Hawking's singularity theorem for Lipschitz Lorentzian metrics

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.

💡 Research Summary

The paper “Hawking’s singularity theorem for Lipschitz Lorentzian metrics” establishes Hawking’s classic singularity theorem for spacetimes whose metric is only locally Lipschitz continuous, i.e. (g\in C^{0,1}). This regularity is substantially weaker than the (C^{2}) or even the (C^{1,1}) assumptions traditionally required for the theorem. The authors achieve this extension by introducing two major technical innovations: (1) new (L^{p})‑type estimates for the Ricci curvature of smooth metrics obtained by a careful regularisation of the original Lipschitz metric, and (2) a world‑volume estimate that replaces the usual focusing arguments based on timelike geodesics.

The paper is organized as follows. Section 1 motivates the problem, emphasizing that many physically relevant solutions of Einstein’s equations—impulsive gravitational waves, thin shells, matched spacetimes—have only (C^{0,1}) regularity. Classical proofs of Hawking’s theorem rely on the existence and uniqueness of geodesics and on Raychaudhuri’s equation, both of which break down when the metric is merely Lipschitz. Consequently, a new analytical framework is required.

Section 2 collects preliminaries. Subsection 2.1 reviews causality theory for continuous metrics, defining timelike, causal, and null locally Lipschitz curves, the relations (I^{\pm}) and (J^{\pm}), global hyperbolicity, and Cauchy hypersurfaces. It also explains how continuous spacetimes fit into the more abstract settings of closed cone structures (Minguzzi) and Lorentzian length spaces (Kunzinger–Sämann), which guarantee the existence of maximizing causal curves even without smoothness. Subsection 2.2 introduces a manifold‑wide convolution procedure (T\star_{M}\rho_{\varepsilon}) for regularising distributional tensor fields. The construction uses a locally finite atlas, a partition of unity, and a non‑negative mollifier; it preserves positivity of scalar distributions. Subsection 2.3 briefly recalls volume comparison tools that will later be used in the world‑volume estimate.

Section 3 is the analytical core. The authors prove a general Friedrichs‑type lemma (Lemma 3.3) that yields, for any Lipschitz metric (g) and any sequence of smooth approximations (g_{\varepsilon}=g\star_{M}\rho_{\varepsilon}), the following properties: (i) (\operatorname{Ric}