In general relativity two classes of initial data are
distinguished. If an isolated body 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 
x
 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 [76]). 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. Regarding global existence questions,
only spherically symmetric spacetimes have been considered for
the EinsteinVlasov system in the asymptotically flat case.
Spacetimes that possess a compact Cauchy hypersurface are
called cosmological spacetimes, and data are accordingly given on
a compact 3manifold. 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.

The EinsteinVlasov System/Kinetic Theory
Håkan Andréasson
http://www.livingreviews.org/lrr20027
© MaxPlanckGesellschaft. ISSN 14338351
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