Displacement convexity of Boltzmann's entropy characterizes the strong energy condition from general relativity

TL;DR

This paper establishes a novel connection between the strong energy condition in general relativity and the geodesic convexity of Boltzmann's entropy on Lorentzian manifolds, extending optimal transport concepts to spacetime geometry.

Contribution

It develops a new framework linking energy conditions in relativity to entropy convexity in a Lorentzian setting, bridging gravity and thermodynamics.

Findings

Strong energy condition is equivalent to entropy convexity.

Extension of optimal transport to Lorentzian manifolds.

Foundation for nonsmooth spacetime analysis.

Abstract

On a Riemannian manifold, lower Ricci curvature bounds are known to be characterized by geodesic convexity properties of various entropies with respect to the Kantorovich-Rubinstein-Wasserstein square distance from optimal transportation. These notions also make sense in a (nonsmooth) metric measure setting, where they have found powerful applications. This article initiates the development of an analogous theory for lower Ricci curvature bounds in timelike directions on a (globally hyperbolic) Lorentzian manifold. In particular, we lift fractional powers of the Lorentz distance (a.k.a. time separation function) to probability measures on spacetime, and show the strong energy condition of Hawking and Penrose is equivalent to geodesic convexity of the Boltzmann-Shannon entropy there. This represents a significant first step towards a formulation of the strong energy condition and exploration of its consequences in nonsmooth spacetimes, and hints at new connections linking the theory of gravity to the second law of thermodynamics.