Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications
Fabio Cavalletti, Andrea Mondino
TL;DR
The paper develops a Lorentzian analogue of synthetic curvature-dimension theory via optimal transport in Lorentzian pre-length spaces, defining the $p$-Lorentz–Wasserstein distance and timelike duality concepts. It introduces entropic timelike curvature-dimension conditions $\mathsf{TCD}^{e}_{p}(K,N)$ and their weak form, along with the Timelike Measure Contraction Property $\mathsf{TMCP}^{e}(K,N)$, and proves stability under convergence, localization results, and geometric consequences such as timelike Bishop–Gromov and Hawking-type singularity theorems. It also establishes the existence and uniqueness of optimal transport maps in timelike non-branching TMCP^e spaces and develops a localization framework via disintegration along transport rays, showing regularity of conditional measures. Together, these results extend volume comparison and singularity theorems to non-smooth Lorentzian settings and provide a robust toolkit for causal OT in synthetic spacetime models with potential relevance to quantum gravity frameworks. The work unifies OT foundations, synthetic timelike curvature bounds, and concrete geometric applications in a broad Lorentzian context.
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.