The stability of abstract boundary essential singularities

TL;DR

This paper demonstrates the conditions under which essential singularities in space-time are stable under metric perturbations, linking abstract boundary theory with geodesic incompleteness stability results.

Contribution

It establishes the $C^{1}$-stability of essential singularities in strongly causal space-times, extending previous work on geodesic incompleteness stability.

Findings

Essential singularities are $C^{1}$-stable under certain conditions.

The stability relates to causal geodesic incompleteness.

Conditions for stability depend on metric perturbations.

Abstract

The abstract boundary has, in recent years, proved a general and flexible way to define the singularities of space-time. In this approach an essential singularity is a non-regular boundary point of an embedding which is accessible by a chosen family of curves within finite parameter distance. Ashley and Scott proved the first theorem relating essential singularities in strongly causal space-times to causal geodesic incompleteness. Linking this with the work of Beem on the $C^{r}$-stability of geodesic incompleteness allows proof of the stability of these singularities. Here I present this result stating the conditions under which essential singularities are $C^{1}$-stable against perturbations of the metric.