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The abstract boundary---a new approach to singularities of manifolds | | Susan M. Scott
; Peter Szekeres
; | | Date: |
26 May 1994 | | Journal: | J.Geom.Phys. 13 (1994) 223-253 | | Subject: | gr-qc | | Abstract: | A new scheme is proposed for dealing with the problem of singularities in General Relativity. The proposal is, however, much more general than this. It can be used to deal with manifolds of any dimension which are endowed with nothing more than an affine connection, and requires a family calc of curves satisfying a {em bounded parameter property} to be specified at the outset. All affinely parametrised geodesics are usually included in this family, but different choices of family calc will in general lead to different singularity structures. Our key notion is the {em abstract boundary/} or {em $a$-boundary/} of a manifold, which is defined for any manifold calm and is independent of both the affine connection and the chosen family calc of curves. The $a$-boundary is made up of equivalence classes of boundary points of calm in all possible open embeddings. It is shown that for a pseudo-Riemannian manifold $(calm,g)$ with a specified family calc of curves, the abstract boundary points can then be split up into four main categories---regular, points at infinity, unapproachable points and singularities. Precise definitions are also provided for the notions of a {em removable singularity} and a {em directional singularity}. The pseudo-Riemannian manifold will be said to be singularity-free if its abstract boundary contains no singularities. The scheme passes a number of tests required of any theory of singularities. For instance, it is shown that all compact manifolds are singularity-free, irrespective of the metric and chosen family calc. | | Source: | arXiv, gr-qc/9405063 | | Services: | Forum | Review | PDF | Favorites | |
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