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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 Ricci scalars (Newman–Penrose formalism) ： ウィキペディア英語版 Ricci scalars (Newman–Penrose formalism) In the Newman–Penrose (NP) formalism of general relativity, independent components of the Ricci tensors of a four-dimensional spacetime are encoded into seven (or ten) Ricci scalars which consist of three real scalars $\backslash ,\; \backslash Phi\_\backslash \}$, three (or six) complex scalars $\backslash \_\backslash ,,\backslash Phi\_=\backslash overline\_\backslash ,,\backslash Phi\_=\backslash overline\_\backslash \}$ and the NP curvature scalar $\backslash Lambda$. Physically, Ricci-NP scalars are related with the energy–momentum distribution of the spacetime due to Einstein's field equation. ==Definitions== Given a complex null tetrad $\backslash$ and with the convention $\backslash$, the Ricci-NP scalars are defined by〔Jeremy Bransom Griffiths, Jiri Podolsky. ''Exact Space-Times in Einstein's General Relativity''. Cambridge: Cambridge University Press, 2009. Chapter 2.〕〔Valeri P Frolov, Igor D Novikov. ''Black Hole Physics: Basic Concepts and New Developments''. Berlin: Springer, 1998. Appendix E.〕〔Abhay Ashtekar, Stephen Fairhurst, Badri Krishnan. ''Isolated horizons: Hamiltonian evolution and the first law''. Physical Review D, 2000, 62(10): 104025. Appendix B. (gr-qc/0005083 )〕 (where overline means complex conjugate) $\backslash Phi\_:=\backslash fracR\_l^a\; l^b\backslash ,,\; \backslash quad\; \backslash Phi\_:=\backslash fracR\_(\backslash ,l^a\; n^b+m^a\backslash bar^b)\backslash ,,\; \backslash quad\backslash Phi\_:=\backslash fracR\_n^a\; n^b\backslash ,,\; \backslash quad\backslash Lambda:=\backslash frac\backslash ,;$ $\backslash Phi\_:=\backslash fracR\_l^a\; m^b\backslash ,,\; \backslash quad\backslash ;\; \backslash Phi\_:=\backslash fracR\_l^a\; \backslash bar^b=\backslash overline\_\backslash ,,$ $\backslash Phi\_:=\backslash fracR\_m^a\; m^b\backslash ,,\; \backslash quad\; \backslash Phi\_:=\backslash fracR\_\backslash bar^a\; \backslash bar^b=\backslash overline\_\backslash ,,$ $\backslash Phi\_:=\backslash fracR\_\backslash bar^a\; n^b\backslash ,,\; \backslash quad\backslash ;\; \backslash Phi\_:=\backslash fracR\_m^a\; n^b=\backslash overline\_\backslash ,.$ Remark I: In these definitions, $R\_$ could be replaced by its trace-free part $Q\_=R\_-\backslash fracg\_R$〔 or by the Einstein tensor $G\_=R\_-\backslash fracg\_R$ because of the normalization (i.e. inner product) relations that :$l\_a\; l^a\; =\; n\_a\; n^a\; =\; m\_a\; m^a\; =\; \backslash bar\_a\; \backslash bar^a=0\backslash ,,$ :$l\_a\; m^a\; =\; l\_a\; \backslash bar^a\; =\; n\_a\; m^a\; =\; n\_a\; \backslash bar^a=0\backslash ,.$ Remark II: Specifically for electrovacuum, we have $\backslash Lambda=0$, thus $24\backslash Lambda\backslash ,=0=\backslash ,R\_g^\backslash ,=\backslash ,R\_\backslash Big(-2l^a\; n^b+2m^a\backslash bar^b\; \backslash Big)\backslash ;\; \backslash Rightarrow\; \backslash ;\; R\_l^a\; n^b\backslash ,=\backslash ,R\_m^a\backslash bar^b\backslash ,,$ and therefore $\backslash Phi\_$ is reduced to $\backslash Phi\_:=\backslash fracR\_(\backslash ,l^a\; n^b+m^a\backslash bar^b)=\backslash fracR\_l^a\; n^b=\backslash fracR\_m^a\backslash bar^a\backslash ,.$ Remark III: If one adopts the convention $\backslash$, the definitions of $\backslash Phi\_$ should take the opposite values;〔Ezra T Newman, Roger Penrose. ''An Approach to Gravitational Radiation by a Method of Spin Coefficients''. Journal of Mathematical Physics, 1962, 3(3): 566-768.〕〔Ezra T Newman, Roger Penrose. ''Errata: An Approach to Gravitational Radiation by a Method of Spin Coefficients''. Journal of Mathematical Physics, 1963, 4(7): 998.〕〔Subrahmanyan Chandrasekhar. ''The Mathematical Theory of Black Holes''. Chicago: University of Chikago Press, 1983.〕〔Peter O'Donnell. ''Introduction to 2-Spinors in General Relativity''. Singapore: World Scientific, 2003.〕 that is to say, $\backslash Phi\_\backslash mapsto-\backslash Phi\_$ after the signature transition. 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「Ricci scalars (Newman–Penrose formalism)」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.