General relativity assumes smooth spacetimes. The metric tensor is differentiable, curvature is well-defined pointwise, geodesics are smooth curves. But physical spacetimes aren't always smooth. Impulsive gravitational waves have metric discontinuities. Thin matter shells create jumps in extrinsic curvature. Matched spacetimes — two solutions glued along a boundary — have metrics that are continuous but not differentiable. These are Lipschitz spacetimes: the metric is continuous and has bounded derivatives almost everywhere, but the curvature is only defined as a distribution.
Braun and Salamo Candal (arXiv: 2603.24195) prove that comparison geometry — the toolkit that bounds geometry above or below by reference to model spaces — works in this setting. They establish the timelike measure contraction property for globally hyperbolic Lipschitz spacetimes with distributional Ricci curvature bounded below. From this, they derive sharp timelike Brunn-Minkowski, Bishop-Gromov, and Bonnet-Myers inequalities.
The technique borrows from optimal transport: localization methods originally developed in convex geometry, adapted to the Lorentzian setting. The results extend prior Riemannian theorems by Petersen-Sprouse and Aubry to spacetimes, and they apply to physically important cases where the standard smooth theory breaks down.
The through-claim: the theorems of general relativity are more robust than the assumptions that derived them. The classical comparison results were proven under smoothness conditions that many physical spacetimes violate. But the results themselves survive at Lipschitz regularity — they needed less than they were given. The physics was always valid where the mathematics hadn't yet ventured.
Braun & Salamo Candal, 2603.24195. General relativity / Lipschitz geometry / comparison theorems / optimal transport / distributional curvature.