Degeneracy of the b-boundary in General Relativity

TL;DR

This paper investigates the degeneracy of the b-boundary in General Relativity, demonstrating non-Hausdorffness under certain curvature conditions and illustrating the issue with well-known solutions.

Contribution

It corrects previous assumptions about the b-boundary's topological properties and shows degeneracy occurs under specific curvature conditions in space-times.

Findings

b-boundary can be non-Hausdorff under certain conditions

Degeneracy illustrated with exact solutions of GR

Clarifies errors in previous topological arguments

Abstract

The b-boundary construction by B. Schmidt is a general way of providing a boundary to a manifold with connection. It has been shown to have undesirable topological properties however. C. J. S. Clarke gave a result showing that for space-times, non-Hausdorffness is to be expected in general, but the argument contains some errors. We show that under somewhat different conditions on the curvature, the b-boundary will be non-Hausdorff, and illustrate the degeneracy by applying the conditions to some well known exact solutions of general relativity.