Polish spaces of causal curves

TL;DR

This paper introduces a new way to topologize spaces of causal curves in spacetimes, making them Polish and useful for optimal transport theory, with applications to causal measure evolution.

Contribution

It proposes a parametrization-based topologization of causal curves, resulting in Polish spaces that facilitate analysis in optimal transport and causal measure evolution.

Findings

Spaces of causal curves are separable and completely metrizable.

The approach enables a well-defined notion of causal time-evolution of measures.

Applications to globally hyperbolic spacetimes demonstrate physical relevance.

Abstract

We propose and study a new approach to the topologization of spaces of (possibly not all) future-directed causal curves in a stably causal spacetime. It relies on parametrizing the curves "in accordance" with a chosen time function. Thus obtained topological spaces of causal curves are separable and completely metrizable, i.e. Polish. The latter property renders them particularly useful in the optimal transport theory. To illustrate this fact, we explore the notion of a causal time-evolution of measures in globally hyperbolic spacetimes and discuss its physical interpretation.