Monge's transport problem in synthetic Lorentzian spacetimes

Abstract

We prove the existence of solutions to the Lorentzian Monge problem on synthetic Lorentzian spacetimes that satisfy the forward timelike measure contraction property $TMC{P}^{+}(K,N)$. The cost for this problem is the time separation function $\ell (x,y)$ (the Lorentz distance), which represents the maximum amount a particle can age when traveling 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 of needle decomposition, we introduce the blended $TMCP$ condition, and obtain a quantitative refinement of the synthetic Hawking singularity theorem. We additionally prove a timelike nonbranching reversibility theorem for $TC{D}_p(K,N)$ spacetimes, showing that forward $p$-essentially nonbranching spacetimes are backward $p$-essentially nonbranching.