The singularity theorems of General Relativity and their low regularity extensions

TL;DR

This paper reviews classical and recent low regularity extensions of the singularity theorems in General Relativity, emphasizing analytical methods and focusing on causal geodesics and distributional curvature.

Contribution

It introduces low regularity Lorentzian metrics and develops $C^1$-singularity theorems using regularisation techniques for distributional curvature.

Findings

Extension of singularity theorems to low regularity metrics

Development of regularisation approach for distributional curvature

Pedagogical overview of classical and modern proofs

Abstract

On the occasion of Sir Roger Penrose's 2020 Nobel Prize in Physics, we review the singularity theorems of General Relativity, as well as their recent extension to Lorentzian metrics of low regularity. The latter is motivated by the quest to explore the nature of the singularities predicted by the classical theorems. Aiming at the more mathematically minded reader, we give a pedagogical introduction to the classical theorems with an emphasis on the analytical side of the arguments. We especially concentrate on focusing results for causal geodesics under appropriate geometric and initial conditions, in the smooth and in the low regularity case. The latter comprise the main technical advance that leads to the proofs of $C^1$-singularity theorems via a regularisation approach that allows to deal with the distributional curvature. We close with an overview on related lines of research and a future outlook.