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On a Riemannian manifold, lower Ricci curvature bounds are known to becharacterized by geodesic convexity properties of various entropies withrespect to the Kantorovich-Rubinstein-Wasserstein square distance from optimaltransportation. These notions also make sense in a (nonsmooth) metric measuresetting, where they have found powerful applications. This article initiatesthe development of an analogous theory for lower Ricci curvature bounds intimelike directions on a (globally hyperbolic) Lorentzian manifold. Inparticular, we lift fractional powers of the Lorentz distance (a.k.a. timeseparation function) to probability measures on spacetime, and show the strongenergy condition of Hawking and Penrose is equivalent to geodesic convexity ofthe Boltzmann-Shannon entropy there. This represents a significant first steptowards a formulation of the strong energy condition and exploration of itsconsequences in nonsmooth spacetimes, and hints at new connections linking thetheory of gravity to the second law of thermodynamics.

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