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

2025/12/23 by Melanie Graf, Graf, Melanie, Yaver Gulusoy +1
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geometric Analysis and Curvature Flows #Noncommutative and Quantum Gravity Theories

paper · doi:10.48550/arxiv.2512.20310

openalex publication_date 2025/12/23 · openalex created_date 2025/12/25 · openalex updated_date 2026/07/28

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 C²-metrics, where McCann's original proof needs to be modified.

Citations

• On the geometry of synthetic null hypersurfaces
• Lorentzian Gromov-Hausdorff convergence and pre-compactness
• Introducing Various Notions of Distances between Space-Times
• Optimal transport on null hypersurfaces and the null energy condition
• Ricci curvature bounds and rigidity for non-smooth Riemannian and semi-Riemannian metrics
• A brief introduction to non-regular spacetime geometry
• Causal convergence conditions through variable timelike Ricci curvature bounds
• A synthetic null energy condition
• Characterization of the null energy condition via displacement convexity of entropy
• Timelike Ricci bounds for low regularity spacetimes by optimal transport
• Rényi's entropy on Lorentzian spaces. Timelike curvature-dimension conditions
• Ricci curvature for metric-measure spaces via optimal transport
• Ricci curvature for metric-measure spaces via optimal transport
• Singularity Theorems and Their Consequences

Related

• A synthetic null energy condition
• Null injectivity estimate under an upper bound on the curvature
• Causal convergence conditions through variable timelike Ricci curvature bounds
• A sharp isoperimetric-type inequality for Lorentzian spaces satisfying timelike Ricci lower bounds
• Timelike Ricci bounds for low regularity spacetimes by optimal transport