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 ${\mathrm{R}\mathrm{i}\mathrm{c}}_N$ is bounded from below by a real number $K$ in every timelike direction satisfies the timelike curvature-dimension condition ${\mathrm{T}\mathrm{C}\mathrm{D}}_q(K,N)$ for all $q\in (0,1)$. A nonpositive-dimensional version ($N\le 0$) of this result is 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.