3 The Asymptotically-Flat Cauchy Problem: Spherically-Symmetric Solutions

In this section, we discuss results on global existence and on the asymptotic structure of solutions of the Cauchy problem in the asymptotically-flat case.

In general relativity two classes of initial data are distinguished in the study of the Cauchy problem: asymptotically-flat initial data and cosmological initial data. The former type of data describes an isolated body. The initial hypersurface is topologically 3 ℝ and appropriate fall-off conditions are imposed to ensure that far away from the body spacetime is approximately flat. Spacetimes, which 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 modeled rather than an isolated body.

The symmetry classes that admit asymptotic flatness are few. The important ones are spherically symmetric and axially symmetric spacetimes. One can also consider a case, which is unphysical, in which spacetime is asymptotically flat except in one direction, namely cylindrically-symmetric spacetimes, cf. [75], where the Cauchy problem is studied. The majority of the work so far has been devoted to the spherically-symmetric case but recently a result on static axisymmetric solutions has been obtained.

In contrast to the asymptotically-flat case, cosmological spacetimes admit a large number of symmetry classes. This provides the possibility to study many special cases for which the difficulties of the full Einstein equations are reduced. The Cauchy problem in the cosmological case is reviewed in Section 4.

The following subsections concern studies of the spherically-symmetric Einstein–Vlasov system. The main goal of these studies is to provide an answer to the weak and strong cosmic censorship conjectures, cf. [191, 61Jump To The Next Citation Point] for formulations of the conjectures.

 3.1 Set up and choice of coordinates
 3.2 Local existence and the continuation criterion
 3.3 Global existence for small initial data
 3.4 Global existence for special classes of large initial data
 3.5 On global existence for general initial data
 3.6 Self-similar solutions
 3.7 Formation of black holes and trapped surfaces
 3.8 Numerical studies on critical collapse
 3.9 The charged case

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