Optimal transport on null hypersurfaces and the null energy condition

2024/08/16 by Cavalletti, Fabio, Manini, Davide, Mondino, Andrea · 2 citations
#83C05 (primary) 49Q22 (secondary) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2408.08986

Abstract

The goal of the present work is to study optimal transport on null hypersurfaces inside Lorentzian manifolds. The challenge here is that optimal transport along a null hypersurface is completely degenerate, as the cost takes only the two values 0 and +∞. The tools developed in the manuscript enable to give an optimal transport characterization of the null energy condition (namely, non-negative Ricci curvature in the null directions) for Lorentzian manifolds in terms of convexity properties of the Boltzmann--Shannon entropy along null-geodesics of probability measures. We obtain as applications: a stability result under convergence of spacetimes, a comparison result for null-cones, and the Hawking area theorem (both in sharp form, for possibly weighted measures, and with apparently new rigidity statements).

Cited by

• Equivalence of the null energy condition to variable lower bounds on the timelike Ricci curvature for C²-Lorentzian metrics
• The Penrose singularity theorem, MOTS stability, and horizon topology in weighted spacetimes

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