Analysis & Applied Math

We prove the existence of solutions to the timelike Lorentzian Monge problem on synthetic spacetimes satisfying the forward timelike measure contraction property $TMCP^+(K,N)$, where $K$ and $N$ correspond to the synthetic curvature and dimension parameters. The cost for this problem is the time separation function (the Lorentz distance), which represents the maximum amount a particle can age while travelling from to $x$ to $y$. The solution is based on the disintegration of measure technique, and amounts to a reduction of the full problem to its one-dimensional counterparts.

As an application, we investigate the intrinsic time asymmetry of synthetic spacetimes satisfying the blended forward $TMCP^+(K_1,N_1)$ and backward $TMCP^-(K_2,N_2)$ conditions, with distinct curvature and dimension parameters in the forward and backward directions. We establish corresponding Bonnet--Myers-type diameter bounds, and a diameter rigidity theorem whose extremal models are asymmetric combinations of de Sitter, Minkowski, and anti-de Sitter-type geometries in the future and past directions.