Optimal transport on null hypersurfaces and the null energy condition

TL;DR

This paper develops a framework for optimal transport on null hypersurfaces in Lorentzian manifolds, linking it to the null energy condition and deriving key geometric and stability results.

Contribution

It introduces a novel approach to optimal transport in degenerate null hypersurfaces, characterizing the null energy condition via entropy convexity along null geodesics.

Findings

Characterization of null energy condition through entropy convexity

Stability of spacetime convergence under null transport

New rigidity statements in Hawking's area theorem

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 $+\infty$. 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).