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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2104.02552 |
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| _version_ | 1866910765164789760 |
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| author | Miller, Tomasz |
| author_facet | Miller, Tomasz |
| contents | 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 $μ_t$ 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 $σ$ on the space of future-directed continuous causal curves endowed with the compact-open topology and (iii) through a causal vector field $X$ of $L^\infty_{\textrm{loc}}$-regularity, with which the map $t \mapsto μ_t$ 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 $σ$. In addition, we discuss the general covariance of descriptions (i)-(iii), unraveling the geometrical, observer-independent notions behind them. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2104_02552 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Causal evolution of probability measures and continuity equation Miller, Tomasz Mathematical Physics General Relativity and Quantum Cosmology Functional Analysis 53C50 (Primary) 53C80, 28E99, 60B05 (Secondary) 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 $μ_t$ 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 $σ$ on the space of future-directed continuous causal curves endowed with the compact-open topology and (iii) through a causal vector field $X$ of $L^\infty_{\textrm{loc}}$-regularity, with which the map $t \mapsto μ_t$ 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 $σ$. In addition, we discuss the general covariance of descriptions (i)-(iii), unraveling the geometrical, observer-independent notions behind them. |
| title | Causal evolution of probability measures and continuity equation |
| topic | Mathematical Physics General Relativity and Quantum Cosmology Functional Analysis 53C50 (Primary) 53C80, 28E99, 60B05 (Secondary) |
| url | https://arxiv.org/abs/2104.02552 |