Comparison theory for Lipschitz spacetimes

TL;DR

This paper establishes new comparison theorems and geometric inequalities for Lipschitz continuous Lorentzian spacetimes with bounded Ricci curvature, extending classical results to lower regularity settings and including impulsive gravitational phenomena.

Contribution

It proves the timelike measure contraction property and related comparison theorems for Lipschitz spacetimes, extending Lorentzian geometry results to non-smooth metrics and impulsive spacetimes.

Findings

Proved timelike measure contraction property for Lipschitz spacetimes.

Derived sharp comparison theorems including d'Alembert and Bishop-Gromov.

Established a timelike Bonnet-Myers inequality and diameter estimate under curvature bounds.

Abstract

We prove a globally hyperbolic spacetime with locally Lipschitz continuous metric and timelike distributional Ricci curvature bounded from below obeys the timelike measure contraction property. The remarkable class of examples of spacetimes that are covered by this result includes impulsive gravity waves, thin shells, and matched spacetimes. As applications, we get new comparison theorems for Lipschitz spacetimes in sharp form: d'Alembert, timelike Brunn-Minkowski, and timelike Bishop-Gromov. Under appropriate nonbranching assumptions (conjectured to hold in even lower regularity), our results also yield the timelike curvature-dimension condition, a volume incompleteness theorem, as well as exact representation formulas and sharp comparison estimates for d'Alembertians of Lorentz distance functions from general spacelike submanifolds. Moreover, we establish the sharp timelike Bonnet--Myers inequality ad hoc using the localization technique from convex geometry. Alongside, we prove a timelike diameter estimate for spacetimes whose timelike Ricci curvature is positive up to a "small" deviation (in an $L^p$-sense). This adapts prior theorems for Riemannian manifolds by Petersen-Sprouse and Aubry to Lorentzian geometry, a transition the former two anticipated almost 30 years ago.