Optimal transport and timelike lower Ricci curvature bounds on Finsler spacetimes.

We prove that a Finsler spacetime endowed with a smooth reference measure whose induced weighted Ricci curvature \mathrm {Ric}_N is bounded from below by a real number K in every timelike direction satisfies the timelike curvature-dimension condition \mathrm {TCD}_q(K,N) for all q\in (0,1). The converse and a nonpositive-dimensional version (N \le 0) of this result are also shown. Our discussion is based on the solvability of the Monge problem with respect to the q-Lorentz–Wasserstein distance as well as the characterization of q-geodesics of probability measures. One consequence of our work is the sharp timelike Brunn–Minkowski inequality in the Lorentz–Finsler case. [ABSTRACT FROM AUTHOR]