A Synthetic Null Energy Condition

We give a simpler approach to Kunzinger and Sämann’s theory of Lorentzian length spaces in the globally hyperbolic case; these provide a nonsmooth framework for general relativity. We close a gap in the regularly localizable setting, by showing consistency of two potentially different notions of timelike geodesic segments used in the literature. In the smooth pseudo-Riemannian setting, we show Penrose’ null energy condition is equivalent to a variable lower bound on the timelike Ricci curvature. This allows us to give a nonsmooth reformulation of the null energy condition using the timelike curvature-dimension conditions of Cavalletti and Mondino (and Braun). Although this definition is consistent with the smooth setting, it proves unstable relative to the notion of pointed measured convergence for which timelike curvature-dimensions conditions are known to be stable. We illustrate this instability using a sequence of smooth weighted Lorentzian manifolds-with-boundary that satisfy it, yet converge to a disconnected pair of timelike related points that violate it in the limit.