On Lorentzian causality with continuous metrics

TL;DR

This paper investigates causality theory on Lorentzian manifolds with continuous metrics, revealing which classical results hold or fail, and establishing conditions under which key causality properties are preserved.

Contribution

It systematically analyzes causality in continuous Lorentzian metrics, showing that many smooth results extend under weaker regularity conditions.

Findings

Light-cones may not be hypersurfaces in continuous metrics

Existence of time functions persists in domains of dependence

C^{0,1} regularity suffices for key causality results

Abstract

We present a systematic study of causality theory on Lorentzian manifolds with continuous metrics. Examples are given which show that some standard facts in smooth Lorentzian geometry, such as light-cones being hypersurfaces, are wrong when metrics which are merely continuous are considered. We show that existence of time functions remains true on domains of dependence with continuous metrics, and that $C^{0,1}$ differentiability of the metric suffices for many key results of the smooth causality theory.