2.1 Spherically symmetric spacetimesThe Einstein-Vlasov System/Kinetic Theory1.3 The Einstein-Vlasov system

2 Global Existence Theorems for the Einstein-Vlasov System

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 tex2html_wrap_inline1747 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 tex2html_wrap_inline1749 on tex2html_wrap_inline1747 is said to be asymptotically flat if there exist global coordinates tex2html_wrap_inline1753 such that as | x | tends to infinity the components tex2html_wrap_inline1618 in these coordinates tend to tex2html_wrap_inline1759, the components tex2html_wrap_inline1622 tend to zero, tex2html_wrap_inline1552 has compact support and certain weighted Sobolev norms of tex2html_wrap_inline1765 and tex2html_wrap_inline1622 are finite (see [76Jump To The Next Citation Point In The Article]). 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.





2.1 Spherically symmetric spacetimesThe Einstein-Vlasov System/Kinetic Theory1.3 The Einstein-Vlasov system

image The Einstein-Vlasov System/Kinetic Theory
Håkan Andréasson
http://www.livingreviews.org/lrr-2002-7
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