where , , , . These are called Schwarzschild coordinates. Asymptotic flatness is expressed by the boundary conditions
A regular centre is also required and is guaranteed by the boundary condition
as spatial coordinate and
as momentum coordinates the, Einstein-Vlasov system reads
The matter quantities are defined by
Let us point out that this system is not the full Einstein-Vlasov system. The remaining field equations, however, can be derived from these equations. See  and the erratum  for more details. Let the square of the angular momentum be denoted by L, i.e.
A consequence of spherical symmetry is that angular momentum is conserved along the characteristics of (30). Introducing the variable
the Vlasov equation for f = f (t, r, w, L) becomes
The matter terms take the form
Let us write down a couple of known facts about the system (31, 32, 35, 36, 37). A solution to the Vlasov equation can be written as
where R and W are solutions to the characteristic system
such that the trajectory (R (s, t, r, w, L), W (s, t, r, w, L), L) goes through the point (r, w, L) when s = t . This representation shows that f is nonnegative for all and that . There are two known conservation laws for the Einstein-Vlasov system: conservation of the number of particles,
and conservation of the ADM mass
Let us now review the global results concerning the Cauchy problem that have been proved for the spherically symmetric Einstein-Vlasov system. As initial data we take a spherically symmetric, nonnegative, and continuously differentiable function with compact support that satisfies
This condition guarantees that no trapped surfaces are present initially. In  it is shown that for such an initial datum there exists a unique, continuously differentiable solution f with on some right maximal interval [0, T). If the solution blows up in finite time, i.e. if , then becomes unbounded as . Moreover, a continuation criterion is shown that says that a local solution can be extended to a global one provided the v -support of f can be bounded on [0, T). (In  they chose to work in the momentum variable v rather than w, L .) This is analogous to the situation for the Vlasov-Maxwell system where the function Q (t) was introduced for the v -support. A control of the v -support immediately implies that and p are bounded in view of (33) and (34). In the Vlasov-Maxwell case the field equations have a regularizing effect in the sense that derivatives can be expressed through spatial integrals, and it follows  that the derivatives of f also can be bounded if the v -support is bounded. For the Einstein-Vlasov system such a regularization is less clear, since depends on in a pointwise manner. However, certain combinations of second and first order derivatives of the metric components can be expressed in terms of matter components only, without derivatives (a consequence of the geodesic deviation equation). This fact turns out to be sufficient for obtaining bounds also on the derivatives of f (see  for details). By considering initial data sufficiently close to zero, Rein and Rendall show that the v -support is bounded on [0, T), and the continuation criterion then implies that . It should be stressed that even for small data no global existence result like this one is known for any other phenomenological matter model coupled to Einstein's equations. The resulting spacetime in  is geodesically complete, and the components of the energy momentum tensor as well as the metric quantities decay with certain algebraic rates in t . The mathematical method used by Rein and Rendall is inspired by the analogous small data result for the Vlasov-Poisson equation by Bardos and Degond . This should not be too surprising since for small data the gravitational fields are expected to be small and a Newtonian spacetime should be a fair approximation. In this context we point out that in  it is proved that the Vlasov-Poisson system is indeed the nonrelativistic limit of the spherically symmetric Einstein-Vlasov system, i.e. the limit when the speed of light . (In  this issue is studied in the asymptotically flat case without symmetry assumptions.) Finally, we mention that there is an analogous small data result using a maximal time coordinate  instead of a Schwarzschild time coordinate.
The case with general data is more subtle. Rendall has shown  that there exist data leading to singular spacetimes as a consequence of Penrose's singularity theorem. This raises the question of what we mean by global existence for such data. The Schwarzschild time coordinate is expected to avoid the singularity, and by global existence we mean that solutions remain smooth as Schwarzschild time tends to infinity. Even though spacetime might be only partially covered in Schwarzschild coordinates, a global existence theorem for general data would nevertheless be very important since weak cosmic censorship would follow from it. A partial investigation for general data was done in , where it is shown that if singularities form in finite Schwarzschild time the first one must be at the centre. More precisely, if f (t, r, w, L)=0 when for some , and for all t, w and L, then the solution remains smooth for all time. This rules out singularities of the shell-crossing type, which can be an annoying problem for other matter models (e.g. dust). The main observation in  is a cancellation property in the term
in the characteristic equation (40). We refer to the original paper for details. In  a numerical study was undertaken. A numerical scheme originally used for the Vlasov-Poisson system was modified to the spherically symmetric Einstein-Vlasov system. It has been shown by Rodewis  that the numerical scheme has the desirable convergence properties. (In the Vlasov-Poisson case convergence was proved in . See also .) The numerical experiments support the conjecture that solutions are singularity-free. This can be seen as evidence that weak cosmic censorship holds for collisionless matter. It may even hold in a stronger sense than in the case of a massless scalar field (see [20, 22]). There may be no naked singularities formed for any regular initial data rather than just for generic data. This speculation is based on the fact that the naked singularities that occur in scalar field collapse appear to be associated with the existence of type II critical collapse, while Vlasov matter is of type I. This is indeed the primary goal of their numerical investigation: to analyze critical collapse and decide whether Vlasov matter is type I or type II.
These different types of matter are defined as follows. Given small initial data no black holes are expected to form and matter will disperse (which has been proved for a scalar field  and for Vlasov matter ). For large data, black holes will form and consequently there is a transition regime separating dispersion of matter and formation of black holes. If we introduce a parameter A on the initial data such that for small A dispersion occurs and for large A a black hole is formed, we get a critical value separating these regions. If we take and denote by the mass of the black hole, then if as , we have type II matter, whereas for type I matter this limit is positive and there is a mass gap. For more information on critical collapse we refer to the review paper by Gundlach .
For Vlasov matter there is an independent numerical simulation by Olabarrieta and Choptuik  (using a maximal time coordinate) and their conclusion agrees with the one in . Critical collapse is related to self similar solutions; Martin-Garcia and Gundlach  have presented a construction of such solutions for the massless Einstein-Vlasov system by using a method based partially on numerics. Since such solutions often are related to naked singularities, it is important to note that their result is for the massless case (in which case there is no known analogous result to the small data theorem in ) and their initial data are not in the class that we have described above.
|The Einstein-Vlasov System/Kinetic Theory
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