The Monge problem, dating back to 1783, is one of the oldest problems in geometric analysis and partial differential equations. In this talk, I will discuss the extension of the Monge problem to synthetic Lorentzian spacetimes, where 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 $x$ to $y$. The solution is based on the needle decomposition technique, and amounts to a reduction of the full problem to its one-dimensional counterparts. As an application, we prove a timelike rigidity theorem that controls the branching of synthetic spacetimes.