The formation of black holes in spherically symmetric gravitational collapse

Abstract

We consider the spherically symmetric, asymptotically flat Einstein–Vlasov system. We find explicit conditions on the initial
data, with ADM mass M, such that the resulting spacetime has the following properties: there is a family of radially outgoing null geodesics where
the area radius r along each geodesic is bounded by 2M, the timelike lines r=c Î [0,2M]{r=c\in [0,2M]} are incomplete, and for r>2M the metric converges asymptotically to the Schwarzschild metric with mass M. The initial data that we construct guarantee the formation of a black hole in the evolution. We give examples of such initial
data with the additional property that the solutions exist for all r≥ 0 and all Schwarzschild time, i.e., we obtain global existence in Schwarzschild coordinates in situations where the initial
data are not small. Some of our results are also established for the Einstein equations coupled to a general matter model
characterized by conditions on the matter quantities.

Full text

Page 1

arXiv:0706.3787v3 [gr-qc] 5 Mar 2008
The formation of black holes in spherically
symmetric gravitational collapse
H˚ akan Andr´ easson
Mathematical Sciences
Chalmers University of Technology
G¨ oteborg University
S-41296 G¨ oteborg, Sweden
email: hand@math.chalmers.se
Markus Kunze
Fachbereich Mathematik
Universit¨ at Duisburg-Essen
D-45117 Essen, Germany
email: markus.kunze@uni-due.de
Gerhard Rein
Mathematisches Institut der Universit¨ at Bayreuth
D-95440 Bayreuth, Germany
email: gerhard.rein@uni-bayreuth.de
March 5, 2008
Abstract
We consider the spherically symmetric,
Einstein-Vlasov system. We find explicit conditions on the initial data,
with ADM mass M, such that the resulting spacetime has the following
properties: there is a family of radially outgoing null geodesics where
the area radius r along each geodesic is bounded by 2M, the time-
like lines r = c ∈ [0,2M] are incomplete, and for r > 2M the metric
converges asymptotically to the Schwarzschild metric with mass M.
The initial data that we construct guarantee the formation of a black
hole in the evolution. We also give examples of such initial data with
the additional property that the solutions exist for all r ≥ 0 and all
Schwarzschild time, i.e., we obtain global existence in Schwarzschild
coordinates in situations where the initial data are not small. Some
asymptotically flat
1

Page 2

of our results are also established for the Einstein equations coupled
to a general matter model characterized by conditions on the matter
quantities.
1Introduction
One of the many striking predictions of General Relativity is the assertion
that under appropriate conditions astrophysical objects like stars or galaxies
undergo a gravitational collapse resulting in a spacetime singularity. This
was first proven by Oppenheimer and Snyder [19] who constructed a semi-
explicit example of a homogeneous spherically symmetric ball of dust, i.e.,
of a pressure-less fluid, which under its self-consistent, general relativistic
gravitational interaction collapses. During this collapse the scalar curvature
of spacetime blows up at the centre of symmetry, and the geometry of space-
time breaks down there. This is referred to as the formation of a spacetime
singularity. An important feature of the Oppenheimer-Snyder solution is
that during the collapse a two-dimensional spacelike sphere evolves which
encloses the singularity and through which no causal curve, i.e., no light
ray or particle trajectory, can pass outward. In this way the spacetime sin-
gularity is isolated from the outside part of spacetime by a so-called event
horizon, and the singularity cannot be seen or in any other way be experi-
enced by observers outside the event horizon. This configuration was later
termed a black hole.
In the 1960s Penrose [20] proved that the formation of spacetime singu-
larities from regular initial data is not restricted to spherically symmetric,
especially constructed or isolated examples but is a genuine, stable feature
of spacetimes. However, this result gives little information about the geo-
metric structure of a spacetime with such a singularity. In particular, it is
in general not known if every spacetime singularity which arises from the
gravitational collapse of regular initial data is covered by an event horizon.
Since the existence of so-called naked singularities (for which, by definition,
the latter is not true) would violate predictability (it would not be possible
to predict from the initial data what an observer would see if he could ob-
serve a singularity), the cosmic censorship conjecture was formulated which
demands that any singularity which arises from the gravitational collapse of
generic regular initial data is indeed hidden behind an event horizon. The
restriction to generic data means that naked singularities are allowed to oc-
cur for a “null set” of the initial data. An important example where naked
singularities do form for a null set, but for which cosmic censorship holds
2

Page 3

true, is the spherically symmetric Einstein-scalar field system, cf. [10, 11].
Actually the above is an informal statement of the so-called weak cosmic
censorship conjecture [34, 12.1]; we will not be concerned with the strong
version in the present paper. For a mathematical discussion and the defini-
tion of the weak cosmic censorship conjecture we refer to [12].
To deal with this conjecture in full generality is out of reach of the
present level of mathematics, but under the assumption of spherical sym-
metry progress has been made in recent years. One important outcome of
these investigations is that the answer is sensitive to which model is chosen
to describe the matter. Christodoulou [7] showed that for dust, i.e., the
matter model used by Oppenheimer and Snyder, cosmic censorship is vio-
lated. On the other hand, in a series of papers Christodoulou investigated
a massless scalar field as matter model and showed in 1999 that weak and
strong cosmic censorship hold true for this matter model; see [11] and the
references therein.
In the present investigation the main example considered as a matter
model is the so-called collisionless gas as described by the Vlasov equation.
It is used extensively in astrophysics, cf. [6], to describe galaxies or globular
clusters which are viewed as large ensembles of mass points which interact
only through the gravitational field that the ensemble creates collectively.
In a relativistic context this leads to the Einstein-Vlasov system. All results
available for this system support the following
Conjecture: Weak cosmic censorship holds for the Einstein-Vlasov system.
We mention explicitly that, in contrast to dust, small, spherically symmet-
ric initial data launch global solutions, i.e., the solutions are geodesically
complete and hence satisfy cosmic censorship, cf. [25]. Also, the numerical
simulations [5, 18, 29] which treat large initial data support the hypothesis
that naked singularities do not form in the evolution. We point out a further
interesting feature of Vlasov matter observed in these numerical studies: In
a one-parameter family of solutions which for large parameters, i.e., large
amplitudes of the initial data, collapse to a black hole the smallest black hole
always has a strictly positive ADM mass, i.e., there is a mass gap. For some
other models, e.g. a scalar field, the mass of the black hole as a function of
the parameter is continuous and arbitrarily small black holes can form, cf.
[15] for a review.
The aim of the present paper is to find explicit conditions on the initial
data which ensure the formation of black holes. This class of initial data
has the important property that, except for “boundary cases”, properly re-
stricted small perturbations of the data lead to solutions with the same
3

Page 4

properties. In this sense the established behaviour of the solutions is stable
and not restricted to especially constructed solutions or initial data, respec-
tively. It turns out that some of our results can be formulated for a general
matter model which satisfies certain specific assumptions, and in order to
give a broader impact to our results we shall do so. At the same time we
emphasize that the Vlasov matter model is the only one which is presently
known to actually satisfy all the assumptions needed for our arguments to
go through.
As an interesting corallary to our main result we show that it is in fact
possible to choose initial data for the Einstein-Vlasov system, which lead to
formation of black holes, such that the solutions exist for all Schwarzschild
time and all r ≥ 0. We thus obtain global existence in Schwarzschild coordi-
nates for initial data which are not small, and to the best of our knowledge
this is the first global existence result in Schwarzschild coordinates for initial
data which lead to gravitational collapse and formation of black holes.
One aspect of our result is that there is a set of initial data which leads
to gravitational collapse such that weak cosmic censorship holds. This point
should be related to an earlier result by Rendall [32], where it is shown that
there exist initial data for the spherically symmetric Einstein-Vlasov system
such that a trapped surface forms in the evolution. The occurrence of a
trapped surface signals the formation of an event horizon. Indeed, Dafermos
[13] has proved that if a spherically symmetric spacetime contains a trapped
surface and the matter model satisfies certain hypotheses then weak cosmic
censorship holds true. In [14] it was then shown that Vlasov matter does
satisfy the required hypotheses. Hence, by combining these results it follows
that initial data exist which lead to gravitational collapse and for which weak
cosmic censorship holds. However, the proof in [32] rests on a continuity
argument, and it is not possible to tell whether or not a given initial data
set will give rise to a black hole. This is in contrast to the explicit conditions
on the initial data, together with the detailed asymptotic structure, that we
obtain in the present work. In this regard it is natural to relate our results
to those of Christodoulou on the spherically symmetric Einstein-scalar field
system [8] and [9]. In [8] it is shown that if the final Bondi mass M is
different from zero, the region exterior to the sphere r = 2M tends to the
Schwarzschild metric with mass M. In Theorem 2.4 below we show that
solutions of the spherically Einstein-Vlasov system, under certain conditions
on the initial data, also converge to the Schwarzschild metric asymptotically.
Furthermore, in [9] explicit conditions on the initial data are specified which
guarantee the formation of trapped surfaces. This paper played a crucial
role in Christodoulou’s proof [11] of the weak and strong cosmic censorship
4

Page 5

conjectures mentioned above. The conditions on the initial data in [9] allow
the ratio of the Hawking mass and the area radius to cover the full range,
i.e., 2m/r ∈ (0,1), whereas our conditions always require 2m/r to be quite
close to one. However, we believe that to understand gravitational collapse
in the case of Vlasov matter the essential situation is when 2m/r is large.
We thus hope that the results in the present paper will lead to progress
on the general understanding of gravitational collapse and the weak cosmic
censorship conjecture in the case of Vlasov matter.
The Vlasov matter model has a further property to recommend it when
compared to other matter models. For the Vlasov-Poisson system, which
arises as the Newtonian limit of the Einstein-Vlasov system in a rigorous
sense [26, 31], and which is used extensively in astrophysics, there is a global
existence and uniqueness result for general, smooth initial data [17, 21]. This
means in particular that any breakdown of a solution of the Einstein-Vlasov
system can be expected to be a genuine, general relativistic effect such as a
spacetime singularity and not only remainder of some bad behaviour which
the matter model exhibits already on the Newtonian level.
To be more specific, consider now a smooth spacetime manifold M
equipped with a spacetime metric gαβ; Greek indices run from 0 to 3. Then
the Einstein equations read
Gαβ= 8πTαβ,(1.1)
where Gαβis the Einstein tensor, a non-linear second order differential ex-
pression in the metric gαβ, and Tαβis the energy-momentum tensor given
by the matter content (or other fields) of the spacetime. To obtain a closed
system, the field equations (1.1) have to be supplemented by
evolution equation(s) for the matter(1.2)
and
the definition of Tαβin terms of the matter and the metric. (1.3)
It is often possible to specify conditions on (1.2) and (1.3) under which one
can establish geometric properties of a spacetime described by the Einstein-
matter system (1.1), (1.2), (1.3). The Penrose singularity theorem men-
tioned above is of this nature, and part of our arguments will also be pre-
sented in this form.
However, in order to verify such general conditions, in particular with
respect to the existence of local or global solutions to the corresponding
5