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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 Solutions of the Einstein field equations ： ウィキペディア英語版 Solutions of the Einstein field equations :''Where appropriate, this article will use the abstract index notation.'' Solutions of the Einstein field equations are spacetimes that result from solving the Einstein field equations (EFE) of general relativity. Solving the field equations actually gives a Lorentz manifold. Solutions are broadly classed as ''exact'' or ''non-exact''. The Einstein field equations are :$G\_\; \backslash ,\; =\; \backslash kappa\; T\_$ or more generally, if one allows a nonzero cosmological constant, :$G\_\; +\; \backslash Lambda\; g\_\; \backslash ,\; =\; \backslash kappa\; T\_$ where $\backslash kappa$ is a constant, and the Einstein tensor on the left side of the equation is equated to the stress–energy tensor representing the energy and momentum present in the spacetime. The Einstein tensor is built up from the metric tensor and its partial derivatives; thus, the EFE are a system of ten partial differential equations to be solved for the metric. ==Solving the equations== It is important to realize that the Einstein field equations alone are not enough to determine the evolution of a gravitational system in many cases. They depend on the stress–energy tensor, which depends on the dynamics of matter and energy (such as trajectories of moving particles), which in turn depends on the gravitational field. If one is only interested in the weak field limit of the theory, the dynamics of matter can be computed using special relativity methods and/or Newtonian laws of gravity and then the resulting stress–energy tensor can be plugged into the Einstein field equations. But if the exact solution is required or a solution describing strong fields, the evolution of the metric and the stress–energy tensor must be solved for together. To obtain solutions, the relevant equations are the above quoted EFE (in either form) plus the continuity equation (to determine evolution of the stress–energy tensor): :$T^\; \backslash ,\; =\; 0$ * Perfect fluid: :$T\_\; \backslash ,\; =\; (\backslash rho\; +\; p)u\_a\; u\_b\; +\; p\; g\_$ where $u^au\_a\; =\; -1\backslash !$ Here $\backslash rho$ is the mass-energy density measured in a momentary co-moving frame, $u\_a$ is the fluid's 4-velocity vector field, and $p$ is the pressure. * Non-interacting dust ( a special case of perfect fluid ): :$T\_\; \backslash ,\; =\; \backslash rho\; u\_a\; u\_b$ For a perfect fluid, another equation of state relating density $\backslash rho$ and pressure $p$ must be added. This equation will often depend on temperature, so a heat transfer equation is required or the postulate that heat transfer can be neglected. Next, notice that only 10 of the original 14 equations are independent, because the continuity equation $T^\; \backslash Gamma^\_\; =\; 0\; \backslash ,.$ In numerical relativity, the preferred gauge is the so-called "3+1 decomposition", based on the ADM formalism. In this decomposition, metric is written in the form :$ds^2\; \backslash ,\; =\; (-N\; +\; N^i\; N^j\; \backslash gamma\_)\; dt^2\; +\; 2N^i\; \backslash gamma\_\; dt\; dx^j\; +\; \backslash gamma\_\; dx^i\; dx^j$, where $i,j\; =\; 1\backslash dots\; 3\; \backslash ,.$ $N$ and $N^i$ are functions of spacetime coordinates and can be chosen arbitrarily in each point. The remaining physical degrees of freedom are contained in $\backslash gamma\_$, which represents the Riemannian metric on 3-hypersurfaces $t=const$. For example, a naive choice of $N=1$, $N\_i=0$, would correspond to a so-called synchronous coordinate system: one where t-coordinate coincides with proper time for any comoving observer (particle that moves along a fixed $x^i$ trajectory.) Once equations of state are chosen and the gauge is fixed, the complete set of equations can be solved for. Unfortunately, even in the simplest case of gravitational field in the vacuum ( vanishing stress–energy tensor ), the problem turns out too complex to be exactly solvable. To get physical results, we can either turn to numerical methods; try to find exact solutions by imposing symmetries; or try middle-ground approaches such as perturbation methods or linear approximations of the Einstein tensor. 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「Solutions of the Einstein field equations」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.