A synthetic null energy condition

TL;DR

This paper introduces a nonsmooth framework for general relativity using Lorentzian length spaces, relates energy conditions to curvature bounds, and discusses stability issues in these conditions under convergence.

Contribution

It simplifies the theory of Lorentzian length spaces, establishes equivalence of geodesic notions, and reformulates the null energy condition in a nonsmooth setting.

Findings

Null energy condition is equivalent to a lower bound on timelike Ricci curvature in smooth settings.

Nonsmooth null energy condition can be formulated via timelike curvature-dimension conditions.

The stability of these conditions under convergence is limited, as shown by counterexamples.

Abstract

We give a simplified approach to Kunzinger & Saemann'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 psuedo-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 \& 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.