PDE aspects of the dynamical optimal transport in the Lorentzian setting

One of the crucial features of optimal transport on Riemannian manifolds is the equivalence of the static', original, formulation of the problem and of the dynamic’ one, based on the study of the continuity equation. This furnishes the key link between Wasserstein geometry and PDEs that has found so many applications in the last 20 years. In this paper we investigate this kind of equivalence on spacetimes. At the PDE level, this requires to transition from the continuity equation to a suitable continuity inequality', to which we shall refer to as causal continuity inequality’. As a direct consequence of our findings we obtain a Lorentzian version of the celebrated Benamou–Brenier formula.