Causal evolution of probability measures and continuity equation

TL;DR

This paper develops a framework for describing the causal evolution of probability measures in a globally hyperbolic spacetime, connecting causality, measure theory, and the continuity equation in a geometrically invariant way.

Contribution

It introduces three equivalent formulations of causal evolution of measures, linking causality, curve measures, and vector fields, with insights into their topological and geometric properties.

Findings

Causality can be characterized via extended precedence relations, measures on causal curves, or causal vector fields.

The compact-open topology aligns with Sobolev space properties of causal curves.

A vector field tangent to the measure evolution can be constructed within this framework.

Abstract

We study the notion of a causal time-evolution of a conserved nonlocal physical quantity in a globally hyperbolic spacetime $\mathcal{M}$. The role of the `global time' is played by a chosen Cauchy temporal function $\mathcal{T}$, whereas the instantaneous configurations of the nonlocal quantity are modeled by probability measures $\mu_t$ supported on the corresponding time slices $\mathcal{T}^{-1}(t)$. We show that the causality of such an evolution can be expressed in three equivalent ways: (i) via the causal precedence relation $\preceq$ extended to probability measures, (ii) with the help of a probability measure $\sigma$ on the space of future-directed continuous causal curves endowed with the compact-open topology and (iii) through a causal vector field $X$ of $L^\infty_{\textrm{loc}}$-regularity, with which the map $t \mapsto \mu_t$ satisfies the continuity equation in the distributional sense. In the course of the proof we find that the compact-open topology is sensitive to the differential properties of the causal curves, being equal to the topology induced from a suitable $H^1_{\textrm{loc}}$-Sobolev space. This enables us to construct $X$ as a vector field in a sense `tangent' to $\sigma$. In addition, we discuss the general covariance of descriptions (i)-(iii), unraveling the geometrical, observer-independent notions behind them.