where , , , . These are called Schwarzschild coordinates. Asymptotic flatness is expressed by the boundary conditions

A regular centre is also required and is guaranteed by the boundary condition

With

as spatial coordinate and

as momentum coordinates the, Einstein-Vlasov system reads

The matter quantities are defined by

Let us point out that this system is not the full
Einstein-Vlasov system. The remaining field equations, however,
can be derived from these equations. See [62] and the erratum [65] for more details. Let the square of the angular momentum be
denoted by
*L*, i.e.

A consequence of spherical symmetry is that angular momentum is conserved along the characteristics of (30). Introducing the variable

the Vlasov equation for
*f*
=
*f*
(*t*,
*r*,
*w*,
*L*) becomes

where

The matter terms take the form

Let us write down a couple of known facts about the system (31, 32, 35, 36, 37). A solution to the Vlasov equation can be written as

where
*R*
and
*W*
are solutions to the characteristic system

such that the trajectory (*R*
(*s*,
*t*,
*r*,
*w*,
*L*),
*W*
(*s*,
*t*,
*r*,
*w*,
*L*),
*L*) goes through the point (*r*,
*w*,
*L*) when
*s*
=
*t*
. This representation shows that
*f*
is nonnegative for all
and that
. There are two known conservation laws for the Einstein-Vlasov
system: conservation of the number of particles,

and conservation of the ADM mass

Let us now review the global results concerning the Cauchy problem that have been proved for the spherically symmetric Einstein-Vlasov system. As initial data we take a spherically symmetric, nonnegative, and continuously differentiable function with compact support that satisfies

This condition guarantees that no trapped surfaces are present
initially. In [62] it is shown that for such an initial datum there exists a
unique, continuously differentiable solution
*f*
with
on some right maximal interval [0,
*T*). If the solution blows up in finite time, i.e. if
, then
becomes unbounded as
. Moreover, a continuation criterion is shown that says that a
local solution can be extended to a global one provided the
*v*
-support of
*f*
can be bounded on [0,
*T*). (In [62] they chose to work in the momentum variable
*v*
rather than
*w*,
*L*
.) This is analogous to the situation for the Vlasov-Maxwell
system where the function
*Q*
(*t*) was introduced for the
*v*
-support. A control of the
*v*
-support immediately implies that
and
*p*
are bounded in view of (33) and (34). In the Vlasov-Maxwell case the field equations have a
regularizing effect in the sense that derivatives can be
expressed through spatial integrals, and it follows [34] that the derivatives of
*f*
also can be bounded if the
*v*
-support is bounded. For the Einstein-Vlasov system such a
regularization is less clear, since
depends on
in a pointwise manner. However, certain combinations of second
and first order derivatives of the metric components can be
expressed in terms of matter components only, without derivatives
(a consequence of the geodesic deviation equation). This fact
turns out to be sufficient for obtaining bounds also on the
derivatives of
*f*
(see [62] for details). By considering initial data sufficiently close to
zero, Rein and Rendall show that the
*v*
-support is bounded on [0,
*T*), and the continuation criterion then implies that
. It should be stressed that even for small data no global
existence result like this one is known for any other
phenomenological matter model coupled to Einstein's equations.
The resulting spacetime in [62] is geodesically complete, and the components of the energy
momentum tensor as well as the metric quantities decay with
certain algebraic rates in
*t*
. The mathematical method used by Rein and Rendall is inspired by
the analogous small data result for the Vlasov-Poisson equation
by Bardos and Degond [10]. This should not be too surprising since for small data the
gravitational fields are expected to be small and a Newtonian
spacetime should be a fair approximation. In this context we
point out that in [63] it is proved that the Vlasov-Poisson system is indeed the
nonrelativistic limit of the spherically symmetric
Einstein-Vlasov system, i.e. the limit when the speed of light
. (In [71] this issue is studied in the asymptotically flat case without
symmetry assumptions.) Finally, we mention that there is an
analogous small data result using a maximal time
coordinate [76] instead of a Schwarzschild time coordinate.

The case with general data is more subtle. Rendall has
shown [70] that there exist data leading to singular spacetimes as a
consequence of Penrose's singularity theorem. This raises the
question of what we mean by global existence for such data. The
Schwarzschild time coordinate is expected to avoid the
singularity, and by global existence we mean that solutions
remain smooth as Schwarzschild time tends to infinity. Even
though spacetime might be only partially covered in Schwarzschild
coordinates, a global existence theorem for general data would
nevertheless be very important since weak cosmic censorship would
follow from it. A partial investigation for general data was done
in [67], where it is shown that if singularities form in finite
Schwarzschild time the first one must be at the centre. More
precisely, if
*f*
(*t*,
*r*,
*w*,
*L*)=0 when
for some
, and for all
*t*,
*w*
and
*L*, then the solution remains smooth for all time. This rules out
singularities of the shell-crossing type, which can be an
annoying problem for other matter models (e.g. dust). The main
observation in [67] is a cancellation property in the term

in the characteristic equation (40). We refer to the original paper for details. In [68] a numerical study was undertaken. A numerical scheme originally used for the Vlasov-Poisson system was modified to the spherically symmetric Einstein-Vlasov system. It has been shown by Rodewis [79] that the numerical scheme has the desirable convergence properties. (In the Vlasov-Poisson case convergence was proved in [81]. See also [28].) The numerical experiments support the conjecture that solutions are singularity-free. This can be seen as evidence that weak cosmic censorship holds for collisionless matter. It may even hold in a stronger sense than in the case of a massless scalar field (see [20, 22]). There may be no naked singularities formed for any regular initial data rather than just for generic data. This speculation is based on the fact that the naked singularities that occur in scalar field collapse appear to be associated with the existence of type II critical collapse, while Vlasov matter is of type I. This is indeed the primary goal of their numerical investigation: to analyze critical collapse and decide whether Vlasov matter is type I or type II.

These different types of matter are defined as follows. Given
small initial data no black holes are expected to form and matter
will disperse (which has been proved for a scalar field [19] and for Vlasov matter [62]). For large data, black holes will form and consequently there
is a transition regime separating dispersion of matter and
formation of black holes. If we introduce a parameter
*A*
on the initial data such that for small
*A*
dispersion occurs and for large
*A*
a black hole is formed, we get a critical value
separating these regions. If we take
and denote by
the mass of the black hole, then if
as
, we have type II matter, whereas for type I matter this limit is
positive and there is a mass gap. For more information on
critical collapse we refer to the review paper by Gundlach [38].

For Vlasov matter there is an independent numerical simulation by Olabarrieta and Choptuik [53] (using a maximal time coordinate) and their conclusion agrees with the one in [68]. Critical collapse is related to self similar solutions; Martin-Garcia and Gundlach [52] have presented a construction of such solutions for the massless Einstein-Vlasov system by using a method based partially on numerics. Since such solutions often are related to naked singularities, it is important to note that their result is for the massless case (in which case there is no known analogous result to the small data theorem in [62]) and their initial data are not in the class that we have described above.

The Einstein-Vlasov System/Kinetic Theory
Håkan Andréasson
http://www.livingreviews.org/lrr-2002-7
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