Timelike Ricci curvature lower bounds via optimal transport for Orlicz-type Lorentzian costs

Abstract

We study the optimal transport problem on globally hyperbolic spacetimes associated with Orlicz-type Lorentzian cost functions of the form $u\circ \ell$, where $u$ is a suitable monotonically increasing and concave function, and $\ell$ is the time separation. Our work encompasses and generalises the case $u(x)=u_p(x)=p^{-1}{x}^p$ for $p\in (0,1)$, as well as the more recent $p<0$, which have been the only examples considered so far in the literature. A fundamental notion for our purposes is the property of $u$-separation for a pair of measures, which generalises McCann's $p$-separation and for which we are able to obtain strong duality to the full Orlicz-type optimization problem. In our main results, we characterise timelike Ricci curvature lower bounds via the convexity of the relative entropy along geodesics arising from the Orlicz-type optimal transport with cost $u\circ \ell$, which is a far-reaching generalisation of McCann's seminal work in the case $u=u_p$, $p\in (0,1)$.

55 pages, some corrections