Stability of Synthetic Timelike Ricci Bounds under $C^0$-Limits and Applications to Impulsive Gravitational Waves

TL;DR

This paper demonstrates the stability of synthetic timelike Ricci curvature bounds under low-regularity limits of Lorentzian metrics, with applications to impulsive gravitational waves and insights into low-regularity Einstein solutions.

Contribution

It proves the stability of the synthetic curvature-dimension condition under low-regularity limits and applies this to impulsive gravitational waves and vacuum spacetimes.

Findings

Synthetic timelike Ricci bounds are stable under locally uniform convergence.

Impulsive gravitational waves with Lipschitz metrics satisfy synthetic Ricci bounds.

Minkowski background admits synthetic upper Ricci curvature bounds.

Abstract

We investigate the stability of timelike Ricci curvature lower bounds under low-regularity limits of Lorentzian metrics. Specifically, we prove that the synthetic curvature-dimension condition $TCD^e_p(K,N)$, which provides an optimal transport formulation of the Hawking-Penrose strong energy condition, is stable under locally uniform convergence of smooth Lorentzian metrics, provided a uniform global hyperbolicity assumption holds. As a consequence, smooth locally uniform limits of vacuum spacetimes satisfy the strong energy condition, even though curvature is not controlled a priori. As a main application, we study impulsive gravitational waves - spacetimes with Lipschitz continuous metrics - and show that large classes of such waves satisfy synthetic timelike Ricci curvature lower bounds. In the case of Minkowski background, we further establish synthetic upper Ricci curvature bounds. Our approach relies on constructing suitable smooth approximations with lower bounds on the timelike Ricci, and analyzing the limiting behavior via Lorentzian optimal transport. These results yield new geometric insights into low-regularity solutions of the Einstein equations and, in particular, provide a counterexample to the extension of the Eschenburg-Galloway-Newman Lorentzian splitting theorem to infinitesimally Minkowskian $TCD^e_p(0,N)$ Lorentzian length spaces. Moreover, our construction shows that a direct Lorentzian analogue of the Cheeger-Colding almost splitting theorem - under assumptions of almost non-negative timelike Ricci curvature and the existence of an almost maximizing line - cannot hold. This highlights a fundamental difference between the geometry of Riemannian and Lorentzian lower Ricci curvature bounds. We also apply the aforementioned stability theorem to weak solutions of the Einstein equations arising from the nonlinear interaction of impulsive gravitational waves.