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 RicN is bounded from below by a real number K in every timelike direction satisfies the timelike curvature-dimension condition TCDq(K, N) for all q is an element of (0, 1). The converse and a nonpositive-dimensional version (N <= 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 qgeodesics of probability measures. One consequence of our work is the sharp timelike Brunn-Minkowski inequality in the Lorentz-Finsler case.