Unusual properties - mathematical and physical - of the a-boundary construction

Abstract

This thesis is written within the framework of the abstract boundary (or a-boundary) of Scott and Szekeres. The a-boundary provides a concept of "boundary" for any n-dimensional, paracompact, connected, Hausdorff manifold, defined in such a way that the boundary is independant of the particular embedding used to display the manifold. This makes it possible to define various types of boundary points of space-time such as "singularities" and "points at infinity". The original research that will be presented in this thesis can be roughly divided up into two categories; results relating to the existence of optimal embeddings of solutions to Einstein's Field Equations and a-boundary singularity theorems. In addition, the implications of the "finite connected neighbourhood region property" and the bounded "acceleration" property are explored. It is also shown that not all space-times are maximally extendable.