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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 Newman–Penrose formalism ： ウィキペディア英語版 Newman–Penrose formalism The Newman–Penrose (NP) formalism〔 The original paper by Newman and Penrose, which introduces the formalism, and uses it to derive example results.〕〔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.〕 is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the space-time, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The most often-used variables in the formalism are the Weyl scalars, derived from the Weyl tensor. In particular, it can be shown that one of these scalars--$\backslash Psi\_4$ in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system. Newman and Penrose introduced the following functions as primary quantities using this tetrad:〔〔 * Twelve complex spin coefficients (in three groups) which describe the change in the tetrad from point to point: $\backslash kappa,\; \backslash rho,\; \backslash sigma,\; \backslash tau\backslash ,;\; \backslash lambda,\; \backslash mu,\; \backslash nu,\; \backslash pi\backslash ,;\; \backslash epsilon,\; \backslash gamma,\; \backslash beta,\; \backslash alpha.$. * Five complex functions encoding Weyl tensors in the tetrad basis: $\backslash Psi\_0,\; \backslash ldots,\; \backslash Psi\_4$. * Ten functions encoding Ricci tensors in the tetrad basis: $\backslash Phi\_,\; \backslash Phi\_,\; \backslash Phi\_,\; \backslash Lambda$ (real); $\backslash Phi\_,\; \backslash Phi\_,\; \backslash Phi\_,\; \backslash Phi\_,\; \backslash Phi\_,\; \backslash Phi\_$ (complex). In many situations—especially algebraically special spacetimes or vacuum spacetimes—the Newman–Penrose formalism simplifies dramatically, as many of the functions go to zero. This simplification allows for various theorems to be proven more easily than using the standard form of Einstein's equations. In this article, we will only employ the tensorial rather than spinorial version of NP formalism, because the former is easier to understand and more popular in relevant papers. One can refer to ref.〔Peter O'Donnell. ''Introduction to 2-Spinors in General Relativity''. Singapore: World Scientific, 2003.〕 for a unified formulation of these two versions. ==Null tetrad and sign convention== The formalism is developed for four-dimensional spacetime, with a Lorentzian-signature metric. At each point, a tetrad (set of four vectors) is introduced. The first two vectors, $l^\backslash mu$ and $n^\backslash mu$ are just a pair of standard (real) null vectors such that $l^a\; n\_a\; =\; -1$. For example, we can think in terms of spherical coordinates, and take $l^a$ to be the outgoing null vector, and $n^a$ to be the ingoing null vector. A complex null vector is then constructed by combining a pair of real, orthogonal unit space-like vectors. In the case of spherical coordinates, the standard choice is :$m^\backslash mu\; =\; \backslash frac\; +\; i\; \backslash hat\; \backslash right)^\backslash mu\backslash \; .$ The complex conjugate of this vector then forms the fourth element of the tetrad. Two sets of signature and normalization conventions are in use for NP formalism: $\backslash$ and $\backslash$. The former is the original one that was adopted when NP formalism was developed〔〔 and has been widely used〔Subrahmanyan Chandrasekhar. ''The Mathematical Theory of Black Holes''. Chicago: University of Chikago Press, 1983.〕〔J B Griffiths. ''Colliding Plane Waves in General Relativity''. Oxford: Oxford University Press, 1991.〕 in black-hole physics, gravitational waves and various other areas in general relativity. However, it is the latter convention that is usually employed in contemporary study of black holes from quasilocal perspectives〔Ivan Booth. ''Black hole boundaries''. Canadian Journal of Physics, 2005, 83(11): 1073-1099. (arXiv:gr-qc/0508107v2 )〕 (such as isolated horizons〔Abhay Ashtekar, Christopher Beetle, Jerzy Lewandowski. ''Geometry of generic isolated horizons''. Classical and Quantum Gravity, 2002, 19(6): 1195-1225. (arXiv:gr-qc/0111067v2 )〕 and dynamical horizons〔Abhay Ashtekar, Badri Krishnan. ''Dynamical horizons: energy, angular momentum, fluxes and balance laws''. Physical Review Letters, 2002, 89(26): 261101. (arXiv:gr-qc/0207080v3 )〕〔Abhay Ashtekar, Badri Krishnan. ''Dynamical horizons and their properties''. Physical Review D, 2003, 68(10): 104030. (arXiv:gr-qc/0308033v4 )〕). In this article, we will utilize $\backslash$ for a systematic review of the NP formalism (see also refs.〔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 )〕). It's important to note that, when switching from $\backslash$ to $\backslash$, definitions of the spin coefficients, Weyl-NP scalars $\backslash Psi\_$ and Ricci-NP scalars $\backslash Phi\_$ need to change their signs; this way, the Einstein-Maxwell equations can be left unchanged. In NP formalism, the complex null tetrad contains two real null (co)vectors $\backslash$ and two complex null (co)vectors $\backslash$. Being ''null'' (co)vectors, ''self''-normalization of $\backslash$ are naturally vanishes, $l\_a\; l^a=n\_a\; n^a=m\_a\; m^a=\backslash bar\_a\; \backslash bar^a=0$, so the following two pairs of ''cross''-normalization are adopted $l\_a\; n^a=-1=l^a\; n\_a\backslash ,,\backslash quad\; m\_a\; \backslash bar^a=1=m^a\; \backslash bar\_a\backslash ,,$ while contractions between the two pairs are also vanishing, $l\_a\; m^a=l\_a\; \backslash bar^a=n\_a\; m^a=n\_a\; \backslash bar^a=0$. Here the indices can be raised and lowered by the global metric $g\_$ which in turn can be obtained via $g\_=-l\_a\; n\_b\; -\; n\_a\; l\_b\; +m\_a\; \backslash bar\_b\; +\backslash bar\_a\; m\_b\backslash ,,\; \backslash quad\; g^=-l^a\; n^b\; -\; n^a\; l^b\; +m^a\; \backslash bar^b\; +\backslash bar^a\; m^b\backslash ,.$ 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「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.