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.