R\'enyi's entropy on Lorentzian spaces. Timelike curvature-dimension conditions

TL;DR

This paper develops synthetic notions of timelike Ricci curvature bounds in Lorentzian spaces using Rényi entropy convexity along specific geodesics, extending geometric analysis to non-smooth spacetimes.

Contribution

It introduces and studies new curvature-dimension and measure-contraction conditions in Lorentzian spaces, establishing their properties and equivalences with entropic conditions.

Findings

Compatibility with smooth Lorentzian geometry

Sharp geometric inequalities derived

Stability and local-to-global properties proven

Abstract

For a Lorentzian space measured by $\mathfrak{m}$ in the sense of Kunzinger, S\"amann, Cavalletti, and Mondino, we introduce and study synthetic notions of timelike lower Ricci curvature bounds by $K\in\boldsymbol{\mathrm{R}}$ and upper dimension bounds by $N\in[1,\infty)$, namely the timelike curvature-dimension conditions $\smash{\mathrm{TCD}_p(K,N)}$ and $\smash{\mathrm{TCD}_p^*(K,N)}$ in weak and strong forms, where $p\in (0,1)$, and the timelike measure-contraction properties $\smash{\mathrm{TMCP}(K,N)}$ and $\smash{\mathrm{TMCP}^*(K,N)}$. These are formulated by convexity properties of the R\'enyi entropy with respect to $\mathfrak{m}$ along $\smash{\ell_p}$-geodesics of probability measures. We show many features of these notions, including their compatibility with the smooth setting, sharp geometric inequalities, stability, equivalence of the named weak and strong versions, local-to-global properties, and uniqueness of chronological $\smash{\ell_p}$-optimal couplings and chronological $\smash{\ell_p}$-geodesics. We also prove the equivalence of $\smash{\mathrm{TCD}_p^*(K,N)}$ and $\smash{\mathrm{TMCP}^*(K,N)}$ to their respective entropic counterparts in the sense of Cavalletti and Mondino. Some of these results are obtained under timelike $p$-essential nonbranching, a concept which is a priori weaker than timelike nonbranching.