The abstract boundary—a new approach to singularities of manifolds

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 C of curves satisfying a bounded parameter property to be specified at the outset. All affinely parametrised geodesics are usually included in this family, but different choices of family C will in general lead to different singularity structures. Our key notion is the abstract boundary or a-boundary of a manifold, which is defined for any manifold M and is independent of both the affine connection and the chosen family C of curves. The a-boundary is made up of equivalence classes of boundary points of M in all possible open embeddings. It is shown that for a pseudo-Riemannian manifold ( M , g) with a specified family C 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 removable singularity and a 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 C . All geodesically complete pseudo-Riemannian manifolds are also singularity-free if the family C simply consists of all affinely parametrised geodesics. Furthermore, if any closed region is excised from a singularity-free manifold then the resulting manifold is still singularity-free. Numerous examples are given throughout the text. Problematic cases posed by Geroch and Misner are discussed in the context of the a-boundary and are shown to be readily accommodated.