Topological properties of the abstract boundary construction for general relativity and their application to space-time extensions

Abstract

The abstract boundary construction of Scott and Szekeres provides a 'boundary' for any n-dimensional, paracompact, connected, Hausdorff, smooth space-time manifold. For a space-time, singularities and points at infinity may then be defined as objects within this boundary. The abstract boundary of a manifold is typically very large. Even so, one does not necessarily have to consider every abstract boundary point in order to provide detailed comments on the structure of the abstract boundary. Towards this end, this thesis develops a number of results concerning the topological contact properties of elements of the abstract boundary. For the first time two topologies, referred to as the attached point topology and the strongly attached point topology, are defined for a manifold together with its abstract boundary. We consider properties of these topologies related to the topological separation of abstract boundary points. In particular, it is demonstrated that the attached point topology is Hausdorff but the strongly attached point topology is not. Even so, it is observed that the strongly attached point topology loses Hausdorff separability in a way that provides further insight into the structure of the abstract boundary. We next consider how regular abstract boundary points of different space-time extensions are related to each other. We conjecture that if two regular abstract boundary points are in contact with each other, then, under certain physically reasonable conditions, such as strong causality holding, they must be equivalent. This has major implications when considering how to best present the geometrical features of a space-time within an embedding. In particular, this result says that the structure of the boundary through which space-time extensions are performed does not depend on the extension. Great progress is made in proving this result. It is rigorously shown that a significant amount of topological information is preserved around the boundary between different envelopments. In order to prove this result a number of reasonable restrictions were placed on the manifold. By considering special cases it is possible to weaken some of these restrictions. A form of the result is proven for isolated abstract boundary points using weakened assumptions. In the final component of the thesis we apply results developed in the previous sections to collections of abstract boundary points called partial cross sections. Partial cross sections allow one to simplify the appearance of the abstract boundary. In particular, they can be used to formulate optimal embeddings which, in essence, are a formal definition for an embedding of a manifold which clearly displays the important geometric features of that manifold. Refined definitions of partial cross sections and optimal embeddings are presented which incorporate the insights gained regarding the contact properties of abstract boundary points. In particular, optimal presentations are defined for the first time. These objects generalise the notion of an optimal embedding of a space-time to include those space-times which require more then one embedding in order to reveal its geometric structure.