In this section we consider a self-gravitating collision-less gas in the framework of general relativity and we
present the Einstein–Vlasov system. It is most often the case in the mathematics literature that the speed of
light and the gravitational constant
are normalized to one, but we keep these constants in the
formulas in this section since in some problems they do play an important role. However, in most
of the problems discussed in the forthcoming sections these constants will be normalized to
one.
Let be a four-dimensional manifold and let
be a metric with Lorentz signature
so that
is a spacetime. The metric is assumed to be time-orientable so that there is a distinction
between future and past directed vectors.
The possible values of the four-momentum of a particle with rest mass
belong to the mass
shell
, defined by
Since we are considering a collisionless gas, the particles follow the geodesics in spacetime. The
geodesics are projections onto spacetime of the curves in defined in local coordinates by
which implies that
This is accordingly the Vlasov equation. We point out that sometimes the density function is considered as a function on the entire tangent bundle In order to write down the Einstein–Vlasov system we need to know the energy-momentum tensor
in terms of
and
. We define
is the induced metric of the submanifold , and that
is invariant under Lorentz
transformations of the tangent space, and it is often the case in the literature that
is written
as
Let us now consider a collisionless gas consisting of particles with different rest masses
, described by
density functions
. Then the Vlasov equations for
the different density functions
, together with the Einstein equations,
form the Einstein–Vlasov system for the collision-less gas. Here is the Ricci tensor,
is the scalar
curvature and
is the cosmological constant.
Henceforth, we always assume that there is only one species of particles in the gas and we write for
its energy momentum tensor. Moreover, in what follows, we normalize the rest mass
of the particles,
the speed of light
, and the gravitational constant
, to one, if not otherwise explicitly stated that this
is not the case.
Let us now investigate the features of the energy momentum tensor for Vlasov matter. We define the particle current density
Using normal coordinates based at a given point and assuming that is compactly supported, it is not
hard to see that
is divergence-free, which is a necessary compatibility condition since the left-hand
side of (2) is divergence-free by the Bianchi identities. A computation in normal coordinates also shows that
is divergence-free, which expresses the fact that the number of particles is conserved. The definitions
of
and
immediately give us a number of inequalities. If
is a future-directed time-like or
null vector then we have
with equality if and only if
at the given point. Hence,
is
always future-directed time-like, if there are particles at that point. Moreover, if
and
are
future-directed time-like vectors then
, which is the dominant energy condition.
This also implies that the weak energy condition holds. If
is a space-like vector, then
. This is called the non-negative pressure condition, and it implies that the strong energy
condition holds as well. That the energy conditions hold for Vlasov matter is one reason that the
Vlasov equation defines a well-behaved matter model in general relativity. Another reason is
the well-posedness theorem by Choquet-Bruhat [55
] for the Einstein–Vlasov system that we
state below. Before stating that theorem we first discuss the conditions imposed on the initial
data.
The initial data in the Cauchy problem for the Einstein–Vlasov system consist of a 3-dimensional
manifold , a Riemannian metric
on
, a symmetric tensor
on
, and a non-negative
scalar function
on the tangent bundle
of
.
The relationship between a given initial data set on
and the metric
on the
spacetime manifold is, that there exists an embedding
of
into the spacetime such that the induced
metric and second fundamental form of
coincide with the result of transporting
with
.
For the relation of the distribution functions
and
we have to note that
is defined on the
mass shell. The initial condition imposed is that the restriction of
to the part of the mass
shell over
should be equal to
, where
sends each point of
the mass shell over
to its orthogonal projection onto the tangent space to
. An
initial data set for the Einstein–Vlasov system must satisfy the constraint equations, which read
Theorem 1 Let be a 3-dimensional manifold,
a smooth Riemannian metric on
,
a
smooth symmetric tensor on
and
a smooth non-negative function of compact support on the
tangent bundle
of
. Suppose that these objects satisfy the constraint equations (33
, 34
). Then
there exists a smooth spacetime
, a smooth distribution function
on the mass shell of this
spacetime, and a smooth embedding
of
into
, which induces the given initial data on
such
that
and
satisfy the Einstein–Vlasov system and
is a Cauchy surface. Moreover, given
any other spacetime
, distribution function
and embedding
satisfying these conditions,
there exists a diffeomorphism
from an open neighborhood of
in
to an open neighborhood of
in
, which satisfies
and carries
and
to
and
,
respectively.
The above formulation is in the case of smooth initial data; for information on the regularity
needed on the initial data we refer to [55] and [118]. In this context we also mention that local
existence has been proven for the Yang–Mills–Vlasov system in [56], and that this problem for the
Einstein–Maxwell–Boltzmann system is treated in [30]. However, this result is not complete, as the
non-negativity of
is left unanswered. Also, the hypotheses on the scattering kernel in this work leave
some room for further investigation. The local existence problem for physically reasonable assumptions on
the scattering kernel does not seem well understood in the context of the Einstein–Boltzmann system, and a
careful study of this problem would be desirable. The mathematical study of the Einstein–Boltzmann
system has been very sparse in the last few decades, although there has been some activity in recent years.
Since most questions on the global properties are completely open let us only very briefly mention some of
these works. Mucha [117] has improved the regularity assumptions on the initial data assumed
in [30]. Global existence for the homogeneous Einstein–Boltzmann system in Robertson–Walker
spacetimes is proven in [125], and a generalization to Bianchi type I symmetry is established
in [124].
In the following sections we present results on the global properties of solutions of the Einstein–Vlasov system, which have been obtained during the last two decades.
Before ending this section we mention a few other sources for more background on the Einstein–Vlasov
system, cf. [156, 158, 73, 176].
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