Let
*M*
be a four-dimensional manifold and let
be a metric with Lorentz signature (-,+,+,+) so that
is a spacetime. We use the abstract index notation, which means
that
is a geometric object and not the components of a tensor.
See [89] for a discussion on this notation. The metric is assumed to be
time-orientable so that there is a distinction between future and
past directed vectors. The worldline of a particle with non-zero
rest mass
*m*
is a timelike curve and the unit future-directed tangent vector
to this curve is the four-velocity of the particle. The
four-momentum
is given by
. We assume that all particles have equal rest mass
*m*
and we normalize so that
*m*
=1. One can also consider massless particles but we will rarely
discuss this case. The possible values of the four-momentum are
all future-directed unit timelike vectors and they constitute a
hypersurface
*P*
in the tangent bundle
*TM*, which is called the mass shell. The distribution function
*f*
that we introduced in the previous sections is a non-negative
function on
*P*
. Since we are considering a collisionless gas, the particles
travel along geodesics in spacetime. The Vlasov equation is an
equation for
*f*
that exactly expresses this fact. To get an explicit expression
for this equation we introduce local coordinates on the mass
shell. We choose local coordinates on
*M*
such that the hypersurfaces
constant are spacelike so that
*t*
is a time coordinate and
,
*j*
=1,2,3, are spatial coordinates (letters in the beginning of the
alphabet always take values 0,1,2,3 and letters in the middle
take 1,2,3). A timelike vector is future directed if and only if
its zero component is positive. Local coordinates on
*P*
can then be taken as
together with the spatial components of the four-momentum
in these coordinates. The Vlasov equation then reads

Here
*a*,
*b*
=0,1,2,3 and
*j*
=1,2,3, and
are the Christoffel symbols. It is understood that
is expressed in terms of
and the metric
using the relation
(recall that
*m*
=1).

In a fixed spacetime the Vlasov equation is a linear
hyperbolic equation for
*f*
and we can solve it by solving the characteristic system,

In terms of initial data the solution to the Vlasov equation can be written as

where and solve (23) and (24), and where and .

In order to write down the Einstein-Vlasov system we need to
define the energy-momentum tensor
in terms of
*f*
and
. In the coordinates
on
*P*
we define

where as usual
and |
*g*
| denotes the absolute value of the determinant of
*g*
. Equation (22) together with Einstein's equations

then form the Einstein-Vlasov system. Here
is the Einstein tensor,
the Ricci tensor and
*R*
is the scalar curvature. We also define the particle current
density

Using normal coordinates based at a given point and assuming
that
*f*
is compactly supported it is not hard to see that
is divergence-free which is a necessary compatability condition
since
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 timelike or null vector then we have
with equality if and only if
*f*
=0 at the given point. Hence
is always future directed timelike if there are particles at
that point. Moreover, if
and
are future directed timelike vectors then
, which is the dominant energy condition. If
is a spacelike vector then
. This is called the non-negative pressure condition. These last
two conditions together with the Einstein equations imply that
for any timelike vector
, which is the strong energy condition. 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 for the Einstein-Vlasov system that we will state
below. Before stating that theorem we will first discuss the
initial conditions imposed.

The data in the Cauchy problem for the Einstein-Vlasov system
consist of the induced Riemannian metric
on the initial hypersurface
*S*, the second fundamental form
of
*S*
and matter data
. The relations between a given initial data set
on a three-dimensional manifold
*S*
and the metric
on the spacetime manifold is that there exists an embedding
of
*S*
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
*f*
and
we have to note that
*f*
is defined on the mass shell. The initial condition imposed is
that the restriction of
*f*
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

Here
and
, where
is the future directed unit normal vector to the initial
hypersurface and
is the orthogonal projection onto the tangent space to the
initial hypersurface. In terms of
we can express
and
by (
satisfies
so it can naturally be identified with a vector intrinsic to
*S*)

Here
is the determinant of the induced Riemannian metric on
*S*
. We can now state the local existence theorem by
Choquet-Bruhat [17] for the Einstein-Vlasov system.

**Theorem 1**
*Let
S
be a 3-dimensional manifold,
a smooth Riemannian metric on
S,
a smooth symmetric tensor on
S
and
a smooth non-negative function of compact support on the tangent
bundle
TS
of
S
. Suppose that these objects satisfy the constraint
equations (26) and (27). Then there exists a smooth spacetime
, a smooth distribution function
f
on the mass shell of this spacetime, and a smooth embedding
of
S
into
M
which induces the given initial data on
S
such that
and
f
satisfy the Einstein-Vlasov system and
is a Cauchy surface. Moreover, given any other spacetime
, distribution function
f
' and embedding
satisfying these conditions, there exists a diffeomorphism
from an open neighbourhood of
in
M
to an open neighbourhood of
in
M
' which satisfies
and carries
and
f
to
and
f
', respectively.*

In this context we also mention that local existence has been proved for the Einstein-Maxwell-Boltzmann system [9] and for the Yang-Mills-Vlasov system [18].

A main theme in the following sections is to discuss special cases for which the local existence theorem can be extended to a global one. There are interesting situations when this can be achieved, and such global existence theorems are not known for Einstein's equations coupled to other forms of phenomenological matter models, i.e. fluid models (see, however, [21]). In this context it should be stressed that the results in the previous sections show that the mathematical understanding of kinetic equations on a flat background space is well-developed. On the other hand the mathematical understanding of fluid equations on a flat background space (also in the absence of a Newtonian gravitational field) is not that well-understood. It would be desirable to have a better mathematical understanding of these equations in the absence of gravity before coupling them to Einstein's equations. This suggests that the Vlasov equation is natural as matter model in mathematical general relativity.

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