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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 Self-dual Palatini action ： ウィキペディア英語版 Self-dual Palatini action Ashtekar variables, which were a new canonical formalism of general relativity, raised new hopes for the canonical quantization of general relativity and eventually led to loop quantum gravity. Smolin and others independently discovered that there exists in fact a Lagrangian formulation of the theory by considering the self-dual formulation of the Tetradic Palatini action principle of general relativity.〔J. Samuel. ''A Lagrangian basis for Ashtekar's formulation of canonical gravity''. Pramana J. Phys. 28 (1987) L429-32〕〔T. Jacobson and L. Smolin. ''The left-handed spin connection as a variable for canonical gravity.'' Phys. Lett. B196 (1987) 39-42.〕〔T. Jacobson and L. Smolin. ''Covariant action for Ashtekar's form of canonical gravity''. Class. Quant. Grav. 5 (1987) 583.〕 These proofs were given in terms of spinors. A purely tensorial proof of the new variables in terms of triads was given by Goldberg〔''Triad approach to the Hamiltonian of general relativity.'' Phys. Rev. D37 (1987) 2116-20.〕 and in terms of tetrads by Henneaux et al.〔M. Henneaux, J.E. Nelson and C. Schomblond. ''Derivation of Ashtekar variables from tetrad gravity.'' Phys. Rev. D39 (1989) 434-7.〕 Here we in particular fill in details of the proof of results for self-dual variables not given in text books. == The Palatini action == The Palatini action for general relativity has as its independent variables the tetrad $e\_I^\backslash alpha$ and a spin connection $\backslash omega\_\backslash alpha^$. Much more details and derivations can be found in the article tetradic Palatini action. The spin connection defines a covariant derivative $D\_\backslash alpha$. The space-time metric is recovered from the tetrad by the formula $g\_\; =\; e^I\_\backslash alpha\; e^J\_\backslash beta\; \backslash eta\_.$ We define the `curvature' by $\backslash Omega\_^\; =\; \backslash partial\_\backslash alpha\; \backslash omega\_\backslash beta^\; -\; \backslash partial\_\backslash beta\; \backslash omega\_\backslash alpha^\; +\; \backslash omega\_\backslash alpha^\; \backslash omega\_^\; -\; \backslash omega\_\backslash beta^\; \backslash omega\_^\; \backslash ;\backslash ;\backslash ;\backslash ;\backslash ;\; Eq(1).$ The Ricci scalar of this curvature is given by $e\_I^\backslash alpha\; e\_J^\backslash beta\; \backslash Omega\_^$. The Palatini action for general relativity reads $S\; =\; \backslash int\; d^4\; x\; \backslash ;\; e\; \backslash ;e\_I^\backslash alpha\; e\_J^\backslash beta\; \backslash ;\; \backslash Omega\_^\; ()$ where $e\; =\; \backslash sqrt$. Variation with respect to the spin connection $\backslash omega\_^$ implies that the spin connection is determined by the compatibility condition $D\_\backslash alpha\; e\_I^\backslash beta\; =\; 0$ and hence becomes the usual covariant derivative $\backslash nabla\_\backslash alpha$. Hence the connection becomes a function of the tetrads and the curvature $\backslash Omega\_^$ is replaced by the curvature $R\_^$ of $\backslash nabla\_\backslash alpha$. Then $e\_I^\backslash alpha\; e\_J^\backslash beta\; R\_^$ is the actual Ricci scalar $R$. Variation with respect to the tetrad gives Einsteins equation $R\_\; -\; g\_\; R\; =\; 0$. 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「Self-dual Palatini action」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.