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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 General covariant transformations ： ウィキペディア英語版 General covariant transformations In physics, general covariant transformations are symmetries of gravitation theory on a world manifold $X$. They are gauge transformations whose parameter functions are vector fields on $X$. From the physical viewpoint, general covariant transformations are treated as particular (holonomic) reference frame transformations in general relativity. In mathematics, general covariant transformations are defined as particular automorphisms of so-called natural fiber bundles. == Mathematical definition == Let $\backslash pi:Y\backslash to\; X$ be a fibered manifold with local fibered coordinates $(x^\backslash lambda,\; y^i)\backslash ,$. Every automorphism of $Y$ is projected onto a diffeomorphism of its base $X$. However, the converse is not true. A diffeomorphism of $X$ need not give rise to an automorphism of $Y$. In particular, an infinitesimal generator of a one-parameter Lie group of automorphisms of $Y$ is a projectable vector field : $u=u^\backslash lambda(x^\backslash mu)\backslash partial\_\backslash lambda\; +\; u^i(x^\backslash mu,y^j)\backslash partial\_i$ on $Y$. This vector field is projected onto a vector field $\backslash tau=u^\backslash lambda\backslash partial\_\backslash lambda$ on $X$, whose flow is a one-parameter group of diffeomorphisms of $X$. Conversely, let $\backslash tau=\backslash tau^\backslash lambda\backslash partial\_\backslash lambda$ be a vector field on $X$. There is a problem of constructing its lift to a projectable vector field on $Y$ projected onto $\backslash tau$. Such a lift always exists, but it need not be canonical. Given a connection $\backslash Gamma$ on $Y$, every vector field $\backslash tau$ on $X$ gives rise to the horizontal vector field : $\backslash Gamma\backslash tau\; =\backslash tau^\backslash lambda(\backslash partial\_\backslash lambda\; +\backslash Gamma\_\backslash lambda^i\backslash partial\_i)$ on $Y$. This horizontal lift $\backslash tau\backslash to\backslash Gamma\backslash tau$ yields a monomorphism of the $C^\backslash infty(X)$-module of vector fields on $X$ to the $C^\backslash infty(Y)$-module of vector fields on $Y$, but this monomorphisms is not a Lie algebra morphism, unless $\backslash Gamma$ is flat. However, there is a category of above mentioned natural bundles $T\backslash to\; X$ which admit the functorial lift $\backslash widetilde\backslash tau$ onto $T$ of any vector field $\backslash tau$ on $X$ such that $\backslash tau\backslash to\backslash widetilde\backslash tau$ is a Lie algebra monomorphism : $(\backslash tau,\backslash widetilde\; \backslash tau\text{'})=\backslash widetilde\; .$ This functorial lift $\backslash widetilde\backslash tau$ is an infinitesimal general covariant transformation of $T$. In a general setting, one considers a monomorphism $f\backslash to\backslash widetilde\; f$ of a group of diffeomorphisms of $X$ to a group of bundle automorphisms of a natural bundle $T\backslash to\; X$. Automorphisms $\backslash widetilde\; f$ are called the general covariant transformations of $T$. For instance, no vertical automorphism of $T$ is a general covariant transformation. Natural bundles are exemplified by tensor bundles. For instance, the tangent bundle $TX$ of $X$ is a natural bundle. Every diffeomorphism $f$ of $X$ gives rise to the tangent automorphism $\backslash widetilde\; f=Tf$ of $TX$ which is a general covariant transformation of $TX$. With respect to the holonomic coordinates $(x^\backslash lambda,\; \backslash dot\; x^\backslash lambda)$ on $TX$, this transformation reads : $\backslash dot\; x\text{'}^\backslash mu=\backslash frac\backslash dot\; x^\backslash nu.$ A frame bundle $FX$ of linear tangent frames in $TX$ also is a natural bundle. General covariant transformations constitute a subgroup of holonomic automorphisms of $FX$. All bundles associated with a frame bundle are natural. However, there are natural bundles which are not associated with $FX$. 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「General covariant transformations」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.