Equivalence of the null energy condition to variable lower bounds on the timelike Ricci curvature for $C^2$-Lorentzian metrics

TL;DR

This paper demonstrates that the null energy condition (NEC) is equivalent to variable lower bounds on the timelike Ricci curvature for $C^2$-Lorentzian metrics, extending classical results to less smooth spacetimes.

Contribution

It proves the equivalence of the NEC and variable lower Ricci curvature bounds for $C^2$-metrics, generalizing McCann's reformulation beyond smooth spacetimes.

Findings

NEC is equivalent to variable lower Ricci bounds for $C^2$-metrics

Extension of McCann's reformulation to less smooth metrics

Supports consistency of generalized energy conditions in Lorentzian geometry

Abstract

The null energy or null convergence condition (NEC) is one of the fundamental assumptions necessary for many celebrated results from Lorentzian Geometry and Mathematical General Relativity. As such there have been several recent efforts to find a good generalization of this condition to the new setting of Lorentzian length spaces or metric measure spacetimes. One important property any such generalization should fulfill is consistency with the classical formulation for a class of spacetimes as large as possible. The purpose of this note is to show that the recent reformulation of the NEC by McCann as variable lower timelike Ricci curvature bounds (arXiv:2304.14341) remains equivalent to the classical NEC not just for smooth but even for $C^2$-metrics, where McCann's original proof needs to be modified.