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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 Stationary spacetime ： ウィキペディア英語版 Stationary spacetime In general relativity, specifically in the Einstein field equations, a spacetime is said to be stationary if it admits a Killing vector that is asymptotically timelike.〔(Ludvigsen, M., ''General Relativity: A Geometric Approach'', Cambridge University Press, 1999, ISBN 052163976X )〕 In a stationary spacetime, the metric tensor components, $g\_$, may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form $(i,j\; =\; 1,2,3)$ : $ds^\; =\; \backslash lambda\; (dt\; -\; \backslash omega\_\backslash ,\; dy^i)^\; -\; \backslash lambda^\; h\_\backslash ,\; dy^i\backslash ,dy^j,$ where $t$ is the time coordinate, $y^$ are the three spatial coordinates and $h\_$ is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field $\backslash xi^$ has the components $\backslash xi^\; =\; (1,0,0,0)$. $\backslash lambda$ is a positive scalar representing the norm of the Killing vector, i.e., $\backslash lambda\; =\; g\_\backslash xi^\backslash xi^$, and $\backslash omega\_$ is a 3-vector, called the twist vector, which vanishes when the Killing vector is hypersurface orthogonal. The latter arises as the spatial components of the twist 4-vector $\backslash omega\_\; =\; e\_\backslash xi^\backslash nabla^\backslash xi^$(see, for example,〔Wald, R.M., (1984). General Relativity, (U. Chicago Press)〕 p. 163) which is orthogonal to the Killing vector $\backslash xi^$, i.e., satisfies $\backslash omega\_\; \backslash xi^\; =\; 0$. The twist vector measures the extent to which the Killing vector fails to be orthogonal to a family of 3-surfaces. A non-zero twist indicates the presence of rotation in the spacetime geometry. The coordinate representation described above has an interesting geometrical interpretation.〔Geroch, R., (1971). J. Math. Phys. 12, 918〕 The time translation Killing vector generates a one-parameter group of motion $G$ in the spacetime $M$. By identifying the spacetime points that lie on a particular trajectory (also called orbit) one gets a 3-dimensional space (the manifold of Killing trajectories) $V=\; M/G$, the quotient space. Each point of $V$ represents a trajectory in the spacetime $M$. This identification, called a canonical projection, $\backslash pi\; :\; M\; \backslash rightarrow\; V$ is a mapping that sends each trajectory in $M$ onto a point in $V$ and induces a metric $h\; =\; -\backslash lambda\; \backslash pi$ *g on $V$ via pullback. The quantities $\backslash lambda$, $\backslash omega\_$ and $h\_$ are all fields on $V$ and are consequently independent of time. Thus, the geometry of a stationary spacetime does not change in time. In the special case $\backslash omega\_\; =\; 0$ the spacetime is said to be static. By definition, every static spacetime is stationary, but the converse is not generally true, as the Kerr metric provides a counterexample. In a stationary spacetime satisfying the vacuum Einstein equations $R\_\; =\; 0$ outside the sources, the twist 4-vector $\backslash omega\_$ is curl-free, : $\backslash nabla\_\backslash mu\; \backslash omega\_\backslash nu\; -\; \backslash nabla\_\backslash nu\; \backslash omega\_\backslash mu\; =\; 0,\backslash ,$ and is therefore locally the gradient of a scalar $\backslash omega$ (called the twist scalar): : $\backslash omega\_\backslash mu\; =\; \backslash nabla\_\backslash mu\; \backslash omega.\backslash ,$ Instead of the scalars $\backslash lambda$ and $\backslash omega$ it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, $\backslash Phi\_$ and $\backslash Phi\_$, defined as〔Hansen, R.O. (1974). J. Math. Phys. 15, 46.〕 : $\backslash Phi\_\; =\; \backslash frac\backslash lambda^(\backslash lambda^\; +\; \backslash omega^\; -1),$ : $\backslash Phi\_\; =\; \backslash frac\backslash lambda^\backslash omega.$ In general relativity the mass potential $\backslash Phi\_$ plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential $\backslash Phi\_$ arises for rotating sources due to the rotational kinetic energy which, because of mass-energy equivalence, can also act as the source of a gravitational field. The situation is analogous to a static electromagnetic field where one has two sets of potentials, electric and magnetic. In general relativity, rotating sources produce a gravitomagnetic field which has no Newtonian analog. A stationary vacuum metric is thus expressible in terms of the Hansen potentials $\backslash Phi\_$ ($A=M$, $J$) and the 3-metric $h\_$. In terms of these quantities the Einstein vacuum field equations can be put in the form〔 : $(h^\backslash nabla\_i\; \backslash nabla\_j\; -\; 2R^)\backslash Phi\_A\; =\; 0,\backslash ,$ : $R^\_\; =\; 2(-\; (1+\; 4\; \backslash Phi^)^\backslash nabla\_\backslash Phi^\backslash nabla\_\backslash Phi^),$ where $\backslash Phi^\; =\; \backslash Phi\_\backslash Phi\_\; =\; (\backslash Phi\_^\; +\; \backslash Phi\_^)$, and $R^\_$ is the Ricci tensor of the spatial metric and $R^\; =\; h^R^\_$ the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics. ==See also== *Axisymmetric spacetime *Static spacetime 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「Stationary spacetime」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.