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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 Bimetric theory ： ウィキペディア英語版 Bimetric gravity Bimetric gravity or bigravity refers to a class of modified mathematical theories of gravity (or gravitation) in which two metric tensors are used instead of one. The second metric may be introduced at high energies, with the implication that the speed of light could be energy-dependent. If the two metrics are dynamical and interact then there will be two graviton modes, one massive and one massless, and thus bimetric theories are closely related to massive gravity. There are several different bimetric theories, such as those attributed to Nathan Rosen (1909–1995)〔〔 or Mordehai Milgrom with Modified Newtonian Dynamics (MOND). More recently, developments in massive gravity have also led to new consistent theories of bimetric gravity. Though none has been shown to account for physical observations more accurately or more consistently than the theory of general relativity, Rosen's theory has been shown to be inconsistent with observations of the Hulse–Taylor binary pulsar.〔 Some of these theories lead to cosmic acceleration at late times and are therefore alternatives to dark energy. == Rosen's bigravity (1940) == In general relativity (GR), it is assumed that the distance between two points in spacetime is given by the metric tensor. Einstein's field equation is then used to calculate the form of the metric based on the distribution of energy and momentum. Rosen (1940) has proposed that at each point of space-time, there is a Euclidean metric tensor $\backslash gamma\_$ in addition to the Riemannian metric tensor $g\_$ . Thus at each point of space-time there are two metrics: :$$ 1.~~~~ds^=g_dx^dx^ :$$ 2.~~~~d\sigma^=\gamma_ dx^ dx^ The first metric tensor, $g\_$ , describes the geometry of space-time and thus the gravitational field. The second metric tensor, $\backslash gamma\_$ , refers to the flat space-time and describes the inertial forces. The Christoffel symbols formed from $g\_$ and $\backslash gamma\_$ are denoted by $\backslash \backslash \}$ and $\backslash Gamma^\_$ respectively. The quantities $\backslash Delta$ are defined such that :$$ \Delta^_=\\}-\Gamma^_~~~~~~~~~~~~~~(1) Two kinds of covariant differentiation then arise: $g$-differentiation based on $g\_$ (denoted by a semicolon), and 3-differentiation based on $\backslash gamma\_$ (denoted by a slash). Ordinary partial derivatives are represented by a comma. Let $R^\_$ and $P^\_$ be the Riemann curvature tensors calculated from $g\_$ and $\backslash gamma\_$, respectively. In the above approach the curvature tensor $P^\_$ is zero, since $\backslash gamma\_$ is the flat space-time metric. From (1) one finds that though and $\backslash Gamma$ are not tensors, but $\backslash Delta$ is a tensor having the same form as except that the ordinary partial derivative is replaced by 3-covariant derivative. A straightforward calculation yields the Riemann curvature tensor :$$ R^_=-\Delta^_+\Delta^_+\Delta^_\Delta^_-\Delta^_\Delta^_ Each term on right hand side is a tensor. It is seen that from GR one can go to the new formulation just by replacing by $\backslash Delta$, ordinary differentiation by 3-covariant differentiation, $\backslash sqrt$ by $\backslash sqrt\}$, integration measure $d^x$ by $\backslash sqrt\; d^x$, where $g\; =\; det(g\_)$, $3\; =\; det(\backslash gamma\_)$ and $d^x\; =\; dx^dx^dx^dx^$. Having once introduced $\backslash gamma\_$ into the theory, one has a great number of new tensors and scalars at one's disposal. One can set up other field equations other than Einstein's. It is possible that some of these will be more satisfactory for the description of nature. The geodesic equation in bimetric relativity (BR) takes the form :$$ \frac+\Gamma^_\frac\frac+\Delta^_\frac\frac=0~~~~~~~~~~~~~~(2) It is seen from equations (1) and (2) that $\backslash Gamma$ can be regarded as describing the inertial field because it vanishes by a suitable coordinate transformation. The quantity $\backslash Delta$, being a tensor, is independent of any coordinate system and hence may be regarded as describing the permanent gravitational field. Rosen (1973) has found BR satisfying the covariance and equivalence principle. In 1966, Rosen showed that the introduction of the space metric into the framework of general relativity not only enables one to get the energy momentum density tensor of the gravitational field, but also enables one to obtain this tensor from a variational principle. The field equations of BR derived from the variational principle are :$$ K^_= N^_-\frac\delta^_N = -8 \pi \kappa T^_~~~~~~~~~~~~~~(3) where :$$ N^_=\frac\gamma^(g^ g_)_ or :$\backslash begin$ N^_=\frac \gamma^\Big\)_ - (g^g_\Gamma^_)_ -\gamma^(\Gamma^_)_ + \Gamma^_(- g^g_\Gamma^_- \Gamma^_ )\\ \qquad -\Gamma^_(- g^g_\Gamma^_ -\Gamma^_ ) + \Gamma^_(- g^g_\Gamma^_ -\Gamma^_ )\Big\} \end :$$ N= g^N_, \qquad \kappa=\sqrt}, and $T^\_$ is the energy-momentum tensor. The variational principle also leads to the relation :$$ T^_=0. Hence from (3) :$$ K^_=0, which implies that in a BR, a test particle in a gravitational field moves on a geodesic with respect to $g\_.$ It is found that the BR and GR theories differ in the following cases: : * propagation of electromagnetic waves : * the external field of a high density star : * the behaviour of intense gravitational waves propagating through a strong static gravitational field. The predictions of gravitational radiation in Rosen's theory have been shown to be in conflict with observations of the Hulse–Taylor binary pulsar.〔 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「Bimetric gravity」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.