Let
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
[117]
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
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
and we normalize so that
. 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
in the tangent bundle
, which is called the mass shell. The distribution function
that we introduced in the previous sections is a non-negative
function on
. Since we are considering a collisionless gas, the particles
travel along geodesics in spacetime. The Vlasov equation is an
equation for
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
such that the hypersurfaces
constant are spacelike so that
is a time coordinate and
,
, are spatial coordinates (letters in the beginning of the
alphabet always take values
and letters in the middle take
). A timelike vector is future directed if and only if its zero
component is positive. Local coordinates on
can then be taken as
together with the spatial components of the four-momentum
in these coordinates. The Vlasov equation then reads
In a fixed spacetime the Vlasov equation is a
linear hyperbolic equation for
and we can solve it by solving the characteristic system,
In order to write down the Einstein-Vlasov
system we need to define the energy-momentum tensor
in terms of
and
. In the coordinates
on
we define
where as usual
, and
denotes the absolute value of the determinant of
. Equation (25
) together with Einstein’s equations,
then form the Einstein-Vlasov system. Here
is the Einstein tensor,
the Ricci tensor,
is the scalar curvature and
is the cosmological constant. In most of this review we will
assume that
, but Section
2.3
is devoted to the case of non-vanishing cosmological constant
(where also the case of adding a scalar field is discussed). We
now 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 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
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
, the second fundamental form
of
and matter data
. The relations between a given initial data set
on a three-dimensional manifold
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
(
29
,
30
).
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 neighbourhood of
in
to an open neighbourhood of
in
which satisfies
and carries
and
to
and
, respectively.
In this context we also mention that local
existence has been proved for the Yang-Mills-Vlasov system
in
[26], and that this problem for the Einstein-Maxwell-Boltzmann system
is treated in
[11]
. This result is however not complete, the non-negativity of
is left unanswered. Also, the hypotheses on the scattering
kernel in this work leave some room for further investigation.
This problem concerning physically reasonable assumptions on the
scattering kernel seems not well understood in the context of the
Einstein-Boltzmann system, and a careful study of this issue
would be desirable.
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, [30]). 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 satisfying. 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.
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