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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 Newton–Cartan theory ： ウィキペディア英語版 Newton–Cartan theory Newton–Cartan theory is a geometrical re-formulation, as well as a generalization, of Newtonian gravity developed by Élie Cartan. In this re-formulation, the structural similarities between Newton's theory and Albert Einstein's general theory of relativity are readily seen, and it has been used by Cartan and Kurt Friedrichs to give a rigorous formulation of the way in which Newtonian gravity can be seen as a specific limit of general relativity, and by Jürgen Ehlers to extend this correspondence to specific solutions of general relativity. ==Geometric formulation of Poisson's equation== In Newton's theory of gravitation, Poisson's equation reads :$$ \Delta U = 4 \pi G \rho \, where $U$ is the gravitational potential, $G$ is the gravitational constant and $\backslash rho$ is the mass density. The weak equivalence principle motivates a geometric version of the equation of motion for a point particle in the potential $U$ :$$ m_t \ddot = - m_g \nabla U where $m\_t$ is the inertial mass and $m\_g$ the gravitational mass. Since, according to the weak equivalence principle $m\_t\; =\; m\_g$, the according equation of motion :$$ \ddot = - \nabla U doesn't contain anymore a reference to the mass of the particle. Following the idea that the solution of the equation then is a property of the curvature of space, a connection is constructed so that the geodesic equation :$$ \frac + \Gamma_^\lambda \frac\frac = 0 represents the equation of motion of a point particle in the potential $U$. The resulting connection is :$$ \Gamma_^ = \gamma^ U_ \Psi_\mu \Psi_\nu with $\backslash Psi\_\backslash mu\; =\; \backslash delta\_\backslash mu^0$ and $\backslash gamma^\; =\; \backslash delta^\backslash mu\_A\; \backslash delta^\backslash nu\_B\; \backslash delta^$ ($A,\; B\; =\; 1,2,3$). The connection has been constructed in one inertial system but can be shown to be valid in any inertial system by showing the invariance of $\backslash Psi\_\backslash mu$ and $\backslash gamma^$ under Galilei-transformations. The Riemann curvature tensor in inertial system coordinates of this connection is then given by :$$ R^\lambda_ = 2 \gamma^ U_\Psi_\Psi_\kappa where the brackets $A\_\; =\; \backslash frac\; (A\_\; -\; A\_)$ mean the antisymmetric combination of the tensor $A\_$. The Ricci tensor is given by :$$ R_ = \Delta U \Psi_\Psi_ \, which leads to following geometric formulation of Poisson's equation :$$ R_ = 4 \pi G \rho \Psi_\mu \Psi_\nu More explicitly, if the roman indices ''i'' and ''j'' range over the spatial coordinates 1, 2, 3, then the connection is given by :$$ \Gamma^i_ = U_ the Riemann curvature tensor by :$$ R^i_ = -R^i_ = U_ and the Ricci tensor and Ricci scalar by :$$ R = R_ = \Delta U where all components not listed equal zero. Note that this formulation does not require introducing the concept of a metric: the connection alone gives all the physical information. 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「Newton–Cartan 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.