Causal evolution of probability measures and continuity equation

This is the authors' abstract. We don't add key points for this paper.

We study the notion of a causal time-evolution of a conserved nonlocal physical quantity in a globally hyperbolic spacetime M. The role of the `global time' is played by a chosen Cauchy temporal function T, whereas the instantaneous configurations of the nonlocal quantity are modeled by probability measures μₜ supported on the corresponding time slices 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 extended to probability measures, (ii) with the help of a probability measure σ 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^∞loc-regularity, with which the map t ↦ μₜ 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 H¹loc-Sobolev space. This enables us to construct X as a vector field in a sense `tangent' to σ. In addition, we discuss the general covariance of descriptions (i)-(iii), unraveling the geometrical, observer-independent notions behind them.