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 An Introduction to General Relativity
and Cosmology

General relativity is a cornerstone of modern physics, and is of major importance in its

applications to cosmology. Experts in the field Plebański and Krasiński provide a thorough
introduction to general relativity to guide the reader through complete derivations of the
most important results.
An Introduction to General Relativity and Cosmology is a unique text that presents
a detailed coverage of cosmology as described by exact methods of relativity and
inhomogeneous cosmological models. Geometric, physical and astrophysical properties
of inhomogeneous cosmological models and advanced aspects of the Kerr metric are all
systematically derived and clearly presented so that the reader can follow and verify all
details. The book contains a detailed presentation of many topics that are not found in
other textbooks.
This textbook for advanced undergraduates and graduates of physics and astronomy will
enable students to develop expertise in the mathematical techniques necessary to study
general relativity.
An Introduction to General Relativity
and Cosmology

Jerzy Plebański
Centro de Investigación y de Estudios Avanzados
Instituto Politécnico Nacional
Apartado Postal 14-740, 07000 México D.F., Mexico

Andrzej Krasiński
Centrum Astronomiczne im. M. Kopernika,
Polska Akademia Nauk, Bartycka 18, 00 716 Warszawa,
Poland
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press

The Edinburgh Building, Cambridge  , UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521856232

© J. Plebanski and A. Krasi nski 2006

This publication is in copyright. Subject to statutory exception and to the provision of

relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.

First published in print format 2006

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 Contents

List of figures page xiii

The scope of this text xvii
Acknowledgements xix

1 How the theory of relativity came into being (a brief historical sketch) 1
1.1 Special versus general relativity 1
1.2 Space and inertia in Newtonian physics 1
1.3 Newton’s theory and the orbits of planets 2
1.4 The basic assumptions of general relativity 4

Part I Elements of differential geometry 7

2 A short sketch of 2-dimensional differential geometry 9

2.1 Constructing parallel straight lines in a flat space 9
2.2 Generalisation of the notion of parallelism to curved surfaces 10

3 Tensors, tensor densities 13

3.1 What are tensors good for? 13
3.2 Differentiable manifolds 13
3.3 Scalars 15
3.4 Contravariant vectors 15
3.5 Covariant vectors 16
3.6 Tensors of second rank 16
3.7 Tensor densities 17
3.8 Tensor densities of arbitrary rank 18
3.9 Algebraic properties of tensor densities 18
3.10 Mappings between manifolds 19
3.11 The Levi-Civita symbol 22
3.12 Multidimensional Kronecker deltas 23
3.13 Examples of applications of the Levi-Civita symbol and of the
multidimensional Kronecker delta 24
3.14 Exercises 25

v
vi Contents

4 Covariant derivatives 26
4.1 Differentiation of tensors 26
4.2 Axioms of the covariant derivative 28
4.3 A field of bases on a manifold and scalar components of tensors 29
4.4 The affine connection 30
4.5 The explicit formula for the covariant derivative of tensor density fields 31
4.6 Exercises 32

5 Parallel transport and geodesic lines 33

5.1 Parallel transport 33
5.2 Geodesic lines 34
5.3 Exercises 35

6 The curvature of a manifold; flat manifolds 36

6.1 The commutator of second covariant derivatives 36
6.2 The commutator of directional covariant derivatives 38
6.3 The relation between curvature and parallel transport 39
6.4 Covariantly constant fields of vector bases 43
6.5 A torsion-free flat manifold 44
6.6 Parallel transport in a flat manifold 44
6.7 Geodesic deviation 45
6.8 Algebraic and differential identities obeyed by the curvature tensor 46
6.9 Exercises 47

7 Riemannian geometry 48
7.1 The metric tensor 48
7.2 Riemann spaces 49
7.3 The signature of a metric, degenerate metrics 49
7.4 Christoffel symbols 51
7.5 The curvature of a Riemann space 51
7.6 Flat Riemann spaces 52
7.7 Subspaces of a Riemann space 53
7.8 Flat Riemann spaces that are globally non-Euclidean 53
7.9 The Riemann curvature versus the normal curvature of a surface 54
7.10 The geodesic line as the line of extremal distance 55
7.11 Mappings between Riemann spaces 56
7.12 Conformally related Riemann spaces 56
7.13 Conformal curvature 58
7.14 Timelike, null and spacelike intervals in a 4-dimensional spacetime 61
7.15 Embeddings of Riemann spaces in Riemann spaces of higher dimension 63
7.16 The Petrov classification 70
7.17 Exercises 72
 Contents vii

8 Symmetries of Riemann spaces, invariance of tensors 74

8.1 Symmetry transformations 74
8.2 The Killing equations 75
8.3 The connection between generators and the invariance transformations 77
8.4 Finding the Killing vector fields 78
8.5 Invariance of other tensor fields 79
8.6 The Lie derivative 80
8.7 The algebra of Killing vector fields 81
8.8 Surface-forming vector fields 81
8.9 Spherically symmetric 4-dimensional Riemann spaces 82
8.10 * Conformal Killing fields and their finite basis 86
8.11 * The maximal dimension of an invariance group 89
8.12 Exercises 91

9 Methods to calculate the curvature quickly – Cartan forms and algebraic

computer programs 94
9.1 The basis of differential forms 94
9.2 The connection forms 95
9.3 The Riemann tensor 96
9.4 Using computers to calculate the curvature 98
9.5 Exercises 98

10 The spatially homogeneous Bianchi type spacetimes 99

10.1 The Bianchi classification of 3-dimensional Lie algebras 99
10.2 The dimension of the group versus the dimension of the orbit 104
10.3 Action of a group on a manifold 105
10.4 Groups acting transitively, homogeneous spaces 105
10.5 Invariant vector fields 106
10.6 The metrics of the Bianchi-type spacetimes 108
10.7 The isotropic Bianchi-type (Robertson–Walker) spacetimes 109
10.8 Exercises 112

11 * The Petrov classification by the spinor method 113

11.1 What is a spinor? 113
11.2 Translating spinors to tensors and vice versa 114
11.3 The spinor image of the Weyl tensor 116
11.4 The Petrov classification in the spinor representation 116
11.5 The Weyl spinor represented as a 3 × 3 complex matrix 117
11.6 The equivalence of the Penrose classes to the Petrov classes 119
11.7 The Petrov classification by the Debever method 120
11.8 Exercises 122
viii Contents
Part II The theory of gravitation 123
12 The Einstein equations and the sources of a gravitational field 125
12.1 Why Riemannian geometry? 125
12.2 Local inertial frames 125
12.3 Trajectories of free motion in Einstein’s theory 126
12.4 Special relativity versus gravitation theory 129
12.5 The Newtonian limit of relativity 130
12.6 Sources of the gravitational field 130
12.7 The Einstein equations 131
12.8 Hilbert’s derivation of the Einstein equations 132
12.9 The Palatini variational principle 136
12.10 The asymptotically Cartesian coordinates and the asymptotically
flat spacetime 136
12.11 The Newtonian limit of Einstein’s equations 136
12.12 Examples of sources in the Einstein equations: perfect fluid and dust 140
12.13 Equations of motion of a perfect fluid 143
12.14 The cosmological constant 144
12.15 An example of an exact solution of Einstein’s equations: a Bianchi
type I spacetime with dust source 145
12.16 * Other gravitation theories 149
12.16.1 The Brans–Dicke theory 149
12.16.2 The Bergmann–Wagoner theory 150
12.16.3 The conformally invariant Canuto theory 150
12.16.4 The Einstein–Cartan theory 150
12.16.5 The bi-metric Rosen theory 151
12.17 Matching solutions of Einstein’s equations 151
12.18 The weak-field approximation to general relativity 154
12.19 Exercises 160

13 The Maxwell and Einstein–Maxwell equations and the

Kaluza–Klein theory 161
13.1 The Lorentz-covariant description of electromagnetic field 161
13.2 The covariant form of the Maxwell equations 161
13.3 The energy-momentum tensor of an electromagnetic field 162
13.4 The Einstein–Maxwell equations 163
13.5 * The variational principle for the Einstein–Maxwell equations 164
13.6 * The Kaluza–Klein theory 164
13.7 Exercises 167

14 Spherically symmetric gravitational fields of isolated objects 168

14.1 The curvature coordinates 168
14.2 Symmetry inheritance 172
 Contents ix

14.3 Spherically symmetric electromagnetic field in vacuum 172

14.4 The Schwarzschild and Reissner–Nordström solutions 173
14.5 Orbits of planets in the gravitational field of the Sun 176
14.6 Deflection of light rays in the Schwarzschild field 183
14.7 Measuring the deflection of light rays 186
14.8 Gravitational lenses 189
14.9 The spurious singularity of the Schwarzschild solution at r = 2m 191
14.10 * Embedding the Schwarzschild spacetime in a flat
Riemannian space 196
14.11 Interpretation of the spurious singularity at r = 2m; black holes 200
14.12 The Schwarzschild solution in other coordinate systems 202
14.13 The equation of hydrostatic equilibrium 203
14.14 The ‘interior Schwarzschild solution’ 206
14.15 * The maximal analytic extension of the Reissner–Nordström
solution 207
14.16 * Motion of particles in the Reissner–Nordström spacetime
with e2 < m2 217
14.17 Exercises 219

15 Relativistic hydrodynamics and thermodynamics 222

15.1 Motion of a continuous medium in Newtonian mechanics 222
15.2 Motion of a continuous medium in relativistic mechanics 224
15.3 The equations of evolution of     and u̇ ;
the Raychaudhuri equation 228
15.4 Singularities and singularity theorems 230
15.5 Relativistic thermodynamics 231
15.6 Exercises 234

16 Relativistic cosmology I: general geometry 235

16.1 A continuous medium as a model of the Universe 235
16.2 Optical observations in the Universe – part I 237
16.2.1 The geometric optics approximation 237
16.2.2 The redshift 239
16.3 The optical tensors 240
16.4 The apparent horizon 242
16.5 * The double-null tetrad 243
16.6 * The Goldberg–Sachs theorem 245
16.7 * Optical observations in the Universe – part II 253
16.7.1 The area distance 253
16.7.2 The reciprocity theorem 256
16.7.3 Other observable quantities 259
16.8 Exercises 260
x Contents

17 Relativistic cosmology II: the Robertson–Walker geometry 261

17.1 The Robertson–Walker metrics as models of the Universe 261
17.2 Optical observations in an R–W Universe 263
17.2.1 The redshift 263
17.2.2 The redshift–distance relation 265
17.2.3 Number counts 265
17.3 The Friedmann equations and the critical density 266
17.4 The Friedmann solutions with  = 0 269
17.4.1 The redshift–distance relation in the  = 0
Friedmann models 270
17.5 The Newtonian cosmology 271
17.6 The Friedmann solutions with the cosmological constant 273
17.7 Horizons in the Robertson–Walker models 277
17.8 The inflationary models and the ‘problems’ they solved 282
17.9 The value of the cosmological constant 286
17.10 The ‘history of the Universe’ 287
17.11 Invariant definitions of the Robertson–Walker models 290
17.12 Different representations of the R–W metrics 291
17.13 Exercises 293

18 Relativistic cosmology III: the Lemaître–Tolman geometry 294

18.1 The comoving–synchronous coordinates 294
18.2 The spherically symmetric inhomogeneous models 294
18.3 The Lemaître–Tolman model 296
18.4 Conditions of regularity at the centre 300
18.5 Formation of voids in the Universe 301
18.6 Formation of other structures in the Universe 303
18.6.1 Density to density evolution 304
18.6.2 Velocity to density evolution 306
18.6.3 Velocity to velocity evolution 308
18.7 The influence of cosmic expansion on planetary orbits 309
18.8 * Apparent horizons in the L–T model 311
18.9 * Black holes in the evolving Universe 316
18.10 * Shell crossings and necks/wormholes 321
18.10.1 E < 0 325
18.10.2 E = 0 327
18.10.3 E > 0 327
18.11 The redshift 328
18.12 The influence of inhomogeneities in matter distribution on the
cosmic microwave background radiation 330
18.13 Matching the L–T model to the Schwarzschild and
Friedmann solutions 332
 Contents xi

18.14 * General properties of the Big Bang/Big Crunch singularities in the

L–T model 332
18.15 * Extending the L–T spacetime through a shell crossing singularity 337
18.16 * Singularities and cosmic censorship 339
18.17 Solving the ‘horizon problem’ without inflation 347
18.18 * The evolution of R t M versus the evolution of t M 348
18.19 * Increasing and decreasing density perturbations 349
18.20 * L&T curio shop 353
18.20.1 Lagging cores of the Big Bang 353
18.20.2 Strange or non-intuitive properties of the L–T model 353
18.20.3 Chances to fit the L–T model to observations 357
18.20.4 An ‘in one ear and out the other’ Universe 357
18.20.5 A ‘string of beads’ Universe 359
18.20.6 Uncertainties in inferring the spatial distribution of matter 359
18.20.7 Is the matter distribution in our Universe fractal? 362
18.20.8 General results related to the L–T models 362
18.21 Exercises 363
19 Relativistic cosmology IV: generalisations of L–T and related geometries 367
19.1 The plane- and hyperbolically symmetric spacetimes 367
19.2 G3 /S2 -symmetric dust solutions with Rr = 0 369
19.3 G3 /S2 -symmetric dust in electromagnetic field, the case Rr = 0 369
19.3.1 Integrals of the field equations 369
19.3.2 Matching the charged dust metric to the Reissner–Nordström
metric 375
19.3.3 Prevention of the Big Crunch singularity by electric charge 377
19.3.4 * Charged dust in curvature and mass-curvature coordinates 379
19.3.5 Regularity conditions at the centre 382
19.3.6 * Shell crossings in charged dust 383
19.4 The Datt–Ruban solution 384
19.5 The Szekeres–Szafron family of solutions 387
19.5.1 The z = 0 subfamily 388
19.5.2 The z = 0 subfamily 392
19.5.3 Interpretation of the Szekeres–Szafron coordinates 394
19.5.4 Common properties of the two subfamilies 396
19.5.5 * The invariant definitions of the Szekeres–Szafron metrics 397
19.6 The Szekeres solutions and their properties 399
19.6.1 The z = 0 subfamily 399
19.6.2 The z = 0 subfamily 400
19.6.3 * The z = 0 family as a limit of the z = 0 family 401
19.7 Properties of the quasi-spherical Szekeres solutions with z = 0 =  403
19.7.1 Basic physical restrictions 403
19.7.2 The significance of  404
xii Contents

19.7.3 Conditions of regularity at the origin 407

19.7.4 Shell crossings 410
19.7.5 Regular maxima and minima 413
19.7.6 The apparent horizons 414
19.7.7 Szekeres wormholes and their properties 418
19.7.8 The mass-dipole 419
19.8 * The Goode–Wainwright representation of the Szekeres solutions 421
19.9 Selected interesting subcases of the Szekeres–Szafron family 426
19.9.1 The Szafron–Wainwright model 426
19.9.2 The toroidal Universe of Senin 428
19.10 * The discarded case in (19.103)–(19.112) 431
19.11 Exercises 435

20 The Kerr solution 438

20.1 The Kerr–Schild metrics 438
20.2 The derivation of the Kerr solution by the original method 441
20.3 Basic properties 447
20.4 * Derivation of the Kerr metric by Carter’s method – from the
separability of the Klein–Gordon equation 452
20.5 The event horizons and the stationary limit hypersurfaces 459
20.6 General geodesics 464
20.7 Geodesics in the equatorial plane 466
20.8 * The maximal analytic extension of the Kerr spacetime 475
20.9 * The Penrose process 486
20.10 Stationary–axisymmetric spacetimes and locally nonrotating
observers 487
20.11 * Ellipsoidal spacetimes 490
20.12 A Newtonian analogue of the Kerr solution 493
20.13 A source of the Kerr field? 494
20.14 Exercises 495
21 Subjects omitted from this book 498
References 501
Index 518
 Figures

1.1 Real planetary orbits. page 3

1.2 A vehicle flying across a light ray. 5
2.1 Parallel straight lines. 9
2.2 Parallel transport on a curved surface. 11
2.3 Parallel transport on a sphere. 11
6.1 One-parameter family of loops. 41
7.1 A light cone. 61
7.2 A non geodesic null line. 62
7.3 The Petrov classification. 71
8.1 A mapping of a manifold. 74
8.2 Surface-forming vector fields. 82
11.1 The Penrose–Petrov classification. 117
12.1 Fermi coordinates. 127
12.2 Gravitational field of a finite body. 157
14.1 Deflection of light rays. 185
14.2 Measuring the deflection of light, Eddington’s method. 187
14.3 Measuring the deflection of microwaves. 188
14.4 A gravitational lens. 189
r 
14.5 Graph of r = r + 2m ln  2m − 1. 193
14.6 The Kruskal diagram. 195
14.7 The surface t = const  = /2 in the Schwarzschild spacetime. 197
14.8 Embedding of the Schwarzschild spacetime in six dimensions projected
onto Z1  Z2  Z3 . 198
14.9 Embedding of the Schwarzschild spacetime in six dimensions projected
onto Z3  Z4  Z5 . 199
14.10 The maximally extended Reissner–Nordström spacetime, e2 < m2 . 211
14.11 The ‘throat’ in the Schwarzschild and in the R–N spacetime. 213
14.12 Embeddings of the v = 0 surface. 214
14.13 Surfaces of Fig. 14.12 placed in correct positions. 214
14.14 Maximal extension of the extreme R–N metric. 216

xiii
xiv Figures

14.15 Embeddings of the t = const  = /2 surface of the extreme

R–N metric. 217
15.1 An everywhere concave function. 231
16.1 Refocussing of light in the Universe. 255
16.2 Reciprocity theorem. 256
17.1 R t in Friedmann models. 270
17.2 Curves Ṙ = 0 in the R  plane. 274
17.3 Recollapsing Friedmann models. 275
17.4  = E Friedmann models. 276
17.5 Remaining Friedmann models. 277
17.6 Illustration to (17.62). 280
17.7 The ‘horizon problem’ in R–W. 283
18.1 Black hole in the E < 0 L–T model. 318
18.2 3-d graph of black hole formation. 319
18.3 Contours of constant R-value. 320
18.4 The compactified diagram of Fig. 18.1. 322
18.5 The event horizon in the frame of Fig. 18.1. 323
18.6 A neck. 326
18.7 Radial rays in around central singularity. 335
18.8 A shell crossing in comoving coordinates. 339
18.9 A shell crossing in Gautreau coordinates. 340
18.10 A naked shell crossing. 343
18.11 Solutions of s = S − sS  . 346
18.12 Solution of the ‘horizon problem’ in L–T. 348
18.13 Evolution of the t r subspace in (18.198). 356
18.14 The model of (18.202)–(18.205). 358
18.15 A ‘string of beads’ Universe. 360
19.1 Stereographic projection to Szekeres–Szafron coordinates. 396
19.2 Circles C1 and C2 projected as disjoint. 417
19.3 Circles C1 and C2 projected one inside the other. 417
19.4 A Szekeres wormhole as a handle. 419
19.5 Szafron–Wainwright model. 428
19.6 A 2-torus. 428
19.7 The 3-torus with the metric (19.311). 429
20.1 Ellipsoids and hyperboloids. 449
20.2 A surface of constant . 450
20.3 Space t = const in the Kerr metric, case a2 < m2 . 460
20.4 Space t = const in the Kerr metric, case a2 = m2 . 461
20.5 Space t = const in the Kerr metric, case a2 > m2 . 462
20.6 Light cones in the Kerr spacetime. 463
20.7 Emin r /0 − 1 for different values of Lz . 468
20.8 Analogue of Fig. 20.7 for null geodesics. 470
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