Inserted: 21 apr 2020
Last Updated: 3 dec 2020
The goal of the present work is three-fold. The first goal is to set foundational results on optimal transport in Lorentzian synthetic 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 Lorentzian 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 Lorentzian synthetic 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 synthetic 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 (e.g. causal Fermion systems).