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We study the notion of a causal time-evolution of a conserved nonlocalphysical quantity in a globally hyperbolic spacetime $\mathcal{M}$. The role ofthe `global time' is played by a chosen Cauchy temporal function $\mathcal{T}$,whereas the instantaneous configurations of the nonlocal quantity are modeledby probability measures $\mu_t$ supported on the corresponding time slices$\mathcal{T}^{-1}(t)$. We show that the causality of such an evolution can beexpressed in three equivalent ways: (i) via the causal precedence relation$\preceq$ extended to probability measures, (ii) with the help of a probabilitymeasure $\sigma$ on the space of future-directed continuous causal curvesendowed with the compact-open topology and (iii) through a causal vector field$X$ of $L^2_{\textrm{loc}}$-regularity, with which the map $t \mapsto \mu_t$satisfies the continuity equation in the distributional sense. In the course ofthe proof we find that the compact-open topology is sensitive to thedifferential properties of the causal curves, being equal to a topology inducedfrom a suitable $H^1_{\textrm{loc}}$-Sobolev space. This enables us toconstruct $X$ as a vector field in a sense `tangent' to $\sigma$. In addition,we discuss the general covariance of descriptions (i)-(iii), unraveling thegeometrical, observer-independent notions behind them.

Causal evolution of probability measures and continuity equation