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

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 $\mathsf{TCD}^e_p(K,N)$, 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. We also apply the aforementioned stability theorem to weak solutions of the Einstein equations arising from the nonlinear interaction of impulsive gravitational waves. These results yield new geometric insights into low-regularity solutions of the Einstein equations.