astrophysically very important, to calculate the precise apparent shape of the black hole. Fifty years
later, with the first image from the Event Horizon Telescope [6, 521], this has become astrophysically
important! For a = m and θO = π/2, Bardeen derived the system (12.106) governing the part of the
critical curve parameterized by r0 . For the NEHK line, he simply noted7 : The non-uniform nature of
the limit a → m allows q to range between 0 and 3m2 at r = m. Bardeen presented then a plot of the
resulting critical curve (Fig. 6 in Ref. [38]) similar to that shown in the top panel of Fig. 12.24.
6
For Bardeen, “black hole” (with quotes) stands for the black spot on the observer’s screen.
7
q is denoted by η by Bardeen.
12.5 Images 495
The term black hole shadow for the interior of the critical curve has been coined by Heino Falcke,
Fulvio Melia and Eric Agol in 2000 [192], cf. the historical note on p. 501.
12.5 Images
12.5.1 Multiples images of a single source
The main features of images of a single luminous source on the screen of a remote observer
in Kerr spacetime are similar to those obtained for Schwarzschild spacetime in Sec. 8.5. In
particular, a single point-like source, wherever localized in the black hole exterior (region MI ),
gives birth to two infinite sequences of images on the screen of the asymptotic inertial observer,
both sequences converging to the critical curve C . Typically, one sequence is formed by null
geodesics having a reduced angular momentum ℓ > 0 and the other one by null geodesics
with ℓ < 0. Each image in a given sequence can be labeled by the number n of half-round trips
around the black hole and is dimmer and dimmer as n increases.
12.5.2 Image of an accretion disk
Beside gravitational waves, the main way of observing black holes is through the electromag-
netic radiation from material orbiting around them, either in the form of stars, as in the case of
Sgr A* [4, 5], or in the form of an accretion disk [2]. In the latter case, most observations are
spectra and time evolution of the global luminosity (the so-called light curve) of the unresolved
accretion disk. However recently, the Event Horizon Telescope team has produced the first
resolved image of an accretion disk around a black hole [6, 521]. In order to discuss this image
in Sec. 12.5.3, let us take a look at the generic properties of (theoretical) images of accretion
flows around a Kerr black hole.
Figures 12.27 and 12.28 present some computed images of an accretion disk around a Kerr
black hole of spin parameter a = 0.5 m and a = 0.95 m, respectively. The accretion disk is a
simple model developed in Ref. [494]. It consists in a geometrically thick and optically thin
accretion disk with an opening angle of 27◦ and an inner boundary located at the prograde
ISCO (cf. Sec. 11.5.3): rin = rISCO
+
, with rISCO
+
≃ 4.233 m for a = 0.5 m (Fig. 12.27) and rISCO
+
≃
1.937 m for a = 0.95 m (Fig. 12.28). The disc is in Keplerian rotation and the electromagnetic
emission is due to thermal synchrotron radiation of electrons in the local magnetic field (see
Ref. [494] for details). The images have been generated by the open-source ray-tracing code
Gyoto [493] (cf. Appendix E).
Figure 12.27 provides images for moderate black hole spin: a = 0.5 m. For a given inclination
θO , the images are rather similar to those of the accretion disk around a Schwarzschild black
hole displayed in Fig. 8.25. In particular, one notices three images of the disk, as in Fig. 8.25:
a broad primary image, a secondary image with a narrow annular shape and a thin tertiary
image appearing as a very thin ring, which is dotted by lack of resolution. The qualitative
explanation of these images is basically the same as in the Schwarzschild case. In particular, for
the disk face-on view (θO = 0), one could use a figure similar to Fig. 8.26, with the orange null
geodesics generating the primary image, the brown ones the secondary image and the red ones
496 Null geodesics and images in Kerr spacetime
10
(r /m) β
0
θO = 0
5
10
10 5 0 5 10
(r /m) α
10
(r /m) β
0
θO = π/6
5
10
10 5 0 5 10
(r /m) α
10
5
(r /m) β
0
θO = π/3
5
10
10 5 0 5 10
(r /m) α
10
5
(r /m) β
0
θO = π/2
5
10
10 5 0 5 10
(r /m) α
Figure 12.27: Images of a thick accretion disk around a Kerr black bole with a = 0.5 m for three observer’s
inclination angles θO . The right column shows the critical curve (red dotted line) superposed on the image.
[Images produced by Gyoto with the input files given in Sec. E.2.2; the addition of the critical curve in the right column
has been performed by means of the SageMath notebook D.5.15]
12.5 Images 497
10
(r /m) β
0
θO = 0
5
10
10 5 0 5 10
(r /m) α
10
(r /m) β
0
θO = π/6
5
10
10 5 0 5 10
(r /m) α
10
5
(r /m) β
0
θO = π/3
5
10
10 5 0 5 10
(r /m) α
10
5
(r /m) β
0
θO = π/2
5
10
10 5 0 5 10
(r /m) α
Figure 12.28: Same as Fig. 12.27 but for a = 0.95 m and the images in the right column using a log scale to
reveal the faint parts. The critical curves are the same as in Fig. 12.20.
498 Null geodesics and images in Kerr spacetime
the tertiary image. The green circle would then mark the location of polar spherical photon
orbits, at rph
pol
≃ 2.883 m for a = 0.5 m, since only the ℓ = 0 critical null geodesics matter for
θO = 0. There is actually an infinite sequence of images, the image of order n being generated
by null geodesics that have performed n half-turns around the sphere r = rph pol
before leaving
toward the observer, with n = 0 corresponding to the primary image. A confirmation of this
interpretation is provided by the superposition of the critical curve C onto the image, as the
red dotted line in the right column of Fig. 12.27. The tertiary image (n = 2) is then almost
indistinguishable from C , in agreement with the exponential convergence of high order images
to C . This exponential convergence has been established for a = 0 by Eq. (8.131), but it holds
for a ̸= 0 as well [235].
Another common feature with the Schwarzschild images of Fig. 8.25 is the left part of the
primary image being brighter than the right part as soon as θO ̸= 0. As discussed in Sec. 8.5.5,
this is due to the Doppler boosting resulting from the rotation of the accretion disk, the left
part moving towards the observer, while the right part is receding.
A difference with the Schwarzschild images discussed in Sec. 8.5.5 is that in the current
case, the disk is not confined to the equatorial plane, being geometrically thick, and moreover it
is optically thin. This means that a given null geodesic touching the screen “carries” not only a
photon from the disk’s surface but also photons from the disk interior, actually all the photons
emitted along the path of the geodesic inside the disk. This cumulative property results in a
enhanced brightness of the secondary image and makes it appear as a relatively bright ring, as
it is clearly seen on the θO = 0 and θO = π/6 images of Fig. 12.27. Another consequence of
optical thinness is that the bottom part of the secondary and tertiary images are not blocked
by the part of the accretion disk standing in the foreground for θO close to π/2, as they are in
the images of Fig. 8.25, which have been obtained for an optically thick disk model.
Another difference between the two sets of images is that in Fig. 12.27 the inner boundary
of the top part of the primary image is closer to the secondary image than in Fig. 8.25 (for
θO = 0, the primary and secondary images in Fig. 12.27 seem even to touch each other). This is
a mere consequence of the inner radius of the accretion disk being closer to the black hole in
Fig. 12.27, since rISCO
+
≃ 4.233 m for a = 0.5 m and rISCO
+
= 6m for a = 0.
In the images of Fig. 12.27, the interior of the critical curve, i.e. the black hole shadow
according to the definition given in Sec. 12.4.2, appears black, except when crossed by the
bottom of the primary image, which arises from the foreground part of the disk. This feature
is similar to the Schwarzschild case presented in Fig. 8.25. However, it does not persist for
highly spinning black holes, as we can see on Fig. 12.28, which shows images for the Kerr
parameter a = 0.95 m. There, the boundary of the central dark spot is clearly distinct from
the critical curve and lies within the latter. This is due to the inner part of the accretion disk
being located well within some spherical photon orbits, including the polar ones. Indeed, for
a = 0.95 m, we have rISCO+
≃ 1.937 m, while the radius of the retrograde circular photon orbit
is rph ≃ 3.995 m and that of polar spherical orbits is rph
− pol
≃ 2.493 m. Given that rph+
≃ 1.386 m,
we have thus rph < rISCO < rph < rph . More generally, using formulas (11.167), (12.62) and
+ + pol −
(12.67a), we have
+
rISCO −
< rph ⇐⇒ a > 0.638 m and +
rISCO pol
< rph ⇐⇒ a > 0.853 m. (12.114)
These properties are illustrated in Fig. 12.29. Hence, for large spins, if the inner boundary of
12.5 Images 499
6 +
rISCO
rph+
rph−
5 rphpol
r/m
3
1
0.0 0.2 0.4 0.6 0.8 1.0
a/m
Figure 12.29: Radius of the timelike prograde ISCO, rISCO
+
[Eq. (11.167)], compared with the radii of the
−
prograde and retrograde outer circular photon orbits, rph
+
and rph [Eq. (12.62)] and the radius of polar spherical
pol
photon orbits, rph [Eq. (12.67a)]. [Figure generated by the notebook D.5.15]
the accretion disk is set by the prograde ISCO, a large part of the primary image lies strictly
inside the critical curve C . Let us for instance consider the image for θO = 0 in Fig. 12.28,
which is the easiest to understand since for θO = 0, C involves only the ℓ = 0 critical null
geodesics (cf. Sec. 12.4.3), i.e. those that are rolling indefinitely around the sphere r0 = rph pol
in their asymptotic past. Since rISCO
+ pol
< rph in the present case, this implies that the part of
the disk located at r ∈ [rISCO , rph ) is an emitting region below the polar spherical photon
+ pol
orbits at r0 = rph
pol
. The emitted photons encounter then the screen scritly inside C . We shall
not provide a full demonstration here, but simply recall that this was established rigorously
for a = 0 in Chap. 8: formula (8.131) along with A(rem ) < 0 for rem < 3m = rph pol
(a = 0) (cf.
Fig. 8.22) yields impact parameters bn that are lower than bc , the latter being nothing but the
±
radius of C for a = 0. Hence we conclude:
Property 12.22: black spot in the image and black hole shadow
If the black hole is surrounded by some emitting material located within some of the
spherical photon orbits, the central black spot in the images does not correspond to the
shadow defined in Sec. 12.4.2 as the interior of the critical curve C . The black spot is
actually smaller than the shadow, lying strictly inside C .
This holds for a = 0 as well, if the emitting matter is not limited inward by the ISCO as in
Fig. 8.25 (see e.g. Fig. 5 in Ref. [234] or Fig. 2 in Ref. [494]). Note however that in the case of a
purely radial inflow, the emitted photons suffer such a strong Doppler effect that the interior of
the critical curve is very dark, so that the effective shadow coincides with the academic shadow
of Sec. 12.4.2, as shown in Refs. [192, 377].
A striking feature of the images shown in Fig. 12.28 is that while the central black spot
departs considerably from spherical symmetry at large inclination angle θO , the secondary
500 Null geodesics and images in Kerr spacetime
and higher order images are pretty circular. This is because they accumulate onto the critical
curve, which is quite close to a circle, as shown in Fig. 12.20. The only major effect of a high
inclination angle is to move the high order images the right.
Besides, we note on Fig. 12.28 that the secondary image (n = 1) is brighter than the primary
one (n = 0). Moreover, the tertiary one (n = 3), which is very thin and superposed onto the
secondary image, is even brighter. This is because the higher the image order, the larger the
number n of half-turns around the black hole of the involved geodesics, and thus the longer
the path length of the geodesics inside the disk, resulting a larger number of photons “carried”
by a given geodesic and accumulating on the same pixel of the screen. Note that this holds only
because the disk is optically thin. Would it be optically thick, a null geodesic impacting the
screen, even if very close to a critical geodesic, would carry only a single photon: the photon
emitted at the first intersection of the geodesic with the disk’s surface when traced backward in
time from the screen. In Sec. 8.5.5 (Schwarzschild case), we considered such an optically thick
disk (Page-Thorne model) and the secondary and tertiary images were not brighter than the
primary one (cf. Fig. 8.25). However in that case, the main reason was that the emitting region
was too far from the black hole for a close-to-critical geodesic to visit various parts of the disk.
For instance, in the lower part of Fig. 8.26, consider the red geodesic that intersects the disk at
the point of coordinates (x, y) = (0, 6m); it contributes to the tertiary image and if one would
extend it to the past beyond the point (0, 6m), it would never encounter the disk again, leaving
the figure at the upper left corner in a more or less straight line. We shall therefore state:
Property 12.23: photon ring
The set constituted by the secondary image and higher order images, i.e. all images of
order n ≥ 1, is called the photon ring. It appears as a bright feature on the observer’s
screen under two conditions: the emitting material must (i) be optically thin and (ii) be
located in the vicinity of some spherical photon orbits.
As shown in Fig. 12.28, the photon ring generally appears as a single ring because the successive
images are superposed on each other and converge exponentially to the the critical curve C .
Only the n = 1 and n = 2 parts of the photon ring are visible in Fig. 12.28 by lack of resolution,
since the thickness of the n-th image decays exponentially with n [235]. We note on Figs. 12.27
and 12.28 that the n = 2 image (tertiary image) is already very close to C . We conclude:
Property 12.24: photon ring and critical curve
In practice the photon ring, and more precisely the part n ≥ 2 of it, materializes the critical
curve.
The photon ring is therefore a potentially observable feature that can provide information on
the spacetime metric independently of the emission model [304]. For instance, by analyzing the
photon ring, one could check that the metric is the Kerr metric and measure the spin parameter
a.
Remark 1: The reader is warned that the term photon ring is sometimes used in the literature for circular
±
photon orbits, as the orbit at r = 3m in Schwarzschild spacetime and the orbits at r = rph in Kerr
12.5 Images 501
spacetime (see e.g. Ref. [84] for a recent example). We are using it here not for an orbit around the black
hole but for a feature on the screen of a remote observer, in agreement with Refs. [47, 302, 304, 235, 237].
Recently the term secondary ring. has been introduced [494] to denote the subpart of the photon ring
that contributes significantly to the flux in the image. Besides, some authors have distinguished the
secondary image (n = 1) by calling it the lensing ring [234].
Historical note : In 1973, Christopher T. Cunningham and James M. Bardeen [146] computed the
multiple images of a star orbiting an extremal Kerr black hole (a = m). In 1979, Jean-Pierre Luminet
[348] predicted that a Schwarzschild black hole illuminated
√ by a parallel light beam would appear to a
remote observer as a black disk of radius Rshad = 3 3m/rO (cf. Eq. (12.91)) surrounded by a sequence
of rings, called ghost rings by Luminet, converging exponentiallty to the rim of the black disk (which is
the critical curve). These rings are fainter and fainter, the brightest ring being the outermost one. In
the very same article [348], Luminet presented the first computed image of an accretion disk around a
Schwarzschild black hole (cf. historical note on p. 280). The first images of an accretion disk around a
Kerr black hole with a ̸= 0 have been computed by S.U. Viergutz in 1993 [491], for a = 0.998 m. In 1997,
Michał Jaroszyński and Andrzej Kurpiewski [299] studied an optically thin accretion disk extending
inwards down to the event horizon of a Kerr black hole with a = 0, 0.5 m and 0.9 m, with some nonzero
radial component of the accretion flow. They showed that the observed intensity just outside the critical
curve is enhanced by the long path length of geodesics orbiting many times around the black hole, while
the intensity inside the critical curve is low, even if there is some emitting material close to the black
hole. In 2000, Heino Falcke, Fulvio Melia and Eric Agol [192] (see also Ref. [191]) have computed images
of an optically thin accretion flow around a Kerr black hole with a = 0.998 m. The accretion flow was
assumed to extend down to the event horizon, infalling with constant angular momentum inside the
ISCO. In agreement with Jaroszyński and Kupriewski’s prediction, they obtained a large intensity just
outside the critical curve and low intensity inside it. For this reason, they introduced the term shadow of
the black hole for the interior of the critical curve, the latter being called by them the apparent boundary
of the black hole. Moreover, in the same article [192], they advanced that, in the case of Galactic Center
black hole Sgr A*, the shadow could be observed by means of very-long-baseline interferometry (VLBI),
thereby proposing what would become the Event Horizon Telescope two decades later [6].
12.5.3 EHT image of M87*
The computed black hole images, as those shown in Figs. 12.27 and 12.28, can be contrasted
with actual observations since the release of the very first observed image by the Event Horizon
Telescope (EHT) collaboration in 2019 [6, 521]. This image, shown in Fig. 12.30, is that of
the supermassive black hole M87* in the nucleus of the giant elliptic galaxy Messier 87 at
the center of the Virgo cluster. It is actually a reconstructed image, after more than one year
of data processing of an incomplete data set (points in the image Fourier plane) obtained by
very-long-baseline interferometry (VLBI) in April 2017. A new image has been published in
2024 from data obtained in April 2018 [9].
In order to interpret the EHT image, it is natural to superpose the (theoretical) critical curve
C of a Kerr black hole onto it. This is all the more meaningful that C is closely related to the
photon ring, as discussed in Sec. 12.5.2. In order to perform the superposition for a given value
of the Kerr spin parameter a, one must know three things: (i) the overall scale factor m/rO , (ii)
the angle θO between the black hole rotation axis and the line of sight and (iii) the orientation
Θ of the projection of the rotation axis onto the plane of the sky. It turns out that for M87*
502 Null geodesics and images in Kerr spacetime
15
10
5 jet
(r /m) βs
0
10
15
15 10 5 0 5 10 15
(r /m) αs
Figure 12.30: Image of the immediate vicinity of the supermassive black hole M87*, as released by the EHT
collaboration in 2019 [6], with two critical curves superposed: that of a Schwarzschild black hole (magenta dotted
circle) and that of an extremal Kerr black hole seen under the inclination θO = 163◦ (green dotted curve), with
the projection of the spin axis onto the screen indicated by the green arrow (position angle Θ = 108◦ with respect
to the βs -axis). The critical curves are scaled by assuming m/rO = 3.7 µas (cf. Table 12.1). (αs , βs ) are screen
angular coordinates, closely related to celestial equatorial coordinates: αs is minus the (relative) right ascension
and βs is the (relative) declination. The white circle in the bottom right corner indicates the EHT resolution
(approx. 20 µas). [Figure generated by the notebook D.5.16; source of the EHT image: Fig. 3 of Ref. [6]]
these three quantities can be estimated. Indeed, the distance from the Earth to M87* has been
measured to rO ≃ 16.7 Mpc and the mass of M87* has been deduced from stellar dynamics
in the inner part of the galactic nucleus, resulting in8 m ≃ 6.2 109 M⊙ (cf. Table 12.1 and
references therein), hence the scale factor m/rO ≃ 3.7 µas. Regarding θO and Θ, both angles
can be determined via reasonable assumptions, thanks to the large relativistic jet emanating
from the close vicinity of the black hole9 , since most (all?) theoretical models predict that the
jet is aligned with the black hole’s rotation axis. There are actually two jets, emitted from both
sides of the black hole in opposite directions. Due to a strong Doppler beaming effect, the jet
moving away from us, usually called the counterjet, is hardly visible, so that the large observed
jet is the one moving towards us. The position angle of the jet in the plane of the sky, measured
from the North axis, is Θjet ∼ −72◦ (see e.g. Fig. 1 in Ref. [509]); it is indicated by the light
blue dashed arrow in Fig. 12.30 (the jet is not visible on the EHT image). The inclination ι of
the jet with respect to the line of sight is estimated from observed motions of “nodes” within
the jet, resulting in a value around ι ∼ 17◦ [365, 509]. In particular, the detection of apparent
superluminal motions, up to vapp ∼ 6c, implies both relativistic velocities and ι being a small
8
Another method, based on the dynamics of the gas in the nucleus disk at r < 40 pc, yields a mass of only
m = 3.5 109 M⊙ [514]; the discrepancy with the value arising from stellar dynamics has not been explained yet.
9
This jet has been discovered in 1918 [148], hence well before one suspects a black hole to lie in the core of the
galaxy M87! It makes M87 belong to the category of galaxies with an active galactic nucleus (AGN). The jet engine
is related to the black hole, possibly via the Penrose process discussed in Sec. 11.3.2, see e.g. Ref. [2].
12.5 Images 503
angle (see e.g. Sec. 5.7.4 of Ref. [228]). Assuming that the jet axis is aligned with the black
holes’s rotation axis, we have either θO = ι or θO = π − ι.
The main feature of the EHT image (Fig. 12.30) is a broad annular region, whose bottom
part is much brighter than its top part. It is natural to interpret this brightness discrepancy
as resulting from relativistic Doppler beaming: the bottom (resp. top) part corresponds to
emitting material moving towards us (resp. receding from us) [7]. Assuming that, so close to
the black hole, the emitting matter is rotating in the same sense as the black hole, due to the
Lense-Thirring effect (cf. Sec. 11.3.4), this implies that the spin of black hole is pointing away
from us, i.e. that θO = π − ι ∼ 163◦ and Θ = π + Θjet ∼ 108◦ . The projection of the resulting
black hole spin vector onto the plane of the sky is shown by the green arrow in Fig. 12.30.
To superpose the critical curves computed in Sec. 12.4 onto the EHT image, it suffices then
to rotate them by the angle Θ and to use the M87* scale factor m/rO = 3.7 µas determined
above. In Fig. 12.30, we have done it for the two extreme values that can be taken by the spin
parameter a: a = 0 (magenta dotted cirle) and a = m (green dotted curve). The two critical
curves are quite close to each other because the Earth observer is located almost on the axis of
rotation, θO being close to π. To have a stronger discrepancy among various possible values of
a, it would have been better to have θO close to π/2 instead, as illustrated in Fig. 12.26.
The main conclusion that one can draw from Fig. 12.30 is that the overall scale of the
EHT image fits well with what could be expected from emitting matter in the close vicinity
of a black hole of mass around 6 109 M⊙ . Unfortunately, the resolution of the EHT image
(∼ 20 µas ∼ 1/6 of the image width!) is not sufficient to draw sharper conclusions. In
particular, one cannot assert whether the annular structure in the image is the photon ring
discussed in Sec. 12.5.2 blurred to the EHT resolution or the primary image of an accretion disk
or some more complicated accretion flow. It follows that it is not possible to infer the (currently
unknown) spin parameter a from that image, not speaking about performing any strong test of
general relativity (see Refs. [233, 224] for an extended discussion of this last point). However,
this is the very first image of this type and the next images, which will result from some EHT
upgrade or from space-based VLBI [305, 350], will undoubtedly be sharper and will certainly
revolutionize black hole astronomy.
12.5.4 Going further
For a detailed study of images of the surroundings of a Kerr black hole, see Refs. [167, 168,
169, 235]. In particular, Ref. [235] provides exact solutions of the null geodesic equation via
elliptic integrals, generalizing those derived in Chap. 8 for the Schwarzschild black hole. We
have discussed only images as seen by a remote observer, which is the “astronomical” setting.
For the images seen by an observer travelling close to, and even inside, a Kerr black hole (the
“science-fiction” setting), see the recent study [434].
504 Null geodesics and images in Kerr spacetime
Chapter 13
Extremal Kerr black hole
Contents
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
13.2 Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . 506
13.3 Maximal analytic extension . . . . . . . . . . . . . . . . . . . . . . . . 513
13.4 Near-horizon extremal Kerr (NHEK) geometry . . . . . . . . . . . . . 523
13.5 Going further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
13.1 Introduction
The Kerr solution of the Einstein equation has been introduced in Sec. 10.2; it depends on
two parameters: the mass m > 0 and the spin parameter a ≥ 0. In Chaps. 10–12, we have
considered the Kerr solution with 0 < a < m and the case a = 0 (Schwarzschild solution) has
been treated in Chaps. 6–9. Here we focus on the case a = m, which is called the extremal Kerr
spacetime. This corresponds to the highest value of a for which the Kerr solution describes a
black hole. Indeed, for a > m, the Kerr metric is still an exact solution of the vacuum Einstein
equation (1.44), but it describes a naked singularity (cf. Sec. 9.6.1): the ring singularity is not
hidden by any horizon to asymptotic observers.
The Kerr spacetime with a = m is not just the maximally spinning black hole of the Kerr
family. As we going to see, it has important properties that are not shared by Kerr spacetimes
with a < m. In particular, the black hole event horizon is degenerate, in the sense defined in
Sec. 3.3.6, i.e. it has a vanishing surface gravity κ. This corresponds to an extremal black hole,
according to the terminology introduced in Sec. 5.4.3. Besides, there is no internal horizon
and the maximal analytic extension, to be constructed in Sec. 13.3 below, is simpler than that
of the Kerr spacetime with a < m. Another specific property of the extremal Kerr spacetime
(actually related to the degeneracy of the horizon) regards the geometry near the horizon: it
admits an enlarged symmetry group, which is generated by four independent Killing vectors,
instead of two for the geometry of the global solution. This is called the NHEK geometry (for
506 Extremal Kerr black hole
near-horizon extremal Kerr) and has received considerable attention in the recent literature. We
shall discuss it in Sec. 13.4.
13.2 Definition and basic properties
13.2.1 The extremal Kerr solution
Let us consider the manifold R2 × S2 described by coordinates (t̃, r, θ, φ̃) such that (t̃, r) cover
R2 and (θ, φ̃) are standard spherical coordinates on S2 . The extremal Kerr spacetime of mass
m > 0 is defined as the pair (M , g) where the manifold M is the following open subset of
R2 × S2 :
M := R2 × S2 \ R (13.1)
with n πo
R := p ∈ R2 × S2 , r(p) = 0 and θ(p) = , (13.2)
2
and the metric g has the following expression in terms of the coordinates (xα̃ ) = (t̃, r, θ, φ̃):
4m2 r sin2 θ
2mr 4mr
g = − 1− 2 dt̃2 + 2 dt̃ dr − dt̃ dφ̃
ρ ρ ρ2
2mr 2mr
+ 1+ 2 2
dr − 2m 1 + 2 sin2 θ dr dφ̃ (13.3)
ρ ρ
3 2
2 2 2 2 2m r sin θ
+ρ dθ + r + m + sin2 θ dφ̃2 ,
ρ2
with
ρ2 := r2 + m2 cos2 θ. (13.4)
In this context, the coordinates (xα̃ ) = (t̃, r, θ, φ̃) are called Kerr coordinates and we recognize
in (13.3) the limit a → m of expression (10.36) for the Kerr metric with a < m.
The metric (13.3) is regular in all M , since the components gα̃β̃ are singular only for ρ = 0,
i.e. for r = 0 and θ = π/2, which defines the set R that has precisely been excluded from M
in the definition (13.1). The Kretschmann curvature invariant K := Rµνρσ Rµνρσ is given by
Eq. (10.22) with a = m; it diverges for ρ → 0. Therefore, as for the Kerr spacetime with a < m
(cf. Sec. 10.2.6), we shall call R the ring singularity of the extremal Kerr spacetime. Note that,
formally, it is not part of the spacetime manifold M [cf. Eq. (13.1)].
Moreover, the Ricci tensor of the metric (13.3) is identically zero in all M (see the note-
book D.5.17 for the computation). Hence, we have:
Property 13.1: extremal Kerr metric as a solution of the vacuum Einstein equation
The metric g of the extremal Kerr spacetime is a solution of the vacuum Einstein equation
R = 0 [Eq. (1.44)].
13.2 Definition and basic properties 507
The inverse metric is
2mr 2mr
−1 − ρ2 ρ2
0 0
2mr (r−m)2 m
ρ2 ρ2
0 ρ2
g α̃β̃ = . (13.5)
1
0 0 ρ2
0
m 1
0 ρ2
0 ρ2 sin2 θ
13.2.2 Boyer-Lindquist coordinates
For a < m, the Kerr manifold M has been split in three open regions, MI , MII and MIII ,
separated by the two Killing √horizons H and Hin [cf. Eqs. (10.29) and (10.2a)]. Since
√ H was
defined by r = r+ := m + m − a [Eq. (10.27)] and Hin by r = r− := m − m2 − a2
2 2
[Eq. 10.28)], we notice that r+ = r− = m in the limit a → m. This implies that H and Hin
coincide when a → m and the region MII , which is bounded by H and Hin , disappears.
Accordingly, we shall split the extremal Kerr manifold M in two open regions only, MI and
MIII , separated by a single hypersurface H :
M = MI ∪ H ∪ MIII , (13.6)
with
MI := {p ∈ M , r(p) > m} (13.7a)
H := {p ∈ M , r(p) = m} (13.7b)
MIII := {p ∈ M , r(p) < m} . (13.7c)
Remark 1: We are using the notation MIII for the “second” region, and not MII , to be consistent with
Chaps. 10–12, i.e. with the limit a → m of the results obtained in these chapters.
The quadratic polynomial in r introduced in Chap. 10, ∆ = r2 −2mr+a2 = (r−r+ )(r−r− ),
reduces to ∆ = (r − m)2 in the limit a → m. Its double root, r = m, defines the hypersurface
H.
In the region MBL := M \ H = MI ∪ MIII , one may introduce the Boyer-Lindquist
coordinates (t, r, θ, φ) such that (r, θ) are the same coordinates as in Kerr coordinates, while t
and φ are related to the Kerr coordinates t̃, r and φ̃ by
2m2 r−m
t = t̃ + − 2m ln (13.8a)
r−m m
m
φ = φ̃ + . (13.8b)
r−m
Differentiating these relations leads to
2mr m
dt̃ = dt + dr and dφ̃ = dφ + dr. (13.9)
(r − m)2 (r − m)2
508 Extremal Kerr black hole
t̃/m
4
r/m
10 5 5 10
Figure 13.1: Trace of the hypersurfaces of constant Boyer-Lindquist time t in the plane (t̃, r). [Figure generated
by the notebook D.5.17]
Remark 2: The differential relations (13.9) can be obtained immediately by substituting a by m in
relations (10.34). However, to get the integrated
√ relations (13.8) from their a < m counterpart (10.35),
one must perform an expansion in ε := m2 − a2 /m, taking into account that r± = m(1 ± ε). One
obtains then (13.8a) up to the additive constant (2 ln 2)m, while (13.8b) is recovered in the same form.
It follows from the transformations (13.8) that the Boyer-Lindquist coordinate frame (∂α )
and the Kerr coordinate frame (∂α̃ ) are related by1
∂t = ∂t̃ (13.10a)
2mr m
∂r = ∂r̃ + ∂ t̃ + ∂φ̃ (13.10b)
(r − m)2 (r − m)2
∂θ = ∂θ (13.10c)
∂φ = ∂φ̃ . (13.10d)
Remark 3: As in Chap. 10, we have denoted by ∂r̃ the second vector of the coordinate frame associated
to the Kerr coordinates (xα̃ ) = (t̃, r, θ, φ̃), in order to distinguish it from the coordinate vector ∂r of
the Boyer-Lindquist coordinates (xα ) = (t, r, θ, φ).
The metric components (gαβ ) with respect to the Boyer-Lindquist coordinates (xα ) =
(t, r, θ, φ) are given by
4m2 r sin2 θ ρ2
2mr
g = − 1− 2 dt2 − dt dφ + dr2
ρ ρ2 (r − m)2
(13.11)
2m3 r sin2 θ
2 2 2 2 2 2
+ρ dθ + r + m + sin θ dφ .
ρ2
This expression can be obtained either by taking the limit a → m of Eq. (10.8) or by using (13.9)
to substitute dt̃ and dφ̃ in Eq. (13.3). We note that grr → +∞ when r → m, which reflects the
1
See also the limit a → m of Eq. (10.38).
13.2 Definition and basic properties 509
singularity of Boyer-Lindquist coordinates on H and explains why the latter was excluded in
the definition of MBL . This singularity is clearly apparent in the coordinate transformations
(13.8), as well as in the spacetime slicing by the hypersurfaces t = const depicted in Fig. 13.1:
the slices accumulate onto H , without crossing it, so that the points on H do not belong to
any hypersurface t = const. That the t = const hypersurfaces do not provide a regular slicing
of the extremal Kerr spacetime (M , g) is also manifest on the Carter-Penrose diagram shown
in Fig. 13.4.
The Boyer-Lindquist components of the inverse metric g −1 are
1 2 2 2m3 r sin2 θ 2m2 r
− (r−m) 2 r + m + ρ2
0 0 − ρ2 (r−m)2
(r−m) 2
0 ρ 2 0 0
αβ
g = . (13.12)
1
0 0 ρ 2 0
2m2 r 1 2mr
− ρ2 (r−m)2 0 0 (r−m)2 sin2 θ 1 − ρ2
They can also be obtained by taking the limit a → m of Eq. (10.16).
13.2.3 Symmetries
The extremal Kerr metric (13.3) is stationary2 and axisymmetric. The corresponding isometry
group is R × SO(2) ≃ R × U(1) and is generated by two commuting Killing vectors ξ and
η. Both the Kerr coordinates and the Boyer-Lindquist ones are adapted to the spacetime
symmetries, i.e. (t̃, φ̃) and (t, φ) are ignorable coordinates, as it is clear from the metric
components (13.3) and (13.11). Accordingly, one can normalize the Killing vectors so that
ξ = ∂t̃ = ∂t and η = ∂φ̃ = ∂φ . (13.13)
13.2.4 Principal null geodesics
As discussed in Sec. 10.4, the Kerr spacetime is endowed with two congruences of null geodesics
tied to the spacetime structure, as described by the Weyl conformal curvature tensor. All the
results of Sec. 10.4 remain valid at the limit a → m. We can summarize them as follows:
Property 13.2: principal null geodesics of the extremal Kerr spacetime
• The ingoing principal null geodesics are the curves
L(v,θ,
in
φ̃) : (v, θ, φ̃) = const ∈ R × [0, π] × [0, 2π), (13.14)
2
Cf. Sec. 5.2.1 for the definition of stationary and some discussion about the terminology.
510 Extremal Kerr black hole
t̃/m
4
k
`
2
r/m
8 6 4 2 2 4 6 8
Figure 13.2: Trace of the principal null geodesics in the plane (t̃, r). The dashed lines correspond to the ingoing
out,I
principal null geodesics L(v,θ,
in
φ̃) and the solid curves to the outgoing principal null geodesics L(u,θ,φ̃)
˜ for r > m
out,III
and L(u,θ, ˜ for r < m. [Figure generated by the notebook D.5.17]
φ̃)
where v is the Kerr advanced time:
2m2 r−m
v := t̃ + r = t + r − + 2m ln . (13.15)
r−m m
Along any geodesic L(v,θ,
in
φ̃) , −r is an affine parameter increasing towards the future;
the corresponding tangent vector is
k = ∂t̃ − ∂r̃ (13.16a)
r2 + m2 m
k= 2
∂t − ∂r + ∂φ in M \ H . (13.16b)
(r − m) (r − m)2
• The outgoing principal null geodesics are the curves
in MI : L(u,θ,
out,I
˜ :
φ̃)
˜ = const ∈ R × [0, π] × [0, 2π),
(u, θ, φ̃) (13.17a)
in MIII : L(u,θ,
out,III
˜ :
φ̃)
˜ = const ∈ R × [0, π] × [0, 2π),
(u, θ, φ̃) (13.17b)
on H : L(θ,ψ)
out,H
: (θ, ψ) = const ∈ [0, π] × [0, 2π), (13.17c)
13.2 Definition and basic properties 511
where u is the Kerr retarded time:
4m2 r−m 2m2 r−m
u := t̃ − r + − 4m ln =t−r+ − 2m ln (13.18)
r−m m r−m m
and φ̃˜ and ψ are defined by
2m m
φ̃˜ := φ̃ + =φ+ (13.19)
r−m r−m
and
t̃
ψ := φ̃ − . (13.20)
2m
Along the geodesics L(u,θ,
out,I
˜ and L(u,θ,φ̃)
φ̃) ˜ , r is an affine parameter increasing towards
out,III
the future, while along L(θ,ψ)
out,H
, such an affine parameter is t̃. The tangent vector
ℓ to the outgoing principal null geodesics that coincides with the Killing vector
ξ + (2m)−1 η on H is
(r + m)2 (r − m)2 m
ℓ= 2 2
∂t̃ + 2 2
∂r̃ + 2 ∂φ̃ (13.21a)
2(r + m ) 2(r + m ) r + m2
1 (r − m)2 m
ℓ = ∂t + 2 2
∂r + ∂φ in M \ H . (13.21b)
2 2(r + m ) 2(r + m2 )
2
Proof. The second equality in Eq. (13.15) follows from Eq. (13.8a). Equation (13.16b) follows
from Eq. (13.16a) via Eq. (13.10). Equations (13.18) and (13.19) are the integrated version of the
system (10.52) with a = m. Equations (13.20), (13.21a) and (13.21b) are the a = m versions
of respectively Eqs. (10.59), (10.57) and (10.61). All the other statements follow from the limit
a → m of results of Sec. 10.4, except for t̃ being an affine parameter along L(θ,ψ)
out,H
, which is
peculiar to the extremal Kerr horizon and will be proven in Sec. 13.2.5.
The principal null geodesic congruences are depicted in terms of the (t̃, r) coordinates in
Fig. 13.2. Note that the outgoing geodesics L(u,θ,
out,I
˜ and L(u,θ,φ̃)
φ̃) ˜ tend to become tangent to H
out,III
for r → m; this agrees with H being generated by some members of the outgoing principal
null congruence, namely the geodesics L(θ,ψ)
out,H
, as we shall see in details in the next subsection.
Another view of the principal null geodesics is provided by the Carter-Penrose diagram of
(M , g) shown in Fig. 13.3, in which both families of geodesics appear as straight lines.
13.2.5 The degenerate horizon
H is the hypersurface of M defined by r = m [Eq. (13.7b)]. Given that the component
g rr = (r − m)2 /ρ2 of the inverse metric with respect to Kerr coordinates [Eq. (13.5)] vanishes
→
−
at r = m, we have g µ̃ν̃ ∂µ̃ r∂ν̃ r = 0 on H , which implies that the gradient ∇r is a null vector
→
−
there and that H is a null hypersurface. Moreover, since the components of ∇r are ∇α̃ r = g α̃r ,
512 Extremal Kerr black hole
we read on Eq. (13.5) that
− H 2m2
→ m H 2m
2
∇r = 2 ∂t̃ + 2 ∂φ̃ = 2 χ, (13.22)
ρ ρ ρ
where χ is the Killing vector field
1
χ := ξ + ΩH η , with ΩH := . (13.23)
2m
It follows immediately that H is a Killing horizon, i.e. a null hypersurface that admits a Killing
vector as null normal (cf. Sec. 3.3.2).
From expression (13.21a) for ℓ, we have immediately
H
ℓ = χ. (13.24)
This means that the null generators of H are the outgoing principal null geodesics L(θ,ψ)
out,H
.
The surface gravity κ of the Killing horizon H has been defined in Sec. 3.3.5 as the non-
H
affinity coefficient of the Killing-vector normal χ to H : ∇χ χ = κ χ [Eq. (3.29)]. Given
the identity (13.24), κ coincides with the value on H of the non-affinity coefficient κℓ of the
tangent ℓ to the outgoing principal null geodesics: ∇ℓ ℓ = κℓ ℓ. A direct computation (cf. the
notebook D.5.17) reveals that
r2 − m2
κℓ = m 2 . (13.25)
(r + m2 )2
In particular, κℓ vanishes for r = m, i.e. on H . Hence κ = κℓ |H leads to3
κ=0. (13.26)
According to the classification introduced in Sec. 3.3.6, it follows that H is a degenerate Killing
horizon. The vanishing of the non-affinity coefficient κ means that ℓ is a geodesic vector on H ,
and not only a pregeodesic one (cf. Remark 1 in Sec. 1.3.2). Equivalently, at any given point
p ∈ H , ℓ is the tangent vector associated to an affine parameter λ of the null geodesic L(θ,ψ)
out,H
through p. Moreover, the affine parameter λ coincides with t̃, up to some additive constant.
Indeed, Eqs. (13.24) and (13.23) imply
dt̃
ℓt̃ = = χt̃ = 1,
dλ
from which λ = t̃ + const. Since the range of t̃ is (−∞, +∞), we conclude that L(θ,ψ) out,H
is a
complete geodesic. This constrasts with the null generators of a non-degenerate Killing horizon,
which are incomplete, as shown in Sec. 3.4.2.
Let us summarize the results obtained above:
3
The vanishing of κ can also be obtained by taking the limit a → m of expression (10.77), which has been
derived for a < m.
13.3 Maximal analytic extension 513
Property 13.3: H as a degenerate Killing horizon
In the extremal Kerr spacetime, the hypersurface H defined by r = m is a degenerate
Killing horizon. Its generators are the outgoing principal null geodesics L(θ,ψ)
out,H
, which
are complete geodesics and which admit the Kerr coordinate t̃ as an affine parameter. The
tangent vector associated to this affine parameter is the Killing vector χ = ξ + ΩH η
[Eq. (13.23)], which coincides on H with the tangent vector ℓ to the outgoing principal
null congruence.
13.2.6 Black hole character
As a Killing horizon, H is a null hypersurface and thus a one-way membrane (cf. Sec. 2.2.2).
Since the ingoing principal null geodesics L(v,θ,
in
φ̃) cross it from MI to MIII (cf. Fig. 13.2), we
conclude that no (massive or null) particle can cross H from MIII to MI . In order to show that
H is actually a black hole event horizon, it suffices to proceed as for the a < m case treated in
Sec. 10.5.2. We shall not repeat the argument here (which is based on the asymptotics of Kerr
spacetime being that of Schwarzschild spacetime — a property that holds for the extremal Kerr
spacetime as well) and jump directly to the conclusion:
Property 13.4: black hole region in the extremal Kerr spacetime
The extremal Kerr spacetime (M , g) can be endowed with a conformal completion at null
infinity such that the future and past null infinities I + and I − are located at the boundary
of MI . The region MIII is then the interior of a black hole, the event horizon of which is
the Killing horizon H . Since H is degenerate, the black hole is extremal, according to the
terminology introduced in Sec. 5.4.3.
The future null infinity I + and the past null infinity I − relative to MI are depicted in
the Carter-Penrose diagram of Figs. 13.3 -13.4. In this diagram, it is clear that MIII is a black
hole region for MI and that H is the corresponding event horizon.
13.3 Maximal analytic extension
13.3.1 Extension of MI for complete outgoing principal null geodesics
Figure 13.3 depicts a Carter-Penrose diagram of the extremal Kerr spacetime (M , g) built by
means of the projection map4 Π : M → R2 , (t̃, r, θ, φ̃) 7→ (T, X) defined by
T = T0 (u, v) and X = X0 (u, v) in MI (13.27a)
T = arctan(v/2) + π/2 and X = arctan(v/2) − π/2 on H (13.27b)
T = T0 (u, v) + π and X = X0 (u, v) − π in MIII (13.27c)
4
cf. the definition of a Carter-Penrose diagram given in Sec. 10.8.1.
514 Extremal Kerr black hole
Figure 13.3: Carter-Penrose diagram of the extremal Kerr spacetime (M , g) constructed via the projection
map Π : M → R2 , (t̃, r, θ, φ̃) 7→ (T, X) defined by Eqs. (13.27)-(13.28). The grey curves represent hypersurfaces
t̃ = const, with t̃ ∈ [−20m, 20m] and the increment δ t̃ = 2m between two successive hypersurfaces. The
hypersurface t̃ = 0 is singled out by a larger thickness. The red dotted curves represent hypersurfaces r = const,
with the increment δr between two successive hypersurfaces being δr = 2m for r < 0 and r > 3m and
δr = 0.2 m for 0 ≤ r ≤ 3m. The hypersurface r = 0 is marked by the brown dashed curve. The green straight
lines depict some selected principal null geodesics (dashed = ingoing, solid = outgoing, as in Fig. 13.2). iint is the
internal infinity discussed in Sec. 13.4.1. [Figure generated by the notebook D.5.17]
13.3 Maximal analytic extension 515
Figure 13.4: Same as Fig. 13.3 but for the time slicing associated to Boyer-Lindquist coordinates (t, r, θ, φ).
The blue curves represent hypersurfaces t = const, with t ∈ [−20m, 20m] and the increment δt = 2m between
two successive hypersurfaces. Note that the spacetime slicing by the hypersurfaces t = const is singular at H ,
contrary to the slicing by the hypersurfaces of constant Kerr time t̃ shown in Fig. 13.3. [Figure generated by the
notebook D.5.17]
with
u v v u
T0 (u, v) := arctan + arctan , X0 (u, v) := arctan − arctan , (13.28)
2 2 2 2
where u and v are the functions of (t̃, r) given by Eqs. (13.18) and (13.15) respectively. Some
outgoing principal null geodesics L(u,θ, out,I
˜ and L(u,θ,φ̃)
φ̃) ˜ are plotted in Fig. 13.3 for selected values
out,III
of u (solid green lines): u = −4m, u = 0 and u = 4m. Since r is an affine parameter along the
null geodesics L(u,θ,
out,I
˜ , it is clear that these geodesics are incomplete, for they all terminate
φ̃)
in the past at the finite value r = m (the South-West boundary of MI in Fig. 13.3), without
any possible extension into MIII from there. To extend MI so that all geodesics L(u,θ, out,I
˜ are
φ̃)
complete, let us introduce a coordinate system on MI that is adapted to the outgoing principal
null geodesics, as the Kerr coordinates (t̃, r, θ, φ̃) were adapted to the ingoing ones. We thus
516 Extremal Kerr black hole
˜ ˜ r, θ, φ̃)
define the outgoing Kerr coordinates (xα̃ ) = (t̃, ˜ by
u = t̃˜ − r ⇐⇒ t̃˜ = u + r, (13.29)
where u is the retarded Kerr time (13.18) and φ̃˜ is related to the angle φ̃ of Kerr coordinates
or to the angle φ of Boyer-Lindquist coordinates by Eq. (13.19). Substituting Eq. (13.18) for u
into t̃˜ = u + r and using Eq. (13.19) linking φ̃˜ to φ̃, we get the transition map between the Kerr
coordinates (t̃, r, θ, φ̃) and the outgoing Kerr coordinates (t̃, ˜ r, θ, φ̃):
˜
4m2
˜ r−m
t̃ = t̃ + − 4m ln
r−m m
on MI , (13.30)
φ̃˜ = φ̃ +
2m
.
r−m
By construction, the tangent vector to L(u,θ,
out,I
˜ associated with the affine parameter r is
φ̃)
then
ℓ′ = ∂t̃˜ + ∂r̃˜, (13.31)
where ∂r̃˜ stands for the vector ∂/∂r of the coordinates (t̃, ˜ r, θ, φ̃).
˜ Indeed ℓ′ α̃˜ = dxα̃˜ /dr =
(1, 1, 0, 0) since along L(u,θ, ˜
˜ , t̃ = r + u, with u constant, and both θ and φ̃ are constant. An
out,I ˜
φ̃)
explicit computation (cf. the notebook D.5.18) shows that ℓ′ is a geodesic vector:
∇ℓ′ ℓ′ = 0, (13.32)
which confirms that r is an affine parameter along L(u,θ,out,I
˜ . ℓ is thus similar to k, which is
φ̃)
′
the tangent vector to the ingoing principal null geodesics L(v,θ, in
φ̃) associated with the affine
parameter −r along them. In this respect, note the symmetry between the relations ℓ′ = ∂t̃˜+∂r̃˜
and k = ∂t̃ − ∂r̃ [Eq. (13.16a)]. The link between ℓ′ and the tangent vector ℓ to L(u,θ, out,I
˜
φ̃)
introduced in Sec. 13.2.4 is easily obtained from the definition of a tangent vector to a curve:
−1
′ dx dx dλ dr
ℓ = = = ℓ,
dr dλ dr dλ
where λ is the (non-affine) parameter of L(u,θ, out,I
˜ associated with ℓ. We read on the Kerr
φ̃)
components (13.21a) of ℓ, as well as on the Boyer-Lindquist ones (13.21b), that dr/dλ = ℓr =
(r − m)2 /(2(r2 + m2 )). Hence
r 2 + m2
ℓ′ = 2 ℓ. (13.33)
(r − m)2
Differentiating Eq. (13.30) leads to
4mr 2m
dt̃˜ = dt̃ − dr and dφ̃˜ = dφ̃ − dr. (13.34)
(r − m)2 (r − m)2
From these relations and the chain rule, we get immediately the link between the outgoing
Kerr coordinate frame and the Kerr coordinate frame:
4mr 2m
∂t̃˜ = ∂t̃ , ∂r̃˜ = ∂r̃ + ∂ t̃ + ∂φ̃ , ∂θ = ∂θ , ∂φ̃˜ = ∂φ̃ . (13.35)
(r − m)2 (r − m)2
13.3 Maximal analytic extension 517
In view of Eq. (13.13), we conclude that ∂t̃˜ and ∂φ̃˜ coincide with the Killing vectors ξ and η of
the Kerr metric:
∂t̃˜ = ξ and ∂φ̃˜ = η. (13.36)
The link between the outgoing Kerr coordinates and the Boyer-Lindquist ones is obtained
by substituting Eq. (13.18) for u into t̃˜ = u + r [Eq. (13.29)]:
2m2 r−m
t̃˜ = t + − 2m ln . (13.37)
r−m m
This relation is to be supplemented by Eq. (13.19) to fully specify the transformation from the
˜ r, θ, φ̃).
Boyer-Lindquist coordinates (t, r, θ, φ) to the outgoing Kerr coordinates (t̃, ˜ Differenti-
ating Eqs. (13.37) and (13.19) leads to
2mr m
dt̃˜ = dt − dr and dφ̃˜ = dφ − dr. (13.38)
(r − m)2 (r − m)2
Remark 1: Equation (13.38) differs from Eq. (13.9) only by the sign + changed to − in the right-hand
side. This reflects the complete symmetry between the Kerr coordinates (t̃, r, θ, φ̃) and the outgoing
˜ r, θ, φ̃)
Kerr coordinates (t̃, ˜ from the point of view of the Boyer-Lindquist coordinates (t, r, θ, φ) (cf. the
discussion at the beginning of Sec. 10.4.2).
The expression of the metric tensor with respect to the outgoing Kerr coordinates is easily
obtained by substituting dt and dφ from Eq. (13.38) into the Boyer-Lindquist expression (13.11)
(see also the notebook D.5.18); we get
4m2 r sin2 θ ˜ ˜
2mr ˜2 4mr ˜ 2mr
g = − 1− 2 dt̃ − 2 dt̃ dr − 2
dt̃ dφ̃ + 1 + 2 dr2
ρ ρ ρ ρ
2m r sin2 θ
3
2mr ˜ 2
+2m 1 + 2 2 2 2 2
sin θ dr dφ̃ + ρ dθ + r + m + 2
2
sin2 θ dφ̃˜ .
ρ ρ
(13.39)
These metric components are very similar to those in Kerr coordinates, as given by Eq. (13.3):
the only differences are gt̃r ˜ and grφ̃
˜ , which have a sign opposite to that of respectively gt̃r
and grφ̃ . Apart from the standard singularities of the spherical coordinates (θ, φ̃)˜ on the axis
θ ∈ {0, π}, the only singularity of the metric components (13.39) would occur at ρ = 0, which
does not happen in MI . In particular there is no divergence for r → m. This can be used to
extend smoothly the spacetime (MI , g) to r ∈ (−∞, m], so that the outgoing principal null
geodesics L(u,θ,
out,I
˜ with θ ̸= π/2 become complete. But the extension to r < m cannot be MIII
φ̃)
as it appears clearly on Fig. 13.3 that the end point of L(u,θ,
out,I
˜ for r → m is not located at the
φ̃)
+
boundary between MI and MIII . We thus introduce a spacetime (M ′ , g) from a new copy of
R2 × S2 with R2 spanned by the coordinates (t̃, ˜ r) and S2 spanned by the coordinates (θ, φ̃),
˜
such that (i) the manifold M is′
n πo
M ′ := R2 × S2 \ R ′ where R ′ := p ∈ R2 × S2 , r(p) = 0 and θ(p) = , (13.40)
2
518 Extremal Kerr black hole
(ii) MI is identified with the part r > m of M ′ and (iii) in all M ′ , g has the components given
by expression (13.39). Furthermore, we define
H ′ := {p ∈ M ′ , r(p) = m} and M ′ III := {p ∈ M ′ , r(p) < m} . (13.41)
We have then MI = M ∩ M ′ . In M ′ III , one can define Kerr coordinates (t̃, r, θ, φ̃) from
˜ r, φ, φ̃)
(t̃, ˜ via formulas (13.18) (with u = t̃˜ − r) and (13.19). It appears then immediately that
(M ′ III , g) is isometric to (MIII , g). It follows that g obeys the vacuum Einstein equation in all
M ′ . The vector field ℓ′ := ∂t̃˜ + ∂r̃˜ [cf. Eq. (13.31)] is a smooth non-vanishing null vector field
on M ′ . Since it is future directed in (MI , g) considered as a part of (M , g), we use it to set
the time orientation in all M ′ .
The tangent vector k to ingoing principal null geodesics has the following components
with respect to the outgoing Kerr coordinates:
(r + m)2 2m
k= ∂t̃˜ − ∂r̃˜ + ∂ ˜. (13.42)
(r − m)2 (r − m)2 φ̃
This follows immediately from k = ∂t̃ − ∂r̃ [Eq. (13.16a)] and using Eq. (13.35) to substitute
∂t̃ and ∂r̃ . Extending Equation (13.42) to all M ′ leads to a vector field that is singular on
H ′ . To get a vector field everywhere regular on M ′ , we rescale it by the inverse of the factor
connecting ℓ to ℓ′ in Eq. (13.33), i.e. we define
(r − m)2
k′ := k. (13.43)
2(r2 + m2 )
Hence
(r + m)2 (r − m)2 m
k′ = ∂ ˜ − ∂r̃˜ + 2 ∂ ˜. (13.44)
2 2
2(r + m ) t̃ 2 2
2(r + m ) r + m2 φ̃
This vector field is clearly regular in all M ′ . Accordingly, it can be used to extend smoothly
the family of ingoing principal null geodesics to H ′ . The price to pay is that k′ is only a
pregeodesic vector field, while k was geodesic, being associated with the affine parameter −r.
The pair (k, k′ ) plays actually the same role as the pair (ℓ′ , ℓ) (note the order!) regarding the
outgoing principal null geodesics.
As H in (M , g), H ′ is a degenerate Killing horizon of (M ′ , g). Indeed, by the same
reasoning as in Sec. 13.2.5, we get that the null normal to H ′ is the Killing vector χ =
ξ + 1/(2m) η [Eq. (13.23) extended to M ′ ]. This normal coincides with k′ on H ′ , as we can
see by setting r = m in Eq. (13.44). This implies that the null geodesic generators of H ′
belong to the ingoing principal null congruence. The non-affinity coefficient of k′ is (cf. the
notebook D.5.18):
m2 − r 2
κk′ = m 2 . (13.45)
(r + m2 )2
We have thus κk′ = 0 on H ′ , so that H ′ is a degenerate Killing horizon.
With M ′ , our extended extremal Kerr spacetime is thus (M0 , g) with
M0 := M ∪ M ′ = MI ∪ MIII ∪ MIII
′
∪H ∪H ′ . (13.46)
13.3 Maximal analytic extension 519
Figure 13.5: Carter-Penrose diagram of the (partially) extended extremal Kerr spacetime (M0 , g). The grey
curves represent hypersurfaces t̃ = const in M , with t̃ ∈ [−10m, 10m] and the increment δ t̃ = 2m between two
successive hypersurfaces. The hypersurface t̃ = 0 is singled out by a larger thickness. The purple curves represent
hypersurfaces t̃˜ = const in M ′ , with t̃˜ ∈ [−10m, 10m] and the increment δ t̃˜ = 2m between two successive
hypersurfaces. The hypersurface t̃˜ = 0 is singled out by a larger thickness. The red dotted curves represent
hypersurfaces r = const, with the increment δr between two successive hypersurfaces being δr = 2m for r < 0
and r > 3m and δr = 0.5 m for 0 ≤ r ≤ 3m. The hypersurfaces r = 0 are marked by brown dashed curves. The
dashed (resp. solid) green straight lines depict some selected ingoing (resp. outgoing) principal null geodesics.
[Figure generated by the notebook D.5.18]
520 Extremal Kerr black hole
A Carter-Penrose diagram of M0 is shown in Fig. 13.5. The projection operator used to build
this diagram (cf. Sec. 10.8.1) is Π : M0 → R2 , with R2 spanned by coordinates (T, X), such
that Π is defined by Eqs. (13.27)-(13.28) on M , while on M ′ , Π is defined by
T = T0 (u, v) and X = X0 (u, v) in MI (13.47a)
T = arctan(u/2) − π/2 and X = − arctan(u/2) − π/2 on H ′ (13.47b)
T = T0 (u, v) − π and X = X0 (u, v) − π in MIII
′
(13.47c)
˜ r) defined
where T0 (u, v) and X0 (u, v) are given by Eq. (13.28) and u and v are functions of (t̃,
by respectively u = t̃˜ − r [Eq. (13.29)] and
4m2 r−m
v = t̃˜ + r − + 4m ln . (13.48)
r−m m
The last equation is obtained by combining Eqs. (13.15) and (13.18).
Remark 2: As we have constructed it, the manifold M0 is covered by two coordinate charts: (t̃, r, θ, φ̃)
on MI ∪ H ∪ MIII and (t̃, ˜ r, θ, φ̃)
˜ on MI ∪ H ′ ∪ M ′ . These two charts overlap in MI , where the
III
transition between them is provided by Eq. (13.30). The Carter-Penrose diagram of Fig. 13.5 might
give the impression that, by means of the coordinates (T, X), one could cover M0 by a single chart,
in a fashion similar to the covering of the entire maximal extension of Schwarzschild spacetime by
the Kruskal-Szekeres coordinates (T, X, θ, φ) (cf. Sec. 9.3). However, this is not possible in a simple
way, due to singularity issues with the azimuthal coordinates: φ̃ diverges on H ′ , φ̃˜ diverges on H
and the Boyer-Lindquist coordinate φ diverges on both H and H ′ . Accordingly, none of (T, X, θ, φ̃),
˜ and (T, X, θ, φ) would provide a regular chart of M0 .
(T, X, θ, φ̃)
Remark 3: The spacetime (M0 , g) is analytic: (i) M0 is an analytic manifold, given that the change of
coordinates (13.30) between the two charts covering M0 is analytic (cf. Remark 4 in Sec. A.2.1) and (ii)
˜ r, θ, φ̃)
the components (13.3) and (13.39) of g are analytic functions of the coordinates (t̃, r, θ, φ̃) and (t̃, ˜
respectively.
The Killing horizon H ′ is actually a white hole horizon from the point of view of MI . More
precisely, we can endow (M0 , g) with the same conformal completion at null infinity as that
used for (M , g) in Sec. 13.2.6, i.e. such that the conformal boundary I is constituted by the
same future and past null infinities I + and I − located at the boundary of MI (cf. Fig. 13.5).
It appears then that
MIII′
∪ H ′ = M0 \ (J + (I − ) ∩ M0 ) (13.49)
In view of the definition (4.41), we conclude:
Property 13.5: white hole region in the extended extremal Kerr spacetime
MIII
′
is the interior of a white hole region, the boundary of which is H ′ .
Since MIII was shown in Sec. 13.2.6 to be the black hole region for the same conformal
completion at null infinity, we may state, according to the terminology introduced in Sec. 4.4.2:
13.3 Maximal analytic extension 521
Property 13.6: domain of outer communications
MI is the domain of outer communications of the spacetime (M0 , g).
13.3.2 Construction of the maximal analytic extension
With (M0 , g), we have achieved our first goal: all the outgoing principal null geodesics crossing
MI and not lying in the equatorial plane (i.e. the geodesics extending the L(u,θ,out,I
˜ family with
φ̃)
θ ̸= π/2 to the past) are complete. However, there remains incomplete geodesics in M0 : the
outgoing principal null geodesics crossing MIII all stop at the value r = m of their affine
parameter, while the ingoing principal null geodesics crossing MIII ′
all start at the value
−r = −m of their affine parameter (cf. Fig. 13.5). To construct a spacetime with complete
geodesics, except for those that encouter the curvature singularity at r = 0 and θ = π/2, one
introduces an infinite number of copies of M0 , (Mn )n∈Z say, and identify the region MIII ′
of
Mn with the region MIII of Mn−1 for all n ∈ Z. The manifold hence obtained,
[
M∗ := Mn , (13.50)
n∈Z
is depicted via a Carter-Penrose diagram in Fig. 13.6. The region MI (resp. MIII ) of Mn is
(n) (n)
denoted by MI (resp. MIII ). Similarly, the Killing horizon H (resp. H ′ ) of Mn is denoted
by H(n) +
(resp. H(n)−
). Note that H(n)
+
is a future event horizon (black hole horizon), while H(n)
−
is a past event horizon (white hole horizon), for the conformal completion at null infinity with
the future and past null infinities I(n) +
and I(n)
−
as copies of I + and I − introduced for M0 .
It is clear that (M∗ , g) is an analytic spacetime, since (M0 , g) is (cf. Remark 3 on p. 520).
By construction, all principal null geodesics are complete in M∗ , except for those that
encounter the curvature singularity at r = 0 and θ = π/2. In particular, this holds for the
outgoing principal null geodesics generating H(n) +
and for the ingoing ones generating H(n) −
.
Contemplating the Carter-Penrose diagram of Fig. 13.6, we might thus perceive each (solid
or dashed) green straight line from an r = −∞ end to an r = +∞ end, as well as each thick
black straight line, as representing a complete principal null geodesic.
It can be shown that actually all timelike or null geodesics of (M∗ , g), and not only the
principal null ones, are complete (see Carter’s article [90] for the proof), so that we can conclude:
Property 13.7: maximal analytic extension of the extremal Kerr spacetime
(M∗ , g) is the maximal analytic extension of the extremal Kerr spacetime.
Remark 4: The construction of the maximal extension of the extremal Kerr spacetime is simpler than
that of the Kerr spacetime with a < m presented in Sec. 10.8. Indeed, for the latter, if one uses only the
˜ r, θ, φ̃),
ingoing and outgoing Kerr coordinate patches (t̃, r, θ, φ̃) and (t̃, ˜ one ends up with a manifold
522 Extremal Kerr black hole
Figure 13.6: Carter-Penrose diagram of the maximal analytic extension (M∗ , g) of the extremal Kerr spacetime.
The red dotted curves represent hypersurfaces r = const, with the increment δr between two successive
hypersurfaces being δr = 2m for r < 0 and r > 2m and δr = 0.2 m for 0 ≤ r ≤ 2m. The hypersurfaces r = 0
are marked by brown dashed curves. The dashed (resp. solid) green straight lines depict ingoing (resp. outgoing)
principal null geodesics with v = 0 (resp. u = 0). [Figure generated by the notebook D.5.18]
13.4 Near-horizon extremal Kerr (NHEK) geometry 523
that is not maximal: the null geodesics generating the Killing horizons at r = r± are not complete,
because the bifurcation spheres (the central dots in Fig. 10.11), accross which these geodesics can be
extended, are missing. Indeed the bifurcation spheres are not covered by Kerr coordinates. To include
them and thus get the full bifurcate Killing horizons (cf. Sec. 3.4), one has to introduce Kruskal-Szekeres-
type coordinates in the vicinity of each bifurcation sphere, in a way similar to the Schwarzschild case
−
treated in Secs. 9.2 and 9.3. In the present case, each Killing horizon H(n) +
or H(n) is degenerate and
−
therefore made of complete null geodesics. In particular, H(n) and H(n) are not part of a bifurcate
+
Killing horizon. In other words, there is no bifurcation sphere in the maximal extension of the extremal
Kerr spacetime and the ingoing and outgoing Kerr coordinate patches are sufficient to cover it entirely.
Historical note : In 1966, Brandon Carter [88] obtained the maximal analytic extension of the rotation
axis A of the extremal Kerr spacetime (cf. historical note on p. 368). In particular, he drew a diagram
(Fig. 1b of Ref. [88]) similar to that of Fig. 13.6, the difference being that Carter’s one is a true conformal
representation, owing to the fact that A is 2-dimensional, while the diagram of Fig. 13.6 is a mere
projection of the 4-dimensional manifold M∗ . In their 1967 classical study entitled Maximal Analytic
Extension of the Kerr metric [72], Robert H. Boyer and Richard W. Lindquist focused on the case a < m.
For a = m, they referred to Carter’s study [88] and stated simply that although his [Carter’s] work was
confined to the symmetry axis (θ = 0, π), it is clear that his conclusions apply with equal force to the full
metric. The detailed construction of the maximal analytic extension of the full extremal Kerr spacetime
was actually presented by Carter himself in 1968 [90], as the special case of vanishing electric charge of
the extremal Kerr-Newman spacetime. The construction was also performed in details in his famous
lectures at Les Houches Summer School in 1972 [95].
13.4 Near-horizon extremal Kerr (NHEK) geometry
13.4.1 The extremal Kerr throat
Let us consider a hypersurface Σt of constant Boyer-Lindquist time t in the external region
MI of the extremal Kerr spacetime, i.e. one of the hypersurfaces shown in blue in Figs. 13.1
and 13.4. Σt is a spacelike hypersurface, since it is clear from the line element (13.11) that
the metric induced by g on Σt is Riemannian (i.e. positive definite) in5 MI . We may thus
visualize the geometry of Σt by some isometric embedding of 2-dimensional slices of Σt into the
Euclidean space R3 , as we did in Sec. 9.5.2 for the hypersurfaces of constant Kruskal-Szekeres
time in Schwarzschild spacetime. It is then natural to consider the slices Σt,θ where θ is held
constant, in addition to t, with Σt, π2 representing the equatorial “plane”. Σt,θ is spanned by the
coordinates (xa ) = (r, φ) and the Riemannian metric q induced on it by the spacetime metric
g is obtained by setting dt = 0 and dθ = 0 in (13.11):
r2 + m2 cos2 θ 2 2m3 r sin2 θ
q= 2
2 2
dr + r + m + 2 2 2
sin2 θ dφ2 . (13.51)
(r − m) r + m cos θ
5
The induced metric would not be Riemannian in the Carter time machine (cf. Sec. 10.2.5), where gφφ < 0; but
this requires r < 0, while r > m in MI .
524 Extremal Kerr black hole
On the other side, the Euclidean space R3 can be described by cylindrical coordinates (X i ) =
(ϖ, z, φ), such that the Euclidean metric f takes the form
f = dϖ2 + dz 2 + ϖ2 dφ2 . (13.52)
Let us define the embedding of the 2-surface Σt,θ into R3 by
Φ : Σt,θ −→ R3
(13.53)
(r, φ) 7−→ (ϖ(r), z(r), φ).
Along Φ(Σt,θ ), we have then dϖ = ϖ′ (r) dr and dz = z ′ (r) dr, so that the metric h induced
by f on Φ(Σt,θ ) is
h = ϖ′ (r)2 + z ′ (r)2 dr2 + ϖ(r)2 dφ2 . (13.54)
Φ performs an isometric embedding iff the two metrics (13.51) and (13.54) coincide6 , i.e. iff
s
2m3 r sin2 θ
ϖ(r) = sin θ r2 + m2 + 2 (13.55a)
r + m2 cos2 θ
r2 + m2 cos2 θ
′ 2 ′ 2
ϖ (r) + z (r) = . (13.55b)
(r − m)2
We deduced from Eq. (13.55b) that
Z r
1 p 2
z(r) = r̄ + m2 cos2 θ − ϖ′ (r̄)2 (r̄ − m)2 dr̄, (13.56)
2m r̄ − m
up to some additive constant, which we set to zero by choosing arbitrarily z(2m) = 0. In the
integrand, ϖ′ (r̄) should be substituted by the value obtained by differentiating Eq. (13.55a).
The functions ϖ(r) and z(r), provided respectively by Eqs. (13.55a) and (13.56), define fully
the isometric embedding (13.53) of the Riemannian 2-manifold (Σt,θ , q) into the Euclidean
space (R3 , f ). The outcome is depicted in Fig. 13.7 for two values of θ: π/2 (left plot) and π/6
(right plot). In both cases, it appears that the embedded surface is infinite in two limits: for
r → +∞ and for r → m.
The first limit, which implies ϖ → +∞ and z → +∞ according to Eqs. (13.55a) and (13.56),
is towards the asymptotic flat end of MI . More precisely, since t is held fixed on Σt,θ , the
asymptotic direction appears as the spacelike infinity point i0 at the right-most corner of the
Carter-Penrose diagram of Fig. 13.4.
The second limit, r → m, corresponds to ϖ → 2m √1+cos sin θ
2 θ and z → −∞, according to
Eqs. (13.55a) and (13.56). This describes an infinite cylinder, since ϖ tends to a constant value,
while z decays to −∞. The divergence of z for r → m is of course due to the factor (r̄ − m)−1
in the integrand of (13.56), which leads to the following behavior:
√
r−m
2
z(r) ∼ m 1 + cos θ ln when r → m. (13.57)
m
6
Formally, one says that the metric q is the pullback of h by Φ: q = Φ∗ h.
13.4 Near-horizon extremal Kerr (NHEK) geometry 525
Figure 13.7: Isometric embeddings of 2-surfaces Σt,θ of constant Boyer-Lindquist time t and constant θ of the
extremal Kerr spacetime in the Euclidean 3-space, for two values of θ: θ = π/2 (equatorial plane) (left figure)
and θ = π/6 (right figure). The drawings are truncated at r = (1 + 10−5 )m (bottom boundary) and at r = 10 m
(top boundary). The blue circle marks the ergosphere, which is located at r = 2m for θ = π/2 and r = 3m/2 for
θ = π/6. The five black circles correspond to r = 1.1 m, 1.01 m, 1.001 m, 1.0001 m and 1.00001 m, from top to
bottom. [Figure generated by the notebook D.5.19]
This logarithmic divergence is clearly apparent in Fig. 13.7: the circles at r − m = 10−1 m,
10−2 m, ..., 10−5 m look equally spaced, all the more than r is close to m. In the Carter-Penrose
diagram of Fig. 13.4, the limit r → m at fixed t appears as the point marked iint . This point is
actually the projection of a 2-sphere (spanned by (θ, φ)) called the internal infinity of the
extremal Kerr spacetime. Let us stress that, similarly to the spacelike infinity i0 and the null
infinities I + and I − , the sphere iint is not part of the physical spacetime (M , g).
The property of iint being located infinitely far from any point at r = r0 > m on a
hypersurface Σt appears clearly on the metric (13.11). Indeed, according to the latter, the
distance along a curve (t, θ, φ) = const between the points at r = r0 and r = m is
Z r0 Z r0 √ 2
√ r + m2 cos2 θ
ℓ= grr dr = dr.
m m r−m
Again, the factor (r − m)−1 makes the integral diverge, so that ℓ = +∞.
Another perspective of iint being “infinitely far” is obtained by noticing that on the Carter-
Penrose diagram of Fig. 13.4, iint is located at the past end of the event horizon H . Since H is
a degenerate Killing horizon (cf. Sec. 13.2.5), it is generated by complete null geodesics: the
principal null geodesics L(θ,ψ)
out,H
introduced in Sec. 13.2.4. Now, since L(θ,ψ)
out,H
is complete, its
526 Extremal Kerr black hole
past end at iint is obtained at the limit t̃ → −∞ of its affine parameter t̃ (see also the grey
curves in Fig. 13.3, which reveal clearly that the limit t̃ → −∞ at r = m is iint ). Hence, in
terms of affine length, iint is infinitely far in the past along any of the null geodesics L(θ,ψ)
out,H
generating H .
The spacetime region in the vicinity of r = m is called the extremal Kerr throat. We are
going to explore it in more details in the next sections.
13.4.2 The NHEK metric
In order to zoom in on the near-horizon region, let us introduce a small parameter ε > 0 and
the Bardeen-Horowitz coordinates7 (T, R, θ, Φ), which are defined on MBL = MI ∪ MIII
and related to the Boyer-Lindquist coordinates (t, r, θ, φ) by
t r−m t
T =ε , R= , Φ=φ− , (13.58)
2m εm 2m
the reverse transformation being
T T
t = 2m , r = m(1 + εR), φ=Φ+ . (13.59)
ε ε
Note that on MBL , the range of T is R, while that of R is (−∞, 0) ∪ (0, +∞), with R > 0 on
MI and R < 0 on MIII . The coordinate choice (13.59) ensures, for a fixed value of (T, R, Φ)
with T > 0,
dφ 1
lim t = +∞, lim r = m, lim φ = +∞, = (13.60)
ε→0 ε→0 ε→0 dt 2m
and is motivated as follows. For ε → 0, the coordinate R clearly implements the zoom on the
region r ≃ m, i.e. the region close to the event horizon H . Regarding the change t → T , we
may notice that the Boyer-Lindquist coordinate t diverges near H — more precisely, t diverges
along the worldline of any particle that reaches H (this is clear on Figs. 13.1 and 13.4). So, we
may keep T = εt/(2m) finite in the vicinity of H , while having t diverging. The division by
2m makes T dimensionless, as R, with the factor 2 being chosen for later convenience. Similarly,
the Boyer-Lindquist coordinate φ diverges along the worldline of any particle that reaches
H . For geodesics, this divergence occurs at the rate dφ/dt = ΩH = 1/(2m) [Eqs. (11.74) and
(13.23)], which is ensured at fixed Φ by the choice (13.59).
Let us denote by Gαβ (T, R, θ, Φ, ε) the components of the metric tensor g with respect to
Bardeen-Horowitz coordinates (X α ) := (T, R, θ, Φ). They depend on ε since the change of
coordinates (13.59)-(13.58) does. The explicit expressions of the Gαβ (T, R, θ, Φ, ε)’s are rather
complicated and can be found in the notebook D.5.20. We are concerned here only by the limit
ε → 0. In other words, we consider the tensor field h on MBL whose components with respect
to the Bardeen-Horowitz coordinates are the limits
hαβ (T, R, θ, Φ) := lim Gαβ (T, R, θ, Φ, ε). (13.61)
ε→0
7
Cf. the historical note on page 536.
13.4 Near-horizon extremal Kerr (NHEK) geometry 527
Evaluating the limits leads to the explicit expression of h (cf. the notebook D.5.20):
dR2 4 sin2 θ
2 2 2 2
h = m (1 + cos θ) −R dT + 2 + dθ + 2 2
(dΦ + R dT ) . (13.62)
R (1 + cos2 θ)2
A priori the limits (13.61) lead only to a symmetric bilinear form. To be a proper metric tensor,
h has to be non-degenerate as well (cf. Sec. A.3.1 in Appendix A). This can be checked by
computing the determinant of the components (13.62); one gets (cf. the notebook D.5.20):
det(hαβ ) = −4m2 (1 + cos2 θ)2 sin2 θ. (13.63)
Hence det(hαβ ) vanishes only for sin θ = 0, i.e. on the rotation axis A . This merely signals
the standard coordinate singularity of spherical-type coordinates on A . Away from A , one
has det(hαβ ) < 0, which shows that h is non-degenerate and has the signature (−, +, +, +)
or (+, −, −, −). Since the (diagonal) (R, θ)-block has clearly the signature (+, +), we get that
only (−, +, +, +) is possible. Hence we may conclude:
Property 13.8: NHEK metric
Equation (13.62) defines a Lorentzian metric tensor h on MBL = MI ∪ MIII . We shall call
it the near-horizon extremal Kerr (NHEK) metric.
The NHEK metric h has been obtained in Bardeen-Horowitz coordinates (T, R, θ, Φ). Let
us express it in terms of Boyer-Lindquist coordinates (t, r, θ, φ). From the transformation law
(13.58), we have
r−m dR dr r − 2m
R dT = dt, = and dΦ + R dT = dφ + dt. (13.64)
2m2 R r−m 2m2
The components of h with respect to Boyer-Lindquist coordinates are then immediately deduced
from Eq. (13.62):
"
(r − m)2 2 dr2
h = m2 (1 + cos2 θ) − dt + + dθ2
4m4 (r − m)2
2 # (13.65)
4 sin2 θ
r − 2m
+ dφ + dt
(1 + cos2 θ)2 2m2
Remark 1: It is remarkable that, while the change of coordinates (t, r, θ, φ) ↔ (T, R, θ, Φ) depends on
the parameter ε [Eqs. (13.59)-(13.58)], the Boyer-Lindquist expression (13.65) of h is independent of ε.
This occurs because ε disappears in the combinations (13.64). In particular, the NHEK metric is unique,
i.e. it does not depend on the value of ε, contrary to the Bardeen-Horowitz coordinates on MBL .
Remark 2: As we have introduced it, the NHEK metric is defined on the part MBL of the Kerr manifold
M , where R ̸= 0. In Sec. 13.4.4, we shall consider instead that h is defined on a manifold distinct from
M , where it can be extended accross R = 0.
528 Extremal Kerr black hole
13.4.3 Anti-de Sitter features
It appears from (13.62) that the NEHK metric h has striking similarities with the metric of
anti-de Sitter spacetime (AdS): compare the dT 2 and dR2 terms with the dt2 and du2 terms of
the components (3.54) of the AdS4 metric in Poincaré coordinates, presented in Example 18 of
Chap. 3, noticing that both the Poincaré horizon in AdS4 (u = 0) and the Kerr horizon (R = 0)
are degenerate Killing horizons. More precisely, the metric induced by h on the rotation axis
A (sin θ = 0), which is a 2-dimensional submanifold of M , is exactly the metric of AdS2 (the
2-dimensional anti-de Sitter spacetime), expressed in the Poincaré coordinates (X a ) = (T, R):
dR2
A 2
h = 2m −R dT + 2 . 2 2
(13.66)
R
√
Moreover, the metric induced by h on the hypersurface θ = θ∗ , with θ∗ := arcsin( 3 − 1) ≃
47.06◦ (so that 4 sin2 θ∗ /(1 + cos2 θ∗ ) = 1), is that of AdS3 :
√ dR2
h θ=θ∗ 2 2 2
= 2( 3 − 1)m −R dT + 2 + (dΦ + R dT ) . 2
(13.67)
R
Note however that the above expression is not that of AdS3 metric in Poincaré coordinates.
Indeed, by expanding the last term, we get
√ dR2
h θ=θ∗ 2
= 2( 3 − 1)m 2R dT dΦ + 2 + dΦ . 2
(13.68)
R
In particular, hθ=θ
TT
∗
= 0, which implies that ∂T is a null vector field, while it should be a
timelike one for Poincaré coordinates. To prove that (13.67) is indeed the metric of AdS3 , let us
introduce the so-called global NHEK coordinates (τ, y, θ, ψ), which, for their (τ, y) part, are
linked to (T, R) by the same relationship as that between global and Poincaré coordinates in
AdS2 , namely
√ τ = arctan2 (2T R2 , R2 − T 2 R2 + 1)
2
1+y sin τ
√
T =
y+ 1+y 2 cos τ +πH(−R)
p
2
R = y + 1 + y cos τ
(1+T 2 )R2 −1
⇐⇒ y = 2R (13.69)
θ = θ
θ=θ
cos τ +y sin τ
Φ = ψ + ln 1+√1+y2 sin τ
2 +R2
(1−T R)
ψ = Φ − ln √
2 2 2 2
,
[(1+T )R −1] +4R
where H stands for the Heaviside function: H(−R) = 1 for R < 0 and 0 for R > 0. Beside the
AdS2 -type transformation for (τ, y) ↔ (T, R), the new azimuthal coordinate ψ is introduced
to insure dΦ + R dT = dψ + y dτ . In terms of the coordinates (τ, y, θ, ψ), the NHEK metric
h reads (cf. the notebook D.5.20):
dy 2 4 sin2 θ
2 2 2 2 2 2
h = m (1 + cos θ) −(1 + y )dτ + + dθ + (dψ + y dτ ) .
1 + y2 (1 + cos2 θ)2
(13.70)
13.4 Near-horizon extremal Kerr (NHEK) geometry 529
On the rotation axis (sin θ = 0), we recover the AdS2 metric expressed in global AdS coordinates
(τ, y). On the hypersurface θ = θ∗ considered above, we get
√ dy 2
h θ=θ∗ 2 2 2
= 2( 3 − 1)m −(1 + y )dτ + 2
+ (dψ + y dτ ) . (13.71)
1 + y2
This is the metric of AdS3 in some standard coordinates8 (compare e.g. with Eq. (17) of Ref. [57]
with y = sinh ω). On constant θ hypersurfaces, with θ ̸= θ∗ or π − θ∗ , the metric induced by h
is not that of AdS3 but that of a so-called warped AdS3 , the θ-term in front of (dψ + y dτ )2 in
Eq. (13.70) being the warp factor (see e.g. Refs. [137, 57, 247]).
√
Remark 3: The value θ∗ = arcsin( 3 − 1) for which the hypersurface θ = θ∗ inherits AdS3 metric
from h is exactly the value θcrit considered in Sec. 12.4.4 [cf. Eq. (12.98)], namely the minimal value
of the colatitude θ of a distant observer for the critical curve (boundary of the black hole shadow) to
contain a vertical straight line segment (the so-called NHEK line; maroon line in Fig. 12.24). A priori,
this is a mere coincidence.
13.4.4 NHEK spacetime
The Bardeen-Horowitz components (13.62) of the NHEK metric h are singular at R = 0. But
this is a mere coordinatep singularity, since, in terms of the global NHEK coordinates (τ, y, θ, ψ),
R = 0 is the locus y + 1 + y 2 cos τ = 0 [cf. Eq. (13.69)] and the components (13.70) of h
are regular there. They are actually regular in all the range τ ∈ R, y ∈ R, θ ∈ (0, π) and
ψ ∈ (0, 2π). As it is introduced in Eq. (13.69), the coordinate τ is actually restricted to (−π, 2π)
on M . We may therefore define an “extended” spacetime as follows:
The NHEK spacetime is the spacetime (N , h), where the manifold N is diffeomorphic
to R2 × S2 and h is the Lorentzian metric defined by Eq. (13.70) in terms of the coordi-
nates (τ, y, θ, ψ), with (τ, y) ∈ R2 and (θ, ψ) ∈ (0, π) × (0, 2π) being standard spherical
coordinates on S2 .
By comparing the metrics (13.70) and (3.49), with the changes of notation y ↔ r and ψ ↔ φ,
we note that (N , h) bears some similarities with AdS4 , but is distinct from it. First of all, r
ranges in (0, +∞) only on AdS4 , while y ranges in9 (−∞, +∞) on N . Furthermore, there is
a global (1 + cos2 θ) term in (13.70), the dθ2 terms differ drastically in the two expressions, the
one in (13.70) being not multiplied by y 2 . The dφ2 and dψ 2 are also truly distinct. Moreover,
the NHEK metric contains the off-diagonal term hτ ψ , while the AdS4 one is purely diagonal.
8
These coordinates are linked to the property of AdS3 being a fiber bundle over AdS2 with fiber R (in the same
way as S3 is a fiber bundle over S2 with fiber S1 (Hopf fibration)); ψ is then the fiber coordinate and spans R.
Since ψ is 2π-periodic on the hypersurface θ = θ∗ , we see that the latter is actually not the whole of AdS3 , but a
quotient of it, obtained by identifying the points ψ and ψ + 2π.
9
Note that there is no coordinate singularity at y = 0 in (13.70), while there is one at r = 0 in (3.49) and the
latter is such that AdS4 cannot be extended analytically to r < 0.
530 Extremal Kerr black hole
The inverse NHEK metric h−1 takes a rather simple form (cf. the notebook D.5.21 for the
computation):
1 y
− 1+y 2 0 0 1+y 2
2
1 0 1 + y 0 0
αβ
h = . (13.72)
2
1 + cos θ 0
0 1 0
y (1+cos2 θ)2 +y 2 (cos4 θ+6 cos2 θ−3)
1+y 2
0 0 4(1+y 2 ) sin2 θ
As AdS4 , the NHEK spacetime is not asymptotically flat. The metrics of both spacetimes
are solutions of the Einstein equation with T = 0 [Eq. (1.43)], but with Λ < 0 for AdS4 metric
and with Λ = 0 for h:
Property 13.9: NHEK metric as a solution of the vacuum Einstein equation
The NHEK metric h is a solution of the vacuum Einstein equation (1.44):
Ric(h) = 0, (13.73)
where Ric(h) stands for the Ricci tensor of h.
Proof. h has been defined on MBL via the limit expression (13.61). Now, for ε > 0, the Gαβ ’s
in the right-hand side of Eq. (13.61) are the components of a metric that has a vanishing Ricci
tensor, since it is nothing but the extremal Kerr metric, expressed in the Bardeen-Horowitz
coordinates (T, R, θ, Φ), which form a perfectly regular coordinate system on MBL for ε > 0
(cf. the transformations (13.59)-(13.58)). By continuity at the limit ε → 0, it follows that h has
a vanishing Ricci tensor as well. A priori this holds only on MBL , but since the components
of h in the global NHEK coordinates (τ, y, θ, ψ) are identical on MBL and N , the result is
immediately extended by analyticity to all the NHEK spacetime.
Remark 4: For the reader skeptical about the limit process and the extension to the whole of N , a
direct computation of the Ricci tensor of h from the global-coordinate components (13.70) is performed
in the notebook D.5.21 and the outcome is indeed identically vanishing.
Total angular momentum
The quantity m2 appears as a global scale factor of the metric h in each of the three expressions
(13.62), (13.70) and (13.65). This is of course the square of the mass of the extremal Kerr
spacetime from which the NHEK spacetime (N , h) arises. However, one cannot associate
intrinsically a mass to the latter. Indeed, one cannot use the Komar integral (5.36), as in the Kerr
case, since the the NHEK spacetime lacks any global asymptotically timelike Killing vector10 .
10
In particular, the Killing vector ∂τ is spacelike for
h θ close to π/2 and large
i enough values of y, since we read
4y 2 sin2 θ m2 y 2
from (13.70) that h(∂τ , ∂τ ) = hτ τ = m (1+cos θ) (1+cos2 θ)2 − 1 − y ∼ 1+cos 2
2 2 2
2 2
2 θ 4 sin θ − (1 + cos θ)
for |y| → +∞, so that h(∂τ , ∂τ ) > 0 for sin θ > (1 + cos2 θ)/2.
13.4 Near-horizon extremal Kerr (NHEK) geometry 531
An alternative concept of mass, which does not require any Killing vector, is that of ADM
mass (cf. Sec. 5.3.5). It is however not applicable either since (N , h) is not asymptotically flat.
Incidentally, this last fact would anyway preclude any unambiguous definition of the Komar
mass, since asymptotic flatness is required to normalize the timelike Killing field at spatial
infinity (scalar square −1).
On the contrary, the angular momentum J of the NHEK spacetime can properly be
defined from the axisymmetry Killing vector η = ∂φ = ∂Φ = ∂ψ , via the Komar formula (5.63).
The Hodge dual ⋆(dη) [cf. Eq. (5.38)], which appears in this formula, has of course to be taken
with respect to the metric h. Since h fulfills the vacuum Einstein equation [Eq. (13.73)], the
value of J does not depend upon the choice of the integration 2-surface S in formula (5.63)
(Property 5.21). One finds (cf. the notebook D.5.21 for the computation):
J = m2 . (13.74)
This is exactly the value J = am = m2 of the angular momentum of the extremal Kerr
spacetime [Eq. (10.80)]. Hence we conclude:
Property 13.10: angular momentum of NHEK spacetime
The NHEK spacetime (N , h) has no well defined mass, but it has a well defined Komar
angular momentum, whose value is nothing but the angular momentum J = m2 of the
extremal Kerr spacetime (M , g) from which (N , h) arises.
Remark 5: For this reason, many authors replace the overall factor m2 of the NHEK metric in Eq. (13.62)
or (13.70) by J.
“Conformal” coordinates
The range of the coordinate y is the whole of R. For pictural purposes (e.g. drawing Carter-
Penrose-like diagrams), it would be convenient to have instead a coordinate spanning some
finite range. As in the AdS4 case presented in Example 18 of Chap. 3, it is quite natural to
introduce
χ := arctan y ⇐⇒ y =: tan χ (13.75)
so that11 χ ∈ (−π/2, π/2). The relation (13.69) to the Bardeen-Horowitz coordinates (T, R, θ, Φ)
can be then rewritten as
τ = arctan2 (2T R2 , R2 − T 2 R2 + 1) + πH(−R)
sin τ
T = cos τ +sin χ
(1+T 2 )R2 −1
cos
R = cos χ
τ +sin χ
χ = arctan 2R
⇐⇒
θ=θ
θ=θ
cos(τ −χ) (1−T R)2 +R2
√
Φ = ψ + ln sin τ +cos χ
ψ = Φ − ln 2 2 2 2
,
[(1+T )R −1] +4R
(13.76)
11
In Example 18, the range of χ was only (0, π/2) because r = tan χ ≥ 0 in AdS4 , while here y = tan χ spans
the whole of R.
532 Extremal Kerr black hole
r=
−
+∞
r=
r=
4
R=−∞
−
III P
R= −1
2
r= 0
r=
0
r=
m
R=
r=
τ
+∞
1
−
R=
∞
R=+∞
0 +
i int I P
∞
r=
−
+∞
r=
2
m
0
r=
2 1 0 1 2
χ
Figure 13.8: 2-dimensional views of the extremal Kerr spacetime (M , g) (left) and of the NHEK spacetime
(0) (0)
(N , h) (right). The left figure is a zoom on the (MI , MIII ) part of the Carter-Penrose diagram shown in
Fig. 13.6. The right figure is a “compactified” view of N based on the coordinates (τ, χ). The near-horizon region
in the Kerr spacetime is visualized by means of red curves, which are curves (actually hypersurfaces) of constant
r, ranging from r = m/2 to r = 3m/2, i.e. within ∆r = m/2 of the horizon value r = m. The r-increment
between two nearby curves is δr = m/10. Grey curves represents hypersurfaces of constant Boyer-Lindquist
coordinate t in the near horizon region, with t varying by steps of δt = 2m in the range [−8m, 8m]. Each of these
constant-r or constant-t curves is mapped to the NHEK spacetime (right figure) by means of the transformation
(13.58) with ε = 1/2, giving birth to constant-R curves ranging from R = −1 to R = 1 (red curves) and to
constant-T curves ranging from T = −2 to T = 2 (grey curves). [Figures generated by the notebooks D.5.18 (left)
and D.5.21 (right)]
These transformations can be used to map the Boyer-Lindquist domain MBL of Kerr spacetime
to the region −π/2 < τ + χ < 3π/2 of the NHEK spacetime N , as illustrated in Fig. 13.8. This
assumes a fixed value of ε > 0 in Eqs. (13.59)-(13.58), so that (T, R, θ, Φ) constitute a regular
coordinate system on MBL = MI ∪ MIII . More precisely, MI , where R > 0, is mapped to the
subregion NP+ defined by −χ − π/2 < τ < χ + π/2, which implies cos τ + sin χ > 0 and
hence R > 0 via (13.76), since cos χ > 0 for χ ∈ (−π/2, π/2). On the other hand, MIII , where
R < 0, is mapped to the subregion NP− defined by χ + π/2 < τ < −χ + 3π/2, which implies
cos τ + sin χ < 0 and hence R < 0 via (13.76). In both NP+ and NP− , the coordinate T spans
the whole of R. We shall call each of the regions NP+ and NP− a Poincaré patch of NHEK
spacetime. They appear as the interiors of triangles having one vertical edge at the NHEK
13.4 Near-horizon extremal Kerr (NHEK) geometry 533
boundary in Fig. 13.8. Note the similarity12 with the Poincaré patch in AdS4 spacetime plotted
in Fig. 3.6.
The NHEK metric components with respect to the coordinates (τ, χ, θ, ψ) are easily deduced
from (13.70):
4 sin2 θ
2 2 1 2 2
2 2
h = m (1 + cos θ) −dτ + dχ + dθ + (dψ + tan χ dτ ) .
cos2 χ (1 + cos2 θ)2
(13.77)
Note the factor 1/ cos2 χ in front of the −dτ 2 + dχ2 term, which is analogous to the conformal
factor 1/ cos2 χ of the AdS4 metric in Eq. (3.50), except that in the present case, 1/ cos2 χ does
not factor all the metric, so that it cannot be considered as a proper conformal factor. Since cos χ
never vanishes for χ ∈ (−π/2, π/2), the factor 1/ cos2 χ is regular in all N . It simply blows
up at the “boundaries” χ → ±π/2 of N , which correspond to y → ±∞ and to R → ±∞ in
the Poincaré patches NP+ and NP− .
One can provide a pictoral view of h being the “near-horizon limit” of g as follows:
The two red strips r ∈ [m/2, 3m/2] and R ∈ [−1, 1] in Fig. 13.8 have been drawn by
setting ε = 1/2 in the transformation (13.58) linking the Boyer-Lindquist coordinates
(t, r, θ, φ) to the Bardeen-Horowitz ones (T, R, θ, Φ). Now, if one maintains the red strip
R ∈ [−1, 1] in the NHEK spacetime unchanged and let ε decays to 0, the red strip in the
extremal Kerr spacetime shrinks to a more and more narrow strip around the horizon. As
ε → 0, this narrow strip, endowed with the Kerr metric g, tends to be isometric to the
fixed NHEK strip R ∈ [−1, 1] endowed with the metric h.
Remark 6: While the extremal Kerr spacetime and the NHEK one are similar in the near-horizon region,
it is clear on Fig. 13.8 that they have very different asymptotics for r → ±∞ and R → ±∞.
13.4.5 NHEK symmetries
Given the similarities with the AdS spacetime, which is a maximally symmetric spacetime13 ,
it should come as no surprise that the NHEK spacetime (N , h) has more symmetries than
the extremal Kerr spacetime (M , g). First of all, it is obvious on the expression (13.62) of h in
Bardeen-Horowitz coordinates that T and Φ are ignorable coordinates, so that the two vector
fields
2m 1
ξ1 := ∂T = ∂t + ∂φ and η = ∂Φ = ∂φ = ∂ψ (13.78)
ε ε
12
In Fig. 3.6, the Poincaré patch (where u > 0) has its vertical edge on the left, while in Fig. 13.8, NP+ (where
R > 0) has its vertical edge on the right, but this results simply from a distinct convention in choosing χ.
13
A spacetime of dimension n is said to be maximally symmetric if, and only if, its group of isometries has
the maximum possible dimension, which is n(n + 1)/2. It is equivalent to say that there are n(n + 1)/2 linearly
independent Killing vector fields when considering only linear combinations with constant coefficients. For n = 4,
one has n(n + 1)/2 = 10, which, among others, is the dimension of the Poincaré group (isometries of Minkowski
spacetime). The group of isometries of AdSn is the pseudo-orthogonal group O(2, n − 1), the dimension of which
is exactly n(n + 1)/2 (cf. e.g. Ref. [390]).
534 Extremal Kerr black hole
are two Killing vectors of h. In the above equation, the second expression of ξ1 is that in the
Boyer-Lindquist coordinate frame and follows directly from the transformation law (13.59).
In particular, note that ξ1 is distinct from the Killing vector ξ = ∂t of the Kerr metric g. It is
rather connected to the horizon-generating Killing vector χ defined by Eq. (13.23):
2m
ξ1 = χ. (13.79)
ε
Besides, in Eq. (13.78), the vector field ∂Φ is identical to the Killing vector field η = ∂φ of the
Kerr metric g as a consequence of the transformation law (13.59). Similarly, the transformation
law (13.69) implies that η = ∂ψ .
A third symmetry apparent on the metric components (13.62) is the invariance under
the transformation (T, R) 7→ (αT, R/α), for any α > 0. Such transformations, which are
called squeeze mappings (or hyperbolic rotations) of the (T, R) plane, form a 1-parameter
group action, whose generator ξ2 is given by formula (3.14) with t = λ := α − 1 (so that the
identity element is for λ = 0). By considering an infinitesimal transformation of parameter
dλ, formula (3.14) leads to the components ξ2T = [(1 + dλ)T − T ]/dλ = T and ξ2R = [(1 −
dλ)R − R]/dλ = −R. Hence the Killing vector field
t
ξ2 = T ∂T − R ∂R = t ∂t + (m − r) ∂r + ∂φ . (13.80)
2m
The second expression stems from the transformation law (13.59) and expresses ξ2 in terms
of the Boyer-Lindquist coordinate frame. We note that this expression is independent of ε,
contrary to the Boyer-Lindquist expression of ξ1 .
Finally, a fourth Killing vector of h shows up clearly on the components (13.70) with respect
to the global NHEK coordinates (τ, y, θ, ψ), since these components are independent of τ .
Hence ∂τ is a Killing vector of h. In what follows, we will use instead the linear combination14
1
ξ3 := ∂τ − ξ1 , (13.81)
2
since it has slightly simpler components with respect to Bardeen-Horowitz coordinates. Given
the transformation law (13.69), we get (cf. the notebook D.5.20)
1 1 1
ξ3 = 2
T + 2 ∂T − RT ∂R − ∂Φ . (13.82)
2 R R
The Boyer-Lindquist expression of ξ3 is found via the transformation law (13.59)-(13.58):
2
m3 1 t2
t t(m − r) m(3m − 2r)
ξ3 = ε + ∂t + ∂r + + ∂φ . (13.83)
4m (r − m)2 2m 2 4m2 (r − m)2
We have thus four Killing vectors of the NEHK metric h: η, ξ1 , ξ2 and ξ3 . Since none of
them is a linear combination with constant coefficients of the three others, the isometry group
G generated by these Killing vectors is 4-dimensional. This is 2 dimensions more than the
14
Let us recall that a linear combination with constant coefficients of Killing vectors is a Killing vector.
13.4 Near-horizon extremal Kerr (NHEK) geometry 535
isometry group of Kerr spacetime, which is R × U(1). In order to fully characterize G, let us
determine its Lie algebra by evaluating the commutator15 of each pair of the four generators η,
ξ1 , ξ2 and ξ3 . We get (cf. the notebook D.5.20)
[η, ξ1 ] = 0, [η, ξ2 ] = 0, [η, ξ3 ] = 0 (13.84)
and
[ξ2 , ξ1 ] = −ξ1 , [ξ2 , ξ3 ] = ξ3 , [ξ1 , ξ3 ] = ξ2 . (13.85)
Equation (13.84) shows that η commutes with all the other generators of G. Since η is the
generator of the axisymmetry group U(1), we may write G = G3 × U(1), where G3 is a
3-dimensional Lie group, generated by ξ1 , ξ2 and ξ3 . The commutation relations (13.85) show16
that the Lie algebra of G3 is the special linear algebra sl(2, R). At this stage, G3 can be one of
the following three Lie groups: SL(2, R), PSL(2, R) = SL(2, R)/Z2 or SL(2, R) (the universal
covering group of SL(2, R)). It cannot be PSL(2, R) because, as it appears clearly on (13.62),
the transformation (T, R) 7→ (−T, −R) is an element of G3 and, in PSL(2, R), this element
would be identified with the identity due to the quotient by Z2 = {Id, −Id}, where Id is the
identity element. G3 is actually the special linear group SL(2, R). We conclude:
Property 13.11: isometry group of NEHK spacetime
The isometry group of the NEHK spacetime is
G = SL(2, R) × U(1) , (13.86)
with U(1) generated by the axisymmetry Killing vector η and SL(2, R) generated by the
three Killing vectors ξ1 , ξ2 and ξ3 given by Eqs. (13.78), (13.80) and (13.82).
This property is often phrased as follows: the near-horizon region of the extremal Kerr
spacetime (the extremal Kerr throat) has some emergent symmetries, i.e. it has more symmetries
than the extremal Kerr spacetime itself. Let us stress that no such thing occurs for non-extremal
Kerr black holes.
Remark 7: SL(2, R) is isomorphic to the spin group Spin(2, 1), which is the isometry group of the
2-dimensional anti-de Sitter spacetime (AdS2 ). The spin group Spin(2, 1) is the double cover of the
pseudo-orthogonal group SO(2, 1), the latter being the isometry group of the “time-cyclic” 2-dimensional
anti-de Sitter spacetime, i.e. the quadric surface X 2 − U 2 − V 2 = −1 in R3 endowed with the flat
metric −dU 2 − dV 2 + dX 2 .
Remark 8: Having an enhanced symmetry group near the horizon is not specific to the extremal Kerr
black hole; this is actually a generic feature, which occurs near degenerate Killing horizons [327, 137].
15
The definition of the commutator of two vector fields is recalled in Appendix A [cf. Eq. (A.27)].
16
More precisely, inthe representation
of sl(2,
R) by
the algebra
of 2 × 2 real matrices with zero trace, a
0 1 0 0 1 0
standard basis is E =
, F =
, H =
. This basis has the following commutation
0 0
1 0
0 −1
relations: [H,
√ E] = 2E, [H, F ] = −2F and [E,√F ] = H. One recovers the last ones from (13.85) by identityfying
ξ1 with F/ 2, ξ2 with H/2 and ξ3 with −E/ 2.
536 Extremal Kerr black hole
Pure geometric arguments suffice to show that a degenerate Killing horizon associated with some Killing
vector ξ1 has necessarily a near-horizon geometry endowed with a second Killing vector, which is
similar to ξ2 [327]. Axisymmetry provides the third Killing vector η and the Einstein equation leads to
the existence of the fourth Killing vector ξ3 [329].
Historical note : The NHEK metric h has been first exhibited by Brandon Carter in 1973 [95]: compare
Eq. (5.63) of Ref. [95] (with Q = P = 0) with Eq. (13.62), taking into account the change of notations
(τ, λ, φ) ↔ (T, −R, Φ). Actually the NHEK metric was found by Carter when searching for the
most general stationary and axisymmetric solutions of the source-free Einstein-Maxwell equations (cf.
Sec. 1.5.2), in order to get the Kerr-Newman solution. The latter appeared then as the generic case
and NHEK-like metrics as special cases. The limit Q = P = 0 of Carter’s metric (5.63) corresponds
to the case of vanishing electric charge Q and magnetic monopole P , so that it is a solution of the
vacuum Einstein equation [Eq. (13.73) above]. Carter stated that the metric h has a 4-dimensional
isometry group and that the hypersurfaces of constant θ are homogeneous and partially isotropic. Even
if he did not get h by some near-horizon limit process, Carter stressed that in the neighborhood of
the extremal Kerr horizon, the Kerr metric g can be approximated arbitrarily closely by h (cf. Fig. 6.3
in Ref. [95], which is similar to our Fig. 13.8). In the related case of the static extremal black hole,
constituted by thep (spherically symmetric) Reissner-Nordström solution with electric charge equal to
the mass (Q = 4π/µ0 m, cf. Eq. (5.107)), Carter exhibited a change of coordinates similar17 to that
between Boyer-Lindquist coordinates and Bardeen-Horowitz coordinates [Eq. (13.59)] and he showed
that the near-horizon limit brings the Reissner-Nordström metric to the Bertotti-Robinson metric. The
latter is the product metric of AdS2 × S2 and constitutes a static solution of the source-free Einstein-
Maxwell equations that is highly symmetric, having the 6-dimensional group SL(2, R) × SO(3) as
isometry group (cf. e.g. Ref. [213]). The NHEK metric has been rediscovered by James Bardeen and Gary
Horowitz in 1999 [41]. They derived it from the Boyer-Lindquist expression of the extremal Kerr metric
by performing the change of coordinates18 (13.58)-(13.59) and taking the limit ε → 0, as presented in
Sec. 13.4.2. In the same article [41], Bardeen and Horowitz introduced the global NHEK coordinates19
(τ, y, θ, ψ) via the transformation (13.69) and obtained expression (13.70) for h [their Eq. (2.9)]. They
have analyzed the global properties of the NHEK spacetime, in particular its geodesic completeness.
Bardeen and Horowitz have also shown that the NHEK isometry group is SL(2, R) × U(1) [Eq. (13.86)]
and have given formulas (13.78), (13.80) and (13.82) for the four Killing vectors ξ1 , ξ2 , ξ3 and η [their
Eq. (2.14)].
13.4.6 Applications
The NHEK geometry is at the core of the so-called Kerr/CFT correspondence, initiated by Monica
Guica, Thomas Hartman, Wei Song and Andrew Strominger in 2009 [247], which conjectures
that quantum gravity in the vicinity of the extremal Kerr horizon is dual to a two-dimensional
conformal field theory (CFT) (see Ref. [137] for a review of this topic).
17
Carter defined the near-horizon coordinates (τ, λ) by t = m2 τ and r = λ + m and took the near-horizon
limit as m → +∞ (cf. the paragraph surrounding Eq. (4.31) in Ref. [95]); this is equivalent to Eq. (13.59) with
ε = 1/m.
18
The coordinates actually used by Bardeen and Horowitz were T̃ := 2mT and R̃ = mR, denoted respectively
t and r by them.
19
ψ is denoted by ϕ in Ref. [41].
13.5 Going further 537
Another domain of application of the NHEK geometry regards the computation of signals
emanating from the neighborhood of an extremal Kerr black hole, either electromagnetic ones
(e.g. [238, 139, 308]) or gravitational-wave ones (e.g. [140, 239]).
13.5 Going further
Apart from the principal null geodesics, we have not discussed the geodesics of the extremal
Kerr black hole in this chapter. Many of their properties can be obtained by taking the limit
a → m of the Kerr geodesics discussed in Chaps. 11 and 12. The specific case of critical null
geodesics, delineating the black hole shadow, has been treated in Sec. 12.4.4 for a = m. For
an extensive study of geodesics close to the horizon, making use of the NHEK geometry, see
Refs. [308, 139].
We have not discussed the stability of extremal Kerr black holes. Contrary to what happens
for subextremal Kerr black holes (a < m), Stefanos Aretakis has shown in 201220 that axisym-
metric scalar perturbations (solutions of the wave equation □g Ψ = 0) have transverse second
derivatives21 that blow up when t̃ → +∞ along the event horizon H of an extremal Kerr
black hole, while Ψ itself decays at least like t̃−1/2 and its transverse first derivative remains
bounded [18] (see also the monograph [19] and Theorem A in Ref. [207]). In 2016, Marc Casals,
Samuel Gralla and Peter Zimmerman [106] have shown that non-axisymmetric scalar pertur-
bations with initial data supported away from H are unstable along H , with a transverse
first derivative growing like t̃ 1/2 , which is a higher rate than the axisymmetric perturbations
considered by Aretakis. This has been recently confirmed by a rigorous mathematical analysis
by Dejan Gajic [207]. Gralla and Zimmerman have shown in 2018 [240] that this instability
extends to electromagnetic and gravitational perturbations and can be considered as a critical
phenomenon associated with the emergent symmetry generated by the NHEK Killing vector
ξ2 (squeeze mappings). Finally let us point out that Aretakis [17] has also shown that solutions
of the non-linear wave equation □g Ψ = N (Ψ, ∇Ψ), where N is a non-linear function, blow
up in a finite time along H .
20
The work [18] appeared on arXiv in 2012, but was published in 2015.
21
By transverse derivative it is meant ∂Ψ/∂r, ∂ 2 Ψ/∂r2 , etc. (recall that H is located at r = const and is
spanned by the coordinates (t̃, θ, φ̃)). If the initial data has support away from H , the divergence occurs only for
the third derivative.
538 Extremal Kerr black hole
Part IV
Dynamical black holes
Chapter 14
Black hole formation 1: dust collapse
Contents
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
14.2 Lemaître-Tolman equations and their solutions . . . . . . . . . . . . . 542
14.3 Oppenheimer-Snyder collapse . . . . . . . . . . . . . . . . . . . . . . . 550
14.4 Observing the black hole formation . . . . . . . . . . . . . . . . . . . . 568
14.5 Going further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
14.1 Introduction
After having investigated black holes in equilibrium, through the Schwarzschild and Kerr
solutions, we move to dynamical black holes, more specifically to the standard process of
black hole formation: gravitational collapse. To deal with analytical solutions, we simplify the
problem as much as possible. First we assume spherical symmetry, which is quite natural as a
first approximation for modeling the gravitational collapse of a stellar core or a gas cloud. A
drawback of this approach is that it forbids the study of gravitational waves, since by virtue
of the Jebsen-Birkhoff theorem (to be proven in Sec. 14.2.5), the exterior of any spherically
symmetric collapsing object is a piece of Schwarzschild spacetime, which does not contain any
gravitational radiation. The second major simplifying approximation is to consider pressureless
matter, commonly referred to as dust.
We start by deriving from the Einstein equation the system of partial differential equations
yielding the metric for spherically symmetric dust (Lemaître-Tolman system) and write down
the most general solutions (Sec. 14.2). Then we specialize these solutions to the case of a
homogeneous ball of dust collapsing from rest (Oppenheimer-Snyder collapse, Sec. 14.3), which
we study in details, in particular regarding the birth of the event horizon and the existence
of trapped surfaces. Finally, we consider the observational appearance of the gravitational
collapse and the black hole formation to a remote observer (Sec. 14.4).
542 Black hole formation 1: dust collapse
14.2 Lemaître-Tolman equations and their solutions
14.2.1 Hypotheses
As mentioned in the Introduction, we shall restrict ourselves to spherically symmetric1 space-
times, and for concreteness, to 4-dimensional ones. The most general spherically symmetric
4-dimensional spacetime (M , g) can be described in terms of coordinates (xα ) = (τ, χ, θ, φ)
such that the metric tensor writes
g = −dτ 2 + a(τ, χ)2 dχ2 + r(τ, χ)2 dθ2 + sin2 θ dφ2 , (14.1)
where a(τ, χ) and r(τ, χ) are generic positive functions. These coordinates are called Lemaître
synchronous coordinates, the qualifier synchronous meaning that τ is the proper time of
a observer staying at fixed value of the spatial coordinates (χ, θ, φ). Note that the function
r(τ, χ) gives the areal radius of the 2-spheres defined by (τ, χ) = const, which are the orbits
of the SO(3) group action (cf. Sec. 6.2.2), i.e. the metric area of these 2-spheres is 4πr(τ, χ)2 .
For simplicity, we consider only pressureless matter, in the form of a perfect fluid with zero
pressure. The matter energy-momentum tensor is then
T = ρ u ⊗ u, (14.2)
where u is the 1-form metric-dual to the fluid 4-velocity u, i.e. the 1-form of components
uα = gαµ uµ [cf. Eq. (A.45)], and ρ is a scalar field that can be interpreted as the fluid energy
density measured in the fluid frame, by virtue of the identity ρ = T (u, u), which follows from
⟨u, u⟩ = g(u, u) = −1. Let us recall that the energy-momentum tensor of a generic perfect
fluid is T = (ρ + p)u ⊗ u + pg, where p is the fluid pressure. Expression (14.2) corresponds
thus to the special case p = 0. Inside the matter, we link the coordinates (τ, χ, θ, φ) to the fluid
by demanding that they are comoving with the fluid, i.e. that any fluid particle stays at fixed
values of (χ, θ, φ). Because the 4-velocity obeys uα = dxα /dτf , where τf is the fluid proper
time [cf. Eq. (1.16)], this amounts to uχ = uθ = uφ = 0, hence
u = ∂τ . (14.3)
A priori, one should only have u = uτ ∂τ , but the synchronous coordinate condition gτ τ = −1
along with the normalization g(u, u) = −1 implies uτ = 1. Since uτ = dτ /dτf , we get
τ = τf (up to some additive constant), which provides the physical interpretation of Lemaître
coordinate τ as the fluid proper time.
14.2.2 Geodesic matter flow
→
−
The equation of energy-momentum conservation ∇ · T = 0 [Eq. (1.45)], which follows from
the Einstein equation (1.40) and the contracted Bianchi identity (A.110) (cf. Sec. 1.5), implies:
1
See Sec. 6.2.2 for the precise definition of spherically symmetric.
14.2 Lemaître-Tolman equations and their solutions 543
Property 14.1: geodesic flow
The worldlines of the fluid particles of the pressureless matter model (14.2) are timelike
geodesics of (M , g).
Proof. If one plugs the energy-momentum tensor (14.2) in the energy-momentum conservation
law (1.45), one obtains ∇µ (ρuµ uα ) = 0, i.e.
∇µ (ρuµ )uα + ρuµ ∇µ uα = 0. (14.4)
Now the two terms in the left-hand side of this equation are orthogonal to each other, as an
immediate consequence of the normalization of the 4-velocity u [Eq. (1.17)]: u · ∇u u = 0. In
particular, u is a timelike vector, while the 4-acceleration ∇u u is a spacelike one. Thus the
only way for Eq. (14.4) to hold is that each term in the left-hand side vanishes separately:
∇µ (ρuµ ) = 0 and uµ ∇µ uα = 0.
The second equation is nothing but the geodesic equation [Eq. (B.1)] for the field lines of u, i.e.
the fluid worldlines.
Each fluid particle is thus in free-fall and moves independently of its neighbors, which is not
surprising since the pressure is zero. This justify the term dust given to the matter model
(14.2).
14.2.3 From the Einstein equation to the Lemaître-Tolman system
Let us write the Einstein equation (1.40) in terms of Lemaître synchronous coordinates (τ, χ, θ, φ)
and with the energy-momentum tensor (14.2)-(14.3) in its right-hand side. As detailed in the
notebook D.6.1, if one disregards the peculiar case2 ∂r/∂χ = 0, the τ χ component yields
1 ∂r
a(τ, χ) = , (14.5)
f (χ) ∂χ
where f (χ) is an arbitrary function of χ. Accordingly, we may rewrite the metric (14.1) as
2
1 ∂r
2
dχ2 + r(τ, χ)2 dθ2 + sin2 θ dφ2 . (14.6)
g = −dτ +
f (χ)2 ∂χ
Property 14.2: Lemaître-Tolman system
The metric (14.6) is a solution of the Einstein equation (1.40) with the energy-momentum
2
For Λ = 0, this case leads to Datt solution [157].
544 Black hole formation 1: dust collapse
tensor (14.2)-(14.3) if, and only if,
2
∂r 2M (χ) Λ
= f (χ)2 − 1 + + r(τ, χ)2 (14.7a)
∂τ r(τ, χ) 3
dM ∂r
= 4πr(τ, χ)2 ρ(τ, χ) , (14.7b)
dχ ∂χ
where M (χ) is an arbitrary function of χ.
Proof. Taking into account (14.5), the χχ and τ τ components of the Einstein equation yield
respectively to (14.7a) and (14.7b), cf. the notebook D.6.1. There is no other independent
component of the Einstein equation.
The function M (χ) is known in the literature as the Misner-Sharp mass or Misner-
Sharp energy, in reference of a study by Misner and Sharp in 1964 [370], although it has been
introduced by Lemaître [346] more than 30 years earlier. This quantity is invariantly defined
for any spherically symmetric spacetime from the areal radius r:
r Λ 2
M := µ
1 − ∇µ r∇ r − r . (14.8)
2 3
It is easy to check that the above relation holds in the present case: we have, thanks to (14.1),
2 2 2 2
µ ∂r ∂r
µν ∂r ∂r ∂r 1 ∂r
∇µ r∇ r = g = gτ τ +g χχ
=− +
∂xµ ∂xν ∂τ ∂χ ∂τ a(τ, χ)2 ∂χ
Using Eq. (14.5), this expression reduces to
2
µ ∂r
∇µ r∇ r = − + f (χ)2 .
∂τ
In view of the Lemaître-Tolman equation (14.7a), we conclude that (14.8) holds for M = M (χ).
Historical note : The Lemaître-Tolman system (14.7) has been first derived in 1932 by Georges Lemaître
[346]: Eqs. (14.6), (14.7a) and (14.7b) are respectively Eqs. (8.1), (8.2) and (8.3) of Ref. [346], up to some
slight change of notations. The system (14.7) however became known as Tolman model or Tolman-Bondi
model, in reference to posterior works by Richard Tolman (1934) [482] and by Hermann Bondi (1947)
[66]. This happened despite Tolman fully acknowledging Lemaître’s work [346] in his article [482]
(Tolman actually met Lemaître in 1932-33 during the latter’s trip to United States [182]) and Bondi [66]
mentioned that “Lemaître studies a problem very closely related to ours and many equations given in the
appendix can be found in the (Lemaître’s) paper”. We refer to Eisenstaedt’s article [182] for a detailed
historical study of Lemaître’s paper [346] (see also Krasiński’s note [323]). We follow the suggestion of
Plebański & Krasiński [414] to call the system (14.7) Lemaître-Tolman, and not merely Lemaître, in order
to distinguish it from other Lemaître contributions to general relativity and cosmology.
14.2 Lemaître-Tolman equations and their solutions 545
14.2.4 Solutions for a vanishing cosmological constant
In the remaining of this chapter, we assume Λ = 0, since we are mainly interested in gravita-
tional collapse in asymptotically flat spacetimes. The Lemaître-Tolman equation (14.7a) can
then be rewritten as
1 2 M (χ)
ṙ − = E(χ), (14.9)
2 r
where ṙ := ∂r/∂τ and
f (χ)2 − 1
E(χ) := . (14.10)
2
For a fixed value of χ, we recognize in (14.9) the equation ruling the 1-dimensional non-
relativistic motion of a particle in a Newtonian potential V = −m/r; E(χ) is then nothing but
the total mechanical energy of the particle per unit mass. As it is well known, the solution of
(14.9) depends on the sign of E(χ):
• if E(χ) > 0, the solution is given in parameterized form (parameter η) by
M (χ)
τ = (2E(χ))3/2 (sinh η − η) + τ0 (χ)
(14.11)
M (χ)
r(τ, χ) =
(cosh η − 1)
2E(χ)
• if E(χ) = 0, the solution is
1/3
9M (χ)
r(τ, χ) = (τ − τ0 (χ))2 (14.12)
2
• if E(χ) < 0, the solution is given in parameterized form (parameter η) by
M (χ)
τ = |2E(χ)|3/2 (η + sin η) + τ0 (χ)
(14.13)
M (χ)
r(τ, χ) =
(1 + cos η)
|2E(χ)|
In the above formulas, τ0 (χ) is an arbitrary function of χ. For E > 0 and E = 0, it sets the
value of τ for which r = 0, while for E < 0, it sets the value of τ for which r takes its maximal
value (m/|E|).
Exercise: prove that each of formulas (14.11)-(14.13) provides a solution of Eq. (14.9).
We may summarize the above results as follows:
Property 14.3: solution for spherical dust collapse
The procedure to get a full solution of spherical dust collapse with Λ = 0 is
1. choose arbitrary functions f (χ), M (χ) and τ0 (χ);
2. evaluate E(χ) via (14.10);
546 Black hole formation 1: dust collapse
3. depending of on the value of E(χ), use (14.11), (14.12) or (14.13) to get the solution
for r(τ, χ);
4. plug this solution into the remaining Lemaître-Tolman equation, Eq. (14.7b), to get
ρ(τ, χ) and into (14.6) to get the metric tensor.
14.2.5 Vacuum solution and the Jebsen-Birkhoff theorem
Let us search for a solution in the vacuum case, i.e. ρ = 0. Equation (14.7b) with ρ = 0 implies
that M (χ) is a constant, which we shall denote by m. Equations (14.9)-(14.10) then yield
r
∂r 2m
= ε f (χ)2 − 1 + , with ε = ±1. (14.14)
∂τ r
Hence r
2m ∂r
dr = ε f (χ)2 − 1 + dτ + dχ.
r ∂χ
Using this identity to substitute (∂r/∂χ) dχ in the metric expression (14.6), we get
" r #
1 2m 2m
g = − 1− dτ 2 − 2ε f (χ)2 − 1 + dτ dr + dr2
f (χ)2 r r
+r2 dθ2 + sin2 θ dφ2 , (14.15)
which can be rearranged as
p !2 −1
2m 1 f (χ)2 − 1 + 2m/r 2m
g = − 1− dτ + ε dr + 1 − dr2
r f (χ) f (χ)(1 − 2m/r) r
+r2 dθ2 + sin2 θ dφ2 . (14.16)
If there exists a scalar function t = t(τ, r) such that
p
1 f (χ)2 − 1 + 2m/r
dt = dτ + ε dr, (14.17)
f (χ) f (χ)(1 − 2m/r)
then Eq. (14.16) would be nothing but Schwarzschild metric expressed in Schwarzschild-Droste
coordinates (t, r, θ, φ) [cf. Eq. (6.14)]. The necessary and sufficient condition for (14.17) to hold
is p !
∂ 1 ∂ f (χ)2 − 1 + 2m/r
= ε ,
∂r f (χ) τ ∂τ f (χ)(1 − 2m/r)
r
this relation expressing the identity ∂ t/∂r∂τ = ∂ t/∂τ ∂r. In it, χ has to be considered as a
2 2
function of (τ, r), which is given implicitly by one of the three expressions (14.11)-(14.13) with
M (χ) = m. Accordingly, the above condition becomes
f ′ (χ) ∂χ ε f ′ (χ) ∂χ
− =p .
f (χ)2 ∂r τ f (χ)2 − 1 + 2m/r f (χ)2 ∂τ r
14.2 Lemaître-Tolman equations and their solutions 547
Simplifying and invoking (14.14), we arrive at a condition independent of f (χ):
∂χ ∂χ ∂r
+ = 0.
∂τ r ∂r τ ∂τ χ
Now this condition is always fulfilled: by the chain rule, the left-hand side is nothing but the
partial derivative ∂χ
∂τ χ
, which is trivially zero if one considers χ as a function of (τ, χ). We
conclude that there exists a function t obeying (14.17), which proves that g is Schwarzschild
metric, whatever the choice of the function f (χ). Since our starting point was the most general
metric for a (possibly non-stationary) spherically symmetric spacetime [Eq. (14.1)], we have
proven a famous theorem of general relativity:
Property 14.4: Jebsen-Birkhoff theorem
In vacuum, the unique spherically symmetric solution of the 4-dimensional Einstein equa-
tion with Λ = 0 [Eq. (1.44)] is the Schwarzschild metric.
In particular, outside any spherically symmetric body, the spacetime is a piece of Schwarz-
schild spacetime. Note that this implies that this part of spacetime is static, even if the central
body is not (for instance it may oscillate radially, keeping its spherical symmetry). In particular,
there are no gravitational waves in spherical symmetry.
Remark 1: The Jebsen-Birkhoff theorem can be viewed as a generalization of the shell theorem version
of Gauss’s law in Newtonian gravity: the gravitational field outside any spherical source is entirely
determined by the total mass m of the source, being identical to that generated by a point of mass m
located at the symmetry center.
Historical note : The Jebsen-Birkhoff theorem has been first proved by the Norwegian physicist Jørg
Tofte Jebsen in 1920 [300]. It was independently established by the American mathematician George
D. Birkhoff in 1923 [62] and became famous under the name Birkhoff’s theorem, the work of Jebsen
remaining largely unknown until recently (see Ref. [303] for details).
14.2.6 Schwarzschild solution in Lemaître coordinates
The above derivation of the Jebsen-Birkhoff theorem offers a way to express the Schwarzschild
metric in new sets of coordinates, not considered in Part II, namely Lemaître and Painlevé-
Gullstrand coordinates. Since we have all freedom on the function f (χ), we will focuss on a
subclass of Lemaître coordinates for which f (χ) = 1. Then E(χ) = 0 and r(τ, χ) is given by
Eq. (14.12). Since M (χ) = m is constant, we cannot choose τ0 (χ) in (14.12) to be a constant,
otherwise we would have ∂r/∂χ = 0 and the metric (14.6) would be degenerate. The simplest
non-constant choice is τ0 (χ) = χ. Equation (14.12) yields then
1/3
9m
r(τ, χ) = (χ − τ )2/3 . (14.18)
2
548 Black hole formation 1: dust collapse
In what follows, we assume χ ≥ τ . Then
r
1 2 3/2
χ−τ = r (14.19)
3 m
and 1/3 r
∂r 4m 2m
= (χ − τ )−1/3 = . (14.20)
∂χ 3 r
Accordingly, Eq. (14.6) yieds
2m
g = −dτ 2 + dχ2 + r(τ, χ)2 dθ2 + sin2 θ dφ2 , (14.21)
r(τ, χ)
where r(τ, χ) is given by Eq. (14.18). This is the expression of the Schwarzschild metric in
terms of the Lemaître coordinates (τ, χ, θ, φ).
The relation between Lemaître coordinates and Schwarzschild-Droste ones is obtained by
setting f (χ) = 1 and3 ε = −1 in Eq. (14.17):
q
2m
r
dτ = dt + dr. (14.22)
1 − 2mr
By integration, we get
r p !
r r/2m − 1
τ = t + 4m + 2m ln p + const. (14.23)
2m r/2m + 1
The expression of χ in terms of (t, r) is then deduced from Eq. (14.19):
r p !
r r r/2m − 1
χ = t + 4m 1+ + 2m ln p + const. (14.24)
2m 6m r/2m + 1
We deduce easily from these formulas the expression of the stationarity Killing vector ξ
of Schwarzschild spacetime in terms of the Lemaître coordinates. Since ξ = ∂t [Eq. (6.6)],
and the above formulas imply ∂τ /∂t = 1 and ∂χ/∂t = 1, we get, applying the chain rule
∂/∂t = ∂/∂τ × ∂τ /∂t + ∂/∂χ × ∂χ/∂t,
ξ = ∂τ + ∂χ . (14.25)
Remark 2: Although very simple, the above relation shows that Lemaître coordinates are not adapted
to the spacetime symmetry generated by the Killing vector ξ: the latter does not coincide with any
coordinate vector ∂α of Lemaître coordinates. This reflects the fact that the metric components (14.21)
depend on τ (via the function r(τ, χ)), in addition to χ.
3
ε = −1 follows from Eqs. (14.14) and (14.18).
14.2 Lemaître-Tolman equations and their solutions 549
The vacuum hypothesis implies that we can no longer interpret Lemaître coordinates as
comoving with some free-falling dust, as in Sec. 14.2.1. However, the geodesic character of
these coordinates remains. Indeed, the vector u := ∂τ is geodesic with respect to the metric
(14.1): ∇u u = 0, as a direct calculation shows (cf. the notebook D.6.1). This implies that the
curves (χ, θ, φ) = const are timelike geodesics. Moreover, the conserved energy per unit mass
along these geodesics is (cf. Sec. B.5)
ε = −ξ · u = −(∂τ + ∂χ ) · ∂τ = − gτ τ − gχτ = 1,
|{z} |{z}
−1 0
where use has been made of Eq. (14.25). ε = 1 means that the geodesics are marginally bound:
they describe the free fall from rest at infinity.
As it is clear on the metric components (14.21), a key feature of Lemaître coordinates is to
be regular at r = 2m, i.e. across the event horizon of Schwarzschild spacetime, contrary to
Schwarzschild-Droste coordinates.
Historical note : The Schwarzschild metric in the form (14.21) has been obtained in 1932 by Georges
Lemaître [346], as a vacuum solution of the Lemaître-Tolman system: cf. Eq. (11.12) of Ref. [346].
Remarkably, Lemaître pointed out that the metric components (14.21) are regular at r = 2m and was the
first author to conclude that the singularity of Schwarzschild’s solution at r = 2m is a mere coordinate
singularity. As pointed out in the historical note on p. 194, eight years before, Arthur Eddington [177]
exhibited a coordinate system that is regular at r = 2m but he did not mention this feature.
14.2.7 Schwarzschild solution in Painlevé-Gullstrand coordinates
The Painlevé-Gullstrand coordinates on Schwarzschild spacetime are defined from the
Lemaître coordinates (τ, χ, θ, φ) by using r, instead of χ, as the radial coordinate. They are
thus the coordinates (τ, r, θ, φ). The expression of the metric tensor in the Painlevé-Gullstrand
coordinates is provided by Eq. (14.15) with f (χ) = 1 and ε = −1:
r
2m 2m
dτ 2 + 2 dτ dr + dr2 + r2 dθ2 + sin2 θ dφ2 . (14.26)
g =− 1−
r r
The Painlevé-Gullstrand coordinates (τ, r, θ, φ) have a noticeable feature: the hypersurfaces
τ = const are flat 3-manifolds, i.e. the metric induced of them by g is the flat Euclidean metric.
This is immediate: by setting dτ = 0 in Eq. (14.26), one gets
g τ =const = dr2 + r2 dθ2 + sin2 θ dφ2 ,
which is nothing but the 3-dimensional Euclidean metric expressed in spherical coordinates
(r, θ, φ). This proves that Schwarzschild spacetime can be sliced by a family of flat hypersurfaces.
The associated 3+1 decomposition of the metric tensor is revealed by rewriting (14.26) as
r !2
2m
g = −dτ 2 + + r2 dθ2 + sin2 θ dφ2 . (14.27)
dr + dτ
r
550 Black hole formation 1: dust collapse
One reads on this expression that the lapse function (see e.g. Ref. [227]) is N = 1 and that the
shift vector is β = 2m/r ∂r . Finding a lapse function equal to one simply reflects that the
p
coordinate time τ is the proper time of some observer, namelly the observer following the
marginally bound radial geodesics discussed above.
Another interesting property of Painlevé-Gullstrand coordinates, which they share with
Lemaître ones, is to be regular at r = 2m: despite the vanishing of gτ τ there, as read on (14.26),
the determinant of the metric components (14.26) is not vanishing, thanks to the off-diagonal
term gτ r . Indeed, det (gαβ ) = −r4 sin2 θ, which is clearly nonzero, except on the axis θ = 0 or
π.
Remark 3: Painlevé-Gullstrand coordinates (τ, r, θ, φ) can be seen as a timelike analog of the ingoing
Eddington-Finkelstein (IEF) coordinates (t̃, r, θ, φ) introduced in Sec. 6.3.2. Both coordinate systems are
based on ingoing radial geodesics of Schwarzschild spacetime, these geodesics being null for IEF and
timelike for Painlevé-Gullstrand.
The reader is referred to Ref. [357] for a more detailed discussion of Painlevé-Gullstrand
coordinates.
Historical note : Painlevé-Gullstrand coordinates have been introduced in 1921 by the French mathe-
matician Paul Painlevé [397], as well as by the Swedish physicist and ophthalmologist Allvar Gullstrand
(1911 laureate of the Nobel Prize in Medicine) in 1922 [248].
14.3 Oppenheimer-Snyder collapse
14.3.1 Pressureless collapse of a star from rest
An important example of dust collapse is that of a spherical star in hydrostatic equilibrium
that suddenly looses pressure support to counterbalance gravity. This roughly models the
astrophysical phenomenon of gravitational collapse of the iron core of a massive star at the
end of thermonuclear evolution — a phenomenon that can ultimately give birth to a supernova
if some bounce occurs. In this astrophysical event, the pressure never vanishes, but after the
collapse has reached a certain stage, it plays a negligible role on the dynamics, so that the
pressureless (dust) approximation developed in this chapter is quite good.
Given the assumed spherical symmetry, the Jebsen-Birkhoff theorem (Property 14.4) implies
that the spacetime metric outside the star is Schwarzschild metric. Let us then characterize the
star by its areal radius r0 (i.e. the Schwarzschild-Droste coordinate r of the star’s surface, cf.
Sec. 6.2.2) at the start of the collapse and its gravitational mass m. The latter is nothing but
the mass parameter of the Schwarzschild metric. It stays constant during all the collapse. We
shall describe the spacetime exterior to the star by means the ingoing Eddington-Finkelstein
coordinates (IEF) (t̃, r, θ, φ) introduced in Sec. 6.3.2, because they are regular on the black hole
event horizon, contrary to Schwarzschild-Droste coordinates. The exterior metric is then given
by Eq. (6.33):
2m 4m 2m
2
dr2 + r2 dθ2 + sin2 θ dφ2 . (14.28)
g =− 1− dt̃ + dt̃ dr + 1 +
r r r
14.3 Oppenheimer-Snyder collapse 551
Let us denote by rs (τ ) the areal radius of the star, i.e. the coordinate r of its surface, as a
function of the proper time τ of a matter particle located at the surface. If the origin of τ is set
to the start of the collapse, we have
rs (0) = r0 . (14.29)
Due to the pressureless hypothesis, the surface particles follow timelike radial geodesics of
Schwarzschild spacetime, as studied in Sec. 7.3.2. Since the collapse starts from rest, the function
rs (τ ) is given by the geodesic solution (7.41):
r
r03
τ= (η + sin η)
8m
0 ≤ η ≤ π, (14.30)
rs (τ ) = r0 (1 + cos η)
2
where the parameter η is zero at the start of the collapse and π at its end. The function rs (τ ) is
plotted in Fig. 14.1 (right). Its graph is an arch of cycloid, the parameter η corresponding to the
angle of rotation of the rolling circle defining the cycloid.
Remark 1: The “cycloidal” evolution law rs (τ ), as given by Eq. (14.30), is identical to that of dust ball
in Newtonian gravity, with τ being then the (universal) Newtonian time. This coincidence follows from
the Lemaître-Tolman equation (14.9) being identical to the first integral of a purely radial motion in
Newtonian gravity.
The IEF coordinate time t̃ corresponding to the surface proper time τ is given by Eq. (7.45):
r −1/2
r0 h r0 i η r0 η
t̃ = 2m −1 η+ (η + sin η) + 2 ln cos + −1 sin , (14.31)
2m 4m 2 2m 2
where t̃ = 0 is assumed at the start of the collapse. Note that by combining Eqs. (14.30) and
(14.31), one can obtain rs = rs (t̃), i.e. the evolution law of the stellar surface in terms of the
IEF coordinate time t̃. The right panel in Fig. 7.4 depicts it for various values of m/r0 .
14.3.2 Oppenheimer-Snyder solution
The metric (14.28) is valid only for r ≥ rs (t̃). The metric inside the star is given by one of the
solutions of the Lemaître-Tolman system given in Sec. 14.2.4. Actually only the solutions with
E(χ) < 0, i.e. those given by Eq. (14.13), are to be considered because they are the only ones
allowing for a collapse starting from rest.
As stressed in Sec. 14.2.4, formula (14.13), in conjunction with Eqs. (14.7b) and (14.6), defines
an infinity of solutions depending on the functions f (χ), M (χ) and τ0 (χ), which can be chosen
arbitrarily. We shall actually focus on the simplest of these solutions, which is that describing
a homogeneous star, i.e. an object whose density ρ in the matter frame, as defined by Eq. (14.2),
is constant at any given proper time τ . In other words, ρ, which is a priori a function of (τ, χ),
is actually a function of τ only. This is certainly a crude approximation of a real star (or stellar
iron core), which has a non-constant density profile, but it captures the essential features
of spherically symmetric collapse giving birth to a black hole. This homogeneous solution
552 Black hole formation 1: dust collapse
is usually called Oppenheimer-Snyder collapse, although the solution originally presented
by J. Robert Oppenheimer and Hartland Snyder in 1939 [392] was slightly different, since it
regards the marginally bound case, i.e. the solution E(χ) = 0 [Eq. (14.12)] instead of E(χ) < 0
[Eq. (14.13)] (cf. the historical note below).
As we are going to see, the homogeneous dust collapse is obtained by the following choice
of the freely specifiable functions:
a0
f (χ) = cos χ, M (χ) = sin3 χ and τ0 (χ) = 0, (14.32)
2
with r
r03
a0 := . (14.33)
2m
The above choice of f (χ) leads to 2E(χ) = − sin2 χ [cf. Eq. (14.10)]; hence E(χ) < 0 for
χ > 0 and the solution to be considered is (14.13). The prefactors are M (χ)/|2E(χ)|3/2 = a0 /2
and M (χ)/|2E(χ)| = (a0 /2) sin χ, so that we get, taking into account the choice τ0 (χ) = 0,
a0
τ=
(η + sin η)
2
0 ≤ η ≤ π. (14.34)
r(τ, χ) = a0 sin χ (1 + cos η)
2
We note that τ depends only on η and not on χ, contrary to what happens for the general
solution (14.13). Moreover, thanks to the choice (14.33), τ matches the proper time at the
surface of the star as given by Eq. (14.30). The areal radii at the surface must match as well.
Let us denote by χs the surface value pof χ. The matching of the surface areal radii given by
Eqs. (14.30) and (14.34) implies r0 = r03 /2m sin χs , from which we get the relation between
χs and (m, r0 ):
r
2m
sin χs = . (14.35)
r0
This implies sin χs > 0 and hence χs ∈ (0, π). There are actually two solutions of Eq. (14.35)
for χs . Given that arcsin is a one-to-one map [0, 1] → [0, π/2], they are
r
2m
χs = arcsin (14.36)
r0
and χs = π − arcsin 2m/r0 . For the physical scenario we have in mind (gravitational
p
collapse of an “ordinary” stellar core), we shall retain only the solution (14.36), which implies
χs ∈ (0, π/2]. The second solution, for which χs ∈ [π/2, π), will be discussed in Remark 3 on
p. 559 below4 . Equation (14.36) show that χs is a function of the dimensionless ratio m/r0 ,
which is the compactness of the initial star (some numerical values can be found in Table 14.1).
4
For the moment, we can note that such a solution would lead to the areal radius r being a decreasing function
of χ near χs , since Eq. (14.34) shows that r(τ, χ) behaves as sin χ, which is decaying on (π/2, π). Such a behavior
is certainly not expected in an “ordinary” star.
14.3 Oppenheimer-Snyder collapse 553
3.0 q
τ 2m/r03 3.0 η
rs (τ)/r0 = a(τ) sinχs /r0 rs (τ)/r0 = a(τ) sinχs /r0
2.5 ρ(τ) πr03 /(6m) 2.5 ρ(τ) πr03 /(6m)
2.0 2.0
r s , τ, ρ
η, r s , ρ
1.5 1.5
1.0 1.0
0.5 0.5
0.0 0.0
0 1π 1π 3π π 0.0 0.2 0.4 0.6q 0.8 1.0 1.2 1.4 1.6
4 2 4
η τ 2m/r03
Figure 14.1: Evolution of the areal radius rs (τ ) the collapsing star [Eq. (14.30); red curve] and of the proper
matter density ρ(τ ) [Eq. (14.39); blue curve] in terms of the conformal time η (left) or the matter proper time τ
(right). The function τ = τ (η) is depicted by the purple dashed curve, while its inverse, η = η(τ ), is depicted by
the orange dashed one. As indicated in the legends, the plot of metric factor a(τ ) is the same as that of rs (τ ) up
to a rescaling by sin χs . [Figures generated by the notebook D.6.2]
The upper limit π/2 can be excluded from the range of χs since it would correspond to r0 = 2m,
which is the areal radius of the black hole horizon in Schwarzschild geometry. The range of
the comoving coordinate χ in the interior of the star is thus
π
0 ≤ χ ≤ χs < , (14.37)
2
with χ = 0 locating the center, since it corresponds to r = 0 by virtue of Eq. (14.34).
By combining Eqs. (14.32), (14.33) and (14.35), we note that M (χ) is an increasing function
of χ in the range (14.37), with M (0) = 0 and
M (χs ) = m. (14.38)
The matter density ρ(τ, χ) is deduced from Eq. (14.7b), with the above values of M (χ) and
r(τ, χ). We get a function of τ only, via η = η(τ ) [cf. Eq. (14.34)], which we rewrite as ρ(τ ):
6m
ρ(τ ) = . (14.39)
πr03 (1 + cos η)3
The independence of ρ from χ confirms that the choice (14.32) leads to a uniform density
object, i.e. a homogeneous star. Let us stress that the homogeneity is maintained during all
the collapse: at each instant of matter proper time τ , the density is uniform is all the star. The
function ρ(τ ) is plotted in Fig. 14.1.
Remark 2: In view of Eq. (14.30), we may rewrite Eq. (14.39) as
m
ρ(τ ) = 4 3
. (14.40)
3 πrs (τ )
554 Black hole formation 1: dust collapse
This formula happens to be identical to that giving the mass density of a uniform ball of mass m and
radius rs (τ ) in Newtonian physics (flat spacetime), but there is no profound physical significance in this
coincidence.
It is instructive to express the initial energy density, ρ0 := ρ(τ = 0), in terms of the
geometrical mass density unit m−2 and the compactness angle χs by combining Eqs. (14.39)
and (14.35):
3 sin6 χs
ρ0 = . (14.41)
32πm2
The metric factor a(τ, χ), which appears in Eq. (14.1), is deduced from Eq. (14.5), with
∂r/∂χ computed from expression (14.34); as for ρ, we get a function of τ only, which we
rewrite as a(τ ):
a0
a(τ ) = (1 + cos η) . (14.42)
2
We may then rewrite the second equation in (14.34) as r(τ, χ) = a(τ ) sin χ, so that the metric
tensor (14.1) inside the star takes the form
g = −dτ 2 + a(τ )2 dχ2 + sin2 χ dθ2 + sin2 θ dφ2 . (14.43)
We recognize the Friedmann-Lemaître-Robertson-Walker (FLRW) metric corresponding to
a closed universe (cf. e.g. Chap. 3 of Ref. [413] or Chap. 17 of Book 3 of Ref. [163]). This is not
so surprising since FLRW metrics are solution of the Einstein equation based on the hypothesis
of homogeneity of constant τ slices. We may use η, instead of τ , as the time coordinate inside
the star, thanks to the relation
a0
dτ = (1 + cos η)dη, (14.44)
2
which follows from (14.34). We then deduce from Eq. (14.43) that
a20
(1 + cos η)2 −dη 2 + dχ2 + sin2 χ dθ2 + sin2 θ dφ2 . (14.45)
g=
4
The term inside square brackets is nothing but the metric g̃ of the Einstein cylinder introduced
in Sec. 4.2.3 [compare Eq. (4.18) with the change of notation η ↔ τ ]. Hence, inside the star, the
metric g is conformal to g̃: g = Ω2 g̃, with the conformal factor Ω := a0 /2(1 + cos η) = a(τ ).
For this reason, η is called the conformal time. Note that while g̃ is a static metric [it obeys
Eq. (5.5)], g is not, since the conformal factor depends on η.
On each hypersurface η = const, or equivalently τ = const, the 3-metric h induced by the
conformal metric g̃ is the standard metric of the hypersphere S3 , expressed in terms of the
hyperspherical coordinates (χ, θ, φ):
h = dχ2 + sin2 χ dθ2 + sin2 θ dφ2 . (14.46)
On the whole S3 , χ ranges from 0 to π. Since in the present case, χ ∈ [0, χs ] with χs < π/2
[Eq. (14.37)], we may view each constant η slice of the interior of the collapsing star as a piece
14.3 Oppenheimer-Snyder collapse 555
of S3 , scaled by the factor a(τ ) given by Eq. (14.42). If one uses the conformal coordinates
(η, χ) to draw a 2-dimensional spacetime diagram of the interior, as in the left part of Fig. 14.2,
then the radial null geodesics appear as straight lines inclined by ±45◦ with the horizontal,
reflecting the Minkowskian part −dη 2 + dχ2 of the metric (14.45).
To show that the FLRW interior (14.45) along with the Schwarzschild exterior (14.28) form
a regular spacetime, one shall check that the Darmois junction conditions are fulfilled at the
interface between the two regions, i.e. on the timelike hypersurface Σ defining the surface
of the collapsing star. These conditions, first enounced by Georges Darmois [154] (see also
Sec. 21.13 of MTW textbook [371], as well as Ref. [332]), are the continuity of the induced
3-metric of Σ and of its extrinsic curvature, when these two tensors are computed on each side
of Σ. These geometric conditions translate the absence of surface layers on Σ. We shall not
check them here; the computation can be found in Sec. 3.8 of Poisson’s textbook [416].
Historical note : As mentioned above, the solution originally obtained by J. Robert Oppenheimer
and Hartland Snyder in 1939 [392] regards pressureless matter evolving on marginally bound timelike
geodesics, i.e. matter in free fall from r → +∞, where it is at rest at τ → −∞. More precisely,
Oppenheimer and Snyder’s original solution corresponds to the choice
r
χ3 2χ3s
f (χ) = 1 M (χ) = m 3 and τ0 (χ) = (14.47)
χs 9m
for the freely specifiable functions determining the solutions of the Lemaître-Tolman system (cf.
Sec. 14.2.4), instead of the choice (14.32) adopted in this chapter. Choosing f (χ) = 1 leads to E(χ) = 0
[cf. Eq. (14.10)] and hence to the solution (14.12), instead of (14.13). This solution is still homogeneous,
since ρ, computed via Eq. (14.7b), is a function of τ only, but it is less adapted to a star of finite size
collapsing from an equilibrium state than the solution (14.13). The latter, which we are using here, has
been first presented by David Beckedorff in 1962 [46] and popularized by B. Kent Harrison, Kip Thorne,
Masami Wakano and John Archibald Wheeler in their 1965 monograph on gravitational collapse [256],
as well as by Misner, Thorne and Wheeler in their 1973 textbook [371].
14.3.3 Spacetime diagrams and maximal extensions
To draw a full spacetime diagram of the collapse, we may extend the ingoing Eddington-
Finkelstein (IEF) coordinates (t̃, r, θ, φ), used to described the Schwarzschild exterior of the
star to the interior region by demanding that relation (14.31), which a priori links t̃ to η at the
stellar surface, holds everywhere in the interior. Regarding the IEF coordinate r, it is naturally
identified with the areal radius r(η, χ) given by Eq. (14.34). The resulting spacetime diagram is
shown in the right part of Fig. 14.2. In this diagram, only the ingoing radial null geodesics in
the exterior appear as straight lines, as a consequence of the definition of the IEF coordinates
(cf. Sec. 6.3.2). The slices of constant proper time τ (dashed red lines) appear as horizontal line
segments by virtue of the above choice for t̃ in the interior. In particular, the star at τ = 0 is
depicted by a horizontal magenta segment, as in the (η, χ) diagram on the left of the figure.
Figure 14.3 presents another view of the Oppenheimer-Snyder collapse. It is built on
Kruskal-Szekeres-like coordinates (T, X) (cf. Sec. 9.2). These coordinates are defined from the
IEF-like ones (t̃, r) via formulas (9.23), with t̃ being replaced there by t̃ − 5m; this translation
in t̃ ensures a better representation of the final stages of the collapse. One may note that the
556 Black hole formation 1: dust collapse
16
π
14
0.50
12
3π
4
1.00 10
1.50
t̃/m
1π 8
η
2 2.00
2.50 6
3.00
1π 4
4
3.50
2
0 0
0 1π
8
1π
4 0 2 4 6 8 10
χ r/m
Figure 14.2: Spacetime diagrams of the Oppenheimer-Snyder collapse in conformal coordinates (η, χ) (left
figure, depicting only the interior) and in Eddington-Finkelstein coordinates (t̃, r) (right figure), for the initial
compactness m/r0 = 1/4, which corresponds to χs = π/4. In both figures, solid red lines are worldlines of matter
particles, uniformly sampling [0, χs ] with δχ = π/32, while dashed red lines denote isosurfaces of constant matter
proper time τ uniformly sampling [0, τend ] with δτ = τend /8. The thick red line is the star’s surface and the thick
magenta one is the star at the initial instant τ = 0. The orange zigzag line marks the curvature singularity. Solid
(resp. dashed) green lines are outgoing (resp. ingoing) radial null geodesics encountering the star’s surface at the
above selected values of τ . The thick black line is the black hole event horizon H and the dotted blue line is the
centered future inner trapping horizon T . In the left figure, thin solid grey to black lines mark isosurfaces of
constant values of the areal radius r(τ, χ) (labeled in units of m). [Figures generated by the notebook D.6.2]
14.3 Oppenheimer-Snyder collapse 557
3m
t̃=8m
r=
2
0
r=
3.6
0 t˜= 6m 3.4 10m
T
t̃ =
3.2
3m
r= 3.0
r=
4m
2 2.8 0
T
t˜= r=
4m
2.6
4m
r=
2.4
r=
4 5m 2.2 t̃ = 8m
t̃ =
2.0
0
2 1 0 1 2 3 4 5 6 2.0 2.5 3.0 3.5
X X
Figure 14.3: Spacetime diagrams of the Oppenheimer-Snyder collapse in Kruskal-Szekeres-type coordinates
(T, X). The colored curves are the same as in Fig. 14.2. In particular, the magenta curve represents the star at
τ = t̃ = 0, when the collapse starts. In addition, solid grey lines are hypersurfaces of constant r, uniformly
sampling [0, 8m] with δr = m and dashed lines are hypersurfaces of constant t̃, uniformly sampling [0, 16m]
with δ t̃ = 2m. The right figure is a zoom on the end of the collapse. [Figures generated by the notebook D.6.2]
surface of the star (thick solid red line) is initially tangent to the hypersurface r = 4m (solide
grey line), which is the geometrical translation of the collapse starting from rest. The part of
the diagram that lies to the right of the thick solid red line is identical to a part of the diagram
shown in Fig. 9.5, which is expected since both represent a piece of Schwarzschild spacetime.
A major difference with the Kruskal-Szekeres diagram of Fig. 9.5 is that the spacetime diagram
shown in Fig. 14.3 is not conformal in the stellar interior: the radial null geodesics are not
straight lines inclined by ±45◦ . The outgoing ones even run backwards in T in the left part of
the diagram, which means that causal relations cannot be simply inferred there.
Finally let us consider a compactified conformal diagram of the Oppenheimer-Snyder col-
lapse, i.e. a Carter-Penrose diagram (cf. Secs. 9.4 and 10.8.1). Thanks to the compactification,
such a diagram offers a view of the entire spacetime. It is then natural to consider maximal
spacetimes, i.e. spacetimes in which each geodesic is inextendible, being complete or termi-
nating at a singularity (cf. Sec. B.3.2). This is clearly not the case of the spacetime depicted in
the diagrams of Figs. 14.2 and 14.3: when run in the past direction, causal geodesics in the star
interior stop on the hypersurface τ = 0 (the magenta line). In particular, the free-falling matter
geodesics, along which τ is an affine parameter, are clearly incomplete and could a priori be
extended to τ < 0.
There are actually two ways to maximally extend the Oppenheimer-Snyder spacetime
considered up to now. The first one corresponds to a star in hydrostatic equilibrium for τ < 0
558 Black hole formation 1: dust collapse
Figure 14.4: Carter-Penrose diagram of the Oppenheimer-Snyder collapse generated by switching off the
pressure at τ = 0 in a star in hydrostatic equilibrium for τ ∈ (−∞, 0). The stellar surface is indicated by the red
curve and some outgoing radial null geodesics are drawn as green straight lines. The black hole event horizon H
is indicated by the black line, the shaded region being the black hole interior. The dashed blue line marks the
trapping horizon T discussed in Sec. 14.3.6, while the orange zigzag line marks the curvature singularity. The
maroon curve is the worldline of the remote static observer O considered in Sec. 14.4.
and in which the pressure drops suddenly to zero at τ = 0. This extension is “astrophysical”
insofar as it models the gravitational collapse of the core of a massive star. Of course the
equilibrium for τ < 0 requires some nonzero pressure inside the star. Therefore it cannot
be described by the Lemaître-Tolman equations discussed in Sec. 14.2; it should rather obey
the Tolman-Oppenheimer-Volkoff equations, which govern hydrostatic equilibria in spherical
symmetry (see e.g. Chap. 23 of Ref. [371]). The Carter-Penrose diagram corresponding to this
extension is drawn schematically (i.e. not by means of explicit coordinates) in Fig. 14.4. The
star seems to expand from the past timelike infinity i− , but this is an artifact of the drawing5 :
the star surface (red line) follows actually a curve r = const (= r0 ) for τ < 0. For τ ≥ 0,
the surface of the star appears as a vertical line, as in the left part of Fig. 14.2, while the
radius is actually decreasing. Actually, one may consider that this part of the Carter-Penrose
diagram is drawn by means of the conformal coordinates (η, χ) used in Fig. 14.2. Note that
the Carter-Penrose diagram of Fig. 14.4 corresponds to a maximally extended spacetime: all
geodesics are inextendible. Note as well that the exterior region is formed by parts of regions
MI and MII of Schwarzschild spacetime (compare with Fig. 9.10).
The second maximal extension of the Oppenheimer-Snyder spacetime is based on the
5
Recall that a Carter-Penrose diagram reflects the causal structure of spacetime but does not respect the metric
distances.
14.3 Oppenheimer-Snyder collapse 559
Figure 14.5: Carter-Penrose diagram of the Oppenheimer-Snyder collapse starting at the instant of time
symmetry (τ = 0) after an expanding phase from a past singularity. The legend is the same as in Fig. 14.4, with in
addition the past curvature singularity (bottom orange zigzag line) and the white hole region (bottom shaded
area), bounded by the past event horizon H − .
assumption that the proper time τ = 0 is an instant of time symmetry of a ball of pressureless
matter that was expanding from an initial past singularity. The spacetime for τ < 0 is then
exactly the time-reversed of that for τ > 0, as illustrated on the Carter-Penrose diagram of
Fig. 14.5. Such an extension is less “astrophysical” than the one considered above, but it has the
advantage to invoke only pressureless matter and hence to be entirely described by an exact
solution of the Einstein equation: the Friedmann-Lemaître-Robertson-Walker solution (14.45)
extended to η ∈ (−π, π) inside the star and matched to the Schwarzschild solution outside the
star. This maximal solution contains a white hole region, i.e. a region in which causal curves
arising from the past null infinity I − cannot penetrate (cf. Sec. 4.4.2). The white hole (null)
boundary is the past event horizon H − (cf. Fig. 14.5). The pressureless matter starts its life
by expanding from a past curvature singularity where r = 0, as a “mini big-bang”, the areal
radius being given by Eq. (14.34) with η ∈ (−π, 0) in the expansion phase and η ∈ (0, π) in the
collapse phase. Note that the exterior region is formed by parts of regions MI , MII and MIV
of the maximally extended Schwarzschild spacetime (compare with Fig. 9.10). Note also that
the star is not visible from the infinite past to the remote static observer O: it only appears
to O after the event denoted by a in Fig. 14.5. On the contrary, in the “astrophysical” model
discussed above, the star is always visible to O (cf. Fig. 14.4), despite it fades very rapidly after
the collapse phase has started to be observed, as we shall see in Sec. 14.4.
Remark 3: The time-symmetric extended solution considered above opens the path to pressureless
solutions with χs ∈ (π/2, π), i.e. with χs = π − arcsin 2m/r0 as the solution of the junction equation
p
(14.35). We disregarded these solutions in Sec. 14.3.2, after having noticed that they lead to the areal
radius r(τ, χ) being a decreasing function of χ for χ ∈ (π/2, χs ]. In other words, starting from the
center of the star at χ = 0 and keeping τ fixed, r(τ, χ) increases up to the maximum value rs (τ )/ sin χs ,
560 Black hole formation 1: dust collapse
Figure 14.6: Carter-Penrose diagram of a semiclosed world. See Fig. 14.5 for the legend.
achieved for χ = π/2, and then decreases to rs (τ ) for χ = χs . Such a behavior is possible if the star’s
surface meets Schwarzschild spacetime in a region of the latter where r decreases when moving away
from the surface (i.e. “to the right” if one places the star at the left of a Carter-Penrose diagram as in
Figs. 14.4-14.5). This happens in the region X̂ < 0 of the compactified Kruskal-Szekeres coordinates (cf.
Fig. 9.10). We may then place the star there; the corresponding Carter-Penrose diagram is shown in
Fig. 14.6. We note that the star exterior comprises the entire region MI of the extended Schwarzschild
spacetime, as well as parts of regions MII , MIII and MIV . Contrary to the solution for χs < π/2
shown in Fig. 14.5, it contains the bifurcation sphere S , where H meets H − (cf. Sec. 9.3.3). Such a
configuration is called a semiclosed world (cf. p. 149 of Ref. [531], Sec. 2.7 of Ref. [387] or Sec. 2.7.2 of
Ref. [202]), because at the limit χs → π, the matter region becomes an entire FRLW closed universe.
Note that the collapse phase (τ > 0) cannot be seen by a remote static observer, like O in Fig. 14.6, for it
is hidden below the event horizon H ; only some initial part of the expanding phase is visible to O on
the segment a − i+ of his worldline.
Historical note : The semiclosed world solutions have been introduced by Oscar Klein in 1961 [317]
and Yakov B. Zeldovich in 1962 [526]. They have been further studied by Igor D. Novikov in 1963 [386].
14.3.4 Final singularity
The collapse ends at η = π, since this value corresponds to the surface areal radius rs (τ ) = 0,
by virtue of Eq. (14.30). Equation (14.34) shows that this corresponds as well to r(τ, χ) = 0
for all particles inside the star. The matter proper time τ at the end of the collapse is given by
η = π in Eq. (14.34):
r r
π r03 r 3/2
0 m 3π
τend = a0 = π =π m=π 3 = , (14.48)
2 8m 2m sin χs 32ρ0
14.3 Oppenheimer-Snyder collapse 561
where the last equality has been obtained by using Eq. (14.41) to express m/ sin3 χs in terms of
the initial proper energy density ρ0 . We note that τend is a function of ρ0 only.
Remark 4: The above formulas for the collapse proper time τend are identical to those giving the
collapse time of a spherical ball of dust of mass m, initial radius r0 and initial mean mass density
ρ0 = 3m/(4πr03 ) in Newtonian gravity. This coincidence arises from the same cycloidal law for rs (τ )
in both general relativity and Newtonian gravity, as already discussed in Remark 1 on p. 551.
Similarly, setting η = π in Eq. (14.31), we get the value of the IEF coordinate t̃ at the end of
the collapse: r
r0 r0 r
0
t̃end = 2m π 1 + − 1 − ln −1 . (14.49)
4m 2m 2m
Example 1: For the compactness
√ m/r0 = 1/4, which is that considered in Figs. 14.2-14.3, the above
formulas yield τend = 2 2πm ≃ 8.89 m and t̃end = 4πm ≃ 12.57 m.
Example 2: Some numerical values of τend are given in Table 14.1 for various astrophysical objects, if
they happen to collapse by a sudden lack of force acting against gravitation. One sees that the Earth
would collapse in 15 minutes, while it would take half an hour for the Sun, 3 seconds for a white dwarf
and a tenth of millisecond for a neutron star. The similarity of the values for the Earth and the Sun,
which have very different masses and radii, is due to the closeness of their values of ρ0 .
Equation (14.39) with η → π immediately yields an infinite proper energy density at the
end of the collapse:
lim ρ(τ ) = +∞. (14.50)
τ →τend
Since ρ is a measurable quantity (by comoving observers), this signals some physical singularity.
This corresponds actually to the apparition of a curvature singularity in the part of spacetime
covering the interior of the star (cf. the discussion in Sec. 6.3.4). Indeed, from the interior metric
(14.45), one can evaluate6 curvature scalars like the Ricci scalar
24 6m
R := Rµµ = = , (14.51)
a20 (1 + cos η)3 rs (τ )3
the “square” of the Ricci tensor
576 36m2
Rµν Rµν = = (14.52)
a40 (1 + cos η)6 rs (τ )6
and the Kretschmann scalar defined by Eq. (6.44):
960 60m2
K := Rµνρσ Rµνρσ = = . (14.53)
a40 (1 + cos η)6 rs (τ )6
Remark 5: Although the components of the Riemann and Ricci tensors with respect to the coordinates
(η, χ, θ, φ) depend on χ (cf. the notebook D.6.3), the above three scalars are independent of it, in full
agreement with the homogeneity of the stellar interior at each instant η.
6
See the notebook D.6.3 for the computations.
562 Black hole formation 1: dust collapse
Earth Sun white dwarf neutron star
m [M⊙ ] 3 10−6 1 0.6 1.4
r0 [km] 6.37 103 6.96 105 8.0 103 12
m/r0 7.0 10−10 2.1 10−6 1.1 10−4 0.17
χs 3.7 10−5 2.1 10−3 1.5 10−2 0.63
ρ0 [kg m−3 ] 5.5 103 1.4 103 5.6 108 3.8 1017
τend 14 min 55 s 29 min 30 s 2.82 s 107 µs
τhb 14 min 55 s 29 min 30 s 2.82 s 75 µs
τend − τhb 6.7 10−11 s 22 µs 13 µs 32 µs
τend − τ∗ 2.0 10−11 s 6.6 µs 3.9 µs 10.4 µs
ρhb [kg m−3 ] 1.8 1029 1.8 1018 4.5 1018 1.4 1018
Table 14.1: Numerical values for the gravitational collapse of various astrophysical bodies, assuming that all
forces counterbalancing gravitation suddenly disappear at the proper time τ = 0. m is the gravitational mass
(constant during the collapse), r0 is the initial areal radius, m/r0 is the initial compactness, χs is the (constant)
value of the hyperspherical angle χ at the surface of the body [Eq. (14.36)], ρ0 is the initial proper energy density,
assuming that the body is homogeneous [Eq. (14.40) with τ = 0, or Eq. (14.41)], τend is the matter proper time at
the end of the collapse, when the central singularity appears [Eq. (14.48)], τhb is the matter proper time when the
black hole event horizon forms at the center of the body [Eq. (14.58)], τ∗ is the proper time when the horizon
encounters the infalling surface [Eq. (14.55)] and ρhb is the proper energy density at τ = τhb [Eq. (14.59)].
It is clear that the above three curvature invariants diverge when η → π, or equivalently,
when rs (τ ) → 0. Hence a curvature singularity appears in the stellar interior at the end of
the collapse. It is depicted by the orange zigzag segment in left spacetime diagram of Fig. 14.2.
This singularity is shrunk to the point (t̃, r) = (t̃end , 0) in the IEF diagram on the right part of
the figure, because rs (τend ) = 0. It is then connected to the curvature singularity at r = 0 of
Schwarzschild spacetime, depicted as the vertical orange zigzag segment.
Remark 6: The reader might be puzzled by the Kretschmann scalar (14.53) being different from that
of Schwarzschild spacetime, as given by Eq. (6.45): KSchwarz = 48m2 /r6 . Both have the same m2 /r6
structure but they differ by the numerical prefactor: 60 versus 48. This actually reflects the discontinuity
of the Riemann tensor between the stellar interior and the Schwarzschild exterior. This discontinuity is
enforced on the Ricci part of the Riemann tensor via the Einstein equation and the uniform density of
the star, the latter yielding a jump of the energy-momentum tensor T between the value (14.2) and zero
(vacuum exterior). The difference is even more dramatic on the Ricci scalar (14.51) and the square of the
Ricci tensor (14.52), since they are identically zero in the Schwarzschild exterior. In other words, the
Einstein equation implies that the metric tensor of the Oppenheimer-Snyder solution is not C 2 . For an
actual stellar core, there is a density gradient so that ρ = 0 at the surface and the metric tensor is at
least C 2 .
14.3 Oppenheimer-Snyder collapse 563
14.3.5 Black hole formation
Since the surface areal radius rs (τ ) is a monotonically decreasing function of τ (cf. Fig. 14.1),
there exists a unique value τ∗ of τ for which rs (τ∗ ) = 2m. For τ > τ∗ , rs (τ ) < 2m and the
exterior Schwarzschild spacetime contains a black hole event horizon H , located at r = 2m in
IEF coordinates (the vertical black segment in Fig. 14.2). The value of τ∗ is determined through
the corresponding conformal time η∗ via Eq. (14.30):
r0 m
(1 + cos η∗ ) = 2m ⇐⇒ cos η∗ = 4 − 1 = 2 sin2 χs − 1 = − cos(2χs ),
2 r0
where use has been made of Eq. (14.35). Given that 0 < η∗ < π and 0 < χs < π/2, the solution
is
η∗ = π − 2χs . (14.54)
Equation (14.30) yields then
π − 2χs + sin(2χs ) 2χs − sin(2χs )
τ∗ = m= 1− τend . (14.55)
sin3 χs π
3: For the compactness 1/4 considered in Figs. 14.2-14.3, one has χs = π/4, so that η∗ = π/2
Example √
and τ∗ = 2(π + 2)m ≃ 7.27 m ≃ 0.82 τend .
Example 4: Values of τ∗ for the various objects considered in Example 2 are given in Table 14.1. We
note that for the collapsing Earth, Sun and white dwarf, τ∗ is very close to τend , while for the collapsing
neutron star, τ∗ ≃ 0.9 τend . This reflects the smallness of χs for the first three objects, since Eq. (14.55)
implies τ∗ /τend ≃ 1 − 4χ3s /(3π) for χs ≪ 1.
For τ < τ∗ , one has rs (τ ) > 2m and the event horizon must be located inside the collapsing
star. To determine it, let us consider the outgoing radial null geodesics, which are drawn as
solid green curves in Fig. 14.2. In view of the metric (14.45), the equation of such a geodesic is
very simple in terms of the coordinates (η, χ): η = χ − χs + ηs , where the constant ηs is the
value of η when the geodesic reaches the surface of the star (χ = χs ). Radial null geodesics
for which ηs < η∗ emerge out of the star in a part of Schwarzschild spacetime where r > 2m,
so that they can espace to infinity. On the contrary, geodesics having ηs > η∗ emerge in the
black hole region of Schwarzschild spacetime; they are thus trapped and eventually end on the
curvature singularity at r = 0. The event horizon is generated by the null radial geodesics of
the limiting case: ηs = η∗ . Hence, inside the star, the equation of the black hole event horizon
H is η = χ − χs + η∗ . Given the value (14.54) for η∗ , we get
H : η = χ + π − 3χs . (14.56)
For a fixed value of (θ, φ), this is the equation of a straight line segment inclined by +45◦ with
respect to the horizontal (cf. Fig. 14.2, left).
The equation of H in terms of the internal IEF coordinate is obtained in parametric form,
with parameter χ ∈ [0, χs ] by substituting (14.56) for η in Eq. (14.31) providing t̃ and in
Eq. (14.34) providing r. One obtains then a curved segment in the (t̃, r) plane, which is drawn
in black in Fig. 14.2, right.
In view of Eq. (14.56), we may state:
564 Black hole formation 1: dust collapse
0.012
0.010
0.008
ρhb m 2
0.006
0.004
0 1π
9
2π
9
1π
3
χs
Figure 14.7: Proper energy density ρhb at the instant of the black hole formation at the center of the star, in
units of m−2 , as a function of the initial compactness angle χs . [Figure generated by the notebook D.6.2]
Property 14.5: birth and growth of the event horizon
• If χs ≤ π/3, or equivalently if m/r0 ≤ 3/8, the event horizon H appears at the
center of the star (χ = 0) at the conformal time
ηhb = π − 3χs (14.57)
or equivalently at the matter proper time
π − 3χs + sin(3χs ) 3χs − sin(3χs )
τhb = m= 1− τend , (14.58)
sin3 χs π
the label ‘hb’ standing for “horizon birth”.
• If χs > π/3, or equivalently if m/r0 > 3/8, the event horizon H is already present
in the star at τ = 0, at the location χ = 3χs − π.
In all cases, the horizon expands according to Eq. (14.56) until it reaches the surface of
the star (χ = χs ) at η = η∗ = π − 2χs or τ = τ∗ given by Eq. (14.55).
Example 5: Continuing with Example 3, i.e. with m/r0 = 1/4, or equivalently with χs = π/4, we
get ηhb = π/4, which corresponds to√the origin of the black straight segment drawn in Fig. 14.2, left.
Equation (14.58) gives τhb = (2 + π/ 2)m ≃ 4.22 m ≃ 0.48 τend .
Example 6: Values of τhb for the various objects considered in Examples 2 and 4 are given in Table 14.1.
As for τ∗ , we note that for the collapsing Earth, Sun and white dwarf, τhb is very close to τend . This
14.3 Oppenheimer-Snyder collapse 565
follows from τhb /τend ≃ 1 − 9χ3s /(2π) for χs ≪ 1. On the contrary, for the collapsing neutron star,
τhb ≃ 0.7 τend .
For χs ≤ π/3, the matter proper energy density at the horizon birth, i.e. at τ = τhb , is
obtained by substituting (14.57) for η into Eq. (14.39); using the identity (14.35) to express
m3 /r03 in terms of χs , we get
3 sin6 χs
ρhb := ρ(τhb ) = . (14.59)
4πm2 (1 − cos(3χs ))3
It is instructive to take the limit of small initial compactness m/r0 ≪ 1, which corresponds to
χs ≪ 1 by virtue of relation (14.36):
2 5 2
ρhb ≃ 2
1 + χs + O(χ4s ). (14.60)
243πm 4
We deduce from Eqs. (14.59)-(14.60) that
2 81
lim ρhb = ≃ 2.62 10−3 m−2 and lim ρhb = ≃ 1.26 10−2 m−2 .
χs →0 243πm2 χs →π/3 2048πm2
Between these two limits, ρhb has a monotonic behavior, as shown in Fig. 14.7. In particular,
one has ρhb < 0.02 m−2 . Hence
Property 14.6: matter density at the horizon’s birth
By selecting a sufficiently large mass m, the proper energy density of matter when the
black hole forms at the center of the collapsing star can be made arbitrarily small.
Example 7: Let us determine the mass m for which ρhb is as small as the density of water: ρhb =
103 kg m−3 . By restoring the G’s and the c’s, the density m−2 becomes c6 /G3 m−2 . We then get
m ≃ 4 107 M⊙ for small initial compactness (χs ≪ 1) and m ≃ 9 107 M⊙ for large initial compactness
(χs ∼ π/3).
The relative density increase at the horizon birth with respect to the initial value ρ0 =
m/(4/3 πr03 ) is obtained by setting η = π − 3χs in Eq. (14.39):
ρhb 8
= . (14.61)
ρ0 (1 − cos(3χs ))3
For small compactness, we get
ρhb 64
∼ . (14.62)
ρ0 s 729χ6s
χ →0
Hence ρhb /ρ0 is diverging very rapidly when χs → 0. On the contrary, for χs large, ρhb /ρ0 can
be moderate. Even, for χs = π/3, Eq. (14.61) leads to ρhb /ρ0 = 1, in agreement with τhb = 0
for that value of χs .
Example 8: Values of ρhb for the various objects considered in Examples 2, 4 and 6 are given in
Table 14.1. These values are all larger than the nuclear density ρnuc = 2 1017 kg m−3 . For the Earth, ρhb
is even ∼ 1012 ρnuc , in agreement with the diverging behavior (14.62). On the contrary, for the neutron
star, for which χs = 0.63, we have only ρhb /ρ0 ≃ 3.7. By comparing with Example 7, we note that all
the examples considered in Table 14.1 have a mass much smaller to that required for a low value of ρhb .
566 Black hole formation 1: dust collapse
13.0
3π
0.50 12.5
4
0.65
12.0
0.70
11.5
1.00
0.75
t̃/m
5π
11.0
η
10.5
10.0
1.50
1π
2
9.5
2.00
9.0
0 1π
8
1π
4 0.0 0.5 1.0 1.5 2.0 2.5 3.0
χ r/m
Figure 14.8: Spacetime diagrams of the Oppenheimer-Snyder collapse, as in Fig. 14.2, but zooming around
the trapped sphere S located at (η, χ) = (11π/16, 3π/16). The left diagram is drawn in terms of the conformal
coordinates (η, χ), while the right one2is
.50based on the Eddington-Finkelstein coordinates (t̃, r). See the legend of
Fig. 14.2 for the meaning of the various curves.
3.00
[Figures generated by the notebook D.6.2]
14.3.6 Trapped surfaces
The concept of trapped surface has been introduced in Sec. 3.2.3 (cf. Fig. 3.1). Let us show
that such surfaces appear at late times in the Oppenheimer-Snyder collapse. To this aim, we
consider a 2-dimensional surface S 3at .50 constant coordinates (η, χ) inside the collapsing star,
with η ̸= π and χ ̸= 0. Such a surface appears as a point in the spacetime diagrams of Fig. 14.8.
The surface S has the topology of a 2-sphere and is spanned by the coordinates (θ, φ). In
particular, it is a closed manifold (compact without boundary). The metric q induced by g on
S is readily obtained by setting dη = 0 and dχ = 0 in Eq. (14.45):
a20
(1 + cos η)2 sin2 χ dθ2 + sin2 θ dφ2 . (14.63)
q=
4
This is clearly a Riemannian metric, i.e. S is a spacelike 2-surface of (M , g). At each point of
S , there are two null directions orthogonal to S : the outgoing one, represented by a solid
green line in Fig. 14.8 and the ingoing one, represented by a dashed green line (see also Fig. 2.10).
Future-directed null vectors along these two directions are respectively
ℓ = ∂η + ∂χ and k = ∂η − ∂χ . (14.64)
The expansions of S along ℓ and k are given by Eq. (2.55) [see also Eq. (3.6) and Fig. 3.1]:
√ √
θ(ℓ) = Lℓ ln q and θ(k) = Lk ln q, (14.65)
14.3 Oppenheimer-Snyder collapse 567
where q is the determinant of the metric q with respect to any coordinate system of S that is
carried along with ℓ or k, such that (θ, φ), since θ and φ are constant along the integral curves of
√
ℓ and k (the radial null geodesics). From Eq. (14.63), we get q = (a20 /4)(1+cos η)2 sin2 χ sin θ.
√ √ √
Furthermore, in view of the components (14.64), Lℓ ln q = ℓµ ∂ ln q/∂xµ = ∂ ln q/∂η +
√ √ √ √ √
∂ ln q/∂χ and Lk ln q = k µ ∂ ln q/∂xµ = ∂ ln q/∂η − ∂ ln q/∂χ. We then get easily
cos (η/2) cos (χ + η/2) cos (η/2) cos (χ − η/2)
θ(ℓ) = 4 and θ(k) = −4 . (14.66)
(1 + cos η) sin χ (1 + cos η) sin χ
Given that η ∈ [0, π) and χ ∈ (0, χs ], with χs < π/2, one has 1 + cos η > 0, cos(η/2) > 0,
sin χ > 0 and cos(χ − η/2) > 0. It follows immediately that
θ(k) < 0 (14.67)
and the sign of θ(ℓ) is that of cos(χ + η/2). Given that the above ranges of η and χ imply
0 < χ + η/2 < π, we have θ(ℓ) < 0 ⇐⇒ χ + η/2 > π/2, i.e.
θ(ℓ) < 0 ⇐⇒ η > π − 2χ. (14.68)
Since S is a closed spacelike 2-surface, we conclude, in view of the definitions given in
Sec. 3.2.3:
Property 14.7: trapped surfaces
At late times, i.e. for η > π − 2χ, the spheres S defined by (η, χ) = const, with η < π
and χ > 0, are trapped surfaces (θ(ℓ) < 0 and θ(k) < 0). These spheres are marginally
trapped (θ(ℓ) = 0 and θ(k) < 0) for η = π − 2χ and untrapped (θ(ℓ) > 0 and θ(k) < 0) for
η < π − 2χ.
Example 9: The trapped surface S corresponding to (η, χ) = (11π/16, 3π/16) in the Oppenheimer-
Snyder collapse with r√ 0 = 4m is plotted as a brown dot in Fig. 14.8. According to Eq. (14.34), the areal
radius of S is rS = 2 2 sin(3π/16)(1 + cos(11π/16)) ≃ 0.70 m. The trapped character of S can be
read on Fig. 14.8: moving to the future along either the ingoing radial null geodesic (dashed green line)
or the outgoing one (solid green line) leads to spheres of smaller areal radius, and hence smaller area.
This is clear on the left diagram when noticing that both geodesics encounter the curve r = 0.65 m in
the immediate future of S . It is even more immediate on the right diagram since r is read directly on
the x-axis.
The set formed by all the marginally trapped spheres of fixed (η, χ) is the hypersurface T
defined by θ(ℓ) = 0. Its equation is obtained by saturating the inequalities in Eq. (14.68):
T : η = π − 2χ. (14.69)
As we shall see in Chap. 18, T is called a future inner trapping horizon. It is drawn as a
dotted blue line in Figs. 14.2 and 14.8. Since the slope of that line is −2 and (η, χ) are conformal
coordinates, it is clear from the left diagrams in these figures that T is a timelike hypersurface.
This is actually a generic feature of future inner trapping horizons, as we shall see in Chap. 18.
568 Black hole formation 1: dust collapse
We note from Figs. 14.2 and 14.8 that T forms at the surface of the collapsing star at the
very same instant where the event horizon H reaches the surface. Indeed, searching for the
intersection H ∩ T from the equations defining the hypersurfaces H and T , i.e. Eqs. (14.56)
and (14.69), we get (η, χ) = (π − 2χs , χs ), which is the 2-sphere representing the surface of
the collapsing star at η = π − 2χs .
The trapped surfaces S are located “above” T , i.e. at η > π − 2χ. The region containing
trapped 2-spheres with r = const actually extend through the collapsing star’s surface up to
the event horizon H . Indeed, the spacetime there is a piece of the region MII of Schwarzschild
spacetime (cf. Sec. 6.2.5), where every 2-sphere at t̃ = const and r = const is trapped (this is
obvious since both ingoing and outgoing radial null geodesics have decreasing areal radius
r towards the future in MII , cf. Fig. 6.3). Outside H , i.e. in the MI region of Schwarzschild
spacetime, there is no trapped surface.
Remark 7: The trapping horizon T is not the boundary of the region where there exist trapped surfaces
in the Oppenheimer-Snyder collapse, but only the boundary for trapped 2-spheres S centered around
χ = 0. Indeed, since the collapsing stellar interior is homogeneous, the trapped 2-spheres S can be
translated in χ at fixed η, leading to other trapped surfaces. One can then show that such “translated”
trapped 2-spheres exist in all the region η > π − χs − χ, which extends to the past of T , since for
χ ∈ [0, χs ), π − χs − χ < π − 2χ. Furthermore, there exist more complicated trapped surfaces even
below this limit, see Ref. [56] for details.
14.4 Observing the black hole formation
Let us consider a remote static observer O who receives electromagnetic radiation from the
surface of a spherically symmetric collapsing star. By static, it is meant that the worldline of O
is a field line of the static Killing vector ξ = ∂t = ∂t̃ in the exterior Schwarzschild region [cf.
Eq. (7.28)]. In particular, O evolves at a fixed value of (rO , θO , φO ) of the coordinates (r, θ, φ)
(cf. Fig. 14.9; see also Figs. 14.4 and 14.5). Without any loss of generality, we choose θO = π/2
and φO = 0. Furthermore, we assume that O is remote: rO ≫ m.
14.4.1 Apparent collapse freezing
Let us suppose that, at the matter comoving proper time τ1 , an electromagnetic signal (“flash”)
is emitted in the radial direction from the surface of the collapsing star at θ = π/2 and φ = 0
(i.e. towards observer O) and that a similar signal is emitted subsequently at proper time τ2 .
Since each signal is carried by an outgoing radial null geodesic (the green curves in Fig. 14.9),
the equation of which is (6.51) with respect to the IEF coordinates, we have
r
rec O
em rs (τi )
t̃i − rO − 4m ln − 1 = t̃ (τi ) − rs (τi ) − 4m ln − 1 , i ∈ {1, 2},
2m 2m
where t̃rec
i is the IEF coordinate t̃ of the reception event of signal no. i by observer O and t̃ (τi )
em
is the IEF coordinate t̃ of the emission event taking place at proper time τi ; t̃ (τi ) is given by
em
Eq. (14.31) with η being the function of τi defined implicitly by Eq. (14.30). We deduce from the
14.4 Observing the black hole formation 569
25
20
15
t̃/m 10
0
0 2 4 6 8 10
r/m
Figure 14.9: Spacetime diagram of the Oppenheimer-Snyder collapse, as in the right panel of Fig. 14.2,
illustrating the increasing delay in the reception by a remote static observer O (maroon worldline at r = 10 m) of
signals emitted radially from the surface of the collapsing star with a constant interval of matter proper time,
δτ = τend /8. See the caption of Fig. 14.2 for the legend of the various curves. [Figure generated by the notebook
D.6.2]
above relation the t̃-interval separating the receptions of signal 1 and 2 by O:
rs (τ1 ) − 2m
rec rec em em
t̃2 − t̃1 = t̃ (τ2 ) − t̃ (τ1 ) + rs (τ1 ) − rs (τ2 ) + 4m ln . (14.70)
rs (τ2 ) − 2m
Since O is remote (rO ≫ m), t̃rec
2 − t̃1 is nothing but the lapse of O’s proper time between the
rec
two reception events. The quantities t̃em (τ2 ) − t̃em (τ1 ) and rs (τ1 ) − rs (τ2 ) are always finite; on
the other side, the logarithm term diverges when rs (τ2 ) approaches 2m, i.e. when τ2 approaches
τ∗ [cf. Eq. (14.55)], so that
t̃rec rec
2 − t̃1 → +∞ when τ2 → τ∗ . (14.71)
In particular, if a sequence of signals is emitted from the star’s surface at a constant rate in
terms of the comoving proper time τ , the signals are received by the remote observer O with an
increasing delay as the collapse proceeds. The delay between two successive signals becoming
infinite, there exists a very last signal received by O. All subsequent signals of the emitted
sequence do not cross the event horizon, i.e. are captured into the black hole.
570 Black hole formation 1: dust collapse
Figure 14.10: Image of the collapsing star on the screen of the remote observer O, at a given instant tO .
Example 10: Consider Fig. 14.9, which depicts the Oppenheimer-Snyder collapse with r0 = 4m, or
equivalently χs = π/4. Starting from τ = 0, a uniform sequence of 8 signals is emitted from the √ star’s
surface, the proper time separation between two successive signals being δτ = τend /8 = π 2m/4
(cf. Example 1 on p. 561). One sees graphically that the signals are received by O with an increasing
separation δ t̃. The last received signal is the 7th one. The 8th signal is trapped in the black hole region
and ends on the curvature singularity.
We conclude from the above analysis:
Property 14.8: apparent freeze of the collapse
For a remote static observer, the collapse, as monitored by the reception of signals regularily
emitted from the star’s surface, appears to slow down and even to freeze completely when
the areal radius of the star tends towards 2m. In particular, the remote observer never
sees the star’s surface crossing the event horizon and does not perceive at all the final
singularity.
Historical note : The above behavior explains why the name frozen star was given to black holes
before the name black hole itself was coined (cf. the historical note on p. 109).
14.4.2 Shape and size of the images of the collapse
The image of the collapsing star is formed on the screen of observer’s O by photons emitted
from the star’s surface and following null geodesics of Schwarzschild spacetime (the spacetime
exterior to the star). In Sec. 14.4.1, we have considered only radial null geodesics. Discussing the
image requires to consider non-radial null geodesics as well. Such geodesics have been studied
in detail in Chap. 8. In particular, it has been shown that they are essentially characterized by a
single parameter: the impact parameter b := |L|/E introduced in Sec. 8.2.3. Radial geodesics
are those for which b = 0. In what follows, we shall work in terms of the Schwarzschild-Droste
14.4 Observing the black hole formation 571
coordinates (t, r, θ, φ), as in Chap. 8. These coordinates are regular in the Schwarzschild exterior
where the null geodesics propagate to the remote observer O and they make computations
slightly simpler than IEF coordinates do.
Due to spherical symmetry, the image of the star at a given instant tO of observer O’s proper
time is a disk and is invariant under rotation around the disk center. Since O is assumed to lie
far away in the weak-field region, tO coincides with the Schwarzschild-Droste time coordinate
at the location of O. As discussed in Sec. 8.5.1, the distance of a screen pixel from the disk center
is proportional to the impact parameter b of the null geodesic carrying the photon hitting this
pixel. Accordingly, all the properties of the image (redshift and intensity at a given point) are
functions of the pair (tO , b), where b can be identified with the radial coordinate in the screen
plane, centered on the image (cf. Fig. 14.10). The pixel at the disk center corresponds to a null
geodesic having b = 0, i.e. traveling radially in the direction θ = θO = π/2 and φ = φO = 0.
The image’s rim, i.e. the circle bounding the disk, is located at some maximal value of b, b⋆ (tO )
say. A great amount of information on b⋆ (tO ) can be inferred from the effective potential
analysis developed in Sec. 8.2.3. Indeed, along a given null geodesic L of impact parameter b,
the following constraint must hold in the exterior of the star:
1 r
b≤ p =p , (14.72)
U (r) 1 − 2m/r
where U (r) is the effective potential (8.14): U (r) := (1 − 2m/r)/r2 . This follows directly from
(dr/dλ̃)2 ≥ 0 in Eq. (8.11). Prior to the collapse, let us assume that the surface of the star is
static and located at r = r0 . Let us then denote by t∗∗O the value of tO at which the disk image
starts to shrink. This corresponds to null geodesics emitted from the star’s surface at the matter
proper time τ = 0 arriving at the disk p boundary. For tO ≤ tO , one has then r ≥ r0 all along
∗∗
L . If we assume r0 > 3m, then 1/ U (r) is increasing with r all along L , given that U (r)
is a decreasing function p on the interval (3m, +∞) (cf. Fig. 8.1). The constraint (14.72) is then
equivalent to b ≤ 1/ U (r0 ), so that the maximal value of b is b0 := 1/ U (r0 ) and we can
p
write:
r0
b⋆ (tO ) = b0 := p for tO ≤ t∗∗O and r0 > 3m. (14.73)
1 − 2m/r0
For tO ≤ t∗∗O , a null geodesic at the image’s rim, i.e having b = b⋆ (tO ), fullfils thus U (r0 ) = b ,
−2
which implies dr/dλ̃ = 0 at r = r0 according to Eq. (8.11). This means that the emission point
(r = r0 ) is a periastron of the geodesic: it corresponds to the minimal value of r along L ,
even if the latter is extended infinitely far in the past. This implies that L is tangent to the
hypersurface r = r0 at the emission point, i.e. that the photon is emitted in a direction tangent
to the star’s surface.
Remark 1: In terms of the compactness angle χs defined by Eq. (14.36), formula (14.73) takes a simple
form:
b⋆ (tO ) = r0 cos χs for tO ≤ t∗∗
O and r0 > 3m. (14.74)
In the non-relativistic limit, χs → 0 and we get b⋆ (tO ) = r0 , as expected (recall that the trully observed
quantity through a telescope is the small angle b̂ = b/rO [Eq. (8.118)], so that b̂ = r0 /rO at the disk’s
rim).
572 Black hole formation 1: dust collapse
One can also use the effective potential analysis to get the value of b⋆ (tO ) at late times,
more precisely for images formed by photons emitted after the collapsing star has shrunk
below the photon sphere p introduced in Sec. 8.2.3, which corresponds to rs (τ ) < 3m. Since
the function r 7→ 1/ U (r) is decreasing on (2m, 3m), reaches a minimum at r = 3m
and increases pon (3m, +∞) (cf. the plot of √ U (r) in Fig. 8.1), the constraint (14.72) becomes
b ≤ min(1/ U (r)) = bc , where bc = 3 3 m [Eq. (8.21)]. An example of photon emitted
at r < 3m and fulfilling this constraint is represented by the trajectory no. 3 in Fig. 8.1.
Furthermore, the constraint is actually a sufficient condition for any photon emitted outward
to reach infinity (cf. Fig. 8.1 once again). However, for b close to bc , photons perform a large
number of orbits very close to the photon sphere, this number being actually infinite for b = bc
(critical null geodesics), as discussed in Sec. 8.3.2 (cf. Figs. 8.7 to 8.9). As a consequence, the
elapsed time coordinate t along the corresponding null geodesic diverges as b → bc , so that the
image boundary b⋆ (tO ) = bc is reached only for tO → +∞:
√
lim b⋆ (tO ) = 3 3 m. (14.75)
tO →+∞
However, one can show (see e.g. Ref. [524] for details) that b⋆ (tO ) converges to bc exponentially
fast: b⋆ (tO ) − bc ∝ exp(−tO /bc ).
To summarize:
Property 14.9: apparent size of the collapsing star
• If the initial star is not ultracompact, i.e. if its arealp radius r0 is larger than 3m,
the radius
√ of the disk image shrinks from b 0 = r 0 / 1 − 2m/r0 [Eq. (14.73)] to
b∞ := 3 3 m [Eq. (14.75)]. The latter value is reached only asymptotically, √ i.e. for
tO → +∞, but the convergence is exponentially fast, so that, b⋆ (tO ) ≃ 3 3 m is a
very good approximation for tO > t∗∗ O + αm, where α ∼ 2bc /m ∼ 10.
• Collapsing stars that are initially ultracompact (r0 < 3m) generate
√ images of constant
size: the radius of the disk image remains fixed at b⋆ = 3 3 m.
Remark 2: The above results are not specific to pressureless collapse: they hold for any spherically
symmetric gravitational collapse.
Example 11: For our favorite√example with r0 =√4m (Figs. 14.2, 14.3, 14.8 and 14.9), the radius of
the image disk evolves from 4 2 m ≃ 5.66 m to 3 3 m ≃ 5.20 m, which constitutes a pretty modest
decrease: only 8%. This is of course because such a star has a large compactness. For the neutron star
considered in Table 14.1, which has a smaller compactness: m/r0 = 0.17, the shrinking factor is higher:
28%. For a concrete example of images of the gravitational collapse of an unstable neutron star into a
black hole computed by means of numerical relativity, see Fig. 4 of Ref. [492], where the modest decrease
of the disk image is very apparent.
The photons that form the star’s image on O’s screen at a given instant tO have impact
parameters b in the range 0 ≤ b ≤ b⋆ (tO ) and have been emitted from the star’s surface
at various proper times τ , depending on the value of b. Indeed, let us evaluate the elapsed
14.4 Observing the black hole formation 573
Schwarzschild-Droste time t between along a null geodesic L of impact parameter b between
the photon emission at the star surface and its reception by O. Thanks to spherical symmetry, it
suffices to consider photons traveling in the plane θ = π/2. The geodesic equations of motion
are then (8.9)-(8.11), from which we get
−1
dt 2m −1/2
= ϵr 1 − 1 − b2 U (r) (14.76a)
dr L r
dφ b −1/2
= ϵr ϵL 2
1 − b2 U (r) , (14.76b)
dr L r
where ϵL = ±1 indicates the sign of the conserved angular momentum L of L and ϵr = ±1 is
+1 (resp. −1) on portions of L where r increases (resp. decreases) towards the future. Note
that ϵL is constant along L . Regarding ϵr , it has to be +1 in the vicinity of observer O. Given
that a null geodesic of Schwarzschild spacetime has at most one r-turning point (cf. Sec. 8.2.4),
one has either ϵr = +1 along all L or ϵr = −1 just around the emission point, switching to
+1 after some periastron passage.
Let E be the event of emission at the star’s surface of a photon that reaches O’s screen at
the radial position b at the instant tO . Let us denote by respectively tem (b, tO ), τem (b, tO ) and
rsem (b, tO ), the Schwarzschild-Droste time coordinate of E , the infalling matter proper time
of E and the areal coordinate r of E . For a pressureless collapse, tem (b, tO ), τem (b, tO ) and
rsem (b, tO ) are related by the free-fall equations (7.41) [or equivalently (14.30)] and (7.42):
(r p r0 η
)
r0 h r0 i − 1 + tan
tem (b, tO ) = 2m −1 η+ (η + sin η) + ln p 2m
r0
2
η (14.77a)
2m 4m 2m
− 1 − tan 2
r
r03
τem (b, tO ) = (η + sin η) (14.77b)
8m
r0
rsem (b, tO ) = (1 + cos η) . (14.77c)
2
Note that we have chosen the origin of the Schwarzschild-Droste coordinate t to coincide with
the start of the collapse: t = 0 ⇐⇒ τ = 0 ⇐⇒ η = 0.
In terms of the star’s areal radius function rs (τ ), one has of course rsem (b, tO ) = rs (τem (b, tO )).
The above relations are valid for τem (b, tO ) ≥ 0, i.e. after the collapse has started. For
τem (b, tO ) < 0, we have, instead of (14.77),
−1/2
2m
tem (b, tO ) = 1 − τem (b, tO ) < 0 (14.78a)
r0
rsem (b, tO ) = r0 . (14.78b)
By integrating Eq. (14.76a) between the emission and reception points, we get
Z rO −1
2m −1/2
tO − tem (b, tO ) = 1− 1 − b2 U (r) dr + ∆, (14.79)
rsem (b,tO ) r
574 Black hole formation 1: dust collapse
where ∆ = 0 if there is no r-turning point (periastron) along the photon geodesic L and
Z rsem (b,tO ) −1
2m −1/2
∆=2 1− 1 − b2 U (r) dr (14.80)
rp r
if there is a one, located at r = rp < rsem (b, tO ). For a given observation instant tO and a
given position on the screen, measured by b, Eq. (14.79) must be solved to get tem (b, tO ). In
general, this is a difficult problem since the lower boundary of the integral, rsem (b, tO ), depends
on tem (b, tO ) in a complicated way, via the parametric system (14.77). However, for b = 0
(radial geodesic), one can compute the integral in Eq. (14.79) in an elementary way; taking into
account that ∆ = 0 for b = 0 (no r-turning point along a radial geodesic), we obtain
em
rs (0, tO ) − 2m
em
tem (0, tO ) = tO − rO + rs (0, tO ) + 2m ln . (14.81)
rO − 2m
This equation must still be solved to get tem (0, tO ) since rsem (0, tO ) is a function of tem (0, tO )
given parametrically by the system (14.77).
Another simplification of the general formula (14.79) occurs for tem (b, tO ) < 0 (emission
before the start of the collapse), since then rsem (b, tO ) = r0 [Eq. (14.78b)]. Equation (14.79)
provides then directly the value of tem (b, tO ) and we see that this is a decreasing function of b.
−1/2
Indeed, at fixed r, b 7→ (1 − b2 U (r)) is an increasing function of b, given that U (r) > 0.
Consequently, the first value of b for which tem (b, tO ) ≥ 0, i.e. for which the collapse starts to
be perceptible, is b = 0. In other words, the collapse manifests itself first at the center of the
disk image. Let us then denote by t∗O the corresponding value of tO . It is obtained by setting
tem (0, tO ) = 0 and rsem (0, tO ) = r0 in Eq. (14.81):
rO − 2m
∗
tO = rO − r0 + 2m ln . (14.82)
r0 − 2m
On the other side, the value of t∗∗
O , defined above as the instant when the disk image starts to
shrink, is obtained by setting b = b0 [the value (14.73)], tem (b, tO ) = 0 and rsem (b, tO ) = r0 in
Eq. (14.79):
Z rO −1
2m −1/2
∗∗
tO = 1− 1 − b20 U (r) dr. (14.83)
r0 r
Note that we have set ∆ = 0 in Eq. (14.79) because the geodesic delimiting the image’s rim is
precisely tangent to the surface in the static (pre-collapse) case, which implies that the photon’s
periastron coincides with the emission point.
By subtracting from Eq. (14.83) the integral expression of t∗O deduced from Eq. (14.79) by
setting b = 0, tem (b, tO ) = 0 and rsem (b, tO ) = r0 , we get, after substitution of Eq. (8.14) for
U (r), Z rO
dr
∗∗ ∗
tO − tO = b0 2
p . (14.84)
r0 r − b0 (1 − 2m/r) + r r2 − b20 (1 − 2m/r)
2 2
Since r2 − b20 (1 − 2m/r) ≥ 0 for r ≥ r0 , this expression shows that the time delay t∗∗
O − tO is
∗
positive.
14.4 Observing the black hole formation 575
14.4.3 Redshift in the image
Let Oem be an observer at the surface of the star comoving with the infalling matter. In particular
the proper time of O is τ and the r-coordinate of Oem is rs (τ ), as given by Eq. (14.30) for a
pressureless collapse. One defines the redshift z of a photon hitting the remote observer
screen’s O by z := (λ∞ − λem )/λem , where λem is the photon wavelength at emission, i.e.
with respect to observer Oem , and λ∞ is the photon wavelength measured by O. Thanks to the
Planck-Einstein relation (1.39): E = hν = h/(cλ), we have
Eem
1+z = , (14.85)
E∞
where Eem (resp. E∞ ) is the photon energy with respect to observer Oem (resp. O).
Frame of the comoving surface observer
The 4-velocity of the surface observer Oem is us = uts ∂t +urs ∂r . In any pressureless collapse from
rest, like the Oppenheimer-Snyderp one, Oem is in radial free fall from rest, so that uts = dt/dτ
is given by Eq. (7.34) with ε = 1 − 2m/r0 [Eq. (7.40)] and us = dr/dτ is given by Eq. (7.36)
r
with ε2 − 1 = −2m/r0 . Hence we get
r −1 r
2m 2m 2m 2m
us = 1 − 1− ∂t − − ∂r , (14.86)
r0 rs rs r0
where rs is a shortcut for rs (τ ). It is useful to expand us onto the frame of the local static
observer whose position coincides with the star surface at a given instant. The orthonormal
frame of the static observer is
−1/2 1/2
2m 2m 1 1
e(t) = 1 − ∂t , e(r) = 1 − ∂r , e(θ) = ∂θ , e(φ) = ∂φ .
r r r r sin θ
(14.87)
We deduce then from (14.86) that
(14.88)
us = Γ e(t) − V e(r) ,
with s s
2m 2m 2m
− 1 1−
V := rs r0
2m and Γ := √ = r0
2m (14.89)
1− r0 1−V2 1− rs
By comparing with the generic formulas (1.35)-(1.36), we conclude that V is the norm of the
velocity V = −V e(r) of Oem with respect to the local static observer and Γ is the corresponding
Lorentz factor.
Remark 3: Formula (14.89) implies limrs →2m V = 1. We recover then a well known result: with respect
to a static observer located just outside the event horizon, a radially free-falling body is falling at the
speed of light.
(t)
Observer Oem is endowed with an orthonormal frame (us , ns , es2 , es3 ), where ns = ns e(t) +
(r)
ns e(r) is a unit spacelike vector normal to the star surface and oriented towards the exterior,
576 Black hole formation 1: dust collapse
while the vectors (es2 , es3 ) are unit spacelike vectors tangent to the star surface. The components
(t) (r)
(ns , ns ) of ns are obtained from the two conditions us · ns = 0 and ns · ns = 1, along with
(14.88):
(14.90)
ns = Γ −V e(t) + e(r) .
Remark 4: Equations (14.88) and (14.90) simply express that the orthonormal pairs (us , ns ) and
(e(t) , e(r) ) are connected by a Lorentz boost of velocity parameter −V .
General formula for the redshift
Let us consider a photon emitted from the surface of the collapsing star. The photon’s energy-
momentum vector p can be decomposed with respect to Oem ’s frame according to Eqs. (1.31)
and (1.30):
p = Eem us + Pem , (14.91)
where Pem is a vector orthogonal to us representing the photon’s linear momentum as measured
by Oem . The energy Eem is then obtained as [cf. Eq. (1.22)]
Eem = −us · p (14.92)
Due to spherical symmetry, we may restrict the study to photons travelling in the equatorial
plane θ = π/2. The photon’s energy-momentum vector p is then given by Eq. (8.121), which
involves the impact parameter b. Once re-expressed in terms of the static observer frame (14.87),
this formula becomes
" −1/2 #
2m b
(14.93)
p
p = E∞ 1− e(t) + ϵr 1 − b2 U (r) e(r) + ϵL e(φ) ,
r r
where ϵr = ±1 and ϵL = ±1. Far from the star, one has of course ϵr = +1, but, as we shall
see below, ϵr = −1 may occur close to the star, which corresponds to a photon emitted in the
inward direction (decreasing r); in this case, the photon will reach a periastron, where ϵr is
switched to +1.
The scalar product in Eq. (14.92) is easily evaluated from expressions (14.88) and (14.93),
owing to the orthonormality of the frame (e(α) ); one gets
−1/2
Eem 2m
(14.94)
p
1+z = =Γ 1− 1 + ϵr V 1 − b2 U (rs ) .
E∞ rs
Substituting expressions (14.89) for V and Γ, as well as (8.14) for U (r), yields
−1 "r r s #
2m 2m 2m 2m b2 2m
1+z = 1− 1− + ϵr − 1− 2 1− . (14.95)
rs r0 rs r0 rs rs
This formula gives the redshift of a photon that hits the screen of the remote observer O at the
impact parameter b at the instant tO . The quantity rs has to be replaced by rsem (b, tO ) as given
by the system (14.77) for the pressureless collapse, where tem (b, tO ) is given by Eq. (14.79).
14.4 Observing the black hole formation 577
Remark 5: Equation (14.94) is valid for any spherically symmetric gravitational collapse, provided that
V is the velocity of the surface with respect to the local static observer as defined by Eq. (14.88) and Γ
is the corresponding Lorentz factor. On the contrary, Eq. (14.95) holds only for a pressureless collapse
starting from rest.
Emission angle
The direction of emission in the frame of the comoving observer Oem is characterized by the
angle Θem between the photon’s linear momentum Pem and the normal to the surface ns .
Considering that the collapsing matter is opaque to radiation, the maximum inclination of an
emitted photon capable to reach the remote observer O is Θem = π/2 (emission tangent to the
surface). Hence any photon received by O has necessarily
π
0 ≤ Θem ≤ . (14.96)
2
By definition, Θem is given by
ns · Pem
cos Θem = √ . (14.97)
Pem · Pem
√
Now, from Eq. (14.91), we have Pem · Pem = Eem (using that p is a null vector) and ns · Pem =
ns · p. Evaluating this last scalar product by means of Eqs. (14.90) and (14.93) and substituting
expression (14.94) for Eem /E∞ , we get
p
V + ϵr 1 − b2 U (rs )
cos Θem = p . (14.98)
1 + ϵr V 1 − b2 U (rs )
This relation can be easily inverted to get
cos Θem − V
(14.99)
p
ϵr 1 − b2 U (rs ) = ,
1 − V cos Θem
which allows us to express b in terms of Θem :
√
1 − V 2 sin Θem
b= p . (14.100)
U (rs )(1 − V cos Θem )
Remark 6: For V = 0, i.e. when the comoving observer coincides with the local static observer (prior
to the start of the collapse), Eqs. (14.98) and (14.100) reduce respectively to Eqs. (8.126) and (8.127) taking
into account the change of notations rs ↔ rem and Θem ↔ |η|.
We may also substitute Eq. (14.99) into Eq. (14.94) to get an expression of the redshift in
terms of Θem :
−1/2
2m 1
1+z = 1− . (14.101)
rs Γ (1 − V cos Θem )
For a pressureless collapse, Γ and V are given by Eq. (14.89) and the above relation simplifies to
1
1+z = q q . (14.102)
2m 2m 2m
1− r0
− rs
− r0
cos Θem
578 Black hole formation 1: dust collapse
Central redshift
The redshift zc at the center of the image of a pressureless collapse is obtained by setting b = 0
(which implies ϵr = +1) in Eq. (14.95):
−1 "r s #
2m 2m 2m 2m
1 + zc = 1 − em 1− + − , (14.103)
rs (0, tO ) r0 rsem (0, tO ) r0
where we have used rsem (0, tO ) for rs . One obtains an equivalent expression if one set Θem = 0
(radial emission) in Eq. (14.102), in agreement with the property b = 0 ⇐⇒ Θem = 0, which
follows from relation (14.100).
For tem (0, tO ) ≤ 0, rsem (0, tO ) = r0 and formula (14.103) reduces to 1+zc = (1−2m/r0 )−1/2 ,
which is the standard expression for the redshift at infinity of a photon emitted by a static
observer at r = r0 in the Schwarzschild spacetime.
For rsem (0, tO ) → 2m, formula (14.103) yields
r −1
2m 2m
1 + zc ∼ 2 1 − 1− , (14.104)
rs →2m r0 rs (0, tO )
which implies z → +∞ for rsem (0, tO ) → 2m.
The elapsed proper time of observer O since the reception of the central photon emitted at
τ = 0 (start of the collapse) is tO − t∗O . We get, from Eqs. (14.81) and (14.82),
r em
∗ em 0
rs (0, tO )
tO − tO = tem (0, tO ) − rs (0, tO ) + r0 + 2m ln − 1 − 2m ln −1 .
2m 2m
When rsem (0, tO ) → 2m, two terms in the right-hand side of the above equation diverge: the
last logarithm and the first term, tem (0, tO ), due to the singularity of Schwarzschild-Droste
coordinates at r = 2m. We can express the latter divergence in terms of a logarithm involving
rsem (0, tO ) thanks to Eq. (6.32): tem (0, tO ) = t̃em (0, tO ) − 2m ln [rsem (0, tO )/2m − 1] + K, where
t̃em (0, tO ) is the IEF coordinate of the emission event of the photon that reaches the center of
O’s screen at the instant tO and K is a constant. t̃em (0, tO ) is finite (cf. Fig. 14.9) and we get
r em
∗ em 0
rs (0, tO )
tO − tO = t̃em (0, tO ) − rs (0, tO ) + r0 + 2m ln − 1 + K −4m ln −1 .
| {z 2m } 2m
A
The terms denoted by A are bounded when rsem (0, tO ) → 2m, so that
em
∗ rs (0, tO ) 2m ∗
tO − tO ∼ −4m ln − 1 ⇐⇒ 1 − em ∼ e−(tO −tO )/(4m) .
rs →2m 2m rs (0, tO ) O
t →+∞
Combining with Eq. (14.104), we obtain the expression of the central redshift at late times:
r
2m (tO −t∗O )/(4m)
1 + zc ∼ 2 1 − e . (14.105)
tO →+∞ r0
Hence we conclude:
14.4 Observing the black hole formation 579
Property 14.10: exponential growth of the central redshift
The photons arriving at the center of the image (b = 0) suffer an increasing redshift. At late
times, the redshift grows exponentially with a characteristic time scale 4m. Equivalently,
the energies of the central photons vanish exponentially fast.
Redshift at the image’s rim
We have the following characterization of the image’s rim:
Property 14.11: photons forming the image’s rim
The image’s rim, or limb, at b = b⋆ (tO ) (cf. Sec. 14.4.2), is formed by photons emitted
tangentially to the star’s surface. In other words, the image’s rim is formed by photons
emitted in the direction Θem = π/2 with respect to the normal in the frame of the comoving
observer Oem .
We have demonstrated this property for tO ≤ t∗∗ O , i.e. prior to the collapse, in Sec. 14.4.2. We
shall admit that it remains true for tO > tO .
∗∗
One has then cos Θem = 0 for b = b⋆ (tO ) and Eq. (14.98) implies ϵr = −1 as soon as V > 0,
i.e. when the collapse has started. Hence these photons are emitted in the inward direction.
Their geodesics have then necessarily a r-turning point (periastron), after which r is increasing
monotonically (ϵr = +1) until reaching observer O.
The redshift z⋆ at the image’s rim is obtained by setting Θem = π/2 in Eq. (14.101):
−1/2
2m
1 + z⋆ = Γ −1
1 − em , (14.106)
rs (b⋆ , tO )
where b⋆ stands for b⋆ (tO ). For pressureless collapse, Γ can be substituted by expression (14.89);
this makes the terms involving rsem (b⋆ , tO ) cancel out. Hence
Property 14.12: constant redshift at the image’s rim
For a pressureless collapse starting from rest, the redshift at the image’s rim keeps a
constant value:
−1/2
2m
1 + z⋆ = 1 − . (14.107)
r0
This rather surprising result can be understood by considering the general expression (14.106):
1+z⋆ is the product of two competing factors: the “static” gravitational redshift (1−2m/rs )−1/2 ,
which is always larger than one and diverges as rs → 2m, and the Doppler blueshift factor
Γ−1 , which is always lower than one and tends to zero as rs → 2m (since V → 1 in that limit).
The Doppler blueshift is of course due to the rim’s photons being emitted inward, which is the
direction of the matter motion.
580 Black hole formation 1: dust collapse
Redshift profile
The redshift across the image, i.e. z as a function of b at fixed tO , is given by Eq. (14.94) for a
generic collapse and by Eq. (14.95) for a pressureless collapse from rest. However, one has to
solve for rs (b, tO ) to make these formulas fully explicit. Doing so (see e.g. Fig. 2 of Ref. [333]
or Fig. 6 of Ref. [524]), one obtains a redshift which is maximal at the image’s center (b = 0),
where it is given by formulas (14.103), and increases exponentially with time [Eq. (14.105)].
The redshift decreases with b until a small annulus near the image’s rim (b = b⋆ (tO )), where it
takes the constant value (14.107). The width of this annulus decays to zero with time.
14.4.4 Image brightness
Beside the redshift, the image is characterized by its brightness. Each point in the image
corresponds to a direction in observer O’s sky. We may then think of a small solid angle dΩ
around this direction as a “pixel” in the image. The brightness of this pixel in the frequency
bandwidth [ν, ν + dν] is represented by the energy flux (energy per unit time per unit area)
dF given by
dF = Iν dΩ dν, (14.108)
where Iν is the specific intensity of the electromagnetic radiation from the collapsing star7 .
The specific intensity Iν depends on the radiation frequency ν as measured by observer O,
as well as on O’s proper time tO and the direction in O’s sky. Taking the origin at the center
of the image, the direction is defined by the small angle b̂ = b/rO and some azimuthal angle
ϕ ∈ [0, 2π). The solid angle increment dΩ can be then written dΩ = sin b̂ db̂ dϕ ≃ b̂ db̂ dϕ.
Due to the rotation symmetry of the image, Iν is independent of ϕ. We may then write
Iν = Iν (b, tO , ν), using b to represent b̂ = b/rO .
A fundamental property of the specific intensity is that the ratio Iν /ν 3 is constant along
a given light ray. In particular, it does not depend on the distance between the source and
the observer. This property follows from Etherington’s reciprocity theorem which basically
states that, in any spacetime, identical area elements carried by two observers in arbitrary
motion subtend identical solid angles when seen by the other observer, up to some redshift
factor (1 + z)2 . This theorem can be proven in the framework of geometrical optics in curved
spacetime (cf. e.g. Sec. 3.6 of Ref. [448], Sec. 6 of Ref. [183] or Sec. 2 of Ref. [412]) or by applying
Liouville’s theorem expressing the constancy of the photon distribution function f along a
photon trajectory in phase space, given that Iν /ν 3 = h4 f , where h is Planck constant (see e.g.
Sec. 22.6 of Ref. [371]). We deduce from the constancy of Iν /ν 3 that
Iν (b, tO , ν) = (1 + z)−3 Iνem (Θem , tem , νem ) , (14.109)
where z = z(b, tO ) is the redshift at the location b on the image at time tO as given by Eq. (14.94)
or (14.95), Θem is the emission angle expressed in terms of (b, tO ) by Eq. (14.98), tem = tem (b, tO )
is the emission coordinate time expressed via Eq. (14.79) and νem is the emission frequency. By
the very definition of the redshift, νem is related to ν by νem = (1 + z)ν [cf. Eq. (14.85)]. The
7
The notation Iν follows a standard convention in the literature, the subscript ν being merely an indicator to
distinguish this quantity from the intensity I, which is integrated over all frequencies.
14.4 Observing the black hole formation 581
function Iνem (Θem , tem , νem ) — the specific intensity at the source — depends on the model of
emission assumed for the surface of the collapsing star.
The intensity I is the specific intensity integrated over all the frequencies:
Z +∞
I(b, tO ) = Iν (b, tO , ν) dν (14.110)
0
Given expression (14.109) and ν = (1 + z) νem , we get
−1
Z +∞
I(b, tO ) = (1 + z)−4
Iνem (Θem , tem , νem ) dνem . (14.111)
0
In view of the central redshift behavior (14.105), we deduce from this formula that the intensity
at the center of the image decreases exponentially with a characteristic time equal to m:
∗
I(0, tO ) ∼ I0 e−(tO −tO )/m . (14.112)
tO →+∞
The specific flux of the image is obtained by integrating the flux element (14.108) over the
solid angle dΩ. Using dΩ = b̂ db̂ dϕ = b/rO2 db dϕ, we get
2π b⋆
Z
Fν (tO , ν) = 2 Iν (b, tO , ν) b db. (14.113)
rO 0
The flux of the image is the specific flux integrated over all the frequencies:
Z +∞
2π b⋆
Z
F (tO ) = Fν (tO , ν) dν = 2 I(b, tO ) b db. (14.114)
0 rO 0
It can be shown (see e.g. Refs. [12, 455]) that, independently of the emission model, the flux is
decaying exponentially according to
∗
√
F (tO ) ∼ F0 e−(tO −tO )/(3 3m)
. (14.115)
tO →+∞
Historical note : The first computation of the appearance of a gravitational collapse as seen by a
remote observer has been performed by Mikhail A. Podurets in 1964 [415]. He obtained the exponential
decay formula (14.115) for the flux, up to an erroneous factor 2 in factor of tO − t∗O . This has been
corrected by William L. Ames and Kip S. Thorne in 1968 [12]. Podurets considered only the image flux,
which is relevant for a remote observer who cannot resolve the collapsing star. The intensity at the
center of the image has been derived by Yakov B. Zeldovich and Igor D. Novikov in 1964 [529], who
obtained the exponential decay formula (14.112). The computation of the intensity at each point of
the image has been achieved by Ames and Thorne in their 1968 study [12], who showed that (i) the
image is brightest at its rim, (ii) the redshift is constant there and (iii) the width of this peripheral bright
region is decaying exponentially with time. In 1969, Jack Jaffe [296] pointed out that Ames and Thorne
considered only photons with an emission angle Θ′em ≤ π/2 in the local static observer frame, while the
limiting condition should be Θem ≤ π/2 in the comoving observer frame [Eq. (14.96)]. This encompasses
extra photons: those that are emitted inward, as shown above. In 1979, Kayll Lake and R. C. Roeder
[333] recomputed the intensity with this hypothesis and concluded that the main properties found by
Podurets, Ames and Thorne are not significantly altered.
582 Black hole formation 1: dust collapse
14.5 Going further
This chapter dealt with the collapse of a spherically symmetric and homogeneous ball of some
pressureless fluid (“dust”). A first generalization amounts to allowing for non-homogeneous
density profiles, keeping the vanishing pressure, see e.g. Ref. [414]. More realistic models
shall take into account pressure, rotation and possibly other physical processes, like neutrino
emission. Although very relevant from an astrophysical point of view, rotation significantly
complicates the problem of gravitational collapse, since gravitational waves are emitted in this
case. This prevents the exterior spacetime to be described by an exact solution, such as the
Schwarzschild one8 . Only numerical solutions are known; see e.g. Chap. 9 of Ref. [456] or
Chaps. 8, 10 and 14 of Ref. [44].
8
Note that there is no equivalent to the Jebsen-Birkhoff theorem (Property 14.4) for a non-spherically symmetric
(e.g. flattened) body; in particular, the vacuum metric outside the body is not Kerr metric.
Chapter 15
Black hole formation 2: Vaidya collapse
Contents
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583
15.2 The ingoing Vaidya metric . . . . . . . . . . . . . . . . . . . . . . . . . 583
15.3 Imploding shell of radiation . . . . . . . . . . . . . . . . . . . . . . . . 588
15.4 Configurations with a naked singularity . . . . . . . . . . . . . . . . . 602
15.5 Going further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
15.1 Introduction
Having investigated the gravitational collapse of a star, modeled as a ball of dust, in the
preceding chapter, we move to a much less astrophysical scenario: the formation of a black
hole by the collapse of a spherical shell of pure radiation (no matter!). Albeit quite academic,
this process illustrates various features of black hole birth and dynamics, in a way somewhat
complementary to the collapse of a dust ball. The model is based on an exact solution of the
Einstein equation sourced by a pure radial electromagnetic flux, known as Vaidya metric, which
we present first (Sec. 15.2). Then, we introduce the model describing the implosion of a radiation
shell and study the black hole formation via such a process (Sec. 15.3). Finally, we focus on a
subclass of models giving birth a naked singularity, i.e. models in which the central curvature
singularity is visible to remote observers located outside the black hole region (Sec. 15.4).
15.2 The ingoing Vaidya metric
15.2.1 General expression
Let us consider a spherically symmetric spacetime (M , g) described by coordinates (v, r, θ, φ)
such that v ∈ R, r ∈ (0, +∞), θ ∈ (0, π) and φ ∈ (0, 2π), (θ, φ) being standard spherical
584 Black hole formation 2: Vaidya collapse
coordinates on S2 and r being the areal radius associated with spherical symmetry (cf. Sec. 6.2.2).
The ingoing Vaidya metric is the metric tensor
2M (v)
dv 2 + 2 dv dr + r2 dθ2 + sin2 θ dφ2 , (15.1)
g =− 1−
r
where M (v) is a real-valued function of v. We immediately notice that this expression strongly
resembles that of the Schwarzschild metric expressed in the null ingoing Eddington-Finkelstein
coordinates, as given by Eq. (6.29). Actually, the only difference is the constant m in Eq. (6.29)
replaced by the function M (v) in Eq. (15.1). We may even say that the Schwarzschild metric is
the special case M (v) = const of the ingoing Vaidya metric.
A key property of the ingoing Vaidya metric is
Property 15.1: null hypersurfaces v = const and their normals
The hypersurfaces v = const are null (i.e. v is a null coordinate); a normal to them is the
null vectora :
−
→ →
−
k := −dv = −∇v ⇐⇒ k := −dv. (15.2)
Moreover, k is equal to minus the vector ∂r of coordinates (v, r, θ, φ):
∂
k=− , (15.3)
∂r v,θ,φ
where we have rewritten ∂r as ∂/∂r|v,θ,φ to distinguish it from the vector ∂r of the IEF
coordinates, to be introduced in Sec. 15.2.2.
a
in index notation: k α := −g αµ ∂µ v = −∇α v ⇐⇒ kα := −∂α v.
Proof. Let Σv be a hypersurface defined by v = const. The metric induced by g on Σv is
obtained by setting dv = 0 in Eq. (15.1): g|Σv = r2 dθ2 + sin2 θ dφ2 . This metric has clearly
the signature (0, +, +), i.e. it is degenerate, hence Σv is a null hypersurface (cf. Sec. 2.2.2).
By construction, k is normal to Σv . It is thus a null vector. Besides, the metric dual of the
coordinate vector field ∂r is the 1-form ∂r = gµr dxµ = gvr dv = dv = −k, which proves
Eq. (15.3).
Property 15.2: time orientation of spacetime
Since dv is nowhere vanishing, k is a nonzero null vector field on M . We then set the time
orientation of (M , g) by declaring that k is future-directed (cf. Sec. 1.2.2).
As shown in the notebook D.6.4, the Ricci tensor of g takes a simple form:
2M ′ (v)
R= dv ⊗ dv, (15.4)
r2
15.2 The ingoing Vaidya metric 585
where M ′ (v) stands for the derivative of the function M (v). The Ricci scalar R = g µν Rµν =
(2M ′ (v)/r2 ) ∇µ v∇µ v = (2M ′ (v)/r2 ) kµ k µ is identically zero, since k is a null vector. A
consequence is
Property 15.3: Vaidya metric as a solution of the Einstein equation
The ingoing Vaidya metric (15.1) is a solution of the Einstein equation (1.40) with Λ = 0
and with the energy-momentum tensor
M ′ (v)
T = k⊗k . (15.5)
4πr2
Remark 1: We have already noticed that Vaidya metric reduces to Schwarzschild metric for M (v) =
const. This corresponds to M ′ (v) = 0, so that Eq. (15.5) reduces to T = 0 and we recover that
Schwarzschild metric is a solution of the vacuum Einstein equation.
The tensor (15.5) has the same structure as the energy-momentum tensor of the dust
model considered in Chap. 14: Tdust = ρ u ⊗ u [Eq. (14.2)]. The main difference is that u is
a timelike vector (the dust 4-velocity), while k is a null vector. For this reason, the energy-
momentum tensor (15.5) is sometimes referred to as a null dust model [416]. It corresponds
physically to the energy-momentum tensor of some monochromatic electromagnetic radiation
in the geometrical optics approximation (see Box 22.4 of Ref. [371]): Trad = q K ⊗ K, where
→
−
q ≥ 0 and K := ∇Φ is the wave vector, Φ being the rapidly-varying phase in the geometrical
optics decomposition A = Re(eiΦ a) of the electromagnetic potential 1-form A. The quantity
q is related to the energy density ε of the electromagnetic field as measured by an observer
O of 4-velocity u by ε = ω 2 q, where ω = −K · u is the frequency of the electromagnetic
radiation as measured by O. Maxwell equations imply that K is a null vector and that it is
geodesic: ∇K K = 0. The geodesic integral curves of K are nothing but the light rays of the
geometrical optics framework. The geodesic character also holds for k as defined by Eq. (15.2),
since k is normal to the null hypersurfaces v = const: the integral curves of k are the null
geodesic generators of these hypersurfaces (cf. Sec. 2.3.3); actually, we have exactly
∇k k = 0. (15.6)
This follows from Eq. (2.22) with ℓ = k and κ = 0 by virtue of Eq. (2.21) with ρ = 0 implied by
→
−
Eq. (2.11) given that u = v and k = −∇v [Eq. (15.2)]. Equations (15.6) and (15.3) imply that
the curves (v, θ, φ) = const are null geodesics, λ := −r is an affine parameter along them and
k is the corresponding tangent vector.
Another condition for the identification of T with Trad is that the coefficient in front
of k ⊗ k in (15.5) is non-negative, since q ≥ 0 in Trad . This constraint can also be seen
as the null energy condition (2.95) introduced in Sec. 2.4.2: for any null vector ℓ, we have
T (ℓ, ℓ) = M ′ (v)/(4πr2 ) (k · ℓ)2 and hence T (ℓ, ℓ) ≥ 0 ⇐⇒ M ′ (v) ≥ 0. In other words,
the function M (v) must be monotically increasing. Then, we can set k = αK, where α is a
constant, to have a perfect match of (15.5) with the electromagnetic radiation energy-momentum
tensor Trad . To summarize:
586 Black hole formation 2: Vaidya collapse
Property 15.4: Vaidya metric sourced by electromagnetic radiation
Provided that M (v) is an increasing functiona , the ingoing Vaidya spacetime (M , g) is
generated by a spherical symmetric electromagnetic radiation within the geometrical
optics approximation. The corresponding light rays are the ingoing radial null geodesics
L(v,θ,φ)
in
defined by v = const, θ = const and φ = const. These geodesics admit k as the
tangent vector associated with their affine parameter λ := −r.
a
by increasing, it is meant strictly increasing (M ′ (v) > 0) or locally constant (M ′ (v) = 0).
Historical note : The Vaidya metric has been actually first derived by Henri Mineur in 1933 [367], as
the solution of the Einstein equation for the exterior of a spherically symmetric body, the “mass” of
which “is varying” due to the radiation of an “energy flux of light”. It thus corresponds to the outgoing
version of the Vaidya metric, while the version (15.1) considered here is ingoing. More precisely, the
solution given by Mineur reads1
du2 − dv 2
2M (x)
2
−ds = 2 dx dr + r 2
− 1− dx2 . (15.7)
u2 r
The coordinate x is the opposite of a retarded time (x = −u in our notations), while (u, v) are coordinates
on the 2-sphere S2 (submanifold (x, r) = const) such that the standard (round) metric of S2 is q =
u−2 −du2 + dv 2 (cf. the unnumbered equation at the top of p. 35 of Mineur’s article [367]). This looks
quite surprising, given that u−2 −du2 + dv 2 is rather the metric of the 2-dimensional anti-de Sitter
space expressed in a variant of Poincaré coordinates (compare Eq. (13.66) with T = v and R = 1/u). In
particular the signature is (−, +), while it should be (+, +) for S2 ! It turns out that the coordinate u
used by Mineur is pure imaginary, i.e. Mineur wrote the metric of S2 via a kind of Wick rotation of the
metric of the hyperbolic plane 2 2
H , the latter being (dX + dY )/Y : setting v = X and u 2= iY , one
2 2 2
gets −q = u −2 du − dv . If one restores the standard spherical coordinates (θ, φ) on S and uses
2 2
the modern notation u = −x as well as the metric signature (−, +, +, +), one can rewrite Mineur’s
solution as3
2M (u)
du2 − 2 du dr + r2 dθ2 + sin2 θ dφ2 . (15.8)
g =− 1−
r
This ressembles the metric (15.1). The only difference is the − sign in front of du dr, while there is a +
sign in front dv dr in (15.1). This results from Mineur’s version being outgoing, given that he considered
a radiating body. On the contrary, the version (15.1) is ingoing, since we are interested in gravitational
collapse and black hole formation.
Twenty years later, in 1953, Prahalad Chunnilal Vaidya presented the metric that bears his name in
the form (15.8) [488]. Previously, in 1943 [486] and in 1951 [487], Vaidya presented the metric outside a
1
This is Mineur’s Eq. (21), p. 47 of Ref. [367]. The minus sign in the left-hand side, which is present Mineur’s
article [367], is fortunate for comparison with the current text since Mineur is using the metric signature
(+, −, −, −), i.e. the opposite of ours.
2
The spaces H2 and S2 are somehow connected as being the only 2-dimensional Riemannian manifolds of
nonzero constant scalar curvature (negative for H2 and positive for S2 ).
3
As one can infer by comparing with other expressions of ds2 in Mineur’s article, the + sign in front of r2
times the S2 line element in Eq. (15.7) is certainly a typo and should be replaced by a − sign; this correction is
performed to get Eq. (15.8).
15.2 The ingoing Vaidya metric 587
spherically symmetric radiating star in an equivalent, but more complicated form:
2M −1 2
2
1 ∂M 2M 2
dr + r2 dθ2 + sin2 θ dφ2 , (15.9)
g=− 1− dT + 1 −
f (M )2 ∂T r r
where f is an arbitrary function and M = M (T, r) fulfills the differential equation ∂M /∂r (1 −
2M /r) = f (M ). By introducing the scalar field u such that du = −f (M )−1 dM and promoting it as a
coordinate, one can bring this solution to Mineur’s form (15.8). In a foreword to Vaidya’s 1951 article
[487], Vishnu Vasudev Narlikar, who was Vaidya’s PhD advisor, wrote: “The treatment as given here is
essentially different from that of Professor H. Mineur as it appears in Ann. de l’Ecole Normal Superieure,
Ser. 3, 5, 1, 1933 (our Ref. [367]). Our attention was kindly drawn to it by Professor Mineur some years
ago.” In a common article by Narlikar and Vaidya published in 1947 [378], one can also read “The line
element (5) [our Eq. (15.9)] was first published by one of us some years ago. A line element equivalent to (5)
but not obviously so was obtained by Mineur several years earlier.” It would probably be fair to call the
metric (15.8) the Mineur-Vaidya metric; we shall however keep here the usual name of Vaidya metric.
The metric (15.8) was popularized and further studied by Richard W. Lindquist, Robert A. Schwartz and
Charles W. Misner in 1965 [347], who apparently were not aware of Mineur’s work.
15.2.2 Expression in ingoing Eddington-Finkelstein coordinates
To deal with black hole formation in Vaidya spacetime, it is quite convenient to work with
the ingoing Eddington-Finkelstein (IEF) coordinates (t, r, θ, φ), which are defined from the
null coordinates (v, r, θ, φ) in the same way as in the Schwarzschild case, i.e. by considering
that v is the advanced time with respect to t [cf. Eq. (6.30)]:
t := v − r ⇐⇒ v = t + r . (15.10)
Remark 2: The coordinate t was denoted by t̃ in Chap. 6, where t was reserved for the Schwarzschild-
Droste time coordinate.
We have dv = dt + dr, from which the IEF expression of the metric tensor is immediately
deduced from Eq. (15.1):
2M (t + r) 2 4M (t + r) 2M (t + r)
g = − 1− dt + dt dr + 1 + dr2
r r r (15.11)
+r2 dθ2 + sin2 θ dφ2 .
The expression of k in terms of the IEF coordinate frame is deduced from Eq. (15.3) and the
chain rule:
k = ∂t − ∂r . (15.12)
Remark 3: The analog equation in Schwarzschild spacetime is Eq. (6.48).
588 Black hole formation 2: Vaidya collapse
15.2.3 Outgoing radial null geodesics
Let us determine the radial null directions at each point by searching for null vectors of the form
ℓ = ∂t + V ∂r . The condition g(ℓ, ℓ) = 0 with the expression (15.11) for g yields immediately
a quadratic equation for V :
2M (t + r) 4M (t + r) 2M (t + r)
1+ V2+ V +1− = 0.
r r r
The two solutions are V = −1 and V = [r − 2M (t + r)]/[r + 2M (t + r)]. The first solution
gives back the ingoing null vector k introduced in Sec. 15.2.1 [cf. Eq. (15.12)]. The null vector ℓ
corresponding to the second solution is
r − 2M (t + r)
ℓ = ∂t + ∂r . (15.13)
r + 2M (t + r)
Property 15.5: radial null geodesics
The integral curves of the vector fields k and ℓ are null geodesics. Those of k are the
ingoing radial null geodesics L(v,θ,φ)
in
already discussed in Sec. 15.2.1, while those of ℓ are
called the outgoing radial null geodesics.
Proof. A direct computation shows that ℓ is a pregeodesic vector field: ∇ℓ ℓ = κℓ, with
κ = −4[rM ′ (t + r) − M (t + r)]/[r + 2M (t + r)]2 (cf. the notebook D.6.4). It follows that the
integral curves of ℓ are geodesics (cf. Sec. B.2.2).
The differential equation governing the outgoing radial null geodesics is obtained by
demanding that ℓ is their tangent vector:
dr r − 2M (t + r)
= . (15.14)
dt r + 2M (t + r)
In what follows, we shall get exact solutions of this equation for M (v) piecewise linear.
Remark 4: As in the Schwarzschild case (cf. Remark 1 on p. 191), the outgoing radial null geodesics are
actually ingoing, i.e. have r decreasing towards the future, as soon as r < 2M (t + r). This corresponds
to the region bounded by the red curve in Fig. 15.1, to be discussed in Sec. 15.3.5.
15.3 Imploding shell of radiation
15.3.1 The imploding shell model
An imploding shell of radiation is defined by the ingoing Vaidya metric with the function
M (v) obeying M ′ (v) ̸= 0 only on a finite interval of v. By choosing properly the origin of
v, we may consider this interval to be [0, v0 ], where the parameter v0 > 0 governs the shell’s
15.3 Imploding shell of radiation 589
thickness. The function M (v) is thus constant outside the interval [0, v0 ]. In order to describe
the formation of a black hole, we choose M (v) = 0 for v < 0. This corresponds to a piece
of Minkowski spacetime, since the metric (15.11) clearly reduces to Minkowski metric (4.3)
wherever M (v) = 0. Denoting by m > 0 the constant value of M (v) for v > v0 , we have then
0 for v < 0 (Minkowski region, MMin )
M (v) = m S(v/v0 ) for 0 ≤ v ≤ v0 (radiation region, Mrad ) (15.15)
m for v > v0 (Schwarzschild region, MSch ),
where S : [0, 1] → [0, 1], x 7→ S(x) is an increasing function obeying S(0) = 0 and S(1) = 1.
The region with v > v0 is qualified as Schwarzschild since for M (v) = m = const, the Vaidya
metric reduces to Schwarzschild metric, as noticed in Sec. 15.2.1. The three regions are shown
in terms of coordinates (t, r) on Fig. 15.1: Mrad (the imploding shell) is the yellow region,
MMin lies below it and MSch lies above it.
The simplest example of a function M (v) obeying (15.15) is obtained for S(x) = x:
v
S(x) = x ⇐⇒ M (v) = m (0 ≤ v ≤ v0 ). (15.16)
v0
It is shown as the blue curve in Fig. 15.2. This choice of S makes M (v) piecewise linear. The
resulting metric tensor (15.1) is continuous but not C 1 at v = 0 and v = v0 . A choice of S
that yields a C 2 metric tensor is S(x) = 6x5 − 15x4 + 10x3 . This choice is depicted by the red
curve in Fig. 15.2.
Historical note : The imploding shell model (15.15) has been introduced by William A. Hiscock, Leslie
G. Williams and Douglas M. Eardley in 1982 [279], as well as by Achilles Papapetrou in 1985 [400]. Both
studies regarded the specific case S(x) = x. Hiscock et al. considered the ingoing Vaidya metric in
the form (15.1) (coordinates (v, r, θ, φ)), while Papapetrou made use of the form (15.11) (coordinates
(t, r, θ, φ)).
15.3.2 Solution for M (v) piecewise linear
Let us consider the simplest choice for M (v), i.e. Eq. (15.16). In the radiation region, the metric
tensor (15.1) takes the form
v 2
dv + 2 dv dr + r2 dθ2 + sin2 θ dφ2 (15.17)
g =− 1−α (0 ≤ v ≤ v0 ),
r
where α is the positive constant defined by
2m
α := . (15.18)
v0
It is immediately apparent on (15.17) that for any λ > 0, the homothety Hλ : (v, r) 7→ (λv, λr)
maps g to λ2 g. Hence Hλ is a conformal isometry of (Mrad , g) with a constant conformal
factor λ2 . The homotheties (Hλ )λ∈R>0 form a 1-dimensional group, the generator of which is
590 Black hole formation 2: Vaidya collapse
t/m
4
2 k `
1 A
r/m
1 2 3 4 5 6
1
B
2
C
3
Figure 15.1: Spacetime diagram of the Vaidya collapse based on the IEF coordinates (t, r) and for the linear
mass function M (v) = mv/v0 with v0 = 3m (homothetic model with α = 2/3). The yellow area is the radiation
region Mrad [cf. Eq. (15.15)], below it lies the Minkowski region MMin and above it, the Schwarzschild region
MSch . The solid (resp. dashed) green curves are outgoing (resp. ingoing) radial null geodesics. The thick black
line marks the event horizon H (Sec. 15.3.3) and the red one the future outer trapping horizon T (Sec. 15.3.5).
Note that H and T coincide in MSch . The curvature singularity is indicated by the orange zigzag line. The part
of the figure corresponding to MMin can be compared with Fig. 4.1, while that corresponding to MSch can be
compared with Fig. 6.3. [Figure generated by the notebook D.6.4]
obtained by considering infinitesimal transformations, i.e. homotheties of ratio λ = 1 + dλ
where dλ is infinitely small. The components of the corresponding displacement vector are
dv = dλ v and dr = dλ r, so that formula (3.14) (with t ↔ λ) leads to the generator
ξ = v ∂v + r ∂r |v,θ,φ = t ∂t + r ∂r . (15.19)
The second equality follows form the change of coordinates (15.10). That ξ has the same
expression with respect to (v, r) and (t, r) coordinates should not be surprising since the
homothety Hλ has the same expression in both coordinate systems: Hλ : (t, r) 7→ (λt, λr),
given that λv = λ(t + r) = λt + λr. The vector field ξ is called a homothetic Killing vector.
The Lie derivative of g along ξ is twice g (cf. the notebook D.6.4 for the computation):
Lξ g = 2g. (15.20)
15.3 Imploding shell of radiation 591
1.0
S(x) = x
S(x) = 6x 5 − 15x 4 + 10x 3
0.8
0.6
M(v)/m
0.4
0.2
0.0
2 0 2 v0 4 6
v/m
Figure 15.2: Function M (v) for the imploding shell model, for v0 = 3m and two different choices of S(x) in
formula (15.15).
We shall refer to the choice M (v) piecewise linear as the homothetic radiation shell model.
Remark 1: The denomination self-similar, in place of homothetic, is also used in the literature (e.g.
[383, 384]).
Remark 2: A homothetic Killing vector is not a Killing vector, for the right-hand side of Eq. (15.20) would
be zero if ξ were a Killing vector [cf. Eq. (3.18)]. In other words, except for λ = 1, the homotheties
Hλ are not isometries, but only conformal isometries. Generally, vector fields generating conformal
isometries are called conformal Killing vectors. They fulfill Lξ g = σg, where σ is a scalar field.
Equation (15.20) constitutes the particular case σ = 2.
Let us introduce the variable
v
, x := (15.21)
r
which is invariant under the homotheties Hλ . The differential equation governing the outgoing
radial null geodesics, Eq. (15.14), can be rewritten as dt/dr = (1+αx)/(1−αx) [cf. Eq. (15.18)].
Given that t = v − r = r(x − 1) implies dt/dr = x − 1 + rdx/dr, we get the equivalent form
dx αx2 − x + 2
r = . (15.22)
dr 1 − αx
Fortunately, this ordinary differential equation is separable, so that its solutions are easily
obtained by quadrature. They depend on whether the quadratic polynomial Pα (x) := αx2 −x+2
admits real roots or not. Let us first focus on the case where Pα has no real root. The discriminant
being 1 − 8α, this occurs if, and only if,
1
α> ⇐⇒ v0 < 16 m. (15.23)
8
By considering the energy-momentum tensor (15.5) with expression (15.16) substituted for
M (v), we get
α
T = k ⊗ k. (15.24)
8πr2
592 Black hole formation 2: Vaidya collapse
Consequently, we may say that the case α > 1/8 corresponds to a large radiation energy
density. We shall discuss the low radiation energy density case (α < 1/8) in Sec. 15.4. For the
moment, assuming (15.23), we have Pα (x) > 0 for any x ∈ R and we may rewrite Eq. (15.22) as
1 − αx
d ln r = dx, (15.25)
αx2 − x + 2
the solution of which is r = r0 fα (x), where
Z x
1 − αx′
ln fα (x) := dx′ (15.26)
0 αx′ 2 − x′ + 2
and the integration constant r0 > 0 is the value of r at x = 0, or equivalently at v = 0, since
fα (0) = 1. Explicitly,
√
2 1 2αx − 1 1
fα (x) = √ exp √ arctan √ + arctan √ ,
2
αx − x + 2 8α − 1 8α − 1 8α − 1
(15.27)
Introducing the dimensionless parameter u := − ln(r0 /m), we conclude:
Property 15.6: outgoing radial null geodesics for the α > 1/8 homothetic shell
For the homothetic model with α > 1/8, the outgoing radial null geodesics in Mrad form
a 3-parameter family of curves L(u,θ,φ) , where the parameter u ∈ R is related to the
out
value r0 of r at the inner edge of the radiation shell (v = 0) by r0 = me−u . The parametric
equation of L(u,θ,φ)
out
in terms of the IEF coordinates is
−u
t = me (x − 1)fα (x)
r = me−u fα (x) 0 ≤ x ≤ xmax , (15.28)
θ = const, φ = const
where the function fα (x) is defined by Eq. (15.27) and either xmax = +∞ (L(u,θ,φ)
out
reaches
r = 0 for some v < v0 ) or xmax is the solution of me xmax fα (xmax ) = v0 (L(u,θ,φ) reaches
−u out
the outer edge of the radiation shell).
The function fα (x) is plotted in Fig. 15.3. It increases from 1 at x = 0 to some maximum
reached for x = α−1 and then decreases to 0 as x → +∞. This behavior follows directly from
the sign of the numerator 1 − αx in Eq. (15.25), given that the denominator αx2 − x + 2 is
always positive for α > 1/8. Note that it could be that xmax < α−1 so that the maximum
of fα (x) is actually not reached along L(u,θ,φ)
out
. In that case, r increases monotonically along
L(u,θ,φ)
out
in Mrad , from r0 to r0 fα (xmax ).
It appears clearly on Eq. (15.28) that the homothety Hλ : (t, r) 7→ (λt, λr) transforms
the geodesic L(u,θ,φ)
out
into the geodesic L(uout ′ ,θ,φ) with u = u − ln λ, in agreement with the
′
homothetic symmetry of (Mrad , g) discussed above.
15.3 Imploding shell of radiation 593
α = 0.25
α = 0.50
α = 1.00
4 α = 2.00
α = 4.00
α = 6.00
3
fα (x)
2
0 1 2 3 4 5 6
x
Figure 15.3: Function fα (x), defined by Eq. (15.27), for some selected values of α > 1/8. For each value of α,
the maximum of fα (x) is achieved for x = α−1 . [Figure generated by the notebook D.6.5]
Some outgoing radial null geodesics are depicted as solid green lines in Fig. 15.1. In Mrad ,
they obey Eq. (15.28). Note that the homothetic symmetry appears clearly on the figure. If a
geodesic L(u,θ,φ)
out
has a r-turning point in Mrad , it must located at x = α−1 , i.e. at t/r = α−1 −1,
or equivalently at v
0
t= − 1 r. (15.29)
2m
The above equation defines a straight line through (t, r) = (0, 0), whose intersection with
Mrad is depicted by a red segment in Fig. 15.1.
Remark 3: The turning point value (15.29) can be obtained directly by setting dr/dt = 0 in Eq. (15.14)
and using the value (15.16) for M (v).
In the Minkowski region, the outgoing radial null geodesics are straight line segments
inclined at +45◦ in Fig. 15.1, while in the Schwarzschild region, they are curves obeying
Eq. (6.51) (with the change of notation t̃ ↔ t).
15.3.3 Black hole formation
In spherical symmetry, the inspection of radial null geodesics provides a direct access to the
black hole event horizon H . Since we have at disposal the exact solution (15.28) for the
outgoing radial null geodesics, let us determine the location of H for the homothetic shell
collapse. We arrive at the following result:
Property 15.7: black hole formation for the α > 1/8 homothetic shell
The homothetic imploding radiation shell with α := 2m/v0 > 1/8 generates a black hole.
The black hole event horizon H is the future light cone of the point of IEF coordinates
594 Black hole formation 2: Vaidya collapse
0.0
0.5
1.0
1.5
thb /m
2.0
2.5
3.0
3.5
4.0
0 2 4 6 8 10 12 14 16
v0 /m
Figure 15.4: Value thb of the coordinate t at the black hole birth [Eq. (15.30)] as a function of the radiation shell
thickness v0 , for the homothetic shell collapse [Eq. (15.16)].
(t, r) = (thb , 0), witha
2 1
thb = −4m exp − √ arctan √ . (15.30)
8α − 1 8α − 1
Outside the radiation shell, H coincides with the Killing horizon of Schwarzschild space-
time located at r = 2m.
a
As in Chap. 14, the subscript ‘hb’ stands for horizon birth.
Proof. In the Schwarzschild region MSch , the Killing horizon is the hypersurface r = 2m.
It is generated by the null geodesics L(θ,φ) out,H
discussed in Sec. 6.3.5 [cf. Eq. (6.52)]. Let us
consider one such geodesic, Lˆ say. It has a fixed value of (θ, φ) and, when followed in the
past direction, it encounters the outer edge of the radiation region Mrad (hypersurface v = v0 )
at the point A such that rA = 2m and tA = v0 − 2m (cf. Fig. 15.1, where Lˆ can be identified
with the black curve). If Lˆ is prolonged into Mrad , still in the past direction, it encounters the
inner edge of Mrad (hypersurface v = 0) at the point B (cf. Fig. 15.1). The portion AB of Lˆ
coincides with the geodesic L(uout B ,θ,φ)
of the outgoing radial null family given by Eq. (15.28),
with uB = − ln(rB /m) (since fα (xB ) = 1, given that xB = 0). We have then rA = rB fα (xA ).
Now, by definition of x [Eq. (15.21)] and α [Eq. (15.18)], xA = vA /rA = v0 /(2m) = α−1 . We
have thus 2m = rB fα (α−1 ). In view of expression (15.27) for fα (x), there comes
2 1
rB = 2m exp − √ arctan √ .
8α − 1 8α − 1
Since B is located on the hypersurface v = 0, we have tB = −rB . If the radial null geodesic Lˆ
is prolonged further to the past in the Minkowski region MMin , it becomes the straight line of
equation t = r − 2rB . Lˆ thus reaches r = 0 at some event C of coordinate t = thb = −2rB ,
hence Eq. (15.30).
15.3 Imploding shell of radiation 595
There remains to prove that the null hypersurface generated by Lˆ when (θ, φ) varies, i.e.
the future light cone H of the point (t, r) = (thb , 0), is indeed the black hole event horizon.
To this aim, let us consider an outgoing radial null geodesic L in MMin such that L crosses
r = 0 at t < thb , i.e. outside H . L arrives then at the inner edge of Mrad with r = r0 > rB .
In Mrad , according to Eq. (15.28), L is homothetic to a part of Lˆ with a ratio r0 /rB > 1.
Hence it emerges at the outer edge of Mrad with r > rA = 2m. L is there in the exterior of
the Schwarzschild black hole, so that it will subsequently reach the future null infinity I + of
MSch . On the contrary, if L crosses r = 0 with t > thb , i.e. inside H , it encounters the inner
edge of Mrad with r = r0 < rB . A part of L is then homothetic to the segment BA of Lˆ with
a ratio r0 /rB < 1. Then either (i) L has a r-turning point and reaches r = 0 in Mrad or (ii) L
reaches the outer edge of Mrad (v = v0 ) at some point A′ . Given that xA = α−1 corresponds to
the maximum of fα (x), one has necessarily fα (xA′ ) ≤ fα (xA ) and thus r0 fα (xA′ ) < rB fα (xA ).
By Eq. (15.28), this implies rA′ < rA = 2m, so that L emerges in the black hole region of
Schwarzschild spacetime. So none of the two possible cases (i) or (ii) leads to L reaching I + .
We conclude that H is a black hole horizon.
The black hole event horizon H is depicted as the thick black curve in Fig. 15.1. Since
thb < 0 [cf. Eq. (15.30)], one immediately notices:
Property 15.8: black hole formation in the Minkowski region
The black hole forms in the Minkowski region of Vaidya spacetime, i.e. in a region where
the spacetime curvature is zero.
This striking feature reflects the non-local character of black holes and will be discussed further
in Chap. 18.
The dependency of thb on the width v0 of the radiation shell is shown in Fig. 15.4. One has
−4m < thb < 0, with
lim thb = −4m and lim thb = 0. (15.31)
v0 →0 v0 →16m
Remark 4: The first limit in (15.31), which corresponds to α → +∞ in Eq. (15.30), is easily recovered
by a pure geometric construction: a zero-width shell implies the equality of the two points A and B
considered in the proof of (15.30), as well as tA = tB = −2m, hence tC = thb = −4m.
Historical note : The imploding radiation shell with M (v) piecewise linear (homothetic model) has
been extensively studied by Achilles Papapetrou in 1985 [400]. He obtained the solution (15.28) for
the outgoing radial null geodesics and determined the location of the event horizon, along the lines
presented above.
15.3.4 Curvature singularity
The Kretschmann scalar K := Rµνρσ Rµνρσ (cf. Sec. 6.3.4) is computed in the notebook D.6.4:
48M (t + r)2
K= . (15.32)
r6
596 Black hole formation 2: Vaidya collapse
K is identically zero in the Minkowski region (M (r + t) = 0), as it should!
It diverges at r = 0 in the radiation and Schwarzschild regions (M (r + t) > 0), tracing the
existence of a curvature singularity there. In other words:
Property 15.9: curvature singularity in Vaidya shell collapse
The Vaidya shell collapse introduced in Sec. 15.3.1 generates a spacetime with a curvature
singularity located at r = 0 and t ≥ 0.
Remark 5: The Kretschmann scalar of Vaidya metric has the same structural form as that of the
Schwarzschild metric, compare Eq. (6.45), while a priori K could have contained some term involving
the derivative of M (v). Indeed M ′ (v) appears in some components of the Riemann tensor, since it is
present in the components of the Ricci tensor, as given by Eq. (15.4).
Remark 6: Since the Ricci tensor of the Vaidya metric is not identically zero [cf. Eq. (15.4)], other
curvature invariants that one might have think of for tracking the curvature singularity are the Ricci
scalar R := g µν Rµν and the Ricci “squared” Rµν Rµν . However, they are both identically zero for k is a
null vector.
The curvature singularity is depicted as the orange broken line in Fig. 15.1, which regards
the homothetic radiation shell with α > 1/8. It is clear on this figure that the singularity bounds
the future of both ingoing and outgoing radial null geodesics (this will be shown rigorously
in Sec. 15.3.6). This implies that, for α > 1/8, the curvature singularity is spacelike, as in
Schwarzschild spacetime. We shall see in Sec. 15.4 that for α < 1/8, the curvature singularity
has a null segment, in addition to the spacelike one.
For the homothetic collapse with α > 1/8 (v0 < 16 m) considered in Secs. 15.3.2 and 15.3.3
and depicted in Fig. 15.1, one has thb < 0 [Eq. (15.30)], so that the curvature singularity is
entirely located in the black hole region. It is therefore hidden to a remote observer. In Sec. 15.4,
we will see that this is no longer the case for α < 1/8: the singularity is then naked.
15.3.5 Trapped surfaces
Let us show that there exist trapped surfaces (cf. Sec. 3.2.3) in the radiation and Schwarzschild
regions of Vaidya spacetime. To this aim, we consider a 2-surface S defined by (t, r) = const. It
is a closed surface with the topology of a 2-sphere, spanned by the coordinates (θ, φ). Moreover,
S is spacelike since the metric induced by g on it is q = r2 dθ2 + sin2 θ dφ2 , as readily seen
on Eq. (15.11). The area of S is simply 4πr2 (r is the areal radius coordinate), so to check
whether S is a trapped surface, it suffices to determine the behavior of r along the two null
directions normal to S , which are nothing but the directions of the ingoing and outgoing
radial null geodesics. Along the ingoing geodesics (tangent vector k), one has dr/dt = −1,
since k = ∂t − ∂r [Eq. (15.12)], so that the expansion is negative: θ(k) < 0. Along the outgoing
radial null geodesics, dr/dt is given by Eq. (15.14). The sign of the expansion θ(ℓ) is then that
of r − 2M (t + r). Hence
15.3 Imploding shell of radiation 597
Property 15.10: trapped surfaces in Vaidya shell collapse
The spherically symmetric surface S defined by (t, r) = const obeys
S is trapped ⇐⇒ r < 2M (t + r). (15.33)
Obviously, the above criterion cannot be fulfilled in the Minkowski region, where M (t +
r) = 0. On the contrary, trapped surfaces exist in the central part of the radiation re-
gion, since M (t + r) > 0 there. They also exist in the black hole part (r < 2m) of the
Schwarzschild region, where M (t + r) = m.
Let us denote by T the hypersurface formed by all the marginally trapped spheres of fixed
(t, r) (vanishing expansion along ℓ). The equation of T is obtained by saturating the inequality
in Eq. (15.33):
T : r = 2M (t + r). (15.34)
As we shall see in Chap. 18, T is called a future outer trapping horizon. Equation (15.34)
implies dr/dt = 0 in Eq. (15.14), hence we get
Property 15.11: trapping horizon and r-turning points of radial null geodesics
If an outgoing radial null geodesic L crosses T , the crossing point is a r-turning point of
L.
This feature appears clearly in Figs. 15.1 and 15.5. Furthermore, T obeys the following
properties:
Property 15.12: causal type of the future outer trapping horizon
In the radiation region, the future outer trapping horizon T is a spacelike hypersurface,
while in the Schwarzschild region, T coincides with the event horizon H and hence is a
null hypersurface there.
Proof. Let us consider (v, θ, φ) as coordinates on the 3-manifold T . We may rewrite Eq. (15.34)
for T as r = 2M (v), so that dr = 2M ′ (v) dv along T . Plugging this relation, as well as
2M (v)/r = 1, into Eq. (15.1) yields immediately′ the metric h induced by g on T : h =
4M (v) dv + r dθ + sin θ dφ . In Mrad , M (v) > 0 [cf. Eq. (15.15)], which implies
′ 2 2 2 2 2
that h is a positive definite metric. Hence T is a spacelike hypersurface in Mrad . In MSch ,
M ′ (v) = 0 and h is a degenerate metric, so that T is a null hypersurface there. Actually, in
MSch , Eq. (15.34) reduces to r = 2m, so that T coincides with the event horizon H there.
Remark 7: The spacelike character of T is actually a generic feature of future outer trapping horizons
in non-stationary spacetimes (such as (Mrad , g)), as we shall see in Chap. 18.
598 Black hole formation 2: Vaidya collapse
t/m
4
2 k `
r/m
1 2 3 4 5 6
Figure 15.5: Same as Fig. 15.1, but for the C 2 mass function M (v) corresponding to the choice S(x) =
6x5 − 15x4 + 10x3 in Eq. (15.15) (cf. the red curve in Fig. 15.2). Note that the trapping horizon T (red curve) is
tangent to the event horizon H (black curve) at the outer edge of radiation region ((t, r) = (m, 2m)). [Figure
generated by the notebook D.6.4]
For the homothetic shell model considered in Sec. 15.3.2, M (v) = mv/v0 = αv/2 and the
equation of T in Mrad is vey simple:
T ∩ Mrad : r = αv. (15.35)
Example 1: For v0 = 3m, as in Fig. 15.1, α = 2/3 and we get r = 2v/3 = 2(t + r)/3 or equivalently
t = r/2. Hence, in Mrad , T appears as the straight line segment of slope 1/2 drawn in red in Fig. 15.1.
It appears clearly there that the outgoing radial null geodesics that cross T do it at a r-turning point. By
considering the light cones delineated by the ingoing and outgoing radial null geodesics, it also appears
clearly on Fig. 15.1 that T is a spacelike hypersurface in Mrad .
Remark 8: As shown above, T coincides with H in MSch , so that the slope of T in Fig. 15.1 changes
abruptly from 1/2 to +∞ at the outer edge of Mrad . This lack of smoothness of the hypersurface T
reflects actually the lack of smoothness of the function M (v) of the homothetic shell model (cf. the
blue curve in Fig. 15.2). Would M (v) be smooth, T would merge smoothly with H at v = v0 . This
is shown in Fig. 15.5, where M (v) is of class C 2 . Besides, we can check on this figure that, while it
has a shape more complicated that in Fig. 15.1, T always lies outside the null cones in Mrad , i.e. T is
spacelike there.
15.3 Imploding shell of radiation 599
15.3.6 Carter-Penrose diagram
In order to draw a Carter-Penrose diagram of the Vaidya collapse, we need first to introduce a
coordinate system (u, v, θ, φ) in the radiation region Mrad such that u is constant along the
outgoing radial null geodesics — v being constant along the ingoing ones by construction. For
the homothetic model (M (v) = αv/2 in Mrad ), the relation between coordinates (u, v, θ, φ)
and (v, r, θ, φ) is provided by Eq. (15.28):
r v
u = − ln + ln fα . (15.36)
m r
From Eqs. (15.36) and (15.26), we get
1 1 − αx v 1
du = − dr + d = − [(αx − 1)dv + 2dr] .
r αx2 − x + 2 r r(αx2 − x + 2)
By comparing with (15.17), we deduce immediately the expression of the metric tensor in terms
of the coordinates (u, v, θ, φ):
g = −r(αx2 − x + 2) du dv + r2 dθ2 + sin2 θ dφ2 (15.37)
(0 ≤ v ≤ v0 ),
where x := v/r and r is to be considered as a function of (u, v) defined implicitly by Eq. (15.36).
Since for α > 1/8, the polynomial αx2 − x + 2 never vanishes, the metric component guv read
on Eq. (15.37) is nonzero in all Mrad . This shows that the coordinates (u, v, θ, φ) are regular
on Mrad (except for the singularities inherent to the spherical coordinates (θ, φ)). Moreover,
they constitute a double-null coordinate system: both u and v are null coordinates. This has
already been shown for v (cf. Property 15.1) and for u, this follows from the component g uu of
the inverse metric deduced from (15.37) being identically zero [cf. Eq. (A.56b)]. Besides, we
read on Eq. (15.37) that guu = 0 and gvv = 0, which means that the coordinate vectors ∂u and
∂v are both null. The vector ∂u is actually tangent to the ingoing radial null geodesics L(v,θ,φ)
in
and the vector ∂v is tangent to the outgoing ones, L(u,θ,φ)
out
. Furthermore, the null vectors ∂u
and ∂v are both future-directed. Indeed, the time orientation of Vaidya spacetime is given by
the vector k (cf. Sec. 15.2.1) and we deduce from Eq. (15.3) that k = −(∂u/∂r) ∂u . Evaluating
∂u/∂r from Eqs. (15.36) and (15.26), we get
2
k= ∂u . (15.38)
r(αx2 − x + 2)
The coefficient in front of ∂u being always positive, we conclude that ∂u is future-directed.
Then, from Eq. (15.37), g(∂u , ∂v ) = guv < 0, so that, according to Lemma 1.2 (Sec. 1.2.2), ∂v is
future-directed as well.
Remark 9: The reader may have noticed a slight asymmetry between the coordinates u and v: u is
dimensionless, while v has the dimension of a time. We could of course make u have the same dimension
as v by introducing an overall factor m in the right-hand side of Eq. (15.36). However, this would make
the formulas slightly more complicate, without any real benefit.
Let us discuss the boundary of Mrad in terms of the (u, v) coordinates. By definition of
Mrad , a part of the boundary consists in the hypersurfaces v = 0 and v = v0 . Another part
600 Black hole formation 2: Vaidya collapse
Figure 15.6: Radiation region Mrad for the homothetic model (M (v) = mv/v0 ) with v0 < 16m, depicted in
terms of the double-null coordinates (u, v). Solid (resp. dashed) green lines are outgoing (resp. ingoing) radial
null geodesics. The orange zigzag curve corresponds to the curvature singularity at r = 0. The black segment
marks the event horizon H , while the red curve marks the trapping horizon T [Eq. (15.42)]. The points A and B
are the same as in Fig. 15.1.
corresponds to the limit r → +∞. In view of Eq. (15.36) and fα (0) = 1 (cf. Eq. (15.26) or
Fig. 15.3), this corresponds to u → −∞. The last part of the boundary of Mrad is set by the
curvature singularity at r = 0 (cf. Sec. 15.3.4). Taking the limit r → 0 in Eq. (15.36), we found
the equation ruling this boundary in terms of (u, v):
v r
0 α 1 π 1
u = ln + u0 , u0 := ln +√ + arctan √ . (15.39)
v 2 8α − 1 2 8α − 1
We conclude that the range of the coordinates (u, v) on Mrad is (cf. Fig. 15.6)
0≤v≤v
0
Mrad : (15.40)
v0
−∞ < u < ln + u0 .
v
In the last inequality, the right-hand side must be replaced by +∞ if v = 0.
Along a given ingoing radial null geodesic L(v,θ,φ)
in
, u can be considered as a (non-affine)
parameter, ∂u being tangent to L(v,θ,φ) . The geodesic hits the curvature singularity for u →
in
ln(v0 /v) + u0 . Similarly, along an outgoing radial null geodesic L(u,θ,φ)
out
, v can be considered
as a (non-affine) parameter, ∂v being tangent to L(u,θ,φ) . If u < u0 , L(u,θ,φ)
out out
reaches the
outer boundary of the radiation shell for v = v0 and is extendible to a null geodesic of the
Schwarzschild region, while if u ≥ u0 , L(u,θ,φ)
out
hits the curvature singularity for v → v0 eu0 −u
(cf. Fig. 15.6).
15.3 Imploding shell of radiation 601
Figure 15.7: Carter-Penrose diagram of Vaidya collapse for the homothetic shell model (M (v) = mv/v0 =
αv/2) with α > 1/8 (v0 < 16m). The Minkowski region MMin , radiation region Mrad and Schwarzschild region
MSch are depicted in respectively pale blue, yellow and grey. Solid (resp. dashed) green lines are outgoing (resp.
ingoing) radial null geodesics. The orange zigzag segment corresponds to the curvature singularity at r = 0. The
hatched area is the black hole region, delimited by the event horizon H . The red dot-filled area, delimited by the
future outer trapping horizon T (red curve), is the region where the spheres (t, r) = const are trapped surfaces.
The points A, B and C along H are the same as in Fig. 15.1.
Since it is generated by outgoing null geodesics, the event horizon H crosses Mrad at a
fixed value of u, uH say. The value of uH is found by setting r = 2m and v = v0 in Eq. (15.36):
2 1
uH = √ arctan √ − ln 2. (15.41)
8α − 1 8α − 1
Given expression (15.39) for u0 , it is easy to check that uH < u0 , as drawn in Fig. 15.6.
Regarding the future outer trapping horizon T introduced in Sec. 15.3.5, it obeys r = αv in
Mrad [Eq. (15.35)], so that the equation of T in terms of the double-null coordinates is easily
obtained by substituting α−1 for v/r in Eq. (15.36) and making use of expressions (15.27) and
(15.41); one gets v
0
T : u = ln + uH . (15.42)
v
Note that this implies u → +∞ for v → 0 and u → uH for v → v0 ; the latter property agrees
with T coinciding with H in MSch , i.e. for v > v0 . T is drawn as a red curve in Fig. 15.6.
The spacetime diagram of Mrad in Fig. 15.6 is already conformal since all radial null geodesics
are straight lines inclined at ±45◦ . To integrate it into a Carter-Penrose diagram, one needs
602 Black hole formation 2: Vaidya collapse
to compactify it along the u direction by introducing a coordinate of the type U = arctan u —
the v direction being already compactified, given the finite range of v in Mrad . One can then
match the obtained diagram to Carter-Penrose diagrams of MMin (cf. Fig. 4.6) and MSch (cf.
Fig. 9.10 or 9.11), thereby getting a Carter-Penrose diagram of the whole Vaidya spacetime, as
shown in Fig. 15.7.
Historical note : The double-null coordinates (u, v), leading to expression (15.37) of the metric tensor
in Mrad , have been introduced by B. Waugh and Kayll Lake in 1986 [516]. A Carter-Penrose diagram
similar to that of Fig. 15.7 except for the radiation region extending to I + (no pure Schwarzschild
exterior but M ′ (v) → 0 for v → +∞) has been exhibited by Yuhji Kuroda in 1984 [330] (cf. his Fig. 2b).
The case of a radiation region bounded by v = v0 , as here, can be found in Fig. 2 of a 2007 article by
Brien C. Nolan [384].
15.4 Configurations with a naked singularity
15.4.1 The low radiation density case
In Sec. 15.3, we have focused on homothetic radiation shells (M (v) = αv/2) with α > 1/8
[Eq. (15.23)]. Let us now discuss the opposite case4 :
1
α< ⇐⇒ v0 > 16 m. (15.43)
8
In view of the form (15.24) of the energy momentum tensor, this corresponds to a low energy
density of the radiation field. When (15.43) is fulfilled, the polynomial Pα (x) := αx2 − x + 2,
which appears in the numerator of the ODE (15.22) ruling outgoing radial null geodesics in
Mrad , admits two real roots:
√ √
1 − 1 − 8α 1 + 1 − 8α
x1 := and x2 := . (15.44)
2α 2α
Useful identities are x1 + x2 = α−1 and x1 x2 = 2α−1 . Note also that 2 < x1 < 4 < x2 < α−1 .
Equation (15.22) can be recast as
dx (x − x1 )(x − x2 )
= r . (15.45)
dr x1 + x 2 − x
This differential equation admits two special solutions, corresponding to outgoing radial null
geodesics with constant value of x:
x = x1 and x = x2 . (15.46)
Let us denote these geodesics by respectively L(θ,φ)∗1
and L(θ,φ)
∗2
. Since x := v/r = 1 + t/r,
their equations in terms of the IEF coordinates (t, r, θ, φ) and in Mrad is simply
L(θ,φ)
∗1
: t = (x1 − 1)r and L(θ,φ)
∗2
: t = (x2 − 1)r. (15.47)
Thus, in Mrad , L(θ,φ)
∗1
and L(θ,φ)
∗2
are straight line segments through the origin (t, r) = (0, 0).
They are depicted as the segments OC and OB respectively in Fig. 15.8.
4
The marginal case α = 1/8 will not be discussed here; it is actually qualitatively similar to the case α < 1/8
insofar as it leads to a naked singularity as well [400].
15.4 Configurations with a naked singularity 603
t/m
20
A
15 B
C
10
r/m
O 2 4 6 8 10 12 14 16
Figure 15.8: Spacetime diagram of the Vaidya collapse based on the IEF coordinates (t, r) and for the linear
mass function M (v) = mv/v0 with v0 = 18 m (homothetic shell model with α = 1/9, which corresponds to
x1 = 3 and x2 = 6). The legend is the same as in Fig. 15.1, with in addition the blue line marking the Cauchy
horizon HC induced by the naked singularity at (t, r) = (0, 0) (Sec. 15.4.4). In the radiation region, the straight
line segments OB and OC are the traces in the (t, r) plane of the homothetic Killing horizons x = x2 and x = x1 ,
the last one being a part of the Cauchy horizon. [Figure generated by the notebook D.6.4]
604 Black hole formation 2: Vaidya collapse
Property 15.13: homothetic Killing horizons
When (θ, φ) varies, L(θ,φ)
∗1
and L(θ,φ)
∗2
generate null hypersurfaces, H1 and H2 respectively,
such that the homothetic Killing vector ξ [cf. Eq. (15.19)] is normal to them in Mrad . One
says that H1 and H2 are homothetic Killing horizonsa .
a
Recall from Sec. 3.3.2 that a Killing horizon is a null hypersurface such that a Killing vector is normal to
it.
Proof. Their equations being t = (x1 − 1)r and t = (x2 − 1)r, L(θ,φ)
∗1
and L(θ,φ)
∗2
are clearly
orbits of the homothety group (Hλ )λ∈R>0 discussed in Sec. 15.3.2. The group generator ξ is
thus tangent to the null curves L(θ,φ)
∗1
on H1 and L(θ,φ)
∗2
on H2 . Hence, ξ is null there. Given
that the only null direction in a null hypersurface is the normal one, we conclude that ξ is
normal to H1 and H2 .
Let us now consider generic outgoing radial null geodesics, i.e. geodesics with x ̸= x1 and
x ̸= x2 ; we may then rewrite Eq. (15.45) as
x1 + x2 − x
d ln r = dx.
(x − x1 )(x − x2 )
This equation is easily integrated to
|x/x2 − 1|x1 /(x2 −x1 )
r = r0 , (15.48)
|x/x1 − 1|x2 /(x2 −x1 )
where r0 is constant along the considered geodesic (but may vary from one geodesic to the
other). If the geodesic intersects Mrad ’s inner edge (x = 0), then r0 represents the value of r at
the intersection point.
Example 2: For α = 1/9, one has x1 = 3 and x2 = 6 [cf. Eq. (15.44)] and Eq. (15.48) reduces to
r = r0 |x/6 − 1|/(x/3 − 1)2 . Such a function of x is plotted in Fig. 15.9.
Having determined the outgoing radial null geodesics, we are in position to establish the
following results:
Property 15.14: black hole and naked singularity for the α < 1/8 model
The homothetic imploding radiation shell with α := 2m/v0 < 1/8 generates a black hole,
the event horizon of which originates at (t, r) = (0, 0), with a slope t/r = x2 −1. Moreover,
the curvature singularity at (t, r) = (0, 0) is naked: it is connected to remote observers by
the outgoing radial null geodesics L(θ,φ)
∗1
, as well as by the family of outgoing radial null
geodesics that cross the outer edge of the radiation shell (v = v0 ) with x1 < x < α−1 ; all
geodesics of this family, which contains L(θ,φ)∗2
, emanate from (t, r) = (0, 0) with a slope
t/r = x2 − 1.
15.4 Configurations with a naked singularity 605
3.0
2.5
2.0
x = α −1
x = x1
x = x2
r/r0
1.5
1.0
0.5
0.0
0 2 4 6 8 10 12
x
Figure 15.9: The coordinate r as a function of x := v/r along outgoing radial null geodesics for the homothetic
Vaidya collapse with α = 1/9 (which implies x1 = 3 and x2 = 6). The graph of r(x) admits a local maximum for
x = α−1 , i.e. x = 9 in the present case, although this is barely noticeable on the figure. [Figure generated by the
notebook D.6.5]
Proof. As in Sec. 15.3.3, let us consider a null generator Lˆ of the Killing horizon of the
Schwarzschild region MSch , which is located at r = 2m. Lˆ intersects the outer edge v = v0
of Mrad at a point A such that rA = 2m and hence xA = α−1 (cf. Fig. 15.8). When prolonged
to the past in Mrad , i.e. to x < α−1 , Lˆ has r decreasing (cf. the graph of r(x) at the left of
x = α−1 in Fig. 15.9), until r = 0 is reached for x = x2 . Since the Kretschmann scalar (15.32) is
K = 12α2 x2 /r4 for the homothetic model, we get K → ∞ for r → 0 when x → x2 > 0. We
thus conclude that Lˆ hits the curvature singularity at x = x2 and cannot be extended further
in the past, contrary to the case α > 1/8 dealt with in Sec. 15.3.3.
Let us now consider any outgoing radial null geodesic L in MSch that intersects the outer
edge v = v0 of Mrad with x such that x1 < x < α−1 , i.e. between the points C and A in
Fig. 15.8. L has r > 2m in MSch , hence it reaches the future null infinity I + . When L
is prolonged backward in Mrad , it starts with r decaying since Eq. (15.14) can be rewritten
dr/dt = (1 − αx)/(1 + αx) for the homothetic model, implying dr/dt < 0. The entire past
of L in Mrad can then be read on Fig. 15.9: if one follows the decaying r direction either for
x1 < x < x2 or x2 < x < α−1 , one ends up to r = 0 for x = x2 , as for Lˆ. Hence the same
conclusion holds: L hits the curvature singularity in the past direction at x = x2 . Given that
L extends to I + in the future, we conclude that the curvature singularity is naked.
Finally, let us consider an outgoing radial null geodesic L in MSch that intersects the outer
edge v = v0 of Mrad with x < x1 , i.e. below the point C in Fig. 15.8. When L is prolonged
backward in Mrad , r decreases along it and we read on the left part of Fig. 15.9 that r reaches
the finite nonzero value r0 at x = 0, i.e. at the inner edge of Mrad . It can then be extended
backward to the Minkowski region MMin , where it reaches r = 0 at t = t0 := −2r0 . Since for
x → x− 1 , r/r0 → +∞ (cf. Fig. 15.9 or Eq. (15.48)), we conclude that r0 , and hence |t0 |, can be
made arbitrarily small by having x close enough to x1 at the outer edge of Mrad . This proves
that the whole of MMin can be connected to I + by this type of null geodesics. It follows that
606 Black hole formation 2: Vaidya collapse
the black hole event horizon is the null hypersurface generated by Lˆ.
Historical note : The formation of naked singularities in spacetimes with a Vaidya region has been
put forward first by B. Steinmüller, Andrew R. King, and Jean-Pierre Lasota in 1975 [463], but this
regards bodies radiating away all their masses, the exterior of which is described by the outgoing Vaidya
metric (15.8). In the context of imploding radiating shells (ingoing Vaidya metric) considered here, the
appearance of naked singularities has been shown first by William A. Hiscock, Leslie G. Williams and
Douglas M. Eardley in 1982 [279] and has been further studied by Yuhji Kuroda in 1984 [330] and Achilles
Papapetrou in 1985 [400]. In particular, the solution (15.48) for the outgoing radial null geodesics has
been exhibited by Papapetrou: compare Eq. (18) in Ref. [400], where X1 = x2 and X2 = x1 .
15.4.2 Analysis in double-null coordinate systems
The above result contains something puzzling at first glance: for a given value of (θ, φ), there
are distinct radial null geodesics emanating from the “point” (t, r) = (0, 0), namely all the
geodesics that cross the outer edge of the radiation shell with a slope t/r = x − 1, x ∈ [x1 , α−1 ).
This appears clearly on Fig. 15.8. It looks like there is an infinity of future light cones emanating
from a single spacetime point! This state of affairs results actually from a bad behavior of the
coordinates (t, r) near (0, 0). To clarify this, let us introduce new coordinates (u, v, θ, φ) such
that u is constant along the outgoing radial null geodesics, v being constant along the ingoing
ones by construction. Let us start by substituting v/r for x in the geodesic equation (15.48); we
get
|v/x2 − r|x1 /(x2 −x1 ) 1
x2 /(x2 −x1 )
= . (15.49)
|v/x1 − r| r0
Contrary to the case α > 1/8 dealt with in Sec. 15.3.6, two coordinate patches (u, v, θ, φ) and
(u′ , v, θ, φ) are actually required to get regular double-null coordinates on Mrad . They are
defined on two overlapping subregions of Mrad :
v v
NI : r > and NII : r < . (15.50)
x2 x1
Since x2 > x1 , one has Mrad = NI ∪ NII . Furthermore, NI (resp. NII ) contains the homothetic
Killing horizon H1 (resp. H2 ) and NI ∩ NII is the region between H1 and H2 .
The NI region
On NI , let us define the parameter u so that
r0 = |u|x2 /(x2 −x1 ) . (15.51)
As r0 , u is constant along a given outgoing radial null geodesic. From Eq. (15.49), we get5
|u| = |v/x1 − r|/(r − v/x2 )x1 /x2 . This equation determines u in terms of v and r up to some
5
We have made use of the identity |v/x2 − r| = r − v/x2 , which holds on NI .
15.4 Configurations with a naked singularity 607
sign. We choose the latter so that u → +∞ at the inner boundary of NI (r → v/x2 ). This
yields
v/x1 − r
u= , −∞ < u < +∞, (15.52)
(r − v/x2 )x1 /x2
with limr→+∞ u = −∞ (cf. Fig. 15.10). We may use (u, v, θ, φ) as a coordinate system on NI ,
instead of the IEF coordinates (t, r, θ, φ).
Example 3: As in Example 2, let us consider the case α = 1/9, for which x1 = 3 and x2 = 6 [cf.
Eq. (15.44)]. Equation (15.52) reduces then to
v/3 − r
u= p . (15.53)
r − v/6
This relation can be inverted, yielding an explicit expression for r(u, v):
r !
v u 2
r= + u − u2 + v . (15.54)
3 2 3
The metric components in coordinates (u, v, θ, φ) are deduced from those in coordinates
(v, r, θ, φ), i.e. Eq. (15.17), via the identity α = 2/(x1 x2 ). We get (see the notebook D.6.6 for
the computation):
x1 /2
2x2 v v
2 2 2 2
(15.55)
g=− r− du dv + r dθ + sin θ dφ r> .
(x2 − x1 )r x2 x2
In this expression, r shall be considered as the function of (u, v) defined implicitly by Eq. (15.52).
The metric component guv is regular and non-vanishing in all NI . Moreover we have a double-
null coordinate system, since v is a null coordinate (Property 15.1) and it can be infered from
(15.55) that g uu = 0, so that u is null as well by Eq. (A.56b). The coordinate vectors ∂u and ∂v
are both null, given that guu = 0 and gvv = 0. The vector ∂u is actually tangent to the ingoing
radial null geodesics, which are defined by (v, θ, φ) = const, and the vector ∂v is tangent to the
outgoing ones, which are defined by (u, θ, φ) = const. Furthermore, the homothetic Killing
horizon H1 , which is defined by v/r = x1 , is the hypersurface u = 0 of NI .
Let us investigate the structure of the inner boundary v = 0 of the radiation shell in
the double-null coordinates (u, v, θ, φ). For r ̸= 0, the limit v → 0 in Eq. (15.52) leads
to u = −r1−x1 /x2 . This implies u < 0. In this part, which is the future boundary of the
Minkowski region of Vaidya spacetime, this relation defines a bijection between r ∈ (0, +∞)
and u ∈ (0, −∞). So we may say that r is a regular coordinate of NI for u < 0. On the other
side, the limit r → 0 in NI implies v → 0 since r > v/x2 in NI . Actually, the following relation
holds:
v ∼ x2 r in NI . (15.56)
r→0
This follows directly from Eq. (15.48), which implies the equivalence
r → 0 ⇐⇒ (x → x2 or x → +∞) (15.57)
608 Black hole formation 2: Vaidya collapse
Figure 15.10: Carter-Penrose diagrams of the subregions NI (right) and NII (left) of the radiation region Mrad .
The region between the homothetic Killing horizons H1 and H2 is common to both NI and NII . Zigzag lines
indicate a curvature singularity. The red curve in NII is the trapping horizon T [Eq. (15.72)].
along any outgoing radial null geodesic, irrespective of the value of u (see also Fig. 15.9). Given
that in NI , x → +∞ is excluded by the constraint x < x2 [Eq. (15.50)], there remains x → x2 ,
which yields the equivalence (15.56). Now, Eq. (15.56) implies that the numerator of Eq. (15.52)
is equivalent to (x2 /x1 − 1)r at the limit r → 0 and thus is positive, so that u > 0 on the part
of the boundary of NI where r → 0. We conclude that the “point” (t, r) = (v, r) = (0, 0) in
IEF coordinates becomes the hypersurface v = 0, u > 0 in the double-null coordinates. This
means that the IEF coordinates are not adapted to describe the vicinity of (t, r) = (0, 0) in
Vaidya spacetime when α < 1/8.
Example 4: Let us perform a first order expansion in v of√ the explicit expression of r(u, v) found in
Example 3 [Eq. (15.54)]. Taking into account the identity u2 = |u|, we get
r = u2 + 1 v + O(v 2 )
for u < 0
1
3 (15.58)
r = v + O(v 2 )
for u > 0.
6
The first equation agrees with the relation u = −r1−x1 /x2 found above for r ̸= 0 and v = 0, given that
x1 /x2 = 1/2 in the present case, while the last equation is exactly (15.56), given that x2 = 6.
As discussed above, the double null coordinates (u, v, θ, φ) are regular coordinates on
NI , because the components (15.55) of the metric tensor are regular (except for the standard
singularities of the spherical coordinates (θ, φ)). Another advantage of these coordinates is
that they allow for an immediate drawing of Carter-Penrose diagrams. It suffices to bring u to
a finite range by setting e.g. U = arctan u and use the coordinates T = U + v and X = v − U
to get such a diagram of NI , as depicted in Fig. 15.10.
Let us describe the (vicinity of the) curvature singularity in the double null coordinates. In
15.4 Configurations with a naked singularity 609
the present case, where M (v) = αv/2, the Kretschmann scalar (15.32) reduces to
v2
K = 12α2 , (15.59)
r6
where r is the function of (u, v) given implicitly by Eq. (15.52) (or explicitly by Eq. (15.54) in the
case α = 1/9). The above expression shows that K diverges only if r → 0. This limit occurs
only for v → 0 in the subregion u ≥ 0 of NI . There, Eq. (15.56) leads to K ∼ 12α2 x62 /v 4 ,
which actually diverges as v → 0. We conclude:
Property 15.15: curvature singularity in region N I
In NI , the curvature singularity corresponds to u ≥ 0 and v → 0.
This curvature singularity corresponds to the zigzag segment in right part of Fig. 15.10.
The NII region
On NII , let us introduce the parameter u′ such that
r0 = |u′ |−x1 /(x2 −x1 ) . (15.60)
Then Eq. (15.49) yields |u′ | = |v/x2 − r|/(v/x1 − r)x2 /x1 . Choosing the sign of u′ such that
u′ → −∞ at the outer boundary of NII (r → v/x1 ), we get
v/x2 − r
u′ = . (15.61)
(v/x1 − r)x2 /x1
We may then consider (u′ , v, θ, φ) as a coordinate system on NII .
Remark 1: Contrary to the range of u on NI , which is (−∞, +∞), the range of u′ on NII is bounded
from above by a function of v, which is given by Eq. (15.68) below.
Example 5: For α = 1/9, i.e. x1 = 3 and x2 = 6, Eq. (15.61) reduces to
v/6 − r
u′ = . (15.62)
(v/3 − r)2
This relation can be inverted, yielding an explicit expression for r(u′ , v):
r !
v 1 2 ′
r= + ′ 1− uv−1 . (15.63)
3 2u 3
As above, the metric components in coordinates (u′ , v, θ, φ) are deduced from those in
coordinates (v, r, θ, φ) [Eq. (15.17)] (cf. the notebook D.6.6 for the computation):
x2 /2
2x1 v v
′ 2 2 2 2
(15.64)
g=− −r du dv + r dθ + sin θ dφ r< ,
(x2 − x1 )r x1 x1
610 Black hole formation 2: Vaidya collapse
where r = r(u′ , v) is defined implicitly by Eq. (15.61). Again, we get a double-null coordinate
system adapted to the ingoing and outgoing radial null geodesics. Moreover, the coordinates
(u′ , v, θ, φ) are regular since gu′ v is finite and non-vanishing in all NII . In addition, the homo-
thetic Killing horizon H2 , which is defined by v/r = x2 , is the hypersurface u′ = 0 of NII . In
NI ∩ NII , we have u > 0 and u′ < 0. The relation between the two coordinates is obtained by
equating the right-hand sides of Eqs. (15.51) and (15.60); one gets
u′ = −u−x2 /x1 in NI ∩ NII , (15.65)
along with the following characterization of the two homothetic Killing horizons (cf. Fig. 15.10):
H1 : u = 0, u′ → −∞ (15.66a)
H2 : u → +∞, u′ = 0. (15.66b)
Example 6: For α = 1/9 (x1 = 3 and x2 = 6), Eq. (15.65) reduces to
1
u′ = − in NI ∩ NII . (15.67)
u2
Substituting this expression for u′ in Eq. (15.63), one recovers Eq. (15.54).
Let us locate the curvature singularity at the boundary of NII . A necessary condition for
the Kretschmann scalar (15.59) to diverge is r(u′ , v) → 0. According to Eq. (15.57), this occurs
along an outgoing radial null geodesic if, and only if, x → x2 or x → +∞. Now, let us rewrite
Eq. (15.61) by substituting xr for v:
x/x2 − 1
u′ = .
(x/x1 − 1)x2 /x1 rx2 /x1 −1
Since u′ is fixed along the outgoing radial null geodesic, we recover from this expression the
two limits x → x2 or x → +∞ in which r(u′ , v) → 0 may occur. The first limit corresponds
to v ∼ x2 r (actually the same behavior as (15.56) in NI ), which implies v → 0. This leads to
K ∼ 12α2 x62 /v 4 , which does diverge for v → 0. For the second limit, x → +∞, we get
x /x
x1 2 1
u′ ∼ .
x2 (xr)x2 /x1 −1
This implies that xr = v tends to a finite value. Then K = 12α2 v 2 /r6 diverges as r−6 there.
We conclude:
Property 15.16: curvature singularity in region N II
In NII , the limit r → 0 always corresponds to a curvature singularity. This singularity is
reached in two places, which constitute parts of the boundary of NII : (i) v → 0 (the zigzag
x /x
segment inclined at 45◦ in Fig. 15.10 left) and (ii) u′ v x2 /x1 −1 → x1 2 1 /x2 (the horizontal
zigzag segment in Fig. 15.10 left). In particular, the ranges of the coordinates (u′ , v) in NII
15.4 Configurations with a naked singularity 611
are
x /x
x1 2 1
0 < v ≤ v0 and − ∞ < u′ < . (15.68)
x2 v x2 /x1 −1
Example 7: For α = 1/9, i.e. x1 = 3 and x2 = 6, Eq. (15.63) implies
lim r = 0 and lim r = 0. (15.69)
v→0 u′ v→3/2
The second limit agrees with the upper boundary for u′ in (15.68), since the latter can be rewritten as
u′ < 3/(2v) for x1 = 3 and x2 = 6.
As a particular case of Eq. (15.68), the maximal value of u′ along an ingoing radial null
geodesic with v = v0 (geodesic on the outer edge of the radiation shell, cf. Fig. 15.10) is
x /x1
x1 2
u′0 = x /x1 −1
. (15.70)
x2 v0 2
15.4.3 Carter-Penrose diagram of the whole spacetime
A Carter-Penrose diagram of the whole Vaidya spacetime can be constructed by assembling the
Carter-Penrose diagrams obtained for the radiation region Mrad (Fig. 15.10) with Carter-Penrose
diagrams of the Minkowski region (v < 0) and of the Schwarzschild region (v > v0 ). To achieve
this, one should locate the black hole event horizon H in Mrad . Given that xA = α−1 > x1
(cf. Sec. 15.4.1), one has H ∩ Mrad ⊂ NII . Since H is generated by outgoing radial null
geodesics, H ∩ Mrad lies a constant value of u′ , which we shall denote by u′H . The value of
u′ H is obtained by setting v = v0 and r = 2m in Eq. (15.61). Using the identities α = 2m/v0
and α = 2/(x1 x2 ), we get
−x2 /x1
2 2
u′H = u′0 1− 1− . (15.71)
x1 x2
From the expressions of x1 and x2 in terms of α [Eq. (15.44)], it is easy to show that u′H < u′0 .
Taking this property into account, one can draw H in the diagram of NII of Fig. 15.10 and
finally obtain the Carter-Penrose diagram of M shown in Fig. 15.11.
Example 8: For v0 = 18m, one has α = 1/9, x1 = 3 and x2 = 6, so that Eqs. (15.70) and (15.72) yield
respectively u′0 = 3/(2v0 ) = 1/(12m) and u′H = 3u′0 /4 = 9/(8v0 ) = 1/(16m).
The Carter-Penrose diagram of Fig. 15.11 enables us to fully solve the puzzle of multiple
outgoing radial null geodesics emanating from the point (t, r) = (0, 0) raised at the beginning
of Sec. 15.4.2: (t, r) = (0, 0), or equivalently, (v, r) = (0, 0), does not define a single point, but
a full segment, denoted by O1 O3 in Fig. 15.11. From each point of this segment, there emanates
a single outgoing radial null geodesic (cf. the green solid lines in Fig. 15.11).
It is instructive to add the future outer trapping horizon T introduced in Sec. 15.3.5 to
the Carter-Penrose diagram. To do so, we need the equation of T in terms of the double null
coordinates. First of all, we notice that in Mrad , T is entirely contained in NII . Indeed, T
612 Black hole formation 2: Vaidya collapse
Figure 15.11: Carter-Penrose diagram of the collapsing Vaidya spacetime with M (v) = αv/2 (homothetic
radiation region) and α < 1/8 (low energy density case). The Minkowski region MMin , radiation region Mrad
and Schwarzschild region MSch are depicted in respectively pale blue, yellow and grey. Solid (resp. dashed) green
lines are outgoing (resp. ingoing) radial null geodesics. Zigzag lines indicate curvature singularities. The hatched
area is the black hole region, delimited by the event horizon H . The red dot-filled area, delimited by the future
outer trapping horizon T (red curve), is the region where the spheres (t, r) = const are trapped surfaces. The
points A, B and C are the same as in Fig. 15.8, while the point O of Fig. 15.8 corresponds to the whole segment
O1 O3 .
obeying r = αv [Eq. (15.35)], the property α < x−1 2 excludes T from NI , by the definition
(15.50) of the latter. It suffices then to express T in terms of the coordinates (u′ , v) of NII . This
is achieved by substituting αv for r in Eq. (15.61) and making use of successively the identity
α = 2/(x1 x2 ), Eq. (15.71) and Eq. (15.70); one gets
v x2 /x1 −1
0
T ∩ Mrad : ′
u = u′H . (15.72)
v
Note that this implies u′ → +∞ for v → 0 and u′ → u′H for v → v0 (cf. Fig. 15.10).
Example 9: For our favorite example α = 1/9 (x1 = 3 and x2 = 6), the equation of T ∩ Mrad reduces
to u′ = u′H v0 /v, which corresponds to a branch of hyperbola in the (u′ , v) plane.
Historical note : The split of Mrad in two regions, denoted here NI and NII , in order to get regular
double-null coordinate systems, has been performed by B. Waugh and Kayll Lake in 1986 [516]. The
double-null coordinates (u, v) and (u′ , v) introduced above are theirs, except for some constant factors.
Four years before, William A. Hiscock, Leslie G. Williams and Douglas M. Eardley [279] exhibited a
15.4 Configurations with a naked singularity 613
Figure 15.12: Same as Fig. 15.11, with in addition the remote observer O, the partial Cauchy surface Σ and its
future Cauchy development D+ (Σ) (blue hatched area, cf. Sec. 10.8.3 for the definition), which is bounded in the
future by the Cauchy horizon HC and the future null infinity I + .
Carter-Penrose diagram similar to that of Fig. 15.11 except that the Cauchy horizon (to be discussed
below) and the event horizon coincide (Fig. 2 in Ref. [279]). Such a configuration is obtained by adding a
radiation surface layer (Dirac delta) at v = v0 . However, as discussed in Ref. [516], these authors have
not set up regular double-null coordinates in Mrad , so that their construction can be seen as heuristic.
In 1984, Yuhji Kuroda [330] exhibited a Carter-Penrose diagram again similar to that of Fig. 15.7, except
for the radiation region extending to I + (no pure Schwarzschild exterior but M ′ (v) → 0 for v → +∞)
(Fig. 3 in Ref. [330]). The case of a radiation region bounded by v = v0 , as here, can be found in Fig. 1 of
a 2007 article by Brien C. Nolan [384].
15.4.4 Naked singularity and Cauchy horizon
We recover on the Carter-Penrose diagram of Fig. 15.11 the naked singularity feature noticed in
Sec. 15.4.1: light rays emitted by the curvature singularity between O1 and O2 reach the future
null infinity I + of the Schwarzschild exterior. This means that the singularity is visible to
remote observers, such as observer O in Fig. 15.12. This contrasts with the curvature singularity
of the Vaidya collapse with α > 1/8 (compare Fig. 15.7 with Fig. 15.11). The latter is indeed
located at the “future boundary” of spacetime, so that no causal geodesic can originate from it.
Remark 2: The part O2 O3 of the curvature singularity (cf. Fig. 15.11) is visible to observers that fall
into the black hole, but remains hidden to remote observers. For this reason, the singularity O2 O3 is
614 Black hole formation 2: Vaidya collapse
called locally naked, while O1 O2 is called globally naked. For remote observers, one may say that the
singularity O2 O3 is “clothed” by the event horizon H .
Remark 3: The curvature singularity of the Vaidya spacetime with v0 < 16m (Fig. 15.7) is not even
locally naked: observers falling into the black hole region never see it until they hit it at the end of their
lives.
The part (v, r) = (0, 0) of the curvature singularity, i.e. the segment O1 O3 in the diagram
of Fig. 15.11 corresponds to a past null boundary of spacetime. Such a singularity is generically
called a shell-focusing singularity. This terms has been coined by Eardley & Smarr [176], who
discovered this type of singularity in spherically symmetric inhomogeneous dust collapse. Due
to its past null character, a shell-focusing singularity is always locally naked: close observers
can see it.
Remote observers see the naked singularity as soon as they cross the null hypersurface HC
generated by the outgoing radial null geodesics emanating from the birth of the singularity
at u = 0 in NI (point O1 in Figs. 15.11 and 15.12 ). HC can be seen as the null hypersurface
extending the homothetic Killing horizon H1 introduced in Sec. 15.4.1 beyond the radiation
region, i.e. one has HC ∩ Mrad = H1 . The hypersurface HC is actually the future Cauchy
horizon6 of any partial Cauchy surface Σ that encounters the Minkowkski region7 . In other
words, the region of spacetime in the future of HC cannot be entirely predicted from initial data
on Σ and the Einstein equation. In particular, data from the naked curvature singularity O1 O3
is required. Since a curvature singularity marks a validity limit of general relativity, one cannot
unambiguously prescribe data there. This seriously hampers the predictability of general
relativity alone for describing the region beyond HC . For instance, a burst of radiation could
come out at any time from the naked singularity and change the fate of observers travelling
in this region. This regards all remote observers at a sufficiently advanced amount of their
proper time, since any such observer necessarily crosses HC at some point (cf. observer O in
Fig. 15.12, noticing that the worldlines of all remote observers terminate at the future timelike
infinity i+ ).
15.5 Going further
For simplicity, we have mostly restricted ourselves to M (v) linear in the radiation region,
except in Secs. 15.3.4 and 15.3.5. This allowed us to benefit from an extra symmetry (homothetic
Killing vector) and to get exact solutions for the outgoing radial null geodesics. The obtained
results are nevertheless representative of those for more general functions M (v). In particular,
the criterion for having a shell-focusing singularity with a generic function M (v) remains
α < 1/8, provided that α is defined as
2M (v)
α := lim+ . (15.73)
v→0 v
6
Cf. Sec. 10.8.3 for the definition.
7
Since it has to be acausal without edge (cf. Sec. 10.8.3), Σ has to meet the spacelike infinity i0 in the
Carter-Penrose diagram of Fig. 15.12.
15.5 Going further 615
See Ref. [330, 194] for details. Of course, for M (v) = mv/v0 , the above definition of α reduces
to (15.18). For α < 1/8, a new possibility may occur: that of a shell-focusing singularity that
lies entirely in the black hole region, i.e. that is naked locally but not globally, contrary to what
happens for M (v) linear (Sec. 15.4). See e.g. Fig. 4 of Ref. [27] or Fig. 9.20 of Ref. [243] for the
corresponding Carter-Penrose diagram. Actually, the Vaidya collapse depicted in Fig. 15.5 lies
in this category: it has M (v) = m(v/v0 )3 [6(v/v0 )2 − 15v/v0 + 10], so that Eq. (15.73) yields
α = 0. The singularity is entirely located in the black hole region and appears to be locally
naked (note the outgoing radial null geodesic emerging from it near (t, r) = (0, 0) in Fig. 15.5).
It is worth to stress that in the literature, many studies regard the outgoing Vaidya metric,
starting from the original studies (cf. the historical note on p. 586), because they are motivated
either by the modelling of radiating bodies or the study of black hole evaporation. Nonetheless,
most results are applicable to Vaidya collapse (which is based on the ingoing version of the
metric) by performing some time reversal. For instance, one can infer from the recent study
[143] that for an imploding Vaidya spacetime without any Minkowksi region (v spanning
the range (−∞, v0 ) in the radiation region), but such that limv→−∞ M (v) = 0, there is a
curvature singularity at the null boundary v → −∞ of spacetime, whatever the choice of the
non-decreasing mass function M (v).
616 Black hole formation 2: Vaidya collapse
Chapter 16
Evolution and thermodynamics of black
holes
Contents
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
16.2 Towards the first law of black hole dynamics . . . . . . . . . . . . . . 617
16.3 Evolution of the black hole area . . . . . . . . . . . . . . . . . . . . . . 635
16.4 The laws of black hole dynamics . . . . . . . . . . . . . . . . . . . . . . 642
16.5 Hawking radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
16.6 Black hole thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 650
16.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
16.1 Introduction
This chapter is in a draft stage.
16.2 Towards the first law of black hole dynamics
16.2.1 Mass variation formula for Kerr black holes
In this section, we assume 4-dimensional general relativity. Let us consider an initially isolated
Kerr black hole of mass and spin parameters (m, a) that is perturbed by the arrival of some
external body or some gravitational radiation. After some transitory dynamical regime (e.g.
absorption of the incoming body and emission of gravitational waves), the black hole relaxes
to a new equilibrium configuration. According to the no-hair theorem (Property 5.38), the final
state has to be a Kerr black hole, of parameters (m + δm, a + δa) say. All global properties of the
black hole (cf. Sec. 10.6) are changed accordingly and we are going to express the change in the
618 Evolution and thermodynamics of black holes
√
Komar mass M = m [Eq. (10.78)] in terms of the change in the area A = 8πm(m + m2 − a2 )
[Eq. (10.83)] and in the angular momentum J = am√ [Eq. (10.80)].
Rewriting formula (10.83) as A = 8π(M + M 4 − J 2 ) and differentiating, we get, for
2
small variations δM and δJ of M and J,
1 2M 3 J
δA = 2M δM + √ δM − √ δJ,
8π M4 − J2 M4 − J2
or equivalently
√
1 M4 − J2 J
δM = √ δA + √ δJ,
8π 2M (M 2 + M 4 − J 2 ) 2M (M 2 + M 4 − J 2 )
| {z } | {z }
κ ΩH
where the identifications of the black hole’s surface gravity κ and rotation velocity ΩH result
from Eqs. (10.77) and (10.69) respectively. Hence we get
κ
δM = δA + ΩH δJ . (16.1)
8π
16.2.2 General mass variation formula
The mass variation formula (16.1) can be derived in a much more general framework, without
assuming that it corresponds to changes between two nearby Kerr solutions and without
restricting the spacetime dimension to 4 or assuming that the event horizon is connected. It
even holds for other gravity theories than general relativity, more specifically for any theory
based on a diffeomorphism-covariant Lagrangian, provided that the area A is replaced by a
quantity named the Wald entropy, which reduces to A for general relativity, as we shall discuss
in Chap. 17.
Here we establish the mass variation formula for two nearby black hole equilibrium config-
urations in general relativity, from integral mass formulas obtained in Chap. 5. More precisely,
we consider a stationary spacetime (M , g) of dimension n ≥ 4 containing a black hole and
a “nearby” black hole spacetime (M , g + δg), which has the same symmetries (stationarity
and possible axisymmetries) as (M , g). We shall call (M , g + δg) the perturbed spacetime,
although we do not require that (M , g + δg) is obtained from (M , g) by some specific physical
perturbation. Note that the same manifold M is used for both spacetimes. There is no loss of
generality in doing so, since the perturbed manifold must be diffeomorphic to the original one,
which allows one to identify the two manifolds. In particular, both manifolds have the same
topology: one would certainly not qualify as “nearby” a manifold with a distinct topology.
Property 16.1: mass variation formula for generic black holes
Let (M , g) be a stationary spacetime of dimension n ≥ 4 (stationary Killing vector ξ) that
contains a black hole, the event horizon of which has K ≥ 1 connected components H1 ,
. . ., HK . We shall assume that each Hk is a Killing horizon with respect to a Killing vector
χk . This is guaranteed with χk = ξ if Hk is non-rotating (ξ null on all Hk ; Property 5.23),
16.2 Towards the first law of black hole dynamics 619
while if Hk is rotating (ξ spacelike on some parts of Hk ), this holds under the hypotheses
of the strong rigidity theorem (Property 5.25), with
Lk
X (i)
χk = ξ + ΩHk ηHk (i) , (16.2)
i=1
(i)
where 1 ≤ Lk ≤ [(n − 1)/2], the ΩHk ’s are constants and the ηHk (i) ’s are axisymmetric
Killing vectors (Property 5.26). We shall encompass both the non-rotating case and the
rotating one by allowing Lk to take the value 0 in Eq. (16.2) so that χk = ξ is recovered
(i)
if Hk is non-rotating. Let JHk be the Komar angular momentum with respect to ηHk (i)
over any cross-section of Hk [Eq. (5.63)]. Being a non-expanding horizon, each Hk has a
well defined area Ak (Property 3.1). Let κk be the surface gravity of Hk , i.e. the coefficient
such that ∇χk χk = κk χk on Hk [cf. Eq. (3.29)]. Let us assume that the null dominance
condition (3.43) is fulfilled on Hk or that Hk is part of a bifurcate Killing horizon; by the
zeroth law of black hole dynamics (Property 3.10 or 3.16), this implies that κk is constant.
Finally let us assume that (M , g) is asymptotically flat with g having the same asymptotic
behavior (5.31) as in general relativitya . Let (M , g + δg) be a nearby stationary spacetime
sharing the same characteristics as (M , g). Then, the change δM∞ in Komar mass at
infinity (cf. Property 5.16) between the two spacetimes is related to the changes δAk in
(i)
area and to the changes δJHk in Komar angular momentum by
K Lk
!
X κk X (i) (i)
δM∞ = δAk + ΩHk δJHk
k=1
8π i=1
Z Z
1 1
+ µν
G δgµν ξρ dV − ρ
δ Gµν ξ µ dV ν , (16.3)
16π Σ 8π Σ
where (i) Σ is any asymptotically flat spacelike hypersurface, the inner boundary of which
is an axisymmetric cross-section S of the event horizon, i.e. S = ∪K k=1 Sk with Sk :=
Σ ∩ Hk and ηHk (i) tangent to Sk for i ∈ {1, . . . , Lk }, (ii) G is the Einstein tensor of g and
(iii) dV is the normal volume element vector of Σ, as defined by Eq. (5.48).
a
Note that g is not required to obey the Einstein equation in M .
Proof. We may consider that the black hole event horizon H is the same hypersurface in
both spacetimes (M , g) and (M , g + δg). If the horizons would differ, we could find a
diffeomorphism M → M that would map the original horizon to the perturbed spacetime’s
horizon. Similarly, there is no loss of generality in considering that the Killing vectors ξ and
ηHk (i) generating the stationarity and axisymmetries are identical, as vector fields on M :
δξ = 0 and δηHk (i) = 0. (16.4)
This amounts to identifying the orbits of the isometry group actions in the two spacetimes.
Given that H is globally invariant by these group actions, this requirement is compatible
with the identification of H in both spacetimes. Let us introduce the short-hand notation
h := δg, or in index notation hαβ = δgαβ . In what follows, indices are raised or lowered with
620 Evolution and thermodynamics of black holes
the unperturbed metric g. In particular, hαβ := g αµ g βν hµν . Note that hαβ ̸= δg αβ . Actually, by
variation of the identity g αµ gµβ = δ αβ , one gets δg αβ = −hαβ . The starting point for proving
(16.3) is the variation of the generalized Smarr formula (5.84):
K
" Lk
#
2(n − 3) X 1 X (i) (i) (i) (i)
δM∞ = (Ak δκk + κk δAk ) + 2 JHk δΩHk + ΩHk δJHk
n−2 k=1
4π i=1
Z
1
− δ Rµν ξ ν dV µ , (16.5)
4π Σ
where Σ is any asymptotically flat spacelike hypersurface, the inner boundary of which is
a cross-section of the event horizon, i.e. some union of cross-sections Sk of the connected
components Hk and dV µ is the normal volume element vector of Σ defined by Eq. (5.48).
Let us start by evaluating the term Ak δκk in the above formula. For the sake of brevity, we
shall drop the index k in what follows, given that we temporarily focus on a single connected
component Hk of the event horizon H . The surface gravity κ (= κk ) is given by Eq. (3.31):
2κ = k µ ∂µ (χν χν ), where χ (= χk ) is the Killing vector normal to the Killing horizon Hk
and k is the future-directed null vector field defined on Hk , transverse to Hk , normal to the
cross-section Sk = Hk ∩ Σ and normalized by χ · k = −1. Note that (χ, k) is a null basis of
the normal plane Tp⊥ Sk at each point p ∈ Sk , χ playing the role of the vector ℓ in Fig. 2.10.
Varying the above expression of κ yields
2δκ = δk µ ∂µ (χν χν ) + k µ ∂µ (δχν χν + χν δχν )
= δk µ ∇µ (χν χν ) + k µ ∇µ (δχν χν + χν δχν )
= 2δk µ χν ∇µ χν + k µ (χν ∇µ δχν + δχν ∇µ χν + δχν ∇µ χν + χν ∇µ δχν )
= 2δk µ χν ∇µ χν + k µ [χν (∇µ δχν + ∇ν δχµ ) + 2δχν ∇µ χν + δχν ∇µ χν + χν ∇µ δχν ]
= 2δk µ χν ∇µ χν + (χµ k ν + k µ χν )∇µ δχν + k µ (2δχν ∇µ χν + δχν ∇µ χν + χν ∇µ δχν ) .
To get the last but one line, we have used the identity χν ∇ν δχµ +δχν ∇µ χν = 0, which expresses
the vanishing of the Lie derivative of the 1-form δχ along χ [Eq. (A.86) with (k, ℓ) = (0, 1)]:
Lχ δχ = 0. (16.6)
The invariance property (16.6) holds because χ is a normal 1-form to Hk (i.e. a vector v is
tangent to Hk iff ⟨χ, v⟩ = 0) and since Hk is the same hypersurface in the original and the
perturbed spacetime, χ + δχ is a normal 1-form of Hk as well. Two normal 1-forms to a given
hypersurface are necessarily collinear: there exists a scalar field λ such that χ + δχ = λχ.
Setting δλ := λ − 1, we get
δχ = δλ χ. (16.7)
Then Lχ δχ = (Lχ δλ) χ + δλ Lχ χ. But Lχ χ = 0 for χ is a Killing vector of (M , g) and,
thanks to Eq. (16.2), Lχ δλ = Lξ δλ + Li=1 Ω(i) Lη(i) δλ = 0 + 0 = 0 because ξ and η(i) are
P
symmetry generators of both (M , g) and (M , g + δg) [cf. Eq. (16.4)]; this establishes (16.6).
On the other side, we have
L
X L
X
α
δχ = (i)
δΩ α
η(i) and δχα = hαµ χ + µ
δΩ(i) η(i)α . (16.8)
i=1 i=1
16.2 Towards the first law of black hole dynamics 621
The first formula readily follows from Eqs. (16.2) and (16.4), while the second one follows
from χα = gαµ χµ , which implies δχα = δgαµ χµ + gαµ δχµ , where δgαµ =: hαµ . By means of
Eq. (16.8), we can rewrite the last two terms in the above expression of δκ as1
L
X L
X
ν ν (i) ν ν
δΩ(i) χν ∇µ η(i)
ν
δχ ∇µ χν + χν ∇µ δχ = δΩ η(i) ∇µ χν + χν ∇µ η(i) =2 .
i=1 i=1
The last equality follows from η(i)
ν ν
∇µ χν = −η(i) ∇ν χµ (Killing equation for χ) = −gµσ η(i)
ν
∇ν χσ =
−gµσ χ ∇ν η(i) (χ and η(i) commute, cf. Property 5.26) = −χ ∇ν η(i)µ = χ ∇µ η(i)ν (Killing
ν σ ν ν
equation for η(i) ). Accordingly, the formula for δκ becomes
L
1 X
δκ = (k µ χν + χµ k ν )∇µ δχν + δΩ(i) k µ χν ∇µ η(i)ν + (δk µ χν + k µ δχν )∇µ χν . (16.9)
2 i=1
| {z }
A
Let us show that A = 0. Thanks to Eq. (16.7), we have A = (δk µ χν + δλ k µ χν )∇µ χν =
(δk µ + δλ k µ )∇µ (χν χν )/2. Now, since Hk is a Killing horizon, we have ∇µ (χν χν ) = 2κχµ on
Hk [Eq. (3.31)]. This yields A = κχµ (δk µ + δλ k µ ) = κ(χµ δk µ − δλ), since χµ k µ = −1 from
the definition of k. Now, the variation of χµ k µ = −1 gives δχµ k µ + χµ δk µ = 0, which in view
of Eq. (16.7), can be rewritten as δλ χµ k µ + χµ δk µ = 0, or equivalently −δλ + χµ δk µ = 0,
hence A = 0. Let us now consider the first term in the right-hand side of Eq. (16.9); we may
rewrite it as
(χµ k ν + k µ χν )∇µ δχν = (q µν − g µν )∇µ δχν ,
where q µν stands for the double metric dual of the metric q induced by g on the cross-section
Sk of Hk : q = g + χ ⊗ k + k ⊗ χ (cf. Eq. (2.34) with ℓ standing for χ). Now, thanks to
Eq. (16.7),
q µν ∇µ δχν = q µν ∇µ (δλχν ) = ∇µ δλ q µν χν +δλ q µν ∇µ χν = 0,
| {z } | {z }
0 0
where q χν = 0 holds for χ is normal to Sk (cf. Eq. (2.48) with ℓ standing for χ) and
µν
q µν ∇µ χν = 0 follows from ∇µ χν = (aµ χν − aν χµ )/2 on Hk [Eq. (3.34)] combined with
q µν χν = 0. Hence
L
!
X
(χµ k ν + k µ χν )∇µ δχν = −g µν ∇µ δχν = −∇µ δχµ = −∇µ hµν χν + δΩ(i) η(i)µ
i=1
L
X
= −χν ∇µ hµν − hµν ∇µ χν − δΩ(i) ∇µ η(i)µ = −χν ∇µ hµν ,
| {z } | {z }
i=1
0 0
where Eq. (16.8) has been used to express δχµ and hµν ∇µ χν = 0 and ∇µ η(i)µ = 0 follow from
the Killing equation for respectively χ and η(i) , given the symmetry of h. Together with A = 0,
this result allows us rewrite Eq. (16.9) as
L
1 X
δκ = − χµ ∇ν hµν + δΩ(i) k µ χν ∇µ η(i)ν . (16.10)
2 i=1
1
Note that ∇µ δΩ(i) = 0 since both Ω(i) and Ω(i) + δΩ(i) are constant.
622 Evolution and thermodynamics of black holes
√
Let us integrate this relation over the cross-section Sk , setting dS := q dn−2 x (the area
element of Sk ). Since δκ and δΩ(i) are constant, we get
Z Z L Z
1 X
δκ dS = − ν µ
χµ ∇ h ν dS + δΩ(i)
∇µ η(i)ν k µ χν dS . (16.11)
2 Sk
| S{z
k
} | {z } i=1 | Sk {z }
Ak ISk −8πJH
(i)
k
(i)
The identification of the last integral with −8πJHk readily follows from formula (5.65) for
the Komar angular momentum, once the area element normal bivector to Sk is expressed as
dS µν = (χµ k ν − k µ χν ) dS [Eq. (5.86)]. As for the integral denoted ISk in Eq. (16.11), it can be
rewritten as an integral involving h and the stationary Killing vector ξ instead of the Killing
vector χ (which depends on Hk , contrary to ξ), namely
Z
1 1
ωµν dS µν , with ωαβ := (16.12)
ISk = ξ ∧ H αβ = (ξα Hβ − Hα ξβ ) ,
Sk 2 2
where H is the vector field defined by
H α := ∇µ hαµ − ∇α hµµ = g αρ g µν (∇ν δgρµ − ∇ρ δgµν ) . (16.13)
To prove (16.12), let us evaluate ωµν dS µν , using successively Eq. (5.86), ξµ χµ = 0 (ξ tangent to
Hk ) and η(i)µ k µ = 0 (η(i) tangent to Sk ):
1
ωµν dS µν = (ξµ Hν − Hµ ξν ) (χµ k ν − k µ χν ) dS = (ξµ χµ Hν k ν − ξµ k µ Hν χν ) dS
2 |{z}
0
L
X
= − χµ k µ − Ω(i) η(i)µ k µ Hν χν dS = χµ H µ dS
| {z } | {z }
i=1
−1 0
= (χµ ∇ν hµν − χµ ∇µ hνν ) dS
X L
= (χµ ∇ν hµν − Lξ hνν − Ω(i) Lη(i) hνν ) dS = χµ ∇ν hµν dS.
| {z }
i=1
| {z }
0 0
In the last line, Lξ hνν = 0 follows from hνν = g µν hµν , Lξ g µν = 0 (ξ Killing vector of g)
and Lξ hµν = 0, the latter being a consequence of ξ being a Killing vector of both g and
g + δg = g + h (cf. Eq. (16.4)), so that Lξ h = Lξ (g + δg) − Lξ g = 0 − 0 = 0. Similarly,
Lη(i) hνν = 0. In view of the above result and the definition of ISk in Eq. (16.11), we conclude
that Eq. (16.12) holds. Therefore, Eq. (16.11) can be rewritten as
Z L
1 X (i) (i)
Ak δκk = − ωµν dS µν
− 8π JHk δΩHk , (16.14)
2 Sk i=1
where we have restored the index k on κ and the subscript Hk on Ω(i) . If we plug this relation
16.2 Towards the first law of black hole dynamics 623
(i) (i)
into Eq. (16.5), the terms JHk δΩHk cancel out and we are left with
K Lk
! K Z
(n − 3) X κk X (i) (i) 1 X
δM∞ = δAk + ΩHk δJHk − ωµν dS µν
n−2 k=1
8π i=1
16π k=1 Sk
Z
1
− δ Rµν ξ ν dV µ .
8π Σ
We note that the inner boundary of the hypersurface Σ is Sint = S = ∪K k=1 Sk , so that the
sum of the integrals over the Sk ’s is actually an integral over Sint and we may apply formula
(5.49) regarding the flux of a 2-form (here ω) to write
K Lk
!
(n − 3)
Z
X κk X (i) (i) 1
δM∞ = δAk + ΩHk δJHk − ωµν dS µν
n−2 k=1
8π i=1
16π S∞
Z Z
1 1
+ ∇ν ωµν dV µ − δ Rµν ξ ν dV µ , (16.15)
8π Σ 8π Σ
where S∞ stands for the outer boundary of Σ, i.e. the limit r → +∞ of a sphere S of
constant value of (x0 , r), (xα ) being an asymptotically Minkowskian coordinate system such
that Σ is a hypersurface x = const, ξ = ∂0 and r := (x ) + · · · + (xn−1 )2 . In view of
0
p
1 2
Eqs. (16.12)-(16.13) and (5.42), we may write
√
Z Z Z
µν µν
ωµν dS = ξµ Hν dS = ξµ (∇σ hνσ − ∇ν hσσ )(sµ nν − nµ sν ) q dn−2 y,
S∞ S∞ S∞
where (ya )1≤a≤n−2 is a coordinate system of S∞ and q is the determinant with respect to it of the
metric q induced by g on S∞ . Now, on S∞ , n = ξ and s = (xi /r)∂i , with ξµ nµ = ξµ ξ µ = −1
and ξµ sµ = 0. Moreover, since (xα ) is asymptotically Minkowskian, we may substitute the
covariant derivatives with partial ones. Hence we get, accounting for ∂0 hi0 = 0 (h = δg is
stationary, since both g and g + δg are),
xi √
Z Z
µν
ωµν dS = (∂j hij − ∂i hσσ ) q dn−2 y.
S∞ S∞ r
By hypothesis, the metric tensor has the asymptotic behavior (5.31); it follows that, for r → +∞,
δM∞ 1 αn δM∞ 1
h00 = αn n−3 + O n−2 and hij = δij + O n−2 ,
r r n − 3 rn−3 r
where αn is given by Eq. (5.32). According to Property 5.11, δM∞ in the above formulas is the
same variation of the Komar mass at infinityPas in the left-hand side of Eq. (16.15). The trace of
h takes the asymptotic value hσσ = −h00 + n−1 i=1 hii = 2αn δM∞ /((n − 3)r
n−3
) + O (1/rn−2 ).
Hence we get
xi ∂ √ n−2
√
Z Z Z
µν αn δM∞ 1 n−2
ωµν dS = − i n−3
q d y = αn δM∞ q̄ d y ,
S∞ n − 3 S∞ r ∂x r |{z}
√ S∞
| {z } rn−2 q̄ | {z }
−(n−3)/rn−2 Ωn−2
624 Evolution and thermodynamics of black holes
where q̄ stands for the determinant of the round metric of the unit sphere Sn−2 with respect to
the coordinates (y a ). Since αn Ωn−2 = 16π/(n − 2) [cf. Eq. (5.32)], there comes
Z
16π
ωµν dS µν = δM∞ .
S∞ n−2
Accordingly, Eq. (16.15) simplifies to
K L
!
k Z Z
X κk X (i) (i) 1 ν 1
µ
δM∞ = δAk + ΩHk δJHk + ∇ ωµν dV − δ Rµν ξ ν dV µ .
k=1
8π i=1
8π Σ 8π Σ
Now, from Eq. (16.12),
2∇ν ωµν dV µ = ∇ν (ξ µ H ν − H µ ξ ν )dVµ = (ξ µ ∇ν H ν + H ν ∇ν ξ µ − ξ ν ∇ν H µ −H µ ∇ν ξ ν )dVµ
| {z } | {z }
0 0
= ∇ν H ν ξ µ dVµ ,
where ∇ν ξ ν = 0 holds because ξ is a Killing vector and we have used H ν ∇ν ξ µ − ξ ν ∇ν H µ =
[H, ξ]µ = −[ξ, H]µ = −Lξ H µ = 0 because Lξ h = 0 and Eq. (16.13) implies Lξ H = 0, for
∇ and Lξ commute, ξ being a Killing vector2 . Hence
K Lk
! Z Z
X κk X (i) (i) 1 ν µ 1
δM∞ = δAk + ΩHk δJHk + ∇ν H ξ dVµ − δ Rµν ξ ν dVµ .
k=1
8π i=1
16π Σ 8π Σ
Let us make the Einstein tensor G appear in the last integral via Eq. (A.111): Rµν ξ ν dVµ =
Gµν ξ ν dVµ + R ξ µ dVµ /2. Given that δξ µ = 0 [Eq. (16.4)], we have δ(R ξ µ dVµ ) = ξ µ (δR dVµ +
R δ(dVµ )), so that we may write
K Lk
! Z
X κk X (i) (i) 1 1
δM∞ = δAk + ΩHk δJHk + I− δ Gµν ξ ν dVµ , (16.16)
k=1
8π i=1
8π 8π Σ
with Z
1
I := ξ µ [∇ν H ν dVµ − (δR dVµ + R δ(dVµ ))] .
2 Σ
√
Now from Eq. (5.48), dVα = −nα γ dn−1 x, where (xi )1≤i≤n−1 are coordinates on Σ, γ is the
determinant with respect to (xi ) of the metric γ induced by g on Σ and the normal 1-form nα
is collinear to the differential of t, the latter being a scalar field defining Σ as a hypersurface
√ √
t = const: nα = −N (dt)α = −N ∂α t. Given the identity −g = N γ (see e.g. Eq. (5.55) of
√
Ref. [227]), we have dVα = ∂α t −g dn−1 x. Since obviously δ(∂α t) = 0 and δ(dn−1 x) = 0, we
get
√ 1 √
δ(dVα ) = ∂α t δ −g dn−1 x = √ δ −g dVα . (16.17)
−g
2
This is easy to see in a coordinate system (xα ) such that ξ = ∂0 , cf. Eq. (A.85); see also Appendix E2 of
Ref. [424] for a covariant proof.
16.2 Towards the first law of black hole dynamics 625
Hence
√ µ
Z
1 ν 1
I= ∇ν H − √ δ R −g ξ dVµ .
2 Σ −g
3
Now, a standard identity , which is at the root of the derivation of the Einstein equation by
extremalizing the Einstein-Hilbert action, is
√ √
δ R −g = (Gµν δg µν + ∇µ H µ ) −g. (16.18)
It’s rather straightforward to establish (16.18) from the Palatini identity 4 :
δRαβ = ∇µ δΓµαβ − ∇β δΓµαµ , (16.19)
where the variations δΓγαβ of the Christoffel symbols are actually tensor fields on M (contrary
to the Christoffel symbols themselves), being expressible via the manifestly tensorial relation5
1
δΓγαβ = g γµ (∇α hµβ + ∇β hαµ − ∇µ hαβ ) . (16.20)
2
Using R := g Rµν , we have indeed
µν
√ √ √ √
δ R −g = δg µν Rµν −g + g µν δRµν −g + g µν Rµν δ −g,
where the second term can be expressed by combining Eqs. (16.19), (16.20) and (16.13):
g µν δRµν = ∇µ (∇ν hµν − ∇µ hνν ) = ∇µ H µ .
On the other side, formula (A.73) for the variation of the determinant g leads to
√ 1√ 1√ 1√
δ −g = −g δ ln |g| = −g g µν δgµν = − −g gµν δg µν . (16.21)
2 2 2
The above three equations establish (16.18). It immediately follows that
Z Z
1 µν ρ 1
I=− Gµν δg ξ dVρ = Gµν δgµν ξρ dV ρ ,
2 Σ 2 Σ
where the second equality results from δg µν = −g µρ g νσ δgρσ . In view of Eq. (16.16), this
completes the proof of the mass variation formula (16.3).
Remark 1: Contrary to the generalized Smarr formula (5.84), the spacetime dimension n does not
appear in the mass variation formula (16.3).
Let us apply the general mass variation formula (16.3) to a solution of the vacuum Einstein
equation (1.44). In this case, the Ricci tensor vanishes identically, and so does the Einstein
tensor: G = 0. Accordingly one gets rid of the integrals in the right-hand side of formula (16.3).
Moreover, the null dominance condition is automatically fulfilled by G = 0, which guarantees
that the surface gravities κk are constant on each connected component of the event horizon.
We may therefore state:
3
See e.g. Eq. (E.1.18) in Wald’s textbook [499], where v = H or Eqs. (4.18)-(4.19) and the unnumbered equation
below (4.20) on p. 437-438 of Deruelle & Uzan’s textbook [163], where V = H.
4
See e.g. Eq. (21.21) in MTW [371], Eq. (3.86) in Straumann’s textbook [464] or Eq. (4.62) in Carroll’s textbook
[87].
5
See e.g. Eq. (3.93) in Straumann’s textbook [464].
626 Evolution and thermodynamics of black holes
Property 16.2: mass variation formula for black holes in vacuum general relativity
Let (M , g) be a stationary spacetime of dimension n ≥ 4 such that g obeys the vacuum
Einstein equation R = 0 [Eq. (1.44)]. Let us suppose that (M , g) contains a black hole,
the event horizon of which has K ≥ 1 connected components H1 , . . ., HK , each of them
being assumed to be a Killing horizon. Let (M , g + δg) be a nearby stationary spacetime
sharing the same characteristics. Using the same notations as in Property 16.1, the change
δM in Komar mass between the two spacetimes is given bya
K L
!
k
X κk X (i) (i)
δM = δAk + ΩHk δJHk . (16.22)
k=1
8π i=1
In the case of a connected event horizon (K = 1) with at most one axisymmetry (0 ≤
L1 ≤ 1, which is the only possibility if n = 4), this formula reduces to
κ
δM = δA + ΩH δJ . (16.23)
8π
In particular, one recovers formula (16.1), which was obtained for a Kerr black hole.
a
We have dropped the subscript ∞ on δM because, in the vacuum case, the Komar mass is independent
of the integration surface (Property 5.15) and hence needs not to be taken at the asymptotically flat end of
the stationary spacetime.
16.2.3 Mass variation formula for charged black holes
For electrically charged black holes in general relativity (electrovacuum spacetimes, cf. Secs. 1.5.2
and 5.5.2), the Einstein tensor G is proportional to the energy-momentum tensor T of the
electromagnetic field F , as given by Eq. (1.52), and the mass variation formula (16.3) can be
transformed into a simple formula:
Property 16.3: mass variation formula for charged black holes in general relativity
Let (M , g) be a stationary spacetime of dimension n ≥ 4, with stationary Killing vector ξ,
endowed with a source-free electromagnetic field F such that (g, F ) obeys the electrovac-
uum Einstein equation (1.54). Let us assume that (M , g) contains a black hole, the event
horizon of which has K ≥ 1 connected components H1 , . . ., HK . Each HkP (1 ≤ k ≤ K)
is assumed to be a Killing horizon with respect to the Killing vector χ = ξ + Li=1 Ω(i) η(i) ,
where 0 ≤ L ≤ [(n − 1)/2], the Ω(i) are constants and the η(i) are axisymmetric Killing
vectorsa . One may have L = 0, i.e. χ = ξ (static configuration). Let (M , g + δg) be a
nearby stationary spacetime sharing the same characteristics. The change δM∞ in Komar
mass at infinity (cf. Property 5.16) between the two spacetimes is related to the changes
(i)
δAk in area, to the changes δJ∞ in Komar angular momentum at infinity (cf. Property 5.22)
16.2 Towards the first law of black hole dynamics 627
and to the changes δQHk in electric charge of each Killing horizon (cf. Property 5.30) by
K L
X κk X
δM∞ = δAk + ΦHk δQHk + Ω(i) δJ∞
(i)
, (16.24)
k=1
8π i=1
where κk is the surface gravity of Hk and ΦHk is the electric potential of Hk (cf. Prop-
erty 5.29).
For a connected event horizon (K = 1) and a single axisymmetry (L = 1, which is
required if n = 4), formula (16.24) reduces to
κ
δM∞ = δA + ΩH δJ∞ + ΦH δQH , (16.25)
8π
where we have dropped the indices k and i on the various quantities and have denoted
ΩH the angular velocity of H with respect to the single axisymmetry.
a
Note that contrary to the more general setting of Property 16.1, all the connected components Hk
are assumed to be Killing horizons with respect to the same Killing vector χ; in other words, the rotation
velocities Ω(i) and axisymmetric Killing vectors η(i) are the same for all the Hk ’s.
Proof. First of all, we note that the null dominant energy condition (3.46) is fulfilled by the
electromagnetic field energy-momentum tensor T [Eq. (1.52)] [321]. By virtue of the zeroth
law (Property 3.10), this guarantees that the surface gravities κk are constant over each Killing
horizon Hk . All the hypotheses of Property 16.1 are thus fulfilled. We shall use the same setup
as in the proof of this property; in particular Eq. (16.4) holds: δξ = 0 and δη(i) = 0. By virtue of
Einstein’s equation (1.41) (with Λ = 0), we may substitute G in the last integral of the general
formula (16.3) by 8πT , where T is given by Eq. (1.52); we get
K Lk
! Z
X κk X (i) 1
δM∞ = δAk + (i)
Ω δJHk + T µν hµν ξ ρ dVρ − δI, (16.26)
k=1
8π i=1
2 Σ
with hµν := δgµν and
Z
1 1
I := Fρµ F ν ξ − Fρσ F ξµ dV µ .
ρ ν ρσ
µ0 Σ 4
Let us introduce an electromagnetic potential A (F = dA) obeying all the spacetime sym-
metries; in particular, Lξ A = 0. Thanks to the Cartan identity (A.95), this is equivalent to
ξ · F + d⟨A, ξ⟩ = 0 or Fαν ξ ν = ∇α (Aν ξ ν ). It follows then that Fρµ F ρν ξ ν = Fρµ ∇ρ (Aν ξ ν ) =
∇ρ (Aν ξ ν Fρµ ), where the last equality stems from the source-free Maxwell equation ∇ρ Fρµ = 0
[Eq. (1.48) with j α = 0]. We may then use Lemma 5.12 to turn the first term in I into a surface
integral: Z Z
1 1
I= σ
Aσ ξ Fµν dS −µν
Fµν F µν ξ ρ dVρ , (16.27)
2µ0 S 4µ0 Σ
where S := ∪K k=1 Sk is the inner boundary of Σ (Sk := Hk ∪ Σ) and we have set the surface
integral at infinity to zero, thanks to the decay of F and A, at least as 1/rn−2 and 1/rn−3
628 Evolution and thermodynamics of black holes
respectively. Let us evaluate the variation of the integral over Σ. We have, using δξ ρ = 0
[Eq. (16.4)],
δ(Fµν F µν ξ ρ dVρ ) = δ(Fµν F µν ) ξ ρ dVρ + Fµν F µν ξ ρ δ(dVρ ).
Now, δ(dVρ ) = (1/2) g µν hµν dVρ [cf. Eqs. (16.17) and (16.21)] and
δ(Fµν F µν ) = δFµν F µν + Fµν δ(g µρ g νσ Fρσ )
= δFµν F µν + Fµν δg µρ g νσ Fρσ + Fµν g µρ δg νσ Fρσ + Fµν g µρ g νσ δFρσ
|{z} |{z} | {z }
−hµρ −hνσ F ρσ
µν
= 2F δFµν − 2Fρµ F ρν hµν
= 2F µν δ(∂µ Aν − ∂ν Aµ ) − 2Fρ µ F ρν hµν
= 2F µν (∂µ δAν − ∂ν δAµ ) − 2Fσ µ F σν hµν
= 2F µν (∇µ δAν − ∇ν δAµ ) − 2Fσ µ F σν hµν
= 4F µν ∇µ δAν − 2Fσ µ F σν hµν .
Hence
µν ρ 1
δ(Fµν F ξ dVρ ) = µ σν
4F ∇µ δAν − 2Fσ F hµν + Fστ F g hµν ξ ρ dVρ
µν στ µν
2
µν µν ρ
= (4F ∇µ δAν − 2µ0 T hµν ) ξ dVρ
= 4∇ν Ωµν dVµ − 2µ0 T µν hµν ξ ρ dVρ , (16.28)
where
Ωµν := δAσ (F σµ ξ ν − F σν ξ µ ) . (16.29)
To prove (16.28), we need to show that F νσ ∇ν δAσ ξ µ = ∇ν Ωµν . This is easily done by a
direct computation of ∇ν Ωµν from (16.29), using ∇ν ξ ν = 0 (Killing equation) and ∇ν F σν = 0
(source-free Maxwell equation):
F σν ξ µ +δAσ (∇ν F σµ ξ ν +F σµ ∇ν ξ ν − ∇ν F σν ξ µ −F σν ∇ν ξ µ ).
∇ν Ωµν = ∇ν δAσ F σµ ξ ν −∇ν δAσ |{z}
νσ
| {z } | {z }
−F 0 0
↠
Expressing the symmetry properties Lξ F = 0 and Lξ δA = 0 as respectively ξ ν ∇ν F σµ −
F σν ∇ν ξ µ = F νµ ∇ν ξ σ and ξ ν ∇ν δAσ = −δAν ∇σ ξ ν [cf. Eq. (A.86)], we get
∇ν Ωµν = −F σµ δAν ∇σ ξ ν + F νσ ∇ν δAσ ξ µ + δAσ F νµ ∇ν ξ σ = F νσ ∇ν δAσ ξ µ .
Hence Eq. (16.28) holds. We may then use expression (16.27) to write the variation of I as
Z Z Z
1 σ µν 1 ν µ 1
δI = δ Aσ ξ Fµν dS − ∇ Ωµν dV + T µν hµν ξ ρ dVρ .
2µ0 S µ0 Σ 2 Σ
In view of the definition (16.29), Ωµν is antisymmetric, so that Ωµν describes a 2-form and
we may invoke Lemma 5.12 with Sint := S to express the second integral as a flux integral
through S (setting to zero the flux through S∞ due to the fast decay of A and F and hence
of Ω). We thus get
Z Z Z
1 σ µν 1 σ µν 1
δI = δ Aσ ξ Fµν dS + δAσ F µ ξν dS + T µν hµν ξ ρ dVρ ,
2µ0 S µ0 S 2 Σ
16.2 Towards the first law of black hole dynamics 629
where we have used Eq. (16.29) and the antisymmetry of dS µν to write Ωµν dS µν = δAσ (F σµ ξν −
F σν ξµ )dS µν = 2δAσ F σµ ξν dS µν . Substituting the above expression for δI into Eq. (16.26) leads
to !
K L
X κk X (i)
δM∞ = δAk + Ω(i) δJHk + (δM∞ )EM , (16.30)
k=1
8π i=1
where the electromagnetic part is defined by
Z Z
EM 1 σ µν 1
(δM∞ ) := − δ Aσ ξ Fµν dS − δAσ F σµ ξν dS µν .
2µ0 S µ 0 S
Let us use ξ = χ − Li=1 Ω(i) η(i) to let appear the null normal χ of the event horizon. Taking
P
into account that η(i)ν dS µν = 0 for η(i) is tangent to S , we get
Z L
! Z
EM 1 σ
X
(i) σ µν 1
(δM∞ ) =− δ Aσ χ − Ω Aσ η(i) Fµν dS − δAσ F σµ χν dS µν .
2µ0 S i=1
µ0 S
Now, on each connected component Sk of S , Eq. (5.99) gives Fµν dS µν = 2µ0 σk dS, where σk
√
is the effective electric charge density of Sk (cf. Property 5.29) and dS := q dn−2 x is the area
element of Sk . Furthermore, thanks to Eq. (5.86), χν dS µν = χν (χµ k ν − k ν χν )dS = −χµ dS.
Hence
K
" Z L Z Z #
X X 1
(δM∞ )EM = −δ Aσ χσ σk dS + δ Ω(i) Aσ η(i)
σ
σk dS + δAσ F σµ χµ dS .
k=1 Sk i=1 Sk µ 0 Sk
But, F σµ χµ = E σ = µ0 σk χσ on Sk [Eqs. (5.93) and (5.96)]. Consequently, the last integral
cancels out with the part δAσ χσ σk dS arising from the first integral and we are left with
K
" Z Z L Z #
X X
(δM∞ )EM = − Aσ δχσ σk dS − Aσ χσ δ(σk dS) + δ Ω(i) Aσ η(i)
σ
σk dS .
k=1 Sk Sk i=1 Sk
Now, from Eq. (16.8), δχσ = Li=1 δΩ(i) η(i)σ
, so that the first integral cancels out with the δΩ(i)
P
part arising from the last integral. Moreover, on Sk , Aσ χσ = −ΦHk = const [Eq. (5.95) with
the constant set to zero by a proper choice of A], so that the second integral can be written
as −ΦHk δQHk , where use has been made of Eq. (5.98) to let appear the electric charge QHk .
Hence, we get
K
" L Z #
X X
(δM∞ )EM = ΦHk δQHk + Ω(i) δ σ
Aσ η(i) σk dS .
k=1 i=1 Sk
Consequently, we may rewrite Eq. (16.30) as
K L K Z
X κk X X (i)
δM∞ = δAk + ΦHk δQHk + Ω (i)
δJHk + δ σ
Aσ η(i) σk dS . (16.31)
k=1
8π i=1 k=1 Sk
630 Evolution and thermodynamics of black holes
Besides, the Komar angular momentum at infinity with respect to the axisymmetry generated
by η(i) is given by formula (5.70):
Z
(i) (i) ν
J∞ = JS + Tµν η(i) dV µ ,
Σ
where we have taken (5.48) into account and have set to zero the term involving the trace T in
Eq. (5.70) since it is always possible to choose Σ so that all the η(i) ’s are tangent to it, ensuring
µ
η(i) dVµ = 0. Substituting the electromagnetic form (1.52) for T and using again η(i) µ
dVµ = 0,
we get Z
(i) (i) 1
J∞ = JS + ν
Fρµ F ρν η(i) dV µ .
µ0 Σ
Now since A obeys the spacetime symmetries, we have Lη(i) A = 0, from which we deduce
that Fρµ F ρν η(i)
ν
= ∇ρ (Aν η(i)ν
Fρµ ), in exactly the same way as that used above to establish
Fρµ F ρν ξ ν = ∇ρ (Aν ξ ν Fρµ ) when evaluating the integral I. Hence
Z Z
(i) (i) 1 ν ρ µ (i) 1 ρ
J∞ = JS − ∇ (Aρ η(i) Fµν ) dV = JS + Aρ η(i) Fµν dS µν ,
µ0 Σ 2µ0 S
(i) (i)
where the second equality stems from Lemma 5.12. Since S = ∪K k=1 Sk , JS = k=1 JHk and
PK
Fµν dS µν = 2µ0 σk dS on Sk [Eq. (5.99)], we may rewrite the above expression as
K
X Z
(i)
(i)
J∞ = JHk + ρ
Aρ η(i) σk dS . (16.32)
k=1 Sk
(i)
Then obviously, the term in factor of Ω(i) in Eq. (16.31) is δJ∞ , which ends the proof of
formula (16.24).
Remark 2: When contrasted with the electrovacuum Smarr formula (5.100), the mass variation formula
(16.24) has the distinctive feature of being independent of the spacetime dimension n.
Remark 3: In the limit of a vanishing electric charge (QH = 0), formula (16.25) reduces to the vacuum
formula (16.23), given that M∞ = M and J∞ = J are independent of the integration surface in vacuum.
Remark 4: An alternative derivation of the mass variation formulas (16.23) and (16.25) can be performed
within the ADM Hamiltonian formulation of general relativity, as shown by Sudarsky and Wald (1992)
[467, 468] (see also Sec. 6.2 of Ref. [503]). This derivation does not rely on the generalized Smarr formula
(used in the proof of Property 16.1, from which formulas (16.23) and (16.25) are derived) but it requires
that the black hole event horizon is part of a bifurcate Killing horizon. This last assumption has been
removed by Gao and Wald in 2001 [212].
16.2.4 Mass variation formula for a black hole surrounded by a fluid
Another interesting application of the general mass variation formula (16.3) regards a black
hole surrounded by some perfect-fluid matter in 4-dimensional general relativity. We first need
some preliminaries about the perfect fluid model:
16.2 Towards the first law of black hole dynamics 631
Property 16.4: simple perfect fluid in axisymmetric spacetimes
In a given 4-dimensional spacetime (M , g), the energy-momentum tensor of a perfect
fluid of 4-velocity u, proper energy density ε and pressure p is
T = (ε + p)u ⊗ u + pg. (16.33)
A simple fluida is defined by an equation of state of the type
ε = ε(s, nb ), (16.34)
where s and nb are respectively the entropy density and the baryon density, both measured
in the fluid’s frame. The fluid’s thermodynamic temperature T and the baryon relativis-
tic chemical potentialb µb are then defined as the partial derivatives T := (∂ε/∂s)nb and
µb := (∂ε/∂nb )s . The resulting identity
dε = T ds + µb dnb (16.35)
expresses the first law of thermodynamics for a small fixed comoving volume. The fluid’s
pressure p is not an independent function of (s, nb ) but rather is given by the thermody-
namic identity
p = T s + µb nb − ε, (16.36)
which, along with (16.35), implies the Gibbs-Duhem relation: dp = s dT + nb dµb .
The law of baryon number conservation asserts that nb u is a conserved current: ∇ ·
(nb u) = 0. Furthermore, if we assume that g obeys the Einstein equation (1.41), the
→
−
energy-momentum conservation (1.45) holds: ∇ · T = 0. Its projection along u lead to
the conservation of the entropy current su, i.e. ∇ · (su) = 0 (no heat exchange between
neighboring elements of a perfect fluid). Given a spacelike hypersurface Σ, the conserved
currents nb u and su give rise to the concepts of number of baryons Nb in Σ and fluid
entropy Sf in Σ:
Z Z
Nb := µ
nb u dVµ and Sf = suµ dVµ , (16.37)
Σ Σ
where dV is Σ’s normal volume element vector [Eq. (5.48)]. If the spacetime (M , g) is
axisymmetric (Killing vector η), a third conserved current is j := −1/(16π)J [η], where
→
−
J [η] is the Komar current of η, as given by Eq. (5.69): J [η] = −2 R · η. Thanks to the
→
−
Einstein equation (1.42), j = T · η − (T µµ /2)η = (ε + p)(u · η) u + (ε − p)/2 η. The
fluid angular momentum in Σ is the integral of j · dV ; assuming that η is tangent to Σ (i.e.
η · dV = 0), we get Z
Jf = (ε + p)uν η ν uµ dVµ . (16.38)
Σ
632 Evolution and thermodynamics of black holes
The three quantities Nb , Sf and Jf do not depend upon the choice of the hypersurface Σ
as long as the latter is large enough to encompass all the fluid, i.e. Σ intersects each fluid
line exactly once (cf. Fig. ??).
a
See Refs. [371] (Chap. 22), [97, 226] or [228] (Chap. 21).
b
By relativistic it is meant that the rest mass is included in the chemical potential.
→
−
Proof. The entropy conservation ∇ · (su) = 0 follows straightforwardly from ∇ · T = 0
projected along u, using the identities (16.35), (16.36) and ∇ · (nb u) = 0 (see e.g. Ref. [226]
for details). Because they are fluxes of conserved currents, the quantities Nb , Sf and Jf take
the same values on two hypersurfaces Σ1 and Σ2 that are connected by a (timelike or null)
hypersurface W with all the currents vanishing on W , which happens if W is located outside
the fluid worldtube.
From an astrophysical point of view, it is natural to consider that the matter surrounding
the black hole forms an accretion disk or torus, so that it can be modeled by a perfect fluid in
circular motion around the black hole rotation axis. One then derives from the general law
(16.3) the following mass variation formula:
Property 16.5: mass variation formula for a black hole surrounded by a fluid
Let (M , g) be a 4-dimensional stationary (Killing vector ξ) and axisymmetric (Killing
vector η) spacetime that contains a black hole with a connected event horizon H . We
assume that H is a Killing horizon with respect to the Killing vector χ = ξ + ΩH η. If
ΩH ̸= 0, this requires the hypotheses of the strong rigidity theorem (Property 5.25). Let us
suppose that g obeys the Einstein equation (1.41) with Λ = 0: G = 8πT , where T is the
energy-momentum tensor of a perfect fluid located in the black hole exterior and whose
features are described in Property 16.4. Moreover, let us assume that the fluid is in circular
motion, i.e. the fluid’s 4-velocity u is a linear combination of the two Killing vectors:
u = ut (ξ + Ωη) , (16.39)
where ut and Ω are two scalar fields. Ω is called the fluid angular velocity because
Ω = dφ/dt along a fluid line, where t and φ are adapted coordinates: ∂t = ξ and
∂φ = η. The notation ut reminds that this factor is the component of u along ∂t in adapted
coordinates. It is fully determined by Ω and the three scalar products formed from the two
Killing vectors:
−1/2
ut = −ξ · ξ − 2Ωξ · η − Ω2 η · η . (16.40)
Let (M , g + δg) be a nearby stationary spacetime sharing the same characteristics. The
change δM∞ in Komar mass at infinity between the two spacetimes is given by
Z Z Z
κ T µb
δM∞ = δA + ΩH δJH + δ(dSf ) + δ(dNb ) + Ω δ(dJf ) , (16.41)
8π Σ ut Σ ut Σ
16.2 Towards the first law of black hole dynamics 633
where κ is H ’s surface gravity, A is H ’s area, JH is H ’s Komar angular momentum, Σ
is any spacelike hypersurface that intersects each fluid line exactly once and that admits η
as a tangent vector, T is the fluid temperature, µb is the baryon chemical potential and dSf ,
dNb and dJf are the infinitesimal increments in the integrals (16.37) and (16.38) defining
the fluid’s entropy, baryon number and angular momentum:
dSf := suµ dVµ , dNb := nb uµ dVµ and dJf := (ε + p)uν η ν uµ dVµ . (16.42)
Proof. Let us specialize the general formula (16.3) to the case n = 4, K = 1, L1 = 1 and g
obeying the Einstein equation (so that the Einstein tensor G can be replaced by 8πT ):
Z Z
κ 1
δM∞ = δA + ΩH δJH + T δgµν ξ dVρ − δ T µν ξ ν dVµ .
µν ρ
(16.43)
8π 2 Σ Σ
Let us rewrite the integrand of the last integral by means of the perfect fluid expression (16.33)
for T :
T µν ξ ν dVµ = [(ε + p)uµ uν ξ ν + pξ µ ] dVµ .
Substituting (ut )−1 u − Ωη for ξ [Eq. (16.39)] and using uν uν = −1 and η µ dVµ = 0 (since η is
tangent to Σ), we get
ε
T µν ξ ν dVµ = − t uµ dVµ − Ω dJf ,
u
where use has been made of Eq. (16.42) to let appear dJf . Let us perform the variation of this
expression and use δε = T δs + µb δnb in agreement with Eq. (16.35):
1 ε δut µ ε
δ (T µν ξ ν dVµ ) = − t
(T δs + µ b δn b )uµ
dVµ + t 2
u dVµ − t δ (uµ dVµ ) − δΩ dJf − Ω δ(dJf ).
u (u ) u
Using Eq. (16.42) to let appear δ(dSf ) and δ(dNb ) and substituting p for T s + µb nb − ε
[Eq. (16.36)], we get
T µb p ε δut µ
δ (T µν ξ ν dVµ ) = − δ(dSf ) − δ(dN b ) − Ω δ(dJf ) + δ(uµ
dV µ ) + u dVµ − δΩ dJf .
ut ut ut (ut )2
Now δ(uµ dVµ ) = δuµ dVµ + uµ δ(dVµ ), with, according to Eqs. (16.39) and (16.4),
δuµ = (δut /ut )uµ + ut δΩ η µ (16.44)
and, according to Eqs. (16.17) and (16.21), δ(dVµ ) = (1/2)g ρσ δgρσ dVµ . Hence, using η µ dVµ = 0,
t
µ δu 1 ρσ
δ(u dVµ ) = + g δgρσ uµ dVµ .
ut 2
The above expression of δ (T µν ξ ν dVµ ) becomes then
T µb
δ (T µν ξ ν dVµ ) = − t
δ(dSf ) − t δ(dNb ) − Ω δ(dJf )
u
t u
δu ν 1 p ρσ
+ t 2
− δΩuν η (ε + p) + t
g δgρσ uµ dVµ .
(u ) 2u
634 Evolution and thermodynamics of black holes
Note that we have used Eq. (16.42) as δΩ dJf = (ε + p)δΩ uν η ν uµ dVµ . Now, from Eq. (16.44),
δut /ut − ut δΩuν η ν = −uν δuν . Hence
T µb hp i 1
δ (T µν ξ ν dVµ ) = − δ(dSf ) − δ(dN b ) − Ω δ(dJf ) + g ρσ
δg ρσ − (ε + p)uν δuν
uµ dVµ .
ut ut 2 ut
Since gρσ uρ uσ = −1 implies uρ δuρ = −(1/2)δgρσ uρ uσ , we may rewrite the above relation as
T µb 1
δ (T µν ξ ν dVµ ) = − t
δ(dSf ) − t δ(dNb ) − Ω δ(dJf ) + T ρσ δgρσ ξ µ dVµ ,
u u 2
where we have used Eq. (16.33) to let appear T ρσ and Eq. (16.39) along with η µ dVµ = 0 to write
(ut )−1 uµ dVµ = ξ µ dVµ . Integrating the above expression on Σ and substituting into Eq. (16.43)
yields formula (16.41).
Historical note : The mass-variation formula (16.1) for Kerr black holes has been first derived by Jacob
Bekenstein in 1972 [51, 52], using the same differentiation procedure as in Sec. 16.2.1. Actually, the
formula derived by Bekenstein applies to Kerr-Newman black holes, being a special case of the generic
formula (16.25) for electrically charged black holes. The extension to generic stationary 4-dimensional
black holes surrounded by a perfect fluid in circular motion [Eq. (16.41)] has been obtained by James
Bardeen, Brandon Carter and Stephen Hawking in 1973 [40]. A different derivation of the same formula
(16.41) can also be found in Bardeen’s lecture at the famous 1972 Les Houches Summer School [39]. The
generalization to include an electromagnetic field has been presented by Carter at the same summer
school [96]. In the pure electrovacuum case (no fluid, no electric current), Carter’s formula (Eq. (9.64) in
Ref. [96]) reduces to the case K = 1, L = 1 of formula (16.24). As for the general form (16.3) involving
the Einstein tensor, it has been exhibited by Carter in 1979 [99] in the case n = 4 and K = 1 (connected
event horizon). The mass variation formula with a fluid (16.41) has been extended to non-circular flows
by Carter in 1979 [98].
16.2.5 A first law?
At this stage, it would be premature to call any of the formulas (16.22), (16.23), (16.24) or
(16.41) the first law of black hole dynamics by analogy with the first law of thermodynamics
dE = T dS − P dV . One can reasonably interpret δM as some energy variation, the term
ΩH δJ as the work6 performed by the torque that is changing by δJ the angular momentum J
of a body rotating at the angular velocity ΩH and the term ΦH δQH as the work performed to
change by an amount δQH the electric charge of a body at the electric potential ΦH . However,
we have not got any argument yet to identify the term (κ/8π) δA with the classical heat
exchange term T dS. For this, we need first to identify the entropy S with the black hole area
A. This is performed in the next section.
6
In Newtonian mechanics, the work done by a torque τ on a body that is rotating by dφ is dW = τ dφ. Given
that τ := dJ/dt, one gets dW = Ω dJ, where Ω = dφ/dt is the body’s angular velocity.
16.3 Evolution of the black hole area 635
16.3 Evolution of the black hole area
16.3.1 Irreversibily in the evolution of a Kerr black hole
Let us consider a particle P in free fall from infinity into a Kerr black hole of mass M and spin
J = aM , with a < M (non-extremal case). Let us denote by E and L the conserved energy
and conserved angular momentum of P (cf. Property 11.1), with E ≪ M and |L| ≪ J. We
shall assume that either P falls directly into the black hole or P splits in the ergoregion into
two particles, P ′ and P∗ , such that P∗ falls into the black hole and P ′ escapes to infinity —
as in the Penrose process described in Sec. 11.3.2. Let (E ′ , L′ ) and (E∗ , L∗ ) be the conserved
energy and angular momentum of P ′ and P∗ , respectively. As shown in Sec. 11.3.2, these
quantities fulfill E = E ′ + E∗ and L = L′ + L∗ . If P falls directly into the black hole, we
shall set P∗ = P, E ′ = 0, E∗ = E, L′ = 0 and L∗ = L. By virtue of the no-hair theorem 5.38,
after P∗ has been captured by the black hole and P ′ has gone sufficiently far, the black hole
appears to the exterior world as a Kerr black hole of mass M + δM and angular momentum
J + δJ, say. Initially, when P was far apart from the black hole, the total energy (ADM mass)
of the spacetime was M + E (recall that E is the energy at infinity of P, cf. Sec. 11.2.2) and
similarly the total angular momentum was J + L. To the first order in E/M , we may neglect
the energy and angular momentum of the gravitational radiation generated in the process, so
that the final spacetime energy is M + δM + E ′ and the final angular momentum is J + δJ + L′ .
By conservation of the total energy and angular momentum, we then get δM = E − E ′ and
δJ = L − L′ , i.e.
δM = E∗ and δJ = L∗ . (16.45)
The evolution of the black hole area A in the process is given by Eq. (16.1):
δA = 8π/κ (δM − ΩH δJ). Given the identities (16.45), one gets
8π
δA = (E∗ − ΩH L∗ ) . (16.46)
κ
Note that the above equation is well-posed since κ > 0 (the black hole is not extremal). Now,
we have established in Sec. 11.3.2 that E∗ > ΩH L∗ [Eq. (11.60)] as a necessary condition for
P∗ to cross the event horizon. It follows immediately that δA > 0:
Property 16.6: area increase for accreting Kerr black holes
Under the accretion of particles of energy sufficiently small to neglect gravitational radia-
tion, the area A of a Kerr black hole necessarily increases:
δA > 0 . (16.47)
Remark 1: The area increase holds even if the mass M decreases, which occurs for a Penrose process
(E ′ > E), since then E∗ < 0 and thus δM < 0 by virtue of Eq. (16.45).
Any transformation of a black hole with δA > 0 is called an irreversible transformation .
For particle accretion, the irreversibility traces back to the inequality E∗ > ΩH L∗ [Eq. (11.60)],
636 Evolution and thermodynamics of black holes
which must hold if P∗ is to cross the event horizon, i.e. is to be actually accreted. Indeed, in view
of Eq. (16.45), one might think naively that the transformation (M, J) → (M + δM, J + δJ)
could be reverted by throwing into the black hole a particle of conserved energy E∗′ = −δM =
−E∗ and conserved angular momentum L′∗ = −δJ = −L∗ . However this is not permitted
because the constraint E∗′ > ΩH L′∗ is violated if E∗ > ΩH L∗ holds.
A reversible transformation is a transformation such that δA = 0. For particle accretion,
it is achieved in the limit case E∗ = ΩH L∗ [cf. Eq. (16.46)]. The particle P∗ reaches then the
event horizon H with a vanishing radial affine velocity dr/dλ, λ being the affine parameter
along P∗ ’s worldline associated with P∗ ’s 4-momentum (cf. Sec. 11.2.1). Indeed, the equation
of geodesic motion (11.43b) yields, on H (r = r+ ),
2
dr r+ + a2
=− 2
(E∗ − ΩH L∗ ). (16.48)
dλ r=r+ r+ + a2 cos2 θ
Note that we have set ϵr = −1 in Eq. (11.43b) to account for the infall motionp of P∗ and
have specialized expression (11.44) for R(r) to ∆ = 0 (r = r+ ), which yields R(r+ ) =
(r+2
+ a2 )|E∗ − ΩH L∗ | = (r+ 2
+ a2 )(E∗ − ΩH L∗ ) thanks to Eq. (11.60) and the identity
a/(r+ + a ) = ΩH [Eq. (10.69)]. Equation (16.48) clearly shows that dr/dλ → 0 in the
2 2
limit ΩH L∗ → E∗ . Moreover, √ from expression (11.29) for R(r), one gets, for a geodesic with
ΩH L∗ = E∗ , R′ (r+ ) = −2 M 2 − a2 (µ2∗ r+ 2
+ K∗ ), where µ∗ is the mass of P∗ and K∗ is the
Carter constant of P∗ . Given that M 2 − a2 ̸= 0 (non-extremal Kerr) and K∗ ≥ 0 [Eq. (11.30)],
we obtain R′ (r+ ) ̸= 0, except if µ∗ = 0 and K∗ = 0. According to Property 11.2, the last case
would occur only if P∗ followed a principal null geodesic of Kerr spacetime. However, this is
excluded since those geodesics do not fulfill ΩH L∗ = E∗ in the black hole exterior: they obey
L∗ = aE∗ sin2 θ [Eqs. (11.6) and (11.9b)], so that ΩH L∗ = E∗ would imply ΩH a sin2 θ = 1, i.e.
r+2
+ a2 cos2 θ = 0, which is impossible. Hence one has always R′ (r+ ) ̸= 0. We conclude that
P∗ has necessarily a r-turning point on H (cf. Sec. 11.2.6). Hence a reversible transformation
corresponds to the idealized case where the particle P∗ is on the verge of being accreted by
the black hole but is actually not. This case is approached when P∗ crosses H with an affine
velocity dr/dλ as small as possible.
The square root of the horizon area A (rescaled by 1/(16π)) is called the black hole’s
irreducible mass:
A
2
Mirr := . (16.49)
16π
Mirr has the dimension of a mass and Eq. (16.47) is equivalent to δMirr > 0, i.e. Mirr cannot
decrease in any accretion process, hence its name. This contrasts with M , which may decrease
(Penrose process). Formula (10.83) for A can be turned into an expression for Mirr in terms of
M and J = aM (cf. Fig. 16.1):
v r !
u
u1 J 2
Mirr = M t 1+ 1− 4 . (16.50)
2 M
This relation is easily inverted to express M in terms of Mirr and J:
J2
M 2 = Mirr
2
+ 2
. (16.51)
4Mirr
16.3 Evolution of the black hole area 637
1.00
0.95
0.90
Mirr /M
0.85
0.80
0.75
0.0 0.2 0.4 0.6 0.8 1.0
J/M 2
Figure 16.1: Irreducible mass Mirr of a Kerr black hole of mass M , as a function of the black hole angular
momentum J [Eq. (16.50)] (J = 0 corresponds to a Schwarzschild black hole and J/M 2 = 1 to an extremal Kerr
one). The shadded area depicts the maximal energy that can be extracted from the black hole by a Penrose process
bringing J down to 0.
Suppose some energy is extracted by a sequence of Penrose processes (cf. Sec. 11.3.2)
from a Kerr black hole of initial mass M and angular momentum J, leading to a final state
corresponding to a Kerr black hole of mass M0 and angular momentum J0 . The maximal energy
∆M = M − M0 that can be extracted is achieved when the final state is a Schwarzschild black
hole (J0 = 0), since then the Penrose process necessarily stops, due to the lack of an outer
ergoregion (cf. Sec. 11.3.2). Since the irreducible mass of a Schwarzschild black hole coincides
with its mass (set J = 0 in Eq. (16.50)), the irreducible mass of the final state is Mirr,0 = M0
and we have ∆M = M − Mirr,0 . The evolution law Mirr,0 ≥ Mirr provides then a upper bound
on the extracted energy:
∆M ≤ M − Mirr , (16.52)
with equality only for a sequence of reversible transformations. Figure 16.1 shows that ∆M/M
is maximal when the initial state is (close to) an extremal Kerr black hole (J = M 2 ). From
Eq. (16.50), one gets
∆M 1
max = 1 − √ ≃ 0.293. (16.53)
M 2
We conclude:
Property 16.7: maximal energy extraction by a Penrose process
A sequence of Penrose processes can extract up to 29% of the mass of a Kerr black hole.
This upper bound is achieved for a maximally spinning black hole (extremal Kerr), by
bringing it down to a non-rotating state (Schwarzschild) via reversible transformations.
Historical note : Demetrios Christodoulou introduced the concepts of irreversible transformation and
irreducible mass of a Kerr black hole in 1970 [112] and established formula (16.51). Actually, he did not
638 Evolution and thermodynamics of black holes
make any argument about the black hole area; rather he integrated the relation dM = ΩH dJ, which
holds when the black hole is accreting a particle with E∗ = ΩH L∗ (cf. Eq. (16.45), i.e. at the saturation
limit of the inequality (11.60). Indeed,
√ by expressing ΩH as a function of (M,√ J) via Eq. (10.69), one
gets dM = JdJ/(2M (M + M − J )), which is equivalent to d(M + M 4 − J 2 ) = 0. The
2 4 2 2
irreducible mass√ Mirr then appears when writing the integration constant for this equation as 2Mirr 2 ,
namely M + M 4 − J 2 = 2Mirr , which is equivalent to Eq. (16.50). The connection with the black
2 2
hole area [Eq. (16.49)] has been performed in a second study, with Remo Ruffini, published in 1971
[114] and which extends Christodoulou’s results to Kerr-Newmann black holes. Meanwhile, Roger
Penrose and Roger Floyd noticed at the end of 1970 (published 1971 [409]) that the area of a Kerr
black hole necessarily increases under particle accretion, i.e. they established Property 16.6. In 1972,
Brandon Carter [94] pointed out that the relation dM = ΩH dJ also holds for a rigidly rotating fluid
star undergoing a thermodynamically reversible transformation (in that case, ΩH stands for the star’s
angular velocity). This strengthened the analogy between irreversibility in black hole processes and
standard thermodynamics.
16.3.2 Hawking’s area theorem
The area increase law turns out to be far more general than established in Property 16.6, which
regards only a Kerr black hole accreting particles: the law actually applies to any black hole
evolving under any process (not necessarily a small perturbation) such that the null convergence
condition [Eq. (2.94)] is satisfied in the vicinity of the event horizon. This is the essence of the
famous area theorem, first established by Hawking in 1971.
The first step towards the area theorem is:
Property 16.8: non-negative expansion of a black hole horizon
Let (M , g) be a n-dimensional spacetime containing a black hole of event horizon H . If
the Ricci tensor R obeys the null convergence condition (2.94) on H , i.e. if R(ℓ, ℓ) ≥ 0
for any null vector ℓ on H — which holds in general relativity if the null energy condition
(2.95) is fulfilled —, then the expansion θ(ℓ) of H along any future-directed null normal ℓ,
as defined in Sec. 2.3.5, is positive or zero:
θ(ℓ) ≥ 0. (16.54)
Proof. Let ℓ be a future-directed null normal vector field of H . ℓ is necessarily tangent to the
null geodesic geodesic generators of H (Property 2.1) and is thus a pregeodesic vector, i.e. it
obeys Eq. (2.22): ∇ℓ ℓ = κ ℓ. If ℓ is not geodesic (κ ̸= 0), it is always possible to rescale it to
ℓ′ = αℓ with α > 0 so that ℓ′ is a future-directed geodesic vector field: ∇ℓ′ ℓ′ = 0 [Eq. (2.24)].
We have then θ(ℓ′ ) = αθ(ℓ) [cf. Eq. (2.68)], so that θ(ℓ) ≥ 0 ⇐⇒ θ(ℓ′ ) ≥ 0. Accordingly, for
proving (16.54), there is no loss of generality in assuming that ℓ is a geodesic vector field. Let
us consider a null geodesic generator L of H . Up to some additive constant, there is a unique
affine parameter λ of L associated to ℓ, i.e. such that ℓ = dx/dλ along L . The evolution
of θ(ℓ) along L is measured by dθ(ℓ) /dλ = ∇ℓ θ(ℓ) and is given by the null Raychaudhuri
16.3 Evolution of the black hole area 639
equation (2.88). Owing to κ = 0 (for ℓ is assumed to be geodesic), it simplifies to
dθ(ℓ) 1
=− θ2 − σab σ ab − R(ℓ, ℓ),
dλ n − 2 (ℓ) | {z } | {z }
≥0 ≥0
where σab σ ab ≥ 0 has been established in Sec. 2.4.2 [Eq. (2.92)] and R(ℓ, ℓ) ≥ 0 holds by virtue
of the null convergence condition on H . Hence
dθ(ℓ) 1
≤− θ2 . (16.55)
dλ n − 2 (ℓ)
Let us assume the negation of (16.54), i.e. that there exists a point p ∈ L ∩ H where
θ(ℓ) = θ0 < 0. By choosing the additive constant in the definition of the affine parameter λ, we
may ensure λ(p) = 0. Equation (16.55) implies then
∀λ ≥ 0, θ(ℓ) (λ) ≤ θ̄(λ), (16.56)
where θ̄(λ) obeys
dθ̄ 1 2
=− θ̄ and θ̄(0) = θ0 .
dλ n−2
The unique solution of this differential equation is
θ0
θ̄(λ) = .
1 + θ0 λ/(n − 2)
It follows that θ̄ → −∞ as λ → −(n − 2)/θ0 > 0. The inequality (16.56) then implies
θ(ℓ) → −∞ as λ → λ∗ with 0 < λ∗ ≤ −(n − 2)/θ0 . Hence the point p∗ ∈ L of parameter λ∗
is a focusing point, i.e. a point where neighboring null geodesic generators of H intersect. But
according to Property 4.8 of black hole event horizons (cf. Sec. 4.4.3), this can happen only
if p∗ is a crossover point, i.e. a point at which the null geodesic L enters H ; however, this
situation is excluded since λ∗ > 0 implies that p∗ lies in the future of p, where L is already in
H . Hence the hypothesis θ0 < 0 leads to a contradiction. It follows that θ0 ≥ 0, i.e. at any
point p ∈ H , θ(ℓ) ≥ 0.
Property 16.9: area theorem (Hawking 1971 [259], Chruściel, Delay, Galloway and
Howard 2001 [120])
Let (M , g) be a spacetime of dimension n ≥ 2 that contains a black hole of event horizon
H . Let us assume that the Ricci tensor R fulfills the null convergence condition (2.94),
i.e. R(ℓ, ℓ) ≥ 0 for any null vector ℓ — which holds in general relativity if the null energy
condition (2.95) is fulfilled. Let Σ1 and Σ2 be spacelike hypersurfaces such that Σ2 lies in
the causal future of Σ1 : Σ2 ⊂ J + (Σ1 ) (cf. Sec. 4.4.1). Let S1 = H ∩ Σ1 and S2 = H ∩ Σ2 .
If
(i) H is smooth between S1 and S2 , and S1 and S2 are cross-sectionsa of H that are
intersected by the same null geodesic generators of H (cf. Fig. 16.2)
or
640 Evolution and thermodynamics of black holes
Figure 16.2: Cross-sections S1 and S2 induced by the spacelike hypersurfaces Σ1 and Σ2 in a smooth part of
the event horizon H , such that S1 and S2 are intersected by the same null geodesic generators of H (green
curves).
(ii) the closure of J − (I + ) ∪ H in the conformal manifold M˜ ⊃ M defining I + is
included in a globally hyperbolic region V of (M˜, g̃) and Σ1 and Σ2 are Cauchy
surfaces of V ,
then the areas A(S1 ) and A(S2 ) obey
A(S2 ) ≥ A(S1 ). (16.57)
a
If H is smooth between S1 and S2 , it is necessarily a null hypersurface there (Property 4.11 on p. 118),
so that the concept of cross-section as defined in Sec. 2.3.4 makes sense.
Proof. For n = 2, the theorem is trivial since S1 and S2 are each reduced to a single point, so
that A(S1 ) = A(S2 ) = 0 and Eq. (16.57) is fulfilled. Let us then assume n ≥ 3. We consider
first the case (i) (H smooth between S1 and S2 ). S1 and S2 are then cross-sections of H
that are connected by null geodesic generators of H (cf. Fig. 16.2). One may introduce a
1-parameter foliation (St )t∈[0,1] of H by cross-sections St such that S0 = S1 and S1 = S2
(cf. the proof of Property 3.1 for details, λ playing the role of t there). The label t can then
be considered as a parameter along the null geodesic generators of H connecting S1 to S2 .
Let ℓ = dx/dt be the associated tangent vector. Given that S2 lies in the future of S1 , ℓ is
future-directed. By the very definition of the expansion θ(ℓ) along ℓ [Eq. (2.49) combined with
Eq. (2.53)], one has
√
Z
d
A(St ) = θ(ℓ) q dx2 · · · dxn−1 ,
dt St
where (x2 , . . . , xn−1 ) is a coordinate system on St and q is the determinant with respect to
these coordinates of the metric induced by g on St . Now, if the null convergence condition
holds, Property 16.8 implies θ(ℓ) ≥ 0; hence
d
A(St ) ≥ 0.
dt
16.3 Evolution of the black hole area 641
Figure 16.3: Surfaces S1 and S2 induced by the spacelike hypersurfaces Σ1 and Σ2 on the event horizon H
corresponding to a black hole merger. All solid dark green and light green curves are null geodesic generators of
H . The light green part of H is generated by null geodesics that entered H at some caustic points (three of
them are indicated as light green dots); the parts of these geodesics outside H are drawn as light green dot-dashed
curves. The dark green dashed curves depict other null geodesics that enter H at the same caustic points.
It follows that t 7→ A(St ) is a nondecreasing function. Consequently, A(S1 ) ≥ A(S0 ) and
Eq. (16.57) holds.
If H is not smooth between S1 and S2 , this is due to the crease set, i.e. the subset of
H where new null geodesics enter H (cf. Sec. 4.4.3). Naively, this reinforces the inequality
A(S2 ) > A(S1 ) since the new geodesics are generating new parts of H and therefore parts
of S2 distinct from those that can be connected to S1 by null geodesic generators (cf. Fig. 16.3).
More precisely, let us assume (ii) and let us consider a point p ∈ S1 . Let L be the null geodesic
generator of H through p. By property 4.8, L stays in H for all its future after p. Since Σ2
is a Cauchy surface lying in the causal future of Σ1 , L necessarily intersects Σ2 at a unique
point q ∈ Σ2 ∩ H = S2 . Hence every point of S1 is mapped to a point of S2 by a null
geodesic generator. Let S2∗ be the part of S2 covered by this map. If we assume that the
part of H between S1 and S2∗ is smooth, we may apply (i) to the pair (S1 , S2∗ ) and get
A(S2∗ ) ≥ A(S1 ). Given that S2∗ ⊂ S2 , one has A(S2 ) ≥ A(S2∗ ) and (16.57) follows. We
refer to the article [120] for the proof in the case where H is not assumed smooth between S1
and S2∗ (70 pages!).
Example 1 (Oppenheimer-Snyder collapse): Let us consider the black hole formed by the collapse of
a ball of pressureless matter, as described by the Oppenheimer-Snyder model studied in Sec. 14.3. Let
St̃ be a cross-section of the event horizon at constant ingoing Eddington-Finkelstein coordinate t̃. The
area of St̃ is simply A = 4πr2 , where r is the areal radius (cf. Sec. 14.3.5). Its evolution can thus be
read directly on Fig. 14.2 (right): it is increasing from A = 0 (at t̃ ≃ 5.6 m) to the Schwarzschild value
A = 16πm2 (at t̃ ≃ 9.7 m). This fully agrees with the area theorem, given that the null energy condition
(2.95) is fulfilled by the energy-momentum tensor (14.2) of the collapsing matter: T (ℓ, ℓ) = ρ(u · ℓ)2 ≥ 0
since ρ ≥ 0.
642 Evolution and thermodynamics of black holes
Example 2 (Vaidya collapse): Similarly, we check on Figs. 15.1 and 15.5 that for the radiation shell
collapse studied in Chap. 15, the area of cross-sections of the event horizon at constant ingoing Eddington-
Finkelstein coordinate t is increasing towards the future, for r is again the areal radius [cf. the metric
(15.11)]. As we have already noticed in Sec. 15.2.1, the radiation energy-momentum tensor (15.5) fulfills
the null energy condition, given that M ′ (v) ≥ 0 (cf. Fig. 15.2).
Historical note : As mentioned in the historical note on p. 637, Roger Penrose and Roger Floyd [409]
have shown in 1970 (published 1971) that the area of a Kerr black hole always increases during particle
accretion, even if its mass is decreasing, as in a Penrose process (Property 16.6). Soon after, in 1971,
Stephen Hawking [259] established the area theorem for generic dynamical black holes and a detailed
proof was given in Hawking & Ellis’ 1973 textbook (Proposition 9.2.7 in [266]). In 2001, Piotr Chruściel,
Erwann Delay, Gregory Galloway and Ralph Howard [120] pointed out that the Hawking & Ellis proof
is valid only for H piecewise smooth (cf. the discussion in Sec. 3.5.1 of Chruściel’s textbook [119]); they
constructed a new proof that does not rely on the smoothness of H .
16.3.3 A second law?
Basically the area theorem 16.9 states that the area of cross-sections of a black hole event
horizon cannot decrease from the past to the future. By its irreversible character, this property
bears some resemblance with the second law of thermodynamics. By itself, this is of course
not sufficient to identify the black hole area with some entropy (any nondecaying physical
quantity has not to be an entropy!). However, we have seen in Sec. 16.2.5 that the candidate
T dS term for a possible first law could be κ/(8π)dA. It is thus tempting to identify A with an
entropy S, up to some constant factor α. Then the temperature T would be κ/(8πα):
S = αA (16.58a)
1
T = κ, (16.58b)
8πα
so that κ/(8π)dA = T dS, which makes the mass variation formula (16.23) look pretty much
like the first law of thermodynamics.
16.4 The laws of black hole dynamics
In Secs. 16.2 and 16.3, we have established black hole properties that are quite analog to the
first and second laws of thermodynamics. Let us now examine the zeroth and third laws of
thermodynamics.
16.4.1 Zeroth law
We have already established a so-called zeroth law of black hole dynamics in Chap. 3, namely the
surface gravity κ of a Killing horizon is constant, provided that the null dominance condition
is fulfilled (Property 3.10) or that the Killing horizon is part of a bifurcate Killing horizon
(Property 3.16). Now, by Properties 5.23 and 5.25, (a connected component of) the event horizon
of a stationary black hole must be a Killing horizon. Hence, we may extend the zeroth law to
black holes:
16.4 The laws of black hole dynamics 643
Property 16.10: zeroth law of black hole dynamics
Let (M , g) be a stationary spacetime of dimension n ≥ 4 containing a black hole. Let H
be a connected component of the black hole event horizon. If H is non-rotating (i.e. if the
stationary Killing vector ξ is null on H ), H is necessarily a Killing horizon (Property 5.23).
If H is rotating (ξ spacelike on some parts of H ), we shall assume that the hypotheses of
the strong rigidity theorem (Property 5.25) hold, so that H is a Killing horizon as well. If
the null dominance condition (3.43) is fulfilled on H — which is guaranteed in general
relativity if the null dominant energy condition (3.46) holds on H — or if H is part of a
bifurcate Killing horizon, then the surface gravity κ is constant over H .
Given that a black hole “in equilibrium” is modeled by a black hole in a stationary spacetime,
this property bears a strong resemblance with the zeroth law of thermodynamics, which states
that the temperature of a body in equilibrium is uniform over the entire body. This strengthens
the interpretation of the surface gravity κ as the temperature T (up to some factor) performed
in Eq. (16.58b).
16.4.2 What about the third law?
The classical Nernst formulation of the third law of thermodynamics states that the entropy
of a system must approach zero (or a universal constant) as its temperature tends to zero. In
this form and with the identification (16.58), this law certainly does not hold for black holes,
because extremal black holes, which have κ = 0, have a non-vanishing area. For instance,
the area of the extremal Kerr black hole of mass m (Chap. 13) is A = 8πm2 (take the limit
a → m in Eq. (10.83)), which is neither zero, nor a universal constant. Similarly, the area of the
extremal Reissner-Nordström black hole of mass m is A = 4πm2 (this is immediate from the
metric (18.1) below). Another formulation of the third law of thermodynamics states that it is
impossible to bring any system to zero temperature by a finite number of operations. This was
the formulation adopted for black holes, as a conjecture, in 1973 by Carter [96] and Bardeen,
Carter and Hawking [40]. It was then reformulated, with a tentative proof, by Israel in 1986
[291], as
No continuous process in which the energy-momentum tensor of accreted matter
remains bounded and satisfies the weak energy condition in a neighborhood of the
apparent horizon can reduce the surface gravity of a black hole to zero within a
finite advanced time.
However, as pointed out recently by Kehle and Unger [311], Israel’s proof is restricted to the case
where outermost trapped surfaces evolve smoothly, while, as we shall discuss in Chap. 18, their
evolution generically presents discontinuous jumps, even if the spacetime and all the matter
fields remain perfectly smooth. In particular, Kehle and Unger [311] have exhibited a regular
spacetime where the collapse of a charged scalar field (obeying the dominant, and hence the
weak, energy condition) turns a Schwarzschild black hole into an extremal Reissner-Nordström
644 Evolution and thermodynamics of black holes
black hole within a finite advanced time7 . Let us note that this result has been recently mitigated
by Reall [427], who has shown that the collapse to a supersymmetric extremal black hole (such
as the extremal Reissner-Nordström one, but not the extremal Kerr one) cannot occur in theories
with an upper bound on the local charge to mass ratio.
From an astrophysical point of view, Bardeen has shown in 1970 [36] that a Schwarzschild
black hole of mass m0 , and hence of surface gravity κ = 1/(4m0 ) > 0 [Eq. (2.29)], can in
principle be spun up to an extremal Kerr black hole (κ = 0) by accreting matter from an
accretion disk terminating at the innermost stable circular orbit (ISCO, cf. Sec. 11.5.3). The total
rest-mass of accreted matter is then 1.86 m0 and the mass of the final black hole is m ≃ 2.45 m0 .
However, if one takes into account the electromagnetic emission of the accretion disk, the
extremal Kerr state cannot be achieved. The reason is that a substantial part of the emitted
photons carry a negative angular momentum ℓ and the black hole absorption cross-section for
photons with ℓ < 0 is larger than for those with ℓ > 0. This clearly appears on the shadow
picture of Fig. 12.20, where the part α > 0, which is due to photons with ℓ < 0 (cf. Eq. 12.88a),
is much larger than the part α < 0 corresponding to ℓ > 0. Consequently, the black hole
absorbs more photons with ℓ < 0 than with ℓ > 0; this diminishes the increase of the black hole
spin a induced by accretion of matter from the disk (which has a positive angular momentum).
Thorne has shown in 1974 [478] that the balance between the two processes leads to a final
spin parameter a ≃ 0.998 m, quite insensitive to the details of the emission mechanism in the
accretion disk. This is close to, but strictly lower than, the critical value a = m that would have
yield a zero surface gravity. Hence, in this respect, we may say that astrophysical black holes
absorbing matter from an accretion disk obey a kind of third law. But this does not appear to
arise from some fundamental properties of black holes; it rather results from the properties of
their environment.
It must be stressed that in standard thermodynamics as well, the third law has not the same
fundamental status as the other laws. In particular it can be violated by some rather simple
systems (see Ref. [504] for a discussion and an example involving a boson (or fermion) gas
confined to a circular ring).
For all the above reasons, we shall no longer discuss the third law here.
16.4.3 Summary: the three laws
Let us collect the results obtained so far, in the form of three laws similar to the laws of
thermodynamics. To stress the analogy, we are using the phrase black hole in equilibrium for
black hole in a stationary spacetime (cf. Chap. 5). Besides, for the sake of brevity, we shall not
repeat the various assumptions underlying these laws, referring to previously stated properties
for all the details. Furthermore, we consider a connected event horizon for simplicity.
Property 16.11: laws of black hole dynamics
Zeroth law: Under the hypotheses of Property 16.10, the surface gravity κ of the event
7
As shown in Sec. 5.5.6 of Poisson’s textbook [416], this cannot happen with a charged null dust (i.e. a charged
generalization of Vaidya collapse discussed in Chap. 15) that obeys the weak energy condition.
16.4 The laws of black hole dynamics 645
horizon of a black hole in equilibrium is uniform over the horizon:
κ = const. (16.59)
First law: Under the hypotheses of Property 16.3 (in particular, general relativity), the
change δM in total mass between two nearby electrovacuum configurations of a
black hole in equilibrium is related to the change δA in horizon area, the change δJ
in total angular momentum and the change δQH in electric charge bya
κ
δM = δA + ΩH δJ + ΦH δQH , (16.60)
8π
where ΩH and ΦH are the horizon’s angular velocity and electric potential respec-
tively.
Second law: Under the hypotheses of Property 16.9, the area of cross-sections of a black
hole event horizon cannot decrease in time:
A(S2 ) ≥ A(S1 ), (16.61)
as soon as the cross-section S2 lies in the causal future of the cross-section S1 .
a
For simplicity, we are considering black holes with a connected horizon and a single angular momentum;
see Eq. (16.24) for the general formula, where M is denoted by M∞ and J by J∞ .
Remark 1: The three laws are valid for any spacetime dimension n ≥ 4, and even n ≥ 2 for the second
law. If one assumes that the event horizon is a Killing horizon, then the zeroth law is valid for n ≥ 2 as
well.
Historical note : The above ensemble of laws, plus a tentative third law, has been first stated in 1973
by James Bardeen, Brandon Carter and Stephen Hawking in a seminal article entitled The Four Laws of
Black Hole Mechanics [40]. This article, which was mostly written during the famous 1972 Les Houches
Summer School [164], exhibits proofs of the zeroth and first laws for n = 4, relies on Hawking’s article
[260] for the second law, and presents the third law as a conjecture, saying that “there are strong reasons
for believing in it” (cf. Sec. 16.4.2).
The three laws listed in Property 16.11 are named the laws of black hole dynamics and not
thermodynamics, because they still appear as a mathematical analogy without any definite
physical content. In particular, if we stick to a pure classical (i.e. non-quantum) point of view
and would like to attribute some thermodynamic temperature T to a black hole, it should be
T = 0. Indeed, if the black hole is placed in contact with a thermal reservoir of temperature
T0 > 0, energy will necessarily flow from the reservoir to the black hole and not in the reverse
way; this implies T < T0 , whatever T0 . For a stationary non-extremal black hole (κ ̸= 0),
T = 0 contradicts the tentative identification (16.58b) between T and κ. It is striking that
quantum field theory in curved spacetime restores the identification, thereby opening the path
to a genuine thermodynamics of black holes. The identification T ∝ κ arises from Hawking
radiation, which we discuss in the next section.
646 Evolution and thermodynamics of black holes
16.5 Hawking radiation
16.5.1 The Hawking radiation phenomenon
Property 16.12: Hawking radiation
Let (M , g) be a n-dimensional (n ≥ 2) asymptotically flat spacetime that is stationary
and contains a black hole, the event horizon of which is a Killing horizon of constant
surface gravity κ. Then, quantum field theory in the curved background (M , g) predicts
that any quantum field gives birth to a thermal radiation from the black hole to infinity,
called Hawking radiation. The radiation temperature as measured by asymptotic inertial
observers at rest with respect to the black hole is
ℏ κ
TH = , (16.62)
kB 2π
where ℏ = 1.054571817 10−34 J s is the reduced Planck constant and kB = 1.380649 10−23
J K−1 is the Boltzmann constant. TH is called the Hawking temperature of the black
hole.
We shall not establish this property here; this is done in many textbooks, e.g. [499, 503, 202,
87, 190], as well in some review articles, e.g. [75, 153, 293]. Rather, we shall limit ourselves to a
few remarks:
Remark 1: Hawking radiation is an outcome of quantum field theory within a fixed classical curved
background (see e.g. [503]). It has not been derived from any quantum theory of gravity, i.e. the
background spacetime (M , g) is not quantized. One says that Hawking radiation is a semiclassical
effect.
Remark 2: The phenomenon of Hawking radiation is essentially kinematic and does not depend on any
field equation for the metric g. In particular it does not rely on the Einstein equation (see e.g. Ref. [496]).
Moreover, the Hawking temperature (16.62) is independent from the spacetime dimension n (see e.g.
Ref. [307]).
Remark 3: Formula (16.62) is given for c = 1 units, which are used throughout the text. If one restores
c and considers that κ has the dimension of an acceleration, the formula should read TH = kℏB 2πc
κ
.
Remark 4: The Hawking temperature TH is the radiation temperature measured infinitely far from
the black hole; it is not a “local” temperature measured in the vicinity of the event horizon H . More
precisely, let us assume for simplicity that the black hole is non-rotating (cf. Property 5.2) and that its
exterior is strictly static. The stationary Killing vector ξ is then null on H and timelike in the black
hole exterior. Let us consider
√ static observers at a finite distance from H , i.e. observers of 4-velocity
u = ξ/V , where V := −ξ · ξ [cf. Eqs. (3.70) and (3.61)]. The temperature of the Hawking radiation
16.5 Hawking radiation 647
measured by these observers is 8
TH
T = . (16.63)
V
For infinitely distant observers, V → 1 [cf. Eq. (5.1)], and we recover T = TH . For static observers
infinitely close to the event horizon, V → 0 and we get T → +∞. Note that those observers are also
experiencing an infinite acceleration [cf. Eq. (3.75)]. On the contrary, a freely-falling observer crossing
the horizon measures T = 0, i.e. she does not detect the Hawking radiation (cf. Remark 5 hereafter).
Remark 5: The Hawking radiation is closely related to the Unruh effect, which is another prediction
of quantum field theory: in vacuum Minkowski spacetime, a uniformly accelerated observer measures a
√
black-body radiation of all particle species at the temperature TU = (ℏ/kB ) a/(2π), where a = a · a
is the norm of the observer’s 4-acceleration [484, 503, 87]. A static observer O close to the black
hole horizon H , as considered in Remark 4, feels an acceleration given by Eq. (3.74): a ∼ κ/V . By
combining formulas (16.63) and (16.62), we may then rewrite the radiation temperature T measured
by O as T ∼ (ℏ/kB ) a/(2π). Hence, it coincides with the Unruh temperature TU of the observer in
Minkowski spacetime sharing the same acceleration a as O. This is somehow expected if one assumes
that freely-falling observers near H perceive quantum states identical to the Minkowski vacuum (no
radiation).
Remark 6: In terms of elementary particles, the Hawking radiation is composed of all existing species
of massless and massive particles. For the latter ones, the contribution is significant only if kB TH > mp ,
where mp is the particle’s mass. For Schwarzschild black holes of mass M ≫ 2 1013 kg (the threshold
for kB TH being lower than the electron mass, see below), the Hawking radiation is composed quasi-
exclusively of neutrinos (87% of the radiated energy), photons (12%) and possibly gravitons (1%)
[394, 481].
Remark 7: The observed radiation spectrum at infinity is not exactly that of a black body of temperature
TH : it is corrected by greybody factors, which originate from the interaction of the radiation with the
spacetime curvature (backscattering).
For a Kerr black hole, the surface gravity κ is given by formula (10.77), so that the Hawking
temperature can be expressed in terms of the black hole mass M = m and the dimensionless
spin parameter ā := a/m = J/M 2 (0 ≤ ā ≤ 1):
ℏ c3 2
TH = . (16.64)
kB 8πGM 1 + (1 − ā2 )−1/2
Kerr
Note that we have restored the gravitational constant G and the speed of light c (cf. Remark 3).
Numerically, we get
M⊙ 2
TH = 6.17 10 −8
K, (16.65)
M 1 + (1 − ā2 )−1/2
or, in units more adapted to small black holes,
1 kg 2
kB TH = 1.057 10 10
GeV. (16.66)
M 1 + (1 − ā2 )−1/2
8
Cf. e.g. Eq. (7.2.10) in [503]; in view of Eq. (16.63), we may say that Hawking radiation obeys Tolman-
Ehrenfest law, which states that the temperature T of a medium in thermal equilibrium in a stationary gravita-
tional field obeys T V = const [444].
648 Evolution and thermodynamics of black holes
Hence, for a Schwarzschild black hole (ā = 0), TH = 6.17 10−8 (M⊙ /M ) K. For astrophysical
black holes, this leads to tiny temperatures: TH ≃ 4 10−9 K for a 15 M⊙ stellar black hole
(e.g. Cyg X-1, cf. Table 7.1), TH ≃ 1.6 10−14 K for the Milky Way central black hole Sgr A*
(M = 4.1 106 M⊙ ) and TH ≃ 9 10−18 K for M87* (M = 6.5 109 M⊙ ). The value TH = 1 K is
achieved for a black hole of mass 6.17 10−8 M⊙ , i.e. 2 % of the Earth’s mass, while the value
kB TH = me c2 (me = 9.11 10−31 kg being the electron mass) is achieved for M = 2.07 1013 kg.
Another conclusion that one can draw from formula (16.64) is that TH = 0 for an extremal
Kerr black hole (ā = 1): such an object does not emit any Hawking radiation.
Remark 8: Formula (16.64) shows that a Schwarzschild black hole (ā = 0) has a negative heat capacity:
its temperature increases while its energy (M ) diminishes. This feature is shared with Newtonian
self-gravitating systems, if one defines the temperature of such systems from the mean kinetic energy
of their constituents.
16.5.2 Black hole evaporation
Since it emits some radiation to infinity, the black hole loses energy, which makes its mass
decrease. For a Schwarzschild black hole, this results in a temperature increase (cf. Remark 8)
and hence an enhanced Hawking radiation. Furthermore, the higher the temperature, the
more quantum fields can join the radiation (cf. Remark 6), which enhances the rate of energy
loss as well. We are thus facing a runaway process, in which the whole black hole eventually
disappears. This phenomenon is called Hawking evaporation or black hole evaporation.
However, the evaporation occurs on extremely long time scales for astrophysical black holes,
as we are going to see.
To get a rough estimate of the evaporation time, let us consider that the total power
(energy radiated per unit time, also called luminosity) L of Hawking radiation received at
infinity obeys the Stefan-Boltzmann law: L = AσTH4 , where σ is Stefan-Boltzmann constant:
σ = π 2 kB4 /(60ℏ3 ) and A is some “emission area”. For simplicity, let us restrict to a Schwarzschild
black hole
√ of mass M . Then, a suitable value of A for an estimate is A = 4πbc , where
2
bc = 3 3M is the critical impact parameter for null geodesics introduced in Chap. 8 [cf.
Eq. (8.21)]. Indeed, 4πb2c can be viewed as the black hole’s area “seen from infinity” (cf. Fig. 8.24).
Moreover, it can be shown that the spectrum of Hawking radiation at high frequencies (i.e.
ν ≫ M −1 , so that quantum fields can be treated within geometrical optics) is that of a black
body of temperature TH and total area 4πb2c (see Fig. 1 of Ref. [394]). Hence A = 108πM 2 .
Given that kB TH = ℏ/(8πM ) [Eq. (16.64) with ā = 0], Stefan-Boltzmann law yields
ℏ
L=C , (16.67)
M2
where C = 9/(20480π) ≃ 1.40 10−4 . Actually, a precise computation, taking into account
the propagation of quantum fields in the Schwarzschild geometry and various particle species,
leads to the same formula (16.67) with [481, 394, 395]
C = 2.83 10−4 . (16.68)
16.5 Hawking radiation 649
This value is valid for M ≫ 2 1013 kg (TH below the electron threshold, cf. Remark 6), so that
the Hawking radiation contains only photons, neutrinos and gravitons9 . The black hole mass
decreases according to dM/dt = −L, i.e.
dM ℏ
= −C 2 . (16.69)
dt M
Here t is the time measured by an inertial observer at infinity. This differential equation is
easily integrated into
1/3
M (t) = M03 − 3Cℏt , (16.70)
where M0 is the mass at t = 0. Defining the evaporation time tevap as the time to reach M = 0
from M = M0 , we deduce from Eq. (16.70) the following property:
Property 16.13: Hawking evaporation of a Schwarzschild black hole
As seen by an inertial observer at infinity, a Schwarzschild black hole of initial mass M0
fully evaporates via Hawking radiation within a time
3
M03
M0
tevap = = 1.54 1066 yr , (16.71)
3Cℏ M⊙
where the value (16.68) of C has been used to get the numerical estimate.
For an astrophysical black hole (M0 > 1M⊙ ), tevap is huge — more than 56 orders of
magnitude larger than the age of the Universe (tUniv ≃ 1.38 1010 yr)! We conclude that
Hawking radiation is ultra-negligible for astrophysical black holes and does not play any role
in their dynamics. Hawking radiation becomes relevant only when tevap < tUniv , say. From
Eq. (16.71), one obtains tevap < tUniv ⇐⇒ M0 < 4.15 1011 kg. Actually, this value is below
2 1013 kg, so that electrons, and even muons, should have been included into the Hawking
radiation (cf. Remark 6). However, the above estimate is pretty good, a precise computation
leading to [352]
tevap < tUniv ⇐⇒ M0 < 5.00 1011 kg. (16.72)
The upper bound is roughly the mass of an asteroid of half-kilometer size. Note that a black
hole of that mass has a Schwarzschild radius of only 0.74 fm (around the proton size!); it would
pertain to the hypothetical class of the so-called primordial black holes, i.e. black holes
of subsolar mass that could have been formed in the primordial universe (see e.g. [86] for a
review).
Remark 9: For a Schwarzschild black hole, the horizon area A is proportional to the square of the mass:
A = 16πM 2 [Eq. (3.4)], so that the evaporation law (16.69) implies dA/dt < 0. At first glance, this seems
9
The original computation, performed by Page in 1976 [394], resulted in C = 2.01 10−4 but it took into
account only the four species of neutrinos known at that time (νe , ν̄e , νµ and ν̄µ ), in addition to photons and
gravitons. It has been updated by adding the tau neutrinos (ντ and ν̄τ ) by Thorne, Zurek & Price in 1986 [481]. The
value (16.68) is also recovered by setting n1/2 = 3, n1 = 1 and n2 = 1 in Eq. (19) of Page’s review article [395].
650 Evolution and thermodynamics of black holes
to contradict the second law of black hole dynamics (Property 16.11), i.e. the area theorem 16.9. Actually
there is no contradiction since one of the hypotheses of the theorem is not fulfilled for evaporating black
holes, namely the null convergence condition. Indeed, the effective energy-momentum tensor10 of the
quantum fields generating the Hawking radiation does not obey the null energy condition (2.95).
Historical note : In 1971, while introducing the superradiant scattering of classical fields by a Kerr
black hole (cf. Sec. 11.3.2 and the historical note on p. 392), Yakov Zeldovich [527] pointed out that the
quantum analog of the field superradiance would be the spontaneous creation of particle/antiparticle
pairs, with one particle falling into the black hole and the other one escaping to infinity. Zeldovich and
his doctoral student Alexei Starobinsky presented a computation of this effect to Stephen Hawking
while he was visiting Moscow in September 1973, accompanied by Kip Thorne (see Hawking’s account
in Chap. 7 of [264] and Thorne’s one in Chap. 12 of [479]). Hawking tried to rederive the effect via a
different approach of quantum field theory in curved spacetime and he realized that superradiance is not
at all required: the spontaneous radiation takes place even for a non-rotating black hole, which has no
outer ergoregion and thus cannot trigger any Penrose/superradiance process. This fantastic discovery
was presented in short article published in early 1974 [262] and the computations were detailed in
a subsequent one published in 1975 [263]. Both articles give formula (16.62) relating the radiation
temperature to the surface gravity and discuss the black hole evaporation.
16.6 Black hole thermodynamics
16.6.1 Bekenstein-Hawking entropy
The phenomenon of Hawking radiation puts on firm grounds the identification of the surface
gravity as the black hole temperature and fully fixes the coefficient α appearing in Eq. (16.58). In-
deed, comparing Eq. (16.58b) with formula (16.62) for T = TH yields α = kB /(4ℏ). Accordingly,
Eq. (16.58a) becomes S = kB A/(4ℏ) and we may state:
Property 16.14: Bekenstein-Hawking entropy
A n-dimensional (n ≥ 3) black hole in equilibrium is endowed with the Bekenstein-
Hawking entropy SBH , which is proportional to the area A of the event horizon according
to the formula:
A
SBH = kB . (16.73)
4ℏ
For n ≥ 4 and Einstein-Maxwell theory (see Property 16.3 for detailed assumptions), the
first law of black hole dynamics (16.60) writes then
δM = TH δSBH + ΩH δJ + ΦH δQH , (16.74)
where TH is the Hawking temperature.
10
See e.g. Ref. [203] for the computation of the renormalized expectation value of the energy-momentum tensor
of a quantum field near the black hole horizon.
16.6 Black hole thermodynamics 651
Remark 1: The subscript “BH” in SBH is meant for Bekenstein-Hawking, not for black hole.
Remark 2: As for the Hawking temperature (16.62), formula (16.73) for the Bekenstein-Hawking
entropy does not depend on the spacetime dimension n. It is often summarized by SBH = A/4, which
holds in units such that kB = 1 and ℏ = 1.
3
Remark 3: For n = 4, if one restores the G’s and c’s, formula (16.73) reads SBH = kB 4Gℏ
c A
. One
recognizes the square of the Planck length: ℓP := Gℏ/c ≃ 1.62 10 −35 m, so that formula (16.73)
p
3
can be rewritten in terms of the dimensionless ratio A/ℓ2P as
A
SBH = kB . (16.75)
4ℓ2P
For a Kerr black hole, the area A is related to the mass M and to the spin parameter a = āM
via Eq. (10.83), so that the Bekenstein-Hawking entropy (16.73) can be expressed as
√
GM 2 1 + 1 − ā2
SBH = 4πkB . (16.76)
ℏc 2 Kerr
Note that, as in formula (16.64) for the Hawking temperature, we have restored G and c in the
above formula. Numerically, we get
2 √
M 1 + 1 − ā2
SBH = 1.05 10 77
kB . (16.77)
M⊙ 2
Hence, for a Schwarzschild black hole (ā = 0), SBH = 1.05 1077 (M/M⊙ )2 kB . This yields huge
values for astrophysical black holes: SBH ≃ 2 1079 kB for a 15 M⊙ stellar black hole, SBH ≃
2 1090 kB for Sgr A* (M = 4.1 106 M⊙ ) and SBH ≃ 4 1096 kB for M87* (M = 6.5 109 M⊙ ).
These numbers are terribly large: besides black holes, the total entropy of the observable
Universe is about 1.1 1090 kB . It originates mostly from the cosmic microwave background and
the cosmic neutrino background (roughly one half each), largely dominating the entropy of the
interstellar and intergalactic medium (∼ 1082 kB ) and that of all the stars (∼ 1081 kB ) [178]. It
follows that the entropy of a single massive black hole, such as Sgr A* or M 87*, is larger than
the entropy of the whole observable Universe!
16.6.2 The generalized second law
In the presence of a black hole, the standard second law of thermodynamics cannot hold in the
Universe outside the black bole: the total entropy of matter and radiation located there can
be decreased by dropping some matter (and its entropy) into the black hole. One then has to
replace the standard second law by
Property 16.15: generalized second law of thermodynamics (GSL) (Bekenstein 1973
[52, 53])
Let us consider a “closed system” formed by a black hole and some matter and electromag-
652 Evolution and thermodynamics of black holes
netic radiation in the black hole exterior. Define the generalized entropy Sgen by
Sgen := Smat + SBH , (16.78)
where Smat is the ordinary entropy of the matter and radiation and SBH is the Bekenstein-
Hawking entropy (16.73) of the black hole. Then, any physical process makes Sgen increase,
or at the very least, stay constant:
∆Sgen ≥ 0. (16.79)
The GSL should be considered more as a postulate ruling thermodynamics rather than a
theorem that can be established. At most it can be proven in some specific cases. For instance,
if one regards the process of Hawking radiation itself, one has ∆SBH < 0, since the black
hole area diminishes during Hawking evaporation, as discussed at the end of Sec. 16.5.1. But
one can show that the entropy of the Hawking radiation compensates for this loss, leading
to ∆Smat ≥ |∆SBH |, so that ∆Sgen ≥ 0 is fulfilled (see Ref. [505] for details). Besides, one can
also show that the GSL is actually obeyed in various thought experiments naively devised to
violate it [85, 503, 505, 510].
Historical note : In 1970-71, the discovery of the irreversible increase of the area of a Kerr black hole
interacting with particles by Roger Penrose, Roger Floyd and Demetrios Christodoulou [409, 112], along
with the general area theorem established by Stephen Hawking [259] (cf. the historical notes on p. 637
and 642), suggested an analogy between the black hole area and the entropy in thermodynamics. In
1972, Jacob Bekenstein [50, 52] was the first to go beyond this formal analogy by considering that a black
hole is endowed with a genuine entropy S proportional to its area A, “as a measure of the inaccessibility
of information (to an exterior observer) as to which particular internal configuration of the black hole is
actually realized in a given case” [52]. Moreover, by means of some heuristic arguments, Bekenstein
found that the relation between S and A should be S = ηkB A/(4πℏ), where η is a dimensionless
coefficient of order unity [50, 52]. This is exactly formula (16.73) provided that η = π. This value of η
was determined only after the black hole temperature was set to the Hawking temperature (16.62). This
was done by Stephen Hawking in the 1975 article detailing the computation of Hawking radiation [263]
(cf. historical note on p. 650). As for the generalized second law of thermodynamics, it was proposed
by Bekenstein himself in the 1972-73 articles introducing the Bekenstein-Hawking entropy [50, 52].
Further arguments for its validity have been provided in a subsequent article [53]. Previously, it was
said that the (standard) second law of thermodynamics was transcended in presence of a black hole
[40, 96], for it could not hold when some matter (and its entropy!) is thrown into the black hole.
16.6.3 Black hole entropy from quantum gravity
By superseding the standard second law of thermodynamics, the GSL provides a genuine
thermodynamic meaning to the Bekenstein-Hawking entropy. In particular, by adding SBH to
the “ordinary” entropy Smat , formula (16.78) puts SBH on the same footing as Smat . It would
be fully satisfactory though to get a “statistical mechanics” origin for SBH , i.e. to interpret
SBH as a measure of the number N of (quantum) microscopic states corresponding to a given
macroscopic state of the black hole, according to Boltzmann’s formula:
SBH = kB ln N . (16.80)
16.7 Concluding remarks 653
Defining the microscopic states a priori requires a quantum theory of gravity, which we do not
have yet. However some partial results have been obtained, based on the two main current
approaches to quantum gravity: string theory (cf. e.g. [402] and Chap. 9 of [246]) and loop
quantum gravity (cf. e.g. [410, 442]). Regarding the former, Strominger and Vafa (1996) [465]
have first established the Bekenstein-Hawking formula (16.73) by counting the microstates
of an extremal supersymmetric black hole in dimension n = 5. By extremal, it is meant
that the event horizon is a degenerate Killing horizon (cf. Sec. 5.4.3), i.e. it has κ = 0, or
equivalently, a vanishing Hawking temperature. By supersymmetric, it is meant that these black
hole solutions have been obtained within a N = 4 supersymmetric Yang-Mills theory, which
corresponds to a low energy limit of some string theory. Further developments are discussed
in the review articles [85, 153, 285, 505, 513, 525]; they are still limited to extremal black holes
or near-extremal ones. Computations are indeed easier for such black holes because their
near-horizon geometries have additional symmetries, as we have seen for the extremal Kerr
case (cf. Sec. 13.4). For the latter, the Bekenstein-Hawking entropy of has been recovered via
some holographic duality11 [247, 137]. Let us stress that the Bekenstein-Hawking entropy of a
Schwarzschild black hole, which is non-extremal, has not been recovered by string theory yet.
Regarding loop quantum gravity, first approaches, devised by Rovelli in 1996 [441] and by
Ashtekar, Baez, Corichi and Krasnov in 1997 [21], resulted in a statistical entropy proportional
to the black hole area, as for the Bekenstein-Hawking entropy (16.73). Subsequent studies lead
to the following formula for the entropy of a spherically symmetric macroscopic black hole of
area A (see e.g. Ref. [188] and Refs. [33, 85, 410] for reviews):
γ0 A 3 A
S = kB − ln , (16.81)
γ 4ℏ 2 ℏ
where γ0 ≃ 0.274 and the dimensionless quantity γ is the so-called Barbero-Immirzi parameter.
Its value is not set by the theory (yet). This unknown parameter is not expected to appear at
the classical limit of loop quantum gravity, which should be general relativity. On the contrary,
in the quantum regime, γ plays a p significant role by determining the
√ quantum of area a0 in
terms of the Planck length ℓP := Gh/c (cf. Remark 3) as a0 = 4 3πγℓP [442]. For large
3 2
values of A, i.e. for A ≫ ℏ, the logarithmic term can be neglected in formula (16.81) and one
recovers the Bekenstein-Hawking entropy (16.73), provided that γ = γ0 .
Other attempts to derive the Bekenstein-Hawking entropy from statistical mechanics
within quantum gravity are described in Sec. 7 of Carlip’s review [85]. See also Ref. [245] for
an historical review up to 2013.
16.7 Concluding remarks
16.7.1 Black hole entropy and the holographic principle
...
11
This is the Kerr/CFT correspondence mentioned in Sec. 13.4.6.
654 Evolution and thermodynamics of black holes
16.7.2 Dependency on general relativity
The first law of black hole dynamics (16.60) strongly depends on the theory of gravity being
general relativity. Indeed, it is derived from the general mass variation formula (16.3) by
substituting the electromagnetic field energy-momentum tensor for the Einstein tensor G (cf.
the proof of Property 16.3), which amounts to assuming the electrovacuum Einstein equation.
On the other hand, the second law of black hole dynamics (cf. Property 16.11), depends less
on general relativity for it requires only the null convergence condition (2.94) to be fulfilled
(cf. the area theorem 16.9). This is of course guaranteed by general relativity with matter and
fields obeying the null energy condition (2.95), but this might hold in some other theories as
well. Similarly, the zeroth law (16.59) depends weakly on general relativity; it is even fully
independent of it if the event horizon is part of a bifurcate Killing horizon (Property 3.16) or
if the spacetime is axisymmetric with orthogonal transitivity of the R × SO(2) group action
(cf. Remark 6 on p. 74); if none of these properties is guaranteed, then the zeroth law requires
the null dominance condition (3.43) (cf. Property 3.10), which is fulfilled by general relativity
combined with the null dominant energy condition (3.46), but may be obeyed in other theories
as well.
Hence, among the three laws listed in Property 16.11, the first one is the most sensitive
to a deviation from general relativity. Contemplating its expression (16.60), one may wonder
whether the term 8π κ
δA could be changed to something else in an alternative theory of gravity.
Would the new term be of the form κ δ(...), we could still interpret it as a “T dS” term, given
that the identification T ↔ κ remains valid beyond general relativity, thanks to the universal
character of Hawking radiation (cf. Remark 2 on p. 646). This would then question the
identification of the entropy S with the black hole area A, i.e. the Bekenstein-Hawking formula
(16.73). We shall discuss this in the next chapter.
Chapter 17
Black hole thermodynamics beyond
general relativity
Contents
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655
17.2 Diffeomorphism-invariant theories of gravity . . . . . . . . . . . . . . 655
17.3 Wald entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666
17.4 Covariant phase space formalism . . . . . . . . . . . . . . . . . . . . . 670
17.5 First law for diffeomorphism-invariant theories . . . . . . . . . . . . 684
17.6 What about the second law? . . . . . . . . . . . . . . . . . . . . . . . . 693
17.1 Introduction
This chapter is in a draft stage.
This chapter relies heavily on exterior calculus, i.e. on calculus on differential forms. The
unfamiliar reader is urged to take a look at Sec. A.4.3 of Appendix A first.
17.2 Diffeomorphism-invariant theories of gravity
17.2.1 Framework
We consider that the n-dimensional spacetime (M , g) is ruled by a metric theory of gravity
built upon a least action principle, with the following action:
√
Z Z Z
S= L= Lϵ = L −g dn x. (17.1)
M M M
656 Black hole thermodynamics beyond general relativity
Here L is the Lagrangian n-form1 , L is the Lagrangian scalar and ϵ is the Levi-Civita
tensor (cf. Sec. A.3.4). The second equality reflects the identity2 L = L ϵ and the third one
provides an expression of the integral in terms of a coordinate system (xα ) on M , g being the
determinant of g with respect to these coordinates [cf. Eq. (A.52)]. For simplicity we disregard
any scalar or matter field, so that L is a function of the metric tensor g only. Actually all that
follows can be easily generalized to account for extra fields, as in scalar-tensor theories for
instance, where L is a function of g and some scalar field ϕ. We limit ourselves here to g,
mostly in order to keep notations simple. More precisely, we assume that at a given point
p ∈ M , L depends only on the values of g and its derivatives at p, up to a certain order. By
derivative, it is meant derivative with respect to some affine connection on M , for instance the
one formed by the partial derivatives with respect to given coordinates. We shall write L(g) to
stress this dependency of L on g, according to the following convention:
Notation: Boldface parentheses are used to denote the local (pointwise) dependency of a
tensor field on M with respect to other ones: A(B, C) means that at a given point p ∈ M ,
A depends only on the values of the tensor fields B and C and their derivatives at p.
We shall consider only gravity theories whose formulation does not depend upon any
specific coordinates on M , nor upon any extra-structure on M , like a privileged vector field.
This amounts to demanding that the Lagrangian n-form L is covariant under the group of
diffeomorphisms of M . More precisely, under the action of a diffeomorphism Φ : M → M ,
the metric tensor is transformed to its pullback (Φ−1 )∗ g by Φ−1 (cf. Sec. A.2.8 for the definition
of the pullback). The inverse of Φ is invoked here because, as g is a tensor of type (0, 2), Φ∗
would carry it from3 Φ(M ) to M , while we are considering the transformation from M to
Φ(M ) (cf. Remark 12 on p. 721). Let L((Φ−1 )∗ g) be the Lagrangian n-form constructed by
keeping the same functional dependence in (Φ−1 )∗ g as L(g) has in g. The invariance of the
theory under diffeomorphisms amounts to demanding that, as a n-form on M , L((Φ−1 )∗ g) is
nothing but the n-form L(g) transformed under the action of Φ, i.e. the pullback of L(g) by
Φ−1 : L((Φ−1 )∗ g) = (Φ−1 )∗ L(g). Given that the spacetimes (M , g) and (Φ(M ), (Φ−1 )∗ g) are
physically equivalent, this means that L(g) and L((Φ−1 )∗ g) are basically the same object. In
particular, the action S given by Eq. (17.1) takes the same value if L((Φ−1 )∗ g) is substituted
for L(g). If one interprets the diffeomorphism action in a “passive” way, i.e. as a change of
coordinates on M , the diffeomorphism-invariance condition simply means that the functional
form of L expressed in Eq. (17.3) below does not depend upon the choice of coordinates. Since
the condition L((Φ−1 )∗ g) = (Φ−1 )∗ L(g) must hold for any diffeomorphism, we may simplify
it by introducing Φ′ := Φ−1 and renaming Φ′ to Φ:
1
As the integral of a n-form over a n-dimensional manifold, S is well-posed and coordinate independent.
2
Since the space of n-forms at any point p ∈ M is 1-dimensional, L and ϵ are necessarily proportional, so
that there is no loss of generality in the writing L = L ϵ.
3
Since Φ is a diffeomorphism M → M , we have of course Φ(M ) = M ; however we keep the notation
Φ(M ) to stress the image character by Φ.
17.2 Diffeomorphism-invariant theories of gravity 657
A gravity theory is said to be diffeomorphism-invariant iff its Lagrangian n-form L
is covariant under the group of diffeomorphisms of M , i.e. iff for any diffeomorphism
Φ : M → M,
L(Φ∗ g) = Φ∗ L(g), (17.2)
where Φ∗ stands for the pullback by Φ, as defined in Sec. A.2.8.
We have then
Property 17.1: Lagrangian of diffeomorphism-invariant theories (Iyer & Wald 1994
[292])
Any diffeomorphism-invariant theory of gravity based on a Lagrangian involving only the
metric tensor g and a finite number of its derivatives (with respect to an arbitrary affine
connection on M ) can be recast in terms of a Lagrangian involving only g, its Riemann
tensor and a finite number of symmetrizeda covariant derivatives of the latter:
(17.3)
L(g) = L gαβ , Rαβγδ , ∇λ Rαβγδ , . . . , ∇(λ1 · · · ∇λm ) Rαβγδ .
There is no loss of generality in considering only symmetrized derivatives, since any antisymmetrized
a
combination can be reexpressed in term of the Riemann tensor itself and lower order derivatives.
The reader is referred to the original article [292] for the proof.
Example 1 (Lagrangian of general relativity): For general relativity, L is the Einstein-Hilbert
Lagrangian:
L = (16π)−1 R = (16π)−1 g µν Rµν = (16π)−1 g µν Rσµσν = (16π)−1 g µν g ρσ Rρµσν . (17.4)
The last expression is clearly of the form (17.3), given that the components (g αβ ) are functions of the
components (gαβ ) only.
17.2.2 Lagrangian variation and presymplectic potential form
A standard computation (see e.g [138, 345]) leads to
Property 17.2: variation of the Lagrangian of a pure metric theory
Let L be a Lagrangian n-form that depends only of the metric tensor g and its derivatives,
as in Eq. (17.3). The variation of L resulting from a generic variation δg of g is expressible
as
δL = E µν δgµν ϵ + dθ, (17.5)
where E µν stands for the components of a symmetric type-(2, 0) tensor field E, called
the Euler-Lagrange tensor, and θ is a (n − 1)-form, called the presymplectic potential
(n − 1)-form (this name will be justified below). The n-form dθ is the exterior derivative
of θ (cf. Sec. A.4.3). E depends on g but not on δg, while θ depends on both g and δg, in a
linear way in terms of δg. All these dependencies are local, so if we want to stress them,
658 Black hole thermodynamics beyond general relativity
we shall write E = E(g) and θ = θ(g, δg) according to the notation introduced on p. 656.
Remark 1: The term dθ collects all the derivatives of δg, which have been gathered by means of the
Leibniz rule, so that the term E µν δgµν ϵ in Eq. (17.5) does not contain any derivative of δg.
It must be stressed immediately that the presymplectic potential (n − 1)-form θ is not
unique. First of all, since d is nilpotent [Eq. (A.92)], θ in formula (17.5) can be substituted by
θ ′ = θ + dY , where Y = Y (g, δg) is an arbitrary (n − 2)-form. Secondly, for a given gravity
theory, the Lagrangian n-form itself is not unique. Indeed the n-form L′ = L + dµ, where
µ = µ(g) is an arbitrary (n − 1)-form with compact support, leads to the same action (17.1)
by virtue of Stokes’ theorem (A.94). The corresponding change in the presymplectic potential
is θ → θ + δµ. From these two facts, we conclude:
Property 17.3: ambiguities in the presymplectic potential form
For a given gravity theory, the presymplectic potential (n − 1)-form θ is not unique: it can
be substituted by
θ ′ = θ + dY + δµ, (17.6)
where (i) Y = Y (g, δg) is an arbitrary (n − 2)-form that is a local function of g and δg,
with a linear dependence on δg and (ii) δµ is the variation triggered by δg of an arbitrary
(n − 1)-form µ = µ(g) that is a local function of g and has compact support. The µ term
in Eq. (17.6) arises from the allowed change
L′ = L + dµ (17.7)
in the Lagrangian n-form.
Example 2 (presymplectic potential form of general relativity): For general relativity, one has
L = (16π)−1 Rϵ [Eq. (17.4)]. Given a coordinate system (xα ) on M , this can be rewritten as L =
√
(16π)−1 R −g e, where g is the determinant of g with respect to (xα ) and e is the n-form whose
components with respect to (xα ) are eα1 ···αn = 1 (resp. −1) if (α1 , . . . , αn ) is an even (resp. odd)
√
permutation of (0, . . . , n) and 0 otherwise [cf. Eq. (A.52)]. One has then δL = (16π)−1 δ(R −g) e,
√
since obviously (xα ), and thus e, is not altered by the variation δg. From the expression of δ(R −g)
given by Eq. (16.18) and using Gµν δg µν = −Gµν δgµν , we get 16πδL = −Gµν δgµν ϵ + ∇µ H µ ϵ, where
Gµν is the double metric dual of the Einstein tensor and H is the vector field defined in terms of δg by
Eq. (16.13), where hαβ = g αµ δgµβ since hαβ := δgαβ . Now, the identity (A.93) yields ∇µ H µ ϵ = d(H·ϵ),
so that 16πδL = −Gµν δgµν ϵ + d(H · ϵ). We conclude that the variation of the Einstein-Hilbert
Lagrangian is indeed of the form (17.5) with
E αβ = −(16π)−1 Gαβ and θ = (16π)−1 H · ϵ = (16π)−1 ⋆H, (17.8)
where the last equality stems from the definition of the Hodge dual of a 1-form [Eq. (5.38) with p = 1]. In
index notation, θα1 ···αn−1 = (16π)−1 H µ ϵµα1 ···αn−1 . Substituting Eq. (16.13) for H µ , we get the explicit
expression of θ(g, δg):
θα1 ···αn−1 = (16π)−1 g µν g ρσ (∇σ δgνρ − ∇ν δgρσ ) ϵµα1 ···αn−1 . (17.9)
We check that θ(g, δg) is linear in δg, as stated in Property 17.2.
17.2 Diffeomorphism-invariant theories of gravity 659
Property 17.4: field equations from the least action principle
The action S defined by (17.1) is extremal with respect to any metric variation δg that has
compact support if, and only if,
E(g) = 0. (17.10)
This constitutes the so-called Euler-Lagrange equations for the considered gravity theory,
also called the field equations or equations of motion.
Proof. Since δg is assumed to have compact support, let D be a n-dimensional submanifold
with boundary of M such that δg = 0 in the neighborhood of the boundary ∂D and outside
D. In view of Eqs. (17.1) and (17.5), we have then
Z Z Z Z Z Z
µν µν
δS = δL = E δgµν ϵ + dθ = E δgµν ϵ + θ= E µν δgµν ϵ,
M D D D D
| ∂D
{z }
0
where Stokes’ theorem (A.94) has been used to let appear the integral of θ on ∂D, which has
been set to zero due to θ(g, δg)|∂D = 0. This last property results from the linearity of θ in δg,
given that δg and all its derivatives are vanishing on ∂D. From the above expression of δS it is
clear that having δS = 0 for any variation δg is equivalent to Eq. (17.10).
Example 3 (field equations of general relativity): For general relativity, E is given by Eq. (17.8).
Hence the field equations (17.10) become G = 0, which is equivalent to R = 0 — the vacuum Einstein
equation (1.44).
17.2.3 Noether charges related to diffeomorphism invariance
Any smooth vector field ξ on M generates a 1-parameter family of infinitesimal diffeomor-
phisms, namely the flow maps along ξ: Φt : M → M , where t is an infinitesimal parameter.
As described in Sec. A.4.2, each Φt displaces any point p ∈ M by the infinitesimal vector t ξ|p .
The flow map Φt is clearly a diffeomorphism, whose inverse is Φ−1 t = Φ−t . Let us consider the
variation δg induced by Φt , or more precisely by Φ−1t in order to simplify expressions; indeed
choosing Φt makes the pullback Φt g appear instead of (Φt ) g, as discussed in Sec. 17.2.1.
−1 ∗ −1 ∗
The variation of g is then
δg = Φ∗t g − g. (17.11)
Since t is infinitesimal, the very definition of the Lie derivative [Eq. (A.82)] leads to Φ∗t g − g =
tLξ g. It follows immediately that
δg = tLξ g. (17.12)
The variation of the Lagrangian n-form induced by δg is δL = L(Φ∗t g)−L(g). Since the theory
is assumed to be diffeomorphism-invariant, the covariance property (17.2) holds: L(Φ∗t g) =
Φ∗t L(g). This yields δL = Φ∗t L(g) − L(g). This is the same form as g’s variation expressed in
660 Black hole thermodynamics beyond general relativity
Eq. (17.11) and, again, the definition (A.82) of the Lie derivative (which is applicable here since
L is a tensor field of type (0, n)) leads to
δL = tLξ L. (17.13)
Remark 2: The variation δg is often defined as a variation rate rather than a genuine variation, namely
as the derivative of a 1-parameter family of metric tensors ĝ(t) such that ĝ(0) = g. Using the symbol δ̂
to distinguish from the variation considered here, the definition is δ̂g := dĝ/dt|t=0 . Since the metric
family induced by the flow maps along ξ is ĝ(t) := Φ∗t g, one gets δ̂g = limt→0 t−1 (Φ∗t g − g). In view
of the definition (A.82) of the Lie derivative, this yields
δ̂g = Lξ g. (17.14)
Similarly, one defines δ̂L := dL(ĝ(t))/dt|t=0 , which yields δ̂L = Lξ L. When comparing with
Eqs. (17.12) and (17.13), we see that the derivative-based definition has the advantage of getting rid of
the (not particularly significant) infinitesimal factor t.
If one plugs Eqs. (17.12) and (17.13) into the generic Lagrangian variation formula (17.5),
one gets tLξ L = tE µν Lξ gµν ϵ + d(θ(g, tLξ g)). Since θ is linear with respect to its second
argument (cf. Property 17.2), one has d(θ(g, tLξ g)) = td(θ(g, Lξ g)), so that there comes
Lξ L = E µν Lξ gµν ϵ + d(θ(g, Lξ g)).
Now, the Cartan identity (A.95) gives Lξ L = ξ · dL + d(ξ · L) = d(ξ · L) since dL = 0, as
for any (n + 1)-form on a n-dimensional manifold. Accordingly, we may rewrite the above
equation as
d (θ(g, Lξ g) − ξ · L) = −E µν Lξ gµν ϵ. (17.15)
If the field equations are fulfilled, i.e. if E = 0 [Eq. (17.10)], the right-hand side of Eq. (17.15)
is zero, which implies that the argument of the exterior derivative in the left-hand side is a
closed (n − 1)-form. This defines a conserved current, which is nothing but the expression
of Noether’s theorem for the invariance of the Lagrangian by the 1-dimensional group of
diffeomorphisms generated by the vector field ξ:
Property 17.5: Noether theorem for diffeomorphisms generated by a vector field
Given a smooth vector field ξ on M , the Noether current (n − 1)-form of ξ is defined by
[502, 292]
J (ξ) := θ(g, Lξ g) − ξ · L, (17.16)
where θ(g, Lξ g) is the (n − 1)-form obtained by substituting Lξ g for δg in the presym-
plectic potential form θ(g, δg) (cf. Property 17.2) and ξ · L is the (n − 1)-form obtained by
setting the first argument of the n-form L to ξ: ξ · L := L(ξ, · · · ). J (ξ) is a local function
of ξ and its derivatives, and depends linearly on ξ. If the gravity theory is diffeomorphism-
invariant, then the (n − 1)-form J (ξ) is closed as soon as the equations of motion (17.10)
are fulfilled or ξ is a Killing vector:
dJ (ξ) = 0 . (17.17)
17.2 Diffeomorphism-invariant theories of gravity 661
The Noether current (n − 1)-form is not unique, some ambiguity J → J ′ resulting from
(17.6) and (17.7), with
J ′ (ξ) = J (ξ) + d [Y (g, Lξ g) + ξ · µ(g)] , (17.18)
where Y (g, Lξ g) is any (n − 2)-form that is linear in Lξ g, µ(g) is any (n − 1)-form with
compact support and ξ · µ is the (n − 2)-form µ(ξ, . . .).
Proof. That J (ξ) is a linear function of ξ follows immediately from the linearity of θ(., .) with
respect to its second argument. Similarly the locality of J (ξ) with respect of ξ is inherited from
that of θ(., .). If the theory diffeomorphism-invariant, Eq. (17.15) holds. Given the definition
of J (ξ), Eq. (17.15) is equivalent to dJ (ξ) = −E µν Lξ gµν ϵ, so that dJ (ξ) = 0 if E = 0 is
fulfilled or ξ is a Killing vector (Lξ g = 0). Finally, both L′ (g) and L(g) in Eq. (17.7) must be
covariant under diffeomorphisms. It follows that µ(g) must be covariant as well. Then the
variation of µ resulting from a diffeomorphism Φt generated by ξ is δµ = tLξ µ, by the very
same argument that led to formula (17.13) for δL. Accordingly, Eq. (17.6) yields θ ′ (g, Lξ g) =
θ(g, Lξ g) + dY (g, Lξ g) + Lξ µ(g). By combining this identity with L′ = L + dµ [Eq. (17.7)],
the definition (17.16) applied to J ′ leads to
J ′ (ξ) = J (ξ) + dY (g, Lξ g) + Lξ µ(g) − ξ · dµ(g).
The Cartan identity (A.95) gives Lξ µ(g) − ξ · dµ(g) = d (ξ · µ(g)) and we obtain (17.18).
Remark 3: From the definition (17.16), it is clear that J (ξ) is a local function of g (and its derivatives),
in addition to ξ. So, following the notation introduced in Sec. 17.2.1, we could have written J (g, ξ).
However for the sake of brevety, we use J (ξ), leaving the dependency on g implicit.
Example 4 (Noether current form of general relativity): From expression (17.9) for θ in general
relativity with δgµν = tLξ gµν = t(∇µ ξν + ∇ν ξµ )[Eq. (A.87)], we get
16πθ(g, Lξ g)α1 ···αn−1 = g µν (∇σ ∇ν ξ σ + ∇σ ∇σ ξν − 2∇ν ∇ρ ξ ρ ) ϵµα1 ···αn−1
= g µν (2Rνσ ξ σ − ∇σ ∇ν ξ σ + ∇σ ∇σ ξν ) ϵµα1 ···αn−1
= (2Rµν ξ ν + ∇ν ∇ν ξ µ − ∇ν ∇µ ξ ν ) ϵµα1 ···αn−1 ,
where the contracted Ricci identity (A.107) has been used to get the second line. Plugging this formula,
as well as L = Rϵ (cf. Example 1) into the definition (17.16) of J (ξ) yields
J(ξ)α1 ···αn−1 = (16π)−1 (2Gµν ξ ν + ∇ν ∇ν ξ µ − ∇ν ∇µ ξ ν ) ϵµα1 ···αn−1 .
If the field equations are obeyed, then G = 0 (cf. Example 3) and the above expression reduces to
J(ξ)α1 ···αn−1 = (16π)−1 (∇ν ∇ν ξ µ − ∇ν ∇µ ξ ν ) ϵµα1 ···αn−1 . (17.19)
We recognize in the term within parentheses the Komar current of ξ as defined by Eq. (5.50): J (ξ)µ :=
∇ν ∇ν ξ µ − ∇ν ∇µ ξ ν . Accordingly, we may write
J (ξ) = (16π)−1 J (ξ) · ϵ = (16π)−1 ⋆ J (ξ), (17.20)
where the last equality stems from Eq. (5.38) with p = 1. Hence, for general relativity, the Noether
current (n − 1)-form of ξ is nothing but the Hodge dual of the (metric dual of the) Komar current of
ξ, up to the factor (16π)−1 . That J (ξ) is closed in this case follows readily from the identity (A.93):
16π dJ (ξ) = d(J (ξ) · ϵ) = (∇ · J (ξ))ϵ, along with the conservation law (5.51): ∇ · J (ξ) = 0.
662 Black hole thermodynamics beyond general relativity
Property 17.6: Noether potential form
Given a vector field ξ on M and a diffeomorphism-invariant theory described by a La-
grangian n-form L, there exists a (n − 2)-form Q(ξ) on M , which is a linear local function
of ξ, such that the Noether current (n − 1)-form of ξ is the exterior derivative of Q(ξ):
J (ξ) = dQ(ξ). (17.21)
Q(ξ) is called the Noether potential (n − 2)-form of ξ. It is not unique, being subject to
the ambiguity Q → Q′ , with
Q′ (ξ) = Q(ξ) + Y (g, Lξ g) + ξ · µ(g) + dZ(g, ξ), (17.22)
where Y (g, Lξ g) is any (n − 2)-form that is linear in Lξ g, µ(g) is any (n − 1)-form with
compact support and Z(g, ξ) is any (n − 3)-form that is linear in ξ.
Proof. Since J (ξ) is a closed differential form [Eq. (17.17)], the Poincaré lemma states that it is
locally exact (on a contractible open set, cf. Sec. A.4.3). However, because dJ (ξ) = 0 holds
for any vector field ξ, it can be shown that J (ξ) is globally exact, i.e. takes the form (17.21)
with Q(ξ) being a local function of ξ [500]. The linearity of Q(ξ) in terms of ξ arises from the
linearity of J (ξ) (cf. Property 17.5). Finally, the Y and µ ambiguities in (17.22) are inherited
from the ambiguities of J [Eq. (17.18)], while the Z ambiguity stems from Eq. (17.21) defining
Q up to the addition of a closed (n − 2)-form (cf. Lemma 1 in Ref. [500]).
Remark 4: As for J (ξ) (cf. Remark 3 on p. 661), Q(ξ) is a local function of g, in addition to ξ, although
the dependency on g is not explicited in the notation Q(ξ).
Remark 5: The (n − 2)-form Q(ξ) has been introduced by Wald (1993) [502] under the name Noether
charge (n − 2)-form. However, as argued by other authors [294, 255], the name Noether potential
(n − 2)-form is more relevant, since Q(ξ) is not conserved: a priori dQ(ξ) ̸= 0 and Eq. (17.21) shows
clearly that Q(ξ) plays the role of a potential of a conserved current. Another name given to Q(ξ) in
the literature is the Noether-Wald charge density [246, 250].
Example 5 (Noether potential form of general relativity): For general relativity, J (ξ) is given by
Eq. (17.19), or Eq. (17.20) in terms of the Komar current of ξ. Let us show that Q(ξ) is then nothing
but the Hodge dual of the exterior derivative of the 1-form ξ (the metric dual to ξ), up to the factor
−(16π)−1 :
Q(ξ) = −(16π)−1 ⋆ (dξ). (17.23)
Given the expression of the Hodge dual of a 2-form [Eq. (5.38) with p = 2], the identity (dξ)αβ =
∇α ξβ − ∇β ξα [Eq. (A.91b)] and the antisymmetry of ϵ, a formula equivalent to (17.23) is
Q(ξ)α1 ···αn−2 = −(16π)−1 ∇µ ξ ν ϵµνα1 ···αn−2 . (17.24)
To prove that Q(ξ) as given by (17.23) obeys dQ(ξ) = J (ξ) [Eq. (17.21)], let us establish the Hodge
dual of this relation: ⋆dQ(ξ) = ⋆J (ξ), which is an equality between 1-forms. From Eq. (17.23), we have
⋆dQ(ξ) = −(16π)−1 ⋆ d ⋆ (dξ). Now, the operator ⋆d⋆ is, up to a sign, the codifferential δ, which
17.2 Diffeomorphism-invariant theories of gravity 663
maps a p-form to a (p − 1)-form and is defined by4 δ := (−1)n(p+1)+1 ⋆ d⋆. Since dξ is a 2-form, we
get ⋆dQ(ξ) = (−1)n (16π)−1 δ(dξ). From the expression of the codifferential in a coordinate chart
(xα ) (cf. e.g. Eq. (14.37) in Ref. [464]), we may write, for any 2-form A:
1 ∂ √
(δA)α = √ −gAµα = −∇µ Aαµ ,
−g ∂xµ
where the last equality follows from the identity (A.75). Applying the above formula to A := dξ,
we get δ(dξ)α = −∇µ (∇α ξ µ − ∇µ ξ α ) = J (ξ)α , i.e. δ(dξ) = J (ξ), where J (ξ) is the Komar
current of ξ defined by Eq. (5.50). Hence ⋆dQ(ξ) = (−1)n (16π)−1 J (ξ). On the other hand, we have
⋆J (ξ) = (16π)−1 ⋆ ⋆J (ξ) from Eq. (17.20). Since ⋆⋆ = (−1)n Id for a 1-form (cf. e.g. formula (14.26) in
Ref. [464]), we immediately get ⋆J (ξ) = ⋆dQ(ξ), which is equivalent to Eq. (17.21).
Property 17.7: Noether charge of a spacelike hypersurface
Let us consider a diffeomorphism-invariant theory described by a Lagrangian n-form L.
Given a vector field ξ on M and a spacelike hypersurface Σ, the Noether charge of Σ
with respect to ξ is Z
QΣ,ξ := J (ξ), (17.25)
Σ
where J (ξ) is the Noether current (n − 1)-form of ξ (Property 17.5) and the integral is
taken with respect to Σ endowed with the future orientation, i.e. the orientation defined
by n · ϵ, where n is the future-directed timelike unit normal to Σ. The Noether charge is
a priori not unique, since any change J → J ′ allowed by (17.18) leads to QΣ,ξ → Q ′ Σ,ξ
with Z
Q Σ,ξ = QΣ,ξ +
′
[Y (g, Lξ g) + ξ · µ(g)] . (17.26)
∂Σ
However, if ξ is a Killing vector and (i) ξ is tangent to ∂Σ or (ii) ∂Σ is located outside the
support of µ(g), the Noether charge is uniquely defined: Q ′ Σ,ξ = QΣ,ξ .
Let Σ1 and Σ2 be two spacelike hypersurfaces, such that Σ2 lies in the future of Σ1
and both hypersurfaces are connected by a hypersurface W , i.e. Σ1 ∪ Σ2 ∪ W is a closed
hypersurface consituting the boundary of a n-dimensional manifold D ⊂ M (cf. Fig. ??).
Then the following conservation law holds:
Z
QΣ2 ,ξ = QΣ1 ,ξ + J (ξ) , (17.27)
W↙
where the integral over W is taken with respect to the inward orientation (indicated by the
arrow), i.e. the orientation defined by v · ϵ, where v is any vector field pointing into D.
Proof. Equation (17.26) readily follows from (17.18) and Stokes’ theorem (A.94). If ξ is a Killing
vector, then Lξ g = 0 by definition (cf. Property 3.6); this implies Y (g, Lξ g) = 0 in the
4
See e.g. Eq. (14.35) of Straumann’s book [464], whose sign convention is followed here. Since the same letter
δ is used to denote the codifferential and the variation operator, we are using a boldface symbol to distinguish the
former.
664 Black hole thermodynamics beyond general relativity
right-hand side of Eq. (17.26), given that Y is linear with respect to Lξ g. Moreover, if ξ is
tangent to ∂Σ (case (i)), the (n − 2)-form ξ · µ = µ(ξ, . . .) restricted to vectors tangent to ∂Σ
is necessarily zero, for µ is a (n − 1)-form. Obviously, this holds as well in the case (ii). Hence
Q ′ Σ,ξ = QΣ,ξ . As for Eq. (17.27), given that ∂D = Σ1 ∪ Σ2 ∪ W , Stokes’ theorem (A.94) and
dJ (ξ) = 0 [Eq. (17.17)] lead to
Z Z Z
J (ξ) + J (ξ) + J (ξ) = 0,
Σ↗
1 Σ↗
2 W↗
where the arrows mean that Σ1 , Σ2 Rand W are oriented according to the outwardRconvention
(with respect to D). Noticing that Σ↗ J (ξ) = −QΣ1 ,ξ , Σ↗ J (ξ) = QΣ2 ,ξ and W ↗ J (ξ) =
R
1 2
− W ↙ J (ξ) yields (17.27).
R
The integral over W in Eq. (17.27) can be interpreted as an “inward flux”, especially if W is
a timelike hypersurface (cf. the electromagnetic analogy detailed in Remark 7 below). If this
inward flux vanishes, Eq. (17.27) expresses the invariance of the Noether charge with respect to
the choice of the hypersurface (conservation during “time evolution”). This occurs if J (ξ) = 0
on W or if W is at some asymptotically flat end of M with J (ξ) decaying sufficiently fast so
that its integral on W is zero.
Property 17.8: Noether charge of a closed spacelike (n − 2)-surface
Let us consider a diffeomorphism-invariant theory described by a Lagrangian n-form L
and let assume that (M , g) is asymptotically flat. Given a vector field ξ on M and a closed
spacelike (n − 2)-surface S ⊂ M , the Noether charge of S with respect to ξ is
Z
QS ,ξ := Q(ξ), (17.28)
S
where Q(ξ) is the Noether potential (n − 2)-form of ξ (Property 17.6) and the orientation
of S for the above integral is the future-outward orientation defined by Eq. (5.39). The
Noether charge is a priori not unique, since any change Q → Q′ obeying (17.22) leads to
QS ,ξ → Q ′ S ,ξ with
Z
Q S ,ξ = QS ,ξ +
′
[Y (g, Lξ g) + ξ · µ(g)] . (17.29)
S
However, if ξ is a Killing vector and (i) ξ is tangent to S or (ii) S is located outside the
support of µ(g), the Noether charge is uniquely defined: Q ′ S ,ξ = QS ,ξ .
Let Σ be a compact spacelike hypersurface bounded by two closed (n − 2)-surfaces: an
“internal” one, Sint , and an “external” one, Sext , the exterior direction being that of the
asymptotic flat end of (M , g). Then
QSext ,ξ = QSint ,ξ + QΣ,ξ . (17.30)
17.2 Diffeomorphism-invariant theories of gravity 665
Proof. Equation (17.29) readily follows from (17.22) along with S dZ = 0 as a result of Stokes’
R
theorem (A.94), given that S is a closed manifold: ∂S = ∅. The proof that Q ′ S ,ξ = QS ,ξ
in the considered cases is identical to the proof of Q ′ Σ,ξ = QΣ,ξ in Property 17.7. Finally, to
prove (17.30), let us substitute dQ for J [cf. Eq. (17.21)] in expression (17.25) of QΣ,ξ . Applying
Stokes’ theorem (A.94) and using ∂Σ = Sint ∪ Sext , we get
Z Z
QΣ,ξ = Q(ξ) + Q(ξ),
↗ ↗
Sint Sext
where the arrows indicate that Sint and R Sext are outward-oriented
R with respect to Σ (which
is itself future-oriented). Noticing that S ↗ Q(ξ) = −QSint ,ξ and S ↗ Q(ξ) = QSext ,ξ yields
int ext
(17.30).
Remark 6: We are using the same letter Q to denote the Noether charge of a hypersurface (Property 17.7)
and the Noether charge of a (n − 2)-surface (Property 17.8). The distinction between the two quantities
occurs only through the subscript: QΣ,ξ versus QS ,ξ . This common notation is justified in so far as
both quantities are strongly related by Eq. (17.30). In particular, if the hypersurface Σ has a single
(external) boundary S , i.e. Sint = ∅, Eq. (17.30) reduces to QS ,ξ = QΣ,ξ .
Remark 7: The name charge in the expression Noether charge comes of course from the analogy with
electromagnetism: considering n = 4 for simplicity, the analog of the Noether current 3-form J is
the Hodge dual of the 1-form j, metric dual of the electric 4-current j: J = ⋆j, while the analog of
the Noether potential 2-form Q is the Hodge dual of the electromagnetic field tensor F divided by
µ0 : Q = µ−10 ⋆F . The relation dQ = J [Eq. (17.21)] is then nothing but Maxwell equation (1.50):
d ⋆F = µ0 ⋆j. Then QΣ,ξ is the total electric charge within the volume Σ (cf. e.g. Eq. (18.2) in Ref. [228])
and QS ,ξ is µ−10 times the flux of the electromagnetic field through the surface S . Accordingly,
Eq. (17.27) expresses the conservation of electric charge and Eq. (17.30) with Sint = ∅ is nothing but
Gauss’s law (cf. e.g. Eq. (18.40) in Ref. [228]).
From Example 5, we note that the (n − 2)-form ⋆(dξ) giving Q(ξ) for general relativity
[cf. Eq. (17.23)] is exactly the differential form that constitutes the integrand in formula (5.36)
defining the Komar mass or in formula (5.63) defining the Komar angular momentum (ξ := η
in that case). In view of the definition (17.28) of the Noether charge, we conclude that
Property 17.9: Komar mass and angular momentum as Noether charges
In a stationary spacetime governed by general relativity, the Komar mass MS over a
(n − 2)-dimensional closed spacelike surface S [Eq. (5.36)] is (n − 2)/(n − 3) times the
Noether chargea of S with respect to the stationary Killing vector ξ:
n−2
MS = QS ,ξ . (17.31)
n−3
Similarly, in an axisymmetric spacetime governed by general relativity, the Komar angular
momentum JS over S [Eq. (5.63)] is minus the Noether charge of S with respect to the
axisymmetric Killing vector η:
JS = −QS ,η . (17.32)
666 Black hole thermodynamics beyond general relativity
a
Since ξ is a Killing vector, the Noether charge is uniquely defined in the present case, provided S is
chosen outside the support of µ if one allows for a Lagrangian change of the type (17.7), cf. Property 17.8.
17.3 Wald entropy
17.3.1 Wald entropy as a Noether charge
Having introduced the concept of Noether charge related to diffeomorphism invariance, we
are in position to define the Wald entropy of a stationary black hole.
Property 17.10: Wald entropy as a Noether charge
Let (M , g) be a stationary spacetime of dimension n ≥ 3 governed by a diffeomorphism-
invariant theory and containing a black hole, whose event horizon H is part of a bifurcate
Killing horizon (cf. Sec. 3.4). Let χ be the Killing vector generating H and κ the corre-
H
sponding surface gravity: ∇χ χ = κχ [Eq. (3.29)]. By virtue of the zeroth law for bifurcate
Killing horizons (Property 3.16), κ is constant over H and κ ̸= 0. Let S be a complete
cross-section of H . The Wald entropy of the black hole is then defined as κ−1 times the
Noether charge of S with respect to χ, up to the constant 2πkB /ℏ:
Z
2πkB 2πkB
SW := QS ,χ = Q(χ) . (17.33)
ℏκ ℏκ S
Here Q(χ) is the Noether potential (n − 2)-form of χ, as defined in Property 17.6. SW is
independent of the choice of the cross-section S of H . Moreover, the Noether charge
QS ,χ does not suffer from the ambiguity (17.29), so that SW is uniquely defined.
Proof. Let S ′ be another complete cross-section of H lying entirely in the future of S . Then
S ∪ S ′ constitutes the boundary of a portion of H , which we shall denote by ∆H . Stokes’
theorem (A.94) and the identity dQ(χ) = J (χ) [Eq. (17.21)] give
Z Z Z
Q(χ) + Q(χ) = J (χ), (17.34)
S↗ S ′↗ ∆H
where the arrows indicate that S and S ′ are outward-oriented with respect to ∆H , the
orientation of the latter being set by a future-directed null vector field k transverse to H .
From the definition (17.16), J (χ) := θ(g, Lχ g) − χ · L = −χ · L, since Lχ g = 0 (χ
Killing) along with θ being linear in Lχ g imply θ(g, Lχ g) = 0. Now, when restricted to
∆H (i.e. applied to (n − 1)-tuples of vectors tangent to ∆H , or equivalently, considered as
the pullback ι∗ (χ · L) of χ · L by the inclusion map ι : ∆H → M , cf. Sec. A.2.8), χ · L is
a vanishing (n − 1)-form, since χ · L := L(χ, . . .), χ is tangent to ∆H and L is a n-form.
Accordingly, the integral of J (χ)R = −χ · L in the right-hand side of Eq. (17.34) is zero. Given
that S ↗ Q(χ) = −QS ,χ and S ′ ↗ Q(χ) = QS ′ ,χ , it follows that QS ′ ,χ = QS ,χ , which
R
17.3 Wald entropy 667
shows that SW does not depend on the choice of S . Regarding the ambiguity (17.29), the
term Y (g, Lχ g) does not contribute since Y (g, Lχ g) is linear in Lχ g (Propery 17.3), so that
Lχ g = 0 (χ Killing) implies Y (g, Lχ g) = 0. The term χ · µ(g) does not contribute either
because, taking advantage of the independence of QS ,χ from S , we are free to choose S as
the bifurcation surface, where χ = 0 (Property 3.12).
If the diffeomorphism-invariant theory is general relativity, the Wald entropy reduces to
the Bekenstein-Hawking entropy defined in Sec. 16.6.1:
Property 17.11: Wald entropy for general relativity
For general relativity, the Wald entropy of a stationary black hole coincides with its
Bekenstein-Hawking entropy, as given by formula (16.73):
A
SW = SBH = kB (general relativity), (17.35)
4ℏ
A being the black hole area.
Proof. For general relativity, the Noether potential (n − 2)-form is given by Eq. (17.23): Q(χ) =
−(16π)−1 ⋆ (dχ). We have then
Z Z Z
1 µν
16πQS ,χ = − ⋆(dχ) = − (dχ)µν dS = − ∇µ χν dS µν ,
S 2 S S
where the second equality follows from Lemma 5.10 and the third one from expression (A.91b)
of the exterior derivative and the antisymmetric character of dS µν . Now, from Eq. (5.87),
√
∇µ χν dS µν = −2κ dS, where dS := q dn−2 x is the area element of S . Since κ is constant,
there comes QS ,χ = κA/(8π). In view of the definition (17.33) of SW , we get (17.35).
17.3.2 Geometric expression of the Wald entropy
The Bekenstein-Hawking entropy does not involve directly the horizon Killing vector χ, but
a geometrical quantity related to a cross-section S of the horizon, namely the area A of S ,
which is independent of S for a Killing horizon. Similarly, one can recast the Wald entropy in
a form that gets rid of χ and κ, refering only to geometrical quantities related to the bifurcation
surface (and to the Lagrangian scalar):
Property 17.12: geometric expression of the Wald entropy (Iyer & Wald 1994 [292])
Let us consider a stationary black hole in a diffeomorphism-invariant theory, whose
Lagrangian scalar L is recast as (17.3). Assuming that the black hole event horizon is part
of a bifurcate Killing horizon, with bifurcation surface Sˆ, the Wald entropy is expressible
as Z
kB δL Sˆ ⊥
SW = −2π ϵ dSµν , (17.36)
ℏ Sˆ δRµνρσ ρσ
668 Black hole thermodynamics beyond general relativity
where
• δL/δRαβγδ stands for the type-(4, 0) tensor field that is the Euler-Lagrange deriva-
tive of L with respect to Rαβγδ , if the latter were considered as a field independent
of gαβ in expression (17.3) of L:
δL ∂L ∂L ∂L
:= − ∇µ + ∇(µ ∇ν) − ··· (17.37)
δRαβγδ ∂Rαβγδ ∂∇µ Rαβγδ ∂∇(µ ∇ν) Rαβγδ
• dSαβ is the metric dual of the area element normal bivector dS αβ to Sˆ, as defined
by Eq. (5.42);
ˆ ˆ
• Sϵ⊥ is the binormal 2-form to Sˆ, i.e. for each point p ∈ Sˆ, Sϵ⊥ |p is the volume
2-form of the metric induced by g in the vector space Tp⊥ Sˆ normal to Sˆ equipped
ˆ
with the future-outward orientationa ; Sϵ⊥ can be expressed as the exterior product
s ∧ n of any pair of vector fields (n, s) along Sˆ forming an orthonormal basis of
Tp⊥ Sˆ at each p ∈ Sˆ, with n future-directed unit timelike and s outward-directed
unit spacelike:
Sˆ ⊥ Sˆ ⊥
ϵ =s∧n ⇐⇒ ϵαβ = s α nβ − nα s β . (17.38)
Note that Eqs. (5.42) and (17.38) yield the following relation between the area element
ˆ
normal bivector dS and the binormal 2-form Sϵ⊥ :
ˆ
dSαβ = Sϵ⊥
αβ dS, (17.39)
√
where dS = q dn−2 x is the area element of Sˆ.
a
See item (iv) on p. 136 for details.
Partial proof. Q(χ) has a linear dependence on χ (cf. Property 17.6). It may contain high order
covariant derivatives of χ but since the latter is a Killing vector, one may use iteratively Kostant
formula (3.83) to express all derivatives of order higher than one in terms of χ and ∇χ, so
that Q(χ) is expressible as
Q(χ)α1 ···αn−2 = χµ Wµα1 ···αn−2 + ∇µ χν X µνα1 ···αn−2 ,
where W = W (g) and X = X(g) are tensor fields of type (0, n−1) and (2, n−2) respectively,
which are antisymmetric with respect to their last n − 2 arguments. For instance, in the case of
general relativity, we read on (17.24) that W = 0 and X µνα1 ···αn−2 = −(16π)−1 ϵµνα1 ···αn−2 . Let
us evaluate the Wald entropy SW by selecting the bifurcation surface Sˆ for S in formula (17.33).
Given that χ = 0 on Sˆ (Property 3.12), the term in W does not contribute to SW . Furthermore,
the following identity holds on Sˆ:
Sˆ ˆ
∇α χβ = κ Sϵ⊥
αβ . (17.40)
17.3 Wald entropy 669
Indeed, since χ = 0 on Sˆ, we have, for any vector v tangent to Sˆ, v µ ∇µ χα = 0. This
shows that, at each point p ∈ Sˆ, the 2-form ∇α χβ = ∇[α χβ] leaves in the vector space Tp⊥ Sˆ
ˆ
normal to Sˆ, as the 2-form Sϵ⊥ ⊥ ˆ
αβ . Since Tp S is 2-dimensional, these 2-forms are necessarily
Sˆ ˆ
proportional: ∇α χβ = λ Sϵ⊥αβ for some scalar field λ on S . By combining Eqs. (5.86) and (17.39),
Sˆ ⊥
we get ϵαβ = χα kβ − kα χβ , where k is the future-directed null vector of Tp⊥ Sˆ such that
k · χ = −1. Then, Eq. (3.29) yields κχα = χµ ∇µ χα = λ(χµ χµ kα − χµ kµ χα ) = λ(0 + χα ), from
Sˆ ˆ
which λ = κ, thereby establishing (17.40). We have then Q(χ)α1 ···αn−2 = κ Sϵ⊥
ρσ X α1 ···αn−2 .
ρσ
Then Eq. (17.33) yields Z
kB Sˆ ⊥
SW = 2π ϵρσ X ρσα1 ···αn−2 . (17.41)
ℏ Sˆ
ˆ
This expression of SW does no longer involve χ, nor κ, but only the geometrical quantity Sϵ⊥
defined on Sˆ and X, which depends on the Lagrangian n-form L. To go further, we must
invoke a result of Wald and Iyer (cf. Eqs (31) and (52) of Ref. [292]), namely
δL
X ρσα1 ···αn−2 = − ϵµνα1 ···αn−2 . (17.42)
δRµνρσ
Accordingly, given the definition (5.38) of the Hodge dual, Eq. (17.41) becomes
Z
kB δL Sˆ ⊥
SW = −4π ⋆A, with Aαβ := gαµ gβν ϵ .
ℏ Sˆ δRµνρσ ρσ
Lemma 5.10 leads then to (17.36).
Let us stress that the integral in expression (17.36) of the Wald entropy has to be taken on
the bifurcation surface Sˆ, while in the Noether charge expression (17.33), the integral can be
taken on any complete cross-section S of the event horizon.
Example 6 (Wald entropy from formula (17.36) for general relativity): For general relativity,
L = (16π)−1 g µν g ρσ Rρµσν [Eq. (17.4)], so that Eq. (17.37) results in
δL ∂L 1 αγ βδ
= = g g .
δRαβγδ ∂Rαβγδ 16π
Accordingly, formula (17.36), combined with Eq. (17.39), yields
Z Z Z
ℏ 1 ˆ Sˆ ⊥ 1 Sˆ ⊥ µν Sˆ ⊥ 1 A
SW = − g µρ g νσ Sϵ⊥ρσ ϵ µν dS = − ( ϵ ) ϵ µν dS = dS = ,
kB 8 Sˆ 8 Sˆ | {z } 4 Sˆ 4
−2
ˆ ˆ
where A is the area of Sˆ and (Sϵ⊥ )µν Sϵ⊥µν = −2 follows from Eq. (17.38) along with nµ n = −1,
µ
nµ s = 0 and sµ s = 1. Hence we recover formula (17.35) — the Bekenstein-Hawking entropy.
µ µ
Of course, being defined by a formula that reduces to the Bekenstein-Hawking entropy for
general relativity (Property 17.11) is not sufficient by itself to assert that the Wald entropy is the
correct concept of black hole entropy for all theories of gravity. This holds because the Wald
entropy naturally enters in the generalization of the first law to all diffeomorphism-invariant
theories that admit a Hamiltonian formulation, which we will discuss in Sec. 17.5, after having
introduced the covariant phase space formalism.
670 Black hole thermodynamics beyond general relativity
17.4 Covariant phase space formalism
The extension of the first law beyond general relativity is naturally achieved within a Hamil-
tonian framework. Indeed, such a framework provides the concept of total mass M — the
variation of which constitutes the left-hand side of the first law — as the canonical energy of the
system (value of the Hamiltonian), thereby generalizing the concept of Kormar or ADM masses
beyond general relativity. The relevant Hamiltonian framework is the co-called covariant phase
space approach, which is adapted to field theories, preserving general covariance. On the
contrary, the standard approach relies on the choice of canonical coordinates and of a time
function t, with respect to which the evolution is governed by Hamilton’s equations.
17.4.1 Hamiltonian formalism without canonical coordinates
In standard mechanics, a system with N degrees of freedom is described by some configuration
coordinates (q i )1≤i≤N and a Lagrangian L = L(q i , q̇ i ) (with q̇ i := dq i /dt). The system’s
Hamiltonian is then H(q i , pi ) = i=1 pi q̇ i − L, where pi ’s are the conjugate momenta: pi :=
N
P
∂L/∂ q̇ i , and the system’s motion is governed by Hamilton’s equations: q̇ i = ∂H/∂pi and
ṗi = −∂H/∂q i , or equivalently,
dq i dpi
= {q i , H} and = {pi , H}, 1 ≤ i ≤ N, (17.43)
dt dt
where {., .} stands for the Poisson bracket:
N
X ∂f ∂g ∂g ∂f
{f, g} := i ∂p
− i . (17.44)
i=1
∂q i ∂q ∂p i
More generally, for any scalar function f = f (q i , pi ), one has
df
= {f, H}. (17.45)
dt
The coordinates (y a ) := (q 1 , . . . , q N , p1 , . . . , pN ) span a smooth manifold F of dimension 2N
called the system’s phase space. The Poisson bracket can be rewritten covariantly, i.e. in a
form independent of the coordinates (y a ), by introducing the following 2-form on F:
N
X
Ω := dpi ∧ dq i , (17.46)
i=1
where d stands for the exterior derivative5 on F, so that dpi (resp. dq i ) is the differential
1-form of pi (resp. q i ) considered as a scalar field on F. Given that q i = y i and pi = y i+N , the
5
On phase space manifolds, we are not using boldface notations for operators and tensor fields, keeping them
for the spacetime manifold only. Hence the exterior derivative on F is denoted by d, and not d, as defined in
Sec. A.4.3, the 2-form (17.46) is denoted Ω and not Ω, etc.
17.4 Covariant phase space formalism 671
components of Ω with respect to the coordinates (y a ) are Ωab = N δb − δai δbi+N =
P i+N i
i=1 δa
δa,b+N − δb,a+N . Hence, we may write, using block matrix notation,
0 −IN
Ωab = , (17.47)
IN 0
where a and b are respectively the row and column indices and IN stands for the identity
matrix of size N . Any coordinate system on F in which the 2-form Ω is expressible as (17.46),
or equivalenty, in which the components of Ω are the matrix (17.47), is called a system of
canonical coordinates. The matrix (Ωab ) is clearly invertible, its inverse being its opposite:
0 IN
(Ω−1 )ab = . (17.48)
−IN 0
This implies that the 2-form Ω is non-degenerate and allows one to define an antisymmetric
type-(2, 0) tensor field on F, called the Poisson bivector, by
Ω−1 := (Ω−1 )ab ∂a ⊗ ∂b . (17.49)
As a type-(2, 0) tensor, Ω−1 acts bilinearly on 1-forms (cf. Sec. A.2.5) and one may define the
Poisson bracket of two scalar fields f and g on F by letting Ω−1 act on the pair of 1-forms
(df, dg):
{f, g} := Ω−1 (df, dg) . (17.50)
Given the components (17.48) of Ω−1 in canonical coordinates, it is easy to check that this
definition reduces to (17.44) for these type of coordinates.
Because dd = 0 (nilpotence of the exterior derivative, cf. Eq. (A.92)), it follows immediately
from Eq. (17.46) that the 2-form Ω is closed: dΩ = 0. This is also obvious from the coordinate
expression (A.90c) of the exterior derivative, given that the components Ωab are constant in
canonical coordinates [Eq. (17.47)]. Generically, a closed non-degenerate 2-form Ω on a smooth
manifold F is called a symplectic form and the pair (F, Ω) is called a symplectic manifold
(see e.g. Chap. 22 of Lee’s textbook [343]). Not only Ω is closed, but it is exact. Indeed, it follows
from expression (17.46) that
Ω = dΘ, (17.51)
where Θ is the 1-form defined by
N
X
Θ := pi dq i . (17.52)
i=1
Because of (17.51), the 1-form Θ is called the symplectic potential. It is also called the
canonical 1-form on F, because its expression (17.52) is independent of the choice of the
canonical coordinates (q i , pi ).
To any scalar field f on F, one associates a unique vector field Xf on F via the Poisson
bivector Ω−1 :
Xf := Ω−1 (., df ) . (17.53)
672 Black hole thermodynamics beyond general relativity
Specifically, for any 1-form σ on F, one has
⟨σ, Xf ⟩ = Xf (σ) = Ω−1 (σ, df ). (17.54)
It follows that the components of Xf with respect to any coordinate system (y a ) are
ab ∂f
Xfa = Ω−1 . (17.55)
∂y b
In particular, for canonical coordinates (y a ) = (q i , pi ), Eq. (17.48) leads to
∂f ∂f
Xf = ∂qi − i ∂pi . (17.56)
∂pi ∂q
As a vector field, Xf is a derivative operator on scalar fields [cf. Eq. (A.10)], which we may
express in terms of the Poisson bracket by combining Eqs. (17.53) and (17.50):
Xf = {., f } . (17.57)
The above identity means that for any scalar field g, Xf (g) = {g, f }, with Xf (g) = Xfa ∂a g.
Formula (17.53) can be inverted to
df = Ω(., Xf ) . (17.58)
Proof. Ω(., Xf ) is a 1-form whose components are Ωab Xfb . Thanks to Eq. (17.55), we get
Ωab Xfb = Ωab (Ω−1 )bc ∂c f = δ ca ∂c f = ∂a f , which gives (17.58).
Remark 1: The vector field Xf and the 1-form df can be considered as being dual to each other
through the symplectic form Ω in the very same way as vector fields and 1-forms are dual through
the metric tensor g on pseudo-Riemannian manifolds: compare formula (17.58) with (A.44) (rewritten
as u = g(., u)) and formula (17.55) with (A.47). These dualities occur because both Ω and g are
non-degenerate bilinear forms.
The Poisson bracket (17.50) can be expressed in terms of the symplectic form Ω acting of
the pair of vector fields (Xf , Xg ):
{f, g} = −Ω(Xf , Xg ) . (17.59)
Proof. Starting from Eq. (17.50) and using Eq. (17.58), we have {f, g} = (Ω−1 )ab ∂a f ∂b g =
(Ω−1 )ab Ωac Xfc Ωbd Xgd = δ ad Ωac Xfc Xgd = Ωdc Xfc Xgd = Ω(Xg , Xf ) = −Ω(Xf , Xg ).
Remark 2: Notice the minus sign in Eq. (17.59), as compared to Eq. (17.50); of course, since Ω is
antisymmetric, Eq. (17.59) can be recast as {f, g} = Ω(Xg , Xf ).
By combining Eqs. (17.58) and (17.59) there comes ⟨df, Xg ⟩ = Ω(Xg , Xf ) = {f, g}. Hence,
we get various equivalent expressions for the Poisson bracket:
{f, g} := Ω−1 (df, dg) = Ω(Xg , Xf ) = ⟨df, Xg ⟩ = Xg (f ). (17.60)
17.4 Covariant phase space formalism 673
2.0
1.5
1.0
b a
0.5
0.0
pθ
0.5
1.0
1.5
2.0
3 2 1 0 1 2 3
θ
Figure 17.1: Phase space F of a simple gravity pendulum, depicted in terms of the canonical coordinates (θ, pθ )
(cf. Example 7). The vertical lines θ = −π and θ = π must be identified, since F = S1 × R. The grey arrows show
the Hamiltonian vector field XH and the grey lines are some of its integral curves. Two of them are highlighted:
the red one corresponds to oscillating solutions (θ having a limited range and pθ taking positive and negative
values), while the blue one corresponds to solutions rotating indefinitely in the anticlockwise direction (pθ > 0).
The points a and b each represent the initial data of a given solution in these families; these points can also be
viewed as defining the solution itself (covariant phase space approach).
The Hamiltonian vector field is the vector field on F associated to the Hamiltonian
differential dH by symplectic duality, i.e. the vector field XH . It obeys Eq. (17.58):
dH = Ω(., XH ) . (17.61)
Remark 3: In the mathematical literature (e.g. [343]), the vector field Xf associated to any scalar
function f on F is called the Hamiltonian vector field of f . This reflects the fact that any scalar function
f can the considered as the Hamiltonian of some theory, even if the latter is unphysical.
Example 7 (phase space of a pendulum): A simple gravity pendulum is a system with N = 1, the
unique degree of freedom being the angle θ ∈ (−π, π] with respect to the vertical direction. For a unit
mass, the Lagrangian is L(θ, θ̇) = (ℓθ̇)2 /2 + gℓ cos θ, where ℓ is the length of the (massless) rod and
g the magnitude of the vertical gravity field. The momentum conjugate to θ is pθ = ∂L/∂ θ̇ = ℓ2 θ̇.
Since the latter can take any real value, the phase space F is the 2-dimensional infinite cylinder S1 × R,
spanned by the canonical coordinates (θ, pθ ). The Hamiltonian is deduced from L by the Legendre
transform H = pθ θ̇ − L, yielding H(θ, pθ ) = p2θ /(2ℓ2 ) − gℓ cos θ. The Hamiltonian vector field is
obtained via Eq. (17.56): XH = ℓ−2 pθ ∂θ − gℓ sin θ ∂pθ . It is depicted in Fig. 17.1.
674 Black hole thermodynamics beyond general relativity
Equation (17.57) with f = H reads {., H} = XH . Hence, for any scalar field f on F,
{f, H} = XH (f ) and the evolution law (17.45) can be rewritten as
df
= XH (f ) . (17.62)
dt
In particular, Hamilton’s equations (17.43) become
dq i dpi
= XH (q i ) and = XH (pi ), 1 ≤ i ≤ N. (17.63)
dt dt
It follows that any solution (q i (t), pi (t)) to the equations of motion is an integral curve of the
Hamiltonian vector field in the phase space (cf. Fig. 7). H is constant along such a curve, since,
from Eq. (17.61), XH (H) = ⟨dH, XH ⟩ = Ω(XH , XH ) = 0.
In view of Eq. (17.62), one may consider the vector field XH as defining the “time evolution”
without having to refer explicitely to the time coordinate t. Of course, we recover from
Eq. (17.62) that dH/dt = 0 since XH (H) = 0. For tensor fields on F of valence higher than a
scalar field, one defines the “time evolution” as the Lie derivative LXH along the Hamiltonian
vector field XH . This is quite natural since the vanishing of LXH T means that the tensor field
T is transported identically to itself along the flow defined by XH (cf. Sec. A.4.2). In particular,
this encompasses Eq. (17.62), since LXH f = XH (f ) for any scalar field f . An important feature
of the evolution thus defined is that the symplectic form is conserved:
LXH Ω = 0 . (17.64)
Proof. The proof relies on the fact that Ω is closed, i.e. dΩ = 0. Indeed, using successively the
Cartan identity (A.95), Eq. (17.61) and the nilpotence property (A.92), we get
LXH Ω = XH · |{z}
dΩ +d(XH · Ω) = −d(Ω(., XH )) = −ddH = 0.
| {z } | {z }
0 Ω(XH ,.) dH
Let us summarize the above results as follows:
Property 17.13: Hamiltonian system
A Hamiltonian system is a triplet (F, Ω, H) where F is a smooth manifold, called the
phase space of the system, Ω is a symplectic form on F, i.e. a closed non-degenerate
2-form, and H is a smooth function F → R called the Hamiltonian. The differential
dH of H is a 1-form on F, whose symplectic dual defines the Hamiltonian vector field,
namely the unique vector field XH on F such that
dH = Ω(., XH ) . (17.65)
By construction, XH preserves the symplectic form: LXH Ω = 0 [Eq. (17.64)]. The evolution
of the underlying physical system is entirely determined by the Hamiltonian vector field:
any solution to the equations of motion is an integral curve of XH . A smooth function
17.4 Covariant phase space formalism 675
f : F → R is called a conserved quantity iff f is constant on every integral curve of XH ,
i.e. iff XH (f ) = 0. In view of Eq. (17.60), this is equivalent to {f, H} = 0. A conserved
quantity is H itself. Its value for a given solution to the equations of motion is called
the energy of the solution. An equilibrium point is a point of F where XH = 0, or
equivalently, given Eq. (17.65), a point where dH = 0.
Example 8 (solutions for the gravity pendulum): For the gravity pendulum considered in Example 7,
the integral curves of XH are depicted in Fig. 17.1. Two of them are represented by the red and the blue
curves. It is clear from the figure that XH vanishes at two points: (θ, pθ ) = (0, 0) and (θ, pθ ) = (π, 0),
which are thus two equilibrium points (respectively stable and unstable).
Remark 4: The above formulation of Hamiltonian systems does not refer to any system of canonical
coordinates (q i , pi ) on the phase space F and, more generally, to any kind of coordinates. It does not
even assume that F is the cotangent bundle T ∗ C of some “configuration space” C, while the cotangent
bundle structure is the standard arena for the Hamiltonian mechanics with a finite number of degrees of
freedom [1].
17.4.2 Covariant phase space
Despite Remark 4, the Hamiltonian formulation sketched in Property 17.13 is not fully covariant.
Indeed, via the Hamiltonian vector field XH , it describes the evolution of the system with
respect to some “privileged time” t [cf. Eq. (17.62)]. To treat covariantly relativistic systems, it
would be desirable to get rid of the choice of a time function. This can be acheived as follows.
There is one, and exactly one, integral curve of XH through each point6 of F. It follows
that a solution is entirely determined by a single point a ∈ F, which can be thought of as
representing the state of the system at some initial instant t = t0 , i.e. the coordinates of a are
(q i (t0 ), pi (t0 )) (cf. Fig. 17.1). If the initial value problem is well posed, there is a one-to-one
correspondance between a point of F and a solution. Consequently, one may view F as the
space of solutions to the equations of motion. This is the essence of the so-called covariant phase
space approach:
Property 17.14: covariant phase space
Let (F, Ω, H) be a Hamiltonian system whose equations of motion constitute a well posed
initial value problem. Each solution to the equations of motion is a curve S : R → F,
t 7→ S(t), which admits the Hamiltonian vector field XH as tangent vector. The set Fsol of
all solutions can be given the structure of a smooth manifold diffeomorphic to F, and for
each t0 ∈ R, an explicit diffeomorphism is the “initial data” map
Φt0 : Fsol −→ F
(17.66)
S 7−→ S(t0 ).
6
At an equilibrium point, the integral curve is reduced to the point itself.
676 Black hole thermodynamics beyond general relativity
Moreover, the pullback of the symplectic form Ω by Φt0 ,
Ωsol := Φ∗t0 Ω, (17.67)
defines a symplectic form on Fsol , which is actually independent of t0 . This makes
(Fsol , Ωsol ) a well-defined symplectic manifold, called the covariant phase space of the
system. Infinitesimal displacement vectors on the manifold Fsol correspond to infinitesimal
variations δS between solutions, so that Eq. (17.67) is equivalent to
−−−−−−−→ −−−−−−−→
Ωsol (δ1 S, δ2 S) := Ω S(t0 )S1 (t0 ), S(t0 )S2 (t0 ) , (17.68)
where S, S1 = S + δ1 S and S2 = S + δ2 S stand for three nearby solutions.
Similarly, the pullback of the Hamiltonian H by Φt0 , i.e Hsol := H ◦ Φt0 , or equivalently,
Hsol (S) := H(S(t0 )), (17.69)
defines a scalar field Hsol on Fsol , which is independent of the choice of t0 .
Proof. The equivalence between Eqs. (17.67) and (17.68) follows readily from the definition
of the pullback map given in Sec. A.2.8. The independence of Ωsol (δ1 S, δ2 S) from the choice
of t0 follows from the constancy of Ω along the Hamiltonian flow, i.e. Eq. (17.64). Indeed
Ωsol (δ1 S, δ2 S), as defined by Eq. (17.68), is independent of t0 iff df /dt = 0, where f (t) :=
−−−−−−→
Ω(Y1 (t), Y2 (t)) and we have introduced the separation vectors Y1 (t) := S(t)S1 (t) and Y2 (t) :=
−−−−−−→
S(t)S2 (t). Now, from Eq. (17.62) and the Leibniz rule,
df
= LXH f = LXH [Ω(Y1 , Y2 )] = LXH Ω (Y1 , Y2 ) + Ω(LXH Y1 , Y2 ) + Ω(Y1 , LXH Y2 ) = 0,
dt | {z } | {z } | {z }
0 0 0
where LXH Ω = 0 is the invariance property (17.64) and LXH Y1 = 0 and LXH Y2 = 0 follows
from the very definition of a Lie derivative (cf. Sec. A.4.2), the separation vectors Y1 and Y2
being naturally Lie-dragged by XH . Finally, the independence of Hsol from the choice of t0
is immediate from Eq. (17.69), which shows that Hsol (S) is the energy of the solution S (cf.
Property 17.13).
The triplet (Fsol , Ωsol , Hsol ) clearly constitutes a Hamiltonian system, as defined in Prop-
erty 17.13, but without reference to any “time evolution”, given that each point S ∈ Fsol is a
fully evolved solution by itself. The fully covariant picture is thus achieved. The Hamiltonian
vector field XHsol is a vector field on Fsol , which generates 1-dimensional families of solutions.
Two solutions S1 and S2 that lie on the same integral curve of XHsol are then solutions to the
original equations of motion that only differ by a time translation, i.e. S2 (t) = S1 (t + t0 ) for
some t0 ∈ R. In particular S1 and S2 have the same energy.
17.4.3 Covariant phase space for gravity theories
In order to apply the covariant phase space approach to a gravity theory governed by a
Lagrangian of the type (17.3), let us consider the space Fsol of solutions g to the field equations
17.4 Covariant phase space formalism 677
(17.10). Since no confusion may arise, for simplicity, we shall drop the index “sol” and denote
this space simply by7 F. A difference with what precedes is that we are dealing with a field
theory, which, by essence, has an infinite number of degrees of freedom. In other words, the
space of solutions F is an infinite-dimensional manifold. More precisely, it can be given the
structure of an infinite-dimensional Banach manifold [345, 255], i.e. a topological space that
locally resembles8 a Banach space of infinite dimension, in the same way as a n-dimensional
manifold locally resembles the vector space Rn . Since any point of F is a metric field g on M ,
a variation δg can be viewed as an infinitesimal displacement vector between two neighboring
points of the manifold F, similar to the displacement vector dx on M introduced in Secs. 1.2.1
and A.2.3. As in the finite dimensional case (cf. Sec. A.2.3), (non-infinitesimal) vectors on F
are defined9 as tangent vectors along curves of F, the latter being the images of smooth maps
P : I ⊂ R → F, λ 7→ P (λ) =: g(λ). Here g(λ) stands for a 1-parameter family of metric
fields on M , this family constituting a parametrization of a given curve L of F. Any vector X
at a point g0 ∈ F can then be written as in Eq. (A.14):
δg
X= , (17.70)
δλ
where δg is the infinitesimal displacement vector connecting g0 =: g(λ) and g(λ + δλ) along
the curve L to which X is tangent to; λ is then the parameter of L associated to X and we
denote by δλ its small increment, instead of dλ as in Sec. A.2.3.
Property 17.15: space of asymptotically flat solutions
In what follows, we assume that the n-dimensional spacetime manifold M is diffeomorphic
to R × Σ̂, where Σ̂ is an orientable manifold of dimension (n − 1) with an asymptotically
Euclidean end, i.e. there exists a compact subset K of Σ̂ such that Σ̂ext := Σ̂ \ K is
diffeomorphic to Rn−1 \ Bn−1 , where Bn−1 is the unit ball of Rn−1 . Furthermore, given two
points p and p′ in M , we shall say that p′ lies in the background future of p iff for any
orientation-preserving diffeomorphism Ψ : M → R × Σ̂, Ψ(p′ ) = (t′ , q ′ ) and Ψ(p) = (t, q)
with t′ > t.
We shall restrict the space F of solutions g of the considered gravity theory to those
that are asymptotically flat, with the asymptotically flat end corresponding to Σ̂ext . The
precise definition of asymptotically flat depends on the gravity theory and should be
such that some integrals indicated below converge and that the asymptotic limit of some
surface integrals is zero.
We are going to see that, from the action (17.1), we can endow F with a (pre)symplectic
form Ω. Let Σ and Σ′ be two non-intersectic hypersurfaces of M , both diffeomorphic to Σ̂ and
such that Σ′ lies in the background future of Σ. Let Σr and Σ′r be compact subparts of Σ and
7
Be aware that in some part of the literature [24, 345, 507], F denotes instead the space of all fields g that are
required to only obey some asymptotic flatness conditions, but not necessarily the field equations; the subspace of
solutions to the field equations — our F — is then denoted by F̄ [345, 507] or Γ [24].
8
Cf. Lang’s book [336] for a precise definition of a Banach manifold.
9
Notice that the definition (A.8) of a tangent vector does not require the manifold to be of finite dimension.
678 Black hole thermodynamics beyond general relativity
Σ′ respectively, extending in the asymptotic flat end and corresponding to balls of Euclidean
radius r in Σ̂ext . Let Wr be the hypercylinder connecting Σr to Σ′r (cf. Fig. ??) and D be the
part of M bounded by Σr , Σ′r and Wr . Let us then consider the action (17.1) restricted to D:
Z
SD [g] := L(g). (17.71)
D
We are using the functional notation SD [g], but, when restricted to solutions g of the field
equations, SD is merely a scalar field on the manifold F, i.e. a function F → R. Accordingly,
the variation δSD of SD due to a small displacement δg on F can be viewed as the action of
the differential 1-form dSD on the infinitesimal vector δg: δSD = ⟨dSD , δg⟩ [cf. Eq. (A.21)].
Now, from the definition (17.71), we get δSD = D δL(g), with δL(g) given by Eq. (17.5),
R
in which E Ris set to zero for elements of F are solutions of the field equations. Hence
⟨dSD , δg⟩ = D dθ(g, δg). By linearity10 , we get for any vector X = δg/δλ at a point g ∈ F,
Z δg
⟨dSD , X⟩ = dθ g, .
D δλ
Thanks to Stokes’ theorem (A.94), we rewrite the right-hand side as a surface integral, taking
into account that ∂D = Σ′r ∪ Σr ∪ Wr :
Z δg Z δg Z δg
⟨dSD , X⟩ = θ g, + θ g, + θ g, , (17.72)
Σ′r ↗ δλ Σ↗r
δλ Wr
↗ δλ
where the arrows indicate the outward orientation of the various hypersurfaces with respect to
D. Since Eq. (17.72) is valid for any vector X on F, we may rewrite it as an identity between
1-forms:
dSD = ΘΣ′r − ΘΣr + ΥΣr ,Σ′r . (17.73)
Here ΘΣr stands the 1-form on F defined by the following action on a vector X in the tangent
space Tg F: Z δg
⟨ΘΣr , X⟩ := θ g, , (17.74)
Σr δλ
where δg/δλ is the relative variation of g defining the vector X by Eq. (17.70) and the orientation
of Σr is that provided by any background-future directed vector transverse to Σr . Note that
ΘΣr is well defined, i.e. the integral in Eq. (17.74) is finite, because Σr is compact and θ is
smooth; moreover, ΘΣr is linear for θ(·, ·) is linear in its second argument. The same definition
holds for ΘΣ′r . Similarly, ΥΣr ,Σ′r stands for the 1-form on F defined by
Z δg
⟨ΥΣr ,Σ′r , X⟩ := θ g, , (17.75)
Wr↗ δλ
with Wr defined as above. Note that the minus sign in front of ΘΣr in Eq. (17.73) occurs because
of the change of orientation of Σr between Eqs. (17.72) and (17.73).
In an attempt to endow F with a symplectic form, thereby turning it into a phase space,
let us consider the 2-form Ω that is the limit Σr → Σ (i.e. the limit r → +∞) of the exterior
derivative of ΘΣr . This defines a closed 2-form, which turns out to be independent of the choice
of Σ:
10
Recall that θ(g, δg) is linear with respect to δg, cf. Property 17.2.
17.4 Covariant phase space formalism 679
Property 17.16: presymplectic form in the space of solutions of a gravity theory
Let F be the (Banach) manifold of solutions g to the field equations that are asymptotically
flat in the asymptotically Euclidean end of the n-dimensional manifold M ∼ R × Σ̂. Let Σ
be a hypersurface of M diffeomorphic to Σ̂ and Σr a compact subpart of Σ corresponding
to a ball of Euclidean radius r in Σ̂ext . Given the 1-form ΘΣr defined on F by Eq. (17.74),
the asymptotically flatness conditions of the gravity theory must insure that the limit
Ω := lim dΘΣr , (17.76)
r→+∞
is well defined, i.e. yields a finite 2-form on F. This 2-form is closed: dΩ = 0. One then
says that Ω is a presymplectic forma on F. Moreover, Ω does not depend on the choice of
the hypersurface Σ. The pair (F, Ω) is called the prephase space of the considered gravity
theory (and asymptotic flatness conditions).
a
Recall that a symplectic form is a 2-form that is closed and non-degenerate; as we shall discuss below, Ω
is actually degenerate.
Proof. For each value of r, dΘΣr is a 2-form on F, so if the limit (17.76) exists, it clearly defines
a 2-form Ω on F. Let us evaluate its exterior derivative, dΩ, by means of formula (A.88)
with p = 2: for any 3-tuple (X1 , X2 , X3 ) of vector fields on F, we have dΩ(X1 , X2 , X3 ) =
X1 (Ω(X2 , X3 )) − X2 (Ω(X1 , X3 )) + X3 (Ω(X1 , X2 )) − Ω([X1 , X2 ], X3 ) + Ω([X1 , X3 ], X2 ) −
Ω([X2 , X3 ], X1 ). Now
X1 (Ω(X2 , X3 )) = X1 lim dΘΣr (X2 , X3 ) = lim X1 (dΘΣr (X2 , X3 )) ,
r→+∞ r→+∞
with similar formulas for X2 (Ω(X1 , X3 )) and X3 (Ω(X1 , X2 )). There comes then dΩ(X1 , X2 , X3 ) =
limr→∞ ddΘΣr (X1 , X2 , X3 ). But ddΘΣr = 0 (nilpotence of the exterior derivative, cf. Eq. (A.92)).
We conclude that dΩ = 0. As for the independence of Ω from Σ, it arises from the follow-
ing identity obtained by taking the exterior derive of Eq. (17.73) and using ddSD = 0 in the
left-hand side:
0 = dΘΣ′r − dΘΣr + dΥΣr ,Σ′r .
Taking the limit r → +∞ yields ΩΣ′ − ΩΣ + lim dΥΣr ,Σ′r = 0. The asymptotic flatness
r→+∞
conditions ensure that lim dΥΣr ,Σ′r = 0 (Lemma 17.18 below), leaving us with ΩΣ′ = ΩΣ .
r→+∞
Remark 5: Instead of the limit (17.76), it could have seemed simpler to define the presymplectic form
as Ω := dΘΣ , where ΘΣ is defined as in Eq. (17.74) but with the integral extended to the whole of Σ.
However, as noticed in Ref. [24] (p. 429), standard asymptotic flatness conditions do not guarantee that
this integral is convergent; hence the 1-form ΘΣ would be ill-defined.
Property 17.17: presymplectic current (n − 1)-form
The action of the presymplectic form Ω on a pair of commuting vector fields (X1 , X2 )
680 Black hole thermodynamics beyond general relativity
of F is expressible as follows. According to Eq. (17.70), each vector field describes the
variation of g along some curves of F. Since they commute, the vector fields (X1 , X2 ) can
be associated to a 2-parameter family g(λ1 , λ2 ) of metric tensors, spanning a 2-surface
S ⊂ F, so that X1 (resp. X2 ) corresponds to variations δg along curves λ2 = const (resp.
λ1 = const) in the 2-surface S, i.e. to partial derivatives:
δg ∂g δg ∂g
X1 = =: and X2 = =: . (17.77)
δλ1 λ2 =const ∂λ1 δλ2 λ1 =const ∂λ2
We have then Z ∂g ∂g
Ω(X1 , X2 ) = ω g, , , (17.78)
Σ ∂λ1 ∂λ2
where ω is the (n − 1)-form on M defined from the presymplectic potential (n − 1)-form
θ of the considered gravity theory (cf. Property 17.2) by
∂g ∂g ∂ ∂g ∂ ∂g
ω g, , := θ g, − θ g, . (17.79)
∂λ1 ∂λ2 ∂λ1 ∂λ2 ∂λ2 ∂λ1
ω is called the presymplectic current (n−1)-form [507]a . It has a local dependence on the
metric tensor g and on the derivatives ∂g/∂λ1 and ∂g/∂λ2 associated to the 2-parameter
family g(λ1 , λ2 ), and is linear with respect to these derivatives.
a
It is called the symplectic current (n − 1)-form in articles by Wald and Iyer [502, 292] preceding [507],
but, as discussed below, presymplectic is a better qualifier than symplectic.
Proof. Let us compute dΘΣr (X1 , X2 ) via the exterior derivative formula (A.89), noticing that (i)
this formula does not involve the manifold dimensionality (contrary to (A.90b), which applies
to finite dimensions only) and (ii) the last term in the right-hand side of (A.89) vanishes since
[X1 , X2 ] = 0. We get
dΘΣr (X1 , X2 ) = X1 (⟨ΘΣr , X2 ⟩) − X2 (⟨ΘΣr , X1 ⟩)
∂ ∂
= ⟨ΘΣr , X2 ⟩ − ⟨ΘΣr , X1 ⟩
∂λ1 Z ∂λ2 Z
∂ ∂g ∂ ∂g
= θ g, − θ g,
∂λ1 Σ ∂λ2 ∂λ2 Σr ∂λ1
Z r
∂ ∂g ∂ ∂g
= θ g, − θ g, .
Σr ∂λ1 ∂λ2 ∂λ2 ∂λ1
The second equality results from X1 (f ) = ∂f /∂λ1 for the scalar field f := ⟨ΘΣr , X2 ⟩ [cf.
Eq. (A.8)], the third equality makes use of Eqs. (17.74) and (17.77) and the last equality stems
from the linearity of the integral. Since Ω(X1 , X2 ) = limr→+∞ dΘΣr (X1 , X2 ) and Σr → Σ for
r → +∞, this proves Eq. (17.78), provided that ω is defined by (17.79). The (n − 1)-form ω
2g
does not depend on the second derivative ∂λ∂1 ∂λ 2
because the second derivatives, which appear
in each of the two terms of the right-hand side of Eq. (17.79), cancel out thanks to the identity
∂2g 2g
∂λ1 ∂λ2
= ∂λ∂2 ∂λ 1
. Hence the notation ω = ω(g, ∂g/∂λ1 , ∂g/∂λ2 ).
17.4 Covariant phase space formalism 681
Example 9 (presymplectic current form of general relativity): For general relativity, one deduces
from expression (17.9) for θ and the definition (17.79) that the presymplectic current (n − 1)-form is
ω(g, δ1 g, δ2 g)α1 ···αn−1 = (32π)−1 P λµντ ρσ (δ2 gµν ∇τ δ1 gρσ − δ1 gµν ∇τ δ2 gρσ ) ϵλα1 ···αn−1 , (17.80)
with
P λµντ ρσ := 2g λρ g σµ g ντ − g λτ g µρ g σν − g λµ g ντ g ρσ − g µν g λρ g στ + g µν g λτ g ρσ . (17.81)
We shall not detail the computation, which is straightforward but rather lengthy. Let us simply mention
that one shall start by expressing the covariant derivatives in formula (17.9) for θ in terms of g’s
√
Christoffel symbols [Eq. (A.64) with (k, ℓ) = (0, 2)] and replace ϵ by −g e as in Example 2, plug the
result in Eq. (17.79) and make use of the identities (16.20) and (16.21). As a check, one can compare with
Eqs. (21)-(23) of Ref. [283] or Eqs. (41)-(43) of Ref. [507].
Lemma 17.18: asymptotic vanishing of dΥΣr ,Σ′r
The 1-form ΥΣr ,Σ′r defined by Eq. (17.75) obeys limr→+∞ dΥΣr ,Σ′r = 0.
Proof. By exactly the same computation as that leading to Eq. (17.78), we have
Z ∂g ∂g
dΥΣr ,Σ′r (X1 , X2 ) = ω g, , .
Wr↗ ∂λ1 ∂λ2
As argued in Ref. [292], if the asymptotic flatness conditions ensure that the integral (17.78) on
the unbounded hypersurface Σ is convergent, then the integral of the same integrand ω over
Wr must tend to zero for r → +∞.
Property 17.19: uniqueness of the presymplectic form
The presymplectic form Ω is unique for a given gravity theory with appropriate asymptotic
flatness conditions. Not only it is does not depend on the choice of the hypersurface Σ
(Property 17.16), but it is unaffected by the ambiguities θ → θ + dY + δµ in the definition
of θ [cf. Eq. (17.6)]. Moreover, the presymplectic current (n − 1)-form ω itself is unaffected
by the ambiguity θ → θ + δµ.
Proof. Let us first show the last point, by evaluating ω from Eq. (17.79) for θ ′ (g, δg) = θ(g, δg)+
δµ(g). For δg = (∂g/∂λ2 ) δλ2 , we get
∂g ∂g ∂
θ ′ g, = θ g, + µ(g),
∂λ2 ∂λ2 ∂λ2
so that
∂2
∂ ∂g
′ ∂ ∂g
θ g, = θ g, + µ(g).
∂λ1 ∂λ2 ∂λ1 ∂λ2 ∂λ1 ∂λ2
682 Black hole thermodynamics beyond general relativity
Thanks to the commutativity of partial derivatives, the terms involving µ cancel each other
when substituting θ ′ for θ in Eq. (17.79). Hence the (n − 1)-form ω computed from θ ′ is the
same as that computed from θ. It follows then from Eq. (17.78) that the presymplectic form Ω
is independent of µ as well. On the other side, the change θ ′ (g, δg) = θ(g, δg) + d (Y (g, δg)),
as permitted by Eq. (17.6), leads to the following change in the presymplectic form:
Z
′ ∂ ∂g ∂ ∂g
Ω (X1 , X2 ) = Ω(X1 , X2 ) + Y g, − Y g, ,
S∞ ∂λ1 ∂λ2 ∂λ2 ∂λ1
where use has been made of Eqs. (17.78)-(17.79), as well as the Stokes’ theorem (A.94), S∞
standing for the “outer boundary” of Σ, i.e. the limit r → +∞ of a (n − 2)-sphere Sr of
radius r in the asymptotically Euclidean end of Σ. Now, given that the (n − 2) form Y must
be constructed from g, δg and their derivatives in a covariant manner, the fall-off conditions
from standard asymptotic flatness are sufficient to make the above integral over S∞ vanish
[292]. Hence Ω′ = Ω.
Property 17.20: closedness of the presymplectic current (n − 1)-form
Provided that the 2-parameter family g(λ1 , λ2 ) involved in the definition (17.79) regards
only solutions to the field equations (i.e. g(λ1 , λ2 ) ∈ F), the presymplectic current ω is a
closed (n − 1)-form:
dω = 0. (17.82)
Proof. The 2-parameter family of metric tensors g(λ1 , λ2 ) gives birth to a 2-parameter family
of Lagrangian n-forms L(λ1 , λ2 ) := L(g(λ1 , λ2 )). The variation δ1 L of L corresponding to a
variation of λ1 is given by Eq. (17.5) with E µν = 0 since g(λ1 , λ2 ) fulfills the field equations.
Hence δ1 L = d (θ(g, δ1 g)). Given that δ1 L = ∂λ∂L
1
δλ1 , δ1 g = ∂λ
∂g
1
δλ1 and θ(., .) is linear with
respect to its second argument, we get ∂L/∂λ1 = d (θ(g, ∂g/∂λ1 )). Taking the derivative of
this expression with respect to λ2 yields
∂ 2L
∂ ∂g ∂ ∂g
= d θ g, =d θ g, .
∂λ2 ∂λ1 ∂λ2 ∂λ1 ∂λ2 ∂λ1
Swapping λ1 and λ2 and subtracting the above formula yields, in view of ω’s definition (17.79),
∂2L 2L
∂λ1 ∂λ2
− ∂λ∂2 ∂λ 1
= dω. But the left-hand side of this equation is trivially zero, hence the result
(17.82).
It turns out that in general, the presymplectic form Ω is degenerate: there exists nonzero
vectors X on F, such that Ω(X, .) = 0. Hence Ω fails to be a symplectic form on F. For
instance, one may conceive a nonzero variation δg of g such that δg = 0 on the hypersurface
Σ involved in Eq. (17.78). Then, by linearity of ω(g, δ1 g, δ2 g) with respect to δ1 g and δ2 g (cf.
Property 17.17), one has ω(g, δg, .) = 0 on Σ and Eq. (17.78) shows that δg defines a degeneracy
direction of Ω. More generally, the degeneracy of Ω is related to the gauge freedom of the
gravity theory, i.e. to the invariance by diffeomorphism. For general relativity, it has been
shown [24] that the degeneracy directions of Ω are exactly those generated by diffeomorphisms
17.4 Covariant phase space formalism 683
of M [cf. Eqs. (17.11)-(17.12)] that reduce to the identity at infinity (i.e. such that ξ in Eq. (17.12)
tends to zero for r → +∞). The gauge freedom is intimately related to the initial value problem
of the theory being not well posed (it becomes well posed only in a fixed gauge, see e.g. Chap. VI
of Ref. [108]). Recall that in the covariant phase space construction of Property 17.14, it was
required that the initial value problem is well posed to endow the space of solutions with a
(non-degenerate) symplectic form.
It is however possible to introduce a quotient space F̄ = F/R on which Ω gives birth to a
true symplectic form. The idea is to “factor out” the degeneracy directions of Ω. This is possible
because these directions are integrable into submanifolds of F. According to Frobenius theorem
(Theorem B.3.1 in Ref. [499]), the integrability is equivalent to the commutator [X1 , X2 ] being a
degeneracy vector field for any two degeneracy vector fields X1 and X2 . To show this property,
let us use the identity (A.79) and the Leibniz rule to write Ω([X1 , X2 ], .) = Ω(LX1 X2 , .) =
LX1 [Ω(X2 , .)] − LX1 Ω (X2 , .) = −LX1 Ω (X2 , .) since Ω(X2 , .) = 0. Now, thanks to Cartan
identity (A.95), LX1 Ω = dΩ(X1 , ., .) + d[Ω(X1 , .)] = 0, given that dΩ = 0 (Property 17.16)
and Ω(X1 , .) = 0. Hence Ω([X1 , X2 ], .) = 0 and the degeneracy directions of Ω are integrable
into submanifods of F. Assuming that these submanifolds provide a regular foliation of F (cf.
the discussion in Ref. [345]), one defines an equivalence relation R on F by setting g1 Rg2 iff
the points g1 and g2 of F belong to the same degeneracy submanifold. The set of equivalence
classes F̄ := F/R can be given the structure of Banach manifold, such that the canonical
projection π : F → F̄ is a smooth map. One may then define a 2-form Ω̄ on F̄ by setting,
for any pair (X̄1 , X̄2 ) of tangent vectors at a point ḡ ∈ F̄, Ω̄(X̄1 , X̄2 ) := Ω(X1 , X2 ), where
(X1 , X2 ) ∈ (Tg F)2 , such that π(g) = ḡ, π∗ X1 = X̄1 and π∗ X2 = X̄2 (pushforward by π, cf.
Sec. A.2.8). Because Ω is preserved along its degeneracy directions (cf. LX1 Ω = 0 in the above
proof), the definition of Ω̄(X̄1 , X̄2 ) is independent of the choice of g in the fiber π −1 (ḡ). It
is also independent of the choice of X1 and X2 in Tg F because two vectors X1 and X1′ such
that π∗ X1 = π∗ X1′ differ only by a vector along a degeneracy direction of Ω. Hence Ω̄(X̄1 , X̄2 )
is unambiguously defined. It can be shown that Ω̄ is a closed non-degenerate 2-form, i.e. a
symplectic form (cf. Refs. [255, 345] for details). The genuine covariant phase space of the
considered gravity theory is then (F̄, Ω̄). However, in what follows, we will deal only with the
prephase space (F, Ω).
17.4.4 Fundamental identity of the covariant phase space formalism
Property 17.20 tells that the (n − 1)-form ω(g, δ1 g, δ2 g) is closed for any pair of variations
(δ1 g, δ2 g) among the solutions to the field equations. Actually, ω is globally 11 exact as soon
as δ2 g results from an infinitesimal diffeomorphism generated by a vector field ξ: ω is then
the exterior derivative of a (n − 2)-form, which involves the variation δ1 Q(ξ) of ξ’s Noether
potential (n − 2)-form. More precisely, we have:
Property 17.21: fundamental identity of the covariant phase space formalism
Let (M , g) be a n-dimensional spacetime governed by a diffeomorphism-invariant theory
11
Let us recall that thanks to Poincaré lemma (cf. Sec. A.4.3), ω closed =⇒ ω locally exact.
684 Black hole thermodynamics beyond general relativity
and let ξ be a vector field on M . Any variation δg among the solutions to the field
equationsa generates a variation δQ(ξ) of the Noether potential (n − 2)-form of ξ (cf.
Property 17.6 and Remark 4 on p. 662), such that
ω(g, δg, Lξ g) = d [δQ(ξ) − ξ · θ(g, δg)] , (17.83)
where ω is the presymplectic current (n − 1)-form (Property 17.17) and θ is the presym-
plectic potential (n − 1)-form (Property 17.2).
a
This means that both g and g + δg are assumed to be solutions to the field equations, which implies
that δg is a solution to the linearized field equation.
Proof. The variation of the Noether current (n − 1)-form J (ξ) induced by δg is obtained from
Eq. (17.16): δJ (ξ) = δθ(g, Lξ g) − ξ · δL. Note that we have set δξ = 0, for the vector field
ξ is independent of g. Here δL is given by Eq. (17.5) with E = 0 since we are dealing with
solutions to the field equations only: δL = dθ(g, δg). Hence
δJ (ξ) = δθ(g, Lξ g) − ξ · dθ(g, δg) = δθ(g, Lξ g) − Lξ θ(g, δg) + d [ξ · θ(g, δg)] ,
where use has been made of Cartan identity (A.95). Now, there is no loss of generality in
considering that there exists a 2-parameter family g(λ1 , λ2 ) of metric fields such that δg =
(∂g/∂λ1 ) δλ1 and ∂g/∂λ2 = Lξ g [cf. Eq. (17.12), rewritten as δ2 g = δλ2 Lξ g]. In view of ω’s
definition (17.79), we have then δθ(g, Lξ g) − Lξ θ(g, δg) = ω(g, δg, Lξ g), so that the above
equation becomes
δJ (ξ) = ω(g, δg, Lξ g) + d [ξ · θ(g, δg)] .
Finally, restoring the explicit dependence of J on g (cf. Remark 3 on p. 661), we have δJ (ξ) :=
J (g + δg, ξ) − J (g, ξ). Since both g and g + δg are solutions to the field equations, we may
invoke Eq. (17.21) to write J (g + δg, ξ) = dQ(g + δg, ξ) and J (g, ξ) = dQ(g, ξ). Hence there
comes δJ (ξ) = d[Q(g + δg, ξ) − Q(g, ξ)] = dδQ(ξ), from which Eq. (17.83) follows.
Remark 6: The denomination fundamental identity of the covariant phase space formalism for Eq. (17.83)
is used in some part of the literature [138, 246, 250], as here, while in another part [283, 284], it is given
to a generalization of it, which holds even if g and g + δg do not satisfy the field equations.
17.5 First law for diffeomorphism-invariant theories
17.5.1 Hamiltonian conjugate to an asymptotic symmetry
A vector field ξ on M is said to be a generator of an asymptotic symmetry iff the dif-
feomorphisms of M generated by ξ preserve the asymptotic flatness conditions stated in
Property 17.15. Equivalently, Φ∗t g ∈ F for any g ∈ F and any diffeomorphism Φt : M → M
generated by ξ. One may then associate to ξ a vector field Xξ on the prephase space F: Xξ is
the vector field defined by Eq. (17.70): Xξ = δg/δλ, with δg given by Eq. (17.12) with t = δλ:
17.5 First law for diffeomorphism-invariant theories 685
δg = δλ Lξ g. The presymplectic product of a generic infinitesimal displacement δg on F by
Xξ is then given by Eq. (17.78):
Z
Ω(δg, Xξ ) = ω(g, δg, Lξ g).
Σ
If we substitute the fundamental identity (17.83) for ω(g, δg, Lξ g) and use Stokes’ theorem
(A.94), we arrive at Z Z
Ω(δg, Xξ ) = δ Q(ξ) − ξ · θ(g, δg), (17.84)
S∞ S∞
where S∞ = limr→+∞ Sr , Sr being the boundary of the hypersurface Σr ⊂ Σ considered in
Sec. 17.4.3. Let us suppose that the right-hand side of Eq. (17.84) is a pure variation, i.e. that
there exists a function Hξ : F → R, g 7→ Hξ [g] such that
Z Z
δHξ = δ Q(ξ) − ξ · θ(g, δg). (17.85)
S∞ S∞
Then we can rewrite Eq. (17.84) as δHξ = Ω(δg, Xξ ). Given that δHξ = ⟨dHξ , δg⟩ [cf.
Eq. (A.21)] and δg is a generic displacement on F, this is actually equivalent to
dHξ = Ω(., Xξ ) . (17.86)
We recognize Eq. (17.65), which defines the Hamiltonian vector field XH in terms of the
Hamiltonian H. The difference is that Eq. (17.65) involves a true symplectic form, while Ω in
Eq. (17.86) is only presymplectic. In particular, we could not invert Ω to define Xξ in terms of
Hξ , but this is not a problem here since Xξ is given a priori and Eq. (17.86) defines Hξ from it
(assuming that the right-hand side of Eq. (17.84) is a pure variation). One says that Hξ is the
Hamiltonian conjugate to the vector field ξ [507]. Hence, if there exists a function Hξ on
F fulfilling (17.85), the dynamics generated on the prephase space (F, Ω) by the vector field
Xξ , i.e. the action of the diffeomorphism group generated by ξ on the solutions g to the field
equations, is a Hamiltonian dynamics.
A natural question is whether or not a given vector field ξ admits a conjugate Hamiltonian.
A sufficient condition is that there exists a (n − 1)-form B(g) on M such that
Z Z
ξ · θ(g, δg) = δ ξ · B(g). (17.87)
S∞ S∞
In particular, this holds if the asymptotic behavior of the presymplectic potential (n − 1)-form
is θ(g, δg) ∼ δB(g) for some (n − 1)-form B(g). This is the case for general relativity:
r→+∞
Example 10 (asymptotic behavior of the presymplectic potential form in general relativity):
Thanks to the asymptotic flatness conditions, let us introduce a flat metric f on M (or at least in the
asymptotic region of M ) such that g ∼ f for r → +∞. In general relativity, θ is given by Eq. (17.8):
θ(g, δg) = (16π)−1 H(δg) · ϵ(g). We may then write
θ(g, δg) ∼ (16π)−1 H̄(δg) · e, (17.88)
r→+∞
686 Black hole thermodynamics beyond general relativity
where e is the Levi-Civita tensor of f and H̄(δg) is the vector defined from δg in the same way as
H(δg) in Eq. (16.13), but substituting f for g: H̄ α := f αρ f µν (Dν δgρµ − Dρ δgµν ), where D stands for
the covariant derivative associated to f . Since δf = 0 and Dδ = δD, for f is independent of g, we
may write H̄(δg) = δK(g), where K(g) is the vector field defined by
K α := f αρ f µν (Dν gρµ − Dρ gµν ) . (17.89)
Since δe = 0 (for f is independent of g), we deduce from Eq. (17.88) that
θ(g, δg) ∼ δB(g), with B(g) := (16π)−1 K(g) · e. (17.90)
r→+∞
The (n − 1)-form B(g) defined above fulfills Eq. (17.87), thereby ensuring that a conjugate Hamiltonian
Hξ exists for any asymptotic symmetry generator ξ.
Remark 1: As the above example shows, the (n − 1)-form B(g) does not need to be covariantly
defined: Eq. (17.90) involves the flat metric f , which is arbitrarily chosen. In particular, B(g) is highly
non-unique; what matters is only its asymptotic behavior.
If condition (17.87) is satisfied, then a function Hξ fulfilling (17.85) is
Z
Hξ [g] := (Q(ξ) − ξ · B(g)) . (17.91)
S∞
More generally, a necessary and sufficient criterion for the existence of Hξ is
Property 17.22: existence of a Hamiltonian conjugate to a vector field (Wald &
Zoupas 2000 [507])
A necessary condition for the existence of a Hamiltonian Hξ : F → R conjugate to the
vector field ξ is that Z
ξ · ω(g, δ1 g, δ2 g) = 0 (17.92)
S∞
for any pair of variations (δ1 g, δ2 g) among the solutions to the field equations. Furthermore,
if the prephase space F is simply connected, this condition is sufficient.
Proof. Let us suppose that Hξ exists. Then, by taking the variation δ1 of Eq. (17.85) written for
δ = δ2 , we get Z
δ1 δ2 Hξ = δ1 δ2 QS∞ ,ξ − ξ · δ1 (θ(g, δ2 g)) ,
S∞
where QS∞ ,ξ is the Noether charge of S∞ with respect to ξ [cf. Eq. (17.28)]. By subtracting
the term 1 ↔ 2 and using (δ1 δ2 − δ2 δ1 )Hξ = 0, as well as (δ1 δ2 − δ2 δ1 )QS∞ ,ξ = 0, we obtain
(17.92), given the definition (17.79) of ω. Hence (17.92) appears as a necessary condition for Hξ
to exist. To prove that it is sufficient when F is simply connected, let us select a point g0 ∈ F
and define for any point g ∈ F and any curve L from g0 to g,
Z
H(g, L ) := α, (17.93)
L
17.5 First law for diffeomorphism-invariant theories 687
where α is the 1-form on F defined by the following action on a vector X = ∂g/∂λ:
Z
∂ ∂g
⟨α, X⟩ := Q(ξ) − ξ · θ g, . (17.94)
S∞ ∂λ ∂λ
Let L ′ be a curve of F, distinct from L , connecting g0 to g. If F is simply connected, there
exists a homotopy from L to L ′ . This defines a compact 2-dimensional surface S ⊂ F, the
boundary of which is L ∪ L ′ , where L ′ is same curve as L ′ , but oriented from g to g0 .
← ←
Stokes’ theorem (A.94) yields then
Z
H(g, L ) − H(g, L ) =
′
dα.
S
To evaluate the 2-form dα on S, let us compute dα(X1 , X2 ) for a pair of vectors (X1 , X2 )
associated to a 2-parameter family ḡ(λ1 , λ2 ) spanning S [cf. Eq. (17.77)]. We may choose
(λ1 , λ2 ) ∈ [0, 1]2 , with ḡ(λ1 , 0) spanning L , ḡ(λ1 , 1) spanning L ′ , ḡ(0, 0) = ḡ(0, 1) = g0
and ḡ(1, 0) = ḡ(1, 1) = g. By a computation similar to that of dΘΣr (X1 , X2 ) in the proof of
Property 17.17, we arrive at
Z ( 2 2
" #)
∂ Q(ξ) ∂ Q(ξ) ∂ ∂ḡ ∂ ∂ḡ
dα(X1 , X2 ) = − −ξ · θ ḡ, − θ ḡ, .
S∞ | ∂λ1 ∂λ2 ∂λ2 ∂λ1 ∂λ ∂λ2 ∂λ ∂λ1
{z } | 1 {z 2 }
0
∂ ḡ ∂ ḡ
ω ḡ, ∂λ , ∂λ
1 2
Hence if Eq. (17.92) holds, dα(X1 , X2 ) = 0. We conclude that dα = 0 on S and thus that
H(g, L ) − H(g, L ′ ) = 0. It follows that H(g, L ) is a function of g only, which we shall
denote by H(g). From the definitions (17.93)-(17.94), as well as the definition of the integral of
a 1-form along a curve, we get
Z Z Z
δH = H(g+δg)−H(g) = ⟨α, δg⟩ = (δQ(ξ) − ξ · θ(g, δg)) = δ Q(ξ)− ξ·θ(g, δg).
S∞ S∞ S∞
Comparing with (17.85), we conclude that H(g) is a Hamiltonian conjugate to ξ.
If the vector field ξ corresponds to an asymptotic time translation, it is natural to define
the energy of the system as the value of the conjugate Hamiltonian (cf. Property 17.13):
Property 17.23: canonical energy relative to an asymptotic time translation
Let (M , g) be an asymptotically flat spacetime governed by some diffeomorphism-invariant
theory of gravity. Let ξ be a vector field on M , generator of an asymptotic symmetry
corresponding to a time translation. Let us assume that the asymptotic flatness conditions
guarantee that a Hamiltonian Hξ conjugate to ξ exists, as well as a (n − 1)-form B such
that Hξ takes the shape (17.91). The canonical energy of the spacetime (M , g) relative to
688 Black hole thermodynamics beyond general relativity
ξ is then the value of Hξ at g:
Z
E := (Q(ξ) − ξ · B) . (17.95)
S∞
Example 11 (canonical energy for general relativity is ADM energy): From expression (17.23) for
Q(ξ) in general relativity, we get
Z Z Z Z
1 1 µν 1
Q(ξ) = − ⋆(dξ) = − (dξ)µν dS = − ∂µ ξν dS µν ,
S∞ 16π S∞ 32π S∞ 16π S∞
where we have used Lemma 5.10 to express the integral in terms of the area element normal bivector
dS to S∞ and have used formula (A.90b) and the antisymmetry of dS to get the last equality. Now,
since ξ generates an asymptotic time translation, we have ξ = ∂t , where t is the first coordinate of an
asymptotically Minkowskian system (xα ) = (t, x1 , . . . , xn−1p ), i.e. a f -Minkowskian coordinate system
(cf. Sec. 5.3.1) for a flat metric f such that g ∼ f for r := (x1 )2 + · · · (xn−1 )2 → +∞. S∞ can then
be considered as the limit r → +∞ of a (n − 2)-sphere St,r defined by (t, r) = const. From Eq. (5.42),
√
we have dS µν := (sµ nν − nµ sν ) dS, with dS := q dn−2 y (the area element of St,r ) and, at the limit
r → +∞, nµ = ξ µ = δ µt and sµ = (0, si ), with si := xi /r for i ∈ {1, . . . , n − 1}. Furthermore,
ξν = gνσ ξ σ = gtν . Hence, there comes
Z Z Z
1 µ ν 1
Q(ξ) = − µ ν
∂µ gtν (s δ t − δ t s ) dS = − (∂i gtt − ∂t gti ) si dS. (17.96)
S∞ 16π S∞ 16π S∞
On the other hand, from the (n−1)-form B obtained in Eq. (17.90), we have ξ·B = (16π)−1 ξ·(K ·e) =
(16π)−1 e(K, ξ, . . .). For r → +∞, e ∼ ϵ and we may write ξ · B ∼ (16π)−1 ϵ(K, ξ, . . .) =
(16π)−1 ⋆ (K ∧ ξ), where K ∧ ξ stands for the 2-form of components (K ∧ ξ)αβ = Kα ξβ − ξα Kβ
and we have used Eq. (5.38) with p = 2 to let appear the Hodge dual of K ∧ ξ. Using again Lemma 5.10
and the antisymmetry property (Kµ ξν − ξµ Kν ) dS µν = 2Kµ ξν dS µν , we arrive at
Z Z Z
1 1
ξ·B = Kµ ξν dS =µν
Kµ gtν (sµ δ νt − δ µt sν ) dS
S∞ 16π S∞ 16π S∞
Z Z
1 1
= (Ki gtt −Kt gti )si dS = − Ki si dS.
16π S∞ |{z} |{z} 16π S∞
−1 0
Now, from Eq. (17.89), we have, on S∞ , Ki = f µν (Dν giµ −Di gµν ) = −(∂t git −∂i gtt )+δ kl (∂l gik −∂i gkl ).
Hence
Z Z Z
1 1
ξ·B =− (∂i gtt − ∂t gti ) si dS − δ kl (∂l gik − ∂i gkl ) si dS.
S∞ 16π S∞ 16π S∞
We notice that the first integral on the right-hand side is exactly that appearing in Eq. (17.96), so that
they cancel each other in formula (17.95) for the canonical energy, leaving
Z
1
E= δ kl (∂l gik − ∂i gkl ) si dS. (17.97)
16π S∞
This is exactly the formula giving the ADM energy (cf. Sec. 5.3.5): compare Eq. (17.97) with e.g. Eq. (8.11)
in Ref. [227].
17.5 First law for diffeomorphism-invariant theories 689
Remark 2: The integral of Q(ξ) over S∞ , which appears in formula (17.95) for E, is the Noether charge
QS∞ ,ξ of S∞ with respect to ξ (cf. Property 17.8). For a stationary spacetime in general relativity,
we have noticed that QS∞ ,ξ is (n − 3)/(n − 2) times the Komar mass at infinity (Property 17.9). The
latter is identical to the ADM energy (Property 5.18). Hence, for a stationary spacetime ruled by general
relativity, the relative contributions to E of the terms Q(ξ) and −ξ · B in Eq. (17.95) are respectively
(n − 3)/(n − 2) and 1/(n − 2), i.e. 1/2 and 1/2 for n = 4.
Let us now consider a vector field η on M that is the generator of an asymptotic sym-
metry corresponding to a spatial rotation, i.e. η ∼r→+∞ x∂y − y∂x in some asymptotically
Minkowskian coordinate system (t, x1 =: x, x2 =: y, x3 , . . . , xn−1 ). Without any loss of
generality, we may choose the (n − 2)-surfaces St,r converging to S∞ for r → +∞ such
that
R η is tangent to each St,r and thus to S∞ . Then, in Eq. (17.84) with ξ := η, we have
S∞
η · θ(g, δg) = 0. Indeed, η · θ := θ(η, ., . . . , .) is a (n − 2)-form that vanishes on S∞ : for
any (n − 2)-tuple of vectors (v1 , . . . , vn−2 ) tangent to S∞ , (η, v1 , . . . , vn−2 ) is (n − 1)-tuple of
vectors all tangent to the (n − 2)-dimensional manifold S∞ ; these vectors are thus necessarily
linearly dependent, so that θ(η, v1 , . . . , vn−2 ) = 0 since θ is fully antisymmetric. It follows
that a Hamiltonian conjugate to η always exists and is given by Hη [g] = S∞ Q(η). The
R
negative of its value for a given solution g to the field equation defines the canonical angular
momentum of that solution relative to η:
Property 17.24: canonical angular momentum relative to an asymptotic spatial
rotation
Let (M , g) be an asymptotically flat spacetime governed by some diffeomorphism-invariant
theory of gravity. Let η be a vector field on M , generator of an asymptotic symmetry
corresponding to a spatial rotation. Then a Hamiltonian Hη conjugate to η always exists
and the canonical angular momentum of the spacetime (M , g) relative to η is the value
of −Hη at g: Z
J := − Q(η). (17.98)
S∞
J is actually the opposite of the Noether charge QS∞ ,η of S∞ with respect to η, as defined
in Property 17.8.
Example 12 (canonical angular momentum of an axisymmetric spacetime in general relativity):
Let (M , g) be an axisymmetric spacetime ruled by general relativity and let η be the Killing vector
generating the axisymmetry. According to Eq. (17.32), J is nothing than the Komar angular momentum
relative to η at infinity.
17.5.2 First law for diffeomorphism-invariant theories
Finally, we arrive at the first law:
Property 17.25: first law of black hole thermodynamics for diffeomorphism-
invariant theories (Iyer & Wald 1994 [292, 502])
690 Black hole thermodynamics beyond general relativity
Let (M , g) be a n-dimensional stationary spacetime governed by a diffeomorphism-
invariant gravity theory. Let us assume that (M , g) contains a black hole, whose event
horizon H is part of a bifurcate Killing horizon (cf. Sec. 3.4) with respect to the Killing
vector
XL
χ=ξ+ Ω(i) η(i) , (17.99)
i=1
where ξ is the stationary Killing vector, the η(i) ’s are L axisymmetric Killing vectors
(1 ≤ L ≤ [(n − 1)/2]) and the Ω(i) ’s are L real constants. By virtue of Property 3.16, the
surface gravity κ of H is constant, so that the Hawking temperature of H is well defined:
TH = ℏκ/(2πkB ) [Eq. (16.62)]. Let us consider an asymptotically flat solution δg to the
linearized field equations around g, i.e. a variation δg in the space F of asymptotically flat
solutions to the field equations (δg is not required to be stationary). Let us define the Wald
entropy of the black hole in the perturbed spacetime (M , g + δg) by the same formula as
in the stationary case, namely Eq. (17.33), but with the integral taken on the bifurcation
surface Sˆ of the unperturbed configurationa :
Z
2πkB
SW := Q(χ), (17.100)
ℏκ Sˆ
where χ is the same vector field on M as in the unperturbed configuration (in other
words, δχ = 0). Finally, let us assume that the asymptotic flatness conditions ensure that
a Hamiltonian conjugate to ξ exists. Then δg generates a variation δE of the canonical
energy relative to ξ [Eq. (17.95)] obeying
L
X
δE = TH δSW + Ω(i) δJ(i) , (17.101)
i=1
where J(i) is the canonical angular momenta relative to η(i) [Eq. (17.98)].
a
As discussed in the proof of Property 16.1, there is no loss of generality in considering that the event
horizons of the stationary spacetime (M , g) and the perturbed one (M , g + δg) are described by the same
hypersurface H of M . A priori H is not a Killing horizon in (M , g + δg), since the latter is not assumed
stationary. The submanifold Sˆ of H is thus not a bifurcation surface in (M , g + δg). However, this does
not prevent ones to define the integral of the (n − 2)-form Q(χ) in (M , g + δg) (i.e. Q(χ) = Q(g + δg, χ),
if the dependence of Q on g is made explicit) on the (n − 2)-surface Sˆ, as in Eq. (17.100). Similarly, in
Eq. (17.100), κ is the surface gravity of H in (M , g), given that there is no well-defined concept of surface
gravity if H is not a Killing horizon.
Proof. Let us consider the fundamental identity (17.83) with ξ = χ, which holds since we
assume that χ is a fixed vector field on M (δχ = 0). We may set ω(g, δg, Lχ g) = 0 in its
left-hand side since ω(g, δg, Lχ g) is linear with respect to Lχ g (Property 17.17) and Lχ g = 0
for χ is a Killing vector of g. There remains then
d [δQ(χ) − χ · θ(g, δg)] = 0.
The left-hand side is a (n − 1)-form. Let us integrate it on a hypersurface Σ extending from
17.5 First law for diffeomorphism-invariant theories 691
the bifurcation surface Sˆ of the bifurcate Killing horizon to the asymptotically flat end. Using
Stokes’ theorem (A.94) with ∂Σ = Sˆ ∪ S∞ , we get
Z Z
δ Q(χ) = [δQ(χ) − χ · θ(g, δg)] . (17.102)
Sˆ S∞
Note that we have ˆ
R used the vanishing of χ on S R (Property 3.12) to write the integral in the
left-hand side as Sˆ δQ(χ), which is equal to δ Sˆ Q(χ). Since Q(χ) is linear with respect to
χ (Property 17.6), Eq. (17.99) leads to
Z Z L
X Z
(i)
δ Q(χ) = [δQ(ξ) − ξ · θ(g, δg)] + Ω δQ(η(i) ) − η(i) · θ(g, δg) ,
ˆ
| S {z } | S∞ {z } i=1 | S∞ {z }
ℏκ
S
2πkB W
δHξ δHη(i)
where we have used Eq. (17.100) to let appear the Wald entropy SW and Eq. (17.85) to let appear
the variations of the Hamiltonians Hξ and Hη(i) conjugate to ξ and η(i) respectively. Given
that δκ = 0 (for κ in Eq. (17.100) is set to the surface gravity in the unperturbed spacetime, cf.
footnote a), ℏκ/(2πkB ) = TH [Eq. (16.62)], Hξ [g] = E (Property 17.23) and Hη(i) [g] = −J(i)
(Property 17.24), we get Eq. (17.101).
At first glance, Property 17.25 generalizes the first law of black hole thermodynamics
obtained for general relativity in Chap. 16 to any diffeomorphism-invariant theory of gravity:
compare Eq. (17.101) with Eq. (16.74), taking into account that, for general relativity, E = M (cf.
Example 11 on p. 688). However, one should notice that δSW in Eq. (17.101) is the variation of the
entropy defined by Eq. (17.100), while in Eq. (16.74), δSBH was the variation of entropy between
two nearby stationary configurations. Given that for a stationary configuration, the Wald
entropy SW =: SW eq
is defined by Eq. (17.33), where κ is the surface gravity corresponding to
that configuration and not of some reference configuration as in Eq. (17.100), one should prove
that δSW = δSW eq
to claim that Eq. (17.101) truly generalizes the first law (16.74). Specifically, let
us use the same framework as in Chap. 16, namely the perturbed configuration is a stationary
black hole, the horizon of which is a Killing horizon with respect to the Killing vector
L
X
χ + δχeq = ξ + (Ω(i) + δΩ(i) )η(i) , (17.103)
i=1
and the corresponding surface gravity is κ+δκ. Note that we are using the same gauge freedom
as in Chap. 16, namely the horizon is the same hypersurface of M and the stationary and
axisymmetric Killing vectors are held fixed: δξ = 0 and δη(i) = 0 [Eq. (16.4)]. We are using
the subscript “eq” in δχeq to avoid any confusion with δχ = 0 assumed in Property 17.25. We
have then, from Eq. (17.33) with S = Sˆ,
Z Z
eq 2πkB 2πkB
δSW = Q(g + δg, χ + δχeq ) − Q(g, χ).
ℏ(κ + δκ) Sˆ ℏκ Sˆ
Using the linearity of Q(g, χ) with respect to χ (cf. Property 17.6) and keeping the linear order
in the perturbation, we get
Z Z
eq 2πkB 2πkB δκ
δSW = [Q(g + δg, χ) + Q(g, δχeq ) − Q(g, χ)] − Q(g, χ).
ℏκ Sˆ ℏκ2 Sˆ
692 Black hole thermodynamics beyond general relativity
Now, the difference between the first and third terms in the first integral corresponds exactly
to δSW with SW defined by Eq. (17.100). Hence there comes
Z Z
eq 2πkB δκ
δSW = δSW + Q(δχeq ) − Q(χ) ,
ℏκ Sˆ κ Sˆ
where we have suppressed the explicit dependence of Q on g. From Eqs. (17.99) and (17.103),
one has δχeq = Li=1 δΩ(i) η(i) (in agreement with Eq. (16.8)). Given that Q(χ) is linear in χ,
P
this yields Q(δχeq ) = Li=1 δΩ(i) Q(η(i) ). On the other hand, we may express δκ in terms of
P
the δΩ(i) ’s as well, thanks to Eq. (16.11), which is purely kinematic and thus independent of
the gravity theory:
L Z
1 X (i)
δκ = − δΩ ∇µ η(i)ν dS µν , (17.104)
2A i=1 Sˆ
where A is the horizon area. Note that we have set to zero the term ISk in Eq. (16.11) since
χ = 0 on Sˆ and ∇ν hµν := ∇ν δgµν is assumed regular on Sˆ. Hence we obtain
L
"Z Z Z #
eq 2πk B
X 1
δSW = δSW + δΩ(i) Q(η(i) ) + ∇µ η(i)ν dS µν × Q(χ) .
ℏκ i=1 Sˆ 2κA Sˆ Sˆ
| {z } | {z } | {z }
(i) (i) κA/(8π)
−J 16πJ
Sˆ Sˆ
The values indicated below the underbraces are those obtained for general relativity, resulting
respectively from Eqs. (17.32), (5.65) and (17.35). The terms between the square brackets then
cancel each other, leaving us with δSW eq
= δSW . We thus conclude:
Property 17.26: first law for general relativity
For vacuum general relativity and for a variation between two stationary black holes, the
first law (17.101) reduces to the “comparison” first law obtained in Property 16.2: Eq. (17.101)
is identical to Eq. (16.22) with K = 1. Morever, still within general relativity, Property 17.25
extends Property 16.2 to non-stationary variations δg.
Remark 3: We have already noticed that the Hawking temperature TH is independent of the gravity
theory (cf. Remark 2 on p. 646). Thus, contrary to the entropy, there was no need to amend it to get the
first law in a generic diffeomorphism-invariant gravity theory.
Remark 4: In the original work by Iyer & Wald [292], the black hole entropy in the perturbed con-
figuration is defined by formula (17.36) instead of formula (17.100). Formula (17.36) makes sense for a
perturbed black hole as well, provided that Sˆ is the bifurcation surface in the unperturbed configuration.
Let us denote by SW ′ the entropy resulting from that definition, keeping S for (17.100). For a stationary
W
black hole, both definitions are equivalent: SW = SW ′ (Property 17.12), but, for a nonstationary one,
this is not obvious. However, it has been proven by Iyer & Wald that δSW = δSW ′ (cf. Eq. (94) in
Ref. [292]), so that the first law (17.101) holds if δSW is substituted by δSW (cf. Eq. (93) in Ref. [292]).
′
Given the definition of SW ′ , it is obvious that, when considering two nearby stationary configurations,
17.6 What about the second law? 693
δS ′ eq
W = δSW , so that using Iyer & Wald’s definition, Property 17.26 would have been immediate. We
′
nethertheless stick to the definition (17.100) since it gives the quantity that naturally appears when
deriving the first law (cf. the proof of Property 17.25), while the derivation for SW
′ is more complicated,
cf. the proof in Ref. [292]. Furthermore, the definition (17.100) is connected with the recently proposed
entropy definitions for dynamical black holes [284, 506, 498], to be discussed below.
17.6 What about the second law?
We have seen above that, for theories distinct from general relativity, the black hole entropy
is a priori not proportional to the black hole area. In particular, to show that SW ∝ A in
Property 17.11, we have used the value (17.23) of Q(χ), which is specific to general relativity.
Thus even in theories for which the null convergence condition holds, the area theorem
(Property 16.9) cannot be viewed as the second law of black hole thermodynamics.
Establishing a second law of black hole thermodynamics for diffeomorphism-invariant
gravity theories requires first to identify some entropy candidate for dynamical black holes.
The Wald entropy (17.100) cannot play this role, even for linear perturbations of a stationary
black hole, for it requires to be evaluated at the bifurcation surface, which is a specific “time”
in the black hole history, and which may even not exist for black holes born in gravitational
collapse, as studied in Chap. 14. The entropy of dynamical black holes has to be defined for
arbitrary cross-sections of the event horizon. In the proof of the first law (Property 17.25),
sticking to the bifurcation surface was crucial to set to zero the term χ · θ(g, δg) in the left-hand
side of Eq. (17.102), so that it becomes the variation of the Noether charge of χ and hence the
variation of the Wald enthalpy. If the integration would have been performed on an arbitrary
cross-section, this would a priori not have lead to an exact variation, contrary to what happens
for the right-hand side of Eq. (17.102), where the integration over S∞ leads to exact variations,
thanks to the existence of Hamiltonians conjugate to ξ and the η(i) ’s.
However, recently, Hollands, Wald and Zhang [284] (hereafter HWZ) have shown that, for
first-order perturbations δg of a stationary black hole, one can define a (n − 1)-form BH on
the event horizon H , such the pullback ι∗ θ(g, δg) of θ(g, δg) on H (i.e. the restriction of the
H
(n − 1)-form θ to vectors tangent to H ) writes ι∗ θ(g, δg) = δBH . This allows one to define
the entropy of an arbitrary cross-section S by
Z
2πkB
SHWZ := (Q(χ) − χ · BH ) , (17.105)
ℏκ S
where κ is the surface gravity of the reference stationary black hole. For S = Sˆ, this formula
reduces to the Wald entropy expression (17.100), given that χ = 0 on Sˆ. Moreover BH =
BH (g, χ, Lχ g) and is linear with respect to Lχ g, so that BH = 0 on a stationary horizon and
SHWZ reduces then to SW as defined by Eq. (17.33). In the proof of Property 17.25, if one use a
hypersurface Σ with inner boundary S , instead of Sˆ, one obtains readily the first law (17.101)
with δSHWZ substituted for δSW . Given that δE and the δJ(i) ’s are the same (for they depend
only on values on S∞ ), it follows that δSHWZ = δSW for first order vacuum perturbations of a
stationary black hole. For non-vacuum perturbations, represented by an energy-momentum
tensor δT , Hollands, Wald and Zhang [284] have shown that δSHWZ (S2 ) − δSHWZ (S1 ) ≥ 0
694 Black hole thermodynamics beyond general relativity
if δT obeys the null energy condition (2.95) and S2 lies in the future of S1 . This constitutes a
perturbative version of the second law.
For general relativity, the HWZ entropy does not reduce to the Bekenstein-Hawking entropy
SBH [Eq. (16.73)], except at the bifurcation surface Sˆ of the unperturbed spacetime, since its
writes Z
kB
SHWZ = SBH + vθ(ℓ) dS (general relativity), (17.106)
4ℏ S
where v is an affine parameter along the null geodesic generators of H such that v = 0 at
Sˆ, ℓ := dx/dv is the associated tangent vector (ℓ is thus a null normal to H ), and θ(ℓ) is the
expansion of H along ℓ. (cf. Property 2.9). Note that the product vθ(ℓ) is independent of the
choice of the affine parameter v. Indeed, for an affine parameter v ′ := αv, one has ℓ′ = α−1 ℓ
and θ(ℓ′ ) = α−1 θ(ℓ) (Property 2.10 with α ↔ α−1 ), so that v ′ θ(ℓ′ ) = vθ(ℓ) . We refer to [506, 498]
for extended discussions about the HWZ entropy, especially in connection with the extension
of the generalized second law (GSL, cf. Property 16.15) beyond general relativity.
Other proposals for the entropy of dynamical black holes in diffeomorphism-invariant
theories of gravity have been put forward in the literature, notably by Dong [171] and Wall
[511]. As for HWZ, they regard perturbations of a stationary black hole (see Ref. [284, 498] for
a comparison with HWZ entropy). Recently, a non-perturbative approach has been introduced
by Davies & Reall [158], leading to a second law for effective field theories.
Chapter 18
The quasi-local approach: trapping
horizons
Contents
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695
18.2 Trapped surfaces and singularity theorems . . . . . . . . . . . . . . . 695
18.3 Trapping horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707
18.1 Introduction
This chapter is in a draft stage.
18.2 Trapped surfaces and singularity theorems
18.2.1 Trapped surfaces
The concept of trapped surfaces has been introduced in Sec. 3.2.3 (see in particular Fig. 3.1).
Let us recall that a submanifold S of a n-dimensional spacetime (M , g) (n ≥ 3) is a trapped
surface iff (i) S is a compact (n − 2)-dimensional manifold (without boundary), (ii) S is
spacelike (positive definite induced metric) and the two systems of null geodesics emerging
orthogonally from S towards the future locally converge, i.e. they have negative expansions
at S : θ(ℓ) < 0 and θ(k) < 0 [Eq. (3.5)], ℓ and k being any future-directed vectors tangent to
these geodesics.
Remark 1: Trapped surfaces are sometimes called closed trapped surfaces (e.g. [405, 266]), to stress their
closed manifold aspect (compact without boundary). We follow here the textbooks [371, 499] and call
them merely trapped surfaces.
Example 1 (trapped surfaces in the Schwarzschild spacetime): Let (M , g) be the Schwarzschild
spacetime as defined in Sec. 6.3.6 [Eq. (6.54)]; it is entirely covered by the ingoing Eddington-Finkelstein
696 The quasi-local approach: trapping horizons
Figure 18.1: Intersection S (red curve) of the past light cones of two points p and q in Minkowski spacetime.
Some light rays emerging from S in the two orthogonal null directions are depicted; both sets of light rays are
converging. Yet S is not trapped because it is not compact. [Figure generated by the notebook D.6.7]
coordinates (t̃, r, θ, φ). Let S be any surface (t̃, r) = const. S is diffeomorphic to S2 and thus compact.
Moreover, from Eq. (6.33), the metric induced by g on S is q = r2 (dθ2 + sin2 θ dφ2 ), which is clearly
positive definite; hence S is spacelike. One also reads on the expression of q that the area of S is
simply A = 4πr2 (i.e. r is the areal radius, cf. Sec. 6.2.2). If S is located in the black hole interior, i.e. if
0 < r < 2m, Property 6.9 implies that A is decreasing along any future-directed null geodesic. It follows
that S is trapped. On the contrary, if located in the black hole exterior (r > 2m), S is untrapped. This
can be checked on Fig. 6.3, where it appears clearly that, in the region r > 2m, r is increasing along the
outgoing radial null geodesics, which are normal to S .
Example 2 (trapped surfaces in the Oppenheimer-Snyder collapse): Spherically symmetric trapped
surfaces in the Oppenheimer-Snyder collapse have been investigated in Sec. 14.3.6 (in particular, see
Fig. 14.8): in terms of the FLRW coordinates (η, χ, θ, φ) spanning the collapsing star [cf. Eq. (14.45)],
they are all the spheres (η, χ) = const such that η > π − 2χ (Property 14.7). Outside the collapsing star,
i.e. in the Schwarzschild region [cf. Eq. (14.28)], they are all the spheres (t̃, r) = const with r < 2m (cf.
Example 1).
Example 3 (trapped surfaces in the Vaidya collapse): Spherically symmetric trapped surfaces in the
Vaidya radiation shell collapse have been investigated in Sec. 15.3.5: in terms of the IEF coordinates
(t, r, θ, φ) [cf. Eq. (15.11)], they are the spheres (t, r) = const such that r < 2M (t + r) (Property 15.10).
The region containing them is depicted as red dot-filled in the Carter-Penrose diagrams of Figs. 15.7,
15.11 and 15.12.
The existence of a trapped surface characterizes strong gravity. More precisely it indicates
that gravity is strong enough to focus all light rays emitted from the surface. This characteriza-
tion is quasi-local, in the sense that it involves a surface. The surface can be “small” but it cannot
be reduced to a point, to which the qualifier local would apply. Note that the hypothesis of
compact manifold is crucial in the definition of a trapped surface. Without it, trapped surfaces
would exist in Minkowski spacetime, as the following example shows.
18.2 Trapped surfaces and singularity theorems 697
Example 4 (a counter-example in Minkowski spacetime): Let us consider the intersection S of
the past light cones Cp and Cq of two points p and q in Minkowski spacetime (cf. Fig. 18.1). Being a
cross-section of the null hypersurface Cp (or Cq ), S is a spacelike surface (Property 2.2). Moreover the
null directions k and ℓ normal to it are given by the null generators of Cp (since S is a cross-section of
Cp ) and of Cq (since S is a cross-section of Cq ). Because Cp and Cq are past light cones, one has clearly
θ(ℓ) < 0 and θ(k) < 0 (cf. Fig. 18.1). However, S fails to be a trapped surface for it is not compact.
Truncating the light cones would not help, because this would make S a manifold with boundary and
hence not a closed one.
It is worth noticing that some (peculiar) black hole spacetimes do not contain any trapped
surface in the strict sense (θ(k) < 0 and θ(ℓ) < 0), but only marginally trapped surfaces (θ(k) < 0
and θ(ℓ) = 0). This is demonstrated by the following example:
Example 5 (no trapped surface in the extremal Reissner-Nordström spacetime): The extremal
Reissner-Nordström black hole is defined by setting Q = 4π/µ0 M and P = 0 in Eq. (5.107).
p
Performing the change of notation M → m, this yields the metric
m 2 2 m −2 2
dr + r2 dθ2 + sin2 θ dφ2 . (18.1)
g =− 1− dt + 1 −
r r
The coordinates (t, r, θ, φ) are singular at r = m. One can introduce Eddington-Finkelstein-like
coordinates (t̃, r, θ, φ) via
r m
t̃ := t + 2m ln −1 + (18.2)
m 1 − r/m
(compare with Eq. (6.32) in the Schwarzschild case) to get a regular coordinate system in the whole
range r ∈ (0, +∞). The hypersurface H defined by r = m appears then as a black hole event horizon.
H is a degenerate Killing horizon with respect to the Killing vector ξ = ∂t = ∂t̃ (cf. Sec. 3.3.6) (Exercise:
prove it!). This is similar to the event horizon of the extremal Kerr spacetime (cf. Chap. 13), which is a
degenerate Killing horizon as well (Property 13.3). From the metric (18.1), it appears that the vector
field ∂t is timelike in both regions r < m (black hole interior) and r > m (black hole exterior), since
gtt < 0 there. Moreover, for the standard time orientation given by ∂t̃ , ∂t is future-directed in these
two regions. Let us consider a 2-sphere S defined by (t, r) = const with r ̸= m. The two families of
null geodesics L± normal to S are ruled by the equations (θ, φ) = const and
dr m 2
=± 1− . (18.3)
dt L± r
This last equation is deduced from Eq. (18.1) by setting ds2 := g(dx, dx) = 0 with dx = dt ∂t + dr ∂r
along L± . One reads on the metric (18.1) that the area of S is A = 4πr2 , as in the Schwarzschild
case (Example 1). Now, since ∂t is future-directed timelike, it appears from Eq. (18.3) that r increases
(resp. decreases) towards to the future along L+ (resp. L− ), in both regions r < m and r > m. It
follows immediately that S is not trapped. Actually, there exists only marginally trapped surfaces in
this spacetime: the cross-sections of the event horizon H .
18.2.2 Penrose’s singularity theorem
Trapped surfaces are the key ingredient of Penrose’s singularity theorem, which basically states
that, under some rather generic assumptions, if gravity is strong enough so that a trapped
698 The quasi-local approach: trapping horizons
surface occurs, then some singularity will appear in the future of it. Before stating the theorem,
let us recall a few concepts on which it relies.
First of all, a Cauchy surface Σ of a spacetime (M , g) is a spacelike hypersurface such
that every inextendible timelike curve of M intersects Σ exactly once (cf. Sec. 5.2.3). If (M , g)
admits a Cauchy surface, it is said to be a globally hyperbolic spacetime. On such a spacetime,
general relativity can be formulated as a well posed Cauchy problem [111], i.e. there exists a
unique solution g of the Einstein equation in M from initial data prescribed on Σ, provided
that the initial data fulfill four components of the Einstein equation known as the constraint
equations (see e.g. Ref. [227]).
The second concept involved in Penrose’s theorem is that of an inextendible incomplete
geodesic. As defined in Appendix B (Sec. B.3.2), a geodesic L of a spacetime (M , g) is
incomplete if some affine parameter λ does not span the whole of R along L . Given that
any two affine parameters are related by an affine transformation, if this holds for a given
affine parameter, this holds for all. More precisely, if L is a causal (i.e. timelike or null)
geodesic and λ is increasing to the future, one says that L is future-incomplete if, and only
if, λ < λmax along L , for some λmax ∈ R. Similarly, one says that L is past-incomplete
if, and only if, λ > λmin along L , for some λmin ∈ R. Furthermore, a geodesic L is said
inextendible if, and only if, there does not exist any geodesic L ′ of (M , g) distinct from L
and such that L ⊂ L ′ . An inextendible incomplete causal geodesic marks the existence of a
singularity of some kind. For instance a free-falling observer having an inextendible future-
incomplete timelike geodesic as worldline has his proper time1 abruptly stopping at some finite
value! Actually, the existence of an inextendible incomplete geodesic is the nowadays adopted
definition of a spacetime singularity. We shall discuss the connection with the concept
of curvature singularity introduced in Sec. 6.3.4 later on. For the moment, we have enough
material to state the famous theorem:
Property 18.1: Penrose’s singularity theorem (Penrose 1965 [405])
Let (M , g) be a n-dimensional time-orientable spacetime (n ≥ 3) such that
• (M , g) admits a non-compact Cauchy surface Σ;
• the null convergence condition (2.94) holds on M : for any null vector ℓ, R(ℓ, ℓ) ≥ 0,
where R is g’s Ricci tensor — for general relativity, such a condition is equivalent to
the null energy condition (2.95): T (ℓ, ℓ) ≥ 0;
• there exists a trapped surface S .
Then at least one inextendible null geodesic emerging orthogonally from S is future-
incomplete.
Proof. We shall prove the theorem by contradiction. Hence let us assume that all inextendible
null geodesics emerging orthogonally from S are future-complete. Let F be the boundary
1
Recall that the proper time is an affine parameter along a timelike geodesic (Property B.7).
18.2 Trapped surfaces and singularity theorems 699
Figure 18.2: Spacetime diagram representing various objects involved in the proof of Penrose’s singularity
theorem (Property 18.1): the trapped surface S (the two red dots), the two null directions ℓ and k normal to S ,
the causal future of S , J + (S ) (pale green region), the boundary F of J + (S ) (dark green curve), the compact
space V containing F (light green curve) and the image Φ(F ) of F (orange segment) into the Cauchy surface Σ
(blue line) by the topological embedding Φ constructed from the field lines of a global timelike vector ξ (maroon
lines). The two dark green dots are crossover points, where null geodesics leave F to the interior of J + (S ).
The compact (n − 2)-dimensional surface S appears disconnected and reduced to two points because this 2D
figure is the cut by a vertical plane Π of the 3D view of Fig. 3.1, where S is drawn as an horizontal circle, whose
intersection with Π is two points.
of the causal future of S (cf. Sec. 4.4.1 and Fig. 18.2): F := ∂J + (S ). Since (M , g) is
globally hyperbolic (it admits Σ as a Cauchy surface), it is necessarily causally simple (cf.
Proposition 6.6.1 of HE [266]). By the very definition of causally simple (cf. Sec. 4.4.1) and the
compacity of S , it follows that F coincides with the future horismos of S : F = E + (S ).
This implies that points of F are connected to S by null geodesics. More precisely, we note
that F is an achronal boundary, namely the boundary of the causal future or past of a given
set. As mentioned in Sec. 4.4.3 (Remark 6), the properties of achronal boundaries are the same
as those established for a black hole event horizon H = ∂J − (I + ) ∩ M (Properties 4.5 to
4.11), provided one changes future to past when appropriate. It follows that F is an achronal
(n − 1)-dimensional topological manifold (Properties 4.5 and 4.7) that is ruled by null geodesics,
called the generators of F and obeying the following properties: when followed in the past
direction, a generator never leaves F until it encounters S and there is a unique generator
through each point of F , except at special points, named crossovers, where the generators
leave F in the future direction, i.e. are future-extended to null geodesics of M that do not
belong2 to F (Property 4.8). Note that two crossovers are shown in Fig. 18.2.
We actually need an additional property of F , which has not been established in Sec. 4.4.3:
on S (which is part of F , cf. Sec. 4.4.1), the generators of F are orthogonal to S (cf. e.g.
Theorem 9.3.11 in Wald’s textbook [499]). They therefore start from S along the two null
2
They actually belong to the interior of J + (S ), i.e. to the chronological future I + (S ) [Eq. (4.33)] (see e.g.
Proposition 4.5.14 in HE [266]).
700 The quasi-local approach: trapping horizons
directions ℓ and k normal to S (cf. Fig. 18.2). Let us denote by L a generic null generator
along ℓ and by λ the affine parameter of L such that, on S , λ = 0 and dx/dλ|L = ℓ. We
shall use the latter relation to extend the definition of ℓ along L away from S . Similarly, let
us denote by L¯ a generic null generator along k and by λ̄ the affine parameter of L¯ such that
λ̄ = 0 on S and dx/dλ̄ L¯ = k. By hypothesis, L and L¯ are future-complete. Since S is a
trapped surface, θ0 := θ(ℓ) S < 0 and θ̄0 := θ(k) S < 0. Then, the null Raychaudhuri equation
(2.88) applied to the null hypersurface generated by the null geodesics of the L family and the
null convergence condition R(ℓ, ℓ) ≥ 0 implies that there exists λ∗ ∈ (0, (n − 2)/|θ0 |] such
that θ(ℓ) → −∞ for λ → λ∗ along L . The reasoning is exactly the same at that used in the
proof of Property 16.8 and we shall not repeat it here. The point λ = λ∗ on L is a caustic point,
where all nearby geodesics starting from S along ℓ converge. Given the properties of F , we
conclude that for λ > λ∗ , L ceases to a be null generator of F , i.e. Lλ>λ∗ ̸∈ F . Similarly,
along L¯, θ(k) → −∞ for λ̄ → λ̄∗ with λ̄∗ ∈ (0, (n − 2)/|θ̄0 |], so that L¯λ̄>λ̄∗ ̸∈ F . Let
n−2 n−2
λmax := sup and λ̄max := sup .
p∈S |θ(ℓ) (p)| p∈S |θ̄(k) (p)|
Since S is compact, λmax and λ̄max are finite. Let us then consider the map f : S ×[0, λmax ] →
M , (p, λ) 7→ q, where q is the point of affine parameter λ on the null geodesic of the L -family
through p. Since all the L geodesics are assumed to be future-complete, f is well defined.
Moreover, f is clearly continuous. Then, given that S ×[0, λmax ] is a compact set, the image set
f (S × [0, λmax ]) is necessarily compact. Similarly, the image set f¯(S × [0, λ̄max ]) is compact,
where f¯ : S × [0, λ̄max ] → M , (p, λ) 7→ q̄ — the point of affine parameter λ̄ on the null
geodesic of the L¯-family through p. The set
V := f (S × [0, λmax ]) ∪ f¯ S × [0, λ̄max ]
is then a compact subset of M , as the union of two such sets. By construction, for each p ∈ S ,
the point q = f (p, λmax ) (resp. q̄ = f (p, λmax )) lies beyond the caustic point on the geodesic
L (resp. L¯) through p. It follows that F ⊂ V (cf. Fig. 18.2). Given that F is closed, being a
topological boundary (F := ∂J + (S )), and V is compact, we conclude that F is compact.
Because (M , g) is time-orientable, there exists a nonvanishing timelike vector field ξ on
M . The field lines of ξ form a congruence of timelike curves C , each of them intersecting Σ in
a single point, for Σ is a Cauchy surface. We may then define a map Φ : F → Σ, p 7→ Cp ∩ Σ,
where Cp is the unique timelike curve of the C congruence through p (cf. Fig. 18.2). The
map Φ is clearly continuous. Moreover, it is injective given that F is achronal (if a point
q ∈ Σ was the image of two distinct points p and p′ of F , this would mean that p and p′ are
connected by a timelike curve, namely a segment of Cp ). Now, there cannot be any injective
continuous map from a compact manifold (without boundary) (here F ) into a non-compact
manifold of the same dimension (here Σ). For instance, there is no injective continuous map
from the sphere S2 into the plane R2 . To show this impossibility in the present case, and
hence to reach the sought contradiction, let us invoke a standard topology theorem (see e.g.
Lemma 4.50 in Lee’s textbook [342]), according to which an injective continuous map from a
compact space to a Hausdorff space is a topological embedding, i.e. a homeomorphism onto its
image. This theorem applies here since Σ is Hausdorff, being a manifold (cf. Sec. A.2.1). Hence
18.2 Trapped surfaces and singularity theorems 701
Φ(F ) is homeomorphic to F . Φ(F ) is therefore a compact subset of Σ. Another standard
result of topology (see e.g. Proposition 4.36b in Lee’s textbook [342]) states that a compact
subset of a Hausdorff space is closed. Hence Φ(F ) is a closed subset of Σ. On the other side,
since F is a (n − 1)-dimensional topological manifold, each point of F has a neighborhood
homeomorphic to an open subset of Rn−1 . The same feature holds then for Φ(F ). Since Σ is a
(n − 1)-dimensional manifold, it follows that Φ(F ) is an open subset of Σ. Hence Φ(F ) is
both open and closed in Σ. Given that Σ is connected, being a Cauchy surface in a connected
spacetime, this implies that Φ(F ) = Σ. Here we reach a contradiction because Σ is assumed
non-compact.
It is worth to stress that Penrose’s singularity theorem is not telling that a curvature
singularity, as described in Sec. 6.3.4, must exist in spacetime (or at some “boundary” of it, since
curvature singularities are usually excluded from the spacetime manifold). The theorem only
stipulates that some inextendible null geodesic is future-incomplete. However, a good reason
for a null geodesic to be future-incomplete is to hit a curvature singularity as the examples
below are going to illustrate.
Another cause of incompleteness could be that the geodesic, L say, has reached some
regular “external boundary” H of the spacetime (M , g) and both M and g can be smoothly
extended beyond this boundary, i.e. there exists a spacetime (M˜, g̃) such that M ⊂ M˜ and
g̃|M = g. The boundary H , which is a part of M˜, not of M (for M is a manifold without
boundary), is then the future Cauchy horizon of the hypersurface Σ in M˜ (cf. Sec. 10.8.3). Note
that since Σ is a Cauchy surface of (M , g), Σ has no Cauchy horizon within M . With respect
to (M˜, g̃), Σ is only a partial Cauchy surface (cf. Sec. 10.8.3 and Fig. 10.12). The geodesic L ,
which was inextendible within M , can then be extended beyond H in a possibly complete
geodesic of (M˜, g̃) and there might be no curvature singularity at all in the spacetime (M˜, g̃).
Example 6 (Schwarzschild spacetime): Let us consider the maximally extended Schwarzschild
spacetime (M , g) discussed in Chap. 9. The hypersurface Σ defined by T = 0 in terms of the Kruskal-
Szekeres coordinates (T, X, θ, φ) is a non-compact Cauchy surface of (M , g) (cf. Fig. 9.8 or Fig. 9.11).
The null convergence condition is trivially fulfilled since the Ricci tensor R is identically zero, g being
a solution of the vacuum Einstein equation (1.44). As detailled in Example 1 (p. 695), any sphere S
defined by (t̃, r) = const is a trapped surface in the region r < 2m of the ingoing Eddington-Finkelstein
patch, i.e. the region MII . All the hypotheses of Penrose’s singularity theorem are then fulfilled, so
there must exist an incomplete null geodesic emerging from S . Actually, all null geodesics emerging
orthogonally from S are incomplete, for they all reach the curvature singularity r = 0 within a finite
affine parameter, as it appears clearly in Fig. 9.7 (remember that r is an affine parameter along these
geodesics, cf. Property 6.3).
Example 7 (Oppenheimer-Snyder collapse): The gravitational collapse of a homogeneous pressure-
less ball, as described by the Oppenheimer-Snyder model discussed in Sec. 14.3, provides an example of a
dynamical spacetime illustrating Penrose’s theorem. The hypersurface Σ defined by t̃ = 0 in terms of the
global IEF coordinates (t̃, r, θ, φ) is a non-compact Cauchy surface of the underlying spacetime (M , g)
(cf. the Carter-Penrose diagram of Fig. 14.4). Since g is a solution of the Einstein equation, the null
convergence condition reduces to the null energy condition: T (ℓ, ℓ) ≥ 0 for any null vector ℓ. Given the
form (14.2) of the energy-momentum tensor T , we get, inside the collapsing star, T (ℓ, ℓ) = ρ(u · ℓ)2 > 0
since ρ > 0, while T (ℓ, ℓ) = 0 outside the star. Hence the null convergence condition is fulfilled. Fi-
nally, as detailed in Example 2 (p. 696), this spacetime contains trapped surfaces. All the hypotheses of
702 The quasi-local approach: trapping horizons
Penrose’s singularity theorem are thus satisfied. Let us check its predictions on the two families L and
L¯ of null geodesics emanating orthogonally from a trapped sphere S that is defined in terms of the
conformal coordinates (η, χ, θ, φ) by (η, χ) = (η0 , χ0 ) = const, with π − η0 < χ0 < χs − (π − η0 ).
This constraint implies that both L and L¯ stay in the interior of the collapsing star until they reach the
curvature singularity η = π (cf. the conformal diagram in Fig. 14.2 left). Let us check that this happens
for a finite value of their affine parameters. The equation of these geodesics is easily deduced from the
conformal form (14.45) of the metric tensor: we get η − η0 = ±(χ − χ0 ) with ± = + for L and ± = −
for L¯. Moreover, η is an affine parameter along ¯
both L and L with respect to the conformal metric
h := −dη + dχ + sin χ dθ + sin θ dφ . According to Eq. (4.13) any affine parameter λ of L or
2 2 2 2 2 2
L¯ with respect to the physical metric g is related to the h-affine parameter η by dλ/dη = aΩ2 , where
a is a positive constant and the conformal factor Ω is read on Eq. (14.45): Ω2 = (a0 /2)2 (1 + cos η)2 .
The integration leads to
λ = α (6η + 8 sin η + sin 2η) + β,
where α and β are two constants such that α > 0. We have then limη→π λ = 6απ + β, which is
obviously finite. We conclude that both L and L¯ are future-incomplete.
Example 8 (Vaidya collapse): Let us consider the collapse of a radiation shell as studied in Sec. 15.3,
assuming that the shell is thin (large radiation density): α := 2m/v0 > 1/8 [cf. Eqs. (15.18) and (15.23)].
As it can be inferred from the Carter-Penrose diagram of Fig. 15.7, a hypersurface t = const in the black
hole exterior is a non-compact Cauchy surface (for instance the hypersurface t = −3m in the α = 2/3
model considered in Fig. 15.1). Besides, we have already noticed in Sec. 15.2.1 that the null energy
condition is fulfilled, for M (v) is an increasing function (cf. Fig. 15.2). Finally, as detailed in Example 3
(p. 696), trapped surfaces forms in the late stages of the collapse. Penrose’s singularity theorem thus
applies and there should be an incomplete inextendible null geodesic emerging from each trapped sphere
S described in Example 3. It is easy to show that this is indeed the case for all ingoing null geodesics3
L¯ = L(v,θ,φ)
in , i.e. geodesics emerging from S along the vector k normal to the null hypersurfaces
v = const (cf. Property 15.1). As it is clear on Fig. 15.7, all the geodesics L(v,θ,φ) in terminate at the
curvature singularity r = 0, v > 0. Since −r is an affine parameter along them (Property 15.4), we
conclude that they are future-incomplete.
Example 9 (Bardeen regular black hole as a counter-example): There exist solutions of the Einstein
equation (with some rather exotic matter sources) corresponding to a black hole without any curvature
singularity. They are called regular black holes (see e.g. the recent book [31]). A well known example
(historically the first one) is the Bardeen black hole [35] (see also Sec. V of Ref. [69]). The underlying
spacetime (M , g) is static and spherically symmetric; in some part of it, the metric is expressed in terms
of the Schwarzschild-Droste-like coordinates (t, r, θ, φ) by
2M (r) −1 2
2M (r) 2
dr + r2 dθ2 + sin2 θ dφ2 , (18.4)
g =− 1− dt + 1 −
r r
where
r3
M (r) := m , (18.5)
(r2 + ℓ2 )3/2
√
m and ℓ being two positive constants obeying ℓ < 4m/(3 3). Would M (r) be a constant function, one
would get the Schwarzschild metric (6.14). Actually the Schwarzschild metric is recovered asymptotically,
3
All outgoing null geodesics from S are also incomplete, but this is harder to show for there is no simple
expression of an affine parameter along them.
18.2 Trapped surfaces and singularity theorems 703
since M (r) → m for r → +∞. For small values of r, one has instead
m 3
M (r) ∼ r . (18.6)
r→0 ℓ3
This implies that g is regular at r = 0, contrary to the Schwarzschild metric. More precisely, setting
Λ := 6m/ℓ3 , we get
Λ 2 −1 2
Λ 2 2
dr + r2 dθ2 + sin2 θ dφ2 . (18.7)
g ∼ − 1− r dt + 1 − r
r→0 3 3
We recognize in the right-hand side the metric of de Sitter spacetime of cosmological constant Λ
expressed in the so-called static coordinates (see e.g. Sec. 4.3 of Ref. [243]). On the other side, the metric
components (18.4) are singular at values of r for which 2M (r) = r. This equation admits two solutions:
r = r± with 0 < r− < r+ . By introducing coordinates (t̃, r, θ, φ) of Eddington-Finkelstein type, one
can show that g is regular at r = r± . The hypersurfaces r = r± turn out to be two Killing horizons; the
outermost one (r = r+ ) is the black hole event horizon, while the innermost one is a Cauchy horizon in
the maximally extended spacetime (see Fig. 1 of Ref. [15]). The Bardeen metric g is a solution of the
Einstein equation (1.40) sourced by a magnetic monopole in some nonlinear electrodynamics [29]. By
nonlinear electrodynamics, it is meant a theory involving an electromagnetic 2-form F and for which
the Lagrangian density L is a nonlinear function of the electromagnetic invariant F := Fµν F µν /4, the
standard Maxwell theory corresponding to the linear case: L = −F/µ0 . Specifically, for the Bardeen
black hole, the nonlinear electrodynamics is defined by [29]
√ !5/2
3m 2ℓ2 F
L=− √ (18.8)
µ0 |ℓ|ℓ2 1 + 2ℓ2 F
and the electromagnetic field is
µ0
F = ℓ sin θ dθ ∧ dφ, (18.9)
4π
which corresponds to a magnetic monopole ℓ located at the center of symmetry (cf. Remark 2 on p. 174).
The energy-momentum tensor4 T derived from the Lagrangian density (18.8) fulfills the weak energy
condition (2.96), which implies the null energy condition and thus the null convergence condition.
Moreover, the Bardeen spacetime contains trapped surfaces in the region r− < r < r+ . Yet all
inextendible null geodesics are complete. The Bardeen spacetime evades Penrose’s singularity theorem
because it does not admit any non-compact Cauchy surface (actually no Cauchy surface at all). More
precisely, in the internal region it admits a partial Cauchy surface Σ to which the boundary F of the
causal future of a given trapped surface S can be mapped injectively. However Σ is compact, having
the topology of S3 (see [70] for details), so that one cannot reach the contradiction completing the proof
of Penrose’s theorem.
18.2.3 Hawking & Penrose’s singularity theorem
Among the three hypotheses of Penrose’s singularity theorem (Property 18.1), the first one,
namely there exists a non-compact Cauchy surface, is an assumption about the global structure
4
For a Lagrangian density L = L(F), the energy-momentum tensor is Tαβ = −L′ (F)Fµα F µβ + L(F) gαβ ,
which generalizes the Maxwellian expression (1.52) to nonlinear electrodynamics.
704 The quasi-local approach: trapping horizons
of spacetime. It cannot be checked by any local physical experiment. As the Bardeen black hole
shows (Example 9), one cannot get rid of this hypothesis without replacing it by other ones. It
turns out that it is possible find more local hypotheses: a stronger convergence condition and
the non-existence of closed timelike curve. More precisely, we have:
Property 18.2: Hawking-Penrose theorem (1970 [267])
Let (M , g) be a n-dimensional time-orientable spacetime (n ≥ 3) such that
1. M does not contain any closed timelike curve;
2. the causal convergence condition holds on M :
R(v, v) ≥ 0 for any timelike or null vector v , (18.10)
where R is g’s Ricci tensor — for general relativity with a cosmological constant Λ,
such a condition is equivalent to the strong energy condition:
1 Λ
T (v, v) − T− v · v ≥ 0 for any timelike or null vector v, (18.11)
n−2 4π
where T is the matter energy-momentum tensor and T := g µν Tµν is its trace;
3. the following generic condition holds: every causal geodesic contains at least one
point at which the (nonzero) tangent vectors v fulfill
β]
v [α R µ ν
µν[γ vδ] v v ̸= 0, (18.12)
where Rαβγδ is g’s Riemann curvature tensor;
4. M contains at least one of the following:
(a) a trapped surface,
(b) a point p such that the expansion of all the future-directed null geodesics, or
of all the past-directed null geodesics, emanating from p becomes negative at
some point along each geodesic,
(c) a spacelike compact hypersurface without boundary (i.e. an embedded compact
manifold of dimension n − 1).
Then M contains at least one incomplete inextendible causal geodesic.
The proof is quite long and we refer to HE [266], p. 267, where it is given for n = 4, the
generalization to any dimension being straightforward (see also Sec. 5 of Ref. [451] and Sec. 5.1.4
of Ref. [454]). Here, we shall limit ourselves to a few comments:
• The non-existence of closed timelike curves is a much weaker condition than the existence
18.2 Trapped surfaces and singularity theorems 705
of a Cauchy surface required in Penrose’s theorem (Property 18.1), the latter condition
implying the former one.
• On the other side, the causal convergence condition (18.10) is stronger than the null
convergence condition (2.94) involved in Penrose’s theorem, for it has to hold for timelike
vectors, in addition to null ones.
• For general relativity, the causal convergence condition (18.10) is equivalent to the strong
energy condition (18.11) thanks to the Einstein equation (1.42). For a perfect fluid of
proper energy density ρ and pressure p, the energy-momentum is T = (ρ + p)u ⊗ u + pg
and the strong energy condition (18.11) is equivalent to
Λ
(n − 3)ρ + (n − 1)p −≥ 0. (18.13)
4π
Note that, even for Λ = 0, the strong energy condition does not imply the weak energy
condition introduced in Sec. 2.4.2 [Eq. (2.96)], contrary to what the names may suggest.
The only logical connection between the two conditions is that both imply the null energy
condition (2.95).
• For timelike geodesics, the generic condition (18.12) is equivalent to5
Rαµνβ v µ v ν ̸= 0. (18.14)
In view of the geodesic deviation equation (B.49), this implies that the relative acceleration
in any timelike geodesic congruence cannot vanish identically along the geodesics. In
physical terms, this means that tidal forces are necessarily felt sometime along the
worldline of any inertial observer. Apart from flat (Minkowski) spacetime, this condition
is violated only in peculiar cases, involving a high degree of symmetry, hence the qualifier
generic. An example is the Einstein static universe considered in Sec. 4.2.3 [Eq. (4.18)]: it
can be shown [451] that the curves (χ, θ, φ) = const are timelike geodesics that have
Rαµνβ v µ v ν = 0 all along them.
• Hypothesis 4 of the Hawking-Penrose theorem is more general than the requirement of
a trapped surface in Penrose’s theorem (Property 18.1), since the trapped surface can be
replaced by a light cone that reconverges in the future (case (b)) or a spacelike compact
hypersurface (case (c)). The latter possibility arises in spatially closed cosmological
models, such as the de Sitter spacetime (cf. Example 3 on p. 103), which admits spacelike
hypersurfaces of S3 topology. This does not occur in Minkowski spacetime: embedded
hypersurfaces with S3 topology are of course possible (the unit sphere of R4 !), but they
cannot be everywhere spacelike.
• The Hawking-Penrose theorem does not specify whether the incomplete geodesic is
timelike or null and whether it is future-incomplete or past-incomplete; it is therefore
less precise than Penrose’s singularity theorem, which tells that the incomplete geodesic
is null and future-incomplete. It can be argued however that in the case (4a) (trapped
surface), the geodesic is future-incomplete (cf. Ref. [451], p. 792).
5
To see it, it suffices to contract (18.12) by vα v δ and use the symmetries of the Riemann tensor (cf. Sec. A.5).
706 The quasi-local approach: trapping horizons
Remark 2: Hypothesis (4c) (existence of a spacelike compact hypersurface) is that stated in the original
article by Hawking & Penrose [267]. It can be generalized to the existence of an achronal compact set
without edge (cf. Theorem 2 on p. 266 of HE [266] and Theorem 5.6 of Ref. [451]).
Example 10 (strong energy condition): Let us check that the strong energy condition (18.11) holds for
the classical spacetimes involved in Examples 6 (Schwarzschild), 7 (Oppenheimer-Snyder) and 8 (Vaidya).
For the Schwarzschild black hole, this is trivial since T = 0 and Λ = 0. For the Oppenheimer-Snyder
collapse, still Λ = 0 and we note that the matter is a perfect fluid with ρ ≥ 0 and p = 0, so that the fluid
strong energy condition (18.13) is fulfilled. For the Vaidya collapse, the energy-momentum tensor T is
given by Eq. (15.5), which yields T (v, v) = M ′ (v)(k · v)2 /(4πr2 ) and T = 0. Given that M ′ (v) ≥ 0
and Λ = 0, it is clear that (18.11) is satisfied.
Example 11 (Bardeen regular black hole as a counter-example): We have seen in Example 9
that the Bardeen black hole, which has no singularity and is geodesically complete, evades Penrose’s
theorem 18.1 because it has no Cauchy surface. It must evade the Hawking-Penrose theorem 18.2 as
well. Hypothesis 1 of that theorem is fulfilled: the Bardeen spacetime does not contain any closed
timelike curve. Hypotheses 3 and 4a are fulfilled as well. The only reason for which the Hawking-
Penrose theorem does not apply is actually hypothesis 2: the strong energy condition is violated by the
nonlinear-electrodynamics energy-momentum tensor sourcing the Bardeen solution (albeit the weak
energy condition is not).
Remark 3: The scope of the Hawking-Penrose theorem goes beyond gravitational collapse and black
hole physics. It encompasses cosmology and can be used to predict the Big-Bang singularity, via
the hypothesis (4b) with the point p being our current position in the Universe and considering the
past-directed null goedesics (our past light-cone) (see Sec. 10.1 of HE [266] for details).
Remark 4: Both Penrose and Hawking-Penrose theorems have been generalized to n-dimensional
spacetimes containing a “trapped” submanifold of any codimension (the codimension is 2 in Penrose’s
theorem 18.1 and 2, n and 1 for respectively cases (4a), (4b) and (4c) of Hawking-Penrose theorem 18.2)
by Gregory Galloway and José Senovilla in 2010 [211]. In this work, the concept of trapped surface, a
priori defined for codimension 2 only (cf. Sec. 18.2.1), is generalized to any codimension by demanding
that the mean curvature vector of the submanifold is everywhere future-directed timelike; moreover,
the null and causal convergence conditions are generalized to inequalities involving the Riemann tensor.
Historical note : The singularity theorem established by Roger Penrose in 1965 [405] (Property 18.1)
was a major breakthrough in general relativity and mathematical physics. It introduced a brand new
concept, that of trapped surfaces, which revealed to be extremely fruitful in relativistic gravity. It also
brought in the characterization of spacetime singularities by geodesic incompleteness. This novel view
point on singularities was subsequently advocated by Robert Geroch in 1968 [216] and Stephen Hawking
and George Ellis in their 1973 textbook [266]; it is nowadays the standard one. From an astrophysical
perspective, Penrose’s theorem showed that once a gravitational collapse has reached a certain stage,
characterized by the appearance of trapped surfaces, then the formation of a singularity is inevitable,
whatever the geometry of the collapse and the stiffness of the matter’s equation of state may be (provided
the very mild null energy condition is obeyed). Previously, it could have been argued that the central
singularity obtained in exact solutions, like that of the Oppenheimer-Snyder collapse (Chap. 14), is an
artifact of spherical symmetry or the assumption of zero pressure, so that e.g. rotation or high pressures
could prevent the singularity from appearing. For this achievement, Roger Penrose was awarded the
Nobel Prize in Physics in 2020. More details about the history of Penrose’s theorem and its impact can be
18.3 Trapping horizons 707
found in the review articles [452, 453, 454, 335]. As for the theorem proven in 1970 by Stephen Hawking
and Roger Penrose [267] (Property 18.2), it is still considered today as the best singularity theorem.
18.3 Trapping horizons
708 The quasi-local approach: trapping horizons
Part V
Appendices
Appendix A
Basic differential geometry
Contents
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
A.2 Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 711
A.3 Pseudo-Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . 721
A.4 The three basic derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 726
A.5 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734
A.1 Introduction
We recall in this appendix basic definitions and results of differential geometry that are used in
the main text. The reader who has some knowledge of general relativity should be familiar
with most of them. It should be clear that this appendix is not intended to be a monograph
on differential geometry. In particular, contrary to the other parts of these notes, we state
many results without proofs, referring the reader to classical textbooks on the topic [331, 343,
344, 390, 59, 110, 189], as well as to the differential geometry sections of the general relativity
textbooks [108, 464, 499].
A.2 Differentiable manifolds
A.2.1 Notion of manifold
Given an integer n ≥ 1, a manifold of dimension n is a topological space M obeying the
following properties:
1. M is a separated space (also called Hausdorff space): any two distinct points of M
admit disjoint open neighborhoods.
712 Basic differential geometry
Figure A.1: Locally a manifold resembles Rn (n = 2 on the figure), but not necessarily at the global level.
2. M has a countable base 1 : there exists a countable family (Uk )k∈N of open sets of M
such that any open set of M can be written as the union (possibly infinite) of some
members of the above family.
3. Around each point of M , there exists a neighborhood which is homeomorphic to an
open subset of Rn .
Property 1 excludes manifolds with “forks” and is very reasonable from a physical point of
view: it allows to distinguish between two points even after a small perturbation. Property 2
excludes “too large” manifolds; in particular it permits setting up the theory of integration on
manifolds. It also allows for a smooth manifold of dimension n to be embedded smoothly into
the Euclidean space R2n (Whitney theorem). Property 3 expresses the essence of a manifold:
it means that, locally, one can label the points of M in a continuous way by n real numbers
(xα )α∈{0,...,n−1} , which are called coordinates (cf. Fig. A.1). More precisely, given an open
subset U ⊂ M , a coordinate system or chart on U is a homeomorphism2
Φ : U ⊂ M −→ V ⊂ Rn
(A.1)
p 7−→ (x0 , . . . , xn−1 ),
where V is an open subset of Rn .
Remark 1: In relativity, it is customary to label the n coordinates by an index ranging from 0 to n − 1.
Actually, this convention is mostly used when M is the spacetime manifold (n = 4 in standard general
relativity). The computer-oriented reader will have noticed the similarity with the index ranging of
arrays in the C/C++ or Python programming languages.
Remark 2: Strictly speaking the definition given above is that of a topological manifold. We are
saying manifold for short.
1
In the language of topology, one says that M is a second-countable space.
2
Let us recall that a homeomorphism between two topological spaces (here U and V ) is a bijective map Φ
such that both Φ and Φ−1 are continuous.
A.2 Differentiable manifolds 713
Usually, one needs more than one coordinate system to cover M . An atlas on M is a set
of pairs (Ui , Φi )i∈I , where I is a set (not necessarily finite), Ui an open set of M and Φi a chart
on Ui , such that the union of all Ui covers M :
[
Ui = M . (A.2)
i∈I
The above definition of a manifold lies at the topological level (cf. Remark 2), meaning that
one has the notion of continuity, but not of differentiability. To get the latter, one should rely
on the smooth structure of Rn , via the atlases: a smooth manifold, is a manifold M equipped
with an atlas (Ui , Φi )i∈I such that for any non-empty intersection Ui ∩ Uj , the map
j : Φj (Ui ∩ Uj ) ⊂ R −→ Φi (Ui ∩ Uj ) ⊂ R
Φi ◦ Φ−1 (A.3)
n n
is smooth (i.e. C ∞ ). Note that the above map is from an open set of Rn to an open set of Rn , so
that the invoked C ∞ differentiability is nothing but that of Rn . Such a map is called a change
of coordinates or, in the mathematical literature, a transition map. The atlas (Ui , Φi )i∈I is
called a smooth atlas. In the following, we consider only smooth manifolds.
Remark 3: Strictly speaking a smooth manifold is a pair (M , A) where A is a (maximal) smooth atlas
on M . Indeed, a given (topological) manifold M can have non-equivalent differentiable structures, as
shown by Milnor (1956) [366] for S7 , the unit sphere of dimension 7: there exist smooth manifolds, the
so-called exotic spheres, that are homeomorphic to S7 but not diffeomorphic to S7 . On the other side,
for n ≤ 6, there is a unique smooth structure for the sphere Sn . Moreover, any manifold of dimension
n ≤ 3 admits a unique smooth structure. Amazingly, in the case of Rn , there exists a unique smooth
structure (the standard one) for any n ̸= 4, but for n = 4 (the standard spacetime case!) there exist
uncountably many non-equivalent smooth structures, the so-called exotic R4 [474].
Remark 4: When discussing the no-hair theorem in Chap. 5, one refers to the concept of analytic
manifold, which is a special case of smooth manifold. Indeed, an analytic manifold is defined as a
manifold equipped with an atlas for which all the changes of coordinates Φi ◦ Φ−1 j are real analytic
functions. Let us recall that an analytic function is a C ∞ function f for which the Taylor series about
any point x in its domain converges to f in some neighborhood of x.
Given two smooth manifolds, M and M ′ , of respective dimensions n and n′ , a map
ϕ : M → M ′ is called a smooth map iff for every p ∈ M , there exist a chart (U , Φ) around
p in the smooth atlas of M and a chart (U ′ , Φ′ ) around ϕ(p) in the smooth atlas of M ′ such
′
that ϕ(U ) ⊂ U ′ and Φ′ ◦ ϕ ◦ Φ−1 is a smooth map Φ(U ) ⊂ Rn → Rn . The map ϕ is said to
be a diffeomorphism iff it is bijective and both ϕ and ϕ−1 are smooth. This implies n = n′ .
A.2.2 Manifold with boundary
A (topological) manifold with boundary M is defined in the same way as a topological
manifold, except that condition 3 in the definition given at the beginning of Sec. A.2.1 is replaced
by
714 Basic differential geometry
3’. Around each point of M , there exists a neighborhood which is homeomorphic to an
open subset3 of the closed half-space
Hn := (x1 , . . . , xn ) ∈ Rn , xn ≥ 0 . (A.4)
A point p ∈ M is said to be a boundary point of M iff there exists a homeomorphism Φ from
an open neighborhood of p to an open subset of Hn such that Φ(p) ∈ ∂Hn , where
∂Hn := (x1 , . . . , xn ) ∈ Rn , xn = 0 . (A.5)
This definition is independent from the choice of Φ (cf. Theorem 1.37 of Ref. [343]). The set of
all boundary points of M is naturally called the boundary of M and is denoted by ∂M .
Remark 5: The boundary ∂M should not be confused with the topological boundary of M , i.e. the
boundary of M as a topological space, which is the closure of M minus the interior of M ; since both
sets coincide with M , the topological boundary of M is obviously ∅.
A smooth manifold with boundary is a manifold with boundary endowed with a smooth
atlas, with the understanding that a transition map
j : Φj (Ui ∩ Uj ) ⊂ H −→ Φi (Ui ∩ Uj ) ⊂ H
Φi ◦ Φ−1 n n
is said to be smooth iff it can be extended around each point of its domain (including the points
of ∂Hn ) into a smooth map from an open subset of Rn to Rn .
A.2.3 Curves and vectors on a manifold
On a smooth manifold, vectors are defined as tangent vectors to a curve. Given an interval
I ⊂ R, a curve is a subset L ⊂ M that is the image of a smooth map I → M :
P : I −→ M
(A.6)
λ 7−→ p = P (λ) ∈ L .
Hence L = P (I) := {P (λ)| λ ∈ I}. The function P is called a parametrization of L
and the real variable λ is called a parameter along L . Given a coordinate system (xα ) in a
neighborhood of a point p ∈ L , the parametrization P is defined by n functions X α : I → R
such that
xα (P (λ)) = X α (λ). (A.7)
Remark 6: In the literature, a curve is often defined as a map P : I → M and not as the image of P .
According to that definition, different parametrizations give birth to different curves.
A scalar field on M is a function f : M → R. Unless specified, we shall always consider
smooth scalar fields, i.e. smooth maps as defined in Sec. A.2.1. At a point p = P (λ) ∈ L , the
3
By open subset of Hn , it is meant a set A ⊂ Hn that is open with respect to the topology of Hn ; A is then not
necessarily open when considered as a subset of Rn (for instance A = Hn ).
A.2 Differentiable manifolds 715
tangent vector to L associated with the parametrization P is the operator v which maps
every scalar field f defined around p to the real number
df 1
v(f ) := = lim [f (P (λ + ε)) − f (P (λ))] . (A.8)
dλ L
ε→0 ε
Given a coordinate system (xα ) around some point p ∈ M , there are n curves Lα through
p associated with (xα ) and called the coordinate lines: for each α ∈ {0, . . . , n − 1}, Lα is
defined as the curve through p parameterized by λ = xα and having constant coordinates xβ
for all β ̸= α. The tangent vector to Lα parameterized by xα is denoted ∂α . From the definition
(A.8), its action on a scalar field f is
df df
∂α (f ) = = .
dxα Lα dxα xβ =const
β̸=α
Considering f as a function of the coordinates (x0 , . . . , xn−1 ) (whereas strictly speaking it is
a function of the points on M ) we recognize in the last term the partial derivative of f with
respect to xα . Hence
∂f
∂α (f ) = . (A.9)
∂xα
Similarly, we may rewrite (A.8) as
1 ∂f dX α
f (X 0 (λ + ε), . . . , X n−1 (λ + ε)) − f (X 0 (λ), . . . , X n−1 (λ)) =
v(f ) = lim ,
ε→0 ε ∂xα dλ
i.e., in view of Eq. (A.9)
v(f ) = v α ∂α (f ) , (A.10)
where
dX α
v α := , 0 ≤ α ≤ n − 1. (A.11)
dλ
In Eq. (A.10), we are using the Einstein summation convention: a repeated index implies
a summation over all the values taken by this index (here, from α = 0 to α = n − 1). The
identity (A.10) being valid for any scalar field f , we conclude that
v = v α ∂α . (A.12)
Since every tangent vector to a curve at p is expressible as (A.12), we conclude that the set of
all tangent vectors to a curve at p is a vector space of dimension n and that (∂α ) constitutes a
basis of it. This vector space is called the tangent space to M at p and is denoted Tp M . The
elements of Tp M are simply called vectors at p. The basis (∂α ) is called the natural basis
associated with the coordinates (xα ) and the coefficients v α in (A.12) are called the components
of the vector v with respect to the coordinates (xα ). The tangent vector space is represented
at two different points in Fig. A.2.
Contrary to what happens for an affine space, one cannot define a vector connecting two
distinct points p and q on a generic manifold, except if p and q are infinitesimally close to each
716 Basic differential geometry
Figure A.2: The vectors at two points p and q on the manifold M belong to two different vector spaces: the
tangent spaces Tp M and Tq M .
other. Indeed, in the latter case, one defines the infinitesimal displacement vector from p to
q as the vector dx ∈ Tp M whose action on a scalar field f is
dx(f ) := df |p→q = f (q) − f (p). (A.13)
Since p and q are infinitesimally close, there is a unique (piece of) curve L going from p to q
and one has
dx = v dλ , (A.14)
where λ is a parameter along L , v the associated tangent vector at p and dλ the parameter
increment from p to q: p = P (λ) and q = P (λ + dλ). The relation (A.14) follows immediately
from the definitions (A.8) and (A.13) for respectively v and dx. Given a coordinate system,
let (xα ) be the coordinates of p and (xα + dxα ) those of q. Then from Eqs. (A.13) and (A.9),
dx(f ) = df = ∂f /∂xα dxα = dxα ∂α (f ). The scalar field f being arbitrary, we conclude that
dx = dxα ∂α . (A.15)
In other words, the components of the infinitesimal displacement vector dx with respect to
the coordinates (xα ) are nothing but the infinitesimal coordinate increments dxα .
A.2 Differentiable manifolds 717
A.2.4 Linear forms
A fundamental operation on vectors consists in mapping them to real numbers, and this in a
linear way. More precisely, at each point p ∈ M , one defines a linear form as a mapping4
ω : Tp M −→ R
(A.16)
v 7−→ ⟨ω, v⟩
that is linear: ⟨ω, λv + u⟩ = λ⟨ω, v⟩ + ⟨ω, u⟩ for all u, v ∈ Tp M and λ ∈ R. The set of all
linear forms at p constitutes a n-dimensional vector space, which is called the dual space
of Tp M and denoted by Tp∗ M . Given the natural basis (∂α ) of Tp M associated with some
coordinates (xα ), there is a unique basis of Tp∗ M , denoted by (dxα ), such that
⟨dxα , ∂β ⟩ = δ αβ , (A.17)
where δ αβ is the Kronecker symbol : δ αβ = 1 if α = β and 0 otherwise. The basis (dxα ) is
called the dual basis of the basis (∂α ). The notation dxα stems from the fact that if we apply
the linear form dxα to the infinitesimal displacement vector (A.15), we get nothing but the
number dxα :
⟨dxα , dx⟩ = ⟨dxα , dxβ ∂β ⟩ = dxβ ⟨dxα , ∂β ⟩ = dxα . (A.18)
| {z }
δα β
Remark 7: Do not confuse the linear form dxα with the infinitesimal increment dxα of the coordinate
xα .
The dual basis can be used to expand any linear form ω, thereby defining its components
ωα with respect to the coordinates (xα ):
ω = ωα dxα . (A.19)
In terms of components, the action of a linear form on a vector takes then a very simple form:
⟨ω, v⟩ = ωα v α . (A.20)
This follows immediately from (A.19), (A.12) and (A.17).
A field of linear forms, i.e. a (smooth) map which associates to each point p ∈ M an element
of the dual space Tp∗ M is called a 1-form. Given a smooth scalar field f on M , there exists
a 1-form canonically associated with it, called the differential of f and denoted df or ∇f .
At each point p ∈ M , df is the unique linear form which, once applied to the infinitesimal
displacement vector dx from p to a nearby point q, gives the change in f between points p and
q:
df := f (q) − f (p) = ⟨df, dx⟩. (A.21)
4
We are using the same bra-ket notation as in quantum mechanics to denote the action of a linear form on a
vector.
718 Basic differential geometry
Since df = ∂f /∂xα dxα , Eq. (A.18) implies that the components of the differential with respect
to the dual basis are nothing but the partial derivatives of f with respect to the coordinates
(xα ) :
∂f
df = α
dxα . (A.22)
∂x
Remark 8: In non-relativistic physics, the concept of gradient of a scalar field is commonly used instead
of the differential, the former being a vector field and the latter a 1-form. This is so because one associates
implicitly a vector →−ω to any 1-form ω via the Euclidean scalar product of R3 : ∀→ −
v ∈ R3 , ⟨ω, →v⟩=→
− −
ω ·→
−v.
→
−
Accordingly, formula (A.21) is rewritten as df = ∇f · dx. But we should keep in mind that, at the
fundamental level, the key quantity is the differential 1-form ∇f = df , since Eq. (A.21) does not require
→
−
any metric on the manifold M to be meaningful. On the contrary, the gradient ∇f is a derived quantity,
obtained from the differential ∇f by metric duality.
Remark 9: For a fixed value of α, the coordinate xα can be considered as a scalar field on M . If we
apply (A.22) to f = xα , we then get dxα = dxα . Hence the dual basis to the natural basis (∂α ) is
formed by the differentials of the coordinates. This justifies the notation dxα used for its elements.
By combining Eqs. (A.22), (A.12) and (A.17), it is easy to see that the 1-form df acting on a
vector v is nothing but v acting on the scalar field f :
⟨df, v⟩ = v(f ) . (A.23)
Natural bases are of course not the only bases in the vector space Tp M . One may use a
basis (eα ) that is not related to any coordinate system on M , for instance an orthonormal
basis with respect to some metric. There exists then a unique basis (eα ) of the dual space Tp∗ M
such that5
⟨eα , eβ ⟩ = δ αβ . (A.24)
(eα ) is called the dual basis to (eα ). The relation (A.17) is a special case of (A.24), for which
eα = ∂α and eα = dxα .
A.2.5 Tensors
Tensors are generalizations of both vectors and linear forms. Let (k, ℓ) ∈ N2 with (k, ℓ) ̸= (0, 0).
At a point p ∈ M , a tensor of type (k, ℓ), also called tensor k times contravariant and ℓ
times covariant, is a map
T : Tp∗ M × · · · × Tp∗ M × Tp M × · · · × Tp M −→ R
(A.25)
| {z } | {z }
k times ℓ times
(ω1 , . . . , ωk , v1 , . . . , vℓ ) 7−→ T (ω1 , . . . , ωk , v1 , . . . , vℓ )
that is linear with respect to each of its arguments. The integer k +ℓ is called the tensor valence,
or sometimes the tensor rank or order. Let us recall the canonical duality Tp∗∗ M = Tp M ,
5
Notice that, according to the standard usage, the symbol for the vector eα and that for the linear form eα
differ only by the position of the index α.
A.2 Differentiable manifolds 719
which means that every vector v can be considered as a linear form on the space Tp∗ M , via
Tp∗ M → R, ω 7→ ⟨ω, v⟩. Accordingly a vector is a tensor of type (1, 0). A linear form is a
tensor of type (0, 1). A tensor of type (0, 2) is called a bilinear form. It maps pairs of vectors
to real numbers, in a linear way for each vector.
Given a basis (eα ) of Tp M and the corresponding dual basis (eα ) in Tp∗ M , any tensor T
of type (k, ℓ) can be expanded as
T = T α1 ...αk β1 ...βℓ eα1 ⊗ . . . ⊗ eαk ⊗ eβ1 ⊗ . . . ⊗ eβℓ , (A.26)
where the tensor product eα1 ⊗ . . . ⊗ eαk ⊗ eβ1 ⊗ . . . ⊗ eβℓ is the tensor of type (k, ℓ) for
which the image of (ω1 , . . . , ωk , v1 , . . . , vℓ ) as in (A.25) is the real number
k
Y ℓ
Y
⟨ωi , eαi ⟩ × ⟨eβj , vj ⟩.
i=1 j=1
Notice that all the products in the above formula are simply products in R. The nk+ℓ real
coefficients T α1 ...αk β1 ...βℓ in (A.26) are called the components of the tensor T with respect to
the basis (eα ). These components are unique and fully characterize the tensor T .
Remark 10: The notations v α and ωα already introduced for the components of a vector v [Eq. (A.12)]
or a linear form ω [Eq. (A.19)] are of course the particular cases (k, ℓ) = (1, 0) or (k, ℓ) = (0, 1) of
(A.26), with, in addition, eα = ∂α and eα = dxα .
A.2.6 Tensor fields on a manifold
A tensor field of type (k, ℓ) is a map T which associates to each point p ∈ M a tensor of type
(k, ℓ) on Tp M . A vector field is naturally a tensor field of type (1, 0). By convention, a scalar
field is considered as a tensor field of type (0, 0). We shall consider only smooth fields. We
shall denote by T |p the tensor representing the value of the tensor field T at a point p ∈ M .
Given a non-negative integer p, a differential form of degree p, or p-form, is a tensor
field of type (0, p), i.e. a field of p-linear forms, that is fully antisymmetric whenever p ≥ 2.
This definition generalizes that of a 1-form given in Sec. A.2.4.
A frame field or moving frame is a n-tuple of vector fields (eα ) such that at each point
p ∈ M , (eα |p ) is a basis of the tangent space Tp M . If n = 4, a frame field is also called a
tetrad and if n = 3, it is called a triad.
Given a vector field v and a scalar field f , the function M → R, p 7→ v|p (f ) clearly defines
a scalar field on M , which we denote naturally by v(f ). We may then define the commutator
of two vector fields u and v as the vector field [u, v] whose action on a scalar field f is
[u, v](f ) := u(v(f )) − v(u(f )). (A.27)
With respect to a coordinate system (xα ), it is not difficult, via (A.12), to see that the components
of the commutator are
∂v α ∂uα
[u, v]α = uµ µ − v µ µ . (A.28)
∂x ∂x
720 Basic differential geometry
A.2.7 Immersions, embeddings and submanifolds
Let M and N be two smooth manifolds and Φ : M −→ N be a smooth map, as defined in
Sec. A.2.1. At a given point p ∈ M , the differential of Φ is the linear map
dΦ|p : Tp M −→ TΦ(p) N (A.29)
that “approximates” Φ in the following sense: if dx ∈ Tp M is the infinitesimal displacement
vector from p to some (infinitesimally close) point q (cf. Sec. A.2.3), then
dΦ|p (dx) = dy, (A.30)
where dy is the infinitesimal displacement vector of TΦ(p) N connecting Φ(p) to Φ(q) (cf.
Fig. ??). If (xα ) is a coordinate chart of M around p and (y β ) a chart of N around Φ(p), such
that Φ is expressed as y β = Y β (xα ), it follows from Eqs. (A.15) and (A.30) that the matrix
of the linear map dΦ|p with respect to the bases (∂α ) of Tp M and (∂β ) of TΦ(p) N is the
Jacobian matrix (∂Y β /∂xα ). Using the characterization of vectors by their action on scalar
fields [Eq. (A.8)], it is then easy to see that
∀v ∈ Tp M , ∀f ∈ C ∞ (N , R), dΦ|p (v)(f ) = v (f ◦ Φ) . (A.31)
This property could be taken as an alternative definition of dΦ|p .
The smooth map Φ is called an immersion iff the differential dΦ|p is injective at any
point p ∈ M . Moreover, Φ is called an embedding iff (i) Φ is an immersion and (ii) Φ is a
homeomorphism M → Φ(M ). Note that an embedding is necessarily injective, contrary to
an immersion.
A submanifold of M is a subset S ⊂ M such that (i) S is a manifold in the subspace
topology and (ii) S has a smooth structure with respect to which the inclusion map ι : S → M
is an embedding. One can show that S is a submanifold of M iff there exists a manifold S0 (a
priori not a subset of M ) and an embedding Φ : S0 → M , such that S = Φ(S0 ).
Remark 11: Strictly speaking, the above definition regards an embedded submanifold; there is also
the wider concept of immersed submanifold (see e.g. Chap 5 of [343]).
One has necessarily dim S ≤ dim M . The non-negative integer m = dim M − dim S is
called the codimension of the submanifold S . A submanifold of codimension 1 is called a
hypersurface. A submanifold of dimension 1 is (the image of) a curve in M , but note that not
all curves are submanifolds: a curve with self-crossing points is not a submanifold.
A.2.8 Pushforwards and pullbacks
Given a smooth map Φ : M → N and a vector v in the tangent space to M at a point p, the
image of v by the differential of Φ at p is called the pushforward of v on N by Φ and is
denoted by Φ∗ v:
Φ∗ v := dΦ|p (v). (A.32)
The pushforward can be geometrically interpreted as follows: consider v as the tangent vector
to a curve L through p associated to some parameter λ (cf. Sec. A.2.3) and let q be the point
A.3 Pseudo-Riemannian manifolds 721
of L separated from p by the infinitesimal parameter increase dλ. One has then v = dx/dλ,
where dx is the infinitesimal vector connecting p to q. Thanks to the linearity of dΦ|p and
the defining relation (A.30), one has Φ∗ v = dy/dλ, i.e. Φ∗ v is the finite vector obtained by
rescaling the infinitesimal vector connecting Φ(p) to Φ(q) by dλ. Note that Φ∗ v is a vector
tangent to the curve Φ(L ) ⊂ N : it is precisely the tangent vector to Φ(L ) at Φ(p) associated
with the parametrization of Φ(L ) by λ.
Let now T be a fully covariant tensor field on N , i.e. a tensor field of type (0, ℓ) on N for
some ℓ ≥ 0. The pullback of T on M by Φ is the tensor field of type (0, ℓ) on M denoted by
Φ∗ T and defined by
∀p ∈ M , ∀(v1 , . . . vℓ ) ∈ (Tp M )ℓ , Φ∗ T |p (v1 , . . . vℓ ) := T |Φ(p) (Φ∗ v1 , . . . , Φ∗ vℓ ) . (A.33)
For a scalar field f : N → R, the above definition with ℓ = 0 reduces to Φ∗ f |p := f |Φ(p) , i.e.
to Φ∗ f (p) := f (Φ(p)), so that the pullback is nothing but the map composition: Φ∗ f = f ◦ Φ.
Remark 12: Via the pushforward operation, a smooth map Φ : M → N naturally carries tangent
vectors from M to N (essentially because it carries curves in M to curves in N , as discussed above),
but it carries covariant tensors, among which linear forms, in the reverse way, i.e. from N to M
(pullback operation). Another difference is that the pushforward is a pointwise operation and does not
extend a priori to vector fields (cf. Fig. ??), while the pullback is well defined for any covariant tensor
field. A case where the pushforward extends to vector fields though is when Φ is a diffeomorphism.
A.3 Pseudo-Riemannian manifolds
A.3.1 Metric tensor
A pseudo-Riemannian manifold is a pair (M , g) where M is a smooth manifold and g is a
metric tensor on M , i.e. a tensor field obeying the following properties:
1. g is a tensor field of type (0, 2): at each point p ∈ M , g|p is a bilinear form acting on
pairs of vectors in the tangent space Tp M :
g|p : Tp M × Tp M −→ R
(A.34)
(u, v) 7−→ g|p (u, v) =: g(u, v)
2. g is symmetric: g(u, v) = g(v, u).
3. g is non-degenerate: at any point p ∈ M , a vector u such that ∀v ∈ Tp M , g(u, v) = 0
is necessarily the null vector.
The properties of being symmetric and non-degenerate are typical of a scalar product. Ac-
cordingly, one says that two vectors u and v are g-orthogonal (or simply orthogonal if there
is no ambiguity) iff g(u, v) = 0. Moreover, when there is no ambiguity on the metric (usually
the spacetime metric), we are using a dot to denote the scalar product of two vectors taken
with g:
∀(u, v) ∈ Tp M × Tp M , u · v := g(u, v) . (A.35)
722 Basic differential geometry
In a given basis (eα ) of Tp M , the components of g is the matrix (gαβ ) defined by formula
(A.26) with (k, ℓ) = (0, 2):
g = gαβ eα ⊗ eβ . (A.36)
For any pair (u, v) of vectors we have then g(u, v) = gαβ uα v β . In particular, considering the
natural basis associated with some coordinate system (xα ), the scalar square of an infinitesimal
displacement vector dx = dxα ∂α [cf. Eqs. (A.13) and (A.15)] is
ds2 := g(dx, dx) = gαβ dxα dxβ . (A.37)
This formula is called the line element expression on the pseudo-Riemannian manifold (M , g).
It is often used to define the metric tensor in general relativity texts. Note that contrary to
what the notation may suggest, ds2 can be negative.
For the dual basis associated with the coordinates (xα ), one has eα = dxα (cf. Sec. A.2.4),
so that Eq. (A.36) can be rewritten as
g = gαβ dxα ⊗ dxβ . (A.38)
One can recast this relation in a form which reminds the line element (A.37) by introducing
the symmetric product notation (cf. e.g. Refs. [344] or [464]):
1
dxα dxβ := dxα ⊗ dxβ + dxβ ⊗ dxα and (dxα )2 := dxα ⊗ dxα . (A.39)
2
Formula (A.38) then becomes
g = gαβ dxα dxβ . (A.40)
Applying this relation to the pair of infinitesimal vectors (dx, dx), one gets the line element
(A.37), by virtue of the identity ⟨dxα , dx⟩ = dxα [Eq. (A.18)].
A.3.2 Signature and orthonormal bases
An important feature of the metric tensor is its signature: in all bases of Tp M where the
components (gαβ ) form a diagonal matrix, there is necessarily the same number, s say, of
negative components and the same number, n − s, of positive components. The independence
of s from the choice of the basis where (gαβ ) is diagonal is a basic result of linear algebra named
Sylvester’s law of inertia. One writes:
sign g = (−, . . . , −, +, . . . , + ). (A.41)
| {z } | {z }
s times n − s times
If s = 0, g is called a Riemannian metric and (M , g) a Riemannian manifold. In this
case, g is positive-definite, which means that g(v, v) ≥ 0 for all v ∈ Tp M and g(v, v) = 0
iff v = 0. A standard example of Riemannian metric is of course the scalar product of the
Euclidean space Rn .
If s = 1, g is called a Lorentzian metric and (M , g) a Lorentzian manifold. One may
then have g(v, v) < 0; vectors for which this occurs are called timelike, whereas vectors for
A.3 Pseudo-Riemannian manifolds 723
which g(v, v) > 0 are called spacelike, and those for which g(v, v) = 0 are called null. The
subset of Tp M formed by all null vectors is termed the null cone of g at p.
A coordinate xα of a coordinate system (x0 , . . . , xn−1 ) is said to be a timelike coordinate
(resp. spacelike coordinate or null coordinate) iff the hypersurfaces defined by xα = const
are spacelike6 (resp. timelike or null).
Remark 1: Being timelike, spacelike or null is a property of the coordinate xα per se (i.e. considering
xα as a scalar field U ⊂ M → R); on the contrary the causal type of the coordinate vector ∂α depends
on the coordinate system (x0 , . . . , xn−1 ) to which xα belongs. More precisely, whatever the causal type
of xα , the vector ∂α can be made spacelike, timelike or null by a proper choice of the complementary
coordinates (xβ )β̸=α .
A basis (eα ) of Tp M is called a g-orthonormal basis (or simply orthonormal basis if
there is no ambiguity on the metric) iff7
g(eα , eα ) = −1 for 0≤α≤s−1
g(eα , eα ) = 1 for s≤α≤n−1 (A.42)
g(eα , eβ ) = 0 for α ̸= β.
A.3.3 Metric duality
Since the bilinear form g is non-degenerate, its matrix (gαβ ) in any basis (eα ) is invertible and
the inverse is denoted by (g αβ ):
g αµ gµβ = δ αβ . (A.43)
The metric g induces an isomorphism between Tp M (vectors) and Tp∗ M (linear forms) which,
in index notation, corresponds to the lowering or raising of the index by contraction with gαβ
or g αβ . In the present book, an index-free symbol will always denote a tensor with a fixed
covariance type (such as a vector, a 1-form, a bilinear form, etc.). We will therefore use a
different symbol to denote its image under the metric isomorphism. In particular, we denote
by an underbar the isomorphism Tp M → Tp∗ M and by an arrow the reverse isomorphism
Tp∗ M → Tp M :
1. For any vector u in Tp M , u stands for the unique linear form such that
∀v ∈ Tp M , ⟨u, v⟩ = g(u, v). (A.44)
However, we will omit the underbar on the components of u, since the position of the
index allows us to distinguish between vectors and linear forms, following the standard
usage: if the components of u in a given basis (eα ) are denoted by uα , the components
of u in the dual basis (eα ) are then denoted by uα and are given by
uα = gαµ uµ . (A.45)
6
Cf. Sec. 2.2.2 for the definition of spacelike, timelike and null hypersurfaces.
7
No summation on α is implied.
724 Basic differential geometry
2. For any linear form ω in Tp∗ M , →
−
ω stands for the unique vector of Tp M such that
∀v ∈ Tp M , g(→
−
ω , v) = ⟨ω, v⟩. (A.46)
As for the underbar, we will omit the arrow on the components of →
−
ω by denoting them
ω ; they are given by
α
ω α = g αµ ωµ . (A.47)
3. We extend the arrow notation to bilinear forms on Tp M (type-(0, 2) tensor): for any
→
−
bilinear form T , we denote by T the tensor of type (1, 1) such that
→
− →
−
∀(u, v) ∈ Tp M × Tp M , T (u, v) = T (u, v) = u · T (v), (A.48)
↠
and by T the tensor of type (2, 0) defined by
↠
∀(u, v) ∈ Tp M × Tp M , T (u, v) =T (u, v). (A.49)
→
−
Note that in the second equality of (A.48), we have considered T as an endomorphism
Tp M → Tp M , which is always possible for a tensor of type (1, 1). If Tαβ are the
→
− ↠
components of T in some basis (eα ), the components of T and T are respectively
→
−
( T )α β = T αβ = g αµ Tµβ (A.50a)
↠
(T )αβ = T αβ = g αµ g βν Tµν . (A.50b)
Remark 2: In mathematical literature, the isomorphism induced by g between Tp M and Tp∗ M is called
the musical isomorphism, because a flat and a sharp symbols are used instead of, respectively, the
underbar and the arrow introduced above:
u♭ = u and ω♯ = →
−
ω.
A.3.4 Levi-Civita tensor
Let us assume that M is an orientable manifold, i.e. that there exists a n-form8 on M (n
being M ’s dimension) that is continuous on M and nowhere vanishing. Then, given a metric
g on M , one can show that there exist only two n-forms ϵ such that for any g-orthonormal
basis (eα ),
ϵ(e0 , . . . , en−1 ) = ±1. (A.51)
Picking one of these two n-forms amounts to choosing an orientation for M . The chosen ϵ is
then called the Levi-Civita tensor associated with the metric g. It is also called the volume
form of g (cf. Sec. A.4.3). The sign in the right-hand side of (A.51) gives then the orientation of
the basis (eα ). More generally, given a (not necessarily orthonormal) basis (eα ) of Tp M , one
8
Cf. Sec. A.2.6 for the definition of a n-form.
A.3 Pseudo-Riemannian manifolds 725
has ϵ(e0 , . . . , en−1 ) ̸= 0 and one says that the basis is right-handed (resp. left-handed) iff
ϵ(e0 , . . . , en−1 ) > 0 (resp. < 0). The components of ϵ with respect to (eα ) are
(A.52)
p
ϵα1 ... αn = ± |g| [α1 , . . . , αn ] ,
where ± must be + (resp. −) for a right-handed (resp. left-handed) basis, g stands for the
determinant of the matrix of g’s components with respect to the basis (eα ):
g := det(gαβ ) (A.53)
and the symbol [α1 , . . . , αn ] takes the value 0 if any two indices (α1 , . . . , αn ) are equal and
takes the value 1 (resp. −1) if (α1 , . . . , αn ) is an even (resp. odd) permutation of (0, . . . , n − 1).
A.3.5 Vector normal to a hypersurface
In a pseudo-Riemannian manifold, one can associate to any hypersurface Σ (cf. Sec. A.2.7) a
unique normal direction, which can be represented by a nonvanishing vector field n defined
on Σ as follows. Locally the hypersurface Σ can be considered as a level set, i.e. there exists a
smooth scalar field f : M → R, such that df ̸= 0 on Σ and for any point p in the considered
local region of M , the following equivalence holds
p ∈ Σ ⇐⇒ f (p) = 0. (A.54)
Then, a vector field v on M is tangent to Σ iff the value of f stays to 0 for a small displacement
dλ along v; thanks to Eqs. (A.13), (A.14) and (A.12), this is equivalent to v(f ) = v µ ∂f /∂xµ = 0,
→
−
or to g(n, v) = 0, where we have let appear the gradient vector n := ∇f ; in terms of
components with respect to a coordinate system (xα ):
∂f
nα = ∇α f = g αµ . (A.55)
∂xµ
The vector field n is called a normal to Σ. All normal vectors to Σ are collinear to each other.
It follows from the definitions given in Sec. A.3.2 that for any coordinate xα of a given chart
(x , . . . , xn−1 ),
0
→
−
xα timelike coordinate ⇐⇒ ∇xα timelike vector ⇐⇒ g αα < 0, (A.56a)
→
−
xα null coordinate ⇐⇒ ∇xα null vector ⇐⇒ g αα = 0, (A.56b)
→
−
xα spacelike coordinate ⇐⇒ ∇xα spacelike vector ⇐⇒ g αα > 0, (A.56c)
where g αα is the component (α, α) of the inverse metric (no summation on α). It appears here be-
→
− →
−
cause, according to Eq. (A.55), g(∇xα , ∇xα ) = gµν g µρ ∂xα /∂xρ g νσ ∂xα /∂xσ = δ ρν δ αρ g νσ δ ασ =
g αα .
726 Basic differential geometry
A.4 The three basic derivatives
Three derivative operators acting on tensor fields can be defined on a pseudo-Riemannian
manifold (M , g). One of them depends on the metric g: the covariant derivative ∇ (Sec. A.4.1).
Another one depends on the choice of a reference vector field: the Lie derivative L (Sec. A.4.2).
The third one, called the exterior derivative and denoted by d, depends only on the smooth-
manifold structure, i.e. it is independent of any (metric or vector) field; on the other side, it is
applicable only to a specific kind of tensor fields, namely differential forms (Sec. A.4.3).
A.4.1 Covariant derivative
Affine connection on a manifold
Let us denote by X(M ) the space of smooth vector fields on M . X(M ) is an infinite-
dimensional vector space over R. Given a vector field v ∈ X(M ), it is not possible from
the manifold structure alone to define its variation between two neighboring points p and q.
Indeed a formula like dv := v|q − v|p is meaningless because the vectors v|q and v|p belong to
two distinct vector spaces, Tq M and Tp M respectively (cf. Fig. A.2), so that their subtraction
is a priori not defined. Note that this issue does not arise for a scalar field [cf. Eq. (A.21)].
The solution is to introduce an extra-structure on M , called an affine connection. The term
connection arises because, by defining the variation of vector fields, this structure connects the
various tangent spaces on the manifold. More precisely, an affine connection on M is a map
∇ : X(M ) × X(M ) −→ X(M )
(A.57)
(u, v) 7−→ ∇u v
that satisfies the following properties:
1. ∇ is bilinear (considering X(M ) as a vector space over R).
2. For any scalar field f and any pair (u, v) of vector fields:
∇f u v = f ∇u v. (A.58)
3. For any scalar field f and any pair (u, v) of vector fields, the following Leibniz rule holds:
∇u (f v) = ⟨df, u⟩ v + f ∇u v, (A.59)
where df is the differential of f as defined in Sec. A.2.4.
The vector ∇u v is called the covariant derivative of v along u.
Remark 1: Property 2 is not implied by property 1, for f is a scalar field, not a real number. Actually,
property 2 ensures that the value of ∇u v at a given point p ∈ M depends only on the vector
u|p ∈ Tp M and not on the behavior of u around p; therefore the role of u is only to give the direction
of the derivative of v.
A.4 The three basic derivatives 727
Given an affine connection, the variation of a vector field v between two neighboring
points, p and q say, is defined as
dv := ∇dx v, (A.60)
where dx is the infinitesimal displacement vector connecting p and q [cf Eq. (A.13)]. One says
that v is parallelly transported from p to q with respect to the connection ∇ iff dv = 0.
Given a frame field (eα ) on M , the connection coefficients of an affine connection ∇ with
respect to (eα ) are the n3 scalar fields Γγ αβ defined by the expansion, at each point p ∈ M , of
the vector ∇eβ eα p onto the basis (eα |p ):
∇eβ eα =: Γµ αβ eµ . (A.61)
An affine connection is entirely defined by its connection coefficients in a given frame field. In
other words, there are as many affine connections on a manifold of dimension n as there are
possibilities of choosing n3 scalar fields Γγ αβ .
Given v ∈ X(M ), the covariant derivative of v with respect to the affine connection
∇ is the tensor field ∇v of type (1, 1) defined by the following action at each point p ∈ M :
∇v|p : Tp∗ M × Tp M −→ R
(A.62)
(ω, u) 7−→ ⟨ω, ∇ũ v|p ⟩,
where ũ is any vector field which performs some extension of u around p: ũ|p = u. As already
noted (cf. Remark 1), ∇ũ v|p is independent of the choice of ũ, so that the mapping (A.62) is
well-defined. By comparing with (A.25), we verify that ∇v|p is a tensor of type (1, 1).
The covariant derivative is extended to all tensor fields by (i) demanding that for a scalar
field it coincides with the differential: ∇f := df and (ii) using the Leibniz rule. As a result,
the covariant derivative of a tensor field T of type (k, ℓ) is a tensor field ∇T of type (k, ℓ + 1).
Its components with respect a given frame field (eα ) are denoted
∇γ T α1 ...αk β1 ...βℓ := (∇T )α1 ...αk β1 ...βℓ γ (A.63)
and are given by
k i
↓
α1 ... σ...αk
α1 ...αk α1 ...αk
X
αi
∇γ T β1 ...βℓ = eγ (T β1 ...βℓ ) + Γ σγ T β1 ...βℓ
i=1
ℓ
X
− Γσ βi γ T α1 ...αk β1 ... σ ...βℓ , (A.64)
↑
i=1 i
where eγ (T α1 ...αk
β1 ...βℓ ) stands for the action of the vector eγ on the scalar field T
α1 ...αk
β1 ...βℓ
resulting from the very definition of a vector (cf. Sec. A.2.3). In particular, if (eα ) is a natural
frame associated with some coordinate system (xα ), then eα = ∂α and eγ (T α1 ...αk β1 ...βℓ ) =
∂T α1 ...αk β1 ...βℓ /∂xγ [cf. Eq. (A.9)].
Remark 2: Notice the position of the index γ in Eq. (A.63): it is the last one on the right-hand side.
According to (A.26), ∇T is then expressed as
∇T = ∇γ T α1 ...αkβ1 ...βℓ eα1 ⊗ . . . ⊗ eαk ⊗ eβ1 ⊗ . . . ⊗ eβℓ ⊗ eγ . (A.65)
728 Basic differential geometry
Because eγ is the last 1-form of the tensorial product on the right-hand side, the notation T α1 ...αkβ1 ...βℓ ;γ
instead of ∇γ T α1 ...αkβ1 ...βℓ would have been more appropriate. The index convention (A.65) agrees
with that of MTW [371] [cf. their Eq. (10.17)].
The covariant derivative of the tensor field T along a vector v is defined by
∇v T := ∇T (., . . . , . , u). (A.66)
| {z }
k+ℓ slots
The components of ∇v T are then v µ ∇µ T α1 ...αk β1 ...βℓ . Note that ∇v T is a tensor field of the
same type as T . In the particular case of a scalar field f , one has ∇v f = ⟨∇f, v⟩ = v(f ).
The divergence with respect to the affine connection ∇ of a tensor field T of type (k, ℓ)
with k ≥ 1 is the tensor field denoted ∇ · T of type (k − 1, ℓ) and whose components with
respect to any frame field are given by
α1 ...αk−1 α1 ...αk−1 µ
(∇ · T ) β1 ...βℓ = ∇µ T β1 ...βℓ . (A.67)
Remark 3: For the divergence, the contraction is performed on the last upper index of T .
Levi-Civita connection
On a pseudo-Riemannian manifold (M , g) there is a unique affine connection ∇ such that
1. ∇ is torsion-free, i.e. for any scalar field f , ∇∇f is a field of symmetric bilinear forms:
∇α ∇β f = ∇β ∇α f. (A.68)
2. The covariant derivative of the metric tensor vanishes identically:
∇g = 0 . (A.69)
∇ is called the Levi-Civita connection associated with g.
With respect to the Levi-Civita connection, the Levi-Civita tensor ϵ introduced in Sec. A.3.4
shares the same property as g:
∇ϵ = 0 . (A.70)
Given a coordinate system (xα ) on M , the connection coefficients of the Levi-Civita
connection with respect to the natural basis (∂α ) are called the Christoffel symbols of g; they
can be evaluated from the partial derivatives of the metric components with respect to the
coordinates:
1 γµ ∂gµβ ∂gαµ ∂gαβ
γ
Γ αβ = g + − . (A.71)
2 ∂xα ∂xβ ∂xµ
Note that the Christoffel symbols are symmetric with respect to the lower two indices.
A.4 The three basic derivatives 729
For the Levi-Civita connection, the expression for the divergence of a vector takes a rather
simple form in a natural basis. Indeed, combining Eqs. (A.67) and (A.64), we get for v ∈ X(M ),
∇ · v = ∇µ v µ = ∂v µ /∂xµ + Γµ σµ v σ . Now, from (A.71), we have
1 ∂gµν 1 ∂ 1 ∂ p
Γµ αµ = g µν α = ln |g| = p |g|, (A.72)
2 ∂x 2 ∂xα |g| ∂xα
where g := det(gαβ ) [Eq. (A.53)]. The last but one equality follows from the general law of
variation of the determinant of any invertible matrix A:
δ(ln | det A|) = tr(A−1 × δA) , (A.73)
where δ denotes any variation that fulfills the Leibniz rule (i.e. a derivation), tr stands for the
trace and × for the matrix product. We conclude that
1 ∂ p
∇·v = p |g| v µ
. (A.74)
|g| ∂xµ
Similarly, for an antisymmetric tensor field of type (2, 0),
∂Aαµ ∂Aαµ 1 ∂ p
∇µ Aαµ = µ
+ Γ α
σµ Aσµ
+Γµ
σµ A ασ
= µ
+ p σ
|g| Aασ ,
∂x | {z } ∂x |g| ∂x
0
where we have used the fact that Γα σµ is symmetric in (σ, µ), whereas Aσµ is antisymmetric.
Hence the simple formula for the divergence of an antisymmetric tensor field of type (2, 0):
1 ∂ p
∇µ Aαµ = p |g| Aαµ
. (A.75)
|g| ∂xµ
A.4.2 Lie derivative
As discussed in Sec. A.4.1, the concept of a derivative of a vector field on a manifold M requires
the introduction of some extra-structure on M . The extra-structure considered in Sec. A.4.1
is an affine connection, possibly provided by some metric tensor (case of the Levi-Civita
connection). Another possible extra-structure is a “reference” vector field, with respect to
which the derivative is to be defined. This leads to the concept of the Lie derivative, which we
discuss here.
Lie derivative of a vector field
Consider a vector field u on M , which shall serve as our “reference flow”. Let v be another
vector field on M , the variation of which is to be studied. One can use u to transport v
from one point p to a neighboring one q and then define the variation of v as the difference
between the actual value v|q of v at q and the transported value via u. More precisely, given
an infinitesimal parameter ε, we define the flow map along u as the map Φε : M → M
730 Basic differential geometry
up’
up
vq
dl vq
dl ℒu vp
q'= (p')
q= (p)
dvp
*
up up’
dl vp vp
p p'
Figure A.3: Geometrical construction of the Lie derivative of a vector field v along a vector field u: given an
infinitesimal parameter dλ, each extremity of the arrow dλ v|p is dragged by some small parameter ε along u, to
form the vector denoted by Φε∗ dλ v|p (cf. Sec. A.2.8). The latter is then compared with the actual value of dλ v
at the point q, the difference (divided by εdλ) defining the Lie derivative Lu v at p.
such that Φε (p) = q, where q is the point connected to p by the infinitesimal displacement
vector →
−
pq = ε u|p (cf. Sec. A.2.3 and Fig. A.3). The natural reference vector to compare v|q
with is then the pushforward of v|p by Φε , i.e. the vector Φε∗ v|p defined in Sec. A.2.8. Since
v|q = v|Φε (p) and Φε∗ v|p belong to the same vector space Tq M , one may subtract the latter
from the former and define the Lie derivative of v along u at p by
1
Lu v|p := lim v|Φε (p) − Φε∗ v|p . (A.76)
ε→0 ε
Remark 4: The term between parentheses in the right-hand side of Eq. (A.76) is the difference between
two vectors of Tq M and thus a vector of Tq M . One gets a vector of Tp M in the left-hand side only
because q = Φε (p) tends to p in the limit ε → 0. Some authors (e.g. [343, 499]) prefer to define the
Lie derivative from the difference of vectors in Tp M , by carrying v|q to Tp M via the pushforward by
ε = Φ−ε and comparing it with v|p ; this results in the formula
Φ−1
1 −1
Lu v|p := lim (Φε )∗ v|Φε (p) − v|p . (A.77)
ε→0 ε
Thanks to the limit ε → 0, formulas (A.76) and (A.77) are equivalent. Formula (A.76) is preferred here in
so far as it corresponds to the geometrical construction displayed in Fig. A.3.
Let (xα ) be a coordinate system adapted to the field u in the sense that the first vector of
the natural basis (∂α ) associated with (xα ) is ∂0 = u. The coordinate expression of the flow
map Φε is then (xα ) → (xα + εδ α0 ). The corresponding Jacobian matrix is the identity matrix
so that the coordinate components of Φε∗ v|p are identical to those of v|p (cf. Sec. A.2.7). It
A.4 The three basic derivatives 731
follows that the component expression of Eq. (A.76) is
α 1h α α
i
Lu v|p = lim (v|Φε (p) ) − (v|p )
ε→0 ε
1
= lim v α (x0 + ε, x1 , . . . , xn−1 ) − v α (x0 , x1 , . . . , xn−1 ) .
ε→0 ε
Hence, in adapted coordinates, the Lie derivative is simply obtained by taking the partial
derivative of the vector components with respect to x0 :
∂v α
Lu v α = , (A.78)
∂x0
where we have used the standard notation for the components of a Lie derivative: Lu v α :=
(Lu v)α . Besides, using the fact that the components of u are uα = (1, 0, . . . , 0) in the adapted
coordinate system, we notice that the components of the commutator of u and v, as given by
(A.28), are [u, v]α = ∂v α /∂x0 . This is exactly (A.78): [u, v]α = Lu v α . We conclude that the
Lie derivative of a vector with respect to another one is actually nothing but the commutator
of these two vectors:
Lu v = [u, v] . (A.79)
Thanks to formula (A.28), we may then express the components of the Lie derivative in an
arbitrary coordinate system:
∂v α µ ∂u
α
L u v α = uµ − v . (A.80)
∂xµ ∂xµ
In view of the symmetry of the Christoffel symbols, the partial derivatives in Eq. (A.80) can
be replaced by the covariant derivative ∇, yielding
Lu v α = uµ ∇µ v α − v µ ∇µ uα . (A.81)
Generalization to any tensor field
Let T be a tensor field of type (0, ℓ) on M , with ℓ ≥ 0. The Lie derivative of T along u is
then defined by comparing the pullback of T by the flow map Φε at some point p (cf. Sec. A.2.8)
to the actual value of T at the same point:
1 ∗
Lu T := lim (Φε T − T ) . (A.82)
ε→0 ε
For a scalar field f , one has Φ∗ε f = f ◦ Φε (cf. Sec. A.2.8), so that the above definition with
ℓ = 0 yields Lu f = u(f ) = ⟨df, u⟩ = uµ ∂µ f [cf. Eq. (A.8)].
Finally, the Lie derivative is extended to any tensor field by means of the Leibniz rule. As a
result, the Lie derivative Lu T of a tensor field T of type (k, ℓ) is a tensor field of the same
type, the components of which with respect to a given coordinate system (xα ) are
k i
α1 ...αk ∂ α1 ...αk
µ
↓
X α1 ... σ...α k ∂uαi
Lu T β1 ...βℓ = u T β1 ...βℓ − T β1 ...βℓ
∂xµ i=1
∂xσ
ℓ
X ∂uσ
+ T α1 ...αk β1 ... σ ...βℓ . (A.83)
i=1
↑ ∂xβi
i
732 Basic differential geometry
In particular, for a 1-form,
∂ωα ∂uµ
Lu ωα = uµ
+ ωµ . (A.84)
∂xµ ∂xα
In coordinates adapted to the vector field u, we have uα = (1, 0, . . . , 0), so that uµ ∂/∂xµ =
∂/∂x0 and ∂uα /∂xβ = 0. Accordingly, Eq. (A.83) reduces to
∂ α1 ...αk
Lu T α1 ...αk β1 ...βℓ =
T β1 ...βℓ (coordinates adapted to u). (A.85)
∂x0
This generalizes Eq. (A.78) obtained for vector fields.
As for the vector case [Eq. (A.80)], the partial derivatives in Eq. (A.83) can be replaced by
the covariant derivative ∇ (or any other connection without torsion), yielding
k i
↓
α1 ... σ...αk
α1 ...αk α1 ...αk
X
µ
Lu T β1 ...βℓ = u ∇µ T β1 ...βℓ − T β1 ...βℓ ∇σ uαi
i=1
ℓ
X
+ T α1 ...αk β1 ... σ ...βℓ ∇βi uσ . (A.86)
↑
i=1 i
A special case of this formula is worth considering, namely T = g (the metric tensor). Since
∇µ gαβ = 0, and gσβ ∇α uσ = ∇α uβ (both thanks to Eq. (A.69)), one gets the so-called Killing
expression of the Lie derivative of the metric tensor:
Lu gαβ = ∇α uβ + ∇β uα . (A.87)
A.4.3 Exterior derivative
In Sec. A.2.6, we have introduced the differential forms or p-forms as tensor fields of type (0, p),
with p ≥ 0, that are antisymmetric in all their arguments as soon as p ≥ 2. Differential forms
play a special role in the theory of integration on manifolds. Indeed, the primary definition
of an integral over a manifold M of dimension n is the integral of a n-form. The Levi-Civita
tensor ϵ introduced in Sec. A.3.4 is a n-form, whose integral over a n-dimensional submanifold
D ⊂ M gives the volume of D with respect to the metric g. The electromagnetic field is a
2-form (cf. Sec. 1.5.2) and the vorticity of a fluid is described by a 2-form as well in relativistic
hydrodynamics (cf. e.g. Ref. [228]).
Being tensor fields, differential forms are subject to the covariant and Lie derivatives
discussed above. In addition, they are subject to a third kind of derivative defined as follows.
Given any p-form ω, its exterior derivative is the (p + 1)-form dω whose action on a (p + 1)-
tuple of vector fields (v1 , . . . , vp+1 ) is given by
p+1
X
dω(v1 , . . . , vp+1 ) := (−1)i+1 vi (ω(v1 , . . . , vi−1 , vi+1 , . . . , vp+1 ))
i=1
p+1
X
+ (−1)i+j ω ([vi , vj ], v1 , . . . , vi−1 , vi+1 , . . . , vj−1 , vj+1 , . . . , vp+1 ) ,
i,j=1
i<j
(A.88)
A.4 The three basic derivatives 733
where in the first line vi (· · · ) stands for the action of the vector field vi on the scalar field
ω(v1 , . . . , vi−1 , vi+1 , . . . , vp+1 ) [cf. Eq. (A.8)]. For p = 0 (scalar field), Eq. (A.88) with ω → f
reduces to ⟨df, v1 ⟩ := v1 (f ), i.e. one recovers Eq. (A.23). Hence, for a scalar field, the exterior
derivative is nothing but the differential. For p = 1, Eq. (A.88) becomes
dω(v1 , v2 ) := v1 (⟨ω, v2 ⟩) − v2 (⟨ω, v1 ⟩) − ⟨ω, [v1 , v2 ]⟩ . (A.89)
In terms of components with respect to a coordinate system (xα ), the definition (A.88) yields
∂ω
0-form: (dω)α = (A.90a)
∂xα
∂ωβ ∂ωα
1-form: (dω)αβ = − β (A.90b)
∂xα ∂x
∂ωβγ ∂ωαγ ∂ωαβ
2-form: (dω)αβγ = α
− + (A.90c)
∂x ∂xβ ∂xγ
p+1
X ∂
p-form: (dω)α1 ···αp+1 = (−1)i+1 αi ωα1 ···αi−1 αi+1 ···αp+1 . (A.90d)
i=1
∂x
Remark 5: The exterior derivative appeals only to the manifold structure; it does not depend upon the
metric tensor g, nor upon any other extra structure on M . Nevertheless, one may replace all partial
derivatives in formulas (A.90) by covariant derivatives taken with respect to the Levi-Civita connection
∇ of g, thanks to the symmetry of the Christoffel symbols on their last two indices:
0-form: (dω)α = ∇α ω (A.91a)
1-form: (dω)αβ = ∇α ωβ − ∇β ωα (A.91b)
2-form: (dω)αβγ = ∇α ωβγ − ∇β ωαγ + ∇γ ωαβ (A.91c)
p+1
X
p-form: (dω)α1 ···αp+1 = (−1)i+1 ∇αi ωα1 ···αi−1 αi+1 ···αp+1 . (A.91d)
i=1
A fundamental property of the exterior derivative is to be nilpotent:
ddω = 0 . (A.92)
A p-form ω is said to be closed iff dω = 0, and exact iff there exists a (p − 1)-form σ such
that ω = dσ. From property (A.92), any exact p-form is closed. The Poincaré lemma states
that the converse is true, at least locally, on a contractible open set.
Given a vector field v on an oriented pseudo-Riemannian manifold (M , g) of dimension
n, the tensor field formed by setting the first argument of the Levi-Civita tensor ϵ to v, while
leaving free the other slots, i.e. ϵ(v, ., . . . , .), is a (n − 1)-form denoted9 v · ϵ. Its exterior
derivative is thus a n-form, which is expressible in terms of the divergence of v [cf. Eq. (A.74)]
as
d(v · ϵ) = (∇ · v) ϵ . (A.93)
9
More generally, the dot notation stands for the contraction on adjacent indices; here: (v · ϵ)α1 ···αn−1 :=
v ϵµα1 ···αn−1 .
µ
734 Basic differential geometry
The exterior derivative enters in the famous Stokes’ theorem: if D is an oriented d-
dimensional manifold with boundary (cf. Sec. A.2.2), then for any (d − 1)-form ω on D,
Z Z
ω= dω , (A.94)
∂D D
where ∂D is D’s boundary oriented according to the outward convention: if ϵD is the d-form
defining the orientation of D (cf. Sec. A.3.4), the (d − 1)-form defining the orientation of ∂D
is chosen to be v · ϵD , where v is an outward-pointing10 vector field along ∂D. Note that
each side of (A.94) is (of course!) well-defined, as the integral of a p-form over an oriented
p-dimensional manifold.
Remark 6: If D is a submanifold of an oriented pseudo-Riemannian manifold (M , g) of dimension
n, the orientation of D is naturally set by choosing ϵD = ϵ if d = n, where ϵ is the Levi-Civita tensor
of g (cf. Sec. A.3.4). If d < n, a free family of n − d vector fields (u1 , . . . , un−d ) nowhere tangent to
D has to be chosen first and ϵD is then defined by ϵD = ϵ(u1 , . . . , un−d , ., . . . , .). Accordingly, the
orientation of ∂D is set by ϵ∂D = v · ϵD = ϵ(u1 , . . . , un−d , v, ., . . . , .).
Another important formula involving the exterior derivative is the Cartan identity, which
expresses the Lie derivative of a p-form ω along a vector field u as
Lu ω = u · dω + d(u · ω) . (A.95)
As above, a dot denotes the contraction on adjacent indices: u · dω is the p-form dω(u, ., . . . , .)
and u · ω is the (p − 1)-form ω(u, ., . . . , .). Notice that if ω is a 1-form, Eq. (A.95) is readily
obtained by combining Eqs. (A.84), (A.90a) and (A.90b). An immediate consequence of the
Cartan identity and the nilpotence property (A.92) is that the Lie derivative and the exterior
derivative commute, i.e. for any vector field u and any p-form ω:
Lu dω = d Lu ω. (A.96)
A.5 Curvature
A.5.1 General definition
The Riemann curvature tensor of an affine connection ∇ (cf. Sec. A.4.1) is defined by
Riem : X∗ (M ) × X(M )3 −→ C ∞ (M , R)
(A.97)
(ω, w, u, v) 7−→ ω, ∇u ∇v w − ∇v ∇u w − ∇[u,v] w ,
where X∗ (M ) stands for the space of 1-forms on M , X(M ) for the space of vector fields on
M and C ∞ (M , R) for the space of smooth scalar fields on M . The above formula does define
a tensor field on M , i.e. the value of Riem(ω, w, u, v) at a given point p ∈ M depends only
10
By outward-pointing, it is meant that at each point p ∈ ∂D, v|p is not tangent to ∂D and there exists a
curve in D starting from p whose tangent vector at p is − v|p .
A.5 Curvature 735
upon the values of the fields ω, w, u and v at p and not upon their behaviors away from p,
as the covariant derivatives in Eq. (A.97) might suggest. We denote the components of this
tensor in a given basis (eα ) by Rγ δαβ , instead of Riemγ δαβ . The definition (A.97) leads to the
following formula, named the Ricci identity:
∀w ∈ X(M ), (∇α ∇β − ∇β ∇α ) wγ = Rγ µαβ wµ . (A.98)
Remark 1: In view of this identity, one may say that the Riemann tensor measures the lack of commu-
tativity of two successive covariant derivatives of a vector field. On the opposite, for a scalar field and a
torsion-free connection, two successive covariant derivatives always commute [cf. Eq. (A.68)].
In a coordinate basis, the components of the Riemann tensor are given in terms of the
connection coefficients by
∂Γα βν ∂Γα βµ
Rα βµν = − + Γα σµ Γσ βν − Γα σν Γσ βµ . (A.99)
∂xµ ∂xν
From the definition (A.97), the Riemann tensor is clearly antisymmetric with respect to its
last two arguments:
Riem(., ., u, v) = −Riem(., ., v, u). (A.100)
In addition, it satisfies the cyclic property
Riem(., u, v, w) + Riem(., w, u, v) + Riem(., v, w, u) = 0. (A.101)
The covariant derivatives of the Riemann tensor obey the Bianchi identity
∇ρ Rα βµν + ∇µ Rα βνρ + ∇ν Rα βρµ = 0 . (A.102)
A.5.2 Case of a pseudo-Riemannian manifold
The Riemann tensor of the Levi-Civita connection obeys the additional antisymmetry:
Riem(ω, w, ., .) = −Riem(w, →
−
ω , ., .). (A.103)
Combined with (A.100) and (A.101), this implies the symmetry property
Riem(ω, w, u, v) = Riem(u, v, →
−
ω , w). (A.104)
A pseudo-Riemannian manifold (M , g) with a vanishing Riemann tensor is called a flat
manifold; in this case, g is said to be a flat metric. If in addition, g has a Riemannian (resp.
Lorentzian) signature, it is called an Euclidean metric (resp. Minkowski metric).
736 Basic differential geometry
A.5.3 Ricci tensor
The Ricci tensor of the affine connection ∇ is the field of bilinear forms R defined by
R : X(M ) × X(M ) −→ C ∞ (M , R)
(A.105)
(u, v) 7−→ Riem(eµ , u, eµ , v),
where (eα ) is a vector frame on M and (eα ) its dual counterpart. This definition is independent
of the choice of (eα ). In terms of components:
Rαβ := Rµ αµβ . (A.106)
Remark 2: Following standard usage, we denote the components of the Riemann and Ricci tensors by
the same letter R, the number of indices allowing us to distinguish between the two tensors. On the
other hand, we are using different symbols, Riem and R, when employing the index-free notation.
The Ricci tensor naturally appears in the contracted Ricci identity:
∀w ∈ X(M ), ∇µ ∇α wµ − ∇α ∇µ wµ = Rµα wµ , (A.107)
which is obtained by taking the trace of the Ricci identity (A.98) on the indices (α, γ) and
relabelling β → α.
For the Levi-Civita connection associated with the metric g, property (A.104) implies that
the Ricci tensor is symmetric:
R(u, v) = R(v, u). (A.108)
In addition, one defines the Ricci scalar (also called scalar curvature) as the trace of the
Ricci tensor with respect to the metric g:
R := g µν Rµν . (A.109)
The Bianchi identity (A.102) implies the divergence-free property
→
−
∇·G =0, (A.110)
→
−
where G in the type-(1, 1) tensor associated by metric duality [cf. (A.48)] to the Einstein
tensor:
1
G := R − R g . (A.111)
2
Equation (A.110) is called the contracted Bianchi identity.
A.5 Curvature 737
A.5.4 Weyl tensor
Let (M , g) be a pseudo-Riemannian manifold of dimension n.
For n = 1, the Riemann tensor vanishes identically, i.e. (M , g) is necessarily flat. The
reader having in mind a curved line in the Euclidean plane R2 might be surprised by the above
statement. This is because the Riemann tensor represents the intrinsic curvature of a manifold.
For a curved line L , the nonzero curvature is the extrinsic curvature, i.e. the curvature resulting
from the embedding of L in R2 .
For n = 2, the Riemann tensor is entirely determined by the Ricci scalar R, according to
the formula:
R γ
Rγ δαβ = δ α gδβ − δ γ β gδα (A.112)
(n = 2).
2
For n = 3, the Riemann tensor is entirely determined by the Ricci tensor, according to
R γ
Rγ δαβ = Rγ α gδβ − Rγ β gδα + δ γ α Rδβ − δ γ β Rδα + δ β gδα − δ γ α gδβ (n = 3). (A.113)
2
For n ≥ 4, the Riemann tensor can be split into (i) a “trace-trace” part, represented by the
Ricci scalar R [Eq. (A.109)], (ii) a “trace” part, represented by the Ricci tensor R [Eq. (A.106)],
and (iii) a “traceless” part, which is constituted by the Weyl conformal curvature tensor, C:
1
Rγ δαβ = C γδαβ + Rγ α gδβ − Rγ β gδα + δ γ α Rδβ − δ γ β Rδα
n−2
1
R δ γ β gδα − δ γ α gδβ . (A.114)
+
(n − 1)(n − 2)
The above relation may be taken as the definition of C. It implies that C is traceless: C µαµβ = 0.
The other possible traces are zero thanks to the symmetries of the Riemann tensor.
Remark 3: The decomposition (A.114) is meaningful for n = 3 as well; by comparing with (A.113), we
see that it results in a vanishing Weyl tensor: C = 0 for n = 3.
738 Basic differential geometry
Appendix B
Geodesics
Contents
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739
B.2 Definition and first properties . . . . . . . . . . . . . . . . . . . . . . . 739
B.3 Existence and uniqueness of geodesics . . . . . . . . . . . . . . . . . . 744
B.4 Geodesics and extremal lengths . . . . . . . . . . . . . . . . . . . . . . 750
B.5 Geodesics and symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 757
B.6 Geodesics and curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . 759
B.1 Introduction
Geodesics play a key role in general relativity, since they represent the worldlines of test particles
and photons (cf. Sec. 1.3). Moreover, in black hole theory, null geodesics play a prominent role,
as the generators of event horizons (cf. Sec. 4.4.3). We review here the definition and main
properties of geodesics on a generic pseudo-Riemannian manifold, i.e. a manifold equipped
with a metric of generic signature, as introduced in Sec. A.3. In particular, the results apply to
pure Riemannian manifolds (positive definite metric), as well as to Lorentzian manifolds, i.e.
spacetimes. Contrary to Appendix A, proofs of most statements will be provided, since they
are quite illustrative.
B.2 Definition and first properties
B.2.1 Geodesics and affine parametrizations
On a Riemannian manifold, i.e. when the metric tensor is positive definite (cf. Sec. A.3.2), a
geodesic is the curve of minimal length between two points (at least for close enough points).
It is also a curve which is “as straight as possible”, in the sense that its tangent vectors are
transported parallelly to themselves along it. A typical example is a geodesic in the Euclidean
740 Geodesics
space: this is nothing but a straight line, for which tangent vectors keep obviously a fixed
direction. In a pseudo-Riemannian manifold, such as the spacetimes of general relativity, one
uses this last property to define geodesics.
Let us first recall the basic definitions given in Sec. A.2.3: a curve 1 is the image L = P (I)
of a map (called a parametrization of the curve) P : I → M , λ 7→ P (λ), where I is an
interval of R and the variable λ is called a parameter of the curve. Moreover, we exclude the
case where L is reduced to a single point of M , i.e. where P is a constant function. We are
now in position to define a geodesic as a “straight” curve:
A smooth curve L of a pseudo-Riemannian manifold (M , g) is called a geodesic iff it
admits a parametrization P whose associated tangent vector field v is transported parallelly
to itself along L :
∇v v = 0 , (B.1)
where ∇ is the Levi-Civita connection of the metric g. The parametrization P is then
called an affine parametrization and the corresponding parameter λ is called an affine
parameter of the geodesic L . Note that the relation between v and λ is
dx
v= , (B.2)
dλ
where dx is the infinitesimal displacement along L corresponding to the change dλ in
the parameter λ (cf. Eq. (A.14)).
The qualifier affine in the above definition stems from:
Property B.1: affine parametrizations of a geodesic
Any two affine parametrizations of a geodesic L are necessarily related by an affine
transformation:
λ′ = aλ + b, (B.3)
where a and b are two real constants such that a ̸= 0.
Proof. Let P : I → L ⊂ M , λ 7→ P (λ) and P ′ : I ′ → L , λ′ 7→ P ′ (λ′ ) be two parametriza-
tions of L . They are necessarily related by a diffeomorphism I → I ′ , λ 7→ λ′ (λ). It follows
from Eq. (B.2) that the tangent vector fields v and v ′ associated with these two parametrizations
are related by
dλ′ ′
v= v. (B.4)
dλ
Using the rules 2 and 3 governing the connection ∇ (cf. Sec. A.4.1), we get then
′ 2
d2 λ′ ′ dλ
∇v v = v + ∇v′ v ′ . (B.5)
dλ2 dλ
1
As already noticed (cf. Remark 6 p. 714), in the mathematical literature, it is common to define a curve as the
parametrization P , and not as its image.
B.2 Definition and first properties 741
If both parametrizations are affine, then ∇v v = 0 and ∇v′ v ′ = 0, so that the above identity
reduces to d2 λ′ /dλ2 = 0, which implies the affine law (B.3).
Remark 1: Because of (B.1), a geodesic is also called an autoparallel curve. It is also sometimes called a
zero-acceleration curve, the vector ∇v v being considered as the “acceleration” of (the parametrization
P of) the curve L ; this is of course by extension of the concept of 4-acceleration a := ∇u u of a
timelike worldline with 4-velocity u, the latter being nothing but the tangent vector associated with the
parametrization of the worldline by its proper time (cf. Sec. 1.3.3).
An important property of geodesics is
Property B.2: constancy of the scalar square of affine tangent vector fields
Let L be a geodesic of (M , g) and v a tangent vector field associated with an affine
parametrization of L . Then the scalar square of v with respect to the metric g is constant
along L :
g(v, v) = const. (B.6)
Proof. The variation of g(v, v) along L is given by
d
(g(v, v)) = v (g(v, v)) = ∇v (g(v, v)) = v µ ∇µ (gρσ v ρ v σ )
dλ
= v µ ∇µ gρσ v ρ v σ + gρσ v µ ∇µ v ρ v σ + gρσ v ρ v µ ∇µ v σ = 0,
| {z } | {z } | {z }
0 0 0
where we have used the fact that ∇ is the Levi-Civita connection of g [Eq. (A.69)] and v obeys
the geodesic equation (B.1).
The constancy of g(v, v) has an interesting corollary: the tangent vector v cannot change its
causal type along L . Hence:
Property B.3: type of geodesics on pseudo-Riemannian manifolds
In a pseudo-Riemannian manifold (M , g), a geodesic L necessarily belongs to one of the
following three categories:
• timelike geodesic: tangent vectors are timelike at all points of L ;
• null geodesic: tangent vectors are null at all points of L ;
• spacelike geodesic: tangent vectors are spacelike at all points of L .
This is in sharp contrast with generic curves, which, for instance, can be timelike on some
portions and null or spacelike on other ones.
In thep timelike case, or the spacelike one, the tangent vector field v can be rescaled by the
constant |g(v, v)| to get a unit tangent vector field, i.e. a tangent vector field u which obeys
g(u, u) = −1 (timelike geodesic) or g(u, u) = 1 (spacelike geodesic). Moreover, in doing
so, the affine character of the parametrization is preserved. Indeed, the rescaling amounts to
choosing the constant a in the affine law (B.3) such that a = |g(v, v)|. Thus, we have
p
742 Geodesics
Property B.4: affine parametrizations with unit tangent vectors
A timelike or spacelike geodesic of a pseudo-Riemannian manifold has an affine parametriza-
tion, the tangent vector of which is a unit vector (i.e. of scalar square ±1 with respect to
g). Moreover, this parametrization is unique up to some choice of origin (choice of b in
(B.3)) and of orientation (a = ±1 in (B.3)).
We shall see in Sec. B.2.2 that for a timelike geodesic, the affine parameter with unit tangent
vector is nothing but the proper time, while for a spacelike geodesic, it is the arc length.
B.2.2 Generic parametrizations of geodesics
Geodesics can be characterized by any of their tangent vectors, i.e. tangent vectors not neces-
sarily associated with an affine parametrization, as follows:
Property B.5: generic tangent vector field along a geodesic
A curve L is a geodesic iff the tangent vector field v associated with any parametrization
of L obeys
∇v v = κ v, (B.7)
where κ is a scalar field along L .
Proof. Let P : I → L , λ 7→ P (λ) be the parametrization of L corresponding to the tangent
vector field v: v = dx/dλ. If L is a geodesic, then there exists a parametrization λ′ 7→
P ′ (λ′ ) whose tangent vector, v ′ say, obeys ∇v′ v ′ = 0. Since the accelerations of any two
parametrizations of L are related by Eq. (B.5), we deduce that v obeys (B.7) with
′ −1 2 ′
dλ dλ
κ= .
dλ dλ2
Conversely, if v obeys (B.7) with κ = κ(λ), then Eq. (B.5) implies that ∇v′ v ′ = 0, i.e. that L
is a geodesic, provided that the change of parametrization λ′ = λ′ (λ) fulfills
dλ′ d2 λ′
κ(λ) − = 0.
dλ dλ2
This differential equation has the following solution:
Z λ" Z λ̃ ! #
λ′ = a exp ˜ λ̃
κ(λ̃)d ˜ dλ̃ + b,
λ1 λ0
where a, b, λ0 and λ1 are constants, with a ̸= 0 and λ0 , λ1 ∈ I.
Property B.5 motivates the following definitions:
B.2 Definition and first properties 743
A vector field v obeying (B.7) is called a pregeodesic vector field. The scalar field κ is then
called the non-affinity coefficient of v. If κ = 0, v is naturally called a geodesic vector
field.
Note that Property B.5 is equivalent to stating that the field lines of a pregeodesic vector field
are geodesics.
An easy consequence of Eq. (B.7) is the following evolution law for the scalar square of the
tangent vector:
Property B.6: evolution of the scalar square of a tangent vector field
Along a geodesic L , the scalar square g(v, v) of the tangent vector v associated with any
parametrization of L evolves according to
∇v [g(v, v)] = 2κ g(v, v), (B.8)
where κ is the non-affinity coefficient of v.
Proof. One has, using ∇g = 0 [Eq. (A.69)] and Eq. (B.7),
v µ ∇µ (gρσ v ρ v σ ) = v µ ∇µ gρσ v ρ v σ + gρσ v µ ∇µ v ρ v σ + gρσ v ρ v µ ∇µ v σ = 2κgρσ v ρ v σ ,
| {z } | {z } | {z }
0 κv ρ κv σ
hence the law (B.8).
We recover of course Eq. (B.6) in the special case κ = 0 (v geodesic vector).
Remark 2: If λ is the parameter associated with v, i.e. v = dx/dλ, we may introduce the scalar
function V (λ) := g(v, v) and rewrite (B.8) as a first-order ordinary differential equation for it [cf.
Eq. (A.8)]:
dV
= 2κ(λ)V (λ). (B.9)
dλ
A consequence of (B.8) is
Property B.7: proper time and arc length as affine parameters
On a Lorentzian manifold, the parametrization of a timelike geodesic by the proper time
(λ = τ ) is an affine parametrization. Similarly, on a Lorentzian or Riemannian mani-
fold, the parametrization of a spacelike geodesic by the arc length (λ = s) is an affine
parametrization.
Proof. The tangent vector associated with the proper time τ along a timelike geodesic is nothing
but the 4-velocity u (cf. Sec. 1.3.3), which is of constant scalar square: g(u, u) = −1, so that
Eq. (B.8) reduces to 0 = −2κ, hence κ = 0, which implies that we are dealing with an affine
parametrization. Similarly, the tangent vector associated with the arc length s along a spacelike
geodesic has a scalar square everywhere equal to 1, leading to the same conclusion.
744 Geodesics
B.3 Existence and uniqueness of geodesics
B.3.1 The geodesic equation
Property B.8: geodesic equation
Let L be a curve in a pseudo-Riemannian manifold (M , g) of dimension n, such that L
is contained in the domain of a coordinate chart (xα )0≤α≤n−1 . Then any parametrization
of L , P : I → L , λ 7→ P (λ), can be described by n functions X α : I → R according to
Eq. (A.7): xα (P (λ)) = X α (λ). The curve L is a geodesic iff there exists a parametrization
of L for which the functions X α fulfills the following system of n second-order differential
equations, called the geodesic equation:
d2 X α µ
α dX dX
ν
+ Γ µν =0, 0 ≤ α ≤ n − 1, (B.10)
dλ2 dλ dλ
where the Γαµν ’s are the Christoffel symbols of the metric g with respect to the coordinates
(xα ), as given by Eq. (A.71).
Proof. Notice first that the components with respect to the chart (xα ) of the tangent vector
field v associated with the parameter λ are [cf. Eq. (A.11)]
dX α
vα = . (B.11)
dλ
On the other side, the components of ∇v v are
∂v α dv α
v µ ∇µ v α = v µ + Γ α
µν v µ ν
v = v(v α
) + Γ α
µν v µ ν
v = + Γαµν v µ v ν ,
∂xµ dλ
where we have used successively Eqs. (A.64), (A.12) and (A.8). The above relation, along with
(B.11), shows that the left-hand side of Eq. (B.10) is nothing but the component α of ∇v v. The
conclusion follows from the very definition of a geodesic given in Sec. B.2.1.
Note that if a solution of the geodesic equation (B.10) is found, the parameter λ is necessarily
an affine parameter. For a generic parameter, the differential equation becomes (B.10) with the
right-hand side replaced by κdX α /dλ, which is the coordinate expression of the right-hand
side κv in Eq. (B.7). Hence, we have
Property B.9: pregeodesic equation
A curve L in the domain of a chart (xα ) is a geodesic iff some (actually all) coordinate
expression xα = X α (λ) of L fulfills the following system of n second-order differential
equations, usually called the pregeodesic equation,
d2 X α µ
α dX dX
ν
dX α
+ Γ µν = κ(λ) , 0 ≤ α ≤ n − 1. (B.12)
dλ2 dλ dλ dλ
B.3 Existence and uniqueness of geodesics 745
for some real-valued function κ(λ).
B.3.2 Existence and uniqueness
We may use the geodesic equation to prove the following theorem:
Property B.10: existence and uniqueness of geodesic from a vector at a point
Given a point p in a pseudo-Riemannian manifold (M , g) and a vector V in the tangent
space to M at p, i.e. V ∈ Tp M , there exists a geodesic L through p such that V is the
value at p of the tangent vector of some affine parametrization λ of L :
dx
V = . (B.13)
dλ p
Moreover, this geodesic is unique, in the sense that any geodesic L ′ sharing the same
property coincides with L in some open neighborhood of p.
Proof. Let (xα ) be a coordinate chart of M around p. Let (V α ) be the components of V in the
basis of Tp M induced by the coordinate frame (∂α ) associated with (xα ):
V = V α ∂α |p .
A geodesic through p having V as tangent vector at p is then obtained as a solution (X α (λ))
of the system (B.10) with the initial conditions [cf. Eq. (B.11)]
dX α
X (0) = x (p) and
α α
(0) = V α . (B.14)
dλ
The system (B.10) + (B.14) constitutes a well-posed Cauchy problem and standard results
about ordinary differential equations, e.g. the Picard-Lindelöf (or Cauchy–Lipschitz) theorem,
guarantee the existence and uniqueness of the solution.
A few definitions follow naturally:
A geodesic L is said to be inextendible or maximal iff there does not exist any geodesic
L ′ ̸= L such that L ⊂ L ′ .
A geodesic L is complete iff the interval spanned by any of its affine parameters is the
whole real line: I = R. A geodesic that is not complete is called incomplete.
It is easy to show:
746 Geodesics
Property B.11: inextendibility of complete geodesics
Any complete geodesic is inextendible.
Proof. Let L be a complete geodesic. Let us consider any geodesic L ′ such that L ⊂ L ′ .
Let λ and λ′ be affine parameters of respectively L and L ′ . Since L ⊂ L ′ , λ′ is also an
affine parameter of L and we must have, along L , λ′ = aλ + b with a ̸= 0 [Eq. (B.3)]. Since
the range of λ is (−∞, +∞), for L is complete, this implies that the range of λ′ on L is
(−∞, +∞) as well, which make impossible to have points in L ′ \ L . Hence L ′ = L , i.e. L
is inextendible.
On physical grounds, one may consider that any timelike geodesic in a given spacetime
must be complete. Otherwise, this would mean that there exists a worldline L of a freely
falling observer that “ends” at some finite proper time. This would be the signature of either
(i) the possibility to extend the spacetime into a larger one or (ii) the ending of worldline L
at some (curvature) singularity. A spacetime in which this does not occur is called timelike
geodesically complete. More generally:
The pseudo-Riemannian manifold (M , g) is said to be geodesically complete iff every
inextendible geodesic is complete.
Remark 1: A well-known theorem of differential geometry, namely the Hopf-Rinow theorem, states
that a connected Riemannian manifold is geodesically complete iff it is complete as a metric space for
the distance function d(p, q) defined as the infimum of the length2 of all curves from p to q (see e.g.
Ref. [344]). However, there is no such theorem for a Lorentzian manifold, for the metric does not induce
any distance function which would make the manifold a metric space.
The following proposition strengthens Property B.10:
Property B.12: existence and uniqueness of inextendible geodesic from a vector
Given a point p in a pseudo-Riemannian manifold (M , g) and a nonzero vector V in
the tangent space to M at p, i.e. V ∈ Tp M , there exists a unique inextendible geodesic
through p, which we shall denote by LV , such that V is the value at p of the tangent
vector of some affine parametrization of LV . We shall then denote by PV the unique affine
parametrization of LV such that
PV (0) = p and v|p = V , (B.15)
where v is the tangent vector field of PV .
We refer to O’Neill’s textbook [390], p. 68 for the proof.
2
The length of a curve is defined by Eq. (B.24) below.
B.3 Existence and uniqueness of geodesics 747
B.3.3 Exponential map
One can make use of geodesics to map a tangent space to the base manifold:
Given a point p in a pseudo-Riemannian manifold (M , g), let Ep be the subset of the
tangent space Tp M defined by V ∈ Ep iff either V = 0 or the affine parametrization PV
of the geodesic LV associated to V by Property B.12 has a domain large enough to include
the interval [0, 1]. The exponential map at p is then defined as
expp : Ep ⊂ Tp M −→ M
p if V = 0 (B.16)
V 7−→
PV (1) ∈ LV if V ̸= 0
In other words, expp maps a nonzero vector V in the tangent space to M at p to the point
of M of affine parameter λ = 1 along the unique geodesic through p such that (i) λ = 0
corresponds to p and (ii) the tangent vector dx/dλ to the geodesic at p is V .
Note that if (M , g) is geodesically complete, Ep = Tp M for every point p ∈ M .
An immediate property of the exponential map is
Property B.13
If V ∈ Ep \ {0}, for any t ∈ [0, 1], expp (tV ) lies on the same geodesic LV as expp (V ), at
the parameter λ = t of the parametrization PV :
∀t ∈ [0, 1], expp (tV ) = PV (t). (B.17)
Proof. For t = 0, the property follows from the definition of expp , since PV (0) = p. If
t ̸= 0, the nonzero vector tV is collinear to V and the uniqueness property of geodesics
(Property B.12) implies that LtV = LV . By virtue of the transformation law (B.4), tV is the
tangent vector to LV corresponding to the affine parameter λ′ = t−1 λ, where λ is the affine
parameter whose tangent vector field obeys v|p = V . From the definition of expp , we have
then expp (tV ) = PtV (λ′ = 1) = PV (λ = t × 1), hence (B.17).
The exponential map realizes a local identification of the manifold with its tangent space at
a given point:
Property B.14: exponential map as a local diffeomorphism
For any p ∈ M , there exists a neighborhood U of 0 in the tangent space Tp M and
a neighborhood U of p in the manifold M such that the exponential map expp is a
diffeomorphism from U to U .
748 Geodesics
Proof. It is clear from its definition that expp is a smooth map, at least on some neighborhood
U ′ of 0 in Tp M . We may then consider the differential of expp at 0, d expp 0 . By virtue of
the inverse function theorem for manifolds (see e.g. Theorem 4.5 in Ref. [343]), it suffices to
show that d expp 0 is invertible to complete the proof. By definition of the differential of a
map (cf. Sec. A.2.7) and since expp : Ep ⊂ Tp M → M and expp (0) = p, d expp 0 carries an
infinitesimal displacement vector of3 T0 (Tp M ), ε say, connecting 0 to a nearby element of
Tp M , ε′ say, to the infinitesimal vector E ∈ Tp M connecting p = expp (0) to q = expp (ε′ ):
d expp 0
: T0 (Tp M ) −→ Tp M
ε 7−→ E = →
−
pq.
Now, since Tp M is a vector space, we have the canonical identification T0 (Tp M ) ≃ Tp M , from
which ε′ = ε. Without any loss of generality, we may write ε = εV , where ε is infinitesimal
small and V ∈ Tp M . Then q = expp (εV ) = PV (ε), where the second identity results from
(B.17). We have thus
−−−−−−−−→
d expp 0 (εV ) = E = → −
pq = PV (0)PV (ε).
−−−−−−−−→
According to the definition of PV , the infinitesimal vector PV (0)PV (ε) along the geodesic LV
is εV , hence
d expp 0 (εV ) = εV .
Since the differential d expp 0 is a linear map, we get d expp 0 (V ) = V . The vector V being
arbitrary, we conclude that d expp 0 is nothing but the identity map of the vector space Tp M :
d expp 0
= idTp M .
In particular, d expp 0
is invertible.
B.3.4 Normal coordinates
Given p ∈ M , a normal neighborhood of p is a neighborhood U of p that is the image
of a starshaped neighborhood of 0 ∈ Tp M under the local diffeomorphism expp given by
Property B.14. By starshaped neighborhood of 0, it is meant a neighborhood U of 0 such
that V ∈ U implies tV ∈ U for any t ∈ [0, 1]. A convex normal neighborhood U is an
open subset of M that is a normal neighborhood of each of its points. It follows that any
two points p and q of a convex normal neighborhood U can be connected by a geodesic
lying entirely in U .
On a normal neighborhood, one may define coordinates linked to geodesics as follows.
3
Here the vector space Tp M is considered as a n-dimensional smooth manifold, and T0 (Tp M ) stands for its
tangent space at 0 (the zero vector of Tp M ).
B.3 Existence and uniqueness of geodesics 749
Let U be a normal neighborhood of p ∈ M and (Eα )0≤α≤n−1 be a basis of Tp M . If (E α )
stands for the basis of Tp∗ M dual to (Eα ), the map
Φ: U −→ Rn
(B.18)
⟨E 0 , exp−1 , exp−1
n−1
q 7−→ p (q)⟩, . . . , ⟨E p (q)⟩
is a coordinate chart on U , which is called geodesic normal coordinates (often shorten
to normal coordinates) associated with the basis (Eα ).
In other words, normal coordinates (xα ) on U are such that the tangent vector xµ (q)Eµ ∈ Tp M
has precisely q as image by the exponential map:
∀q ∈ U , expp (xµ (q)Eµ ) = q. (B.19)
Remark 2: Some authors, e.g. [390], add the condition that the basis (Eα ) is orthonormal (with respect
to the metric g) in the definition of normal coordinates. We follow here the more general definition
of [318, 63, 266, 464]. The name Riemann normal coordinates is also commonly encountered in the
literature, either for normal coordinates as defined here (e.g. [371, 499]) or for those with the basis
orthonormality requirement (e.g. [418]).
A characteristic feature of normal coordinates is that, in terms of them, geodesics through
p look like straight lines through 0 in Rn :
Property B.15: geodesics in terms of normal coordinates
In a normal coordinate system (xα ), the equation of the unique geodesic LV through
p admitting V ∈ Tp M as tangent vector at p is (as long as LV remains in the normal
neighborhood U )
xα = X α (λ) = λV α , (B.20)
where the V α ’s are the components of V with respect to the basis (Eα ) defining the normal
coordinates: V = V µ Eµ .
Proof. Let λ be the affine parameter of LV corresponding to the parametrization PV . The
coordinate equation of LV is then xα = X α (λ) with [cf. Eq. (B.18)]
X α (λ) = xα (PV (λ)) = ⟨E α , exp−1
p (PV (λ))⟩.
Now, according to Eq. (B.17), PV (λ) = expp (λV ). Hence
X α (λ) = ⟨E α , exp−1 α α α
p ◦ expp (λV )⟩ = ⟨E , λV ⟩ = λ⟨E , V ⟩ = λV .
750 Geodesics
Property B.16: metric components and Christoffel symbols in terms of normal
coordinates
Let (xα ) be a normal coordinate system around p ∈ M associated with a basis (Eα ) of
Tp M . Then
• the coordinate frame (∂α ) associated with (xα ) coincides with (Eα ) at p:
∂α |p = Eα ; (B.21)
• the values at p of the components (gαβ ) of the metric tensor g with respect to (xα )
are
gαβ (p) = g|p (Eα , Eβ ); (B.22)
• the Christoffel symbols of g with respect to the coordinates (xα ) vanish at p:
Γαβγ (p) = 0. (B.23)
Proof. Let U be the normal neighborhood covered by (xα ) and V ∈ exp−1 p (U ) ⊂ Tp M . The
tangent vector field to the geodesic LV corresponding to the parametrization PV is v = Ẋ µ ∂µ
with Ẋ µ obtained by deriving (B.20) with respect to λ: Ẋ µ = V µ . Hence v = V µ ∂µ . Now,
from the very definition of PV , v|p = V = V µ Eµ . We have therefore
V µ ∂µ |p = V µ Eµ .
This identity being fulfilled for any V µ , Eq. (B.21) follows. Equation. (B.22) is an immediate
consequence of Eq. (B.21), since gαβ = g(∂α , ∂β ). Finally, with the functions X α (λ) given by
(B.20), the geodesic equation (B.10) reduces to Γαµν V µ V ν = 0. In particular, at p, we get
Γαµν (p)V µ V ν = 0.
This identity must hold for any V α . It expresses therefore that, for each value of α, the quadratic
form V 7→ Γαµν (p)V µ V ν is identically zero on Tp M . Since the Christoffel symbols Γαµν are
symmetric in µν, it is equivalent to say that, for each value of α, the symmetric bilinear form
(U , V ) 7→ Γαµν (p)U µ V ν is identically zero, which amounts to Γαµν (p) = 0, i.e. Eq. (B.23).
B.4 Geodesics and extremal lengths
B.4.1 Length of a curve
Geodesics in a pseudo-Riemannian manifold (M , g) have been defined in Sec. B.2.1 as the
“straightest lines”, i.e. as autoparallel curves with respect to the Levi-Civita connection of g.
Here, we make some attempt to connect them with the first feature mentioned in Sec. B.2.1,
namely, in a pure Riemannian manifold, geodesics are locally the curves of minimal length. We
have first to define the length of a curve. Of course, when the metric is not positive definite,
B.4 Geodesics and extremal lengths 751
one cannot
p use the integral of the norm of infinitesimal displacements along the curve, i.e.
ds := g(dx,p dx), since g(dx, dx) can be negative. Rather, it is quite natural to employ
instead ds := |g(dx, dx)|. Using dx = v dλ [Eq. (A.14)], we end up with the following
definition:
The length of a curve L connecting two points p and q in a pseudo-Riemannian manifold
(M , g) is the real number
Z λq
(B.24)
p
L(p,q) (L ) := |g(v, v)| dλ,
λp
where λ is some parameter along L , λp (resp. λq ) being its value at p (resp. q), v = dx/dλ
is the associated tangent vector field, and we assume λq ≥ λp .
Thanks to the transformation law (B.4), it is easy to check that the value of L(p,q) (L ) is
independent from the choice of the parametrization of L , i.e. for a fixed pair of points (p, q), it
is a function of L only.
When L is included in the domain of a coordinate chart (xα ), so that its equation is
xα = X α (λ), we may rewrite (B.24) as [cf. Eq. (B.11)]
Z λq r
L(p,q) (L ) := gµν (X ρ (λ))Ẋ µ Ẋ ν dλ, (B.25)
λp
where Ẋ α := dX α /dλ and gµν (X ρ (λ)) stands for the components of the metric tensor g with
respect to the coordinates (xα ) at the point of coordinates X ρ (λ).
From the very definition of L(p,q) (L ), it is obvious that
L(p,q) (L ) ≥ 0. (B.26)
Moreover, if it exists, any null curve from p to q achieves the absolute minimum of the length,
without having to be a geodesic:
L null =⇒ L(p,q) (L ) = 0. (B.27)
B.4.2 Timelike and spacelike geodesics as stationary points of the
length functional
The property (B.27) implies that, in a pseudo-Riemannian manifold, the curve that minimizes
the length between two points is not necessarily a geodesic. A typical example is the null helix
in Minkowski spacetime, discussed in Remark 2 on p. 35. Moreover, when g is not positive
definite, it could be relevant to consider curves of maximal length between two points, i.e. to
search for an extremum, be it a minimum or a maximum.
To find the curves of extremal length, it is quite natural to study the behavior of the length as
a variational problem, i.e. to consider L(p,q) (L ) as an “action” and to write the Euler-Lagrange
equation for the “Lagrangian” defined as the integrand of (B.25):
r
L(X α , Ẋ α ) = gµν (X ρ )Ẋ µ Ẋ ν . (B.28)
752 Geodesics
Before proceeding, a few caveats must be made. First of all, the Euler-Lagrange equation locate
only stationary points of the action (here the length L(p,q) (L )), i.e. points where the action
does not vary to first order in small changes of the curve. Such points are not necessarily
extrema: they can be saddle points, as we shall see. Secondly, because of the square root in
(B.28), the Lagrangian is not differentiable at points where gµν Ẋ µ Ẋ ν = 0. This corresponds to
points where the curve L is null. We shall therefore exclude such curves in our analysis (we
shall return to null curves in Sec. B.4.3). But then gµν Ẋ µ Ẋ ν has to be either always positive
along L (i.e. L is spacelike) or always negative (i.e. L is timelike); indeed, by continuity it
cannot change sign without going through zero. We shall then apply the variational principle
separately to two subsets of curves connecting p and q: the timelike ones and the spacelike
ones. The calculations will be conducted in parallel by introducing the sign parameter ϵ = −1
for timelike curves and ϵ = +1 for spacelike ones. One can then get rid of the absolute value
in the Lagrangian, which becomes
q
α α
L(X , Ẋ ) = ϵgµν (X ρ )Ẋ µ Ẋ ν . (B.29)
Asking that the length (B.25) is stationary with respect to small changes in the curve connecting
p and q is equivalent to the Euler-Lagrange equation:
d ∂L ∂L
− = 0. (B.30)
dλ ∂ Ẋ α ∂X α
We have
∂ ∂g
µν
α
ρ µ ν
gµν (X )Ẋ Ẋ = α
Ẋ µ Ẋ ν , (B.31)
∂X ∂x
with the understanding that ∂gµν /∂xα shall be taken at the point X ρ (λ). Hence, given the
Lagrangian (B.29),
∂L ϵ ∂gµν µ ν
= Ẋ Ẋ . (B.32)
∂X α 2L ∂xα
Besides,
∂
gµν (X ρ )Ẋ µ Ẋ ν = gαν Ẋ ν + gµα Ẋ µ = 2gαµ Ẋ µ . (B.33)
∂ Ẋ α
Hence
∂L ϵ
= gαµ Ẋ µ ,
∂ Ẋ α L
from which,
d ∂L ϵ dL ϵ ∂gαµ ν µ ϵ
=− 2 gαµ Ẋ µ + ν
Ẋ Ẋ + gαµ Ẍ µ . (B.34)
dλ ∂ Ẋ α L dλ L ∂x L
In view of (B.32) and (B.34), the Euler-Lagrange equation (B.30) becomes, after multiplication
by L/ϵ,
1 dL ∂gαµ µ ν 1 ∂gµν µ ν
− gαµ Ẋ µ + ν
Ẋ Ẋ + gαµ Ẍ µ − Ẋ Ẋ = 0.
L dλ ∂x 2 ∂xα
B.4 Geodesics and extremal lengths 753
Now, playing with the names of repeated indices and using the symmetry of gαβ , we can rewrite
the second term as
∂gαµ µ ν 1 ∂gαν ν µ ∂gµα µ ν 1 ∂gαν ∂gµα
ν
Ẋ Ẋ = µ
Ẋ Ẋ + ν
Ẋ Ẋ = µ
+ ν
Ẋ µ Ẋ ν . (B.35)
∂x 2 ∂x ∂x 2 ∂x ∂x
Accordingly, we get
1 ∂gαν ∂gµα ∂gµν
µ
gαµ Ẍ + + − Ẋ µ Ẋ ν = κgαµ Ẋ µ , (B.36)
2 ∂xµ ∂xν ∂xα
where we have introduced
1 dL
κ := . (B.37)
L dλ
If we multiply Eq. (B.36) by the matrix g αβ (the components of the inverse metric) and use
g αβ gαµ = δ βµ as well as the expression (A.71) of the Christoffel symbols, we get exactly the
pregeodesic equation (B.12). Hence we conclude:
Property B.17: timelike and spacelike geodesics as stationary points of the length
Among all timelike (resp. spacelike) curves connecting two points p and q, a curve has a
stationary length iff it is a timelike (resp. spacelike) geodesic.
For a timelike geodesic, and for points p and q not too far (in the same normal neighborhood),
the stationary length corresponds actually to a maximum:
Property B.18: timelike geodesics as maximal length curves
Let (M , g) be a Lorentzian manifold, p ∈ M and U some normal neighborhooda of p.
For any point q ∈ U such that there exists a timelike curve in U from p to q, the geodesic
from p to q is the unique timelike curve of largest length in U connecting p to q.
a
See Sec. B.3.4.
We shall not provide a full proof here but refer instead to the proof of Proposition 5.34 in
O’Neill’s textbook [390]. We shall only illustrate the property on a specific example in flat
spacetime (Example 1 below).
If one interprets timelike curves as worldlines and length as proper time (cf. Sec. 1.3.3),
Property B.18 can be viewed as a generalization of the standard “twin paradox” of special
relativity: when they meet again, the twin who followed the geodesic (i.e. some inertial
trajectory) is older than his brother or her sister, who made a round trip.
Example 1 (Timelike geodesic in Minkowski spacetime): Let us suppose that (M , g) is the 4-
dimensional Minkowski spacetime. All geodesics are then (segments of) straight lines. If p and q
are connected by a timelike geodesic L , we may consider a Minkowskian coordinate system (xα ) =
(t, x, y, z) such that xα (p) = (0, 0, 0, 0) and xα (q) = (T, 0, 0, 0), for some T > 0. t is then the proper
754 Geodesics
t/T
1 q
0.8
0.6
0.4
0.2
p x/T
-0.4 -0.2 0.2 0.4
Figure B.1: Timelike curves Lh connecting the point p of coordinates (0, 0, 0, 0) to the point q of coordi-
nates (T, 0, 0, 0) in Minkowski spacetime. From the left to right, the depicted curves correspond to h spanning
[−3/4, 3/4], with the step δh = 1/4.
time along L and L(p,q) (L ) = T . Let us consider the one-parameter family of curves (Lh )h∈(−1,1)
defined by xα = X α (λ) with λ ∈ [0, T ] and
h
X 0 (λ) = λ, X 1 (λ) = λ(T − λ), X 2 (λ) = 0, X 3 (λ) = 0.
T
Note that X 0 (λ) = λ means that the curve parameter coincides with the time coordinate: λ = t. We
have L0 = L and for h ̸= 0, L is an arc of parabola from p to q in the (t, x) plane (cf. Fig. B.1); the
dimensionless parameter h is related to the curve’s maximal extension along x by xmax = hT /4. We
have
0 1 λ
Ẋ (λ) = 1, Ẋ (λ) = h 1 − 2 , Ẋ 2 (λ) = 0, Ẋ 3 (λ) = 0.
T
Given that (gαβ ) = diag(−1, 1, 1, 1), it follows that gµν Ẋ µ Ẋ ν = −1+h2 (1−2λ/T )2 . Since λ ∈ [0, T ],
this shows that Lh is a timelike curve as long as −1 ≤ h ≤ 1. Its length is
s
λ 2
Z T Z hp Z arcsin h
T T
L(p,q) (Lh ) = 2
1−h 1−2 dλ = 2
1 − u du = cos2 θ dθ.
0 T 2h −h 2h −arcsin h
Evaluating the integral leads to
p
T arcsin h
L(p,q) (Lh ) = 1 − h2 + .
2 h
Note that arcsin h/h is well-defined at h = 0, since limh→0 arcsin h/h = 1. The graph of L(p,q) (Lh )
as a function h is plotted in Fig. B.2. We see clearly that h = 0, i.e. the geodesic L , corresponds to the
maximal length.
For a spacelike geodesic in a Lorentzian manifold, the stationary length corresponds neither
to a maximum nor a minimum, but rather to a saddle point, as the example below illustrates.
B.4 Geodesics and extremal lengths 755
L(p, q) ( h )/T
1
0.95
0.9
0.85
0.8
-1 -0.5 0 0.5 1
h
Figure B.2: Length of the timelike curve Lh connecting the point p of coordinates (0, 0, 0, 0) to the point q of
coordinates (T, 0, 0, 0) in Minkowski spacetime, as a function of the parameter h measuring the deviation from
the timelike geodesic L = L0 .
Example 2 (Spacelike geodesic in Minkowski spacetime): As in Example 1, we consider Minkowski
spacetime, but this time, L is assumed to be a spacelike geodesic from p to q. Since L is necessarily
a straight line segment, without any loss of generality, we may introduce a Minkowskian coordinate
system (xα ) = (t, x, y, z) such that xα (p) = (0, 0, 0, 0) and xα (q) = (0, L, 0, 0) for some L > 0, which
is nothing but the length L(p,q) (L ) of the geodesic L . Any spacelike curve L ′ connecting p and q and
lying in the hyperplane Σ defined by t = 0 obeys L(p,q) (L ′ ) ≥ L(p,q) (L ) since Σ, equipped with the
metric induced by g, is a 3-dimensional Euclidean space.
Let us consider some one-parameter family of curves (Lh )h∈(−1,1) lying in the orthogonal comple-
ment of Σ through p and q, namely the curves defined xα = X α (λ) with λ ∈ [0, L] and
h
X 0 (λ) = λ(L − λ), X 1 (λ) = λ, X 2 (λ) = 0, X 3 (λ) = 0.
L
As in Example 1, we have L0 = L and for h ̸= 0, the Lh ’s are arcs of parabola from p to q, which
remain spacelike as long as −1 < h < 1 (cf. Fig. B.3). The computations are similar to those of Example 1,
leading to
L p 2
arcsin h
L(p,q) (Lh ) = 1−h + .
2 h
L(p,q) (Lh )/L is exactly the same of function of h as L(p,q) (Lh )/T in Example 1. In view of Fig. B.2, we
therefore assert that L(p,q) (Lh ) ≤ L(p,q) (L ).
We conclude that the spacelike geodesic L corresponds to a saddle point of the length functional:
it is a minimum among the curves lying in the (x, y, z) hyperplane but a maximum among those lying
in the (t, x) plane.
B.4.3 All geodesics as stationary points of some action
We have excluded null geodesics from the above variational analysis by invoking the necessary
smoothness of the Lagrangian (B.28). We may further convince ourselves that null geodesics
756 Geodesics
t/L
0.15
0.1
0.05 p q x/L
-0.05 0.2 0.4 0.6 0.8 1
-0.1
-0.15
Figure B.3: Spacelike curves Lh connecting the point p of coordinates (0, 0, 0, 0) to the point q of coordinates
(0, L, 0, 0) in Minkowski spacetime. From the bottom to the top, the depicted curves correspond to h spanning
[−3/4, 3/4], with the step δh = 1/4.
would not have fit in the analysis by noticing the division by L in Eq. (B.37), which excludes
L = 0. However, it is possible to get all geodesics, including the null ones, from a variational
principle; one has to start from a different action, namely
Z λq
1
S(p,q) (P ) := gµν (X ρ )Ẋ µ Ẋ ν dλ , (B.38)
2 λp
where P is a parametrization of the curve L , λ the corresponding parameter and xα = X α (λ)
the coordinate expression of P .
The Lagrangian in (B.38) is
1
L2 (X α , Ẋ α ) = gµν (X ρ )Ẋ µ Ẋ ν . (B.39)
2
We notice that it is always differentiable, even when gµν Ẋ µ Ẋ ν = 0, i.e. it allows for null
curves. However, the price to pay is that, contrary to the length (B.25), the action depends
on the parametrization of the curve, hence the notation S(p,q) (P ) rather than S(p,q) (L ). For
this reason, S(p,q) (P ) is not expected to have any significant physical meaning, contrary to
L(p,q) (L ), which is the proper time along timelike curves.
Searching for stationary points of the action (B.38) is straightforward. Indeed, given
Eqs. (B.31) and (B.33), we have
∂L2 1 ∂gµν µ ν ∂L2
α
= Ẋ Ẋ and = gαµ Ẋ µ ,
∂X 2 ∂xα ∂ Ẋ α
so that
d∂L2 ∂gαµ ν µ
= Ẋ Ẋ + gαµ Ẍ µ .
dλ∂ Ẋ α ∂xν
Using the identity (B.35), the Euler-Lagrange equation (B.30) (with L substituted by L2 ) turns
out to be equivalent to the geodesic equation (B.10). We conclude:
B.5 Geodesics and symmetries 757
Property B.19: geodesics as stationary points of the action (B.38)
In a pseudo-Riemannian manifold (M , g), a curve L equipped with a parametrization P
is a stationary point of the action (B.38) iff L is a geodesic and P an affine parametrization
of it.
Remark 1: The variational principle applied to the action (B.38) leads directly to the geodesic equation
(B.10), which implies that the involved parametrization is affine. On the contrary, the variation of the
length functional (B.25), leads only to the pregeodesic equation (B.12) (cf. the computation in Sec. B.4.2),
which permits a generic parametrization of the geodesic, in agreement with the fact that the length is
parametrization-independent, contrary to the action (B.38).
Remark 2: The factor 1/2 in Eq. (B.38) does not play any role in the variational principle, so we could
have dropped it. However, thanks to it, the momentum conjugate to X α takes a simple form:
∂L
Πα := = gαµ Ẋ µ . (B.40)
∂ Ẋ α
The Lagrangian (B.39) can be then written L2 = 1/2 Πµ Ẋ µ and the Hamiltonian deduced from it by
the standard Legendre transformation is H = Πµ Ẋ µ − L2 = 1/2 Πµ Ẋ µ , i.e.
1
H(X α , Πα ) = g µν (X ρ )Πµ Πν . (B.41)
2
Such a Hamiltonian has been used by Carter [90] to study the geodesics in Kerr spacetime, discovering
the famous Carter constant.
B.5 Geodesics and symmetries
B.5.1 Geodesics in presence of a Killing vector
As a reminiscence of Noether’s theorem, symmetries in a pseudo-Riemannian manifold lead to
conserved quantities along geodesics. Let us first recall that 1-dimensional isometry groups
and the related concept of Killing vector field have been introduced in Sec. 3.3.1. In terms of
them, we may state the following conservation law:
Property B.20: constancy of the scalar product of an affine tangent vector with a
Killing vector
If the pseudo-Riemannian manifold (M , g) admits a 1-dimensional isometry group of
generator ξ, i.e. if ξ is a Killing vector field of (M , g), then along any geodesic L , the
g-scalar product of ξ by any tangent vector field v = dx/dλ associated with an affine
parameter λ of L is constant:
g(ξ, v) = const. (B.42)
758 Geodesics
Proof. The variation of g(ξ, v) along L is, according to Eq. (A.8),
d
g(ξ, v) = v (g(ξ, v)) = ∇v (g(ξ, v))
dλ
= v σ ∇σ (gµν ξ µ v ν ) = v σ ∇σ (ξν v ν ) = ∇σ ξν v σ v ν + ξν v σ ∇σ v ν
1
= (∇σ ξν + ∇ν ξσ )v σ v ν + ξν v σ ∇σ v ν = 0, (B.43)
2 | {z } | {z }
0 0
where the first zero holds because ξ obeys the Killing equation (3.19) and the second one holds
thanks to Eq. (B.1), which expresses that L is a geodesic and v the tangent vector associated
with some affine parameter.
Remark 1: If the tangent vector v is associated with a generic (not necessarily affine) parameter of L ,
the second zero in Eq. (B.43) must be replaced by κv ν , where κ is the non-affinity coefficient of v [cf.
Eq. (B.7)]. Accordingly g(ξ, v) is no longer constant along L but rather evolves according to
d
g(ξ, v) = κ g(ξ, v). (B.44)
dλ
Note that κ a priori varies along L , so that the integration of this first-order differential equation
depends of the precise form of the function κ(λ).
B.5.2 Geodesics in presence of a Killing tensor
While the concept of Killing vector is by definition tight to a spacetime symmetry (isometry),
there is a generalization of the Killing equation (3.19) to tensors of higher ranks, which is not
directly related to any symmetry of the metric tensor. It is however interesting since it leads to
conserved quantities along geodesics.
A Killing tensor (also called a Stäckel-Killing tensor [95]) of rank p ≥ 1 in the pseudo-
Riemannian manifold (M , g) is a tensor field K of type (0, p) that is fully symmetric and
whose covariant derivative obeys
∇(α1 Kα2 ...αp+1 ) = 0. (B.45)
Example 3: A trivial example of Killing tensor is the metric tensor g itself. If (M , g) admits a Killing
vector ξ, another example is K = ξ (the 1-form associated to ξ by metric duality), since for p = 1,
Eq. (B.45) reduces to the Killing equation (3.19). An example for p = 2 is then K = ξ ⊗ ξ (by the Leibniz
rule + the Killing equation). Similarly, K = ξ ⊗ ξ ⊗ ξ is a Killing tensor with p = 3, etc. A less trivial
example is the Walker-Penrose Killing tensor of Kerr spacetime discussed in Sec. 11.2.4.
If a spacetime is endowed with a Killing tensor that is not trivial, i.e. neither formed from
g nor any Killing vector as in Example 3, one often says that this spacetime has a hidden
symmetry (see e.g. the review article [201] for an extended discussion). This is because, as
Killing vectors, Killing tensors give birth to conserved quantities along geodesics:
B.6 Geodesics and curvature 759
Property B.21: conserved quantity induced by a Killing tensor along a geodesic
Let K be a Killing tensor of rank p on the pseudo-Riemannian manifold (M , g). Along
along any geodesic L , the scalar K(v, . . . , v), where v = dx/dλ is the tangent vector to
L associated to some affine parameter λ, is constant:
K(v, . . . , v) = const. (B.46)
Proof. The variation of K(v, . . . , v) along L is given by
d
(K(v, . . . , v)) = ∇v (K(v, . . . , v)) = v µ ∇µ (Kν1 ...νp v ν1 · · · v νp )
dλ
= ∇µ Kν1 ...νp v µ v ν1 · · · v νp
| {z }
0
+Kν1 ...νp v ∇µ v ν1 · · · v νp + · · · + Kν1 ...νp v ν1 · · · v µ ∇µ v νp
µ
| {z } | {z }
0 0
= 0,
where the first zero results from (B.45), while the zeros in the line below arise from the geodesic
equation (B.1) obeyed by v.
Example 4: Since we have already noticed that the metric tensor g is a Killing tensor (Example 3),
Property B.2 (constancy of g(v, v)) appears as a special case of Property B.21. For the Walker-Penrose
Killing tensor K of Kerr spacetime, the conserved quantity K(v, v) is called the Carter constant (cf.
Sec. 11.2.4).
B.6 Geodesics and curvature
Let (Ls )s∈R be a smooth 1-parameter family of non-intersecting geodesics in a pseudo-
Riemannian manifold (M , g). Let λ be an affine parameter along each geodesic Ls , such that
λ varies smoothly from one geodesic to the next one, i.e. such that the map Φ : I × R → M ,
(λ, s) 7→ Ps (λ) is smooth, PS s : I → M being the parametrization of Ls associated to λ (cf.
Sec. B.2.1). Note that S := s∈R Ls is a 2-dimensional surface of M , which is parametrized
by (λ, s) (cf. Fig. B.4); S is actually the image of the map Φ defined above. On S , let us denote
by v and z the vector fields of the natural basis associated to the coordinates (λ, s):
v := ∂λ and z := ∂s . (B.47)
By construction, v coincides with the tangent vector field dx/dλ associated to the affine
parameter λ along each geodesic Ls [Eq. (B.2)]. The vector field z is called the separation
vector at fixed λ of the family (Ls )s∈R because at a fixed value of λ, two infinitely close
geodesics Ls and Ls+ds are connected by the vector dx = ds z (cf. Fig. B.4). Note that the
separation vector depends upon the choice of the affine parameter λ.
760 Geodesics
Figure B.4: 1-parameter family of geodesics (Ls )s∈R affinely parametrized by λ. The 2-surface S formed by
the geodesic family is spanned by the coordinates (λ, s). The infinitesimal vector connecting a point of Ls to the
point of Ls+ds having the same value of λ is dx = ds z, where z is the separation vector at fixed λ.
When the geodesics Ls are timelike or null, v is the evolution vector along Ls associated
to the affine parameter λ. One may then consider the covariant derivative ∇v z as the relative
velocity of neighboring geodesics and the second order derivative ∇v ∇v z as the relative
acceleration of neighboring geodesics. Even if Ls is spacelike, we keep the terms relative
velocity and relative acceleration.
Property B.22: geodesic deviation equation
The relative acceleration of nearby members of a 1-parameter family (Ls )s∈R of geodesics
in a pseudo-Riemannian manifold (M , g) is given by
∇v ∇v z = Riem (., v, v, z) , (B.48)
where v is the evolution vector along Ls associated to some affine parameter λ, z is
the separation vector at fixed λ, ∇ is the Levi-Civita connection of g and Riem is the
aassociated Riemann curvature tensor (cf. Sec. A.5). Equation (B.48) is the called the
geodesic deviation equation or the Jacobi equation. In index notation, Eq. (B.48) becomes
v µ ∇µ (v ν ∇ν z α ) = Rαρµν v ρ v µ z ν . (B.49)
Proof. Because they are vector fields associated to the coordinate system (λ, s) of S , it is
obvious that the vector fields v and z commute: [v, z] = 0. This can also be seen from
the identity [v, z] = Lv z [Eq. (A.79)] and the very definition of the Lie derivative Lv z (cf.
Sec. A.4.2), which in the present case yields Lv z = 0. Hence we have v ν ∇ν z α = z ν ∇ν v α .
B.6 Geodesics and curvature 761
Taking another derivative along v, we get successively
v µ ∇µ (v ν ∇ν z α ) = v µ ∇µ (z ν ∇ν v α ) = v µ ∇µ z ν ∇ν v α + z ν v µ ∇µ ∇ν v α
= v µ ∇µ z ν ∇ν v α + z ν v µ (∇ν ∇µ v α + Rαρµν v ρ )
= z µ ∇µ v ν ∇ν v α + z µ v ν ∇µ ∇ν v α + Rαρµν v ρ v µ z ν
= z µ ∇µ (v ν ∇ν v α ) + Rαρµν v ρ v µ z ν = Rαρµν v ρ v µ z ν ,
| {z }
0
where the Ricci identity (A.98) has been used to get the second line and again the commutation
property v µ ∇µ z ν = z µ ∇µ v ν to get the third line. Finally, v ν ∇ν v α = 0 in the last line follows
from the geodesic character of v [Eq. (B.1)], given that it is the tangent vector corresponding
to an affine parametrization of Ls .
The geodesic deviation equation (B.48) is a second order linear differential equation for the
vector field z. Thanks to it, the curvature of the pseudo-Riemannian manifold (M , g) can be
interpreted as the obstruction for neighboring geodesics that are initially parallel (zero relative
velocity ∇v z at λ = 0) to stay parallel for λ ̸= 0. Actually, Eq. (B.48) can even be used to define
the Riemann curvature tensor Riem, instead of Eq. (A.97) (see e.g. Chap. 11 of MTW [371] for
details).
762 Geodesics
Appendix C
Kerr-Schild metrics
Contents
C.1 Generic Kerr-Schild spacetimes . . . . . . . . . . . . . . . . . . . . . . 763
C.2 Case of Kerr spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . 765
C.1 Generic Kerr-Schild spacetimes
C.1.1 Definition
A spacetime (M , g) is said to have a Kerr-Schild metric iff the metric tensor g can be written
g = f + 2Hk ⊗ k , (C.1)
or equivalently (in index notation):
gαβ = fαβ + 2Hkα kβ , (C.2)
where f is a flat Lorentzian metric on M (Minkowski metric), H is a scalar field on M and k
is a 1-form on M such that the vector associated to it via f is a null vector of the metric f :
f µν kµ kν = 0, (C.3)
where f µν stands for the components of the inverse of the metric f (i.e. f αµ fµβ = δ αβ ).
A motivation for studying Kerr-Schild metrics is that the inverse metric has a simple
expression:
g αβ = f αβ − 2Hk α k β , (C.4)
where
k α := f αµ kµ . (C.5)
764 Kerr-Schild metrics
Proof: we have successively:
(f αµ − 2Hk α k µ )gµβ = (f αµ − 2Hk α k µ )(fµβ + 2Hkµ kβ )
= f αµ fµβ +2H f αµ kµ kβ − 2Hk α k µ fµβ −4H 2 k α k µ kµ kβ
| {z } | {z } | {z } |{z}
δ αβ kα kβ 0
= δ αβ ,
which establishes Eq. (C.4).
Given (C.4), it is easy to see that the vector field k associated to the 1-form k by g-duality
(cf. Sec. A.3.3) is the same as the vector field obtained by f -duality:
g αµ kµ = (f αµ − 2Hk α k µ )kµ = f αµ kµ −2Hk α k µ kµ = k α .
| {z } |{z}
kα 0
Accordingly, we may write the components of k simply as k α without having to specify whether
the index has been raised with g or f :
k α = f αµ kµ = g αµ kµ . (C.6)
It follows immediately that k is a null vector field for both metrics:
g(k, k) = f (k, k) = 0 . (C.7)
If (M , g) is a spacetime of Kerr-Schild type, then Kerr-Schild coordinates are coordinates
(x ) = (t, x, y, z) that are Minkowskian for f , i.e. coordinates in which the flat metric f takes
α
the form
f = −dt2 + dx2 + dy 2 + dz 2 . (C.8)
C.1.2 Basic property
Property C.1: null vector of Kerr-Schild metric as a geodesic vector
Let g be a Kerr-Schild metric. If g obeys the vacuum Einstein equation (1.44), i.e. if the
Ricci tensor of g vanishes identically, then the scalar field H appearing in Eq. (C.1) can be
chosen so that k is a geodesic vector fielda :
k µ ∇µ k α = 0 , (C.9)
where ∇ stands for the covariant derivative associated with g.
a
See Sec. B.2.2 for the definition of a geodesic vector field.
The proof of the above proposition can be found in Ref. [314].
C.2 Case of Kerr spacetime 765
C.2 Case of Kerr spacetime
C.2.1 Kerr-Schild form
Let consider the Kerr spacetime (M , g), where M is the manifold (10.29): M = R2 × S2 \ R
and g is the metric tensor given by Eq. (10.36) in terms of the Kerr coordinates (t̃, r, θ, φ̃)
introduced in Sec. 10.3.3. Let us show that g is a Kerr-Schild metric, with the associated null
vector field k being nothing but the vector field generating the ingoing principal null geodesics
L(v,θ,
in
φ̃) discussed in Sec. 10.4. Its expression in terms of the Kerr coordinates is given by
Eq. (10.43):
k = ∂t̃ − ∂r̃ . (C.10)
In other words, the components of k with respect to the Kerr coordinates (t̃, r, θ, φ̃) are
k α = (1, −1, 0, 0). The 1-form k associated to k by g-duality is given by Eq. (10.44):
k = −dt̃ − dr + a sin2 θ dφ̃. (C.11)
Equivalently, kα = (−1, −1, 0, a sin2 θ). Let us then introduce the symmetric bilinear form
f := g − 2Hk ⊗ k, (C.12)
where H is the following scalar field on M :
mr
H := , (C.13)
ρ2
with ρ2 := r2 + a2 cos2 θ [Eq. (10.9)]. The expression of f in terms of Kerr coordinates is
deduced from that of g [Eq. (10.36)] and that of k [Eq. (C.11)]:
f = −dt̃2 + dr2 − 2a sin2 θ dr dφ̃ + ρ2 dθ2 + (r2 + a2 ) sin2 θ dφ̃2 . (C.14)
It is easy to check that f αβ := g αβ + 2Hk α k β defines an inverse of f : f αµ fµβ = δ αβ (compu-
tation similar to that in Sec. C.1.1). Hence the symmetric bilinear form f is non-degenerate;
this implies that f is a metric tensor on M (cf. Sec. A.3.1). Given the components (C.11), it is
immediate to check that k is a null vector for f as well: f (k, k) = 0. Moreover, f is a flat
metric, since a direct computation of its Riemann tensor (cf. the notebook D.5.4) reveals that
Riem(f ) = 0. (C.15)
In view of the definition given in Sec. C.1.1, we conclude:
Property C.2: Kerr metric as a Kerr-Schild metric
The Kerr metric g is a Kerr-Schild metric, i.e. it can be written in the form (C.1) with the
flat metric f given by Eq. (C.14), the scalar field H given by Eq. (C.13) and the null vector k
given by Eq. (C.10), k being the tangent vector field to the ingoing principal null geodesics.
766 Kerr-Schild metrics
In Sec. 10.4, we have already noticed that k is a geodesic vector: ∇k k = 0 [Eq. (10.45)], in
agreement with (C.9).
Remark 1: The Kerr metric can also be brought to the Kerr-Schild form by using the tangent vector
field to the outgoing principal null geodesics. Hence the Kerr-Schild decomposition (C.1) is not unique
for the Kerr metric.
C.2.2 Kerr-Schild coordinates on Kerr spacetime
It is not immediately obvious that the metric f given by Eq. (C.14) is a flat Lorentzian metric,
except at the limit a = 0. Let us introduce coordinates in which f takes a manifest Minkowskian
form, i.e. Kerr-Schild coordinates, according to the nomenclature introduced in Sec. C.1.1.
Actually, if a ̸= 0, one cannot introduce a Kerr-Schild coordinate system on the whole
spacetime manifold M = R2 × S2 \ R as defined by Eq. (10.29). One has to split it in two parts:
M := M+ ∪ M− , (C.16a)
M+ := R × [0, +∞) × S2 \ R (C.16b)
M− := R × (−∞, 0] × S2 \ R. (C.16c)
In other words, M+ is the part r ≥ 0 of M , while M− is the part r ≤ 0. Note that M+ and
M− are submanifolds with boundaries (cf. Sec. A.2.2), which overlap at r = 0. In terms of
the domains introduced in Sec. 10.2.1, M+ contains MI , MII and a part of MIII , while M− is
entirely included in MIII . The Kerr-Schild coordinates (t̃, x, y, z) on M+ are defined from
the Kerr coordinates (t̃, r, θ, φ̃) by the following formulas:
t̃ = t̃ (C.17a)
x = (r cos φ̃ − a sin φ̃) sin θ (C.17b)
y = (r sin φ̃ + a cos φ̃) sin θ (C.17c)
z = r cos θ. (C.17d)
Remark 2: As we shall see in Sec. C.2.3, the Kerr-Schild coordinates are singular at r = 0, so strictly
speaking, we should have omitted r = 0 from the definition of M+ and M− .
Remark 3: For a = 0, Eqs. (C.17b)-(C.17d) reduce to the standard relations between Cartesian and
spherical coordinates in Euclidean space.
Remark 4: Equations (C.17b)-(C.17c) can be combined into a single relation:
x + iy = (r + ia)eiφ̃ sin θ. (C.18)
From Eqs. (C.17b)-(C.17c), we get
x2 + y 2 = (r2 + a2 ) sin2 θ. (C.19)
C.2 Case of Kerr spacetime 767
Combining with Eq. (C.17d) yields:
x2 + y 2 z 2
+ 2 =1. (C.20)
r 2 + a2 r
This is a quadratic equation in r2 . Solving it results in
r
1 2
(C.21)
p
r= 2 2 2 2 2 2 2 2
x + y + z − a + (x + y + z − a ) + 4a z . 2 2
2
The components of f in terms on the coordinates (xα ) = (t̃, x, y, z) are obtained via the
tensor change-of-components formula with the transformation (C.17) (cf. the notebook D.5.4):
f = −dt̃2 + dx2 + dy 2 + dz 2 . (C.22)
This proves that (t̃, x, y, z) are Kerr-Schild coordinates, as announced.
The expression of the vector k in terms of the Kerr-Schild coordinates is obtained similarly:
rx + ay ry − ax z
k = ∂t̃ − 2 2
∂x − 2 2
∂y − ∂z . (C.23)
r +a r +a r
In this formula, r is to be considered as the function of (x, y, z) given by Eq. (C.21). For the
associated 1-form, we get
rx + ay ry − ax z
k = −dt̃ − 2 2
dx − 2 2
dy − dz. (C.24)
r +a r +a r
The scalar factor H can be re-expressed from Eq. (C.13) in terms of z and r:
mr3
H= 4 . (C.25)
r + a2 z 2
Remark 5: If a = 0 (Schwarzschild limit), we get
p m x y z
r = x2 + y 2 + z 2 , H = and k = −dt̃ − dx − dy − dz. (C.26)
r r r r
p 6: For a ̸= 0, the relations (C.26) hold at first order in the limit r ≫ a, or equivalently in the
Remark
limit x2 + y 2 + z 2 ≫ a.
The explicit form of the components (gαβ ) of the Kerr metric in Kerr-Schild coordinates
can be read off by expanding the following expression of g:
g = −dt̃2 + dx2 + dy 2 + dz 2
2mr3
rx + ay ry − ax z
2 , (C.27)
+ 4 dt̃ + 2 dx + 2 dy + dz
r + a2 z 2 r + a2 r + a2 r
which is obtained by combining Eqs (C.2), (C.22), (C.25) and (C.24).
768 Kerr-Schild metrics
Figure C.1: Surface t̃ = const, φ̃ = 0 or π and r ≥ 0 of the a = 0.9 m Kerr spacetime depicted in terms of the
Kerr-Schild coordinates (x, y, z). The drawing is limited to r ≤ 3m. The vertical thick green line is the axis of
rotation. On the right of it φ̃ = 0, while on the left of it φ̃ = π. The red lines are curves r = const, while the
green ones are curves θ = const, which can be thought of as the traces of the ingoing principal null geodesics.
The thick black curve marks the black hole event horizon and the thick blue curve the Cauchy horizon. The thick
red segment along the y-axis marks the intersection of the surface with the disk r = 0. [Figure produced with the
notebook D.5.4]
Remark 7: It is clear on (C.27) that all metric components
√ in Kerr-Schild coordinates are regular both
at the the black
√ hole event horizon (r = m + m2 − a2 , cf. Sec. 10.5.2) and the Cauchy horizon
(r = m − m2 − a2 , cf. Sec. 10.8.3). This property, which is shared by the Kerr coordinates, is in
sharp contrast with the metric components in Boyer-Lindquist coordinates, which are singular at both
horizons (cf. Sec. 10.2.6).
Finally, the axisymmetry Killing vector η = ∂φ̃ of Kerr spacetime [Eq. (10.39)] has the
following expression in terms of Kerr-Schild coordinates:
η = −y ∂x + x ∂y . (C.28)
This is easily established from the chain rule and the partial derivative with respect to φ̃ of
expressions (C.17). We notice on Eq. (C.28) that η is also a Killing vector for the flat metric f ,
namely the Killing vector generating spatial rotations about the z-axis.
The identity (C.20) shows that, in the Euclidean space spanned by the (x, y, z) coordinates,
the surfaces of constant r ̸= 0 are confocal1 ellipsoids of revolution. This is depicted in Fig. C.1,
which represents a slice t̃ = const and φ̃ = 0 or π in terms of the (x, y, z) coordinates. Note that
1
In any plane containing the axis of symmetry x2 + y 2 = 0, the trace of the ellipsoids are ellipses that share
the same foci, located at the abscissas ±a along the z = 0 axis.
C.2 Case of Kerr spacetime 769
Figure C.2: Surfaces (t̃, θ) = const of the a = 0.9 m Kerr spacetime depicted in terms of the Kerr-Schild
coordinates (x, y, z). The disk-like surface in the plane z = 0 is for θ = π/2, while the cone-like surface is for
θ = π/6. The pale brown lines are curves (r, θ) = const, while the green ones are curves (θ, φ̃) = const. The
latter can be thought of as the traces of the ingoing principal null geodesics. The central pink disk is the (double)
disk r = 0, the boundary of which is the curvature singularity. [Figure produced with the notebook D.5.4]
this slice is not a plane but a warped surface, with a kink along the red segment −a < y < a at
(x, z) = (0, 0), which is the intersection of the slice with the double disk r = 0 (to be√discussed
below). Peculiar r = const surfaces are the black hole event √ horizon (r = r+ = m + m − a ,
2 2
cf. Sec. 10.5.2) and the Cauchy horizon (r = r− = m − m2 − a2 , cf. Sec. 10.8.3). They are
depicted in respectively black and blue in Fig. C.1.
Since the ingoing principal null geodesics L(v,θ,in
φ̃) (cf. Sec. 10.4) are curves (θ, φ̃) = const,
their traces in the 3-space of Fig. C.1 are the green lines that terminates at the disk r = 0 (the
red segment along the y-axis). Another view of the ingoing principal null geodesics is provided
by Fig. C.2, which shows two surfaces (t̃, θ) = const in terms of the Kerr-Schild coordinates
(x, y, z), namely the surfaces θ = π/6 and θ = π/2. We notice that, although they are straight
lines in terms of the Kerr-Schild coordinates, the ingoing principal null geodesics are winding
around the rotation axis in the direction of the black hole rotation, which is indicated by η [cf.
Eq. (C.28)].
C.2.3 The double-disk r = 0
For r = 0, the system (C.17) reduces to
t̃ = t̃ (C.29a)
x = −a sin θ sin φ̃ (C.29b)
y = a sin θ cos φ̃ (C.29c)
z = 0 (C.29d)
For a ̸= 0 and a fixed value of t̃, the subset r = 0 of the Kerr spacetime M is the set S0,t
discussed in Sec. 10.2.2 [cf. Eq. (10.14)], where t is related to t̃ via Eq. (10.35a), which for r = 0,
is basically t̃ = t + const. S0,t is topologically a 2-sphere minus its equator. It is therefore
770 Kerr-Schild metrics
0.5
y/m
0
-0.5
-1
-1 -0.5 0 0.5 1
x/m
Figure C.3: Disk (actually double disk) r = 0 of the a = 0.9 m Kerr spacetime depicted in terms of the
Kerr-Schild coordinates (x, y). The pale brown circles are the curves θ = const, while the red segments are the
curves φ̃ = const. The disk boundary at x + y 2 = a is the curvature singularity of Kerr spacetime. [Figure
p
2
produced with the notebook D.5.4]
disconnected, with two connected components: the two open hemispheres, S0,t +
and S0,t
−
say. As
shown in Sec. 10.2.2, S0,t+
and S0,t
−
are actually two disks that, with respect to the metric induced
by g, (i) have radius a and (ii) are flat. The disk S0,t
+
is spanned by the coordinates (θ, φ̃) with
θ ∈ [0, π/2), while S0,t is spanned by the coordinates (θ, φ̃) with θ ∈ (π/2, π]. Let p ∈ S0,t
− +
be a
point of Kerr coordinates (t̃, 0, θ, φ̃). The point q of coordinates (t̃, 0, π − θ, φ̃) belongs to S0,t
−
;
it is therefore distinct from p, since S0,t and S0,t are two disjoint sets. Now, the transformation
+ −
(C.29) maps both p and q to the same value (t̃, −a sin θ sin φ̃, a sin θ cos φ, 0) of the Kerr-Schild
coordinates (t̃, x, y, z). We conclude that, for r = 0, the Kerr-Schild coordinate system fails to
establish a one-to-one correspondence between the manifold points and some open subset of
R4 . Actually, as it is clear from (C.29), the two disks S0,t +
and S0,t
−
are mapped by Kerr-Schild
coordinates to a single disk, namely the disk of radius a centered at (x, y, z) = (0, 0, 0) in
the plane z = 0. This disk is shown in Fig. C.2 (pink central disk) and in Fig. C.3; its section
p= 0 or π is depicted as the red segment in Fig. C.1. The disk boundary is the circle z = 0,
φ̃
x2 + y 2 = a; it is not part of M , being the curvature singularity of Kerr spacetime (cf.
Sec. 10.2.6).
C.2.4 Kerr-Schild coordinates on the r ≤ 0 part
On the domain M− , i.e. for r ≤ 0, one can introduce another patch of Kerr-Schild coordinates,
(t̃, x′ , y ′ , z ′ ) say, by formulas similar to (C.17). A difference is when expressing the square root
for r as in Eq. (C.21), one has to take the minus sign, so that
s
1
q
r := − ′ 2 ′ 2 ′ 2 2 ′ 2 ′ 2 ′ 2 2 2 2 ′
x + y + z − a + (x + y + z − a ) + 4a z .2
(C.30)
2
C.2 Case of Kerr spacetime 771
Figure C.4: Three views of the immersion of the full t̃ = const and φ̃ = 0 or π surface of the a = 0.9 m
Kerr spacetime in the Euclidean space R3 , using the Kerr-Schild coordinates (x, y, z) for the r ≥ 0 part (drawn
in grey) and the Kerr-Schild coordinates (x′ , y ′ , z ′ ) for the r ≤ 0 part (drawn in pink). The red lines are curves
(r, φ̃) = const, while the green straight lines are ingoing principal null geodesics, which obey (θ, φ̃) = const.
The thick (resp. blue) black curve marks the black hole event horizon (resp. Cauchy horizon). [Figure produced
with the notebook D.5.4; see this notebook for an interactive 3D view]
The two Kerr-Schild coordinate domains M+ and M− are connected through the r = 0
hypersurface. On a t̃ = const slice, this means being connected through the double disk S0,t .
Such a connection is depicted in Fig. C.4, which represents the (non-isometric) immersion in R3
of the surface t̃ = const and φ̃ ∈ {0, π}, with r ranging from −∞ to +∞ (actually from −3m
to 3m on the figure). The immersion is not an embedding2 because the sheet r ≥ 0 (in grey)
intersects the sheet r ≤ 0 (in pink) along the rotation axis, while the only intersection of M+
and M− is along the double disk S0,t (reduced to the segment −a < y < a at (x, z) = (0, 0)
in Fig. C.4). In other words, the intersection of the grey and pink sheets along the z-axis in
Fig. C.4 is spurious (does not correspond to an intersection in the physical spacetime), while
the intersection along the y-axis is physical. From the central and right views in Fig. C.4, one
sees clearly that the ingoing principal null geodesics go smoothly from the r > 0 region to the
r < 0 region through S0,t .
C.2.5 Link with Boyer-Lindquist coordinates
The link between the Kerr-Schild coordinates (t̃, x, y, z) and the Boyer-Lindquist coordinates
(t, r, θ, φ) is obtained by combining Eqs. (C.17) with Eqs. (10.35).
Historical note : Kerr-Schild coordinates on Kerr spacetime have been introduced by Roy Kerr in the
famous 1963 paper [312] announcing the discovery of the Kerr metric. They have been discussed further
by Robert Boyer and Richard Lindquist in 1967 [72], Brandon Carter in 1968 [90] and Stephen Hawking
2
See Sec. A.2.7 for the definitions of immersion and embedding.
772 Kerr-Schild metrics
and George Ellis in 1973 [266]. Generic Kerr-Schild metrics have been introduced and studied by Roy
Kerr and Alfred Schild in 1965 [314].
Appendix D
SageMath computations
Contents
D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773
D.2 Minkowski spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774
D.3 Anti-de Sitter spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . 774
D.4 Schwarzschild spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . 774
D.5 Kerr spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778
D.6 Evolution and thermodynamics . . . . . . . . . . . . . . . . . . . . . . 781
D.1 Introduction
SageMath (https://www.sagemath.org/) is a modern free, open-source mathematics soft-
ware system, which is based on the Python programming language. It makes use of over 90
open-source packages, among which are Maxima, Pynac and SymPy (symbolic calculations),
GAP (group theory), PARI/GP (number theory), Singular (polynomial computations), and
matplotlib (high quality 2D figures). SageMath provides a uniform Python interface to all
these packages. However, SageMath is more than a mere interface: it contains a large and
increasing part of original code (more than 750,000 lines of Python and Cython, involving 5344
classes). SageMath was created in 2005 by William Stein [462] and since then its development
has been sustained by more than a hundred researchers (mostly mathematicians). Very good
introductory textbooks about SageMath are [532, 306, 34].
The SageManifolds project (https://sagemanifolds.obspm.fr/) provides SageMath
with tools for differential geometry and tensor calculus, which are used here to perform
computations relative to black hole spacetimes.
There are basically two ways to use SageMath:
• Install it on your computer, by downloading the sources or a binary version from https:
//www.sagemath.org/ (the SageManifolds extensions towards differential geometry are
fully integrated in version 7.5 and higher)
774 SageMath computations
• Use it online via CoCalc: https://cocalc.com/
The SageMath notebooks (Jupyter format) accompanying these lecture notes are available
at the nbviewer.jupyter.org links provided below. Clicking on the link leads to a read-only view
of the notebook. Then, by clicking on the Execute on Binder button (the three interlaced circles
in the top right menu), one gets access to a freely modifiable and executable version. All the
notebooks are also collected on the page
https://relativite.obspm.fr/blackholes/sage.html
D.2 Minkowski spacetime
D.2.1 Conformal completion of Minkowski spacetime
This notebook accompanies Chap. 4; in particular, it provides many figures for Sec. 4.2.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/conformal_Minkowski.
ipynb
D.3 Anti-de Sitter spacetime
D.3.1 Poincaré horizon in AdS4 as a degenerate Killing horizon
This notebook accompanies Example 18 of Chap. 3.
https://nbviewer.jupyter.org/github/sagemanifolds/SageManifolds/blob/master/Notebooks/SM_anti_
de_Sitter_Poincare_hor.ipynb
D.3.2 Conformal completion of AdS4
This notebook introduces AdS4 in various coordinate systems and provides the figure of Exam-
ple 1 in Chap. 4.
https://nbviewer.jupyter.org/github/sagemanifolds/SageManifolds/blob/master/Notebooks/SM_anti_
de_Sitter.ipynb
D.4 Schwarzschild spacetime
D.4.1 The Schwarzschild horizon
This notebook accompanies Chap. 2 in treating the future event horizon of Schwarzschild
spacetime in Eddington-Finkelstein coordinates as an example of null hypersurface:
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Schwarzschild_horizon.
ipynb
D.4 Schwarzschild spacetime 775
D.4.2 Solving the Einstein equation: Kottler solution
This notebook accompanies Chap. 6: it computes the Kottler solution by solving the Einstein
equation for vacuum spherically symmetric spacetimes with a cosmological constant Λ, yield-
ing Schwarzschild solution in the special case Λ = 0.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kottler_solution.
ipynb
D.4.3 Kretschmann scalar of Schwarzschild spacetime
This notebook accompanies Chap. 6: it computes the Riemann curvature tensor of Schwarzschild
metric and evaluates the Kretschmann scalar as defined by Eq. (6.44).
https://nbviewer.jupyter.org/github/sagemanifolds/SageManifolds/blob/master/Notebooks/SM_basic_
Schwarzschild.ipynb
D.4.4 Radial null geodesics in Schwarzschild spacetime
This notebook accompanies Chap. 6: it provides figures based on Schwarzschild-Droste coordi-
nates and ingoing Eddington-Finkelstein coordinates.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Schwarz_radial_
null_geod.ipynb
D.4.5 Radial timelike geodesics in Schwarzschild spacetime
This notebook accompanies Chap. 7: it provides figures as well as the computation of the
integral leading to of Eq. (7.42).
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/ges_radial_free_
fall.ipynb
D.4.6 Timelike orbits in Schwarzschild spacetime
This notebook accompanies Chap. 7: it provides figures of timelike orbits in the equatorial plane.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/ges_orbits.ipynb
D.4.7 Effective potential for null geodesics in Schwarzschild spacetime
This notebook accompanies Chap. 8, providing the plot of the effective potential U (r).
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/ges_effective_potential_
null.ipynb
776 SageMath computations
D.4.8 Null geodesics in Schwarzschild spacetime
This notebook accompanies Chap. 8, computing and plotting various null geodesics.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/ges_null_geod.ipynb
D.4.9 Periastron and apoastron of null geodesics in Schwarzschild
spacetime
This notebook accompanies Chap. 8, computing periastrons and apoastrons along null geodesics.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/ges_null_periastron.
ipynb
D.4.10 Critical null geodesics in Schwarzschild spacetime
This notebook accompanies Chap. 8, plotting critical null geodesics.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/ges_null_critical_
geod.ipynb
D.4.11 Elliptic integrals for null geodesics in Schwarzschild spacetime
This notebook accompanies Chap. 8, computing the trace of null geodesics in the equatorial
plane via elliptic integrals.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/gis_elliptic_int.
ipynb
D.4.12 Null geodesics in Schwarzschild spacetime with b < bc
This notebook accompanies Chap. 8, computing various quantities that are relevant for null
geodesics with an impact parameter lower than the critical one.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/gis_paramaters_
b_lt_bc.ipynb
D.4.13 Multiple images in Schwarzschild spacetime
This notebook accompanies Chap. 8, computing null geodesics that depart in a fixed direction
(the observer one).
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/ges_null_images.
ipynb
D.4 Schwarzschild spacetime 777
D.4.14 Emission from a point source in Schwarzschild spacetime
This notebook accompanies Chap. 8, computing quantities related to the emission by a static
observer.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/gis_emission.ipynb
D.4.15 Images of an accretion disk around a Schwarzschild black hole
This notebook accompanies Chap. 8, computing null geodesics illustrating the formation of
images of an accretion disk.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/gis_disk_image.
ipynb
D.4.16 Kruskal-Szekeres coordinates in Schwarzschild spacetime
This notebook accompanies Chap. 9: it provides the figures based on Kruskal-Szekeres coordi-
nates.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Schwarz_Kruskal_
Szekeres.ipynb
D.4.17 Standard (singular) Carter-Penrose diagram of Schwarzschild
spacetime
This notebook accompanies Chap. 9: it provides the standard Carter-Penrose diagram shown
in Fig. 9.10.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Schwarz_conformal_
std.ipynb
D.4.18 Regular Carter-Penrose diagram of Schwarzschild spacetime
This notebook accompanies Chap. 9: it provides the regular Carter-Penrose diagram shown in
Fig. 9.11.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Schwarz_conformal.
ipynb
D.4.19 Einstein-Rosen bridge in Schwarzschild spacetime
This notebook accompanies Chap. 9: it provides the isometric embedding diagrams shown in
Figs. 9.14 to 9.16, as well as the associated Kruskal diagram of Fig. 9.12.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Einstein-Rosen_
bridge.ipynb
778 SageMath computations
D.5 Kerr spacetime
D.5.1 Kerr metric as a solution of the Einstein equation
This notebook accompanies Chap. 10: the Kerr metric, expressed in Boyer-Lindquist coordi-
nates, is shown to be a solution of the vacuum Einstein equation. Moreover, the Kretschmann
scalar is computed.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_solution.ipynb
D.5.2 Kerr spacetime in Kerr coordinates
These two notebooks accompany Chap. 10: the Kerr metric is expressed in advanced Kerr
coordinates (v, r, θ, φ̃) (notebook 1) and in Kerr coordinates (t̃, r, θ, φ̃) (notebook 2), the vacuum
Einstein equation is checked, the outgoing and ingoing principal null geodesics are considered
and the black hole surface gravity is computed.
1. https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_in_advanced_
Kerr_coord.ipynb
2. https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_in_Kerr_
coord.ipynb
D.5.3 Plot of principal null geodesics in Kerr spacetime
This notebook provides some figures for Chap. 10.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_princ_null_
geod.ipynb
D.5.4 Kerr-Schild coordinates on Kerr spacetime
This notebook accompanies Appendix C: the Kerr metric is expressed in a Kerr-Schild form
and Kerr-Schild coordinates are introduced.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_Schild.ipynb
D.5.5 ZAMO frame on Kerr spacetime
This notebook accompanies Chap. 10, providing various formulas relative to the orthonormal
frame carried by a Zero-angular-momentum observer (ZAMO).
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_ZAMO_frame.
ipynb
D.5 Kerr spacetime 779
D.5.6 Carter frame on Kerr spacetime
This notebook accompanies Chap. 10, providing various formulas relative to the orthonormal
frame carried by a Carter observer.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_Carter_frame.
ipynb
D.5.7 Walker-Penrose Killing tensor on Kerr spacetime
This notebook accompanies Chap. 11; it shows that the tensor K defined by Eq. (11.16) is
actually a Killing tensor.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_Killing_tensor.
ipynb
D.5.8 Timelike and null geodesics in Kerr spacetime
This notebook accompanies Chap. 11, computing and plotting null and timelike geodesics.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_geod_plots.
ipynb
D.5.9 Circular equatorial orbits in Kerr spacetime
This notebook provides some figures for Chap. 11.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_circular_orbits.
ipynb
D.5.10 Zero-energy null geodesics in Kerr spacetime
This notebook provides some figures for Chap. 12.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_null_geod_
zero_ener.ipynb
D.5.11 Existence and stability of spherical photon orbits in Kerr space-
time
This notebook provides some figures for Chap. 12.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_spher_photon_
existence.ipynb
D.5.12 Plots of spherical photon orbits geodesics in Kerr spacetime
This notebook provides some figures for Chap. 12.
780 SageMath computations
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_spher_null_
geod.ipynb
D.5.13 Plots of null geodesics in Kerr spacetime
This notebook provides some figures for Chap. 12.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_null_geod_
plots.ipynb
D.5.14 Shadow and critical curve of a Kerr black hole
This notebook provides some figures for Chap. 12.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_shadow.ipynb
D.5.15 Images of an accretion disk around a Kerr black hole
This notebook provides some figures for Chap. 12.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_images.ipynb
D.5.16 Critical curve of a Kerr black hole onto the EHT image of M87*
This notebook generates Fig. 12.30.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_image_M87.
ipynb
D.5.17 Extremal Kerr spacetime
This notebook accompanies Chap. 13.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_extremal.ipynb
D.5.18 Maximal extension of the extremal Kerr spacetime
This notebook accompanies Chap. 13.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_extremal_extended.
ipynb
D.6 Evolution and thermodynamics 781
D.5.19 Isometric embbedings of (t, θ) sections of the extremal Kerr
spacetime
This notebook accompanies Chap. 13.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Kerr_extremal_throat_
emb.ipynb
D.5.20 Near-horizon extremal Kerr geometry
This notebook accompanies Chap. 13.
https://nbviewer.jupyter.org/github/sagemanifolds/SageManifolds/blob/master/Notebooks/SM_extremal_
Kerr_near_horizon.ipynb
D.5.21 NHEK spacetime
This notebook accompanies Chap. 13.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/NHEK_spacetime.
ipynb
D.6 Evolution and thermodynamics
D.6.1 Lemaître-Tolman equations
This notebook accompanies Chap. 14: it provides the derivation of the Lemaître-Tolman equa-
tions from the Einstein equation expressed in Lemaître synchronous coordinates.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Lemaitre_Tolman.
ipynb
D.6.2 Oppenheimer-Snyder collapse: spacetime diagrams
This notebook generates some figures for Chap. 14.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Oppenheimer_Snyder.
ipynb
D.6.3 Oppenheimer-Snyder collapse: curvature
This notebook accompanies Chap. 14: it evaluates the curvature tensor and some associated
scalars.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Oppenheimer_Snyder_
curvature.ipynb
782 SageMath computations
D.6.4 Vaidya spacetime
This notebook accompanies Chaps. 15 and 18: the Vaidya metric is expressed in Eddington-
Finkelstein coordinates, the Einstein equation is checked, the outgoing and ingoing radial
null geodesics are computed and the trapping horizon and the event horizon are drawn in a
spacetime diagram.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Vaidya.ipynb
D.6.5 Solving the ODE for outgoing radial null geodesics in Vaidya
spacetime
This notebook accompanies Chap. 15.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Vaidya_solve_ode_
out.ipynb
D.6.6 Naked singularity in Vaidya collapse
This notebook accompanies Chap. 15; it corresponds to the case where a naked singularity
appears in the collapse of a radaition shell.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/Vaidya_nk_sing.
ipynb
D.6.7 False trapped surface in Minkowski spacetime
This notebook generates Fig. 18.1.
https://nbviewer.jupyter.org/github/egourgoulhon/BHLectures/blob/master/sage/loc_cone_intersect.
ipynb
Appendix E
Gyoto computations
Contents
E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783
E.2 Image computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783
E.1 Introduction
Gyoto (https://gyoto.obspm.fr) is a free open-source C++ code for computing orbits and
ray-traced images in general relativity [493]. It has a Python interface and has the capability to
integrate geodesics not only in analytical spacetimes (such as Kerr) but also in numerical ones,
i.e. in spacetimes arising from numerical relativity.
E.2 Image computations
Here we provide the Gyoto input files, in XML format, that have been used to produce the
images shown in Chaps. 8 and 12. To generate the images, it suffices to run Gyoto as
gyoto input.xml output.fits
By default, the computation is performed in parallel on 8 threads; you can adapt to your CPU
by changing the field NThreads in the file input.xml. The output image is in FITS format and
can be converted to PNG or JPEG by most image processing programs, such as GIMP.
E.2.1 Accretion disk around a Schwarzschild black hole
The input XML files for generating the images shown in Fig. 8.25 are the files gis_disk*.xml
in the directory
https://github.com/egourgoulhon/BHLectures/tree/master/gyoto
784 Gyoto computations
E.2.2 Accretion disk around a Kerr black hole
The input XML files for generating the images shown in Figs. 12.27 and 12.28 are the files
gik_a*.xml in the directory
https://github.com/egourgoulhon/BHLectures/tree/master/gyoto
Appendix F
On the Web
Here is a selection of scientific web pages related to black holes:
• Movies of binary black holes mergers computed by the SXS team:
https://www.black-holes.org/explore/movies
• Movies from computations of the Center for Computational Relativity and Gravitation,
Rochester Institute of Technology:
https://ccrg.rit.edu/movies
• Journey around a black hole (Alain Riazuelo)
http://www2.iap.fr/users/riazuelo/bh/
• Kerr black holes images and videos (David Madore)
http://www.madore.org/~david/math/kerr.html
• Spherical photon orbits around a Kerr black hole (Edward Teo):
http://phyweb.physics.nus.edu.sg/~phyteoe/kerr/
• Kerr Spherical Photon Orbits (Leo C. Stein):
https://duetosymmetry.com/tool/kerr-circular-photon-orbits/
• Scratch pad for visualizing bound, timelike geodesics in Kerr spacetime:
http://nielswarburton.net/geodesics/interactive/Kerr_geodesic.html
• Gyoto gallery (images and movies of accretion flows around black holes and alternative
compact objects; see also Appendix E):
https://gyoto.obspm.fr/gallery/
• Movie of Oppenheimer-Snyder collapse (Hirotaka Yoshino):
http://ysnhrtk.web.fc2.com/animations.html
786 On the Web
Bibliography
[1] R. Abraham and J.E. Marsden: Foundations of Mechanics, second edition, Addison-Wesley,
Redwood City (1978).
[2] M.A. Abramowicz and P.C. Fragile: Foundations of Black Hole Accretion Disk Theory,
Living Rev. Relativ. 16, 1 (2013).
[3] M. Abramowitz and I. Stegun: Handbook of Mathematical Functions with Formulas, Graphs,
and Mathematical Tables, NBS, 10th printing (1972); updated in the NIST Digital Library
of Mathematical Functions, available at https://dlmf.nist.gov/.
[4] R. Abuter et al. (GRAVITY Collaboration): Detection of the gravitational redshift in the
orbit of the star S2 near the Galactic centre massive black hole, Astron. Astrophys. 615,
L15 (2018); arXiv: 1807.09409.
[5] R. Abuter et al. (GRAVITY Collaboration): Detection of the Schwarzschild precession in
the orbit of the star S2 near the Galactic centre massive black hole, Astron. Astrophys. 636,
L5 (2020); arXiv: 2004.07187.
[6] K. Akiyama et al. (the Event Horizon Telescope Collaboration): First M87 Event Horizon
Telescope Results. I. The Shadow of the Supermassive Black Hole, Astrophys. J. Lett. 875,
L1 (2019); arXiv: 1906.11238.
[7] K. Akiyama et al. (the Event Horizon Telescope Collaboration): First M87 Event Horizon
Telescope Results. V. Physical Origin of the Asymmetric Ring, Astrophys. J. Lett. 875, L5
(2019); arXiv: 1906.11242.
[8] K. Akiyama et al. (the Event Horizon Telescope Collaboration): First M87 Event Horizon
Telescope Results. VI. The Shadow and Mass of the Central Black Hole, Astrophys. J. Lett.
875, L6 (2019); arXiv: 1906.11243.
[9] K. Akiyama et al. (the Event Horizon Telescope Collaboration): The persistent shadow of
the supermassive black hole of M 87. I. Observations, calibration, imaging, and analysis,
Astron. Astrophys. 681, A79 (2024).
[10] M. Alcubierre: Introduction to 3+1 Numerical Relativity, Oxford Univ. Press, Oxford (2008)
DOI: 10.1093/acprof:oso/9780199205677.001.0001
[11] S. Alexakis, A. D. Ionescu, and S. Klainerman: Rigidity of stationary black holes with small
angular momentum on the horizon, Duke Math. J. 163, 2603 (2014); arXiv: 1304.0487.
[12] W.L. Ames and K. Thorne: The Optical Appearance of a Star that is Collapsing Through its
Gravitational Radius, Astrophys. J. 151, 659 (1968).
788 BIBLIOGRAPHY
[13] L. Andersson, T. Bäckdahl, and P. Blue: Geometry of Black Hole Spacetimes, in Asymptotic
Analysis in General Relativity, edited by T. Daudé, D. Häfner and J.-P. Nicolas, Cambridge
Univ. Press, Cambridge (2018), p. 9; DOI: 10.1017/9781108186612.002; arXiv: 1610.03540.
[14] N. Andersson: Gravitational-Wave Astronomy: Exploring the Dark Side of the Universe,
Oxford Univ. Press, Oxford (2020); DOI: 10.1093/oso/9780198568032.001.0001.
[15] S. Ansoldi: Spherical black holes with regular center: a review of existing models including
a recent realization with Gaussian sources, in Proceedings of the second edition of the
international workshop Dynamics and Thermodynamics of Blackholes and Naked Sin-
gularities, edited by L. Fatibene, M. Francaviglia, R. Giambò and G. Magli, published as
arXiv:1205.2839 (2008); arXiv: 0802.0330.
[16] T. Anson, E. Babichev, C. Charmousis and M. Hassaine: Disforming the Kerr metric,
J. High Energ. Phys. 01(2021), 18 (2021); arXiv: 2006.06461.
[17] S. Aretakis: Nonlinear instability of scalar fields on extremal black holes, Phys. Rev. D 87,
084052 (2013); arXiv: 1304.4616.
[18] S. Aretakis: Horizon instability of extremal black holes, Adv. Theor. Math. Phys. 19, 507
(2015); arXiv: 1206.6598.
[19] S. Aretakis: Dynamics of Extremal Black Holes, Springer, Cham (2018); DOI: 10.1007/978-
3-319-95183-6.
[20] R. Arnowitt, S. Deser and C.W. Misner: The dynamics of general relativity, in Gravitation:
an introduction to current research, edited by L. Witten, Wiley, New York (1962), p. 227;
reprinted in Gen. Relat. Grav. 40, 1997 (2008); arXiv: gr-qc/0405109.
[21] A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov: Quantum Geometry and Black Hole
Entropy, Phys. Rev. Lett. 80, 904 (1998); arXiv: gr-qc/9710007.
[22] A. Ashtekar, C. Beetle, and J. Lewandowski: Mechanics of rotating isolated horizons, Phys.
Rev. D 64, 044016 (2001); arXiv: gr-qc/0103026.
[23] A. Ashtekar, C. Beetle, and J. Lewandowski: Geometry of generic isolated horizons, Class.
Quantum Grav. 19, 1195 (2002); arXiv: gr-qc/0111067.
[24] A. Ashtekar, L. Bombelli, and O. Reula: The Covariant Phase Space of Asymptotically
Flat Gravitational Fields, in Mechanics, Analysis and Geometry: 200 Years After Lagrange,
edited by M. Francaviglia, North Holland, Amsterdam (1991), p. 417; DOI: 10.1016/B978-
0-444-88958-4.50021-5.
[25] A. Ashtekar, S. Fairhurst, and B. Krishnan: Isolated horizons: Hamiltonian evolution and
the first law, Phys. Rev. D 62, 104025 (2000); arXiv: gr-qc/0005083.
[26] A. Ashtekar, N. Khera, M. Kolanowski, and J. Lewandowski: Non-expanding horizons: mul-
tipoles and the symmetry group, J. High Energ. Phys. 2022, 28 (2022); arXiv: 2111.07873.
[27] A. Ashtekar and B. Krishnan: Isolated and dynamical horizons and their applications,
Living Rev. Relativ. 7, 10 (2004).
[28] A. Ashtekar and A. Magnon-Ashtekar: On conserved quantities in general relativity,
J. Math. Phys. 20, 793 (1979).
[29] E. Ayón-Beato and A. García: The Bardeen model as a nonlinear magnetic monopole, Phys.
Lett. B 493, 149 (2000); arXiv: gr-qc/0009077.
BIBLIOGRAPHY 789
[30] C. Bambi: Black Holes: A Laboratory for Testing Strong Gravity, Springer, Singapore
(2017); DOI: 10.1007/978-981-10-4524-0.
[31] C. Bambi (Ed.) : Regular Black Holes: Towards a New Paradigm of Gravitational Collapse,
Springer, Singapore (2023); DOI: 10.1007/978-981-99-1596-5.
[32] R. Banerjee, B.R. Majhi, S.K. Modak, and S. Samanta: Killing symmetries and Smarr
formula for black holes in arbitrary dimensions, Phys. Rev. D 82, 124002 (2010); arXiv:
1007.5204.
[33] J.F. Barbero G. and D. Pranzetti: Black Hole Entropy in Loop Quantum Gravity, in Handbook
of Quantum Gravity, edited by C. Bambi, L. Modesto, and I. Shapiro, I., Springer, Singapore
(2023); DOI: 10.1007/978-981-19-3079-9_104-1; arXiv: 2212.13469.
[34] G.V. Bard: Sage for Undergraduates, Americ. Math. Soc. (2015); preprint freely available
from http://www.gregory-bard.com/Sage.html.
[35] J.M. Bardeen: Non-singular general relativistic gravitational collapse, in Abstracts of the
5th international conference on gravitation and the theory of relativity, edited by V.A. Fock
et al., Tbilisi University Press (1968).
[36] J.M. Bardeen: Kerr Metric Black Holes, Nature 226, 64 (1970).
[37] J.M. Bardeen: A Variational Principle for Rotating Stars in General Relativity, Astrophys. J.
162, 71 (1970).
[38] J.M. Bardeen: Timelike and Null Geodesics in the Kerr Metric, in Black Holes – Les astres
occlus, edited by C. DeWitt and B. DeWitt, Gordon and Breach, New York (1973), p. 215.
[39] J.M. Bardeen: Rapidly Rotating Stars, Disks, and Black Holes, in Black Holes – Les astres
occlus, edited by C. DeWitt and B. DeWitt, Gordon and Breach, New York (1973), p. 241.
[40] J.M. Bardeen, B. Carter, and S.W. Hawking: The Four Laws of Black Hole Mechanics,
Commun. Math. Phys., 31, 161 (1973).
[41] J. Bardeen and G.T. Horowitz: Extreme Kerr throat geometry: A vacuum analog of AdS 2 ×
S 2 , Phys. Rev. D 60, 104030 (1999); arXiv: hep-th/9905099.
[42] J.M. Bardeen, W.H. Press, and S.A. Teukolsky: Rotating Black Holes: Locally Nonrotating
Frames, Energy Extraction, and Scalar Synchrotron Radiation, Astrophys. J. 178, 347 (1972).
[43] H. Barzegar, P.T. Chruściel, and M. Hörzinger: Energy in higher-dimensional spacetimes,
Phys. Rev. D 96, 124002 (2017); arXiv: 1708.03122.
[44] T.W. Baumgarte and S.L. Shapiro: Numerical Relativity: Solving Einstein’s Equations on the
Computer, Cambridge Univ. Press, Cambridge (2010); DOI: 10.1017/CBO9781139193344
[45] C.A.B. Bayona and N.R.F. Braga: Anti-de Sitter boundary in Poincaré coordinates, Gen.
Relativ. Gravit. 39, 1367 (2007); arXiv: hep-th/0512182.
[46] D.L. Beckedorff: Terminal Configurations of Stellar Evolution, undergraduate senior
thesis, Princeton University (1962); URL: http://arks.princeton.edu/ark:/88435/
dsp019593tx34r.
[47] K. Beckwith and C. Done: Extreme gravitational lensing near rotating black holes, Month.
Not. Roy. Astron. Soc. 359, 1217 (2005); arXiv: astro-ph/0411339.
[48] J.K. Beem, P.E. Ehrlich and K.L. Easley: Global Lorentzian Geometry, 2nd edition, Marcel
Dekker, New York (1996); DOI: 10.1201/9780203753125.
790 BIBLIOGRAPHY
[49] R. Beig: Arnowitt-Deser-Misner energy and g00 , Phys. Lett. A 69, 153 (1978).
[50] J.D. Bekenstein: Black holes and the second law, Lett. Nuovo Cimento 4, 737 (1972).
[51] J.D. Bekenstein: Extraction of Energy and Charge from a Black Hole, Phys. Rev. D 7, 949
(1973).
[52] J.D. Bekenstein: Black Holes and Entropy, Phys. Rev. D 7, 2333 (1973).
[53] J.D. Bekenstein: Generalized second law of thermodynamics in black-hole physics, Phys.
Rev. D 9, 3292 (1974).
[54] J. Ben Achour, H. Liu, H. Motohashi, S. Mukohyama, and K. Noui: On rotating black holes
in DHOST theories, J. Cosmo. Astropart. Phys. 11(2020), 001 (2020); arXiv: 2006.07245.
[55] R. Bender et al.: HST STIS Spectroscopy of the Triple Nucleus of M31: Two Nested Disks
in Keplerian Rotation around a Supermassive Black Hole, Astrophys. J. 631, 280 (2005);
arXiv: astro-ph/0509839.
[56] I. Bengtsson, E. Jakobsson, and J.M.M. Senovilla: Trapped surfaces in Oppenheimer-Snyder
black holes Phys. Rev. D 88, 064012 (2013)
DOI: 10.1103/PhysRevD.88.064012 arXiv: 1306.6486
[57] I. Bengtsson and P. Sandin: Anti-de Sitter space, squashed and stretched, Class. Quantum
Grav. 23, 971 (2006); arXiv: gr-qc/0509076.
[58] I. Bengtsson and J.M.M. Senovilla: Note on trapped surfaces in the Vaidya solution, Phys.
Rev. D 79, 024027 (2009); arXiv: 0809.2213.
[59] M. Berger: A Panoramic View of Riemannian Geometry, Springer, Berlin (2003).
[60] J. Bičák and Z. Stuchlík: On the Latitudinal and Radial Motion in the Field of a Rotating
Black Hole, Bul. Astron. Inst. Czechoslovakia 27, 129 (1976).
[61] D. Bini, A. Geralico, and R.T. Jantzen: Gyroscope precession along general timelike geodesics
in a Kerr black hole spacetime, Phys. Rev. D 95, 124022 (2017); arXiv: 1703.09525.
[62] G.D. Birkhoff: Relativity and Modern Physics, Harvard University Press, Cambridge (1923).
[63] R.L. Bishop and S.I. Goldberg: Tensor Analysis on Manifolds, Macmillan, New York (1968);
reprinted by Dover, New York (1980).
[64] G.S. Bisnovatyi-Kogan and O.Yu. Tsupko: Strong gravitational lensing by Schwarzschild
black holes, Astrofizika 51, 125 (2008); English translation in Astrophysics 51, 99 (2008);
arXiv: 0803.2468.
[65] L. Blanchet: Post-Newtonian theory for gravitational waves, Living Rev. Relativ. 27, 4
(2024).
[66] H. Bondi: Spherically Symmetrical Models in General Relativity, Month. Not. Roy. Astron.
Soc. 107, 410 (1947); reprinted in Gen. Relativ. Gravit. 31, 1783 (1999).
[67] I. Booth, L. Brits, J.A Gonzalez, and C. Van Den Broeck: Marginally trapped tubes and
dynamical horizons, Class. Quantum Grav. 23, 413 (2005); arXiv: 0506119.
[68] I. Booth and S. Fairhurst: Isolated, slowly evolving, and dynamical trapping horizons:
Geometry and mechanics from surface deformations, Phys. Rev. D 75, 084019 (2007); arXiv:
gr-qc/0610032.
BIBLIOGRAPHY 791
[69] A. Borde: Open and closed universes, initial singularities, and inflation, Phys. Rev. D 50,
3692 (1994)
DOI: 10.1103/PhysRevD.50.3692 arXiv: gr-qc/9403049
[70] A. Borde: Regular black holes and topology change, Phys. Rev. D 55, 7615 (1997); arXiv:
gr-qc/9612057
[71] R.H. Boyer: Geodesic Killing orbits and bifurcate Killing horizons, Proc. Roy. Soc. Lond. A
311, 245 (1969).
[72] R.H. Boyer and R.W. Lindquist: Maximal Analytic Extension of the Kerr Metric, J. Math.
Phys. 8, 265 (1967).
[73] R.H. Boyer and T.G. Price: An interpretation of the Kerr metric in general relativity, Math.
Proc. Cambridge Phil. Soc. 61, 531 (1965).
[74] D. Brill: History of a black hole horizon, Grav. Cosmo. 20, 165 (2014).
[75] R. Brout, S. Massar, R. Parentani, and P. Spindel: A Primer for Black Hole Quantum Physics,
Phys. Rep. 260, 329 (1995); arXiv: 0710.4345.
[76] G.L. Bunting: Proof of the uniqueness conjecture for black holes, Ph.D. thesis, University of
New England, Armidale, Australia (1983).
[77] G.L. Bunting and A.K.M. Masood-ul-Alam: Nonexistence of multiple black holes in asymp-
totically Euclidean static vacuum space-time, Gen. Relat. Grav. 19, 147 (1987).
[78] L.M. Burko, G. Khanna, and A. Zenginoğlu: Cauchy-horizon singularity inside perturbed
Kerr black holes, Phys. Rev. D 93, 041501(R) (2016); arXiv: 1601.05120.
[79] P.F. Byrd and M.D. Friedman: Handbook of Elliptic Integrals for Engineers and Scientists,
2nd ed., Springer (Berlin) (1971); DOI: 10.1007/978-3-642-65138-0
[80] A. Čadež and U. Kostić: Optics in the Schwarzschild spacetime, Phys. Rev. D 72, 104024
(2005); arXiv: gr-qc/0405037.
[81] L.-M. Cao: Deformation of codimension-2 surfaces and horizon thermodynamics, J. High
Energ. Phys. 2011, 112 (2011); arXiv: 1009.4540.
[82] L. Capuano, L. Santoni, and E. Barausse: Black hole hairs in scalar-tensor gravity and the
lack thereof, Phys. Rev. D 108, 064058 (2023); arXiv: 2304.12750.
[83] V. Cardoso: The hole picture, Nature Review Phys. 1, 701 (2019); arXiv: 1910.04173.
[84] V. Cardoso, F. Duque, and A. Foschi: The light ring and the appearance of matter accreted
by black holes, Phys. Rev. D 103, 104044 (2021); arXiv: 2102.07784.
[85] S. Carlip: Black hole thermodynamics, Int. J. Mod. Phys. D 23, 1430023 (2014); arXiv:
1410.1486.
[86] B. Carr, K. Kohri, Y. Sendouda, and J. Yokoyama: Constraints on primordial black holes,
Rep. Prog. Phys. 84, 116902 (2021); arXiv: 2002.12778.
[87] S.M. Carroll: Spacetime and Geometry: An Introduction to General Relativity, Addison Wes-
ley (Pearson Education), San Fransisco (2004); https://www.preposterousuniverse.
com/spacetimeandgeometry/.
[88] B. Carter: Complete Analytic Extension of the Symmetry Axis of Kerr’s Solution of Einstein’s
Equations, Phys. Rev. 141, 1242 (1966).
792 BIBLIOGRAPHY
[89] B. Carter: Stationary Axisymmetric Systems in General Relativity, Ph. D. Thesis, Cambridge
University (1967); available at https://luth.obspm.fr/~luthier/carter/Thesis/.
[90] B. Carter: Global Structure of the Kerr Family of Gravitational Fields, Phys. Rev. 174, 1559
(1968).
[91] B. Carter: Hamilton-Jacobi and Schrodinger Separable Solutions of Einstein’s Equations,
Commun. Math. Phys. 10, 280 (1968).
[92] B. Carter: Killing horizons and orthogonally transitive groups in space-time, J. Math. Phys.
10, 70 (1969).
[93] B. Carter: Axisymmetric Black Hole Has Only Two Degrees of Freedom, Phys. Rev. Lett. 26,
331 (1971).
[94] B. Carter: Rigidity of a Black Hole, Nature Physical Science 238, 71 (1972).
[95] B. Carter: Black hole equilibrium states Part I. Analytic and Geometric Properties of the Kerr
Solutions, in Black Holes – Les astres occlus, edited by C. DeWitt and B. DeWitt, Gordon
and Breach, New York (1973), p. 57; reprinted (with corrections) in Gen. Relativ. Gravit.
41, 2873 (2009).
[96] B. Carter: Black hole equilibrium states Part II. General theory of stationary black hole
states, in Black Holes – Les astres occlus, edited by C. DeWitt and B. DeWitt, Gordon and
Breach, New York (1973), p. 125; reprinted (with corrections) in Gen. Relativ. Gravit. 42,
653 (2010).
[97] B. Carter: Perfect fluid and magnetic field conservation laws in the theory of black hole
accretion rings, in Active Galactic Nuclei, edited by C. Hazard and S. Mitton, Cambridge
Univ. Press, Cambridge (1979), p. 273; available at https://luth.obspm.fr/~luthier/
carter/trav/Carter79.pdf.
[98] B. Carter: Generalised mass variation formula for a stationary axisymmetric star or black
hole with surrounding accretion disc, in Journées Relativistes 1979, edited by I. Moret-Bailly
and C. Latrémolière, Faculté des Sciences, Angers (1979), p. 166; arXiv: astro-ph/0206113.
[99] B. Carter: The general theory of the mechanical, electromagnetic and thermodynamic
properties of black holes, in General Relativity: An Einstein Centenary Survey, edited by
S.W. Hawking and W. Israel, Cambridge Univ. Press, Cambridge (1979), p. 294.
[100] B. Carter: Bunting identity and Mazur identity for non-linear elliptic systems including the
black hole equilibrium problem, Commun. Math. Phys. 99, 563 (1985).
[101] B. Carter: Mathematical foundations of the theory of relativistic stellar and black hole
configurations, in Gravitation in Astrophysics, edited by B. Carter and J.B. Hartle, Plenum
Press, New York (1987), p. 63.
[102] B. Carter: Extended tensorial curvature analysis for embeddings and foliations, Contemp.
Math. 203, 207 (1997).
[103] B. Carter: Has the black hole equilibrium problem been solved?, in The Eighth Marcel
Grossmann Meeting, edited by T. Piran, World Scientific, Singapore (1999), p. 136; arXiv:
gr-qc/9712038.
BIBLIOGRAPHY 793
[104] B. Carter: Energy dominance and the Hawking Ellis vacuum conservation theorem, in The
Future of Theoretical Physics and Cosmology — Celebrating Stephen Hawking’s Contribu-
tions to Physics, edited by G.W. Gibbons, E.P.S. Shellard, and S.J. Rankin, Cambridge Univ.
Press, Cambridge (2003), p. 177; arXiv: gr-qc/0205010.
[105] B. Carter: Half Century of Black-Hole Theory: From Physicists’ Purgatory to Mathemati-
cians’ Paradise, in A Century of Relativity Physics: ERE 2005; XXVIII Spanish Relativity
Meeting, edited by L. Mornas and J. Diaz Alonso, AIP Conference Proceedings 841, 29
(2006); arXiv: gr-qc/0604064.
[106] M. Casals, S.E. Gralla, and P. Zimmerman: Horizon instability of extremal Kerr black holes:
Nonaxisymmetric modes and enhanced growth rate, Phys. Rev. D 94, 064003 (2016); arXiv:
1606.08505.
[107] S. Chandrasekhar: The Mathematical Theory of Black Holes, Oxford Univ. Press, New
York (1983); DOI: 10.1093/oso/9780198503705.001.0001.
[108] Y. Choquet-Bruhat: General Relativity and Einstein’s Equations, Oxford Univ. Press, New
York (2009); DOI: 10.1093/acprof:oso/9780199230723.001.0001.
[109] Y. Choquet-Bruhat: Introduction to General Relativity, Black Holes, and Cosmology, Oxford
Univ. Press, New York (2015); DOI: 10.1093/oso/9780199666454.001.0001.
[110] Y. Choquet-Bruhat, C. De Witt-Moretten, and M. Dillard-Bleick: Analysis, Manifolds and
Physics, North-Holland, Amsterdam (1977).
[111] Y. Choquet-Bruhat and R. Geroch: Global aspects of the Cauchy problem in general
relativity, Commun. Math. Phys. 14, 329 (1969).
[112] D. Christodoulou: Reversible and Irreversible Transformations in Black-Hole Physics, Phys.
Rev. Lett. 25, 1596 (1970).
[113] D. Christodoulou: The Formation of Black Holes in General Relativity, European Mathe-
matical Society, Zürich (2009); DOI: 10.4171/068; arXiv: 0805.3880.
[114] D. Christodoulou and R. Ruffini: Reversible Transformations of a Charged Black Hole, Phys.
Rev. D 4, 3552 (1971).
[115] P.T. Chruściel: “No Hair” Theorems – Folklore, Conjectures, Results, in Differential Geometry
and Mathematical Physics, Contemporary Mathematics 170; edited by J.K. Beem and K.L.
Duggal, American Mathematical Society, Providence (1994), p. 23; arXiv: gr-qc/9402032.
[116] P.T. Chruściel: On Rigidity of Analytic Black Holes, Commun. Math. Phys. 189, 1 (1997);
arXiv: gr-qc/9610011.
[117] P.T. Chruściel: Black holes, in Proceedings of the Tübingen conference on conformal structure
of space-time, edited by J. Frauendiener, H. Friedrich, Lecture Notes in Physics 604, 61,
Springer, Berlin (2002); arXiv: gr-qc/0201053.
[118] P.T. Chruściel: Black holes – An Introduction, in 100 Years of Relativity, edited by A.
Ashtekar, World Scientific, Singapore (2005), p. 93; DOI: 10.1142/9789812700988_0004.
[119] P.T. Chruściel: Geometry of Black Holes, Oxford Univ. Press, Oxford (2020); https://
global.oup.com/academic/product/geometry-of-black-holes-9780198855415.
[120] P.T. Chruściel, E. Delay, G. Galloway, and R. Howard: Regularity of horizons and the area
theorem, Annales Henri Poincaré 2, 109 (2001); arXiv: gr-qc/0001003.
794 BIBLIOGRAPHY
[121] P.T. Chruściel and G.J. Galloway: Uniqueness of static black holes without analyticity,
Class. Quantum Grav. 27, 152001 (2010); arXiv: 1004.0513.
[122] P.T. Chruściel and J. Lopes Costa: On uniqueness of stationary vacuum black holes,
Astérisque 321, 195 (2008); arXiv: 0806.0016.
[123] P.T. Chruściel, J. Lopes Costa, and M. Heusler: Stationary Black Holes: Uniqueness and
Beyond, Living Rev. Relativ. 15, 7 (2012).
[124] P.T. Chruściel, M. Maliborski, and N. Yunes: Structure of the singular ring in Kerr-like
metrics, Phys. Rev. D 101, 104048 (2020); arXiv: 1912.06020.
[125] P.T. Chruściel and L. Nguyen: A Uniqueness Theorem for Degenerate Kerr–Newman Black
Holes, Ann. Henri Poincaré 11, 585 (2010); arXiv: 1002.1737.
[126] P.T. Chruściel, C.R. Ölz, and S.J Szybka: Space-time diagrammatics, Phys. Rev. D 86,
124041 (2012); arXiv: 1211.1718.
[127] P.T. Chruściel, H.S. Reall and P. Tod: On non-existence of static vacuum black holes with
degenerate components of the event horizon, Class. Quantum Grav. 23, 549 (2006); arXiv:
gr-qc/0512041.
[128] P.T. Chruściel and P. Tod: The Classification of Static Electro–Vacuum Space–Times Con-
taining an Asymptotically Flat Spacelike Hypersurface with Compact Interior, Commun.
Math. Phys. 271, 577 (2007); arXiv: gr-qc/0512043.
[129] P.T. Chruściel and R.M. Wald: Maximal hypersurfaces in stationary asymptotically flat
spacetimes, Commun. Math. Phys. 163, 561 (1994); arXiv: gr-qc/9304009.
[130] P.T. Chruściel and R.M. Wald: On the topology of stationary black holes, Class. Quantum
Grav. 11, L147 (1994); arXiv: gr-qc/9410004.
[131] T. Chu, H.P. Pfeiffer, and M.I. Cohen: Horizon dynamics of distorted rotating black holes,
Phys. Rev. D 83, 104018 (2011); arXiv: 1011.2601.
[132] M.I. Cohen, J.D. Kaplan, and M.A. Scheel: Toroidal horizons in binary black hole inspirals,
Phys. Rev. D 85, 024031 (2012); arXiv: 1110.1668.
[133] M.I. Cohen, H.P. Pfeiffer, and M.A. Scheel: Revisiting event horizon finders, Class. Quantum
Grav. 26, 035005 (2009); arXiv: 0809.2628.
[134] P. Collas and D. Klein: Embeddings and time evolution of the Schwarzschild wormhole,
Amer. J. Phys. 80, 203 (2012); arXiv: 1107.4871.
[135] S. Collin-Zahn: Des quasars aux trous noirs, EDP Sciences, Les Ulis (2009).
[136] G. Compère: An introduction to the mechanics of black holes, Lecture notes, Second
Modave Summer School in Mathematical Physics (2006); arXiv: gr-qc/0611129.
[137] G. Compère: The Kerr/CFT correspondence and its extensions, Living Rev. Relativ. 20, 1
(2017).
[138] G. Compère: Advanced Lectures on General Relativity, Springer, Cham (2019); DOI:
10.1007/978-3-030-04260-8; arXiv: 1801.07064.
[139] G. Compère and A. Druart: Near-horizon geodesics of high spin black holes, Phys. Rev. D
101, 084042 (2020); erratum in Phys. Rev. D 102, 029901 (2020); arXiv: 2001.03478.
BIBLIOGRAPHY 795
[140] G. Compère, K. Fransen, T. Hertog, and J. Long: Gravitational waves from plunges into
Gargantua, Class. Quantum Grav. 35, 104002 (2018); erratum in Class. Quantum Grav.
36, 179501 (2019); arXiv: 1712.07130.
[141] G. Contopoulos: Orbits through the ergosphere of a Kerr black hole, Gen. Relat. Grav. 16,
43 (1984).
[142] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, and D.E. Knuth: On the Lambert W
function, Adv. Comput. Math. 5, 329 (1996).
[143] A. Coudray and J.-P. Nicolas: Geometry of Vaidya spacetimes, 1Gen. Relat. Grav. 53, 73
(2021); arXiv: 2101.06544.
[144] P.V.P. Cunha and C.A.R. Herdeiro: Shadows and strong gravitational lensing: a brief review,
Gen. Relat. Grav. 50, 42 (2018); arXiv: 1801.00860.
[145] P.V.P. Cunha, C.A.R. Herdeiro and E. Radu: Fundamental photon orbits: Black hole shadows
and spacetime instabilities, Phys. Rev. D 96, 024039 (2017); arXiv: 1705.05461.
[146] C.T. Cunningham and J.M. Bardeen: The Optical Appearance of a Star Orbiting an Extreme
Kerr Black Hole Astrophys. J. 183, 237 (1973).
[147] E. Curiel: A Potted Early History of “black hole”, appendix to K. Landsman’s article [334]
(2021).
[148] H.D. Curtis: Descriptions of 762 Nebulae and Clusters Photographed with the Crossley
Reflector, Publications of Lick Observatory 13, 9 (1918).
[149] M. Dafermos and I. Rodnianski: Lecture on black holes and linear waves, in Clay
Math. Proc. 17: Evolution Equations, edited by D. Ellwood, I. Rodnianski, G.
Staffilani and J. Wunsch, AMS / Clay Mathematical Institute, Cambridge (2013),
p. 97; available at https://people.maths.ox.ac.uk/cmi/library/proceedings/
cmip017c.pdf; arXiv: 0811.0354.
[150] T. Damour: Black-hole eddy currents, Phys. Rev. D 18, 3598 (1978).
[151] T. Damour: Quelques propriétés mécaniques, électromagnétiques, thermodynamiques et
quantiques des trous noirs, Thèse de doctorat d’État, Université Paris 6 (1979); available
at https://pagesperso.ihes.fr/~damour/Articles/.
[152] T. Damour: Surface effects in black hole physics, in Proceedings of the Second Marcel
Grossmann Meeting on General Relativity, edited by R. Ruffini, North Holland, Amsterdam
(1982), p. 587; available at https://pagesperso.ihes.fr/~damour/Articles/.
[153] T. Damour: The Entropy of Black Holes: A Primer, in Poincaré Seminar 2003: Bose-Einstein
Condensation – Entropy, edited by J. Dalibard, B. Duplantier and V. Rivasseau, Birkhaäuser,
Basel (2004), p. 227; URL: https://link.springer.com/book/9783764371067; arXiv:
hep-th/0401160.
[154] G. Darmois: Les équations de la gravitation einsteinienne, Mémorial des sciences mathé-
matiques, no. 25 (1927); available at https://www.numdam.org/item/?id=MSM_1927_
_25__1_0.
[155] C. G. Darwin: The gravity field of a particle, Proc. Roy. Soc. Lond. A 249, 180 (1959).
[156] C. G. Darwin: The gravity field of a particle. II, Proc. Roy. Soc. Lond. A 263, 39 (1961).
796 BIBLIOGRAPHY
[157] B. Datt: On a Class of Solutions of the Gravitation Equations of Relativity, Zeitschrift für
Physik 108, 314 (1938); reprinted in Gen. Relativ. Gravit. 31, 1619 (1999).
[158] I. Davies and H.S. Reall: Nonperturbative Second Law of Black Hole Mechanics in Effective
Field Theory, Phys. Rev. Lett. 132, 171402 (2024); arXiv: 2312.07659.
[159] F. de Felice: Equatorial Geodesic Motion in the Gravitational Field of a Rotating Source,
Nuovo Cimento B 57, 351 (1968).
[160] C. De Jans: Sur le mouvement d’une particule matérielle dans un champ de gravitation à
symétrie sphérique, Mem. Acad. Roy. Belgique Cl. Sci. 7, 1 (1923).
[161] N. Deruelle: Black Holes in General Relativity, lectures at Institut de Physique Théorique,
CEA, Saclay (France) (2009); slides and notes available at http://ipht.cea.fr/
Docspht/search/article.php?id=t09/346.
[162] N. Deruelle and J.-P. Uzan: Théories de la Relativité, Belin, Paris (2014).
[163] N. Deruelle and J.-P. Uzan: Relativity in Modern Physics, Oxford Univ. Press, Oxford
(2018); DOI: 10.1093/oso/9780198786399.001.0001.
[164] C. DeWitt and B. DeWitt (eds.): Black Holes – Les astres occlus, Gordon and Breach, New
York (1973).
[165] J. Dexter and E. Agol: A Fast New Public Code for Computing Photon Orbits in a Kerr
Spacetime, Astrophys. J. 696, 1616 (2009); arXiv: 0903.0620.
[166] V.I. Dokuchaev: Is there life inside black holes?, Class. Quantum Grav. 28, 235015 (2011);
arXiv: 1103.6140.
[167] V.I. Dokuchaev: To see the invisible: Image of the event horizon within the black hole
shadow, Int. J. Mod. Phys. D 28, 1941005 (2019); arXiv: 1812.06787.
[168] V.I. Dokuchaev and N.O. Nazarova: Event Horizon Image within Black Hole Shadow, J. Exp.
Theor. Phys. 128, 578 (2019); arXiv: 1804.08030.
[169] V.I. Dokuchaev and N.O. Nazarova: Silhouettes of invisible black holes, Physics-Uspekhi
63, 583 (2020); arXiv: 1911.07695.
[170] V.I. Dokuchaev and N.O. Nazarova: Visible shapes of black holes M87* and SgrA*, Universe
6, 154 (2020); arXiv: 2007.14121.
[171] X. Dong: Holographic entanglement entropy for general higher derivative gravity, J. High
Energ. Phys. 01(2014), 44 (2014); arXiv: 1310.5713.
[172] A.G. Doroshkevich, Ya.B. Zeldovich and I.D. Novikov: Gravitational Collapse of Non-
symmetric and Rotating Masses (in Russian), Zhurnal Éksperimental’noı̆ i Teoreticheskoı̆
Fiziki 49, 170 (1965); English translation in Soviet Physics JETP 22, 122 (1966).
[173] S. Drasco and S.A. Hughes: Rotating black hole orbit functionals in the frequency domain,
Phys. Rev. D 69, 044015 (2004); arXiv: astro-ph/0308479.
[174] S. Drasco and S.A. Hughes: Gravitational wave snapshots of generic extreme mass ratio
inspirals, Phys. Rev. D 73, 024027 (2006); arXiv: gr-qc/0509101.
[175] J. Droste: The Field of a Single Centre in Einstein’s Theory of Gravitation, and the Motion
of a Particle in that Field, Koninklijke Nederlandsche Akademie van Wetenschappen,
Proceedings 19, 197 (1917); reprinted in Gen. Relativ. Gravit. 34, 1545 (2002).
BIBLIOGRAPHY 797
[176] D.M. Eardley and L. Smarr: Time functions in numerical relativity: Marginally bound dust
collapse, Phys. Rev. D 19, 2239 (1979).
[177] A.S. Eddington: A comparison of Whitehead’s and Einstein’s Formulæ, Nature 113, 192
(1924).
[178] C.A. Egan and C.H. Lineweaver: A Larger Estimate of the Entropy of the Universe, Astro-
phys. J. 710, 1825 (2010); arXiv: 0909.3983.
[179] A. Einstein and N. Rosen: The Particle Problem in the General Theory of Relativity, Phys.
Rev. 48, 73 (1935).
[180] J. Eisenstaedt: Histoire et Singularités de la Solution de Schwarzschild (1915-1923), Archive
for History of Exact Sciences 27, 157 (1982).
[181] J. Eisenstaedt: Trajectoires et Impasses de la Solution de Schwarzschild, Archive for History
of Exact Sciences 37, 275 (1987).
[182] J. Eisenstaedt: Lemaître and the Schwarzschild Solution, in The Attraction of Gravitation:
New Studies in the History of General Relativity, Proc. Third International Conference
on the History and Philosophy of General Relativity. Einstein Studies, Vol. 5, edited by
J. Earman, M. Janssen, and J.D. Norton, Birkhäuser, Boston (1993), p. 353.
[183] G.F.R. Ellis: Relativistic cosmology, in Proceedings of the International School of Physics
“Enrico Fermi”, Course 47: General relativity and cosmology, edited by B. K. Sachs, Academic
Press, New York and London (1971), p. 104; reprinted in Gen. Relat. Grav. 41, 581 (2009).
[184] H. Elvang and P. Figueras: Black saturn, J. High Energ. Phys. 05(2007), 050 (2007); arXiv:
hep-th/0701035.
[185] R. Emparan, S. Ohashi and T. Shiromizu: No-dipole-hair theorem for higher-dimensional
static black holes, Phys. Rev. D 82, 084032 (2010); arXiv: 1007.3847.
[186] R. Emparan and H.S. Reall: A Rotating Black Ring Solution in Five Dimensions, Phys. Rev.
Lett. 88, 101101 (2002); arXiv: hep-th/0110260.
[187] R. Emparan and H.S. Reall: Black Holes in Higher Dimensions, Living Rev. Relativ. 11, 6
(2008).
[188] J. Engle, K. Noui, and A. Perez: Black Hole Entropy and SU(2) Chern-Simons Theory, Phys.
Rev. Lett. 105, 031302 (2010); arXiv: 0905.3168.
[189] H. Eschrig: Topology and Geometry for Physics, Springer, Berlin (2011); DOI: 10.1007/978-
3-642-14700-5.
[190] A. Fabbri and J. Navarro-Salas: Modeling Black Hole Evaporation, Imperial College Press,
London, and World Scientific, Singapore (2005); DOI: 10.1142/p378.
[191] H. Falcke: Imaging black holes: past, present and future, J. Phys. Conf. Ser. 942, 012001
(2017); arXiv: 1801.03298.
[192] H. Falcke, F. Melia, and E. Agol: Viewing the Shadow of the Black Hole at the Galactic
Center, Astrophys. J. 528, L13 (2000); arXiv: astro-ph/9912263.
[193] J.R. Farah, D.W. Pesce, M.D. Johnson, and L. Blackburn: On the Approximation of the Black
Hole Shadow with a Simple Polar Curve, Astrophys. J. 900, 77 (2020); arXiv: 2007.06732.
[194] F. Fayos and R. Torres: A class of interiors for Vaidya’s radiating metric: singularity-free
gravitational collapse, Class. Quantum Grav. 25, 175009 (2008).
798 BIBLIOGRAPHY
[195] D. Finkelstein: Past-Future Asymmetry of the Gravitational Field of a Point Particle, Phys.
Rev. 110, 965 (1958).
[196] A. Fischer and J. Marsden: The initial value problem and the dynamical formulation of
general relativity, in General Relativity: an Einstein Centenary Survey, edited by S.W.
Hawking and W. Israel, Cambridge Univ. Press, Cambridge (1979), p. 138.
[197] L. Flamm: Beiträge zur Einsteinschen Gravitationstheorie, Physikalische Zeitschrift 17,
448 (1916); English translation in Gen. Relativ. Gravit. 47, 72 (2015).
[198] J. Frauendiener: Conformal Infinity, Living Rev. Relativ. 7, 1 (2004).
[199] J.L. Friedman, K. Schleich, and D.M. Witt: Topological censorship, Phys. Rev. Lett. 71, 1486
(1993); erratum in Phys. Rev. Lett. 75, 1872 (1995); arXiv: gr-qc/9305017.
[200] H. Friedrich, I. Rácz, and R.M. Wald: On the Rigidity Theorem for Spacetimes with a
Stationary Event Horizon or a Compact Cauchy Horizon, Commun. Math. Phys. 204, 691
(1999); arXiv: gr-qc/9811021.
[201] V.P. Frolov, P. Krtouš, and D. Kubizňák: Black holes, hidden symmetries, and complete
integrability, Living Rev. Relativ. 20, 6 (2017).
[202] V.P. Frolov and I.D. Novikov: Black Hole Physics: Basic Concepts and New Developments,
Kluwer Academic Publisher, Dordrecht (1998); reprinted by Springer, Dordrecht (1998);
DOI: 10.1007/978-94-011-5139-9.
[203] V.P. Frolov and K.S. Thorne: Renormalized stress-energy tensor near the horizon of a slowly
evolving, rotating black hole Phys. Rev. D 39, 2125 (1989).
[204] V.P. Frolov and A. Zelnikov: Introduction to Black Hole Physics, Oxford Univ. Press, New
York (2011); DOI: 10.1093/acprof:oso/9780199692293.001.0001.
[205] C. Fronsdal: Completion and Embedding of the Schwarzschild Solution, Phys. Rev. 116,
778 (1959).
[206] R. Fujita & W. Hikida: Analytical solutions of bound timelike geodesic orbits in Kerr
spacetime, Class. Quantum Grav. 26, 135002 (2009); arXiv: 0906.1420.
[207] D. Gajic: Azimuthal instabilities on extremal Kerr, submitted (2023); arXiv: 2302.06636.
[208] G.J. Galloway: Least area tori, black holes and topological censorship, in Differential
Geometry and Mathematical Physics, Contemporary Mathematics 170; edited by J.K. Beem
and K.L. Duggal, American Mathematical Society, Providence (1994), p. 113.
[209] G.J. Galloway: Null Geometry and the Einstein Equations, in The Einstein Equations and
the Large Scale Behavior of Gravitational Fields, edited by P.T. Chruściel and H. Friedrich,
Birkhäuser, Basel (2004), p. 379; DOI: 10.1007/978-3-0348-7953-8_11; available at https:
//www.math.miami.edu/~galloway/papers/cargproc.pdf.
[210] G.J. Galloway and R. Schoen: A Generalization of Hawking’s Black Hole Topology Theorem
to Higher Dimensions, Commun. Math. Phys. 266, 571 (2006); arXiv: gr-qc/0509107.
[211] G.J. Galloway and J.M.M. Senovilla: Singularity theorems based on trapped submanifolds
of arbitrary co-dimension, Class. Quantum Grav. 27, 152002 (2010); arXiv: 1005.1249.
[212] S. Gao and R.M. Wald: “Physical process version” of the first law and the generalized
second law for charged and rotating black holes, Phys. Rev. D 64, 084020 (2001); arXiv:
gr-qc/0106071.
BIBLIOGRAPHY 799
[213] D. Garfinkle and E.N. Glass: Bertotti-Robinson and Melvin spacetimes, Class. Quantum
Grav. 28, 215012 (2011); arXiv: 1109.1535.
[214] K. Gebhardt, J. Adams, D. Richstone, T.R. Lauer, S.M. Faber, K. Gültekin, J. Murphy, and
S. Tremaine: The Black Hole Mass in M87 from Gemini/NIFS Adaptive Optics Observations,
Astrophys. J. 729, 119 (2011); arXiv: 1101.1954.
[215] R. Genzel, F. Eisenhauer and S. Gillessen: The Galactic Center massive black hole and
nuclear star cluster, Rev. Mod. Phys. 82, 3121 (2010); arXiv: 1006.0064.
[216] R. Geroch: What is a singularity in general relativity? Ann. Phys. 48, 526 (1968).
[217] R. Geroch and J.B. Hartle: Distorted black holes, J. Math. Phys. 23, 680 (1982).
[218] G.W. Gibbons, D. Ida and T. Shiromizu: Uniqueness of (dilatonic) charged black holes and
black p-branes in higher dimensions, Phys. Rev. D 66, 044010 (2002); arXiv: hep-th/0206136.
[219] G.W. Gibbons, D. Ida and T. Shiromizu: Uniqueness and Non-Uniqueness of Static Vacuum
Black Holes in Higher Dimensions, Prog. Theor. Phys. Suppl. 148, 284 (2002); arXiv:
gr-qc/0203004.
[220] G.W. Gibbons and M. Vyska: The application of Weierstrass elliptic functions to
Schwarzschild null geodesics, Class. Quantum Grav. 29, 065016 (2012); arXiv: 1110.6508.
[221] G. Gibbons and C.M. Will: On the Multiple Deaths of Whitehead’s Theory of Gravity,
Studies in History and Philosophy of Science Part B: Studies in History and Philosophy
of Modern Physics 39, 41 (2008); arXiv: gr-qc/0611006.
[222] V.L. Ginzburg: The Magnetic Fields of Collapsing Masses and the Nature of Superstars (in
Russian), Doklady Akademii Nauk SSSR 156, 43 (1964); English translation in Soviet
Physics Doklady 9, 329 (1964).
[223] K. Glampedakis and D. Kennefick: Zoom and whirl: Eccentric equatorial orbits around
spinning black holes and their evolution under gravitational radiation reaction, Phys. Rev.
D 66, 044002 (2002); arXiv: gr-qc/0203086.
[224] K. Glampedakis and G. Pappas: Can supermassive black hole shadows test the Kerr metric?,
Phys. Rev. D 104, L081503 (2021); arXiv: 2102.13573.
[225] L. Gou, J.E. McClintock, R.A. Remillard, J.F. Steiner, M.J. Reid, J.A. Orosz, R. Narayan, M.
Hanke, and J. García: Confirmation via the Continuum-Fitting Method that the Spin of the
Black Hole in Cygnus X-1 is Extreme, Astrophys. J. 790, 29 (2014); arXiv: 1308.4760.
[226] E. Gourgoulhon : An introduction to relativistic hydrodynamics, in Stellar Fluid Dynamics
and Numerical Simulations: From the Sun to Neutron Stars, edited by M. Rieutord and
B. Dubrulle, EAS Publications Series 21, 43-79 (2006); arXiv: gr-qc/0603009.
[227] E. Gourgoulhon: 3+1 Formalism in General Relativity; Bases of Numerical Relativity,
Springer, Berlin (2012); URL: https://relativite.obspm.fr/3p1.
[228] E. Gourgoulhon: Special Relativity in General Frames: From Particles to Astrophysics,
Springer, Berlin (2013); URL: https://relativite.obspm.fr/sperel.
[229] E. Gourgoulhon: Relativité générale, Master course lecture notes; URL: https://luth.
obspm.fr/~luthier/gourgoulhon/fr/master/relat.html.
[230] E. Gourgoulhon and J.L. Jaramillo: A 3+1 perspective on null hypersurfaces and isolated
horizons, Phys. Rep. 423, 159 (2006); arXiv: gr-qc/0503113.
800 BIBLIOGRAPHY
[231] E. Gourgoulhon and J.L. Jaramillo: New theoretical approaches to black holes, in Proceed-
ings of the Conference Jean-Pierre Lasota, X-ray Binaries, Accretion Disks and Compact
Stars held in Trzebieszowice, Poland (7-13 October 2007), edited by M. Abramowicz and
O. Straub, New Astronomy Reviews 51, 791 (2008); arXiv: 0803.2944.
[232] I.S. Gradshteyn, I.M. Ryzhik, Y.V. Geronimus, M.Y. Tseytlin, and A. Jeffrey: Table of
Integrals, Series, and Products (8 ed.), Academic Press (2015).
[233] S.E. Gralla: Can the EHT M87 results be used to test general relativity?, Phys. Rev. D 103,
024023 (2021); arXiv: 2010.08557.
[234] S.E. Gralla, D.E. Holz, and R.M. Wald: Black hole shadows, photon rings, and lensing rings,
Phys. Rev. D 100, 024018 (2019); arXiv: 1906.00873.
[235] S.E. Gralla and A. Lupsasca: Lensing by Kerr black holes, Phys. Rev. D 101, 044031 (2020);
arXiv: 1910.12873.
[236] S.E. Gralla and A. Lupsasca: Null geodesics of the Kerr exterior, Phys. Rev. D 101, 044032
(2020); arXiv: 1910.12881.
[237] S.E. Gralla and A. Lupsasca: Observable shape of black hole photon rings, Phys. Rev. D
102, 124003 (2020); arXiv: 2007.10336.
[238] S.E. Gralla, A. Lupsasca, and A. Strominger: Observational signature of high spin at the
Event Horizon Telescope, Month. Not. Roy. Astron. Soc. 475, 3829 (2018); arXiv: 1710.11112.
[239] S.E. Gralla, A.P. Porfyriadis, and N. Warburton: Particle on the innermost stable circular
orbit of a rapidly spinning black hole, Phys. Rev. D 92, 064029 (2015); arXiv: 1506.08496.
[240] S.E. Gralla and P. Zimmerman: Critical exponents of extremal Kerr perturbations, Class.
Quantum Grav. 35, 095002 (2018); arXiv: 1711.00855.
[241] P. Grandclément: Fully consistent rotating black holes in the cubic Galileon theory, Class.
Quantum Grav. 41, 025012 (2024); arXiv: 2308.11245.
[242] A.A. Grib and Y.V. Pavlov: Black holes and particles with zero or negative energy, Theor.
Math. Phys. 190, 268 (2017); arXiv: 1601.02592.
[243] J.B. Griffiths and J. Podolský: Exact Space-Times in Einstein’s General Relativity, Cambridge
Univ. Press, Cambridge (2009); DOI: 10.1017/CBO9780511635397.
[244] R. Grossman, J. Levin, and G. Perez-Giz: Harmonic structure of generic Kerr orbits, Phys.
Rev. D 85, 023012 (2012); arXiv: 1105.5811.
[245] D. Grumiller, R. McNees and J. Salzer: Black Holes and Thermodynamics: The First Half
Century, in Quantum Aspects of Black Holes, edited by X. Calmet, Springer, Cham (2015),
p. 27; DOI: 10.1007/978-3-319-10852-0_2; arXiv: 1402.5127.
[246] D. Grumiller and M.M. Sheikh-Jabbari: Black Hole Physics: From Collapse to Evaporation,
Springer, Cham (2022); DOI: 10.1007/978-3-031-10343-8.
[247] M. Guica, T. Hartman, W. Song, and A. Strominger: The Kerr/CFT correspondence, Phys.
Rev. D 80, 124008 (2009); arXiv: 0809.4266.
[248] A. Gullstrand: Allgemeine Lösung des statischen Einkörperproblems in der Einsteinschen
Gravitationstheorie, Arkiv. Mat. Astron. Fys. 16, 1 (1922).
[249] Y. Hagihara: Theory of the Relativistic Trajectories in a Gravitational Field of Schwarzschild,
Japanese Journal of Astronomy and Geophysics 8, 67 (1931).
BIBLIOGRAPHY 801
[250] K. Hajian and M.M. Sheikh-Jabbari: Solution phase space and conserved charges: A general
formulation for charges associated with exact symmetries, Phys. Rev. D 93, 044074 (2016);
arXiv: 1512.05584.
[251] P. Hájíček: Exact models of charged black holes. I. Geometry of totally geodesic null
hypersurface, Commun. Math. Phys. 34, 37 (1973).
[252] P. Hájíček: Exact models of charged black holes. II. Axisymmetric stationary horizons,
Commun. Math. Phys. 34, 53 (1973).
[253] P. Hájíček: Can outside fields destroy black holes ?, J. Math. Phys. 15, 1554 (1974).
[254] J. Haláček and T. Ledvinka: The analytic conformal compactification of the Schwarzschild
spacetime, Class. Quantum Grav. 31, 015007 (2014); arXiv: 1401.1337.
[255] D. Harlow and J. Wu: Covariant phase space with boundaries, J. High Energ. Phys.
10(2020), 146 (2020); arXiv: 1906.08616.
[256] B.K. Harrison, K.S. Thorne, M. Wakano and J.A. Wheeler: Gravitation Theory and Gravi-
tational Collapse, University of Chicago Press, Chicago (1965); URL: https://archive.org/
details/GravitationalTheoryandCollapseHarrison65-17293.
[257] J.B. Hartle: Gravity: An Introduction to Einstein’s General Relativity, Addison Wes-
ley (Pearson Education), San Fransisco (2003); URL: https://archive.org/details/
JBHartleGravityAnIntroductionToEinsteinsGeneralRelativityAddisonWesley20031.
[258] J.B. Hartle and S.W. Hawking: Solutions of the Einstein-Maxwell equations with many
black holes, Commun. Math. Phys. 26, 87 (1972); available at https://projecteuclid.org/
journalArticle/Download?urlId=cmp%2F1103858037.
[259] S.W. Hawking: Gravitational radiation from colliding black holes, Phys. Rev. Lett. 26, 1344
(1971).
[260] S.W. Hawking: Black holes in general relativity, Commun. Math. Phys. 25, 152 (1972).
[261] S.W. Hawking: The event horizon, in Black Holes – Les astres occlus, edited by C. DeWitt
and B. DeWitt, Gordon and Breach, New York (1973), p. 1.
[262] S.W. Hawking: Black hole explosions?, Nature 248, 30 (1974).
[263] S.W. Hawking: Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975);
erratum in Commun. Math. Phys. 46, 206 (1976).
[264] S.W. Hawking: A Brief History of Time: From the Big Bang to Black Holes, Bantam Books,
Toronto and New York (1988).
[265] S.W. Hawking: The Nature of Space and Time, part of a series of six lectures with R. Penrose
sponsored by Princeton University Press (1994); arXiv: hep-th/9409195.
[266] S.W. Hawking and G.F.R. Ellis: The large scale structure of space-time, Cambridge Univ.
Press, Cambridge (1973); DOI: 10.1017/CBO9780511524646.
[267] S.W. Hawking and R. Penrose: The singularities of gravitational collapse and cosmology,
Proc. Roy. Soc. Lond. A 314, 529 (1970).
[268] S.W. Hawking and R. Penrose: The Nature of Space and Time, Princeton Univ. Press,
Princeton (2015).
[269] S.A. Hayward: General laws of black-hole dynamics, Phys. Rev. D 49, 6467 (1994); arXiv:
gr-qc/9303006.
802 BIBLIOGRAPHY
[270] C. Heinicke and F.W. Hehl: Schwarzschild and Kerr solutions of Einstein’s field equation:
An Introduction, Int. J. Mod. Phys. D 24, 1530006 (2015); arXiv: 1503.02172.
[271] F. Hélein: Multisymplectic formalism and the covariant phase space, in Variational Prob-
lems in Differential Geometry, edited by R. Bielawski, K. Houston and M. Speight, Cam-
bridge Univ. Press, Cambridge (2011), p. 94; DOI: 10.1017/CBO9780511863219.007; arXiv:
1106.2086.
[272] C.A.R. Herdeiro and J.P.S. Lemos: O buraco negro cinquenta anos depois: A génese do
nome, Gazeta de Fisica 41, 2 (2018); English translation in The black hole fifty years after:
Genesis of the name, arXiv: 1811.06587.
[273] W.C. Hernandez and C.W. Misner: Observer Time as a Coordinate in Relativistic Spherical
Hydrodynamics, Astrophys. J. 143, 452 (1966).
[274] M. Heusler: On the uniqueness of the Reissner-Nordström solution with electric and magnetic
charge, Class. Quantum Grav. 11, L49 (1994).
[275] M. Heusler: Black Hole Uniqueness Theorems, Cambridge Univ. Press, Cambridge (1996);
DOI: 10.1017/CBO9780511661396.
[276] M. Heusler: On the uniqueness of the Papapetrou-Majumdar metric, Class. Quantum Grav.
14, L129 (1997); arXiv: gr-qc/9607001.
[277] D. Hilbert: Die Grundlagen der Physik (Zweite Mitteilung), Nachrichten von der
Gesellschaft der Wissenscahften zu Göttingen, Mathematisch-physikalishe Klasse, Jahr
1917, 53 (1917).
[278] D. Hilbert: Die Grundlagen der Physik II, lectures given at Göttingen University in winter
1916-1917, lecture notes written by R. Bär; reprinted in [445], p. 162-307.
[279] W.A. Hiscock, L.G. Williams, and D.M. Eardley: Creation of particles by shell-focusing
singularities, Phys. Rev. D 26, 751 (1982).
[280] S. Hollands and A. Ishibashi: Black hole uniqueness theorems in higher dimensional
spacetimes, Class. Quantum Grav. 29, 163001 (2012); arXiv: 1206.1164.
[281] S. Hollands, A. Ishibashi, and H.S. Reall: A Stationary Black Hole Must be Axisymmetric
in Effective Field Theory, Commun. Math. Phys. 401, 2757 (2023); arXiv: 2212.06554.
[282] S. Hollands, A. Ishibashi, and R.M. Wald: A Higher Dimensional Stationary Rotating Black
Hole Must be Axisymmetric, Commun. Math. Phys. 271, 699 (2007); arXiv: gr-qc/0605106.
[283] S. Hollands and R.M. Wald: Stability of Black Holes and Black Branes, Commun. Math.
Phys. 321, 629 (2013); arXiv: 1201.0463.
[284] S. Hollands, R.M. Wald and V.G. Zhang: Entropy of dynamical black holes, Phys. Rev. D
110, 024070 (2024); arXiv: 2402.00818.
[285] G.T. Horowitz: Quantum States of Black Holes, in Black Holes and Relativistic Stars, edited
by R.M. Wald, Univ. Chicago Press, Chicago (1998), p. 241; arXiv: gr-qc/9704072.
[286] S. Husa and J. Winicour: Asymmetric merger of black holes, Phys. Rev. D 60, 084019
(1999); arXiv: gr-qc/9905039.
[287] S. Hwang: A Rigidity Theorem for Ricci Flat Metrics, Geometriae Dedicata 71, 5 (1998).
[288] A.D. Ionescu and S. Klainerman: Rigidity results in general relativity: A review, Surveys
in Differential Geometry 20, 123 (2015); arXiv: 1501.01587.
BIBLIOGRAPHY 803
[289] W. Israel: Event Horizons in Static Vacuum Space-Times, Phys. Rev. 164, 1776 (1967).
[290] W. Israel: Event horizons in static electrovac space-times, Commun. Math. Phys. 8, 245
(1968).
[291] W. Israel: Third Law of Black-Hole Dynamics: A Formulation and Proof, Phys. Rev. Lett.
57, 397 (1986).
[292] V. Iyer and R.M. Wald: Some properties of the Noether charge and a proposal for dynamical
black hole entropy, Phys. Rev. D 50, 846 (1994); arXiv: gr-qc/9403028.
[293] T. Jacobson: Introductory Lectures on Black Hole Thermodynamics, lectures given at the
University of Utrecht (1996); URL: https://www.physics.umd.edu/grt/taj/776b/
lectures.pdf.
[294] T. Jacobson, G. Kang, and R.C. Myers: On black hole entropy, Phys. Rev. D 49, 6587 (1994);
arXiv: gr-qc/9312023.
[295] T. Jacobson and Y.-A. Soong: Slices of the Kerr ergosurface, Class. Quantum Grav. 26,
055014 (2009); arXiv: 0809.2369.
[296] J. Jaffe: Collapsing objects and the backward emission of light, Ann. Phys. 55, 374 (1969).
[297] J.L. Jaramillo: An Introduction to Local Black Hole Horizons in the 3+1 Approach of Gen-
eral Relativity, in Black Holes: New Horizon, edited by S.A. Hayward, World Scientific,
Singapore (2013); DOI: 10.1142/S0218271811020366; arXiv: 1108.2408.
[298] J.L. Jaramillo and E. Gourgoulhon: Mass and Angular Momentum in General Relativity, in
Mass and Motion in General Relativity, edited by L. Blanchet, A. Spallicci and B. Whiting,
Fundamental Theories of Physics 162, 87, Springer, Dordrecht (2011); arXiv: 1001.5429.
[299] M. Jaroszyński and A. Kurpiewski: Optics near Kerr black holes: spectra of advection
dominated accretion flows, Astron. Astrophys. 326, 419 (1997); arXiv: astro-ph/9705044.
[300] J.T. Jebsen: Über die allgemeinen kugelsymmetrischen Lösungen der Einsteinschen Gravita-
tionsgleichungen im Vakuum, Arkiv för Matematik, Astronomi och Fysik 15, Nr. 18, p. 1
(1921); English translation in Gen. Relativ. Grav. 37, 2253 (2005).
[301] T. Johannsen: Sgr A* and general relativity, Class. Quantum Grav. 33, 113001 (2016);
arXiv: 1512.03818.
[302] T. Johannsen and D. Psaltis: Testing the No-Hair Theorem with Observations in the Electro-
magnetic Spectrum: II. Black-Hole Images, Astrophys. J. 718, 446 (2010); arXiv: 1005.1931.
[303] N.V. Johansen and F. Ravndal: On the discovery of Birkhoff’s theorem, Gen. Relativ. Grav.
38, 537 (2006); arXiv: physics/0508163.
[304] M.D. Johnson, A. Lupsasca, A. Strominger, G.N. Wong, S. Hadar, D. Kapec, R. Narayan,
A. Chael, C.F. Gammie, P. Galison, D.C. Palumbo, S.S. Doeleman, L. Blackburn, M. Wielgus,
D.W. Pesce, J.R. Farah, and J.M. Moran: Universal interferometric signatures of a black
hole’s photon ring, Science Advances 6, eaaz1310 (2020); arXiv: 1907.04329.
[305] M.D. Johnson et al: The Black Hole Explorer: motivation and vision, Proc. of SPIE 13092,
130922D (2024); arXiv: 2406.12917.
[306] D. Joyner and W. Stein: Sage Tutorial, CreateSpace (2014).
804 BIBLIOGRAPHY
[307] P. Kanti and E. Winstanley: Hawking Radiation from Higher-Dimensional Black Holes, in
Quantum Aspects of Black Holes, edited by X. Calmet, Fundamental Theories of Physics
178, 229, Springer, Cham (2015); arXiv: 1402.3952.
[308] D. Kapec and A. Lupsasca: Particle motion near high-spin black holes, Class. Quantum
Grav. 37, 015006 (2020)
DOI: 10.1088/1361-6382/ab519e arXiv: 1905.11406
[309] J. Katz: A note on Komar’s anomalous factor, Class. Quantum Grav. 2, 423 (1985)
DOI: 10.1088/0264-9381/2/3/018
[310] B.S. Kay and R.M. Wald: Theorems on the uniqueness and thermal properties of stationary,
nonsingular, quasifree states on spacetimes with a bifurcate killing horizon, Phys. Rep. 207,
49 (1991)
DOI: 10.1016/0370-1573(91)90015-E
[311] C. Kehle and R. Unger: Gravitational collapse to extremal black holes and the third law of
black hole thermodynamics, J. Eur. Math. Soc., in press (2025); arXiv: 2211.15742.
[312] R.P. Kerr: Gravitational Field of a Spinning Mass as an Example of Algebraically Special
Metrics, Phys. Rev. Lett. 11, 237 (1963)
DOI: 10.1103/PhysRevLett.11.237
[313] R.P. Kerr: Discovering the Kerr and Kerr-Schild metrics, in The Kerr spacetime, edited by
D.L. Wiltshire, M. Visser, and S.M. Scott, Cambridge Univ. Press, Cambridge (2009), p. 38
arXiv: 0706.1109
[314] R.P. Kerr and A. Schild: A New Class of Vacuum Solutions of the Einstein Field Equations,
in Atti del Convegno sulla Relativita Generale: Problemi dell’Energia e Onde Gravitazionali,
G. Barbèra Editore, Firenze (1965), p. 1; reprinted in Gen. Relativ. Gravit. 41, 2485 (2009)
DOI: 10.1007/s10714-009-0857-z
[315] I. Khavkine: Covariant phase space, constraints, gauge and the Peierls formula, Int. J. Mod.
Phys. A 29, 1430009 (2014)
DOI: 10.1142/S0217751X14300099 arXiv: 1402.1282
[316] A. King: Supermassive Black Holes, Cambridge Univ. Press, Cambridge (2023)
DOI: 10.1017/9781108768849
[317] O. Klein: Einige Probleme der allgemeinen Relativitätstheorie, in Werner Heisenberg und
die Physik unserer Zeit, edited by F. Bopp, Vieweg+Teubner Verlag, Wiesbaden (1961),
p. 58
DOI: 10.1007/978-3-663-05439-9_5
[318] S. Kobayashi and K. Nomizu: Foundations of Differential Geometry, vol. 1, Interscience,
New York (1963).
[319] A. Komar: Covariant Conservation Laws in General Relativity, Phys. Rev. 113, 934 (1959)
DOI: 10.1103/PhysRev.113.934
[320] A. Komar: Asymptotic Covariant Conservation Laws for Gravitational Radiation, Phys.
Rev. 127, 1411 (1962)
DOI: 10.1103/PhysRev.127.1411
BIBLIOGRAPHY 805
[321] E.-A. Kontou and K. Sanders: Energy conditions in general relativity and quantum field
theory, Class. Quantum Grav. 37, 193001 (2020)
DOI: 10.1088/1361-6382/ab8fcf arXiv: 2003.01815
[322] F. Kottler: Über die physikalischen Grundlagen der Einsteinschen Gravitationstheorie,
Annalen der Physik 56, 401 (1918)
DOI: 10.1002/andp.19183611402
[323] A. Krasiński: Editor’s Note: The Expanding Universe, Gen. Relativ. Gravit. 29, 637 (1997)
DOI: 10.1023/A:1018803604510
[324] A. Krasiński, E. Verdaguer and R.P. Kerr: Editorial note to: R. P. Kerr and A. Schild, A new
class of vacuum solutions of the Einstein field equations, Gen. Relativ. Gravit. 41, 2469
(2009)
DOI: 10.1007/s10714-009-0856-0
[325] B. Krishnan: Quasi-local Black Hole Horizons, in Springer Handbook of Spacetime, edited
by A. Ashtekar and V. Petkov, Springer, Berlin, Heidelberg (2014), p. 527
DOI: 10.1007/978-3-642-41992-8_25 arXiv: 1303.4635
[326] M.D. Kruskal: Maximal Extension of Schwarzschild Metric, Phys. Rev. 119, 1743 (1960)
DOI: 10.1103/PhysRev.119.1743
[327] H.K. Kunduri and J. Lucietti: Classification of Near-Horizon Geometries of Extremal Black
Holes, Living Rev. Relativ. 16, 8 (2013)
DOI: 10.12942/lrr-2013-8
[328] H.K. Kunduri and J. Lucietti: No static bubbling spacetimes in higher dimensional Einstein-
Maxwell theory, Class. Quantum Grav. 35, 054003 (2018)
DOI: 10.1088/1361-6382/aaa744 arXiv: 1712.02668
[329] H.K. Kunduri, J. Lucietti and H.S. Reall: Near-horizon symmetries of extremal black holes,
Class. Quantum Grav. 24, 4169 (2007)
DOI: 10.1088/0264-9381/24/16/012 arXiv: 0705.4214
[330] Y. Kuroda: Naked Singularities in the Vaidya Spacetime, Prog. Theor. Phys. 72, 63 (1984)
DOI: 10.1143/PTP.72.63
[331] J. Lafontaine: An Introduction to Differential Manifolds, Springer, Cham (2015)
DOI: 10.1007/978-3-319-20735-3
[332] K. Lake: Revisiting the Darmois and Lichnerowicz junction conditions, Gen. Relativ. Gravit.
49, 134 (2017)
DOI: 10.1007/s10714-017-2300-1 arXiv: 1705.01090
[333] K. Lake and R.C. Roeder: Note on the optical appearance of a star collapsing through its
gravitational radius, Astrophys. J. 232, 277 (1979)
DOI: 10.1086/157286
[334] K. Landsman: Singularities, Black Holes, and Cosmic Censorship: A Tribute to Roger Penrose,
Foundations of Physics 51, 42 (2021)
DOI: 10.1007/s10701-021-00432-1 arXic: 2101.02687
[335] K. Landsman: Penrose’s 1965 singularity theorem: from geodesic incompleteness to cosmic
censorship, Gen. Relat. Grav. 54, 115 (2022)
DOI: 10.1007/s10714-022-02973-w arXiv: 2205.01680
806 BIBLIOGRAPHY
[336] S. Lang: Differential and Riemannian Manifolds, Springer-Verlag, New York (1995).
[337] D. Langlois: Relativité générale, Vuibert, Paris (2013).
[338] D. Langlois: Dark energy and modified gravity in degenerate higher-order scalar–tensor
(DHOST) theories: A review, Int. J. Mod. Phys. D 28, 1942006 (2019)
DOI: 10.1142/S0218271819420069 arXiv: 1811.06271
[339] J.-P. Lasota, E. Gourgoulhon, M. Abramowicz, A. Tchekhovskoy, and R. Narayan: Ex-
tracting black-hole rotational energy: The generalized Penrose process, Phys. Rev. D 89,
024041 (2014)
DOI: 10.1103/PhysRevD.89.024041 arXiv: 1310.7499
[340] M. von Laue: Die Relativitätstheorie. Zweiter Band: Die Allgemeine Relativitätstheorie
Und Einsteins Lehre Von Der Schwerkraft, Friedr. Vieweg & Sohn, Braunschweig (1921);
available at
https://archive.org/details/dierelativitts02laueuoft/page/n5/mode/2up
[341] M. Le Bellac: Quantum Physics, Cambridge Univ. Press, Cambridge (2006)
DOI: 10.1017/CBO9780511616471
[342] J. M. Lee: Introduction to Topological Manifolds, 2nd edition, Springer, New-York (2011)
DOI: 10.1007/978-1-4419-7940-7
[343] J. M. Lee: Introduction to Smooth Manifolds, 2nd edition, Springer, New-York (2013)
DOI: 10.1007/978-1-4419-9982-5
[344] J. M. Lee: Introduction to Riemannian Manifolds, 2nd edition, Springer, Cham (2018)
DOI: 10.1007/978-3-319-91755-9
[345] J. Lee and R.M. Wald: Local symmetries and constraints, J. Math. Phys. 31, 725 (1990)
DOI: 10.1063/1.528801
[346] G. Lemaître: L’univers en expansion, Publication du Laboratoire d’Astronomie et de
Géodésie de l’Université de Louvain 9, 171 (1932); reprinted in Annales de la Société
Scientifique de Bruxelles A 53, 51 (1933); English translation in Gen. Relativ. Gravit. 29,
641 (1997)
DOI: 10.1023/A:1018855621348
[347] R. W. Lindquist, R. A. Schwartz, and C. W. Misner: Vaidya’s Radiating Schwarzschild
Metric, Phys. Rev. 137, B1364 (1965)
DOI: 10.1103/PhysRev.137.B1364
[348] J.-P. Luminet: Image of a spherical black hole with thin accretion disk, Astron. Astrophys.
75, 228 (1979).
[349] J.-P. Luminet: An Illustrated History of Black Hole Imaging: Personal Recollections (1972-
2002), preprint (2019); arXiv: 1902.11196.
[350] A. Lupsasca, A. Cárdenas-Avendaño, D.C.M. Palumbo, M.D. Johnson, S.E. Gralla, D.P. Mar-
rone, P. Galison, P. Tiede, and L.Keeble: The Black Hole Explorer: photon ring science,
detection, and shape measurement, Proc. of SPIE 13092, 130926Q (2024); arXiv: 2406.09498
[351] D. Lynden-Bell: Galactic Nuclei as Collapsed Old Quasars, Nature 223, 690 (1969)
DOI: 10.1038/223690a0
BIBLIOGRAPHY 807
[352] J.H. MacGibbon, B.J. Carr, and D.N. Page: Do evaporating black holes form photospheres?,
Phys. Rev. D 78, 064043 (2008)
DOI: 10.1103/PhysRevD.78.064043 arXiv: 0709.2380
[353] S.D. Majumdar: A Class of Exact Solutions of Einstein’s Field Equations, Phys. Rev. 72, 390
(1947)
DOI: 10.1103/PhysRev.72.390
[354] J. Maldacena and L. Susskind: Cool horizons for entangled black holes, Fortschritte der
Physik 61, 781 (2013)
DOI: 10.1002/prop.201300020 arXiv: 1306.0533
[355] J.-A. Marck: Journey to a black hole, movie (1991)
https://www.youtube.com/watch?v=5Oqop50ltrM
see also https://images.cnrs.fr/video/46
[356] J.-A. Marck: Short-cut method of solution of geodesic equations for Schwarzchild black hole,
Class. Quantum Grav. 13, 393 (1996)
DOI: 10.1088/0264-9381/13/3/007 arXiv: gr-qc/9505010
[357] K. Martel and E. Poisson: Regular coordinate systems for Schwarzschild and other spherical
spacetimes, Amer. J. Phys. 69, 476 (2001)
DOI: 10.1119/1.1336836 arXiv: gr-qc/0001069
[358] A.K.M. Masood-ul-Alam: Uniqueness proof of static charged black holes revisited, Class.
Quantum Grav. 9, L53 (1992)
DOI: 10.1088/0264-9381/9/5/001
[359] R.A. Matzner, H.E. Seidel, S.L. Shapiro, L. Smarr, W.-M. Suen, S.A. Teukolsky, and J.
Winicour: Geometry of a Black Hole Collision, Science 270, 941 (1995)
DOI: 10.1126/science.270.5238.941
[360] N.E. Mavromatos and V.A. Mitsou: Magnetic monopoles revisited: Models and searches at
colliders and in the Cosmos, Int. J. Mod. Phys. A 35, 2030012 (2020)
DOI: 10.1142/S0217751X20300124 arXiv: 2005.05100
[361] P.O. Mazur: Proof of uniqueness of the Kerr-Newman black hole solution, J. Phys. A: Math.
Gen. 15, 3173 (1982)
DOI: 10.1088/0305-4470/15/10/021
[362] P.O. Mazur: Black hole uniqueness from a hidden symmetry of Einstein’s gravity, Gen.
Relat. Grav. 16, 211 (1984)
DOI: 10.1007/BF00762536
[363] P.O. Mazur: A global identity for nonlinear σ-models, Phys. Lett. A 100, 341 (1984)
DOI: 10.1016/0375-9601(84)91084-3
[364] P.O. Mazur: Black Hole Uniqueness Theorems, unpublished report (2001)
arXiv: hep-th/0101012
[365] F. Mertens, A.P. Lobanov, R.C. Walker, and P.E. Hardee: Kinematics of the jet in M87 on
scales of 100-1000 Schwarzschild radii, Astron. Astrophys. 595, A54 (2016)
DOI: 10.1051/0004-6361/201628829 arXiv: 1608.05063
[366] J. W. Milnor: On manifolds homeomorphic to the 7-sphere, Ann. Math. 64, 399 (1956).
808 BIBLIOGRAPHY
[367] H. Mineur: La mécanique des masses variables. Le problème des deux corps, Annales
scientifiques de l’École Normale Supérieure, Ser. 3, 50, 1 (1933)
DOI: 10.24033/asens.825
[368] Y. Mino: Perturbative approach to an orbital evolution around a supermassive black hole,
Phys. Rev. D 67, 084027 (2003)
DOI: 10.1103/PhysRevD.67.084027 arXiv: gr-qc/0302075
[369] C.W. Misner: Interpretation of Gravitational-Wave Observations, Phys. Rev. Lett. 28, 994
(1972)
DOI: 10.1103/PhysRevLett.28.994
[370] C.W. Misner and D.H. Sharp: Relativistic Equations for Adiabatic, Spherically Symmetric
Gravitational Collapse, Phys. Rev. 136, B571 (1964)
DOI: 10.1103/PhysRev.136.B571
[371] C.W. Misner, K.S. Thorne, and J.A. Wheeler: Gravitation, Freeman, New York (1973);
reprinted by Princeton University Press, Princeton (2017)
URL: https://press.princeton.edu/books/ebook/9781400889099/gravitation
[372] V. Moncrief and J. Isenberg: Symmetries of higher dimensional black holes, Class. Quantum
Grav. 25, 195015 (2008)
DOI: 10.1088/0264-9381/25/19/195015 arXiv: 0805.1451
[373] H. Müller zum Hagen, D.C. Robinson, and H.J. Seifert: Black holes in static vacuum
space-times, Gen. Relat. Grav. 4, 53 (1973)
DOI: 10.1007/BF00769760
[374] G. Muñoz: Orbits of massless particles in the Schwarzschild metric: Exact solutions, Amer.
J. Phys 82, 564 (2014)
DOI: 10.1119/1.4866274
[375] R.C. Myers: Higher-dimensional black holes in compactified space-times, Phys. Rev. D 35,
455 (1987)
DOI: 10.1103/PhysRevD.35.455
[376] R.C. Myers and M.J. Perry: Black holes in higher dimensional space-times, Ann. Phys. 172,
304 (1986)
DOI: 10.1016/0003-4916(86)90186-7
[377] R. Narayan, M.D. Johnson, Michael D., and C.F. Gammie: The Shadow of a Spherically
Accreting Black Hole, Astrophys. J. Lett. 885, L33 (2019)
DOI: 10.3847/2041-8213/ab518c arXiv: 1910.02957
[378] V.V. Narlikar and P.C. Vaidya: A Spherically Symmetrical Non-Static Electromagnetic Field,
Nature 159, 642 (1947)
DOI: 10.1038/159642a0
[379] J. Nash: The Imbedding Problem for Riemannian Manifolds, Ann. Math. 63, 20 (1956)
DOI: 10.2307/1969989
[380] E.T. Newman, E. Couch, K. Chinnapared, A. Exton, A. Prakash, and R. Torrence: Metric
of a Rotating, Charged Mass, J. Math. Phys. 6, 918 (1965)
DOI: 10.1063/1.1704351
BIBLIOGRAPHY 809
[381] R.P.A.C. Newman: Topology and stability of marginal 2-surfaces, Class. Quantum Grav. 4,
277 (1987)
DOI: 10.1088/0264-9381/4/2/011
[382] J.-P. Nicolas: The Conformal Approach to Asymptotic Analysis, in From Riemann to Differ-
ential Geometry and Relativity, edited by L. Ji, A. Papadopoulos and S. Yamada, Springer,
Cham (2017), p. 571
DOI: 10.1007/978-3-319-60039-0_19 arXiv: 1508.02592
[383] B.C. Nolan: Sectors of spherical homothetic collapse, Class. Quantum Grav. 18, 1651 (2001)
DOI: 10.1088/0264-9381/18/9/304 arXiv: gr-qc/0010032
[384] B.C. Nolan: Odd-parity perturbations of self-similar Vaidya spacetime, Class. Quantum
Grav. 24, 177 (2007)
DOI: 10.1088/0264-9381/24/1/010 arXiv: gr-qc/0608040
[385] G. Nordström: On the Energy of the Gravitational Field in Einstein’s Theory, Verhandl.
Koninkl. Ned. Akad. Wetenschap., Afdel. Natuurk. 20, 1238 (1918); available at
https://archive.org/details/proceedingsofsec202koni/page/1238
[386] I.D. Novikov: On the evolution of a semiclosed world (in Russian), Astronomicheskii
Zhurnal 40, 772 (1963); English translation in Soviet Astronomy 7, 587 (1964)
URL: https://ui.adsabs.harvard.edu/abs/1964SvA.....7..587N/abstract
[387] I.D. Novikov and V.P. Frolov: Physics of Black Holes, Kluwer, Dordrecht (1989)
DOI: 10.1007/978-94-017-2651-1
[388] I.D. Novikov and K.S. Thorne: Astrophysics of Black Holes, in Black Holes – Les astres
occlus, edited by C. DeWitt and B. DeWitt, Gordon and Breach, New York (1973), p. 343.
[389] M. Nozawa, T. Shiromizu, K. Izumi and S. Yamada: Divergence equations and uniqueness
theorem of static black holes, Class. Quantum Grav. 35, 175009 (2018)
DOI: 10.1088/1361-6382/aad206 arXiv: 1805.11385
[390] B. O’Neill: Semi-Riemannian Geometry, with Applications to Relativity, Academic Press,
New York (1983).
[391] B. O’Neill: The geometry of Kerr black holes, A.K. Peters, Wellesley (MA) (1995); reprinted
by Dover, Mineola (NY) (2014).
[392] J.R. Oppenheimer and H. Snyder: On Continued Gravitational Contraction, Phys. Rev. 56,
455 (1939)
DOI: 10.1103/PhysRev.56.455
[393] J.A. Orosz, J.E. McClintock, J.P. Aufdenberg, R.A. Remillard, M.J. Reid, R. Narayan, and L.
Gou: The Mass of the Black Hole in Cygnus X-1, Astrophys. J. 742, 84 (2011)
DOI: 10.1088/0004-637X/742/2/84 arXiv: 1106.3689
[394] D.N. Page: Particle emission rates from a black hole: Massless particles from an uncharged,
nonrotating hole, Phys. Rev. D 13, 198 (1976)
DOI: 10.1103/PhysRevD.13.198
[395] D.N. Page: Hawking radiation and black hole thermodynamics, New J. Phys. 7, 203 (2005)
DOI: 10.1088/1367-2630/7/1/203 arXiv: hep-th/0409024
810 BIBLIOGRAPHY
[396] D.N. Page and K.S. Thorne: Disk-Accretion onto a Black Hole. Time-Averaged Structure of
Accretion Disk, Astrophys. J. 191, 499 (1974)
DOI: 10.1086/152990
[397] P. Painlevé: La mécanique classique et la théorie de la relativité, Comptes Rendus Acad.
Sci. (Paris) 173, 677 (1921)
URL: http://www.bibnum.education.fr/physique/relativite/
la-mecanique-classique-et-la-theorie-de-la-relativite
[398] E. Papantonopoulos (ed.): Physics of Black Holes, Springer, Berlin (2009).
[399] A. Papapetrou: A Static Solution of the Gravitational Field for an Arbitrary Charge-
Distribution, Proc. R. Irish Acad. A, 51, 191 (1947)
URL: https://www.jstor.org/stable/20488481
[400] A. Papapetrou : Formation of a Singularity and Causality, in A Random Walk in Relativity
and Cosmology, edited by N. Dadhich, J. Krishna Rao, J.V. Narlikar and C.V. Vishveshwara,
Wiley Eastern, New Dehli (1985), p. 184.
[401] R. Parentani and P. Spindel: Hawking radiation, Scholarpedia 6, 6958 (2011)
DOI: 10.4249/scholarpedia.6958
[402] A.W. Peet: TASI Lectures on Black Holes in String Theory, in Strings, Branes and Gravity,
edited by J.A. Harvey, S. Kachru, and E. Silverstein, World Scientific, Singapore (2001),
p. 353
DOI: 10.1142/9789812799630_0003 arXiv: hep-th/0008241
[403] R. Penrose: Asymptotic Properties of Fields and Space-Times, Phys. Rev. Lett. 10, 66 (1963)
DOI: 10.1103/PhysRevLett.10.66
[404] R. Penrose: Conformal treatment of infinity, in Relativity, groups and topology, edited by
B. DeWitt and C. DeWitt, Gordon and Breach, New York (1964), p. 565; reprinted in Gen.
Relativ. Gravit. 43, 901 (2011)
DOI: 10.1007/s10714-010-1110-5
[405] R. Penrose: Gravitational collapse and space-time singularities, Phys. Rev. Lett. 14, 57
(1965)
DOI: 10.1103/PhysRevLett.14.57
[406] R. Penrose: Zero Rest-Mass Fields Including Gravitation: Asymptotic Behaviour, Proc. Roy.
Soc. Lond. A 284, 159 (1965)
http://www.jstor.org/stable/2415306
[407] R. Penrose: Structure of Space-Time, in Battelle Rencontres: 1967 Lectures in Mathematics
and Physics, edited by C.M. DeWitt and J.A. Wheeler, Benjamin, New York (1968), p. 121.
[408] R. Penrose: Gravitational Collapse: The Role of General Relativity, Rivista del Nuovo
Cimento, Numero Speziale 1, 252 (1969); reprinted in Gen. Relativ. Gravit. 34, 1141 (2002)
DOI: 10.1023/A:1016578408204
[409] R. Penrose and G.R. Floyd: Extraction of Rotational Energy from a Black Hole, Nature
Phys. Sci. 229, 177 (1971)
DOI: 10.1038/physci229177a0
[410] A. Perez: Black holes in loop quantum gravity, Rep. Prog. Phys. 80, 126901 (2017)
DOI: 10.1088/1361-6633/aa7e14 arXiv: 1703.09149
BIBLIOGRAPHY 811
[411] G. Perez-Giz: From Measure Zero to Measure Hero: Periodic Kerr Orbits and Gravitational
Wave Physics, PhD thesis, Columbia University, New York (2011)
DOI: 10.7916/D8P84JVB
[412] V. Perlick: Gravitational Lensing from a Spacetime Perspective, Living Rev. Relativ. 7, 9
(2004)
DOI: 10.12942/lrr-2004-9
[413] P. Peter and J.-P. Uzan: Primordial Cosmology, Oxford Univ. Press, Oxford (2009).
[414] J. Plebański and A. Krasiński: An Introduction to General Relativity and Cosmology,
Cambridge Univ. Press, Cambridge (2006)
DOI: 10.1017/CBO9780511617676
[415] M.A. Podurets: Asymptotic behavior of the optical luminosity of a star in gravitational
collapse in terms of general relativity theory (in Russian), Astronomicheskii Zhurnal 41,
1090 (1964); English translation in Soviet Astronomy 8, 868 (1965)
URL: https://ui.adsabs.harvard.edu/abs/1965SvA.....8..868P/abstract
[416] E. Poisson: A Relativist’s Toolkit, The Mathematics of Black-Hole Mechanics, Cambridge
Univ. Press, Cambridge (2004)
DOI: 10.1017/CBO9780511606601
[417] E. Poisson and W. Israel: Internal structure of black holes, Phys. Rev. D 41, 1796 (1990)
DOI: 10.1103/PhysRevD.41.1796
[418] E. Poisson and C.M. Will: Gravity: Newtonian, Post-Newtonian, Relativistic, Cambridge
Univ. Press, Cambridge (2014)
DOI: 10.1017/CBO9781139507486
[419] A. Pound and B. Wardell: Black Hole Perturbation Theory and Gravitational Self-Force, in
Handbook of Gravitational Wave Astronomy, edited by C. Bambi, S. Katsanevas, and K.D.
Kokkotas, Springer, Singapore (2021)
DOI: 10.1007/978-981-15-4702-7_38-1 arXiv: 2101.04592
[420] W.H. Press and S.A. Teukolsky: Floating Orbits, Superradiant Scattering and the Black-hole
Bomb, Nature 238, 211 (1972)
DOI: 10.1038/238211a0
[421] I. Rácz: A simple proof of the recent generalizations of Hawking’s black hole topology
theorem, Class. Quantum Grav. 25, 162001 (2008)
DOI: 10.1088/0264-9381/25/16/162001 arXiv: 0806.4373
[422] I. Rácz and R.M. Wald: Global extensions of spacetimes describing asymptotic final states
of black holes, Class. Quantum Grav. 13, 539 (1996)
DOI: 10.1088/0264-9381/13/3/017 arXiv: gr-qc/9507055
[423] A. Rajantie: Introduction to magnetic monopoles, Contemporary Physics 53, 195 (2012)
DOI: 10.1080/00107514.2012.685693 arXiv: 1204.3077
[424] P. Ramond and A. Le Tiec: Multipolar particles in helically symmetric spacetimes, Class.
Quantum Grav. 38, 135022 (2021)
DOI: 10.1088/1361-6382/abebef arXiv: 2005.00602
812 BIBLIOGRAPHY
[425] H.S. Reall: Higher-Dimensional Black Holes, in General Relativity, Cosmology and As-
trophysics; Perspectives 100 years after Einstein’s stay in Prague, edited by J. Bičák and
T. Ledvinka, Springer, Cham (2014), p. 245
DOI: 10.1007/978-3-319-06349-2_12
[426] H.S. Reall: Black Holes, lecture notes for the course on Black Holes in Part III of the
Cambridge Mathematical Tripos (2020); available at
http://www.damtp.cam.ac.uk/user/hsr1000/teaching.html
[427] H.S. Reall: Third law of black hole mechanics for supersymmetric black holes and a quasilo-
cal mass-charge inequality, Phys. Rev. D 110, 124059 (2024); arXiv: 2410.11956.
[428] M.J. Reid, J.E. McClintock, R. Narayan, L. Gou, R.A. Remillard, and J.A. Orosz: The
trigonometric parallax of Cygnus X-1, Astrophys. J. 742, 83 (2011)
DOI: 10.1088/0004-637X/742/2/83 arXiv: 1106.3688
[429] H. Reissner: Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen
Theorie, Annalen der Physik 50, 106 (1916)
DOI: 10.1002/andp.19163550905
[430] C.S. Reynolds: Observational Constraints on Black Hole Spin, Ann. Rev. Astron. Astrophys.
59, 117 (2021)
DOI: 10.1146/annurev-astro-112420-035022 arXiv: 2011.08948
[431] L. Rezzolla and O. Zanotti: Relativistic Hydrodynamics, Oxford Univ. Press, Oxford (2013)
DOI: 10.1093/acprof:oso/9780198528906.001.0001
[432] A. Riazuelo: Trous noirs et trous de vers, Pour la Science 372, 36 (2008).
[433] A. Riazuelo: Seeing relativity – I. Ray tracing in a Schwarzschild metric to explore the
maximal analytic extension of the metric and making a proper rendering of the stars, Int. J.
Mod. Phys. D 28, 1950042 (2019)
DOI: 10.1142/S0218271819500421 arXiv: 1511.06025
[434] A. Riazuelo: Seeing relativity – III. Journeying within the Kerr metric toward the negative
gravity region, Int. J. Mod. Phys. D 29, 2050109 (2020)
DOI: 10.1142/S0218271820501096 arXiv: 2008.04384
[435] W. Rindler: Visual Horizons in World-Models, Month. Not. Roy. Astron. Soc. 116, 662
(1956); reprinted in Gen. Relativ. Gravit. 34, 133 (2002)
DOI: 10.1023/A:1015347106729
[436] D.C. Robinson: Classification of black holes with electromagnetic fields, Phys. Rev. D 10,
458 (1974)
DOI: 10.1103/PhysRevD.10.458
[437] D.C. Robinson: Uniqueness of the Kerr Black Hole, Phys. Rev. Lett. 34, 905 (1975)
DOI: 10.1103/PhysRevLett.34.905
[438] D.C. Robinson: A simple proof of the generalization of Israel’s theorem, Gen. Relat. Grav.
8, 695 (1977)
DOI: 10.1007/BF00756322
[439] D.C. Robinson: Four decades of black hole uniqueness theorems, in The Kerr spacetime,
edited by D.L. Wiltshire, M. Visser, and S.M. Scott, Cambridge Univ. Press, Cambridge
BIBLIOGRAPHY 813
(2009), p. 115; avalaible at
https://nms.kcl.ac.uk/david.robinson/web_page/blackholes.pdf
[440] I. Robinson and A. Trautman: Some Spherical Gravitational Waves in General Relativity,
Proc. Roy. Soc. Lond. A 265, 463 (1962)
DOI: 10.1098/rspa.1962.0036
[441] C. Rovelli: Black Hole Entropy from Loop Quantum Gravity, Phys. Rev. Lett. 77, 3288
(1996)
DOI: 10.1103/PhysRevLett.77.3288 arXiv: gr-qc/9603063
[442] C. Rovelli: Quantum Gravity, Cambridge Univ. Press, Cambridge (2004); available at
https://www.cpt.univ-mrs.fr/~rovelli/book.pdf
DOI: 10.1017/CBO9780511755804
[443] R. Ruffini and J.A. Wheeler: Introducing the black hole, Physics Today 24, 30 (1971)
DOI: 10.1063/1.3022513
[444] J. Santiago and M. Visser: Tolman temperature gradients in a gravitational field, Eur. J.
Phys. 40, 025604 (2019)
DOI: 10.1088/1361-6404/aaff1c arXiv: 1803.04106
[445] T. Sauer and U. Majer (eds.): David Hilbert’s Lectures on the Foundations of Physics, 1915-
1927, Springer, Berlin (2009)
DOI: 10.1007/b12915
[446] M.A. Scheel, M. Boyle, T. Chu, L.E. Kidder, K.D. Matthews, and H.P. Pfeiffer: High-
accuracy waveforms for binary black hole inspiral, merger, and ringdown, Phys. Rev. D 79,
024003 (2009)
DOI: 10.1103/PhysRevD.79.024003 arXiv: 0810.1767
[447] W. Schmidt: Celestial mechanics in Kerr spacetime, Class. Quantum Grav. 19, 2743 (2002)
DOI: 10.1088/0264-9381/19/10/314 arXiv: gr-qc/0202090
[448] P. Schneider, J. Ehlers, and E.E. Falco: Gravitational Lenses, Springer-Verlag, Berlin (1992).
[449] K. Schwarzschild: Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen
Theorie, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu
Berlin, Phys.-Math. Klasse 1916, 189 (1916); English translation in Gen. Relativ. Gravit.
35, 951 (2003)
DOI: 10.1023/A:1022971926521 arXiv: physics/9905030
[450] O. Semerák: Stationary frames in the Kerr field, Gen. Relat. Grav. 25, 1041 (1993)
DOI: 10.1007/BF00763554
[451] J.M.M. Senovilla: Singularity Theorems and Their Consequences, Gen. Relat. Grav. 30, 701
(1998)
DOI: 10.1023/A:1018801101244 arXiv: 1801.04912
[452] J.M.M. Senovilla: A critical appraisal of the singularity theorems, Phil. Trans. R. Soc. A.
380, 20210174 (2022)
DOI: 10.1098/rsta.2021.0174 arXiv: 2108.07296
[453] J.M.M. Senovilla: The influence of Penrose’s singularity theorem in general relativity, Gen.
Relat. Grav. 54, 151 (2022)
DOI: 10.1007/s10714-022-03038-8 arXiv: 2206.13925
814 BIBLIOGRAPHY
[454] J.M.M. Senovilla and D. Garfinkle: The 1965 Penrose singularity theorem, Class. Quantum
Grav. 32, 124008 (2015)
DOI: 10.1088/0264-9381/32/12/124008 arXiv: 1410.5226
[455] S.L. Shapiro: Thermal radiation from stellar collapse to a black hole, Phys. Rev. D 40, 1858
(1989)
DOI: 10.1103/PhysRevD.40.1858
[456] M. Shibata: Numerical Relativity, World Scientific, Singapore (2016); DOI: 10.1142/9692.
[457] M. Siino: Topological Appearance of Event Horizon: What Is the Topology of the Event
Horizon That We Can See?, Prog. Theor. Phys. 99, 1 (1998)
DOI: 10.1143/PTP.99.1 arXiv: gr-qc/9803099
[458] M. Siino: Topology of event horizons Phys. Rev. D 58, 104016 (1998)
DOI: 10.1103/PhysRevD.58.104016 arXiv: gr-qc/9701003
[459] L. Smarr: Mass Formula for Kerr Black Holes, Phys. Rev. Lett. 30, 71 (1973); errata in Phys.
Rev. Lett. 30, 521 (1973)
DOI: 10.1103/PhysRevLett.30.71 and 10.1103/PhysRevLett.30.521 (errata)
[460] A.A. Starobinsky: Amplification of waves during reflection from a rotating “black hole” (in
Russian), Zhur. Eksp. Teor. Fiz. 64, 48 (1973); English translation in Soviet Physics JETP
37, 28 (1973)
URL: http://jetp.ras.ru/cgi-bin/e/index/e/37/1/p28?a=list
[461] L.C. Stein and N. Warburton: Location of the last stable orbit in Kerr spacetime, Phys. Rev.
D 101, 064007 (2020)
DOI: 10.1103/PhysRevD.101.064007 arXiv: 1912.07609
[462] W. Stein and D. Joyner: SAGE: System for Algebra and Geometry Experimentation, Com-
mun. Comput. Algebra 39, 61 (2005).
[463] B. Steinmüller, A.R. King, and J.-P. Lasota: Radiating bodies and naked singularities, Phys.
Lett. A 51, 191 (1975)
DOI: 10.1016/0375-9601(75)90525-3
[464] N. Straumann: General Relavity, 2nd edition, Springer, Dordrecht (2013)
DOI: 10.1007/978-94-007-5410-2
[465] A. Strominger and C. Vafa: Microscopic origin of the Bekenstein-Hawking entropy, Phys.
Lett. B 379, 99 (1996)
DOI: 10.1016/0370-2693(96)00345-0 arXiv: hep-th/9601029
[466] Z. Stuchlík: The Radial Motion of Photons in Kerr Metric, Bul. Astron. Inst. Czechoslovakia
32, 40 (1981)
https://ui.adsabs.harvard.edu/abs/1981BAICz..32...40S
[467] D. Sudarsky and R.M. Wald: Extrema of mass, stationarity, and staticity, and solutions to
the Einstein–Yang–Mills equations, Phys. Rev. D 46, 1453 (1992)
DOI: 10.1103/PhysRevD.46.1453
[468] D. Sudarsky and R.M. Wald: Mass formulas for stationary Einstein-Yang-Mills black holes
and a simple proof of two staticity theorems, Phys. Rev. D 47, R5209(R) (1993)
DOI: 10.1103/PhysRevD.47.R5209 arXiv: gr-qc/9305023
BIBLIOGRAPHY 815
[469] J.L. Synge: The Gravitational Field of a Particle, Proc. Roy. Irish Acad. A 53, 83 (1950)
http://www.jstor.org/stable/20488511
[470] J.L. Synge: The Escape of Photons from Gravitationally Intense Stars, Month. Not. Roy.
Astron. Soc. 131, 463 (1966)
DOI: 10.1093/mnras/131.3.463
[471] L.B. Szabados: Quasi-Local Energy-Momentum and Angular Momentum in General Rela-
tivity, Living Rev. Relativ. 12, 4 (2009)
DOI: 10.12942/lrr-2009-4
[472] G. Szekeres: On the Singularities of a Riemannian Manifold, Publicationes Mathematicae
Debrecen 7, 285 (1960); reprinted in Gen. Relativ. Gravit. 34, 2001 (2002)
DOI: 10.1023/A:1020744914721
[473] F.R. Tangherlini: Schwarzschild field in n dimensions and the dimensionality of space
problem, Il Nuovo Cimento 27, 636 (1963)
DOI: 10.1007/BF02784569
[474] C. H. Taubes: Gauge theory on asymptotically periodic 4-manifolds, J. Differential Geom.
25, 363 (1987).
[475] E. Teo: Spherical Photon Orbits Around a Kerr Black Hole, Gen. Relativ. Gravit. 35, 1909
(2003)
DOI: 10.1023/A:1026286607562
[476] S.A. Teukolsky: The Kerr metric, Class. Quantum Grav. 32 124006 (2015)
DOI: 10.1088/0264-9381/32/12/124006 arXiv: 1410.2130
[477] S.A. Teukolsky and W.H. Press: Perturbations of a rotating black hole. III. Interaction of
the hole with gravitational and electromagnetic radiation, Astrophys. J. 193, 443 (1974)
DOI: 10.1086/153180
[478] K.S. Thorne: Disk-Accretion onto a Black Hole. II. Evolution of the Hole, Astrophys. J. 191,
507 (1974)
DOI: 10.1086/152991
[479] K.S. Thorne: Black Holes and Time Warps, Norton, New York (1994).
[480] K.S. Thorne and D. MacDonald: Electrodynamics in curved spacetime: 3 + 1 formulation,
Month. Not. Roy. Astron. Soc. 198, 339 (1982)
DOI: 0.1093/mnras/198.2.339
[481] K.S. Thorne, W.H. Zurek, and R.H. Price: The Thermal Atmosphere of a Black Hole, in Black
holes: The Membrane Paradigm, edited by K.S. Thorne, R.H. Price and D.A. MacDonald,
Yale University Press, New Haven (1986), p. 280.
[482] R.C. Tolman: Effect of Inhomogeneity on Cosmological Models, Proc. Nat. Acad. Sci. USA
20, 169 (1934); reprinted in Gen. Relativ. Gravit. 29, 935 (1997)
DOI: 10.1023/A:1018891418565
[483] P.K. Townsend: Black Holes, Lecture notes for a course taught in Part III of the Cambridge
University Mathematical Tripos (1997)
arXiv: gr-qc/9707012
[484] W.G. Unruh: Notes on black-hole evaporation, Phys. Rev. D 14, 870 (1976)
DOI: 10.1103/PhysRevD.14.870
816 BIBLIOGRAPHY
[485] W.G. Unruh: Universal coordinates for Schwarzschild black holes (2014)
arXiv: 1401.3393
[486] P.C. Vaidya: The External Field of a Radiating Star in General Relativity, Current Science
12, 183 (1943); reprinted in Gen. Relativ. Gravit. 31, 119 (1999)
DOI: 10.1023/A:1018871522880
[487] P.C. Vaidya: The Gravitational Field of a Radiating Star, Proc. Indian Acad. Sci. A33, 264
(1951); reprinted in Gen. Relativ. Gravit. 31, 121 (1999)
DOI: 10.1023/A:1018875606950
[488] P.C. Vaidya: ’Newtonian’ Time in General Relativity, Nature 171, 260 (1953); reprinted in
Gen. Relativ. Gravit. 31, 137 (1999)
DOI: 10.1023/A:1018827723788
[489] K. Van Aelst: Note on equatorial geodesics in circular spacetimes, Class. Quantum Grav.
37, 207001 (2020)
DOI: 10.1088/1361-6382/aba80c
[490] K. Van Aelst, E. Gourgoulhon, P. Grandclément, and C. Charmousis: Hairy rotating black
holes in cubic Galileon theory, Class. Quantum Grav. 37, 035007 (2020)
DOI: 10.1088/1361-6382/ab6391 arXiv: 1910.08451
[491] S.U. Viergutz: Image generation in Kerr geometry. I. Analytical investigations on the
stationary emitter-observer problem
Astron. Astrophys. 272, 355 (1993)
https://ui.adsabs.harvard.edu/abs/1993A&A...272..355V
[492] F.H. Vincent, E. Gourgoulhon, and J. Novak: 3+1 geodesic equation and images in numerical
spacetimes, Class. Quantum Grav. 29, 245005 (2012)
DOI: 10.1088/0264-9381/29/24/245005 arXiv: 1208.3927
[493] F.H. Vincent, T. Paumard, E. Gourgoulhon, and G. Perrin: GYOTO: a new general rela-
tivistic ray-tracing code, Class. Quantum Grav. 28, 225011 (2011)
DOI: 10.1088/0264-9381/28/22/225011 arXiv: 1109.4769
[494] F. H. Vincent, M. Wielgus, M. A. Abramowicz, E. Gourgoulhon, J.-P. Lasota, T. Paumard,
and G. Perrin: Geometric modeling of M87* as a Kerr black hole or a non-Kerr compact
object, Astron. Astrophys. 646, A37 (2021)
DOI: 10.1051/0004-6361/202037787 arXiv: 2002.09226
[495] C. V. Vishveshwara: Generalization of the “Schwarzschild Surface” to Arbitrary Static and
Stationary Metrics, J. Math. Phys. 9, 1319 (1968)
DOI: 10.1063/1.1664717
[496] M. Visser: Hawking Radiation without Black Hole Entropy, Phys. Rev. Lett. 80, 3436 (1998)
DOI: 10.1103/PhysRevLett.80.3436 arXiv: gr-qc/9712016
[497] M. Visser: The Kerr spacetime — a brief introduction, in The Kerr spacetime, edited by
D.L. Wiltshire, M. Visser, and S.M. Scott, Cambridge Univ. Press, Cambridge (2009), p. 3
arXiv: 0706.0622
[498] M. Visser and Z. Yan: Properties of dynamical black hole entropy, J. High Energ. Phys.
10(2024), 029 (2024)
DOI: 10.1007/JHEP10(2024)029 arXiv: 2403.07140
BIBLIOGRAPHY 817
[499] R.M. Wald: General relativity, University of Chicago Press, Chicago (1984).
[500] R.M. Wald: On identically closed forms locally constructed from a field, J. Math. Phys. 31,
2378 (1990).
[501] R.M. Wald: Black Holes and Thermodynamics, in Black Hole Physics, edited by V. De Sab-
bata and Z. Zhang, Kluwer Academic Publisher, Dordrecht (1992), p. 55; DOI: 10.1007/978-
94-011-2420-1_2.
[502] R.M. Wald: Black hole entropy is the Noether charge, Phys. Rev. D 48, R3427 (1993); arXiv:
gr-qc/9307038.
[503] R.M. Wald: Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics,
University of Chicago Press, Chicago (1994)
URL: https://press.uchicago.edu/ucp/books/book/chicago/Q/bo3684008.html
[504] R.M. Wald: “Nernst theorem” and black hole thermodynamics, Phys. Rev. D 56, 6467 (1997);
arXiv: gr-qc/9704008.
[505] R.M. Wald: The thermodynamics of black holes, Living Rev. Relativ. 4, 6 (2001).
[506] R.M. Wald: The Entropy of Black Holes, in Black Holes Inside and Out 2024: visions for the
future of black hole physics, edited by L. Buoninfante, R. Carballo-Rubio, V. Cardoso, F. Di
Filippo and A. Eichhorn (2024), p. 29; arXiv: 2410.14414.
[507] R.M. Wald and A. Zoupas: General definition of “conserved quantities” in general relativity
and other theories of gravity, Phys. Rev. D 61, 084027 (2000); arXiv: gr-qc/9911095.
[508] M. Walker and R. Penrose: On Quadratic First Integrals of the Geodesic Equations for Type
{22} Spacetimes, Commun. Math. Phys. 18, 265 (1970).
[509] R.C. Walker, P.E. Hardee, F.B. Davies, C. Ly, and W. Junor: The Structure and Dynamics
of the Subparsec Jet in M87 Based on 50 VLBA Observations over 17 Years at 43 GHz,
Astrophys. J. 855 128 (2018); arXiv: 1802.06166.
[510] A.C. Wall: Ten proofs of the generalized second law, J. High Energ. Phys. 06(2009), 021
(2009); arXiv: 0901.3865.
[511] A.C. Wall: A Second Law for higher curvature gravity, Int. J. Mod. Phys. D 24, 1544014
(2015); arXiv: 1504.08040.
[512] D. Wallace: The case for black hole thermodynamics part I: Phenomenological thermo-
dynamics, Studies in History and Philosophy of Modern Physics 64, 52 (2018); arXiv:
1710.02724.
[513] D. Wallace: The case for black hole thermodynamics part II: Statistical mechanics, Studies
in History and Philosophy of Modern Physics 66, 103 (2019); arXiv: 1710.02725.
[514] J.L. Walsh, A.J. Barth, L.C. Ho, and M. Sarzi: The M87 Black Hole Mass from Gas-dynamical
Models of Space Telescope Imaging Spectrograph Observations, Astrophys. J. 770, 86 (2013);
arXiv: 1304.7273.
[515] N. Warburton, L. Barack, and N. Sago: Isofrequency pairing of geodesic orbits in Kerr
geometry, Phys. Rev. D 87, 084012 (2013); arXiv: 1301.3918.
[516] B. Waugh and K. Lake: Double-null coordinates for the Vaidya metric, Phys. Rev. D 34,
2978 (1986).
[517] H. Weyl: Zur Gravitationstheorie, Annalen der Physik 54, 117 (1917).
818 BIBLIOGRAPHY
[518] H. Weyl: Über die statischen kugelsymmetrischen Lösungen von Einsteins “kosmologischen”
Gravitationsgleichungen, Physikalische Zeitschrift 20, 31 (1919).
[519] J.A. Wheeler: Feynman and Jacob Bekenstein, Web of Stories (1996); URL: https://www.
webofstories.com/play/john.wheeler/84.
[520] J.A. Wheeler: Geons, Black Holes, and Quantum Foam: a Life in Physics, Norton, New
York (1998).
[521] M. Wielgus et al.: Monitoring the Morphology of M87* in 2009-2017 with the Event Horizon
Telescope, Astrophys. J. 901, 67 (2020); arXiv: 2009.11842.
[522] D.C. Wilkins: Bound Geodesics in the Kerr Metric, Phys. Rev. D 5, 814 (1972).
[523] J.W. York: Kinematics and dynamics of general relativity, in Sources of gravitational
radiation, edited by L.L. Smarr, Cambridge University Press, Cambridge (1979), p. 83.
[524] H. Yoshino, K. Takahashi, and K. Nakao: How does a collapsing star look?, Phys. Rev. D
100, 084062 (2019); arXiv: 1908.04223.
[525] A. Zaffaroni: AdS black holes, holography and localization, Living Rev. Relativ. 23, 2 (2020);
arXiv: 1902.07176.
[526] Ya.B. Zeldovich: Semiclosed Worlds in the General Theory of Relativity (in Russian), Zhur.
Eksp. Teor. Fiz. 43, 1037 (1962); English translation in Soviet Physics JETP 16, 732 (1963).
[527] Ya.B. Zeldovich: Generation of Waves by a Rotating Body (in Russian), Pisma Red. Zhur.
Eksp. Teor. Fiz. 14, 270 (1971); English translation in Soviet Physics JETP Letters 14, 180
(1971).
[528] Ya.B. Zeldovich: Amplification of Cylindrical Electromagnetic Waves Reflected from a
Rotating Body (in Russian), Zhur. Eksp. Teor. Fiz. 62, 2076 (1972); English translation in
Soviet Physics JETP 35, 1085 (1972)
URL: http://jetp.ras.ru/cgi-bin/e/index/e/35/6/p1085?a=list
[529] Ya.B. Zeldovich and I.D. Novikov: Relativistic Astrophysics. I. (in Russian), Uspekhi
Fizicheskikh Nauk 84, 377 (1964); English translation in Soviet Physics Uspekhi 7, 763
(1965).
[530] Ya.B. Zeldovich and I.D. Novikov: Relativistic Astrophysics, vol. 1: Stars and Relativity,
University of Chicago Press, Chicago (1971).
[531] Ya.B. Zeldovich and I.D. Novikov: Relativistic Astrophysics, vol. 2: The Structure and
Evolution of the Universe, University of Chicago Press, Chicago (1983).
[532] P. Zimmermann et al.: Computational Mathematics with SageMath, SIAM (2018); freely
available from https://www.sagemath.org/sagebook/.
[533] R.L. Znajek: Black hole electrodynamics and the Carter tetrad, Month. Not. Roy. Astron.
Soc. 179, 457 (1977).
Index
N + 1 decomposition, 146 AGN, 436, 502
p-form, 719 algebraically special metric, 326
1-form, 717 analytic
3+1 formalism, 358 function, 713
3+1 slicing, 340 manifold, 713
4-momentum, 17 angular
4-rotation vector, 360 frequency
4-velocity, 19 Mino –, 414
momentum
acausal at infinity, 208, 375
hypersurface, 368 canonical –, 689
acceleration conserved –, 208, 375
of a curve, 741 Komar –, 148, 352, 531, 665
accretion Komar – at infinity, 152
disk, 277, 434, 495, 632, 644 total –, 152
achronal, 706 screen – coordinates, 454, 483
boundary, 119, 699 velocity, 225
set, 109 anti-de Sitter spacetime, 76, 101, 123, 528
active galactic nucleus, 436, 502 apoapsis, 228
adapted coordinates, 123 apoastron, 228, 239, 412, 459
ADM apocenter, 228
energy, 147, 688 approach the singularity, 408
energy-momentum, 147 arc length, 742, 743
Hamiltonian formulation, 154, 630 area
mass, 147, 531, 670 element bivector, 137
momentum, 147 of a cross-section, 58
advanced of a non-expanding horizon, 59
Kerr coordinates, 334 of the Kerr black hole, 353
time, 93, 190, 192, 587 theorem, 639
affine areal
connection, 726 coordinates, 185
induced –, 62 radius, 184, 542
parameter, 19, 740 asymptotic
parametrization, 740 θ-value, 398
820 INDEX
r-value, 398 bivector
inertial observer, 364 area element –, 137
symmetry Poisson –, 671
generator, 684 black
asymptotically body, 647
approach the singularity, 408 hole, 25, 106
Euclidean end, 677 evaporation, 648
flat, 104, 677 in equilibrium, 644
simple, 104 primordial –, 649
atlas, 713 region, 106
smooth –, 713 regular –, 702
autoparallel, 741 rotation velocity, 161, 349, 391, 458
average angular frequency, 417 shadow, 276, 482, 501
axisymmetric spacetime, 147, 160 spacetime, 106
ring, 128, 179
background saturn, 161, 164, 179
future, 677 Blandford-Znajek mechanism, 390
Banach manifold, 677 Boltzmann formula, 652
Barbero-Immirzi parameter, 653 boost, 67
Bardeen black hole, 702 bound
Bardeen-Horowitz coordinates, 526 geodesic, 406
baryon orbit, 225, 406
density, 631 boundary
number conservation, 631 achronal, 699
basis achronal –, 119
dual, 717, 718 conformal –, 100
left-handed –, 725 of a manifold, 714
natural, 715 topological –, 714
null –, 42 Boyer’s theorem, 86
orthonormal –, 42 Boyer-Lindquist coordinates, 324, 507
right-handed –, 725 Bunting-Mazur theorem, 177
Bekenstein-Hawking entropy, 650
Bertotti-Robinson spacetime, 536 canonical
Bianchi identity, 735 1-form, 671
contracted, 736 angular momentum, 689
bifurcate coordinates, 671
Killing horizon, 83, 295, 368, 369 energy, 687
bifurcation cardioid, 492, 493
sphere, 184, 296, 369, 523, 560 Cartan
surface, 83, 368 identity, 734
bilinear form, 719 structure equations, 326
binding energy, 144, 434 Carter
binormal 2-form, 668 coframe, 362
Birkhoff’s theorem, 547 constant, 378, 382, 757, 759
INDEX 821
reduced –, 410 orbit, 223, 419
frame, 361, 393 innermost stable –, 224, 430, 644
observer, 361 marginally bound –, 439
time machine, 332, 356, 394 photon orbit, 238, 473
Carter-Israel conjecture, 180 vortical –, 474
Carter-Penrose diagram, 298, 301, 364, 367, spacetime, 161
557 closed
regular –, 304 differential form, 733
Carter-Robinson theorem, 176 manifold, 36, 58
Cartesian trapped surface, 695
Boyer-Lindquist coordinates, 396 codifferential, 662
oval, 493 codimension, 720
Cauchy coframe
development ZAMO –, 358
future –, 368 collapse
past –, 368 gravitational –, 318
horizon, 368, 614, 701 commutator, 719, 731
problem, 368, 745 comoving coordinates, 542
surface, 127, 368, 698 compactification, 101
partial –, 368, 701 compactness, 552
causal complete
convergence condition, 704 cross-section, 36
curve, 17 elliptic integral, 251
future, 105 future null infinity, 106
past, 105 geodesic, 745
type, 26 geodesically – spacetime, 290, 746
vector, 15 completion
causally simple, 106, 699 conformal –, 100
caustic, 112 component
centrifugal of a linear form, 717
barrier, 224 of a tensor, 719
CFT, 536 w.r.t. a coordinate system, 715
change cone
of coordinates, 713 light –, 29
chart, 712 conformal, 94
chemical potential, 631 boundary, 100
Christoffel symbols, 374, 728, 744 compactification, 101
chronological completion
future, 105 at infinity, 100
past, 105 at null infinity, 103, 306
circular curvature, 737
geodesic diagram, 301
observer, 437 factor, 94
motion, 632 field theory, 536
822 INDEX
isometry, 589 counterjet, 502
Killing vector, 591 covariant, 657, 718
metric, 317 derivative, 727
time, 554 along a vector, 726, 728
transformation, 94 phase space, 670, 675, 676
conformally related metrics, 94 crease set, 113, 641
congruence, 54, 119, 190 critical
connection curve, 482
1-form, 62 null geodesic, 572
affine –, 726 external –, 245
coefficients, 727 in Kerr spacetime, 479
induced –, 62 internal –, 245
Levi-Civita –, 728 cross-section, 36
conservation complete –, 36
of energy-momentum, 23, 631 crossover point, 112, 699
conserved cubic Galileon, 75, 145
angular momentum, 208, 375 current
energy, 208, 375 electric –, 23
quantity, 675 Komar –, 143, 151, 631, 661
constraint equations, 698 magnetic –, 174
contracted curvature
Bianchi identity, 736 extrinsic –, 142, 153, 555, 737
Ricci identity, 736 Gaussian –, 126
contravariant, 718 intrinsic, 737
convergence scalar, 736
condition singularity, 197, 333, 701
causal –, 704 tensor, 734, 760
null –, 55, 61, 127, 638, 639, 654, 698 curve, 714, 740
convex normal neighborhood, 748 causal –, 17
coordinate, 712 cycloid, 551
canonical –, 671 Cygnus X-1, 223, 485
change, 713 cylinder
ignorable –, 66, 123 Einstein –, 96
line, 715
singularity, 196, 334 d’Alembertian, 129
system, 712 dark energy, 56
cosmic Darmois junction conditions, 555
censorship de Sitter spacetime, 103, 703, 705
weak –, 318 deflection
microwave background, 651 angle, 257
neutrino background, 651 of light, 260, 460
cosmological constant, 22 deformation rate, 50
cotangent bundle, 675 degenerate
countable base, 712 Killing horizon, 76, 512, 697
INDEX 823
derivative timelike geodesic, 215
covariant –, 726, 727 Einstein
exterior –, 135, 732 cylinder, 96, 554
Lie –, 730, 731 equation, 22
deviation electrovacuum –, 24, 173, 175, 177,
geodesic –, 197, 705, 760 178
DHOST, 160 vacuum –, 23, 76, 104, 171, 176, 178,
diffeomorphism, 713 326, 506, 530, 659, 764
invariance, 657 metric, 23
differential, 717 relation, 20
form, 719, 732 ring, 275
of a smooth map, 720 static universe, 96, 705
dimension of a manifold, 711 summation convention, 715
disformed Kerr solution, 160 tensor, 22, 73, 736
divergence linearized –, 130
tensor, 728 Einstein-Hilbert action, 22, 137, 625, 657
vector, 729 Einstein-Maxwell system, 24
domain of outer communications, 109, 127, Einstein-Rosen bridge, 313
153, 521 electric
dominance charge, 23
null – condition, 73, 125, 643 of a Killing horizon, 167
dominant energy condition, 73, 168, 627 current density, 23
Doppler beaming, 502 field, 166, 361
double-null coordinates, 599 Killing – potential, 166
dragging potential of a Killing horizon, 166
frame –, 395 electromagnetic field, 23
of inertial frames, 395 source-free –, 24
dual electrovacuum Einstein equation, 24, 173,
basis, 717, 718 175, 177, 178
Hodge –, 24, 136, 391 elliptic
of Weyl tensor, 341 integral
vector space, 717 complete –, 251
dust, 541, 543 incomplete –, 214, 251
null –, 585 Jacobi – function, 214
dynamical sine, 252
horizon, 28 Weierstrass – function, 213, 231
emanate from the singularity, 408
eccentricity, 417 embedded
Eddington-Finkelstein submanifold, 720
coordinates, 30, 192, 202, 334, 584, 587 embedding, 720
effective isometric –, 310
θ-potential, 398 Nash – theorem, 311
potential emergent symmetry, 535, 537
null geodesic, 236 emission
824 INDEX
angle, 271 Euler-Lagrange
energy derivative, 668
at infinity, 208, 375 equation, 659, 752
canonical –, 687 tensor, 657
condition Eulerian observer, 358
dominant –, 73 evaporation (black hole), 648
null –, 55, 61, 585, 638, 639, 654, 694, event
698 horizon, 26, 28, 107
null dominant –, 73, 125, 643 Event Horizon Telescope, 233, 485, 501
strong –, 704 exact diff. form, 733
weak –, 56, 643, 705 exotic
conserved –, 208, 375 R4 , 713
flux, 580 sphere, 713
in Hamiltonian framework, 675 expansion
of a particle, 20 of a null hypersurface, 49
energy-momentum positive – theorem, 638
conservation, 23, 631 exponential map, 747
tensor, 22 extension
vector, 17 maximal –, 293
entropy exterior
Bekenstein-Hawking –, 650 derivative, 135, 732
density, 631 product, 33, 138
generalized –, 652 external
Wald –, 618, 666 critical null geodesic, 245
EPR paradox, 319 extremal
equation black hole, 162, 643, 653
of state, 631 Kerr
equatorial spacetime, 506, 643, 644, 648
circular photon orbit, 475 throat, 526
orbit, 413 Kerr-Newman spacetime, 178
ER = EPR, 319 Reissner-Nordström
ergoregion, 162 black hole, 643, 697
inner –, 331 spacetime, 174, 697
of Kerr spacetime, 331 extrinsic curvature, 142, 153, 555, 737
outer –, 331, 394, 440
ergosphere Fermi-Walker
inner –, 331 operator, 360
outer –, 331, 440 transport, 360
ergosurface, 331 Ferrari’s method, 428
eternal Schwarzschild black hole, 318 field
Etherington’s reciprocity theorem, 580 equations, 659
Euclidean frame –, 719
metric, 735 scalar –, 719
Euler characteristic, 22, 126 tensor –, 719
INDEX 825
vector –, 719 event horizon, 107
first law horismos, 106, 699
of BH dynamics, 645 incomplete geodesic, 698
of thermodynamics, 631 infinity, 103
fixed point, 64 inner trapping horizon, 567
Flamm paraboloid, 312–314 light cone, 29
flat null infinity, 99, 103
asymptotically –, 104, 677 outer trapping horizon, 597
manifold, 735 timelike infinity, 99
metric, 735 future-directed, 15
flow map, 729
FLRW metric, 554 g-orthogonal, 721
fluid g-orthonormal basis, 723
perfect –, 55, 144, 542, 631, 705 gauge freedom, 129
flux Gauss’s law, 135, 167, 547, 665
integral, 137 Gauss-Bonnet theorem, 22, 126
of an image, 581 Gaussian
specific, 581 curvature, 126
form null coordinates, 159
p-form, 719 general relativity, 22
bilinear –, 719 generalized
differential –, 719, 732 entropy, 652
linear –, 717 second law, 651
volume –, 724 Smarr formula, 163, 620, 625
frame generator
dragging, 395 of a null hypersurface, 35
field, 719 of an achronal boundary, 699
of an observer, 20 of an event horizon, 112, 118
ZAMO –, 357 generic
free fall, 208 condition, 704
Friedmann-Lemaître-Robertson-Walker geodesic, 18
metric, 554 complete –, 745
Frobenius deviation equation, 197, 705, 760
theorem, 33, 70, 81, 328 equation, 191, 212, 744
frozen star, 109, 570 future-incomplete –, 698
fundamental in Kerr spacetime, 373
photon orbit, 461 in Schwarzschild spacetime, 207
future incomplete –, 86, 281, 698, 745
background –, 677 inextendible –, 698, 745
Cauchy maximal –, 745
development, 368 normal coordinates, 749
horizon, 368, 614, 701 null –, 234, 741
causal –, 105 past-incomplete –, 698
chronological –, 105 spacelike –, 741
826 INDEX
timelike –, 741 evaporation, 648
vector, 512 radiation, 646
vector field, 18, 34, 743 temperature, 646
with a bound orbit, 406 Hawking-Penrose theorem, 704
geodesically complete, 290, 746 heat capacity, 648
geometrical optics, 585 hidden symmetry, 758
global Hilbert gauge, 129
NHEK coordinates, 528 hit the singularity, 408
globally Hodge dual, 24, 136, 391
hyperbolic, 127, 698 homeomorphic, 125
naked singularity, 614 homeomorphism, 712
gradient, 32, 718 homothetic
gravitational Killing
binding energy, 144 horizon, 604
collapse, 318, 541, 550 vector, 590
lensing, 275 radiation shell, 591
potential, 144 Hopf-Rinow theorem, 746
potential energy, 144 horismos, 106, 699
waves, 541, 582 horizon
gravity bifurcate Killing –, 83, 295, 368, 369
strong –, 696 Cauchy –, 368, 614, 701
surface –, 82, 351 dynamical –, 28
group event –, 26, 28
action, 63 inner –, 348
isometry –, 64 isolated, 63
rotation –, 327 Killing –, 66, 348
symmetry –, 64 local isometry –, 68
translation –, 327 non-expanding –, 58, 125
GSL, 651 Hubble Space Telescope, 485
Gyoto, 278, 495, 783 hyperbolic
gyroscope, 360 plane, 586
rotation, 534
Hamilton’s equations, 670 hypersphere, 78, 554
Hamilton-Jacobi equation, 388 hypersurface, 26, 720
Hamiltonian hypersurface-orthogonal, 123
conjugate to a vector field, 685
for geodesic motion, 757 IEF, 192, 587
formulation of general relativity, 154, ignorable coordinate, 66, 123
630 immersed
system, 674 submanifold, 720
vector field, 673, 674 immersion, 720
harmonic coordinates, 130 impact parameter, 236
Hausdorff space, 711 imploding
Hawking shell of radiation, 588
INDEX 827
incomplete stable circular orbit, 224, 430, 644
elliptic integral, 214, 251 integral
future – geodesic, 698 flux –, 137
past – geodesic, 698 of motion
incomplete geodesic, 86, 281, 698, 745 reduced –, 410
index intensity, 581
lowering, 723 specific –, 580
raising, 723 internal
induced critical null geodesic, 245
affine connection, 62 infinity, 525
metric, 26, 38 intrinsic curvature, 737
inertial invariance
observer under a group action, 64
asymptotic –, 364 inverse metric, 723
inextendible geodesic, 698, 745 irreducible mass, 636
infinite redshift surface, 331 irreversible transformation, 635
infinitesimal ISCO, 224, 278
displacement vector, 14, 716 isolated
infinity horizon, 63
future null –, 99 isometric
future timelike –, 99 embedding, 310
internal –, 525 isometry, 64
past null –, 99 conformal –, 589
past timelike –, 99 group, 64
spacelike –, 99 horizon, 68
ingoing isotropic
Eddington-Finkelstein coordinates, 316
coordinates, 192, 587 Israel uniqueness theorem
domain, 195 electrovacuum, 173, 174
null geodesic, 189, 253, 588 vacuum, 171
principal null geodesic, 340
Jacobi
radial
elliptic function, 214
null geodesic, 588
elliptic sine, 252
null geodesics, 586 Jacobi equation, 760
Vaidya metric, 584 Jebsen-Birkhoff theorem, 371, 541, 547
inner jet, 502
circular
photon orbit, 475 Keplerian
timelike orbit, 426 orbit, 231
ergoregion, 331 Kerr
ergosphere, 331 black hole, 323
horizon, 348 coordinates, 338, 506
spherical photon orbit, 465 advanced –, 334
innermost outgoing –, 516
828 INDEX
retarded –, 342 Kruskal-Szekeres
extremal – spacetime, 506, 643, 648 coordinates, 284, 292, 520
metric, 326
disformed –, 160 Lagrangian
spacetime, 177, 335 form, 656
Kerr-Newman scalar, 656
black hole, 170, 634, 638 Lambert function, 285
metric, 536 Laplace operator, 137
spacetime, 177, 179, 387 lapse function, 146, 357, 550
extremal –, 178 least action, 655
Kerr-Schild left-handed basis, 725
coordinates, 340, 764 Lemaître synchronous coordinates, 542
metric, 194, 763 Lemaître-Tolman system, 543
Kerr-star coordinates, 334 length of a curve, 751
Kerr/CFT correspondence, 536, 653 Lense-Thirring effect, 395, 469, 503
Killing lensing ring, 501
equation, 65, 378 level set, 28
expression, 732 Levi-Civita
horizon, 66, 348 connection, 728
bifurcate –, 83, 295, 368, 369 tensor, 44, 724
degenerate –, 76, 512, 697 Lie
electric potential of a –, 166 algebra, 535
homothetic –, 604 derivative, 65, 730, 731
non-degenerate –, 76 dragging, 43
tensor, 378, 758 light
Walker-Penrose –, 378, 758 cone, 15, 29
vector, 65 curve, 495
homothetic –, 590 ray, 585
Komar ring, 473
angular momentum, 148, 352, 531, 665 line
at infinity, 152 coordinate –, 715
current, 143, 151, 631, 661 element, 14, 722
mass, 135, 187, 351, 530, 665, 670 linear form, 717
at infinity, 145 Lipschitz submanifold, 112
Kostant formula, 88 local isometry horizon, 68
Kottler metric, 186 locally
Kretschmann scalar, 196 naked singularity, 614
of Kerr metric, 333, 506 non-rotating observer, 361
of Oppenheimer-Snyder metric, 561 loop quantum gravity, 653
of Schwarzschild metric, 196 Lorentz
of Vaidya metric, 595 boost, 576
Kronecker symbol, 717 factor, 21
Kruskal Lorentzian
diagram, 293, 294 manifold, 722
INDEX 829
metric, 722 extension, 293
Lorenz gauge, 129 geodesic, 745
lowering an index, 723 hypersurface, 153
maximally symmetric, 533
M87*, 223, 485, 501 Maxwell equations, 18, 23, 585, 665
magnetic mechanical
current density, 174 energy, 216
field, 361 membrane
monopole, 174 one-way –, 26
Majumdar-Papapetrou metric, 721
black hole, 127, 152, 173, 176, 177 induced –, 26, 38
manifold, 14, 711 space, 746
analytic –, 713 tensor, 721
Banach –, 677 Minkowski
pseudo-Riemannian, 721 metric, 735, 763
smooth –, 713 spacetime, 29, 92
smooth – with boundary, 714 Minkowskian coordinates, 29, 92, 128, 764
symplectic –, 671 Mino
topological –, 712 angular frequency, 414
with boundary, 98, 713 parameter, 384
marginally time, 411
bound Misner-Sharp
circular orbit, 225, 439 energy, 544
geodesic, 225, 407, 549 mass, 544
orbit, 407 modulus
outer trapped surface, 59 of an elliptic integral, 251
stable circular orbit, 431 momentum
stable spherical orbit, 476 of a particle, 20
trapped surface, 60, 697 momentum constraint, 153
mass MOTS, 59
ADM –, 531 moving
gravitational –, 187, 227 frame, 719
inflation instability, 371 musical isomorphism, 724
irreducible –, 636 Myers-Perry black hole, 128, 179
Komar –, 135, 187, 351, 530, 665
Komar – at infinity, 145 naked singularity, 174, 318, 505, 604, 613
Newtonian –, 144 globally –, 614
parameter of Kerr solution, 324 locally –, 614
parameter of Schwarzschild solution, Nash embedding theorem, 311
187, 227 natural basis, 715
total –, 145 near-horizon
massive particle, 18 metric, 527
massless particle, 18 negative-energy particle, 388, 389, 477
maximal neighborhood
830 INDEX
normal –, 748 639, 654, 698
Newtonian coordinate, 192, 723
mass, 144 dominance condition, 73, 125, 643
mechanical energy, 216 dominant energy condition, 73, 125, 643
NHEK dust, 585
line, 490, 529 energy condition, 55, 61, 585, 638, 639,
metric, 527 654, 694, 698
spacetime, 529 future – infinity, 103
NIEF, 192 geodesic, 234, 741
no-hair theorem, 171, 178 generator, 35
Noether infinity, 99
charge ingoing Eddington-Finkelstein
of a hypersurface, 663 coordinates, 192, 334, 584
of a surface, 664 outgoing Eddington-Finkelstein
current form, 660 coordinates, 202
potential past – infinity, 103
form, 662 Raychaudhuri equation, 54, 700
Noether’s theorem, 660, 757 vector, 723
Noether-Wald charge density, 662
non-affinity coefficient, 34, 84, 351, 743 O’Neill
non-degenerate coordinates, 325, 330, 448, 471
bilinear form, 721 observer, 20
Killing horizon, 76 Carter –, 361
non-expanding horizon, 58, 125 circular geodesic –, 437
non-extremal black hole, 162 frame, 20
non-polar orbit, 413 static –, 215, 356
non-rotating stationary –, 81, 354, 437
horizon, 124 one-way membrane, 26
observer, 361 Oppenheimer-Snyder collapse, 552
nonlinear electrodynamics, 703 orbit
normal bound –, 225
area element, 137 circular –, 223, 419
bivector, 137 Keplerian –, 231
coordinates, 749 marginally bound –, 225, 407
Riemann –, 749 photon –, 238, 460
neighborhood, 748 under a group action, 64
convex –, 748 orbital
to a hypersurface, 725 period, 228
volume order of a tensor, 718
element, 141 orientable
null manifold, 724
basis, 42 time –, 15, 199
cone, 15, 723 orientation
convergence condition, 55, 61, 127, 638, of a manifold, 724
INDEX 831
orthogonal, 721 infinity, 103
complement, 38 null infinity, 99, 103
projector, 42, 146 timelike infinity, 99
orthogonally transitive, 75, 161 past-directed, 15
orthonormal basis, 42, 723 Penrose
outer diagram, 301
ergoregion, 331, 394, 440 process, 390, 635, 637
ergosphere, 331, 440 singularity theorem, 698
spherical photon orbit, 466 Penrose-Carter diagram, 301
trapped surface, 59 Penrose-Frolov-Novikov coordinates, 301
outgoing perfect fluid, 55, 144, 542, 631, 705
Kerr coordinates, 516 periapsis, 228
null geodesic, 189, 253 periastron, 228, 238, 412, 459
principal null geodesic, 342, 346 advance, 231
outward-pointing, 734 passage, 231
oval pericenter, 228
Cartesian –, 493 perturbed
of Descartes, 493 spacetime, 618
Petrov type, 343
Painlevé-Gullstrand coordinates, 549 phase space, 670, 674
Palatini identity, 625 covariant –, 675, 676
parallel transport, 727 photon, 234
parallelly transported, 727 orbit
parameter circular –, 238, 473
affine –, 740 fundamental –, 461
Mino –, 384 inner spherical –, 465
parameter along a curve, 714 outer spherical –, 466
parametrization, 714 spherical –, 460
affine –, 740 region, 479
partial ring, 500
Cauchy surface, 368, 701 shell, 479
particle sphere, 238, 572
massive –, 18 Planck
massless –, 18 length, 651
negative-energy –, 388, 389, 477 Planck-Einstein relation, 22
zero-energy –, 389 Poincaré
past coordinates on AdSn , 78, 528
Cauchy group, 533
development, 368 half-space, 78
causal –, 105 horizon of AdS4 , 78
chronological –, 105 lemma, 733
event horizon, 108, 367 patch
horismos, 106 of AdS4 , 78
incomplete geodesic, 698 of NHEK, 532
832 INDEX
transformation, 129 quasi-local, 696
Poisson
bivector, 671 radial
bracket, 670 geodesic, 212
Poisson equation, 137, 144 radiation
polar energy density, 592
orbit, 413 radius
spherical photon orbits, 466 of circular orbit, 419
positive mass theorem, 172 raising an index, 723
potential rank of a tensor, 718
chemical –, 631 Raychaudhuri
pregeodesic null – equation, 700
equation, 744 null – equation, 54
vector field, 18, 34, 512, 743 reciprocity theorem, 580
prephase space, 679 redshift
presymplectic in gravitational collapse, 575
current form, 680 infinite – surface, 331
form, 679 reduced
potential form, 657 Carter constant, 410
primordial black hole, 649 integral of motion, 410
principal reflection
null direction, 341 time –, 123
null geodesic, 765, 769 regular black hole, 702
ingoing –, 340, 362 Reissner-Nordström
outgoing –, 342, 346, 362 black hole, 173, 175
product extremal –, 643, 697
tensor –, 719 metric, 536
prograde spacetime, 175
orbit, 424 Reissner-Nordström-Tangherlini spacetime,
outer circular 175
photon orbit, 475, 485 retarded
timelike orbit, 426 Kerr coordinates, 342
projection time, 29, 93, 190
diagram, 365 retrograde
proper orbit, 424
time, 19, 742 outer circular
pseudo-Riemannian manifold, 721 photon orbit, 475, 485
pseudo-stationary, 122 timelike orbit, 426
pullback, 65, 136, 310, 721 reversible transformation, 636
pushforward, 64, 310, 720, 730 Ricci
identity, 735
quantum contracted –, 736
gravity, 536 scalar, 22, 736
quasar, 436 tensor, 22, 736
INDEX 833
Ricci-flat metric, 23 Schwarzschild-Droste
Riemann coordinates, 186
curvature, 734, 760 domain, 188
normal coordinates, 749 Schwarzschild-Tangherlini spacetime, 172
Riemannian screen, 268
manifold, 38, 722 angular coordinates, 454, 483
metric, 722 scri, 98
right-handed basis, 725 second law
rigidity theorem, 161 generalized –, 651
strong –, 156, 619 of BH dynamics, 645
weak –, 161 secondary
ring ring, 501
Einstein –, 275 self-similar Vaidya spacetime, 591
lensing –, 501 semiclassical effect, 646
light –, 473 semiclosed world, 318, 560
photon –, 500 separated space, 711
secondary –, 501 separation vector, 759
singularity, 333, 408, 506 shadow
rotating horizon, 124 black hole –, 276, 482, 501
rotation shear
1-form, 62, 74 tensor, 51
axis, 148, 368 shell
group, 327 imploding – of radiation, 588
velocity, 161, 349, 391, 458 shell theorem, 547
shell-focusing singularity, 614
SageManifolds, 773 shell-focusing singulariy, 614
SageMath, 773 shift vector, 146, 358, 550
Sagittarius A*, 223, 485, 495, 501 signature, 722
scalar simple
curvature, 736 causally –, 106, 699
field, 714, 719 singularity, 698, 746
product, 721 coordinate –, 196, 334
scattering, 238, 460 curvature –, 197, 333, 701
Schwarzschild globally naked –, 614
AdS metric, 186 Hawking-Penrose – theorem, 704
anti-de Sitter metric, 186 locally naked –, 614
black hole, 107, 183 naked –, 318, 613
coordinates, 186 Penrose’s – theorem, 698
de Sitter metric, 186 ring –, 333, 506
horizon, 108, 195 shell-focusing –, 614
metric, 186, 327 spacetime –, 698, 706
radius, 188 Smarr formula, 354
spacetime, 29, 124, 172, 199 4-dimensional –, 165
wormhole, 313 for charged black holes, 168
834 INDEX
generalized –, 163, 620, 625 observer, 81
smooth in Kerr spacetime, 354, 437
atlas, 713 spacetime, 122
manifold, 713 strictly –, 122
manifold with boundary, 714 Stefan-Boltzmann law, 648
map, 713 Stokes
source-free theorem, 734
electromagnetic field, 24 strictly
spacelike static, 123
coordinate, 723 stationary, 122
geodesic, 741 string theory, 653
infinity, 99, 524 strong
vector, 723 energy condition, 704
spacetime, 13 rigidity theorem, 156, 619
singularity, 698, 706 Stäckel-Killing tensor, 758
specific submanifold
conserved embedded –, 720
angular momentum, 215, 410 immersed –, 720
energy, 215, 410 superluminal motion, 502
flux, 581 supernova, 550
intensity, 580 superradiance, 391, 650
sphere superradiant scattering, 391, 650
photon –, 238, 572 surface
spherical gravity, 74, 82, 84, 351, 512
photon orbit, 460 Sylvester’s law of inertia, 722
inner –, 465 symmetric, 721
outer –, 466 symmetry
spin group, 64
parameter of Kerr solution, 324 hidden –, 758
squeeze mapping, 534 symplectic
stable form, 671, 674
circular orbit, 224, 278, 430 manifold, 671
starshaped neighborhood, 748 potential, 671
static synchronous
limit, 331, 356 coordinates, 542
observer Synge
in Kerr spacetime, 356 coordinates, 295
in Schwarzschild spacetime, 215 diagram, 295
spacetime, 123, 153, 184
strictly –, 123 tachyon, 17
universe (Einstein), 96, 705 tangent
staticity theorem, 153 space, 14, 715
stationary vector, 715
black hole, 121 telescope, 267
INDEX 835
aperture, 267 transcended law, 652
temperature transition map, 713
thermodynamic –, 631 translation
tensor, 718 group, 327
field, 719 trapped
product, 719 surface, 59, 566, 695, 706
tetrad, 719 closed –, 695
thermodynamic temperature, 631 marginally –, 60, 697
third law marginally outer –, 59
of thermodynamics, 643 outer –, 59
throat trapping horizon
extremal Kerr –, 526 future
tidal force, 197, 705 inner –, 567
time outer –, 597
advanced –, 190, 192, 587 traversable
conformal –, 554 wormhole, 313
machine (Carter), 332, 356, 394 triad, 719
Mino –, 411 turning point
proper –, 19, 742, 743 θ-turning point, 384
reflection symmetry, 123 r-turning point, 383
retarded –, 190 twin paradox, 753
time-orientable, 15, 199 twist 3-form, 80, 328
timelike type of a tensor, 718
coordinate, 723
ultracompact, 572
geodesic, 741
Unruh effect, 647
geodesically complete, 746
infinity, 99 vacuum
vector, 722 Einstein equation, 23, 76, 104, 171, 176,
Tolman model, 544 178, 326, 506, 530, 659, 764
Tolman-Bondi model, 544 permeability, 23
Tolman-Ehrenfest law, 647 Vaidya
Tolman-Oppenheimer-Volkoff equations, metric
558 ingoing –, 584
topological valence, 718
censorship theorem, 128 vector, 715
manifold, 712 field, 719
torsion-free, 728 infinitesimal –, 14, 716
tortoise coordinate, 190, 284 tangent –, 715
total tangent – space, 715
angular momentum, 152 Viète
mass, 145 formulas, 239
totally geodesic, 62 substitution, 244
trajectory VLBI, 501
of a geodesic, 213 volume
836 INDEX
form, 724 white
normal – element, 141 fountain, 108
vortical white hole, 108, 307, 367, 446, 520, 559
circular photon orbit, 474 Whitney theorem, 712
geodesic, 404, 455 winding
vorticity, 54 number
2-form, 361 of null geodesic, 257
worldline, 17
Wald entropy, 618, 666
wormhole
Walker-Penrose Killing tensor, 378, 758
Schwarzschild –, 313
warped AdS3 , 529
traversable –, 313
wave
vector, 585 Yamabe invariant, 128
weak
cosmic censorship, 318 ZAMO, 357
energy condition, 56, 643, 705 coframe, 358
rigidity theorem, 161 frame, 357
weakly zero-acceleration, 741
asymptotically simple, 104 zero-angular-momentum observer, 357
asymptotically simple and empty, 104 zero-energy particle, 389
relativistic, 128, 147 zeroth law
Weierstrass elliptic function, 213, 231 of BH dynamics, 74, 87, 351, 642, 644
Weyl curvature tensor, 737 of thermodynamics, 643
dual, 341 zoom-whirl, 414
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