# The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics

## TL;DR

This paper extends the Hawking-Penrose singularity theorem to Lorentzian metrics with lower regularity ($C^{1,1}$), by formulating weak energy and genericity conditions and analyzing geodesic behavior through regularisation techniques.

## Contribution

It demonstrates that the singularity theorem holds for $C^{1,1}$-metrics, introducing weak conditions and a detailed Riccati equation analysis for approximating metrics.

## Key findings

- Singularity theorem valid for $C^{1,1}$-Lorentzian metrics.
- Weak energy and genericity conditions formulated for low-regularity metrics.
- Causal geodesics become non-maximising under these conditions.

## Abstract

We show that the Hawking--Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of $C^{1, 1}$-regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for $C^{1,1}$-metrics, and of $C^0$-trapped submanifolds. By regularisation, we show that, under these weak conditions, causal geodesics necessarily become non-maximising. This requires a detailed analysis of the matrix Riccati equation for the approximating metrics, which may be of independent interest.