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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 Raychaudhuri equation ： ウィキペディア英語版 Raychaudhuri equation In general relativity, the Raychaudhuri equation, or Landau–Raychaudhuri equation,〔Spacetime as a deformable solid, M. O. Tahim, R. R. Landim, and C. A. S. Almeida, .〕 is a fundamental result describing the motion of nearby bits of matter. The equation is important as a fundamental lemma for the Penrose-Hawking singularity theorems and for the study of exact solutions in general relativity, but has independent interest, since it offers a simple and general validation of our intuitive expectation that gravitation should be a universal attractive force between any two bits of mass-energy in general relativity, as it is in Newton's theory of gravitation. The equation was discovered independently by the celebrated Indian physicist Amal Kumar Raychaudhuri and the Soviet physicist Lev Landau.〔p. 84, ''The large scale structure of space-time'', Stephen W. Hawking and G. F. R. Ellis, Cambridge University Press, 1973, ISBN 0-521-09906-4.〕 ==Mathematical statement== Given a timelike unit vector field $\backslash vec$ (which can be interpreted as a family or congruence of nonintersecting world lines, not necessarily geodesics), Raychaudhuri's equation can be written :$\backslash dot\; =\; -\; \backslash frac\; -\; 2\; \backslash sigma^2\; +\; 2\; \backslash omega^2\; -\; ">)^a\}\_a\; +\; \_$ where :$\backslash sigma^2\; =\; \backslash frac\; \backslash sigma\_\; \backslash ,\; \backslash sigma^,\; \backslash ;\; \backslash omega^2\; =\; \backslash frac\; \backslash omega\_\; \backslash ,\; \backslash omega^$ are (non-negative) quadratic invariants of the ''shear tensor'' :$\backslash sigma\_\; =\; \backslash theta\_\; -\; \backslash frac\; \backslash ,\; \backslash theta\; \backslash ,\; h\_$ and the ''vorticity tensor'' :$\backslash omega\_\; =\; \_a\; \backslash ,\; \_b\; X\_$ respectively. Here, :$\backslash theta\_\; =\; \_a\; \backslash ,\; \_b\; X\_$ is the ''expansion tensor'', $\backslash theta$ is its trace, called the ''expansion scalar'', and :$h\_\; =\; g\_\; +\; X\_a\; \backslash ,\; X\_b$ is the ''projection tensor'' onto the hyperplanes orthogonal to $\backslash vec$. Also, dot denotes differentiation with respect to proper time counted along the world lines in the congruence. Finally, the trace of the tidal tensor $E()\_$ can also be written :$">)^a\}\_\; =\; R\_\; \backslash ,\; X^m\; \backslash ,\; X^n$ This quantity is sometimes called the ''Raychaudhuri scalar''. 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「Raychaudhuri equation」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.