Foundations of and Applications for the Abstract Boundary Construction for Space-Time

Abstract

The original content of this thesis is comprised of three parts. First, we investigate the foundations of the Abstract Boundary. We start by presenting a one-to-one correspondence between the set of envelopments and a subset of the set of distances on our manifold. This correspondence allows us to define the Abstract Boundary in terms of mathematical structures defined on the manifold, rather than having to use structures additional to the manifold. We take the ideas used in the correspondence and generalise the Abstract Boundary to be applicable to any first countable topological space. Then, using the correspondence and the generalisation we give two alternative constructions for the Abstract Boundary. These new methods of construction allow us to bring many new tools to the analysis of the Abstract Boundary and thus enrich the subject and provide new avenues for research. Second, we discuss how the limiting behaviour of curves relates to the Abstract Boundary. We restrict our attention to the manifold itself and give a classification of the behaviour of curves via the number of limit points they possess. As an application of the classification we weaken the causality assumption of the Abstract Boundary singularity theorem. As an illustration of the problems that curves in a certain class of the classification can cause we give a definition of causality for Abstract Boundary points. In the process of doing so we generalise the distinguishing and strong causality conditions for the boundaries of envelopments and the Abstract Boundary itself. Third, we investigate the link between the Penrose-Hawking singularity theorems and the Krolak strong curvature condition. We review the singularity theorems and analyse their proofs to determine what can be said about the predicted incomplete geodesics. We see that the conclusions that can be made and the criteria for the Krolak strong curvature condition do not mesh easily. For this reason we present two necessary and sufficient conditions for a geodesic to satisfy the Krolak strong curvature condition, that provide a link between the conclusions and the Krolak condition. The result is that we need to investigate the limiting behaviour of jaix