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

TL;DR

This paper extends the optimal transport framework on spacetimes to Orlicz-type Lorentzian costs, characterizing timelike Ricci curvature bounds through entropy convexity, generalizing previous p-parameter cases.

Contribution

It introduces a new class of Lorentzian costs based on Orlicz functions, establishing duality and curvature characterizations beyond prior p-parameter models.

Findings

Generalized optimal transport costs to Orlicz-type functions.

Established strong duality for the new class of costs.

Characterized Ricci curvature bounds via entropy convexity.

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 $u = u_p$, $p \in (0,1)$.