Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications

Abstract

The goal of the present work is three-fold. The first goal is to set foundational results on optimal transport in Lorentzian (pre-)length spaces, including cyclical monotonicity, stability of optimal couplings and Kantorovich duality (several results are new even for smooth Lorentzian manifolds). The second one is to give a synthetic notion of “timelike Ricci curvature bounded below and dimension bounded above” for a measured Lorentzian pre-length space using optimal transport. The key idea being to analyse convexity properties of Entropy functionals along future directed timelike geodesics of probability measures. This notion is proved to be stable under a suitable weak convergence of measured Lorentzian pre-length spaces, giving a glimpse on the strength of the approach we propose. The third goal is to draw applications, most notably extending volume comparisons and Hawking singularity Theorem (in sharp form) to the synthetic setting. The framework of Lorentzian pre-length spaces includes as remarkable classes of examples: space-times endowed with a causally plain (or, more strongly, locally Lipschitz) continuous Lorentzian metric, closed cone structures, some approaches to quantum gravity.