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 R i c N Ric_N is bounded from below by a real number K K in every timelike direction satisfies the timelike curvature-dimension condition T C D q ( K , N ) TCD_q(K,N) for all q ∈<!-- ∈ --> ( 0 , 1 ) q∈ (0,1) . The converse and a nonpositive-dimensional version ( N ≤<!-- ≤ --> 0 N ≤ 0 ) of this result are also shown. Our discussion is based on the solvability of the Monge problem with respect to the q q -Lorentz–Wasserstein distance as well as the characterization of q q -geodesics of probability measures. One consequence of our work is the sharp timelike Brunn–Minkowski inequality in the Lorentz–Finsler case.