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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Lemaître–Tolman metric ： ウィキペディア英語版 Lemaître–Tolman metric In mathematical physics, the Lemaître–Tolman metric is the spherically symmetric dust solution of Einstein's field equations was first found by Lemaître in 1933 and then Tolman in 1934. It was later investigated by Bondi in 1947. This solution describes a spherical cloud of dust (finite or infinite) that is expanding or collapsing under gravity. It is also known as the Lemaître-Tolman-Bondi metric and the Tolman metric. The metric is: : $\backslash mathrms^\; =\; \backslash mathrmt^2\; -\; \backslash frac\; \backslash mathrmr^2\; -\; R^2\; \backslash ,\; \backslash mathrm\backslash Omega^2$ where: : $\backslash mathrm\backslash Omega^2\; =\; \backslash mathrm\backslash theta^2\; +\; \backslash sin^2\backslash theta\; \backslash ,\; \backslash mathrm\backslash phi^2$ : $R\; =\; R(t,r)~,~~~~~~~~\; R\text{'}\; =\; \backslash partial\; R\; /\; \backslash partial\; r~,~~~~~~~~\; E\; =\; E(r)\; >\; -\backslash frac$ The matter is comoving, which means its 4-velocity is: : $u^a\; =\; \backslash delta^a\_0\; =\; (1,\; 0,\; 0,\; 0)$ so the spatial coordinates $(r,\; \backslash theta,\; \backslash phi)$ are attached to the particles of dust. The pressure is zero (hence ''dust''), the density is : $8\; \backslash pi\; \backslash rho\; =\; \backslash frac$ and the evolution equation is : $\backslash dot^2\; =\; \backslash frac\; +\; 2\; E$ where : $\backslash dot\; =\; \backslash partial\; R\; /\; \backslash partial\; t$ The evolution equation has three solutions, depending on the sign of $E$, : $E\; >\; 0:~~~~~~~~\; R\; =\; \backslash frac\; (\backslash cosh\backslash eta\; -\; 1)~,~~~~~~~~\; (\backslash sinh\backslash eta\; -\; \backslash eta)\; =\; \backslash frac~;$ : $E\; =\; 0:~~~~~~~~\; R\; =\; \backslash left(\; \backslash frac\; \backslash right)^~;$ : which are known as ''hyperbolic'', ''parabolic'', and ''elliptic'' evolutions respectively. The meanings of the three arbitrary functions, which depend on $r$ only, are: * $E(r)$ – both a local geometry parameter, and the energy per unit mass of the dust particles at comoving coordinate radius $r$, * $M(r)$ – the gravitational mass within the comoving sphere at radius $r$, * $t\_B(r)$ – the time of the big bang for worldlines at radius $r$. Special cases are the Schwarzschild metric in geodesic coordinates $M\; =$ constant, and the Friedmann–Lemaître–Robertson–Walker metric, e.g. $E\; =\; 0~,~~\; t\_B\; =$ constant for the flat case. ==See also== *Lemaître coordinates *Introduction to the mathematics of general relativity *Stress–energy tensor *Metric tensor (general relativity) *Relativistic angular momentum 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lemaître–Tolman metric」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.