SOME ASPECTS OF SINGULARITIES . IN GENERAL RELATIVITY

Abstract

One of the most important subjects in General Relativity is the singularities of space-times. Most exact solutions of the Einstein's field equations have singularities. Many space-time singularities are recognized as divergence of the curvature towards the boundaries of space-times as described by simple exact solutions, for example the Friedmann solution and Schwarzschild solution. These considerations do not guarantee that the occurrence of singularities in all the exact solutions are always associated with the divergence of curvature. For example, the components of the curvature tensor in the Taub-NUT space-time are bounded, but this space-time is still singular. Various exact solutions of the Einstein equations have been inspected in order to formulate a good definition of a singularity which can be used in' a reasonably straightforward way on sorts of examples that commonly arise in General Relativity. Unfortunately, most of these definitions encounter some difficults. Finally, Clarke [31] in his book "the analy­ sis of space-time singularities" gave an appropriate definition for a space-time to be singular. A space-time is singular if it contains an incomplete geodesic (or more general curve) 'Y such that there is no extension for which 'Y is extendible. Singularities are not points of the physical space-time, but just imply that in a maximal space-time some inextendible non-spacelike curve end at a finit value of the (generalized) affine parameter.

Historically there have been several approaches to define the set of singular points by considering them as a boundary to provide ex­ plicit boundary constructions for space-times. The first approach is the so called geodesic-boundary (g-boundary), given by Geroch [41]. It represents the limit points of all incomplete geodesics. The second is the causal or c-boundary of Geroch, Kronheimer and Penrose [45].