Curvature singularities and abstract boundary singularity theorems for space-time

TL;DR

This paper explores the abstract boundary approach to defining and understanding singularities in General Relativity, establishing new theorems linking essential singularities with geodesic incompleteness and curvature conditions.

Contribution

It introduces singularity theorems within the abstract boundary framework and relates strong curvature singularities to boundary essential singularities.

Findings

Established a link between abstract boundary essential singularities and geodesic incompleteness.

Derived new singularity theorems for maximally extended space-times.

Connected strong curvature singularities with abstract boundary essential singularities.

Abstract

The abstract boundary construction of Scott and Szekeres is a general and flexible way to define singularities in General Relativity. The abstract boundary construction also proves of great utility when applied to questions about more general boundary features of space-time. Within this construction an essential singularity is a non-regular boundary point which is accessible by a curve of interest (e.g. a geodesic) within finite (affine) parameter distance and is not removable. Ashley and Scott proved the first theorem linking abstract boundary essential singularities with the notion of causal geodesic incompleteness for strongly causal, maximally extended space-times. The relationship between this result and the classical singularity theorems of Penrose and Hawking has enabled us to obtain abstract boundary singularity theorems. This paper describes essential singularity results for maximally extended space-times and presents our recent efforts to establish a relationship between the strong curvature singularities of Tipler and Krolak and abstract boundary essential singularities.