Singularity theorems for $C^1$-Lorentzian metrics

TL;DR

This paper extends the classical singularity theorems of General Relativity to $C^1$-Lorentzian metrics, providing complete proofs and stability results in low regularity settings.

Contribution

It offers the first complete proofs of Hawking and Penrose singularity theorems for $C^1$-metrics and establishes stability of geodesic completeness in this regularity.

Findings

Proved singularity theorems for $C^1$-Lorentzian metrics.

Established $C^1$-fine stability of geodesic completeness.

Extended techniques to prove Myers Theorem for $C^1$-metrics.

Abstract

Continuing recent efforts in extending the classical singularity theorems of General Relativity to low regularity metrics, we give a complete proof of both the Hawking and the Penrose singularity theorem for $C^1$-Lorentzian metrics - a regularity where one still has existence but not uniqueness for solutions of the geodesic equation. The proofs make use of careful estimates of the curvature of approximating smooth metrics and certain stability properties of long existence times for causal geodesics. On the way we also prove that for globally hyperbolic spacetimes with a $C^1$-metric causal geodesic completeness is $C^1$-fine stable. This improves a similar older stability result of Beem and Ehrlich where they also used the $C^1$-fine topology to measure closeness but still required smoothness of all metrics. Lastly, we include a brief appendix where we use some of the same techniques in the Riemannian case to give a proof of the classical Myers Theorem for $C^1$-metrics.