Static spherically symmetric Einstein-Vlasov bifurcations of the

Schwarzschild spacetime
∗
Fatima Ezzahra Jabiri
arXiv:2001.08645v1 [math.AP] 23 Jan 2020

January 24, 2020

Abstract
We construct a one-parameter family of static and spherically symmetric solutions to the Einstein-
Vlasov system bifurcating from the Schwarzschild spacetime. The constructed solutions have the property
that the spatial support of the matter is a finite, spherically symmetric shell located away from the black
hole. Our proof is mostly based on the analysis of the set of trapped timelike geodesics and of the effective
potential energy for static spacetimes close to Schwarzschild. This provides an alternative approach to the
construction of static solutions to the Einstein-Vlasov system in a neighbourhood of black hole vacuum
spacetimes.

Contents
1 Introduction 2

2 Preliminaries and basic background material 4

2.1 The Einstein-Vlasov system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Static and Spherically symmetric solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Metric ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 Vlasov field on static and spherically symmetric spacetimes . . . . . . . . . . . . . . . 6
2.3 The Schwarzschild spacetime and its timelike geodesics . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Ansatz for the distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Reduced Einstein equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Rein’s work on solutions with a Schwarzschild-like black hole . . . . . . . . . . . . . . . . . . 15

3 Statement of the main result 18

4 Solving the reduced Einstein Vlasov system 19

4.1 Set up for the implicit function theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Radii of the matter shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.3 Regularity of the matter terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 The condition (2.61) is satisfied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.5 Solving for µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.5.1 G is well defined on U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.5.2 G is continuously Fréchet differentiable on U . . . . . . . . . . . . . . . . . . . . . . . 33
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Appendix 36

A Study of the geodesic motion in the exterior of Scwharzschild spacetime 36

∗ Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France,

jabiri@ljll.math.upmc.fr

1
 2

1 Introduction
The relativistic kinetic theory of gases plays an important role in the description of matter fields surround-
ing black holes such as plasmas or distribution of stars [22], [6]. In particular, Vlasov matter is used to
analyse galaxies or globular galaxies where the stars play the role of gas particles and where collisions be-
tween these are sufficiently rare to be neglected, so that the only interaction taken into account is gravitation.

In the geometric context of general relativity, the formulation of relativistic kinetic theory was developed
by Synge [34], Israel [15] and Tauber and Weinberg [35]. We also refer to the introduction of [1] for more
details. In this work, we are interested in the Vlasov matter model. It is described by a particle distribution
function on the phase space which is transported along causal geodesics, resulting in the Vlasov equation.
The latter is coupled to the equations for the gravitational field, where the source terms are computed from
the distribution function. In the non-relativistic setting, i.e. the Newtonian framework, the resulting nonlin-
ear system of partial diffrential equations is the Vlasov-Poisson (VP) system [26], while its general relativistic
counterpart forms the Einstein-Vlasov (EV) system. Collisionless matter possesses several attractive features
from a partial differential equations viewpoint. On any fixed background, it avoids pathologies such as shock
formation, contrary to more traditional fluid models. Moreover, one has global solutions of the VP system
in three dimensions for general initial data [23], [20].

The local well-posedness of the Cauchy problem for the EV system was first investigated in [9] by Choquet-
Bruhat. Concerning the nonlinear stability of the Minkowski spacetime as the trivial solution of the EV
system, it was proven in the case of spherically symmetric initial data by Rendall and Rein [25] in the
massive case and by Dafermos [10] for the massless case. The general case was recently shown by Fajman,
Joudioux and Smulevici [13] and independently by Lindblad-Taylor [19] for the massive case, and by Taylor
[36] for the massless case. Nonlinear stability results have been given by Fajman [12] and Ringtröm [29]
in the case of cosmological spacetimes. See also [33], [5], [11], [38], [32] for several results on cosmological
spacetimes with symmetries.

In stellar dynamics, equilibrium states can be described by stationary solutions of the VP or EV system.
Static and spherically symmetric solutions can be obtained by assuming that the distribution function has
the following form
f (x, v) = Φ(E, ℓ),
where E and ℓ are interpreted as the energy and the total angular momentum of particles respectively. In
fact, in the Newtonian setting, the distribution function associated to a stationary and spherically symmetric
solution to the VP system is necessarily described by a function depending only on E and ℓ. Such statement
is referred to as Jean’s theorem [16], [17], [7]. However, it has been shown that its generalisation to general
relativity is false in general [31].
A particular choice of Φ, called the polytropic ansatz, which is commonly used to construct static and
spherically symmetric states for both VP and EV system is given by
(
(E0 − E)µ ℓk , E < E0 ,
Φ(E, ℓ) := (1.1)
0, E ≥ E0 .

where E0 > 0, µ > −1 and k > −1. In [27], Rein and Rendall gave the first class of asymptotically flat,
static, spherically symmetric solutions to EV system with finite mass and finite support such that Φ depends
only on the energy of particles. In [24], Rein extended the above result for distribution functions depending
also on ℓ, where Φ is similar to the polytropic ansatz (1.1): Φ(E, ℓ) = φ(E)(ℓ − ℓ0 )k+ , φ(E) = 0 if E > E0 ,
k > −1 and ℓ0 ≥ 0. Among these, there are singularity-free solutions with a regular center, and also solutions
with a Schwarzschild-like black hole. Other results beyond spherical symmetry were recently established. In
[3], static and axisymmetric solutions to EV system were constructed by Rein-Andréasson-Kunze and then
extended in [4] to establish the existence of rotating stationary and axisymmetric solutions to the Einstein-
Vlasov system. The constructed solutions are obtained as bifurcations of a spherically symmetric Newtonian
steady state. We note that the proof is based on the application of the implicit function theorem and it
 3

can be traced back to Lichtenstein [18] who proved the existence of non-relativistic, stationary, axisymmetric
self-gravitating fluid balls. Similar arguments were adapted by Andréasson-Fajman-Thaller in [2] to prove the
existence of static spherically symmetric solutions of the Einstein-Vlasov-Maxwell system with non-vanishing
cosmological constant. Recently, Thaller has constructed in [37] static and axially symmetric shells of the
Einstein-Vlasov-Maxwell system.

In this paper, we construct static spherically symmetric solutions to the EV system which contain a matter
shell located in the exterior region of the Schwarzschild black hole. This provides an alternative construction
to that of Rein [24], See remarks 1 and 2 below for a comparison of our works and results. Our proof is based
on the study of trapped timelike geodesics of spacetimes closed to Schwarzschild. In particular, we show (and
exploit) that for some values of energy and total angular momentum (E, ℓ), the effective potential associated
to a particle moving in a perturbed Schwarzschild spacetime and that of a particle with same (E, ℓ) moving
in Schwarzschild are similar. Our distribution function will then be supported on the set of trapped timelike
geodesics, and this will lead to the finiteness of the mass and the matter shell’s radii.

Recall that the effective potential energy EℓSch of a particle of rest mass m = 1 and angular momentum
ℓ, moving in the exterior of region of the Schwarzschild spacetime is given by
r = 2M

1 l = 16M²
l = 12M²
l = 7M²
E = 0.97
0.95

0.9
ESch

0.85
l

0.8

0.75

0.7
0 5 10 15 20 25 30
r

Figure 1: Shape of the effective potential energy EℓSch for three cases of ℓ, with M = 1, E = 0.97.

  
2M ℓ
EℓSch (r) = 1− 1+ 2 . (1.2)
r r
 4

In particular, trapped geodesics occur when the equation

EℓSch (r) = E 2 , (1.3)

admits three distinct roots r0Sch (E, ℓ) < r1Sch (E, ℓ) < r2Sch (E, ℓ).
We state now our main result:
Theorem 1. There exists a 1−parameter family of smooth, static, spherically symmetric asymptotically flat
spacetimes (M, gδ ) and distribution functions f δ : Γ1 → R+ solving the Einstein-Vlasov system given by
equations (2.1), (2.6) and (2.8), such that f δ verifies

∀(x, v) ∈ Γ1 , f δ (x, v) = Φ(E δ , ℓ; δ)Ψ(r, (E δ , ℓ), gδ ). (1.4)

where Φ(·, ·; δ) is supported on a compact set Bbound of the set of parameters (E, ℓ) corresponding to trapped
timelike trajectories, Ψ is a positive cut-off function, defined below (cf. (2.45)) which selects the trapped
geodesics with parameters (E, ℓ) ∈ Bbound , Γ1 is the mass shell of particles with rest mass m = 1, and E δ is
the local energy with respect to the metric gδ .
Moreover, the resulted spacetimes contain a shell of Vlasov matter in the following sense: there exist Rmin <
Rmax such that the metric is given by Schwarzschild metric of mass M in the region ]2M, Rmin ], by Schwarzschild
metric of mass M δ in the region [Rmax , ∞[, and f δ does not identically vanish in the region [Rmin , Rmax ].
Remark 1. Compared to the work of Rein [24], the solutions constructed here are bifurcating from Schwarzschild
and therefore, a priori small. Moreover, the ansatz used in this work is different from that used by Rein.
In particular, it is not polytropic. Another difference is that we do not use the Tolman-Oppenheimer-Volkov
equation (2.82) in our argument. See Theorem 2 for the exact assumptions on the profile Φ.
Remark 2. A posteriori, in the small data case, one can check that the distribution functions constructed by
Rein are supported on trapped timelike geodesics. However, not all the trapped geodesics are admissible in his
construction. In our case, we include a more general support, cf. Section 2.6 for more details on this part.
Remark 3. The support of Φ(E, ℓ; δ) has two connected components: one corresponds to geodesics which
start near r0 (E, ℓ) and reach the horizon in a finite proper time, and the other one corresponds to trapped
geodesics. Ψ is introduced so that it is equal to 0 outside Bbound and equal to a cut-off function depending
on the r variable (cf (2.46)), χ on Bbound . The latter is equal to 0 on the first connected component of the
support of Φ(E, ℓ; δ) and to 1 on the second component. This allows to eliminate the undesired trajectories.
The reason behind the use of this cut-off function is related to the non-validity of Jean’s theorem in general
relativity [31].
The paper is organised as follows. In Section 2, we present basic background material on the Einstein-
Vlasov system and the geodesic motion in the Schwarzschild exterior. We also present the ansatz for the
metric and the distribution function and we reduce the Einstein equations to a system of ordinary differential
equations in the metric components. We end this section with a short description of Rein’s construction
to compare it with ours. In Section 3, we give a detailed formulation of our main result. In Section 4, we
prove Theorem 1. To this end, we control quantitatively the effective potential and the resulting trapped
timelike geodesics for static spherically symmetric spacetimes close to Schwarzschild. The main theorem
is then obtained by application of the implicit function theorem. Finally, Appendix A contains a proof of
Proposition 1 concerning the classification of timelike geodesics in the Schwarzschild spacetime.
Acknowledgements. I would like to thank my advisor Jacques Smulevici for suggesting this problem to me,
as well for many interesting discussions and crucial suggestions. I would also like to thank Nicolas Clozeau for
many helpful remarks. This work was supported by the ERC grant 714408 GEOWAKI, under the European
Union’s Horizon 2020 research and innovation program.

2 Preliminaries and basic background material

In this section, we introduce basic material necessary for the rest of the paper.
2.1 The Einstein-Vlasov system 5

2.1 The Einstein-Vlasov system

In this work, we study the Einstein field equations for a time oriented Lorentzian manifold (M, g) in the
presence of matter
1
Ric(g) − gR(g) = 8πT (g), (2.1)
2
where Ric denotes the Ricci curvature tensor of g, R denotes the scalar curvature and T denotes the energy-
momentum tensor which must be specified by the matter theory. The model considered here is the Vlasov
matter. It is assumed that the latter is represented by a scalar positive function f : T M → R called the
distribution function. The condition that f represents the distribution of a collection of particles moving
freely in the given spacetime is that it should be constant along the geodesic flow, that is
L[f ] = 0, (2.2)
where L denotes the Liouville vector field. The latter equation is called the Vlasov equation. In a local
coordinate chart (xα , v β ) on T M, where (v β ) are the components of the four-momentum corresponding to
xα , the Liouville vector field L reads
∂ ∂
L = vµ µ
− Γµαβ (g)v α v β µ (2.3)
∂x ∂v
and the corresponding integral curves satisfy the geodesic equations of motion
 µ
dx
 (τ ) = v µ ,
dτ

µ (2.4)
 dv (τ ) = −Γµ v α v β ,

αβ
dτ
where Γµαβ are the Christoffel symbols given in the local coordinates xα by

1 µν ∂gβν ∂gαβ
 
µ ∂gαν
Γ αβ = g + −
2 ∂xα ∂xβ ∂xν
and where τ is an affine parameter which corresponds to the proper time in the case of timelike geodesics.
The trajectory of a particle in T M is an element of the geodesic flow generated by L and its projection onto
the spacetime manifold M corresponds to a geodesic of the spacetime.
1
It is easy to see that the quantity L(x, v) := v α v β gαβ is conserved along solutions of (2.4) 1 . In the case of
2
timelike geodesics this corresponds to the conservation of the rest mass m > 0 of the particle. In the following,
m2
we will consider only particles with the same rest mass so that we can set L(x, v) = − . Furthermore,
2
for physical reasons, we require that all particles move on future directed timelike geodesics. Therefore, the
distribution function is supported on the seven dimensional manifold of T M 2 , called the the mass shell,
denoted by Γm and defined by
Γm := (x, v) ∈ T M : gx (v, v) = −m2 , and v α is future pointing .

(2.5)
We note that by construction Γm is invariant under the geodesic flow.

We assume that there exist local coordinates on M, denoted by (xα )α=0···3 such that x0 = t is a smooth
strictly increasing function along any future causal curve and its gradient is past directed and timelike. Then,
the condition
gαβ v α v β = −1 where v α is future directed
allows to write v 0 in terms of (xα , v a ). It is given by
 q 
v 0 = −(g00 )−1 g0j v j + (g0j v j )2 − g00 (m2 + gij v i v j ) .
1 We note that L is the Lagrangian of a free-particle.
2 See [30] Lemma.7
2.2 Static and Spherically symmetric solutions 6

Therefore, Γm can be parametrised by (x0 , xa , v a ). Hence, the distribution function can be written as a
function of (x0 , xa , v a ) and the Vlasov equation has the form

∂f v a ∂f a v α v β ∂f
+ − Γ αβ = 0. (2.6)
∂x0 v 0 ∂xa v 0 ∂v a
In order to define the energy-momentum tensor which couples the Vlasov equation to the Einstein field equa-
tions, we introduce the natural volume element on the fibre Γm,x := v α ∈ Tx M : g αβ v α v β = −m2 , v 0 > 0


of Γm at a point x ∈ M given in the adapted local coordinates (x0 , xa , v a ) by

q
− det (gαβ )
dvolx (v) := dv 1 dv 2 dv 3 . (2.7)
−v0
The energy momentum tensor is now defined by
Z
∀x ∈ M Tαβ (x) := vα vβ f (x, v) dvolx (v), (2.8)
Γm,x

where f = f (x0 , xa , v a ) and dvolx (v) = dvolx (v a )3 . In order for (2.8) to be well defined, f has to have
certain regularity and decay properties. One sufficient requirement would be to demand that f have compact
support on Γx , ∀x ∈ M and it is integrable with respect to v. In this work, we assume that all particles have
the same rest mass and it is normalised to 1. From now on, the mass shell Γ1 will be denoted by Γ. Finally,
we refer to (2.1) and (2.6) with T given by (2.8) as the Einstein-Vlasov system.

2.2 Static and Spherically symmetric solutions

2.2.1 Metric ansatz
We are looking for static and spherically symmetric asymptotically flat solutions to the EV system. Therefore,
we consider the following ansatz for the metric, written in standard (t, r, θ, φ) coordinates:

g = −e2µ(r) dt2 + e2λ(r) dr2 + r2 (dθ2 + sin2 θdφ2 ). (2.9)

with boundary conditions at infinity:

lim λ(r) = lim µ(r) = 0. (2.10)

r→∞ r→∞

Since we consider solutions close to a Schwarzschild spacetime of mass M > 0 outside from the black hole
region, we fix a manifold of the form

O :=R×]2M, ∞[×S 2 . (2.11)

t r θ, φ (2.12)

2.2.2 Vlasov field on static and spherically symmetric spacetimes

The distribution function f is conserved along the geodesic flow. Hence, any function of the integrals of
motion will satisfy the Vlasov equation. In this context, we look for integrals of motion for the geodesic
equation (2.4) on a static and spherically symmetric background. By symmetry assumptions, the vector
fields
∂
ξt = , (2.13)
∂t
generating stationarity, and
∂ ∂
ξ1 = − cos φ cot θ − sin φ , (2.14)
∂φ ∂θ
3 The latin indices run from 1...3.
2.3 The Schwarzschild spacetime and its timelike geodesics 7

∂ ∂
ξ2 = − sin φ cot θ + cos φ , (2.15)
∂φ ∂θ
∂
ξ3 = (2.16)
∂φ
generating spherical symmetry are Killing. Therefore, the quantities
q
E(x, v) := −gx (v, ξt ) = eµ(r) m2 + (eλ(r) v r )2 + (rv θ )2 + (r sin θv φ )2 , (2.17)

ℓ1 (x, v) := gx (v, ξ1 ) = r2 − sin φv θ − cos θ sin θ cos φv φ ,

(2.18)
ℓ2 (x, v) := gx (v, ξ2 ) = r2 cos φv θ − cos θ sin θ sin φv φ ,


(2.19)
2 2 φ
ℓz (x, v) := gx (v, ξ3 ) = r sin θv , (2.20)
are conserved. In particular, E and

ℓ := ℓ(r, θ, v θ , v φ ) = ℓ21 + ℓ22 + ℓ2z = r4 (v θ )2 + sin2 θ(v φ )2

(2.21)

are conserved. We recall that m := −gαβ v α v β is also a conserved quantity. Since all particles have the same
rest mass m = 1, E becomes
q
E = E(r, v r , v θ , v φ ) = eµ(r) 1 + (eλ(r) v r )2 + (rv θ )2 + (r sin θv φ )2 . (2.22)

Note that E and ℓ can be interpreted respectively as the energy and the total angular momentum of the
particles.

2.3 The Schwarzschild spacetime and its timelike geodesics

The Schwarzschild family of spacetimes is a one-parameter family of spherically symmetric Lorentzian man-
ifolds, indexed by M > 0, which are solutions to the Einstein vacuum equations

Ric(g) = 0. (2.23)

The parameter M denotes the ADM mass. The domain of outer communications of these solutions can be
represented by O and a metric which takes the form
Sch Sch
gSch = −e2µ (r)
dt2 + e2λ (r)
dr2 + r2 (dθ2 + sin2 θdφ2 ), (2.24)

where  −1
Sch 2M Sch 2M
e2µ (r)
=1− and e2λ (r)
= 1− . (2.25)
r r
In this work, we shall be interested in particles moving in the exterior region. We note that the study of the
geodesic motion is included in the classical books of general relativity. See for example [8, Chapter 3] or [21,
Chapter 33]. We recall here a complete classification of timelike geodesics in order to be self contained.
The delineation of Schwarzschild’s geodesics requires solving the geodesic equations given by (2.4). There-
fore, we need to integrate a system of 8 ordinary differential equations. However, the symmetries of the
Schwarzschild spacetime and the mass shell condition

m2
L(x, v) = − (2.26)
2
imply the complete integrability of the system.

Let (xα , v α ) = (t, r, θ, φ, v t , v r , v θ , v φ ) be a local coordinate system on T M. The quantities

E = −vt , ℓ = r4 (v θ )2 + sin2 θ(v φ )2 , ℓz = r sin θv φ .

2.3 The Schwarzschild spacetime and its timelike geodesics 8

are then conserved along the geodesic flow. Besides, the mass shell condition on Γ takes the form
Sch Sch
−e2µ (r) (v t )2 + e2λ (r) (v r )2 + r2 (v θ )2 + sin2 θ(v φ )2 = −1

which is equivalent to
 
2 λSch r 2 2µSch (r) ℓ
E = EℓSch (r) + (e v ) , where EℓSch (r) =e 1+ 2 . (2.27)
r
and implies
EℓSch (r) ≤ E 2 (2.28)
for any geodesic moving in the exterior region.
Let γ : I → M be a timelike geodesic in the spacetime, defined on interval I ⊂ R. In the adapted local
coordinates (xα , v α ), we have
γ(τ ) = (t(τ ), r(τ ), θ(τ ), φ(τ )) and γ ′ (τ ) = (v t (τ ), v r (τ ), v θ (τ ), v φ (τ )).
Besides, γ satisfies the geodesic equations of motion (2.4). One can easily see from the equations
vt = −E, (2.29)
φ
r sin θv = ℓz , (2.30)
ℓ2z
(rv θ )2 + = ℓ, 2 (2.31)
 sin θ
Sch Sch ℓ
(eλ (r) v r )2 − e−2µ (r) E 2 + 1 + 2 = 0 (2.32)
r
that if we solve the geodesic motion in the radial direction r(τ ) then we can integrate the remaining equations
of motion. More precisely, if one solves for r then we get t from (2.29), φ from (2.30) and θ from (2.31).
Therefore, we will study the geodesic equation projected only in the radial direction i.e we consider the
reduced system
dr
= vr , (2.33)
dτ
dv r
= −Γrαβ v α v β . (2.34)
dτ
Straightforward computations of the right hand side of the second equation lead to
 −1  −1  
M 2M M 2M 1 2M
Γrαβ v α v β = 2 1 − E2 − 2 1 − (v r )2 − 3 1 − ℓ.
r r r r r r
We combine the latter expression with (2.32), to obtain
dv r
   
M ℓ ℓ 2M
=− 2 1+ 2 + 3 1− .
dτ r r r r
Now, by (2.27), it is easy to see that
dv r 1 ′
= − EℓSch (r).
dτ 2
In order to lighten the notations, we denote v r by w. Thus, we consider the differential system
dr
= w, (2.35)
dτ
dw 1 ′
= − EℓSch (r). (2.36)
dτ 2
We classify solutions of (2.35)-(2.36) based on the shape of EℓSch as well as the roots of the equation
E 2 = EℓSch (r). (2.37)
More precisely, we state the following proposition. We include a complete proof of this classical result in
Appendix A.
2.3 The Schwarzschild spacetime and its timelike geodesics 9

Proposition 1. Consider a timelike geodesic γ : I → M of the Schwarzschild exterior of mass M > 0

parametrised by γ(τ ) = (t(τ ), r(τ ), θ(τ ), φ(τ )) and normalised so that gSch (γ̇, γ̇) = −1. Let E and ℓ be the as-
sociated energy and total angular momentum defined respectively by E := −vt and ℓ := r4 (v θ )2 + sin2 θ(v φ )2 .

Then, we have the following classification

1. If ℓ ≤ 12M 2, then
r
8
(a) if < E < 1, then the orbit starts at some point r0 > 2M , the unique root of the equation
9
(2.37), and reaches the horizon r = 2M in a finite proper time.
(b) if E ≥ 1, then the equation (2.37) admits no positive roots, and the orbit goes to infinity r = +∞
in the future, while in the past, it reaches the horizon r = 2M in finite affine time.
r
8
(c) if E = , then ℓ = 12M 2. In this case, the equation (2.37) admits a unique triple root, given
9
by rc = 6M . The orbit is circular of radius rc .

2. If ℓ > 12M 2, then

Sch
(a) if ℓ = ℓlb (E), the geodesic is a circle of radius rmax (ℓ), given by
r !
Sch ℓ 12M 2
rmax (ℓ) = 1− 1− . (2.38)
2M ℓ

Here, ℓlb (E) is given by

12M 2 9 2
ℓlb (E) := √ , α := E − 1. (2.39)
1 − 4α − 8α2 + 8α α2 + α 8

(b) if ℓ < ℓlb (E),

i. if E ≥ 1, then the equation (2.37) admits no positive roots. Hence, the orbit goes to infinity
r = +∞ in the future, while in the past, it reaches the horizon r = 2M in finite affine time.
ii. otherwise, the orbit starts at some point r0 > 2M , the unique root of the equation (2.37), and
reaches the horizon (r = 2M ) in a finite proper time.
(c) otherwise, if ℓ > ℓlb (E),
i. if E ≥ 1, then the equation (2.37) admits two positive distinct roots 2M < r0 < r2 and two
orbits are possible
A. The geodesic starts at r0 and reaches the horizon in a finite proper time.
B. The orbit is hyperbolic, so unbounded. The orbit starts from infinity, hits the potential
barrier at r2 , the biggest root of the equation (2.37), and goes back to infinity.
ii. otherwise, if E < 1, and necessarily ℓ ≤ ℓub (E), where ℓub (E) is given by

12M 2
ℓub (E) := √ , (2.40)
1 − 4α − 8α2 − 8α α2 + α
Sch
A. if ℓ = ℓub (E), the geodesic is a circle of radius rmin (ℓ), given by
r !
ℓ 12M 2
Sch
rmin (ℓ) = 1+ 1− . (2.41)
2M ℓ

B. If ℓ < ℓub (E), then the equation (2.37) admits three distinct roots 2M < r0Sch (E, ℓ) <
r1Sch (E, ℓ) < r2Sch (E, ℓ), and two orbits are possible
• The geodesic starts from r0Sch (E, ℓ) and reaches the horizon in a finite time.
2.4 Ansatz for the distribution function 10

• The geodesic is periodic. It oscillates between an aphelion r2Sch (E, ℓ) and a perihelion
r1Sch (E, ℓ).
r
8
Remark 4. In view of (4.18), the lower bound E ≥ holds along any orbit.
9
In view of the above proposition, we introduce Abound , the set of (E, ℓ) parameters corresponding to
trapped (non-circular) geodesics.
Definition 1.
( #r " )
8
, ∞ × 12M 2 , ∞ : E < 1,
 
Abound := (E, ℓ) ∈ ℓlb (E) < ℓ < ℓub (E) . (2.42)
9

Note in view of the orbits described in 2.c.ii.B that given (E, ℓ) ∈ Abound , two cases are possible: the
orbit is either trapped or it reaches the horizon in a finite proper time.

2.4 Ansatz for the distribution function

We are interested in static and spherically symmetric distribution functions. For simplicity, we fix the rest
mass of the particles to be 1. In this context, we assume that f : Γ1 → R+ takes the form

f (t, r, θ, φ, v r , v θ , v φ ) = Φ(E, ℓ)Ψη (r, (E, ℓ), µ). (2.43)

where
• Φ : Abound → R+ is a C 2 function on its domain and it is supported on Bbound , a set of the form
[E1 , E2 ] × [ℓ1 , ℓ2 ], where E1 , E2 , ℓ1 and ℓ2 verify
r
8
< E1 < E2 < 1 and ℓlb (E2 ) < ℓ1 < ℓ2 < ℓub (E1 ). (2.44)
9

• η > 0 is constant that will be specified later, Ψη (·, ·, µ) ∈ C ∞ (R × Abound , R+ ) is a cut-off function
depending on the metric coefficient µ, such that

 χη (· − r1 (µ, (E, ℓ))), (E, ℓ) ∈ Bbound

Ψη (·, (E, ℓ), µ) := (2.45)
0 (E, ℓ) ∈
/ Abound ,


where r1 is a positive function of (µ, E, ℓ) which will be defined later 4 and χη ∈ C ∞ (R, R+ ) is a cut-off
function defined by

 1 s ≥ 0,

χη (s) :=  ≤ 1 s ∈ [−η, 0], (2.46)
0 s < −η.


• E and ℓ are defined respectively by (2.22), (2.21).

Sch Sch
Based on the monotonicity properties of ℓub , ℓub rmin and rmax , the set Bbound is included in Abound . More
precisely, we state the following lemma
Lemma 1. Let ℓ ∈]12M 2 , ∞[
Sch
• rmax decreases monotonically from 6M to 3M on ]12M 2 , ∞[.
 
ℓ
4r
1 will be the second largest root of the equation e2µ(r) 1+ = E 2 corresponding to the metric g.
r2
2.4 Ansatz for the distribution function 11

Sch
• rmin increases monotonically from 6M to ∞ on ]12M 2 , ∞[.
#r "
8
Let E ∈ ,1 ,
9
r
2 2 8
• ℓlb grows monotonically from 12M to 16M when E grows from to 1.
9
r
2 8
• ℓub grows monotonically from 12M to ∞ when E grows from to 1.
9
Moreover, ∀(E, ℓ) ∈ [E1 , E2 ] × ]ℓlb (E2 ), ℓlb (E1 )[

ℓlb (E) < ℓlb (E2 ) < ℓ < ℓub (E1 ) < ℓub (E).

Proof. The proof is straightforward in view of (2.38) , (2.41), (2.39) and (2.40).
Hence, by the above lemma, [E1 , E2 ] × ]ℓlb (E2 ), ℓlb (E1 )[ ⊂ Abound and in particular, Bbound ⊂⊂ Abound .
Finally, we note that all trapped geodesics lie in the region ]4M, ∞[:
Lemma 2. Let (E, ℓ) ∈ Bbound . Let riSch (E, ℓ)5 , i ∈ {0, 1, 2} be the three roots of the equation
 
2µSch (r) ℓ
e 1 + 2 = E2. (2.47)
r

Then,
• r0Sch (E, ·) decreases monotonically on ]ℓ1 , ℓ2 [ and r0Sch (·, ℓ) increases monotonically on ]E1 , E2 [.
• r1Sch (E, ·) increases monotonically on ]ℓ1 , ℓ2 [ and r1Sch (·, ℓ) decreases monotonically on ]E1 , E2 [.
• r2Sch (E, ·) decreases monotonically on ]ℓ1 , ℓ2 [ and r2Sch (·, ℓ) increases monotonically on ]E1 , E2 [.
Moreover,
∀(E, ℓ) ∈ Abound : r1Sch (E, ℓ) > 4M.
1
Proof. First we note that the equation (2.47) is cubic in . Its roots can be obtained analytically by Cardan’s
r
formula and in particular, depend smoothly on (E, ℓ). Besides, the roots are simple. Otherwise, the orbit
would be circular. Therefore,
(EℓSch )′ (riSch (E, ℓ)) 6= 0.
riSch verify
E 2 = EℓSch (riSch (E, ℓ)).
We derive the latter equation with respect to E and ℓ to obtain
∂E Sch
Sch
∂riSch ∂ℓ (ri
ℓ
(E, ℓ)) ∂riSch 2E
(E, ℓ) = − Sch ′ Sch
, and (E, ℓ) = − Sch ′ Sch
∂ℓ (Eℓ ) (ri (E, ℓ)) ∂E (Eℓ ) (ri (E, ℓ))

We use the monotonicity properties of EℓSch on ]2M, ∞[ to determine the sign of the above derivatives. For
the second point, we use the monotonicity properties of r1Sch , rmax
Sch
and ℓub to obtain

∀(E, ℓ) ∈ Abound : r1Sch (E, ℓ) > r1Sch (E, ℓlb (E)) Sch
and rmax Sch
(ℓlb (E)) > rmax (ℓlb (1)).

Besides, by the definition on ℓub (the value of the angular momentum such that E max (ℓ) = EℓSch (rmax
Sch
(ℓ)) =
2
E ), we have
r1Sch (E, ℓlb (E)) = rmax
Sch
(ℓlb (E)).
5 We note that the dependence of ri in (E, ℓ) is smooth.
2.5 Reduced Einstein equations 12

Therefore,
r1Sch (E, ℓ) > rmax
Sch
(ℓlb (1)).
To conclude, it suffices to note that ℓlb (1) = 16M 2 so that, when using (2.38), we obtain
Sch
rmax (ℓlb (1)) = 4M.

This ends the proof.

Remark 5. The choice of an open set for the range of ℓ allow us eliminate the critical values leading to circu-
lar orbits which correspond to double roots of the equation (2.37): the points (E1 , ℓub (E1 )) and (E2 , ℓlb (E2 )).

2.5 Reduced Einstein equations

After specifying the ansatz for the metric and for the distribution function, we compute the energy-momentum
tensor associated to (2.9) and the distribution function (2.43). In order to handle the mass shell condition,
we introduce the new momentum variables

p0 := −eµ(r) v t , p1 := eλ(r) v r , p2 := rv θ , p3 := r sin θv φ , (2.48)

so that the associated frame

∂ ∂ 1 ∂ 1 ∂
e0 := e−µ(r) , e1 := e−λ(r) , e2 := , e3 := , (2.49)
∂t ∂r r ∂θ r sin θ ∂φ
forms an orthonormal frame of Tx M. In terms of the new momentum coordinates, the mass shell condition
becomes p
−(p0 )2 + (p1 )2 + (p2 )2 + (p3 )2 = −1 i.e p0 = 1 + |p|2 ,
where p = (p1 , p2 , p3 ) ∈ R3 and | · | is the Euclidean norm on R3 . It is convient to introduce an angle variable
χ ∈ [0, 2π[ such that
√ √
θ ℓ cos χ φ ℓ sin χ
v = 2
, v = 2 ,
r r sin θ
so that
r4 (v θ )2 + r4 sin2 θ(v φ )2 = ℓ.
We also introduce the effective potential in the metric (2.9)
 
2µ(r) ℓ
Eℓ (r) := e 1+ 2 . (2.50)
r

In local coordinates, the energy momentum tensor reads

Z
Tαβ (x) = vα vβ f (xµ , v a )dx vol(v a ),
Γx
Z
= vα vβ Φ(E(xµ , v a ), ℓ(xµ , v a ))Ψη (r, (E(xµ , v a ), ℓ(xµ , v a )), µ)dx vol(v),
Γx

where dx vol(v) is given by (2.7), E(xµ , v a ) and ℓ(xµ , v a ) are respectively given by (2.22) and (2.21). Here,
xµ = (t, r, θ, φ) and v a = (v r , v θ , v φ ). We make a first change of variables (v r , v θ , v φ ) → (p1 , p2 , p3 ) to get

d3 p
Z
Tαβ (x) = vα (p)vβ (p)Φ (E(r, |p|), ℓ(r, p)) Ψη (r, (E(r, |p|), ℓ(r, p)), µ) p
R3 1 + |p|2
d3 p
Z
=2 vα (p)vβ (p)Φ (E(r, |p|), ℓ(r, p)) Ψη (r, (E(r, |p|), ℓ(r, p)), µ) p .
[0,+∞[×R2 1 + |p|2
2.5 Reduced Einstein equations 13

We perform a second change of variables (p1 , p2 , p3 ) 7→ (E(r, |p|), ℓ(r, p), χ). We compute the pi s in terms of
the new variables:
p p
1 2 2 −2µ(r) ℓ(r, p) 2 ℓ(r, p) cos χ 3 ℓ(r, p) sin χ
(p ) = E (r, |p|)e −1− , p = , p = .
r2 r r
Now it is easy to see that the domain of (E, ℓ, χ) is given by Dr × [0, 2π[ where
Dr := (E, ℓ) ∈]0, ∞[×[0, ∞[ : Eℓ (r) ≤ E 2 .

(2.51)
By straightforward computations of the Jacobian, we get
d3 p 1
p = p dEdℓdχ.
1 + |p|2 2 2
2r E − Eℓ (r)
Therefore, the energy momentum tensor becomes
1
Z Z
Tαβ (x) = uα uβ Φ(E, ℓ)Ψη (r, (E, ℓ, µ) p dEdℓdχ,
Dr [0,2π[ r2 2
E − Eℓ (r)
where
u2t = E 2 , u2r = e2λ(r)−2µ(r) E 2 − Eℓ (r) , , u2θ = ℓ cos2 χ, u2φ = ℓ sin2 θ sin2 χ.

Hence, the non-vanishing energy-momentum tensor components are given by

2π Φ(E, ℓ)
Z
Ttt (r) = 2 E 2 Ψη (r, (E, ℓ, µ) p dEdℓ,
r Dr E 2 − Eℓ (r)
2π
Z p
Trr (r) = 2 e2(λ(r)−µ(r)) Ψη (r, (E, ℓ, µ)Φ(E, ℓ) E 2 − Eℓ (r) dEdℓ,
r Dr
π ℓ
Z
Tθθ (r) = 2 Ψη (r, (E, ℓ, µ)Φ(E, ℓ) p dEdℓ,
r Dr E 2 − Eℓ (r)
Tφφ (r, θ) = sin2 θTθθ (r).
It remains to compute the Einstein tensor
1
Gαβ = Ricαβ − gαβ R(g)
2
with respect to the metric (2.9). Straightforward computations lead to
e2µ(r)  −2λ(r) ′

Gtt = e (2rλ (r) − 1) + 1 ,
r2
e2λ(r)  −2λ(r) ′

Grr = e (2rµ (r) + 1) − 1 ,
r2
Gθθ = re−2λ(r) rµ′2 (r) − λ′ (r) + µ′ (r) − rµ′ (r)λ′ (r) + rµ′′ (r) ,

Gφφ = sin2 θGθθ ,

while the remaining components vanish. Therefore, the Einstein-Vlasov system is reduced to the following
system of differential equations with respect the radial variable r:
Φ(E, ℓ))
Z
e−2λ(r) (2rλ′ (r) − 1) + 1 = 8π 2 e−2µ(r) E 2 Ψη (r, (E, ℓ, µ)Φ(E, ℓ) p dEdℓ,
E 2 − E (r)
Dr ℓ
(2.52)
Z p
e−2λ(r) (2rµ′ (r) + 1) − 1 = 8π 2 e−2µ(r) Φ(E, ℓ)Ψη (r, (E, ℓ, µ) E 2 − Eℓ (r)) dEdℓ,
Dr
(2.53)

4π 2 ℓ
Z
rµ′2 (r) − λ′ (r) + µ′ (r) − rµ′ (r)λ′ (r) + rµ′′ (r) = 3 e2λ(r) Φ(E, ℓ)Ψη (r, (E, ℓ, µ) p dEdℓ.
r Dr E 2 − Eℓ (r))
(2.54)
2.5 Reduced Einstein equations 14

We perform a last change of variable E = eµ(r) ε and we set

∞ r 2 (ε2 −1)
2π ε2
Z Z
GΦ (r, µ) := Φ(eµ(r) ε, ℓ)Ψη (r, (eµ(r) ε, ℓ, µ)) q dℓdε, (2.55)
r2 1 0 ε2 − 1 − ℓ
r2
r 2 (ε2 −1)
∞
r
2π ℓ
Z Z
HΦ (r, µ) := 2 Φ(eµ(r) ε, ℓ)Ψη (r, (eµ(r) ε, ℓ, µ)) ε2 − 1 − dℓdε. (2.56)
r 1 0 r2

The equations (2.52) and (2.53) become

e−2λ(r) (2rλ′ (r) − 1) + 1 = 8πr2 GΦ (r, µ), (2.57)

−2λ(r) ′ 2
e (2rµ (r) + 1) − 1 = 8πr HΦ (r, µ). (2.58)

We call the latter nonlinear system for µ and λ the reduced Einstein-Vlasov system.
We note that when inserting the ansatz of f (2.43) in the definition of the energy momentum tensor, the
matter terms GΦ and HΦ become functionals of the yet unknown metric function µ and of the radial position
r. Besides, from the reduced Einstein-Vlasov system (2.57)-(2.58), we will show that one can express λ in
terms of the unknown metric µ so that we are left with the problem of solving only for µ. Therefore, we will
define the solution operator used in the implicit function theorem for µ only . For this, let ρ > 0 and R > 0
be such that:
R> max r2Sch (E, ℓ),
(E,ℓ)∈Bbound

0 < 2M + ρ < min r0Sch (E, ℓ)

(E,ℓ)∈Bbound

and set I :=]2M + ρ, R[. We will solve (2.57)-(2.58) on I. Then, we extend the solution to ]2M, ∞[ by the
Schwarzschild solution. In this context, we state the following lemma:
Lemma 3. Let ρ > 0 and R > 2M + ρ. Let (λ, µ) be a solution of the reduced Einstein Vlasov system such
that f is on the form (2.43), with boundary conditions
 
1 2M
λ(ρ + 2M ) = − log 1 − := λ0 , (2.59)
2 2M + ρ

and  
1 2M
µ(ρ + 2M ) = log 1 − := µ0 . (2.60)
2 2M + ρ
Moreover, we assume that
∀r ∈ I , 2m(µ)(r) < r. (2.61)
where Z r
m(µ)(r) := M + 4π s2 GΦ (s, µ) ds. (2.62)
2M+ρ

Then, we have
2m(µ)(r)
e−2λ(r) = 1 − . (2.63)
r
Besides,
r  
1 1
Z
µ(r) = µ0 + 2m(µ)(s)
4πsHΦ (s, µ) + 2 m(µ)(s) ds. (2.64)
2M+ρ 1− s
s

Proof. We integrate (2.57) between 2M + ρ and some r ∈ I to obtain

Z r Z r Z r
s(2λ′ e−2λ(s) ) ds − e−2λ(s) ds + (r − (2M + ρ)) = 8π s2 GΦ (s, µ) ds.
2M+ρ 2M+ρ 2M+ρ
2.6 Rein’s work on solutions with a Schwarzschild-like black hole 15

We integrate by parts the first term of the left hand side to have
2M + ρ  −2λ(2M+ρ)  8π Z r
e−2λ(r) = 1 + e −1 − s2 GΦ (s, µ) ds,
r r 2M+ρ
Since
2M
e−2λ(2M+ρ) = 1 − ,
2M + ρ
we have
2m(µ)(r)
∀r ∈ I , e−2λ(r) = 1 − (2.65)
r
where Z r
∀r ∈ I , m(µ)(r) := M + 4π s2 GΦ (s, µ) ds. (2.66)
2M+ρ

Thus, the solution of (2.57) is given by (2.65). In order to find µ, we use (2.65) so that we can write (2.58)
on the form  
′ 2λ(r) 1
µ (r) = e 4πrHΦ (r, µ) + 2 m(µ)(r) . (2.67)
r
Now, we integrate the latter equation between 2M + ρ and r ∈ I to obtain
Z r  
1 1
µ(r) = µ0 + 4πsH Φ (r, µ) + m(µ)(s) ds. (2.68)
2M+ρ 1 −
2m(µ)(s) s2
s

2.6 Rein’s work on solutions with a Schwarzschild-like black hole

In order to contrast it with our approach, we give an overview of Rein’s proof [24] concerning the construction
of static spherically symmetric solutions to the Einstein-Vlasov system with a Schwarzschild-like black hole
such that the spacetime has a finite mass and the matter field has a finite radius. Firstly, one starts with an
ansatz of the form
f (x, p) = Φ(E, ℓ) = φ(E)(ℓ − ℓ0 )l+ , E > 0, ℓ ≥ 0, (2.69)
where ℓ0 ≥ 0, l > 1/2 and φ ∈ L∞ (]0, ∞[) is positive with φ(E) = 0, ∀E > E0 for some E0 > 0.
In this context, we note that a necessary condition to obtain a steady state for the Einstein-Vlasov system
with finite total mass is that Φ must vanish for energy values larger than some cut-off energy E0 ≥ 0 [28].
The same result was proven in [7] for the Vlasov-Poisson system. This motivates the choice of the cut-off
function φ.
Under the above ansatz with a metric on the form (2.9), the Einstein Vlasov system becomes
e−2λ(r) (2rλ′ (r) − 1) + 1 = 8πr2 GΦ (r, µ(r)), (2.70)
−2λ(r) ′ 2
e (2rµ (r) + 1) − 1 = 8πr H Φ (r, µ(r)), (2.71)
 
−2λ(r) ′′ ′ 1 ′ ′
e µ + (µ + )(µ − λ ) = 8πK Φ (r, µ(r)) (2.72)
r
where
r !
2l −(2l+4)u u ℓ0
GΦ (r, u) = cl r e gφ e 1+ 2 , (2.73)
r
r !
cl ℓ0
H Φ (r, u) = r2l e−(2l+4)u hφ e u
1+ 2 , (2.74)
2l + 3 r
r !
cl ℓ0
K Φ (r, u) = (l + 1)HΦ (r, u) + ℓ0 r2l−2 e−(2l+2)u kφ e u
1+ 2 , (2.75)
2 r
(2.76)
2.6 Rein’s work on solutions with a Schwarzschild-like black hole 16

where gφ , hφ and kφ are defined by

Z ∞
1
gφ (t) := φ(ε)ε2 (ε2 − t2 )l+ 2 dε, (2.77)
t
Z ∞
3
hφ (t) := φ(ε)(ε2 − t2 )l+ 2 dε, (2.78)
t
Z ∞
1
kφ (t) := φ(ε)(ε2 − t2 )l+ 2 dε (2.79)
t
(2.80)

and cl is defined by
1
sl
Z
cl := 2π √ ds. (2.81)
0 1−s
More precisely, we have
Proposition 2 (Rein, [24]). Let Φ satisfy the assumptions stated above. Then for every r0 ≥ 0, λ0 ≥ 0
and µ0 ∈ R with λ0 = 0 if r0 = 0, there exists a unique solution λ, µ ∈ C 1 ([r0 , ∞[) of the reduced Einstein
equations (2.70)-(2.71) with
λ(r0 ) = λ0 , µ(r0 ) = µ0 .
Local existence is first proven for λ, µ ∈ C 1 ([r0 , R[) where R > r0 . Then, solutions are shown to extend
to R = ∞. To this end, we note the crucial use of the Tolman-Oppenheimer-Volkov (TOV) equation:
2
p′ (r) = −µ′ (r)(p(r) + ρ(r)) − (p(r) − pT (r)), (2.82)
r
ρ(r) = GΦ (r, µ(r)), p(r) = H Φ (r, µ(r)), and pT (r) = K Φ (r, µ(r)).
ρ, p and pT are interpreted respectively as the energy density, the radial pressure and the tangential pressure.
In order to construct solutions outside from the Schwarzschild black hole, the assumption ℓ0 > 0 plays an
important role in having vacuum between 2M and some r0 > 2M , to be defined below. The construction is
then based on gluing a vacuum region and a region containing Vlasov matter. A posteriori, one can check
that the spacetime has a finite total ADM mass and the matter has a finite radius.
For the gluing,
• one starts by fixing a Schwarzschild black hole of mass M , and one then imposes a vacuum region until
r0 > 2M . The position r0 will be chosen in the following way:
1. First, one can fix E0 = 1. We note that, according to Proposition 1, there can be no bounded
orbits for E0 > 1 in the Schwarzschild spacetime, which motivates this value of E0 6 . This implies
r
u ℓ0
GΦ (r, u) = H Φ (r, u) = 0 if e 1 + 2 ≥ 1,
r
r
u ℓ0
GΦ (r, u), H Φ (r, u) > 0 if e 1 + 2 < 1.
r

In particular, the Schwarzschild metric with mass M solves the reduced Einstein-Vlasov system
for all r > 2M such that r r
2M ℓ0
1− 1 + 2 ≥ 1.
r r
6 In the small data regime, one expects that the range of parameters leading to trapped geodesics to be close to that of

Schwarzschild.
2.6 Rein’s work on solutions with a Schwarzschild-like black hole 17

1 1.5
2
l = 15M l = 17M 2

1.4
0.95

1.3

0.9
1.2
ESch

0.85 1.1
l

1
0.8

0.9

0.75
0.8

0.7 0.7
0 10 20 30 0 10 20 30
r

Figure 2: The three roots of the equation EℓSch (r) = E 2 with E = 0.97 and M = 1 for two cases of ℓ:
Left panel ℓ < 16M 2 , which corresponds to the case where E max (ℓ) < 1. Right panel : ℓ > 16M 2 , where
E max (ℓ) > 1.

This imposes ℓ0 > 16M 2 and we have vacuum region when r ∈ [r− , r+ ], where
p
ℓ0 ± ℓ20 − 16M 2 ℓ0
r± := .
4M
Define ( r )
8
ARein
bound = (E, ℓ) ∈] , 1[×[ℓ0 , ∞[ . (2.83)
9

Note that ARein

bound is strictly included in Abound . Indeed, there exists trapped geodesics of the
2
Schwarzschild spacetime such that E < 1 and ℓ < 16M #r , cf " Figure 2.6 and Proposition 1. In
8
contrast, our solutions can be possibly supported on , 1 ×]12M 2, +∞[.
9
2. Now, one can impose the distribution function to be zero in the region ]2M, r− [ and extend the
metric with a Schwarzschild metric up to 2M . Note that in this context, the ansatz on f is no
longer valid for r > 2M so that the distribution function is not purely a function of E and ℓ. This
is related to our cut-off function χη . In fact, we recall from Proposition 1 that for a particle with
 18

(E, ℓ) ∈ Abound , two orbits are possible. The cut-off function selects only the trapped ones which
are located in the region r ≥ r1 (E, ℓ), "similar to" the region r ≥ r+ in Rein’s work.
• Next, one can impose initial data at r0 := r+ to be
r
2M 1
µ0 := 1 − , λ0 := q .
r0 1− 2M
r0

and apply Proposition 2. The solution to the Einstein Vlasov system is then obtained on [r0 , ∞[ and
extended to the whole domain by gluing it to the Schwarzschild metric at r0 .

3 Statement of the main result

In this section, we give a more detailed formulation of our result. More precisely, we have
Theorem 2. Let M > 0 and let O = R×]2M, ∞[×S 2 be the domain of outer communications parametrised
by the standard Schwarzschild coordinates (t, r, θ, φ). Let Bbound ⊂⊂ Abound be a compact subset of the set
Abound defined in the following way:
r !
8
• Fix E1 , E2 ∈ , 1 such that E1 < E2 ,
9

• Let [ℓ1 , ℓ2 ] be any compact subset of ]ℓlb (E2 ), ℓub (E1 )[, where ℓlb and ℓub are respectively defined by
(2.39) and (2.40).
• Set
Bbound := [E1 , E2 ] × [ℓ1 , ℓ2 ]. (3.1)

Let Φ : Abound × R+ → R+ be a C 2 function with respect to the first two variables, C 1 with respect to the
third variable and such that
• ∀δ ∈ [0, ∞[ , Φ(·, ·; δ) is supported in Bbound .
• ∀(E, ℓ) ∈ Abound , Φ(E, ℓ; 0) = ∂ℓ Φ(E, ℓ; 0) = 0, does not identically vanish and ∀δ > 0, Φ(·, ·, δ) does
not identically vanish on Bbound .
Then, there exists δ0 > 0 and a one-parameter family of functions

(λδ , µδ )δ∈[0,δ0 [ ∈ (C 2 (]2M, ∞[))2 , f δ ∈ C 2 (O × R3 )

with the following properties

1. (λ0 , µ0 ) = (λSch , µSch ) corresponds to a Schwarzschild solution with mass M .
2. For all (E, ℓ) ∈ Bbound , the equation
 
µδ (r) ℓ
e 1 + 2 = E2
r

admits three distinct positive roots 2M < r0 (µδ , E, ℓ) < r1 (µδ , E, ℓ) < r2 (µδ , E, ℓ). Moreover, there
exists η > 0 depending only on δ0 such that

r0 (µδ , E, ℓ) + η < r1 (µδ , E, ℓ).

 19

3. The function f δ takes the form

f δ (x, v) = Φ(E δ , ℓ; δ)Ψη r, (E δ , ℓ), µδ ,

for (x, v) ∈ O × R3 with coordinates (t, r, θ, φ, v r , v θ , v φ ) and where

q
δ
E δ := e2µ (r) 1 + (eλδ (r) v r )2 + (rv θ )2 + (r sin θv φ )2 , ℓ := r4 (v θ )2 + sin2 θ(v φ )2 ,


(3.2)

and Ψη is defined by (2.45)-(2.46).

4. Let g δ be defined by
δ δ
δ
g(t,r,θ,φ) = −e2µ (r)
dt2 + e2λ (r)
dr2 + r2 (dθ2 + sin2 θdφ2 ), (3.3)

then (O, gδ , f δ ) is a static and spherically symmetric solution to the Einstein-Vlasov system (2.1) -
(2.6) - (2.8) describing a matter shell orbiting a Schwarzschild like black hole in the following sense:
δ δ
• ∃Rmin , Rmax ∈]2M, ∞[ called the matter shell radii, which satisfy
δ δ δ
Rmin < Rmax , and Rmin > 4M.

δ δ
• ∀r ∈]2M, Rmin [∪]Rmax , ∞[,
f δ (x, v) = 0,
δ δ
• ∃r̃ ∈]Rmin , Rmax [,
f δ (r̃, ·) > 0.
• The metric g δ is given by the Schwarzschild metric with mass M in the region ]2M, Rmin
δ
[.
• ∃Mδ > M such that the metric g δ is again given by the Schwarzschild metric with mass M δ in the
δ
region ]Rmax , ∞[.

4 Solving the reduced Einstein Vlasov system

4.1 Set up for the implicit function theorem
In this section, we define the solution mapping on which we are going to apply the implicit function theorem.
Theorem 3 (Implicit function theorem 7 ). Let B1 , B2 and X be Banach spaces and G a mapping from an
open subset U of B1 × X into B2 . Let (u0 , σ0 ) be a point in U satisfying
1. G[u0 , σ0 ] = 0,
2. G is continuously Fréchet differentiable on U,
3. the partial Fréchet derivative with respect to the first variable L = G1(u0 ,σ0 ) is invertible.

Then, there exists a neighbourhood N of σ0 in X such that the equation G[u, σ] = 0 is solvable for each
σ ∈ N , with solution u = uσ ∈ B1 .
We recall that a mapping G : U ⊂ B1 × X → B2 is said to be Fréchet differentiable at a point (u, σ) ∈ U
if there exits a continuous linear map L(u, σ) : B1 × X → B2 such that

||G(u + δu, σ + δσ) − G(u, σ) − L(u, σ) · (δu, δσ)||B2

lim = 0.
||(δu,δσ)||B1 ×X →0 ||(δu, δσ)||B1 ×X
7 See Theorem 17.6 , Ch. 17 of [14] for a proof.
4.1 Set up for the implicit function theorem 20

G is Fréchet differentiable if it is Fréchet differentiable at every point (u, σ) ∈ U. It is continuously Fréchet

differentiable if the mapping

L : U → L(B1 × X, B2 ),
(u, σ) 7→ L(u, σ)

is continuous. For every (u, σ) ∈ U such that G is Fréchet differentiable, the map L(u, σ) is called the Fréchet
differential at (u, σ) of G and it is noted DG(u,σ) .
By the partial Fréchet derivatives of G, denoted G1(u,σ) , G2(u,σ) at (u, σ), we mean the bounded linear mappings
from B1 , X respectively, into B2 defined by

G1(u,σ) (h) := DG(u,σ) (h, 0) , G2(u,σ) (k) := DG(u,σ) (0, k),

for h ∈ B1 , k ∈ X.

Solutions to the reduced Einstein-Vlasov system will be obtained by perturbing the Schwarzschild space-
time using a bifurcation parameter δ ≥ 0. The latter turns on in the presence of Vlasov matter supported
on Bbound ⊂⊂ Abound . To this end, we transform the problem of finding solutions to the differential equa-
tion (2.67) into the problem of finding zeros of an operator G, for which we will apply the implicit function
theorem. The Schwarzschild metric and the parameter δ = 0, more precisely (µSch , 0), will be a zero of this
operator.
For this, we adjust the ansatz (2.43) to make the dependence on δ explicit:

f = Φ(E, ℓ; δ)χη (r − r1 (µ, E, ℓ)), (4.1)

such that
∀(E, ℓ) ∈ Abound , Φ(E, ℓ; 0) = 0
where Φ : Abound × R+ → R+ . We will impose in the following some regularity conditions on Φ so that the
solution operator is well defined. Assuming that (λ, µ, f ) solve the EV-system, we can apply Lemma 3 with
the ansatz (4.1) to obtain
2m(µ; δ)(r)
e−2λ(r) = 1 − , (4.2)
r
where Z r
m(µ; δ)(r) := M + 4π s2 GΦ (s, µ; δ) ds, (4.3)
2M+ρ
 
′ 2λ(r) 1
µ (r) = e 4πrHΦ (r, µ; δ) + 2 m(µ; δ)(r) (4.4)
r
and r  
1 1
Z
µ(r) = µ0 + 2m(µ;δ)(s)
4πsHΦ (s, µ; δ) + 2 m(µ; δ)(s) ds (4.5)
2M+ρ 1− s
s
where
∞ r 2 (ε2 −1)
π ε2
Z Z
GΦ (r, µ; δ) := Φ(eµ(r) ε, ℓ; δ)Ψη (r, (eµ(r) ε, ℓ), µ) q dℓdε, (4.6)
r2 1 0 ε2 − 1 − ℓ
r2
r 2 (ε2 −1)
∞
r
π ℓ
Z Z
HΦ (r, µ; δ) := 2 Φ(eµ(r) ε, ℓ; δ)Ψη (r, (eµ(r) ε, ℓ), µ) ε2 − 1 − dℓdε. (4.7)
r 1 0 r2

As we mention before, we will apply Theorem 3 to solve (4.4). Once obtained, we deduce λ and f through
(4.30) and (4.1). We define now the function space in which we will obtain the solutions of (4.4). We consider
the Banach space  
X := C 1 (I), || · ||C 1 (I) , I =]2M + ρ, R[ (4.8)
4.2 Radii of the matter shell 21

and we recall the definition of the C 1 norm on C 1 (I).

∀g ∈ C 1 (I) : ||g||C 1 (I) := ||g||∞ + ||g ′ ||∞ . (4.9)

We define U(δ̃0 ) ⊂ X × [0, δ̃0 [ for some δ̃0 ∈]0, δmax [ as

n o
U(δ̃0 ) := (µ; δ) ∈ X × [0, δ̃0 [ : µ − µSch C 1 (I) < δ̃0 = B(µSch , δ̃0 ) × [0, δ̃0 [,
 
(4.10)

where B(µSch , δ̃0 ) is the open ball in X of centre µSch and radius δ̃0 and δmax is defined by
 
Sch Sch
δmax := min min r0 (E, ℓ) − (2M + ρ), R − max r2 (E, ℓ) .
(E,ℓ)∈Bbound (E,ℓ)∈Bbound

δ̃0 will be chosen small enough so that the three roots of the equation (2.50) exist and so that the condition

∀r ∈ I , 2m(µ; δ)(r) < r (4.11)

is satisfied. Besides, we make the following assumptions on Φ:

(Φ1 ) ∀δ ∈ [0, δ̃0 [, supp Φ(· ; δ) ⊂ Bbound ,
(Φ2 ) Φ is C 1 with respect to δ and ∃C > 0, ∀(E, ℓ) ∈ Bbound , ∀δ ∈ [0, δ̃0 [, |Φ(E, ℓ; δ)| ≤ C,
(Φ3 ) Φ is C 2 with respect to (E, ℓ).
In view of (4.5), we define the solution operator G corresponding to µ by

G : U(δ̃0 ) → X
(µ̃; δ) 7→ G(µ̃; δ)

where ∀r ∈ I,
r   !
1 1
Z
G(µ̃; δ)(r) := µ̃(r) − µ0 + 2m(µ̃;δ)(s)
4πsHΦ (s, µ̃; δ) + 2 m(µ̃; δ)(s) ds . (4.12)
2M+ρ 1− s
s

We verify the steps allowing to apply Theorem 3. In Subsection 4.2 below, we show that we have a well-
defined ansatz for f , that is the existence of a matter shell surrounding the black hole after a well chosen
δ̃0 . In Subsection 4.3, we investigate the regularity of the matter terms and in Subsection 4.4, we check that
(2.61) is always satisfied after possibly shrinking δ̃0 .

4.2 Radii of the matter shell

In this section, we show that for a suitable choice of δ̃0 , ∀µ ∈ B(µSch , δ̃0 ), there exists Rmin (µ), Rmax (µ) ∈ I
satisfying Rmin (µ) < Rmax (µ), such that suppr f ⊂ [Rmin (µ), Rmax (µ)]. We also show that the ansatz for f
(2.43) is well defined. More precisely, we state the following result
Proposition 3. Let 0 < δ̃0 < δmax . Then, there exists δ0 ∈]0, δ̃0 ] such that ∀µ ∈ B(µSch , δ0 ), ∀(E, ℓ) ∈
Bbound , there exist unique ri (µ, (E, ℓ)) ∈ B(riSch (E, ℓ), δ0 )8 , i ∈ {0, 1, 2} such that ri (µ, (E, ℓ)) solve the
equation  
2µ(r) ℓ
e 1 + 2 = E2. (4.13)
r
Moreover, there are no other roots for the above equation outside the balls B(riSch (E, ℓ), δ0 ).
Remark 6. A direct application of the implicit function theorem would yield the existence of the three roots
but for neighbourhoods of µSch which a priori may depend on (E, ℓ). Thus, we revisit its proof in order to
obtain bounds uniform in (E, ℓ).
8 B(r Sch (E, ℓ), δ ) = r ∈ I : |r − riSch (E, ℓ)| < δ0 .

i 0
4.2 Radii of the matter shell 22

Proof. 1. Existence and uniqueness: Let (E, ℓ) ∈ Bbound . Consider the mapping

F E,ℓ : B(µSch , δ̃0 ) × I → R

 
ℓ
( µ ; r ) 7→ Eℓ (r) − E 2 = e2µ(r) 1 + 2 − E 2 .
r
We have
• F E,ℓ is continuously Fréchet differentiable on B(µSch , δ̃0 ) × I. In fact,
♦ ∀µ ∈ B(µSch , δ̃0 ), the mapping r 7→ F E,ℓ (µ, r) is continuously differentiable on I with deriva-
tive
∂F E,ℓ
  
ℓ ℓ
(µ, r) = 2e2µ(r) 1 + 2 µ′ (r) − 3 .
∂r r r
E,ℓ E,ℓ
♦ It is easy to see that the mapping F : µ 7→ F [µ](r) := F E,ℓ (µ, r) is continuously Fréchet
differentiable on B(µSch , δ̃0 ) with Fréchet derivative:
 
E,ℓ ℓ
∀µ ∈ B(µSch , δ̃0 ) , ∀µ̃ ∈ X , DF (µ)[µ̃](r) = 2µ̃(r)e2µ(r) 1 + 2 .
r
♦ Since the Fréchet differentials with respect to each variable exist and they are continuous,
F E,ℓ is continuously Fréchet differentiable on B(µSch , δ̃0 ) × I.
• The points (µSch , r0Sch (E, ℓ)), (µSch , r1Sch (E, ℓ)) and (µSch , r2Sch (E, ℓ)) are zeros for F E,ℓ .
• The differential of F E,ℓ with respect to r at these points does not vanish. Otherwise, the trajectory
is circular which is not possible since (E, ℓ) ∈ Abound .
We define the following mapping on I: ∀µ ∈ B(µSch , δ̃0 ),
FµE,ℓ (r) := r − φ(E, ℓ)F E,ℓ (µ, r),
where φ is defined by
−1
∂F E,ℓ Sch Sch

φ(E, ℓ) = (µ , ri (E, ℓ)) .
∂r
We will show that there exists 0 < δ0 < δ̃0 , uniform in (E, ℓ) such that ∀µ ∈ B(µSch , δ̃0 ), FµE,ℓ is a
contraction on B(riSch (E, ℓ), δ0 ).
Since F E,ℓ is differentiable with respect to r, we have
∂FµE,ℓ ∂F E,ℓ
∀r ∈ I , (r) = 1 − φ(E, ℓ) (µ, r).
∂r ∂r
Since the mappings (E, ℓ) 7→ riSch and (E, ℓ) 7→ F E,ℓ (µ, r) depend smoothly on (E, ℓ) and since Bbound
is compact, there exists C > 0 such that ∀(E, ℓ) ∈ Bbound
|φ(E, ℓ)| ≤ C.
Moreover, there exists δ0 < δ̃0 such that ∀(E, ℓ) ∈ Bbound , ∀(µ, r) ∈ B(µSch , δ0 ) × B(riSch (E, ℓ), δ0 ),
 E,ℓ
∂F E,ℓ Sch Sch

 ∂F  1
 ∂r (µ, r) − ∂r (µ , ri (E, ℓ)) ≤ 2(C + 1) .
 

In fact,
∂F E,ℓ ∂F E,ℓ Sch Sch
(µ, r) − (µ , ri (E, ℓ)) =
∂r ∂r ! !
 
ℓ ℓ ℓ ℓ
2eµ(r) 1 + 2 µ′ (r) − 3 − 1 + ′ Sch

2 µ (ri (E, ℓ)) + Sch
3
r r riSch (E, ℓ) ri (E, ℓ)
! !
ℓ ′ Sch ℓ 
2µ(r) 2µSch (r)

+2 1+
2 µ (ri (E, ℓ)) −
3 e − e
riSch (E, ℓ) riSch (E, ℓ)
4.2 Radii of the matter shell 23

Since Bbound is compact and the different quantities depend continuously on (E, ℓ), there exists C0 > 0
such that ∀(E, ℓ) ∈ Bbound ,
 ! ! 
 ℓ ′ Sch ℓ 
2µ(r)

2µSch (r) 

 2µ(r)

2µSch (r) 
2 1+ µ (r (E, ℓ)) − e − e ≤ C − e


2 i
3  0 e 
 riSch (E, ℓ) riSch (E, ℓ) 
≤ C0 C1 δ̃0 .

Similarly, we have
 r2 − 2rrSch (E, ℓ) + rSch (E, ℓ)
2  
   
ℓ ℓ 
i i Sch

 3 −  = ℓ r − r (E, ℓ) ≤ C0 δ̃0
   
3 Sch i
r r3 ri (E, ℓ)3


rSch (E, ℓ) 
i
 

and
  !    
 ℓ ℓ  ℓ
 1 + 2 µ′ (r) − ′
(riSch (E, ℓ)) ′ ′ Sch


1+ µ ≤ 1+ 2 µ (r) − µ (ri (E, ℓ) 
  

2
 r riSch (E, ℓ)   r
 !

 ′ Sch 1 1 
+ ℓµ (ri (E, ℓ) −

r2 Sch

2 
 r (E, ℓ)
i
≤ C0 δ̃0 .

Moreover, r 7→ e2µ(r) is bounded on I. Therefore, we choose δ0 so that

 E,ℓ
∂F E,ℓ Sch Sch 

 ∂F 1
 ∂r (µ, r) − ∂r (µ , ri ) ≤ 2(C + 1) .


Hence, we obtain
∂FµE,ℓ
 
 ∂F E,ℓ Sch
 C 1
 µ Sch 
(r) − (ri ) ≤ ≤ .
 ∂r ∂r  2(C + 1) 2


∂FµE,ℓ
Sch
Since (riSch (E, ℓ)) = 0, we get
∂r  
 ∂F E,ℓ  1
 µ
(r) ≤ .

 ∂r  2


Now, we show that

∀r ∈ B(riSch (E, ℓ), δ0 ) ,
 E,ℓ
F (r) − rSch (E, ℓ) ≤ δ0 .

µ i

We have ∀(E, ℓ) ∈ Bbound , ∀r ∈ B(riSch (E, ℓ), δ0 )

FµE,ℓ (r) − riSch (E, ℓ) = FµE,ℓ (r) − FµSch riSch

= FµE,ℓ (r) − Fµ (riSch (E, ℓ)) + FµE,ℓ (riSch (E, ℓ)) − FµSch (riSch (E, ℓ)).

By the mean value theorem,

Fµ (r) − Fµ (riSch (E, ℓ)) ≤ 1 |r − riSch (E, ℓ)| ≤ δ0 .

 E,ℓ 
2 2
Moreover, by the same argument above, we show that there exists C > 0 independent from (E, ℓ) such
that  E,ℓ Sch
Fµ (ri ) − FµSch (riSch (E, ℓ)) ≤ Cδ0 .


Therefore, we can update δ0 so that

 E,ℓ
Fµ (r) − riSch  ≤ δ0 .

4.2 Radii of the matter shell 24

It remains to show that FµE,ℓ is a contraction on B(riSch (E, ℓ), δ0 ), ∀µ ∈ B(µSch , δ0 ). For this, we have
by the mean value theorem,

F (r1 ) − F E,ℓ (r2 ) ≤ 1 |r1 − r2 |,

 E,ℓ
∀r1 , r2 ∈ B(riSch (E, ℓ), δ0 ).

µ µ
2

Thus, we can apply the fixed point theorem to obtain : there exists δ0 < δ̃0 such that

∀(E, ℓ) ∈ Bbound , ∀µ ∈ B(µSch , δ0 ) ∃!riE,ℓ : B(µSch , δ0 ) → B(riSch (E, ℓ), δ0 ) such that F E,ℓ (µ, riE,ℓ (µ)) = 0.

(E,ℓ)
2. Regularity: It remains to show that ∀(E, ℓ) ∈ Bbound , the mapping ri is continuously Fréchet
Sch
differentiable on B(µ , δ0 ). In order to lighten the expressions, we will not write the dependence of
ri on (E, ℓ).
First, we show that ri is Lipschitz. For this, let µ, µ ∈ B(µSch , δ0 ) and set

r = ri (µ), r = ri (µ).

We have

r − r = ri (µ) − ri (µ)
= Fµ (r) − Fµ (r)
= Fµ (r) − Fµ (r) + Fµ (r) − Fµ (r)
= Fµ (r) − Fµ (r) + φ(E, ℓ) F E,ℓ (µ, r) − F E,ℓ (µ, r) .

We have
1
|Fµ (r) − Fµ (r)| ≤ |r − r|
2
and
φ(E, ℓ) F E,ℓ (µ, r) − F E,ℓ (µ, r)  ≤ C ||µ − µ|| .


X

Therefore,
|r − r| ≤ 2C ||µ − µ||X .
Thus, ri is Lipschitz, so continuous on B(µSch , δ0 ). Since
 
 ∂F E,ℓ  1
Sch  µ
∀r ∈ B(ri , δ0 ),  (r) ≤ ,

 ∂r  2
!k !−1
∂FµE,ℓ ∂FµE,ℓ 1
Σ ∂r
(r) converges to 1−
∂r
(r) = E,ℓ . Hence,
φ(E, ℓ) ∂F∂r

∂F E,ℓ
∀r ∈ B(riSch , δ0 ), ∀µ ∈ B(µSch , δ0 ), (r, µ) 6= 0.
∂r
Since F E,ℓ is differentiable at (µ, r), we have

0 = F E,ℓ (µ, r) − F E,ℓ (µ, r) = Dµ F E,ℓ (µ, r) · (µ − µ) + ∂r F E,ℓ (µ, r)(r − r) + o (||µ − µ||X + |r − r|) .

By the above estimates we have

o(|r − r|) = o(||µ − µ||X ).
Therefore,

−1
r − r = − ∂r F E,ℓ (µ, r) Dµ F E,ℓ (µ, r) · (µ − µ) + o(||µ − µ||).
4.2 Radii of the matter shell 25

3. Finally, we note that after updating δ0 , we obtain

Eℓ − EℓSch  1 < δ0 .
 
C (I)
(4.14)

In fact, ∀r ∈ I,
 
ℓ  2µ Sch

Eℓ (r) − EℓSch (r) = 1+ 2 e (r) − e2µ (r)
r

We have  Sch

 2µ
e (r) − e2µ (r) ≤ Cδ0 ,


ℓ
and r 7→ 1 + is bounded. Therefore, we can control ||Eℓ − EℓSch ||∞ . Now, we compute ∀r ∈ I
r2
       
′ ℓ ℓ Sch ′ ℓ ℓ
Eℓ′ (r) − EℓSch (r) = 2e2µ(r) µ′ (r) 1 + 2 − 3 − 2e2µ (r) µSch (r) 1 + 2 − 3
r r r r
 
2ℓ  Sch
 ℓ ′ Sch

= − 3 e2µ(r) − e2µ (r) + 2 1 + 2 µ′ (r)e2µ(r) − µSch (r)e2µ (r)
r r
We control the last term in the following way
     
′ ′
Sch
 Sch
 Sch
 ′
µ (r)e2µ(r) − µSch (r)e2µ (r)  ≤ µ′ (r) e2µ(r) − e2µ (r)  +  µ′ (r) − µSch (r) e2µ (r) 
    

≤ Cδ0 .

4. Uniqueness of the roots on I: We show, after possibly shrinking δ0 , that ∀µ ∈ B(µSch , δ0 ) ∀(E, ℓ) ∈
Bbound , there are no others roots of the equation (4.13). First, we set

∀r ∈ I , PE,ℓ (r) := Eℓ (r) − E 2

and
∀r ∈ I , Sch
PE,ℓ (r) := EℓSch (r) − E 2 .
We claim
[ that for δ0 sufficiently small, there exists C > 0 such that ∀(E, ℓ) ∈ Bbound , ∀r ∈ J :=
I\ B(riSch (E, ℓ), δ0 ) we have
i=0,1,2
 Sch 
PE,ℓ (r) > Cδ0 .
Sch
In fact, by the monotonicity properties of PE,ℓ , it is easy to see that
 Sch   Sch Sch  Sch Sch
∀(E, ℓ) ∈ Bbound , ∀r ∈ J : PE,ℓ (r) ≥ max PE,ℓ (r0 (E, ℓ) − δ0 ) , PE,ℓ (r0 (E, ℓ) + δ0 )
Sch Sch Sch Sch
, PE,ℓ (r1 (E, ℓ) − δ0 ), PE,ℓ (r1 (E, ℓ) + δ0 ),
 Sch Sch  Sch Sch
P
E,ℓ(r (E, ℓ) − δ0 ) , P
2 (r (E, ℓ) + δ0 ) .
E,ℓ 2

Now, we show that for all |h| sufficiently small, there exist C(h) > 0 uniform in (E, ℓ) such that
Sch Sch
|PE,ℓ (ri (E, ℓ) + h)| > C(h).

We have
Sch Sch Sch Sch Sch ′ Sch
PE,ℓ (ri (E, ℓ) + h) = PE,ℓ (ri (E, ℓ)) + h(PE,ℓ ) (ri (E, ℓ)) + o(h),
Sch (k)
where o(h) is uniform in (E, ℓ) by continuity of (PE,ℓ ) with respect to (E, ℓ) and compactness
Sch Sch ′ Sch
of Bbound . Moreover, ri (E, ℓ) are simple roots so that (PE,ℓ ) (ri (E, ℓ)) 6= 0 and (E, ℓ) 7→
Sch ′ Sch
(PE,ℓ ) (ri (E, ℓ)) is continuous. Hence, for h sufficiently small, we obtain
Sch Sch
 Sch ′ Sch 
|PE,ℓ (ri (E, ℓ) + h)| > |h| (PE,ℓ ) (ri (E, ℓ)) > C|h|,
4.2 Radii of the matter shell 26

where C is some constant which uniform in δ0 and (E, ℓ). Therefore, we update δ0 so that
 Sch Sch 
PE,ℓ (ri (E, ℓ) ± δ0 ) > Cδ0 .

Now, let δ1 < δ0 . Then, ∀µ ∈ B(µSch , δ1 ) ⊂ B(µSch , δ0 ), ∀(E, ℓ) ∈ Bbound , ri (µ, E, ℓ) is unique in the
ball B(riSch (E, ℓ), δ0 ). Moreover, ∀r ∈ J, we have
Sch Sch
PE,ℓ (r) = PE,ℓ (r) − PE,ℓ (r) + PE,ℓ (r).

By the triangular inequality, the latter implies for δ1 < Cδ0 that

∀r ∈ J, |PE,ℓ (r)| > −δ1 + Cδ0 > 0.

Therefore, after updating δ1 , PE,ℓ does not vanish outside the balls B(riSch (E, ℓ), δ1 ). This yields the
uniqueness.

Lemma 4. ∀µ ∈ B(µSch , δ0 ), ri (µ, ·) have the same monotonicity properties as riSch . More precisely,
∀(E, ℓ) ∈ Bbound ,
• r0 (µ, E, ·) decreases on ]ℓ1 , ℓ2 [ and r0 (µ, ·, ℓ) increases on ]E1 , E2 [,
• r1 (µ, E, ·) increases on ]ℓ1 , ℓ2 [ and r1 (µ, ·, ℓ) decreases on ]E1 , E2 [,
• r2 (µ, E, ·) decreases on ]ℓ1 , ℓ2 [ and r2 (µ, ·, ℓ) increases on ]E1 , E2 [.
The proof of the latter lemma is straightforward.
Therefore, using the above monotonicity properties of r1 (µ, ·) and r2 (µ, ·) one can define the radii of the
matter shell in the following way:
Rmin (µ) := r1 (µ, E2 , ℓ1 ) (4.15)
and
Rmax (µ) := r2 (µ, E2 , ℓ1 ). (4.16)
Since ∀(E, ℓ) ∈ Abound , r1Sch (E, ℓ) > 4M , we can update δ0 such that

∀(E, ℓ) ∈ Bbound , r1 (µ, E, ℓ) > 4M.

In particular,
Rmin (µ) > 4M. (4.17)
Now, we need to define η > 0 appearing in the ansatz of f , (2.43). We choose η > 0 independent of (µ, E, ℓ)
such that
r1 (µ, E, ℓ) − r0 (µ, E, ℓ) > η > 0.
Lemma 5. There exists η > 0 independent of µ, E, ℓ such that for all µ ∈ B(µSch , δ0 ) and for all (E, ℓ) ∈
Bbound we have r1 (µ, E, ℓ) − r0 (µ, E, ℓ) > η > 0.
Proof. For this, we set

hSch : Bbound → R
(E, ℓ) 7→ r1Sch (E, ℓ) − r0Sch (E, ℓ)

By monotonicity properties of riSch , it is easy to see that hSch (·, ℓ) is decreasing and hSch (E, ·) is increasing.
Therefore, one can easily bound h :

∀(E, ℓ) ∈ Bbound , hSch (E, ℓ) ≥ hSch (E2 , ℓ) ≥ hSch (E2 , ℓ1 ) > 0.

Thus, ∃η > 0 such that ∀(E, ℓ) ∈ Bbound

hSch (E, ℓ) > 2η.
4.3 Regularity of the matter terms 27

Now we set
h : B(µSch , δ0 ) × Bbound → R
(µ; (E, ℓ)) 7→ h(µ, (E, ℓ)) := r1 (µ, E, ℓ) − r0 (µ, E, ℓ).

We have, ∀µ ∈ B(µSch , δ0 ), ∀(E, ℓ) ∈ Bbound

h(µ, E, ℓ) = h(µ, (E, ℓ)) − hSch (E, ℓ) + hSch (E, ℓ) > −2δ0 + 2η.
It remains to update δ0 so that the latter inequality is greater than η.
Now that we have justified the ansatz for f , it remains to check that suppr f ⊂ [Rmin (µ), Rmax (µ)]. To
this end, we state the following result
Lemma 6. Let µ ∈ B(µSch , δ0 ) and f be a distribution function of the form (2.43). Then
suppr f ⊂ [Rmin (µ), Rmax (µ)].
Proof. Let (xµ , v a ) ∈ Γ and denote by r its radial component. Let E(xµ , v a ) and ℓ(xµ , v a ) be defined by
(2.22) and (2.21). If r ∈ suppr f , then
Φ(E, ℓ) > 0, and Ψη (r, E, ℓ, µ) > 0.
Therefore, (E, ℓ) ∈ Bbound and
Ψη (r, E, ℓ, µ) = χη (r − r1 (µ, (E, ℓ))) > 0.
Hence,
r ≥ r1 (µ, E, ℓ) − η > r0 (µ, E, ℓ).
The last inequality is due to the definition of η. Now since (E, ℓ) ∈ Bbound , the equation Eℓ (r̃) = E 2 admits
three distinct positive roots. Moreover, we have
Eℓ (r) ≤ E 2 (4.18)
for any geodesic moving in the exterior region. Therefore,
r ∈ ]2M + ρ, r0 (µ, E, ℓ)] ∪ [r1 (µ, E, ℓ), r2 (µ, E, ℓ)] .
Hence, r must lie in the region [r1 (µ, E, ℓ), r2 (µ, E, ℓ)]. By construction of Rmin (µ) and Rmax (µ), we conclude
that
r ∈ [Rmin (µ), Rmax (µ)] .

4.3 Regularity of the matter terms

In this section, we show that the matter terms Gφ and HΦ given by respectively (4.6) and (4.7) are well
defined on I × U(δ0 ). Then, we investigate their regularity with respect to each variable.
Let (r, µ; δ) ∈ I × U(δ0 ) and let ε ∈ [1, ∞[ and ℓ ∈ [0, r2 (ε2 − 1)[. Note that if r2 (ε2 − 1) < ℓ1 , then ℓ < ℓ1
and
Φ(eµ(r) ε, ℓ; δ) = 0,
since suppℓ Φ ⊂ [ℓ1 , ℓ2 ]. Therefore,
∞ r 2 (ε2 −1)
2π ε2
Z Z
GΦ (r, µ; δ) = Φ(eµ(r) ε, ℓ; δ)Ψη (r, (eµ(r) ε, ℓ), µ) q dℓdε,
r2 1 0 ε2 − 1 − ℓ
r2
e−µ(r) E2 r 2 (ε2 −1)
2π ε2
Z Z
= Φ(eµ(r) ε, ℓ; δ)Ψη (r, (eµ(r) ε, ℓ), µ) q dℓdε.
r2
q
ℓ ℓ
1+ r12 ℓ1 ε2 − 1 − r2
4.3 Regularity of the matter terms 28

We make a first change of variable from ℓ to ℓ̃ := ℓ − ℓ1 ,

e−µ(r) E2 r 2 (ε2 −1)−ℓ1
2π ε2
Z Z
GΦ (r, µ; δ) = 2 Φ(eµ(r) ε, ℓ̃+ℓ1 ; δ)Ψη (r, (eµ(r) ε, ℓ̃+ℓ1 ), µ) q dℓ̃dε.
r
q
ℓ
1+ r12 ℓ1 ℓ̃


0 ε2 − 1 − −
r2 r2

ℓ̃
We make a second change of variable from ℓ̃ to s := ,
r2 (ε2 − 1) − ℓ1
e−µ(r) E2    12
ℓ1
Z
2 2
GΦ (r, µ; δ) = 2π ε ε − 1+ 2 gΦ (r, µ, eµ(r) ε; δ)dε,
r
q
ℓ
1+ r12

where gΦ is defined by
Z 1
ds
gΦ (r, µ, E; δ) := Φ(E, sr2 (e−2µ(r) E 2 − 1) + (1 − s)ℓ1; δ)Ψη (r, E, sr2 (e−2µ(r) E 2 − 1) + (1 − s)ℓ1), µ) √ .
0 1−s
(4.19)
We make a last change of variable from ε to E := eµ(r) ε to obtain
E2    21
ℓ1
Z
−4µ(r) 2 2 2µ(r)
GΦ (r, µ; δ) = 2πe E E −e 1+ 2 gΦ (r, µ, E; δ)dE. (4.20)
r
q
ℓ
eµ(r) 1+ r12

With the same change of variables we compute

r 2 (ε2 −1)
∞
r
2π ℓ
Z Z
µ(r) µ(r)
HΦ (r, µ; δ) = 2 Φ(e ε, ℓ; δ)Ψη (r, (e ε, ℓ), µ) ε2 − 1 − dℓdε
r 1 0 r2
e−µ(r) E2 r 2 (ε2 −1)
r
2π ℓ
Z Z
µ(r) µ(r)
= 2 Φ(e ε, ℓ; δ)Ψη (r, (e ε, ℓ), µ) ε2 − 1 − dℓdε
r r2
q
ℓ
1+ r12 ℓ1
s
e−µ(r)
E2 r 2 (ε2 −1)−ℓ1 
2π ℓ1 ℓ̃
Z Z
= Φ(eµ(r) ε, ℓ̃ + ℓ1 ; δ)Ψη (r, (eµ(r) ε, ℓ̃ + ℓ1 ), µ) ε2 − 1 − − dℓ̃dε
r2 r2 r2
q
ℓ
1+ r12 0

e−µ(r) E2    23
ℓ1
Z
2
= 2π (ε − 1 + 2 hΦ (r, µ, eµ(r) ε; δ)dε,
r
q
ℓ
1+ r12

where hΦ is defined by
Z 1
√
hΦ (r, µ, E; δ) := Φ(E, sr2 (e−2µ(r) E 2 −1)+(1−s)ℓ1 ; δ)Ψη (r, (E, sr2 (e−2µ(r) E 2 −1)+(1−s)ℓ1 ), µ) 1 − s ds.
0
(4.21)
Therefore,
E2    23
ℓ1
Z
−4µ(r) 2 2µ(r)
HΦ (r, µ; δ) = 2πe E −e 1+ 2 hΦ (r, µ, E; δ) dE. (4.22)
r
q
ℓ
eµ(r) 1+ r12

1
It is clear that hφ is well defined on I × C 1 × U(δ0 ). Since s 7→ √ is integrable, gφ is well defined on
1−s
I × C 1 × U(δ0 ). Now we state a regularity result of GΦ and HΦ .
Proposition 4. 1. gΦ and hΦ defined respectively by (4.19) and (4.21) are C 2 with respect to r and E
respectively on I, ]E1 , E2 [ and C 1 with respect to δ on [0, δ0 [. Furthermore, they are continuously
Fréchet differentiable with respect to µ on B(µSch , δ0 ).
2. Similarly, GΦ and HΦ are continuously Fréchet differentiable with respect to µ on B(µSch , δ0 ), C 2 with
respect to r and C 1 with respect to δ.
4.3 Regularity of the matter terms 29

Proof. The proof is based on the regularity of the ansatz function as well as on the regularity of r1 .
1. • By the dominated convergence theorem, regularity of Φ, its compact support and the assumption
(Φ2 ), it is easy to see that gΦ and hΦ are C 2 with respect to r and E on their domain and C 1
with respect to δ.
• For the differentiability of gΦ and hΦ with respect to µ: let (r, E, δ) ∈ I×]E1 , E2 [×[0, δ[ and let
µ ∈ B(µSch , δ0 ), we have

gΦ (r, µ, E; δ) =
Z 1
ds
Φ(E, sr2 (e−2µ(r) E 2 − 1) + (1 − s)ℓ1 ; δ)Ψη (r, (E, sr2 (e−2µ(r) E 2 − 1) + (1 − s)ℓ1 ), µ) √ =
0 1−s
Z 1
ds
Φ(E, sr2 (e−2µ(r) E 2 − 1) + (1 − s)ℓ1 ; δ)χη (r − r1 (µ, E, sr2 (e−2µ(r) E 2 − 1) + (1 − s)ℓ1 )) √ .
0 1−s
Since r1 is continuously Fréchet differentiable on B(µSch , δ0 ), χη is smooth on R and Φ(·, ·; δ) is
C 2 on Bbound , the mappings
φ : µ 7→ Φ(E, sr2 (e−2µ(r) E 2 − 1) + (1 − s)ℓ1 ; δ) =: Φ(s, µ)
and
ψ : µ 7→ χη (r − r1 (µ, E, sr2 (e−2µ(r) E 2 − 1) + (1 − s)ℓ1 ))
are continuously Fréchet differentiable on B(µSch , δ0 ). Their Fréchet derivatives are respectively
given by : ∀µ̃ ∈ X ,
Dφ(µ)[µ̃] = −2sr2 E 2 µ̃(r)∂ℓ Φ(E, sr2 (e−2µ(r) E 2 − 1) + (1 − s)ℓ1 ; δ)e−2µ(r)
and
 
Dψ(µ)[µ̃] = − Dµ r 1 (s, µ)[µ̃] − 2sr2 E 2 µ̃(r)∂ℓ r 1 (s, µ)e−2µ(r) χ′η (r − r 1 (s, µ))

where
r1 (s, µ) := r1 (µ, eµ(r) ε, sr2 (e−2µ(r) E 2 − 1) + (1 − s)ℓ1 ).
Since χη is either 0 or 1 on the support of Φ(E, ℓ; δ), we have
Dψ(µ)[µ̃] = 0.
Therefore, for all µ̃ ∈ X sufficiently small,
Z 1
gΦ (r, µ + µ̃; δ)(r) = Φ(s, µ + µ̃)ψ(µ + µ̃)(s) ds
0
Z 1      
= φ(µ)(s) + Dφ(µ)[µ̃](s) + O ||µ̃||2C 1 (I) ψ(µ)(s) + O ||µ̃||2C 1 (I) ds
0
Z 1  
= Φ(s)ψ(µ)(s) + ψ(µ)(s)Dφ(µ)[µ̃](s) + O ||µ̃||2C 1 (I) ds
0
1
ds
Z  
= gΦ (µ)(r) + ψ(µ)(s)Dφ(µ)[µ̃](s) √ + O ||µ̃||2C 1 (I) ds
0 1−s
Note that we didn’t write the dependence of φ, ψ and gΦ on the remaining variables in order to
lighten the expressions. Now we define
Z 1
ds
DgΦ (µ)[µ̃](r) := − ψ(µ)(s)Dφ(µ)[µ̃](s) √ .
0 1−s
It is clear that DgΦ (µ) is a linear mapping from X to R. Besides, since r1 is continuously Fréchet
differentiable on B(µSch , δ0 ), gΦ is also continuously Fréchet differentiable.
4.4 The condition (2.61) is satisfied 30

• In the same way, we obtain regularity for hΦ .

2. Regularity for GΦ and HΦ are straightforward. In particular, their Fréchet derivatives with respect to
µ are respectively given by ∀µ̃ ∈ X
E2    21
ℓ1
Z
−4µ(r) 22 2µ(r)
Dµ GΦ (r, µ; δ)[µ̃] = 2πe −4µ̃(r) E E −e 1+ 2 gΦ (r, µ, E; δ)dE
r
q
ℓ
eµ(r) 1+ r12

E2 2 2µ(r) ℓ1

 21
E µ̃(r)e 1+
 
ℓ1
Z
r2 2 2 2µ(r)
+ −q
gΦ (r, µ; δ) + E E − e 1+ 2 Dµ gΦ (r, µ, E; δ)[µ̃]dE  .
r
q
ℓ ℓ1
eµ(r) 1+ r12 E 2 − e2µ(r) 1 + r2
(4.23)
and
E2   23
ℓ1
Z
−4µ(r) 2 2µ(r) 2
Dµ HΦ (r, µ; δ)[µ̃] = 2πe −4µ̃(r) E E −e 1+ 2 hΦ (r, µ, E; δ)dE
r
q
ℓ
eµ(r) 1+ r12
Z E2  s  
2 2µ(r) ℓ1 ℓ1
+ −3E µ̃(r)e 1+ 2 E 2 − e2µ(r) 1 + 2 hΦ (r, µ, E; δ)
r r
q
ℓ
eµ(r) 1+ r12
   21 !
ℓ1
+E 2 E 2 − e2µ(r) 1 + 2 Dµ hΦ (r, µ, E; δ)[µ̃]dE .
r
(4.24)

4.4 The condition (2.61) is satisfied

In this section, we show that we can choose δ0 even smaller so that the condition (2.61) is satisfied. More
precisely, we state the following lemma
Lemma 7. There exists δ0 ∈]0, δ̃0 ] such that ∀(µ; δ) ∈ U(δ0 ), ∀r ∈ I

2m(µ; δ)(r) < r. (4.25)

Moreover, there exists C > 0 such that ∀r ∈ I,

 −1
2m(µ; δ)(r)
1− < C.
r

Proof. 1. First, we recall Z r

2m(µ; δ)(r) := 2M + 8π s2 GΦ (s, µ; δ) ds
2M+ρ

where GΦ (s, µ; δ) is given by

E2    12
ℓ1
Z
−4µ(r) 2 2 2µ(r)
GΦ (r, µ; δ) = 2πe E E −e 1+ 2 gΦ (r, µ, E; δ)dE.
r
q
ℓ
eµ(r) 1+ r12

2. If 2M + ρ < r ≤ Rmin (µ), then

2m(µ; δ)(r) = 2M < r.
Hence, the condition (2.61) is always satisfied. Besides, ∀r ∈ ]2M + ρ, Rmin (µ)[, we have
 −1
2m(µ; δ)(r) 2M
1− <1+ .
r ρ
4.4 The condition (2.61) is satisfied 31

3. Now, let r ≥ Rmin (µ). We claim that ∃ C > 0, independent of (µ, δ) such that
||r 7→ 8πGΦ (r, µ; δ)||C 1 (I) ≤ Cδ0 .
In fact, for a fixed r ∈ I we write
 
GΦ (r, µ; δ) = GΦ (r, µSch ; 0) + Dµ GΦ (r, µSch ; 0)[µ − µSch ] + δ∂δ GΦ (r, µSch ; 0) + O ||µ − µSch ||2C 1 (I) + δ 2
= δ∂δ GΦ (r, µSch ; 0) + O(δ0 ),
where we used
GΦ (r, µSch ; 0) = Dµ GΦ (r, µSch ; 0)[µ − µSch ] = 0.
Now we have
∂δ GΦ (r, µSch ; 0) =
 −2 Z E2     21
2M 2 2 2M ℓ1
2π 1 − E E − 1 − 1 + 2
∂δ gΦ (r, µSch , E; 0) dE.
r r r
q
ℓ1
(1− r )(1+ 2 )
2M
r

Since r ∈ I and by the the (Φ2 ) assumption and the definition of χη , there exists C > 0 independent
of (r, µ; δ) such that
∀E ∈ [E1 , E2 ], ∀r ∈ I ∂δ gΦ (r, µSch , E; 0) ≤ C.
 

Therefore,
∂δ GΦ (r, µSch ; 0)
 
 −2 Z E2     21
2M 2 2 2M ℓ1
≤ 2Cδπ 1 − E E − 1− 1+ dE + O(δ0 )
2M + ρ 2M + ρ (2M + ρ)2
q
ℓ1
r )(1+ r2 )
(1− 2M
Z E2
≤ Cδ E 3 dE + O(δ0 )
0
≤ Cδ0 .
This yields the result.
We have, by (4.17), ∀µ ∈ B(µSch , δ0 ),
Rmin (µ) > 4M.
Therefore,
Rmin (µ) Cδ0 3
2
2m(µ; δ)(r) ≤ + r − Rmin (µ)3
2 3
Rmin (µ) CR2 δ0
≤ + r
 2 2
3
1 CR δ0
≤ + r.
2 3
It remains to update δ0 so that
1 CR2 δ0
+ < 1.
2 3
We take for example  
3
δ0 = min δ0 , .
4CR2
Therefore, ∀r ≥ Rmin (µ), we have
 −1
2m(µ; δ)(r)
1− < 4.
r
This concludes the proof.
4.5 Solving for µ 32

4.5 Solving for µ

We check in a number of steps that the mapping G defined by (4.12) satisfies the conditions for applying the
implicit function theorem:
1. First, we need to check that G is well defined on U(δ0 ).
2. It is clear that (µSch ; 0) is a zero for G.
3. Next, we need to check that G is continuously Fréchet differentiable on U(δ0 ).

4. Finally, we have to show that the partial Fréchet derivative with respect to the first variable µ at the
point (µSch ; 0):

L : C 1 (I) → C 1 (I)
µ 7→ DG(µSch ;0) (µ; 0)

is invertible.
Provided the above facts hold, we can now apply Theorem 3 to the mapping G : U → X to obtain:
Theorem 4. There exists δ1 , δ2 ∈]0, δ0 [ and a unique differentiable solution map

µ : [0, δ1 [→ B µSch , δ2 ⊂ C 1 (I),

such that S(0) = µSch and

G(µ(δ); δ) = 0, ∀δ ∈ [0, δ1 [.
The remainder of this section is devoted to the proofs of steps 1 to 4.

4.5.1 G is well defined on U

Let (µ; δ) ∈ U(δ0 ). Recall the definition of G
r   !
1 1
Z
G(µ; δ)(r) := µ(r) − µ0 + 2m(µ;δ)(s)
4πsHΦ (s, µ; δ) + 2 m(µ; δ)(s) ds .
2M+ρ 1− s
s

By Lemma 7, there exists C > 0 such that

 −1
2m(µ; δ)(r)
∀r ∈ I, 1 − < C. (4.26)
r

Besides, ∀r ∈ I,
r   Z Rmax (µ)  
1 ds 1 ds
Z
4πsHΦ (s, µ; δ) + 2 m(µ; δ)(s) 2m(µ;δ)(s)
≤ 4πsHΦ (s, µ; δ) + 2 m(µ; δ)(s) 2m(µ;δ)(s)
.
2M+ρ s 1− 2M+ρ s 1−
s s

The integral of the right hand side is finite. Therefore, G(µ; δ0 ) is well defined on I. Moreover, by Proposition
4 and lemma 7 we have
• r 7→ HΦ (r, µ; δ) is C 1 on I,
• r 7→ m(µ; δ)(r) is C 1 on I,
Hence, G(µ; δ) is well defined and it is C 1 on I. Moreover, it easy to control its C 1 norm thanks to (4.26)
and the compact support of r 7→ HΦ (r, µ; δ) and r 7→ GΦ (r, µ(r), µ; δ).
4.5 Solving for µ 33

4.5.2 G is continuously Fréchet differentiable on U

Let (µ; δ) ∈ U(δ0 ). First, we show the differentiability with respect to µ. In this context, we drop the
dependence on δ in order to lighten the expressions. Our first claim is that G has the Fréchet derivative with
respect to µ given by ∀µ̃ ∈ C 1 (I) , ∀r ∈ I,
Z r  
1 2 1 1
D G(µ)[µ̃](r) := µ̃(r) − 2 Dm(µ)[µ̃](s) 4πsHΦ (s, µ) + 2 m(µ)(s)
2M+ρ s s

2m(µ)(s)
1−
s (4.27)
 
1 1
+ 4πsDµ HΦ (s, µ)[µ̃] + 2 Dm(µ)[µ̃](s) ds,
2m(µ)(s) s
1−
s
where Dm(µ) is the Fréchet derivative of m with respect to µ, given by
Z r
Dm(µ)[µ̃](r) = 4π s2 Dµ GΦ (s, µ)[µ̃]ds. (4.28)
2M+ρ

We need to prove  
G(µ + µ̃) − G(µ) − D1 G(µ)[µ̃]
C 1 (I)
lim = 0.
||µ̃||C 1 (I) →0 ||µ̃||C 1 (I)
We will only prove in details

||m(µ + µ̃) − m(µ) − Dm(µ)[µ̃]||C 1 (I)

lim = 0.
||µ̃||C 1 (I) →0 ||µ̃||C 1 (I)

The remaining terms are treated in the same way thanks to the estimate (4.26), the compact support in r of
the matter terms as well as their regularity.
Let r ∈ I. We compute
Z r
m(µ + µ̃)(r) − m(µ)(r) − Dm(µ)[µ̃](r) = 4π s2 (GΦ (s, µ + µ̃) − GΦ (s, µ) − Dµ GΦ (s, µ)[µ̃]) ds.
2M+ρ

Since GΦ is continuously Fréchet differentiable with respect to µ, we write in a neighbourhood of the point
µ with a fixed s
 
GΦ (s, µ + µ̃) = GΦ (s, µ̃) + Dµ GΦ (s, µ)[µ̃] + O ||µ̃||2C 1 (I) .

Therefore,
|m(µ + µ̃)(r) − m(µ)(r) − Dm(µ)[µ̃](r)| . ||µ̃||2C 1 (I) ,
so that
||m(µ + µ̃) − m(µ) − Dm(µ)[µ̃]||∞
lim = 0.
||µ̃||C 1 (I) →0 ||µ̃||C 1 (I)

It remains to control the C 1 norm of m(µ + µ̃) − m(µ) − Dm(µ)[µ̃]. It is clear that the latter is C 1 on I and
the derivative is given by ∀r ∈ I :

m(µ + µ̃)′ (r) − m(µ)′ (r) − Dm(µ)[µ̃]′ (r) = 4πr2 (GΦ (r, µ + µ̃) − GΦ (r, µ) − Dµ GΦ (r, µ)[µ̃]))
 
= O ||µ̃||2C 1 (I) ,

which implies
||m(µ + µ̃)′ − m(µ)′ − Dm(µ)[µ̃]′ ||∞
lim = 0.
||µ̃||C 1 (I) →0 ||µ̃||C 1 (I)
4.6 Conclusions 34

To conclude, we show that the Fréchet derivative is continuous on B(µSch , δ0 ). Let µ1 , µ2 ∈ B(µSch , δ0 ) and
let µ̃ ∈ C 1 (I) ||µ̃||C 1 (I) ≤ 1, we compute ∀r ∈ I
Z r
Dm(µ1 )(µ̃)(r) − Dm(µ2 )(µ̃)(r) = 4π s2 (Dµ GΦ (s, µ1 )[µ̃]) − Dµ GΦ (s, µ2 )[µ̃])) ds.
2M+ρ

We use the regularity results of Proposition 4 and the estimate ||µ̃||C 1 (I) ≤ 1 to estimate each term by
||µ1 − µ2 ||C 1 (I) . Similarly, we control ||Dm(µ1 )(µ̃)′ − Dm(µ2 )(µ̃)′ ||∞ . By taking the supremum on ||µ̃||C 1 (I) ,
we obtain
||Dm(µ1 ) − Dm(µ2 )||L(C 1 (I),C 1 (I)) . ||µ1 − µ2 ||C 1 (I) .

Finally, when µ1 → µ2 in C 1 (I) we obtain Dm(µ1 ) → Dm(µ2 ) in L(C 1 (I), C 1 (I)). This proves the continuity.

We check now the differentiability with respect to δ. We claim that

Z r  
2 1 ′ 1
Dδ G(δ)(r) = − 2 m (δ)(s) 4πsH Φ (s, δ) + m(µ)(s)
2M+ρ s s2

2m(µ)(s)
1−
s (4.29)
 
1 1
+ 4πs∂δ HΦ (s; δ) + 2 m′ (δ)(s) ds,
2m(µ)(s) s
1−
s
where Z r
′
m (δ)(s) = 4π s2 ∂δ GΦ (s, µ; δ) ds,
2M+ρ

and
∂δ HΦ (s; δ) = ∂δ HΦ (s, µ; δ).
Again, we dropped the dependance on µ in order to lighten the expressions. By regularity assumptions
on Φ with respect to the third variable, the compact support of the matter terms and by the dominated
convergence theorem, G is C 1 with respect to the parameter δ.

We check the third step: we evaluate the derivative of G with respect to µ at the point (µSch , 0). For
this we use Proposition 4 and the fact that Φ(·, ·; 0) = ∂ℓ Φ(·, ·; 0) = 0 to obtain

Dµ GΦ (s, µSch ; 0) = Dµ HΦ (s, µSch ; 0) = 0.

Therefore, Dµ G(µSch ; 0) is reduced to the identity, which is invertible. Since all the assumptions are satisfied,
we apply Theorem 3 to obtain Theorem 4.

4.6 Conclusions
In this section, we deduce from Theorem 4 the remaining metric component and the expression of the matter
terms. More precisely, we set ∀δ ∈ [0, δ1 [
 
1 2m(µ(δ); δ)
λδ (r) := − log 1 − , ∀r ∈ I. (4.30)
2 r

λδ is well defined since

∀r ∈ I, r > 2m(µ(δ); δ).
α a 3
As for the matter field, we set ∀(x , v ) ∈ O × R

f δ (xα , v a ) := Φ(Eδ , ℓ; δ)Ψη r − r1 (µ(δ), E δ , ℓ) ,

(4.31)
4.6 Conclusions 35

where q
E δ = e2µδ (r) 1 + (eλδ (r) v r )2 + (rv θ )2 + (r sin θv φ )2
and ℓ is given by (2.21).
Such f δ is a solution to the Vlasov equation (2.2) since Φ is a solution and Ψη is constant on the support of
Φ. Furthermore, we set the energy density to be

ρδ (r) := GΦ (r, µ(δ); δ), (4.32)

We define the radii of the matter shell to be

δ
Rmin := Rmin (µ(δ)) (4.33)

and
δ
Rmax := Rmax (µ(δ)), (4.34)
where Rmin and Rmax are defined by (4.15) and (4.16). Finally, we define the total mass by
Z Rδmax
δ
M := M + 4π r2 ρδ (r) dr. (4.35)
Rδmin

It is easy to see that µ(δ) and λδ are actually C 2 on I. In fact, since µ(δ) is C 1 on I, m(µ(δ); δ) given by
(4.3) is C 2 on I and r 7→ H(r, µ(δ); δ) is C 1 on I by Proposition 4. Moreover,

G(µ(δ); δ) = 0.

That is µ(δ) is implicitly given by

r   !
1 1
Z
µ(δ)(r) = µ0 + 2m(µ(δ);δ)(s)
4πsHΦ (s, µ(δ); δ) + 2 m(µ(δ); δ)(s) ds .
2M+ρ 1− s
s

From the above equation it is clear that µ(δ) is C 2 on I. Therefore, λδ is also C 2 on I and f δ is also C 2 with
respect to r.
It remains to extend the solution on ]2M, ∞[. It suffices to extend (µ, λ) by (µSch , λSch ) of mass M on the
domain ]2M, 2M + ρ], and by (µSch , λSch ) of mass M δ on the domain [R, ∞[. We recall that the matter field
vanishes in these regions thanks to its compact support. Thus, (µ, λ) are still C 2 on ]2M, ∞[. This ends the
proof of Theorem 2.
 36

A Study of the geodesic motion in the exterior of Scwharzschild

spacetime
We present a detailed study of the geodesic motion in the exterior region of a fixed Schwarzschild spacetime.
We will classify the geodesics based on the study of the effective energy potential. Such a classification is of
course classical and we refer to [8, Chapter 3] or [21, Chapter 33] for more details. In this section, we prove
Proposition 1.
1
• First, note that EℓSch is a cubic function in . Its derivative is given by
r
′ 2
EℓSch (r) = M r2 − ℓr + 3M ℓ .


∀r > 2M , 4
r
Three cases are possible :
′
1. EℓSch has two distinct roots rmax
Sch Sch
< rmin Sch
, where rmax Sch
and rmin correspond respectively to the
Sch
maximiser and the minimiser of Eℓ . They are given by
r !
Sch ℓ 12M 2
rmax (ℓ) = 1− 1− ,
2M ℓ
r !
Sch ℓ 12M 2
rmin (ℓ) = 1+ 1− .
2M ℓ

The extremums of EℓSch are given by

8 ℓ − 12M 2
E min (ℓ) = EℓSch (rmin
Sch
)= + Sch (ℓ)
, (A.1)
9 9M rmin

8 ℓ − 12M 2
E max (ℓ) = EℓSch (rmax
Sch
)= + Sch (ℓ)
. (A.2)
9 9M rmax
In fact, we compute
r
2M 4M 2 2 1 12M 2
1 − Sch =1−  = − 1− ,
rmin (ℓ) 3 3 ℓ
q
12M 2
ℓ 1− 1− ℓ

and r !2
ℓ ℓ 12M 2
1 + Sch 2 = 1 + 1+ 1− .
rmin (ℓ) 36M 2 ℓ
Therefore,
  
2M ℓ
EℓSch (rmin
Sch
(ℓ)) = 1− Sch (ℓ)
1+
Sch (ℓ)2
rmin rmin
r ! r !
2 ℓ ℓ 12M 2 2 1 12M 2
= + + 1− − 1−
3 18M 2 18M 2 ℓ 3 3 ℓ
8 ℓ − 12M 2
= + Sch (ℓ)
,
9 9M rmin

where the last expression is obtained by straightforward computations. We do the same thing for
E max (ℓ). This case occurs if and only if ℓ > 12M 2 .
 37

′
2. EℓSch has one double root at rc = 6M . The extremum of EℓSch is given by
8
Eℓc = .
9
This case occur if and only if ℓ = 12M 2 .
′
3. EℓSch has no real roots. Then, EℓSch is monotonically increasing from 0 to +∞. This case occurs
if and only if ℓ < 12M 2 .
We refer to Figure 1 in the introduction for the shape of the potential energy in the three cases.
• Note that by the mass shell condition (2.27), we have

E 2 ≥ EℓSch (r)

for any timelike geodesic moving in the exterior region. In particular,

8
E 2 ≥ E min (ℓ) ≥ .
9
Therefore, we obtain a lower bound on E:
r
8
E≥ .
9

• Now, we claim that the trajectory is a circle of radius r0Sch > 2M if and only if
′
EℓSch (r0Sch ) = E 2 and EℓSch (r0Sch ) = 0. (A.3)

Indeed, if the motion is circular of radius r0Sch , then ∀τ ∈ R : r(τ ) = r0Sch . Thus,

∀τ ∈ R , w(τ ) = ṙ(τ ) = 0 and ẇ(τ ) = 0.

Besides, it is a solution to the system (2.35)-(2.36). Therefore

′ ′
∀τ ∈ R , EℓSch (r0Sch ) = EℓSch (r(τ )) = 0.

By (2.27), we have
∀τ ∈ R , w(τ )2 + EℓSch (r(τ )) = E 2 .
In particular,
EℓSch (r0Sch ) = E 2 .
Now, let us suppose that there exists r0Sch > 2M such that
′
EℓSch (r0Sch ) = E 2 and EℓSch (r0Sch ) = 0.
′
By the above assumption, we have w = 0 and EℓSch (r0Sch ) = 0 so that the point (r0Sch , 0) is a stationary
point ∀ℓ ≥ 12M 2 . Furthermore,
1. r0Sch = 6, if ℓ = 12M 2 ,
2. r0Sch ∈ rmin (ℓ) , if ℓ > 12M 2 .
 Sch Sch

(ℓ), rmax
The circular orbits are thus characterised by (A.3).
• We consider now the case ℓ > 12M 2 and the equation

E 2 = EℓSch (r). (A.4)


8
Let (ℓ, E) ∈ 12M 2 , ∞ × , ∞ . Four cases may occur:
 
9
 38

1. If E 2 = Eℓmin or E 2 = Eℓmax , then rmin

Sch Sch
or rmax satisfy (A.3), so that they are double roots.
Besides, we note that ℓ = ℓub where

12M 2 9 2
ℓub (E) := √ α := E −1 (A.5)
1 − 4α − 8α2 − 8α α2 + α 8

satisfies E 2 = Eℓmin . In fact, we solve the equation below for ℓ

8 ℓ − 12M 2
+   = E2.
9
q
ℓ 12M 2
92 1 + 1 − ℓ

We make the following change of variables

r
9 ℓ − 12M 2
α := E 2 − 1 and X := .
8 ℓ
The equation becomes
X2
= α,
4 (1 + X)
which is easily solvable for X. We can then obtain ℓub (E). Similarly, we obtain ℓlb defined by

12M 2
ℓlb (E) := √ , (A.6)
1 − 4α − 8α2 + 8α α2 + α
which solves the equation E 2 = Eℓmax .
2. If E 2 > Eℓmax , then two cases occur
(a) E 2 < 1. The equation (A.4) has one simple root r2Sch (E, ℓ) > rmin
Sch
(ℓ)
2
(b) E ≥ 1. The equation (A.4) has no positive roots. The trajectories in this case are similar to
the trajectories in case 3 (where ℓ < 12M 2 ).
2
3. E ∈ Eℓmin , Eℓmax . Again, two cases occur
 

(a) E 2 < 1. Then the equation (A.4) admits three simple positive roots riSch (E, ℓ)

r0Sch (E, ℓ) < rmax

Sch
(ℓ) < r1Sch (E, ℓ) < rmin
Sch
(ℓ) < r2Sch (E, ℓ). (A.7)

(b) E 2 ≥ 1. The equation (A.4) admits two simple positive roots riSch (E, ℓ)

r0Sch (E, ℓ) < rmax

Sch
(ℓ) < r1Sch (E, ℓ) < rmin
Sch
(ℓ). (A.8)

• We consider now the case ℓ ≤ 12M 2 . Three cases may occur:

8
1. If E 2 = , then the equation (A.4) has one triple positive root rc = 6M .
9
8
2. If < E 2 < 1, then the equation (A.4) has one simple positive root r1Sch (E, ℓ).
9
3. If E 2 ≥ 1, then the equation (A.4) no positive roots.

• Based on the above cases, we define the following parameters sets

(r ) ( #r " )
8 2
[ 8
, ∞ × 12M 2 , ∞ : E 2 < 1,
 
Acirc := ( , 12M ) (E, ℓ) ∈ ℓ = ℓub (E)
9 9
( #r " ) (A.9)
[ 8
, ∞ × 12M 2, ∞ : ℓ = ℓlb (E) ,
 
(E, ℓ) ∈
9
 39

( #r " )
8
, ∞ × 12M 2 , ∞ : E 2 < 1,
 
Abound := (E, ℓ) ∈ ℓlb (E) < ℓ < ℓub (E) , (A.10)
9
( #r " )
8
, ∞ × 12M 2 , ∞ : E 2 ≥ 1,
 
Aunbound := (E, ℓ) ∈ ℓ > ℓlb (E) , (A.11)
9

( #r " ) ( #r " )
8 [ 8
, ∞ × 12M 2 , ∞ : E 2 < 1, 2
   
Aabs := (E, ℓ) ∈ ℓ < ℓlb (E) (E, ℓ) ∈ , 1 × 0, 12M
9 9
( #r " )
[ 8
E∈ ,1 , ℓ = 12M 2 .
9
(A.12)
• Now, we determine the nature of orbits (circular, bounded, unbounded, "absorbed by the black hole")
in terms of the parameters (E, ℓ) as well as the initial position and velocity. Let ℓ ∈ [0, ∞[, r̃ ∈]2M, ∞[
and w̃ ∈ R. We compute q
E= EℓSch (r̃) + w̃2 .

1. If (E, ℓ) ∈ Abound , then there exists riSch := riSch (E, ℓ), i ∈ {0, 1, 2} solutions of (A.4) and satisfying
(A.7). Now recall that
EℓSch (r) ≤ E 2 , ∀r > 2M. (A.13)
Sch Sch Sch
 
This implies that r̃ must lie in the region 2M, r0 ∪ [r1 , r2 ]. Two cases are possible
– either the geodesic starts at r0Sch and reaches the horizon r = 2M in a finite proper time,
– or the geodesic is trapped between r1Sch and r2Sch .
2. If (E, ℓ) ∈ Aunbound , then there exists riSch := riSch (E, ℓ), i ∈ {0, 1, } solutions of (A.4) and
satisfying (A.8). By (A.13), r̃ must lie in the region ]2M, r0Sch ] ∪ [r1Sch , ∞[. Therefore, two cases
are possible
– either the geodesic starts at r0Sch and reaches the horizon r = 2M in a finite proper time,
– or the geodesic stars from infinity, hits the potential barrier at r1Sch and comes back to infinity.
3. If (E, ℓ) ∈ Aabs , then the equation (A.4) has at most one positive root r0Sch . Two cases are possible:
– The geodesic starts at r0Sch and reaches the horizon in a finite proper time.
– The geodesic starts from infinity and reaches the horizon in a finite time.
r
8
4. If (E, ℓ) ∈ Acirc , then the equation (A.4) admits one triple root if E = or a double root. In
9
this case, the geodesic is a circle.
References 40

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