Hawking's singularity theorem for Lipschitz Lorentzian metrics

TL;DR

This paper extends Hawking's singularity theorem to Lorentzian metrics with Lipschitz regularity by developing new Ricci curvature estimates and a novel volume control method for spacelike surfaces.

Contribution

It introduces a new approach to prove singularity theorems under weaker regularity assumptions on spacetime metrics, using innovative estimates and inequalities.

Findings

Proves Hawking's singularity theorem for Lipschitz continuous metrics.

Develops new Ricci curvature estimates for regularised smooth metrics.

Introduces a segment-type inequality for volume control of spacelike surfaces.

Abstract

We prove Hawking's singularity theorem for spacetime metrics of local Lipschitz regularity. The proof rests on (1) new estimates for the Ricci curvature of regularising smooth metrics that are based upon a quite general Friedrichs-type lemma and (2) the replacement of the usual focusing techniques for timelike geodesics -- which in the absence of a classical ODE-theory for the initial value problem are no longer available -- by a worldvolume estimate based on a segment-type inequality that allows one to control the volume of the set of points in a spacelike surface that possess long maximisers.