  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.
