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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 Riemann–Cartan geometry ： ウィキペディア英語版 Einstein–Cartan theory In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory, is a classical theory of gravitation similar to general relativity but relaxing the assumption that the affine connection has vanishing antisymmetric part (torsion tensor), so that the torsion can be coupled to the intrinsic angular momentum (spin) of matter, much in the same way in which the curvature is coupled to the energy and momentum of matter. In fact, the spin of matter in curved spacetime requires that torsion is not constrained to be zero but is a variable in the principle of stationary action. Regarding the metric and torsion tensors as independent variables gives the correct generalization of the conservation law for the total (orbital plus intrinsic) angular momentum to the presence of the gravitational field. The theory was first proposed by Élie Cartan in 1922〔Élie Cartan. "Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion." C. R. Acad. Sci. (Paris) 174, 593–595 (1922).〕 and expounded in the following few years.〔Élie Cartan. "Sur les variétés à connexion affine et la théorie de la relativité généralisée." Part I: Ann. Éc. Norm. 40, 325–412 (1923) and ibid. 41, 1–25 (1924); Part II: ibid. 42, 17–88 (1925).〕 Dennis Sciama〔Dennis W. Sciama. ("The physical structure of general relativity" ), Rev. Mod. Phys. 36, 463-469 (1964).〕 and Tom Kibble〔Tom W. B. Kibble. ("Lorentz invariance and the gravitational field" ), J. Math. Phys. 2, 212-221 (1961).〕 independently revisited the theory in the 1960s, and an important review was published in 1976.〔Friedrich W. Hehl, Paul von der Heyde, G. David Kerlick, and James M. Nester. "General relativity with spin and torsion: Foundations and prospects." Rev. Mod. Phys. 48, 393–416 (1976). http://link.aps.org/doi/10.1103/RevModPhys.48.393〕 Albert Einstein became affiliated with the theory in 1928 during his unsuccessful attempt to match torsion to the electromagnetic field tensor as part of a unified field theory. This line of thought led him to the related but different theory of teleparallelism.〔Hubert F. M. Goenner. ("On the History of Unified Field Theories." ) Living Rev. Relativity, 7, 2 (2004).〕 Einstein–Cartan theory has been historically overshadowed by its torsion-free counterpart and other alternatives like Brans–Dicke theory because torsion seemed to add little predictive benefit at the expense of the tractability of its equations. Since the Einstein–Cartan theory is purely classical, it also does not fully address the issue of quantum gravity. In the Einstein–Cartan theory, the Dirac equation becomes nonlinear〔F. W. Hehl and B. K. Datta. ("Nonlinear spinor equation and asymmetric connection in general relativity" ), J. Math. Phys. 12, 1334–1339 (1971).〕 and therefore the superposition principle used in usual quantization techniques would not work. Recently, interest in Einstein–Cartan theory has been driven toward cosmological implications, most importantly, the avoidance of a gravitational singularity at the beginning of the universe. The theory is considered viable and remains an active topic in the physics community.〔Friedrich W. Hehl. ("Note on the torsion tensor." ) Letter to Physics Today. March 2007, page 16.〕 ==Field equations== The Einstein field equations of general relativity can be derived by postulating the Einstein–Hilbert action to be the true action of spacetime and then varying that action with respect to the metric tensor. The field equations of Einstein–Cartan theory come from exactly the same approach. Let $\backslash mathcal\_\backslash mathrm$ represent the Lagrangian density of matter and $\backslash mathcal\_\backslash mathrm$ represent the Lagrangian density of the gravitational field. The Lagrangian density for the gravitational field in the Einstein–Cartan theory is proportional to the Ricci scalar: :$\backslash mathcal\_\backslash mathrm=\backslash fracR\; \backslash sqrt$ :$S=\backslash int\; \backslash left(\; \backslash mathcal\_\backslash mathrm\; +\; \backslash mathcal\_\backslash mathrm\; \backslash right)\; \backslash ,\; d^4x\; ,$ where $g$ is the determinant of the metric tensor, and $\backslash kappa$ is a physical constant $8\backslash pi\; G/c^4$ involving the gravitational constant and the speed of light. By Hamilton's principle, the variation of the total action $S$ for the gravitational field and matter vanishes: :$\backslash delta\; S\; =\; 0.$ The variation with respect to the metric tensor $g^$ yields the Einstein equations: :$\backslash frac\}P\_=0$ :R g_=\kappa P_ |} where $R\_$ is the Ricci tensor and $P\_$ is the ''canonical'' energy-momentum tensor. The Ricci tensor is no longer symmetric because it contains the nonzero torsion tensor. The right-hand side of the equation cannot be symmetric either, so $P\_$ must contain the nonzero spin tensor. This canonical energy-momentum tensor is related to the more familiar ''symmetric'' energy-momentum tensor by the Belinfante–Rosenfeld procedure. The variation with respect to the torsion tensor $\_\backslash mathrm\}\; -\backslash frac\}^c\; +\; ^c^c\; \}^c$ |} where $\{\backslash sigma\_\{ab\}\}^c$ is the spin tensor. 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「Einstein–Cartan theory」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.