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- W4287239187 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 $L^2_{textrm{loc}}$-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 a 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." @default.
- W4287239187 created "2022-07-25" @default.
- W4287239187 creator A5056101803 @default.
- W4287239187 date "2021-04-06" @default.
- W4287239187 modified "2023-09-24" @default.
- W4287239187 title "Causal evolution of probability measures and continuity equation" @default.
- W4287239187 doi "https://doi.org/10.48550/arxiv.2104.02552" @default.
- W4287239187 hasPublicationYear "2021" @default.
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