翻訳と辞書 Words near each other ・ Metric modulation ・ Metric outer measure ・ Metric Pixel Canvas ・ Metric prefix ・ Metric signature ・ Metric space ・ Metric space aimed at its subspace ・ Metric Structures for Riemannian and Non-Riemannian Spaces ・ Metric system ・ Metric tensor ・ Metric tensor (general relativity) ・ Metric time ・ Metric Today ・ Metric tree ・ Metric typographic units ・ Metric-affine gravitation theory ・ Metrica ・ Metrical phonology ・ Metrical psalter ・ Metrical task system ・ Metrication ・ Metrication Board ・ Metrication in Australia ・ Metrication in Barbados ・ Metrication in Canada ・ Metrication in Chile ・ Metrication in Guatemala ・ Metrication in India ・ Metrication in Ireland ・ Metrication in Jamaica Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Metric-affine gravitation theory ： ウィキペディア英語版 Metric-affine gravitation theory In comparison with General Relativity, dynamic variables of metric-affine gravitation theory are both a pseudo-Riemannian metric and a general linear connection on a world manifold $X$. Metric-affine gravitation theory has been suggested as a natural generalization of Einstein–Cartan theory of gravity with torsion where a linear connection obeys the condition that a covariant derivative of a metric equals zero. Metric-affine gravitation theory straightforwardly comes from gauge gravitation theory where a general linear connection plays the role of a gauge field. Let $TX$ be the tangent bundle over a manifold $X$ provided with bundle coordinates $(x^\backslash mu,\backslash dot\; x^\backslash mu)$. A general linear connection on $TX$ is represented by a connection tangent-valued form : $\backslash Gamma=dx^\backslash lambda\backslash otimes(\backslash partial\_\backslash lambda\; +\backslash Gamma\_\backslash lambdadx^\backslash mu\backslash otimes\; dx^\backslash nu$ on $TX$ is defined as a global section of the quotient bundle $FX/SO(1,3)\backslash to\; X$, where $SO(1,3)$ is the Lorentz group. Therefore, on can regard it as a classical Higgs field in gauge gravitation theory. Gauge symmetries of metric-affine gravitation theory are general covariant transformations. It is essential that, given a pseudo-Riemannian metric $g$, any linear connection $\backslash Gamma$ on $TX$ admits a splitting : $\backslash Gamma\_=\backslash \; +S\_\; +\backslash frac12\; C\_$ in the Christoffel symbols : $\backslash =\; -\backslash frac12(\backslash partial\_\backslash mu\; g\_\; +\; \backslash partial\_\backslash alpha$ g_-\partial_\nu g_), a nonmetricity tensor : $C\_=C\_=\backslash nabla^\backslash Gamma\_\backslash mu\; g\_=\backslash partial\_\backslash mu\; g\_\; +\backslash Gamma\_\; +\; \backslash Gamma\_$ and a contorsion tensor : $S\_=-S\_=\backslash frac12(T\_\; +T\_\; +\; T\_+\; C\_\; -C\_),$ where : $T\_=\backslash frac12(\backslash Gamma\_\; -\; \backslash Gamma\_)$ is the torsion tensor of $\backslash Gamma$. Due to this splitting, metric-affine gravitation theory possesses a different collection of dynamic variables which are a pseudo-Riemannian metric, a non-metricity tensor and a torsion tensor. As a consequence, a Lagrangian of metric-affine gravitation theory can contain different terms expressed both in a curvature of a connection $\backslash Gamma$ and its torsion and non-metricity tensors. In particular, a metric-affine f(R) gravity, whose Lagrangian is an arbitrary function of a scalar curvature $R$ of $\backslash Gamma$, is considered. A linear connection $\backslash Gamma$ is called the ''metric connection'' for a pseudo-Riemannian metric $g$ if $g$ is its integral section, i.e., the metricity condition : $\backslash nabla^\backslash Gamma\_\backslash mu\; g\_=0$ holds. A metric connection reads : $\backslash Gamma\_=\backslash \; +\; \backslash frac12(T\_\; +T\_\; +\; T\_).$ For instance, the Levi-Civita connection in General Relativity is a torsion-free metric connection. A metric connection is associated to a principal connection on a Lorentz reduced subbundle $F^gX$ of the frame bundle $FX$ corresponding to a section $g$ of the quotient bundle $FX/SO(1,3)\backslash to\; X$. Restricted to metric connections, metric-affine gravitation theory comes to the above-mentioned Einstein – Cartan gravitation theory. At the same time, any linear connection $\backslash Gamma$ defines a principal adapted connection $\backslash Gamma^g$ on a Lorentz reduced subbundle $F^gX$ by its restriction to a Lorentz subalgebra of a Lie algebra of a general linear group $GL(4,\backslash mathbb\; R)$. For instance, the Dirac operator in metric-affine gravitation theory in the presence of a general linear connection $\backslash Gamma$ is well defined, and it depends just of the adapted connection $\backslash Gamma^g$. Therefore, Einstein – Cartan gravitation theory can be formulated as the metric-affine one, without appealing to the metricity constraint. In metric-affine gravitation theory, in comparison with the Einstein - Cartan one, a question on a matter source of a non-metricity tensor arises. It is so called hypermomentum, e.g., a Noether current of a scaling symmetry. ==References== * F.Hehl, J. McCrea, E. Mielke, Y. Ne'eman, Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilaton invariance, ''Physics Reports'' 258 (1995) 1-171; (arXiv: gr-qc/9402012 ) * V. Vitagliano, T. Sotiriou, S. Liberati, The dynamics of metric-affine gravity, ''Annals of Physics'' 326 (2011) 1259-1273; (arXiv: 1008.0171 ) * G. Sardanashvily, Classical gauge gravitation theory, ''Int. J. Geom. Methods Mod. Phys.'' 8 (2011) 1869-1895; (arXiv: 1110.1176 ) * C. Karahan, A. Altas, D. Demir, Scalars, vectors and tensors from metric-affine gravity, ''General Relativity and Gravitation'' 45 (2013) 319-343; (arXiv: 1110.5168 ) 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「Metric-affine gravitation 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.