2 Global Existence Theorems for the Einstein–Vlasov System

In general relativity two classes of initial data are distinguished. If an isolated matter distribution is studied, the data are called asymptotically flat. The initial hypersurface is topologically and (since far away from the body one expects spacetime to be approximately flat) appropriate fall off conditions are imposed. Roughly, a smooth data set on is said to be asymptotically flat if there exist global coordinates such that as tends to infinity the components in these coordinates tend to , the components tend to zero, has compact support and certain weighted Sobolev norms of and are finite (see [101]). The symmetry classes that admit asymptotical flatness are few. The important ones are spherically symmetric and axially symmetric spacetimes. One can also consider a case in which spacetime is asymptotically flat except in one direction, namely cylindrical spacetimes. A lot of work has been done on the spherically symmetric case and this will be reviewed below. In the case of cylindrical symmetry it has been shown that if singularities form, then the first one must occur at the axis of symmetry [39].

Spacetimes that possess a compact Cauchy hypersurface are called cosmological spacetimes, and data are accordingly given on a compact 3-manifold. In this case the whole universe is modelled and not only an isolated body. In contrast to the asymptotically flat case, cosmological spacetimes admit a large number of symmetry classes. This gives one the possibility to study interesting special cases for which the difficulties of the full Einstein equations are strongly reduced. We will discuss below cases for which the spacetime is characterized by the dimension of its isometry group together with the dimension of the orbit of the isometry group.