## 2 The Einstein–Vlasov System

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

Hence, if , is the set of all future-directed time-like vectors with length , and if it is the set of all future-directed null vectors. On we take and (letters in the beginning of the alphabet always take values and letters in the middle take ) as local coordinates, and is expressed in terms of and the metric in view of Equation (25). Thus, the density function is a non-negative function on . Below we drop the index on and simply write .

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

Here are the Christoffel symbols. Along a geodesic the density function is invariant so that

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 rather than on the mass shell . The Vlasov equation for then takes the form
This equation follows from (26) if we take the mass shell condition into account. Indeed, by abuse of notation, we have
Here is considered as a function on in the left-hand side, and on in the right-hand side. From the mass shell condition we derive
Inserting these relations into (26) we obtain (27). If we let , and divide the Vlasov equation (26) by we obtain the most common form in the literature of the Vlasov equation
In a fixed spacetime the Vlasov equation (28) is a linear hyperbolic equation for and we can solve it by solving the characteristic system,
In terms of initial data the solution of the Vlasov equation can be written as
where and solve Equations (29, 30), and where

In order to write down the Einstein–Vlasov system we need to know the energy-momentum tensor in terms of and . We define

where, as usual, , and denotes the absolute value of the determinant of . We remark that the measure

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

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 )
We can now state the local existence theorem by Choquet-Bruhat [55], for the Einstein–Vlasov system.

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].