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  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 , and that this problem for the Einstein–Maxwell–Boltzmann system is treated in . 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, ). 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.
© Max Planck Society and the author(s)