Time functions and $K$-causality between measures

TL;DR

This paper extends the concept of $K^+$ causality from spacetime points to probability measures using optimal transport, showing that key properties like antisymmetry are preserved in the measure-theoretic setting.

Contribution

It generalizes Minguzzi's characterization of $K^+$ to measures and proves the antisymmetry of the extended relation in stably causal spacetimes.

Findings

$K^+$ extends to probability measures via optimal transport.

The measure-theoretic $K^+$ retains antisymmetry in stably causal spacetimes.

The generalization preserves key causal properties from points to measures.

Abstract

Employing the notion of a coupling between measures, drawn from the optimal transport theory, we study the extension of the Sorkin-Woolgar causal relation $K^+$ onto the space $\mathscr{P}(\mathcal{M})$ of Borel probability measures on a given spacetime $\mathcal{M}$. We show that Minguzzi's characterization of $K^+$ in terms of time functions possesses a "measure-theoretic" generalization. Moreover, we prove that the relation $K^+$ extended onto $\mathscr{P}(\mathcal{M})$ retains its property of antisymmetry for $\mathcal{M}$ stably causal.