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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 Vaidya metric ： ウィキペディア英語版 Vaidya metric In general relativity, the Vaidya metric describes the non-empty external spacetime of a spherically symmetric and nonrotating star which is either emitting or absorbing null dusts. It is named after the Indian physicist Prahalad Chunnilal Vaidya and constitutes a simplest non-static generalization of the non-radiative Schwarzschild solution to Einstein's field equation, and therefore is also called the "radiating(/shining) Schwarzschild metric". == From Schwarzschild to Vaidya metrics == The Schwarzschild metric as the static and spherically symmetric solution to Einstein's equation reads $(1)\backslash quad\; ds^2=-\backslash Big(\; 1-\backslash frac\; \backslash Big)\; dt^2+\backslash Big(\; 1-\backslash frac\; \backslash Big)^dr^2+r^2(d\backslash theta^2+\backslash sin^2\backslash theta\backslash ,d\backslash phi^2)\backslash ;.$ To remove the coordinate singularity of this metric at $r=2M$, one could switch to the Eddington–Finkelstein coordinates. Thus, introduce the "retarded(/outgoing)" null coordinate $u$ by $(2)\backslash quad\; t=u+r+2M\backslash ln\backslash Big(\backslash frac-1\backslash Big)\backslash qquad\backslash Rightarrow\backslash quad\; dt=du+\backslash Big(\; 1-\backslash frac\; \backslash Big)^dr\backslash ;,$ and Eq(1) could be transformed into the "retarded(/outgoing) Schwarzschild metric" $(3)\backslash quad\; ds^2=-\backslash Big(\; 1-\backslash frac\; \backslash Big)\; du^2-2dudr+r^2(d\backslash theta^2+\backslash sin^2\backslash theta\backslash ,d\backslash phi^2)\backslash ;;$ or, we could instead employ the "advanced(/ingoing)" null coordinate $v$ by $(4)\backslash quad\; t=v-r-2M\backslash ln\backslash Big(\backslash frac-1\backslash Big)\backslash qquad\backslash Rightarrow\backslash quad\; dt=dv-\backslash Big(\; 1-\backslash frac\; \backslash Big)^dr\backslash ;,$ so Eq(1) becomes the "advanced(/ingoing) Schwarzschild metric" $(5)\backslash quad\; ds^2=-\backslash Big(\; 1-\backslash frac\; \backslash Big)\; dv^2+2dvdr+r^2(d\backslash theta^2+\backslash sin^2\backslash theta\backslash ,d\backslash phi^2)\backslash ;.$ Eq(3) and Eq(5), as static and spherically symmetric solutions, are valid for both ordinary celestial objects with finite radii and singular objects such as black holes. It turns out that, it is still physically reasonable if one extends the mass parameter $M$ in Eqs(3) and Eq(5) from a constant to functions of the corresponding null coordinate, $M(u)$ and $M(v)$ respectively, thus $(6)\backslash quad\; ds^2=-\backslash Big(\; 1-\backslash frac\; \backslash Big)\; du^2-2dudr+r^2(d\backslash theta^2+\backslash sin^2\backslash theta\backslash ,d\backslash phi^2)\backslash ;,$ $(7)\backslash quad\; ds^2=-\backslash Big(\; 1-\backslash frac\; \backslash Big)\; dv^2+2dvdr+r^2(d\backslash theta^2+\backslash sin^2\backslash theta\backslash ,d\backslash phi^2)\backslash ;.$ The extended metrics Eq(6) and Eq(7) are respectively the "retarded(/outgoing)" and "advanced(/ingoing)" Vaidya metrics.〔Eric Poisson. ''A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics''. Cambridge: Cambridge University Press, 2004. Section 4.3.5 and Section 5.1.8.〕〔Jeremy Bransom Griffiths, Jiri Podolsky. ''Exact Space-Times in Einstein's General Relativity''. Cambridge: Cambridge University Press, 2009. Section 9.5.〕 It is also interesting and sometimes useful to recast the Vaidya metrics Eqs(6)(7) into the form $(8)\backslash quad\; ds^2\; =\backslash fracdu^2\; +ds^2(\backslash text)=\backslash fracdv^2\; +ds^2(\backslash text)\backslash ,,$ where $ds^2(\backslash text)=-\; du^2-2dudr+r^2\; (d\backslash theta^2+\backslash sin^2\backslash theta\backslash ,\; d\backslash phi^2\; )=-\; dv^2+2dvdr+r^2\; (d\backslash theta^2+\backslash sin^2\backslash theta\backslash ,\; d\backslash phi^2\; )=-dt^2+dr^2+r^2\; (d\backslash theta^2+\backslash sin^2\backslash theta\backslash ,\; d\backslash phi^2\; )$ represents the metric of flat spacetime. 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「Vaidya 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.