Equivalence of the null energy condition to variable lower bounds on the timelike Ricci curvature for C²-Lorentzian metrics

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 reformulation of the NEC by McCann as variable lower timelike Ricci curvature bounds in [16] remains equivalent to the classical NEC not just for smooth but even for C² C 2 -metrics, where McCann’s original proof needs to be modified.