Causal convergence conditions through variable timelike Ricci curvature bounds

Abstract

We describe a nonsmooth notion of globally hyperbolic, regular length metric spacetimes $(\mathrm{M},l)$. It is based on ideas of Kunzinger-S\"amann, but does not require Lipschitz continuity of causal curves. We study geodesics on $\mathrm{M}$ and the space of probability measures over $\mathrm{M}$ in detail. Furthermore, for such a spacetime endowed with a reference measure $\mathfrak{m}$, a lower semicontinuous function $k\colon \mathrm{M} \to \textbf{R}$, and constants $0<p<1$ and $N\geq 1$, we introduce and study the entropic timelike curvature dimension condition $\smash{\mathrm{TCD}_p^e(k,N)}$ with variable Ricci curvature bound $k$. This provides a unified synthetic approach to general relativistic energy conditions, including $\bullet$ the Hawking-Penrose strong energy condition $\mathrm{Ric}\geq 0$, or more generally $\mathrm{Ric}\geq K$ for constant $K\in\textbf{R}$, in all timelike directions, $\bullet$ the weak energy condition $\mathrm{Ric} \geq \mathrm{scal} - \Lambda$ in all timelike directions, and $\bullet$ the null energy condition $\smash{\mathrm{Ric} \geq 0}$ in all null directions. Our approach also allows for the synthetic quantification of asymptotic conditions or integral controls on the timelike Ricci curvature. For example, we give a nonsmooth generalization of a timelike diameter estimate of Frankel-Galloway (and Schneider), and of a Hawking-type singularity theorem which requires only that the negative Ricci curvature have small enough integral relative to the maximal mean curvature of an achronal slice. As further applications, we discuss the stability of our notion and provide timelike geometric inequalities. To obtain sharp constants in the latter, we develop the localization paradigm in the variable $k$ framework.