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  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  for the Einstein-Vlasov system.
be a 3-dimensional manifold,
a smooth Riemannian metric on
a smooth symmetric tensor on
a smooth non-negative function of compact support on the tangent
. Suppose that these objects satisfy the constraint
equations (26) and (27). Then there exists a smooth spacetime
, a smooth distribution function
on the mass shell of this spacetime, and a smooth embedding
which induces the given initial data on
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
to an open neighbourhood of
' which satisfies
In this context we also mention that local existence has been proved for the Einstein-Maxwell-Boltzmann system  and for the Yang-Mills-Vlasov system .
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, ). 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
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