Timelike Ricci bounds for low regularity spacetimes by optimal transport

Abstract

We prove that a globally hyperbolic smooth spacetime endowed with a [Formula: see text]-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 [Formula: see text], in fact the stronger timelike curvature-dimension condition holds. In this regularity, we also obtain uniqueness of chronological optimal couplings and chronological geodesics.