Generalizations of the Abstract Boundary singularity theorem

TL;DR

This paper extends the Abstract Boundary singularity theorem to broader classes of curves and conditions, enhancing the understanding of singularities in spacetime manifolds.

Contribution

It introduces two generalizations of the theorem: one to continuous causal curves with the distinguishing condition, and another to locally Lipschitz curves under specific imprisonment conditions.

Findings

Extended affine parameters to locally Lipschitz curves.

Generalized the theorem to continuous causal curves.

Proved the theorem for manifolds with no totally imprisoned inextendible locally Lipschitz curves.

Abstract

The Abstract Boundary singularity theorem was first proven by Ashley and Scott. It links the existence of incomplete causal geodesics in strongly causal, maximally extended spacetimes to the existence of Abstract Boundary essential singularities, i.e., non-removable singular boundary points. We give two generalizations of this theorem: the first to continuous causal curves and the distinguishing condition, the second to locally Lipschitz curves in manifolds such that no inextendible locally Lipschitz curve is totally imprisoned. To do this we extend generalized affine parameters from $C^1$ curves to locally Lipschitz curves.