Timelike Ricci bounds for low regularity spacetimes by optimal transport

TL;DR

This paper establishes timelike Ricci bounds for low regularity spacetimes using optimal transport, leading to geometric inequalities and curvature conditions in a synthetic setting.

Contribution

It proves timelike measure-contraction and curvature-dimension conditions for low regularity spacetimes, extending geometric analysis in Lorentzian geometry.

Findings

Spacetimes with C^1 metrics satisfy timelike measure-contraction property.

C^{1,1} spacetimes obey stronger curvature-dimension conditions.

Results include sharp geometric inequalities without nonbranching assumptions.

Abstract

We prove that a globally hyperbolic smooth spacetime endowed with a $\smash{\mathrm{C}^1}$-Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions, in a distributional sense, obeys the timelike measure-contraction property. This result includes a class of spacetimes with borderline regularity for which local existence results for the vacuum Einstein equation are known in the setting of spaces with timelike Ricci bounds in a synthetic sense. In particular, these spacetimes satisfy timelike Brunn-Minkowski, Bonnet-Myers, and Bishop-Gromov inequalities in sharp form, without any timelike nonbranching assumption. If the metric is even $\smash{\mathrm{C}^{1,1}}$, in fact the stronger timelike curvature-dimension condition holds. In this regularity, we also obtain uniqueness of chronological optimal couplings and chronological geodesics.