Examples of covering properties of boundary points of space-times

TL;DR

This paper explores the complex classification of boundary points in space-times, highlighting the dependence on embedding and providing examples to illustrate various covering relations among boundary sets.

Contribution

It systematically examines the possible covering relations of boundary points in space-times, demonstrating the range of configurations consistent with the definitions.

Findings

All theoretically possible covering relations are demonstrated through examples.

Boundary point classification depends on embedding choices, making it inherently subtle.

The paper clarifies the intrinsic nature of manifold completion with respect to pseudo-Riemannian metrics.

Abstract

The problem of classifying boundary points of space-time, for example singularities, regular points and points at infinity, is an unexpectedly subtle one. Due to the fact that whether or not two boundary points are identified or even "nearby" is dependant on the way the space-time is embedded, difficulties occur when singularities are thought of as an inherently local aspect of a space-time, as an analogy with electromagnetism would imply. The completion of a manifold with respect to a pseudo-Riemannian metric can be defined intrinsically, [SS94]. This is done via an equivalence relation, formalising which boundary sets cover other sets. This paper works through the possibilities, providing examples to show that all covering relations not immediately ruled out by the definitions are possible.