The Hawking-Penrose singularity theorem for $C^1$-Lorentzian metrics

TL;DR

This paper extends the Hawking-Penrose singularity theorem to Lorentzian metrics with only $C^1$ regularity, addressing challenges of distributional Ricci tensor and geodesic branching, and introduces new frameworks for low regularity causality and tensor analysis.

Contribution

It develops a theory of tensor distributions of finite order and introduces maximal causal non-branching as an alternative to geodesic incompleteness for $C^1$-metrics.

Findings

Extended Hawking-Penrose theorem to $C^1$-metrics

Developed a distributional tensor framework

Introduced maximal causal non-branching condition

Abstract

We extend both the Hawking-Penrose Theorem and its generalisation due to Galloway and Senovilla to Lorentzian metrics of regularity $C^1$. For metrics of such low regularity, two main obstacles have to be addressed. On the one hand, the Ricci tensor now is distributional, and on the other hand, unique solvability of the geodesic equation is lost. To deal with the first issue in a consistent way, we develop a theory of tensor distributions of finite order, which also provides a framework for the recent proofs of the theorems of Hawking and of Penrose for $C^1$-metrics [7]. For the second issue, we study geodesic branching and add a further alternative to causal geodesic incompleteness to the theorem, namely a condition of maximal causal non-branching. The genericity condition is re-cast in a distributional form that applies to the current reduced regularity while still being fully compatible with the smooth and $C^{1,1}$-settings. In addition, we develop refinements of the comparison techniques used in the proof of the $C^{1,1}$-version of the theorem [8]. The necessary results from low regularity causality theory are collected in an appendix.