A review of Lorentzian synthetic theory of timelike Ricci curvature bounds

Abstract

The scope of this survey is to give a self-contained introduction to synthetic timelike Ricci curvature bounds for (possibly non-smooth) Lorentzian spaces via optimal transport & entropy tools, including a synthetic version of Hawking's singularity theorem and a synthetic characterisation of Einstein's vacuum equations. We will also discuss some motivations arising from the smooth world and some possible directions for future research.

Causal bubbles in globally hyperbolic spacetimes

“…The entropic convexity condition defining TCD(K, N )-spaces asks that the Rényi entropy Ent(f vol) := f log f d vol is (K, N )-convex along timelike geodesics in a space of probability measures. We refer to [13, Definition 2.8] and [3, Definition 2.17] (see also [5,6]) for the definitions and properties of Lorentzian pre-length spaces and (K, N )-convexity, respectively. On smooth spacetimes, such convexity properties characterize the strong energy condition, cf.…”