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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 Bel decomposition ： ウィキペディア英語版 Bel decomposition In semi-Riemannian geometry, the Bel decomposition, taken with respect to a specific timelike congruence, is a way of breaking up the Riemann tensor of a pseudo-Riemannian manifold into lower order tensors with properties similar to the electric field and magnetic field. Such a decomposition was partially described by Alphonse Matte in 1953 and by Lluis Bel in 1958. This decomposition is particularly important in general relativity. This is the case of four-dimensional Lorentzian manifolds, for which there are only three pieces with simple properties and individual physical interpretations. == Decomposition of the Riemann tensor == In four dimensions the Bel decomposition of the Riemann tensor, with respect to a timelike unit vector field $\backslash vec$, not necessarily geodesic or hypersurface orthogonal, consists of three pieces #the ''electrogravitic tensor'' $E()\_\; =\; R\_\; \backslash ,\; X^m\; \backslash ,\; X^n$ #the ''magnetogravitic tensor'' $B()\_\; =\; ^\backslash star\; R\}\_\; \backslash ,\; X^m\; \backslash ,\; X^n$ #the ''topogravitic tensor'' $L()\_\; =\; ^\backslash star\; R^\backslash star\}\_\; \backslash ,\; X^m\; \backslash ,\; X^n$ Because these are all ''transverse'' (i.e. projected to the spatial hyperplane elements orthogonal to our timelike unit vector field), they can be represented as linear operators on three-dimensional vectors, or as three by three real matrices. They are respectively symmetric, traceless, and symmetric (6,8,6 linearly independent components, for a total of 20). If we write these operators as E, B, L respectively, the principal invariants of the Riemann tensor are obtained as follows: *$K\_1/4$ is the trace of E2 + L2 - 2 B BT, *$-K\_2/8$ is the trace of B ( E - L ), *$K\_3/8$ is the trace of E L - B2. 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「Bel decomposition」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.