What actually happens when you approach a gravitational singularity?

TL;DR

This paper proposes a coordinate-independent method to identify gravitational singularities associated with incomplete null geodesics, enabling standard analysis of curvature divergence.

Contribution

It introduces a concrete relationship linking incomplete null geodesics to gravitational singularities, facilitating curvature divergence studies with standard techniques.

Findings

Established a coordinate-independent definition of singularities.

Enabled analysis of curvature divergence using standard methods.

Connected null geodesic incompleteness with physical singularities.

Abstract

Roger Penrose's 2020 Nobel Prize in Physics recognises that his identification of the concepts of "gravitational singularity" and an "incomplete, inextendible, null geodesic" is physically very important. The existence of an incomplete, inextendible, null geodesic doesn't say much, however, if anything, about curvature divergence, nor is it a helpful definition for performing actual calculations. Physicists have long sought for a coordinate independent method of defining where a singularity is located, given an incomplete, inextendible, null geodesic, that also allows for standard analytic techniques to be implemented. In this essay we present a solution to this issue. It is now possible to give a concrete relationship between an incomplete, inextendible, null geodesic and a gravitational singularity, and to study any possible curvature divergence using standard techniques.