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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 Gibbons–Hawking–York boundary term ： ウィキペディア英語版 Gibbons–Hawking–York boundary term In general relativity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary. The Einstein–Hilbert action is the basis for the most elementary variational principle from which the field equations of general relativity can be defined. However, the use of the Einstein–Hilbert action is appropriate only when the underlying spacetime manifold $\backslash mathcal$ is closed, i.e., a manifold which is both compact and without boundary. In the event that the manifold has a boundary $\backslash partial\backslash mathcal$, the action should be supplemented by a boundary term so that the variational principle is well-defined. The necessity of such a boundary term was first realised by York and later refined in a minor way by Gibbons and Hawking. For a manifold that is not closed, the appropriate action is :$\backslash mathcal\_\backslash mathrm\; +\; \backslash mathcal\_\backslash mathrm\; =\; \backslash frac\; \backslash int\_\backslash mathcal\; \backslash mathrm^4\; x\; \backslash ,\; \backslash sqrt\; R\; +\; \backslash frac\; \backslash int\_^3\; y\; \backslash ,\; \backslash epsilon\; \backslash sqrtK,$ where $\backslash mathcal\_\backslash mathrm$ is the Einstein–Hilbert action, $\backslash mathcal\_\backslash mathrm$ is the Gibbons–Hawking–York boundary term, $h\_$ is the induced metric (see section below on definitions) on the boundary, $h$ its determinant, $K$ is the trace of the second fundamental form, $\backslash epsilon$ is equal to $+1$ where $\backslash partial\; \backslash mathcal$ is timelike and $-1$ where $\backslash partial\; \backslash mathcal$ is spacelike, and $y^a$ are the coordinates on the boundary. Varying the action with respect to the metric $g\_$, subject to the condition $$ \delta g_ \big|_ is fixed (see section below). There remains ambiguity in the action up to an arbitrary functional of the induced metric $h\_$. That a boundary term is needed in the gravitational case is due to the fact that $R$, the gravitational Lagrangian density, contains second derivatives of the metric tensor. This is a non-typical feature of field theories, which are usually formulated in terms of Lagrangians that involve first derivatives of fields to be varied over only. The GHY term is desirable, as it possesses a number of other key features. When passing to the hamiltonian formalism, it is necessary to include the GHY term in order to reproduce the correct Arnowitt-Deser-Misner energy (ADM energy). The term is required to ensure the path integral (a la Hawking) for quantum gravity has the correct composition properties. When calculating black hole entropy using the euclidean semiclassical approach, the entire contribution comes from the GHY term. This term has had more recent applications in loop quantum gravity in calculating transition amplitudes and background-independent scattering amplitudes. In order to a finite value for the action, we may have to subtract off a surface term for flat spacetime: $$ S_ + S_ = \frac \int_\mathcal \mathrm^4 x \, \sqrt R + \frac \int_^3 y \, \epsilon \sqrt K - \int_^3 y \, \epsilon \sqrt K_0, where $K\_0$ is the extrinsic curvature of the boundary imbedded flat spacetime. As $\backslash sqrt$ is invariant under variations of $g\_$ this addition term does not effect the field equations. = Introduction to hyper-surfaces = == Defining hyper-surfaces == In a four-dimensional spacetime manifold, a hypersurface is a three-dimensional submanifold that can be either timelike, spacelike, or null. A particular hyper-surface $\backslash Sigma$ can be selected either by imposing a constraint on the coordinates $$ f (x^\alpha) = 0, or by giving parametric equations, $$ x^\alpha = x^\alpha (y^a), where $y^a$ ($a=1,2,3$) are coordinates intrinsic to the hyper-surface. For example, a two-sphere in three-dimensional Euclidean space can be described either by $$ f (x^\alpha) = x^2 + y^2 + z^2 - r^2 = 0, where $r$ is the radius of the sphere, or by $$ x = r \sin \theta \cos \phi , \quad y = r \sin \theta \sin \phi , \quad and \; z = r \cos \theta , where $\backslash theta$ and $\backslash phi$ are intrinsic coordinates. 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「Gibbons–Hawking–York boundary term」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.