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 Einstein-Vlasov 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 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.
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The Einstein-Vlasov System/Kinetic Theory
Håkan Andréasson http://www.livingreviews.org/lrr-2002-7 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |