Fixing Einstein's equations
Fixing Einstein's equations
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修正爱因斯坦方程
DOI：
10.1103/physrevlett.82.4384
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发表时间：
1999
影响因子：
8.6
通讯作者：
Jr.
中科院分区：
文献类型：
--
作者：
A. Anderson;J. York;Jr.
当爱因斯坦的广义相对论方程被视为演化初始数据的动力系统时，它有一个严重的缺陷：它们不能被证明是适定的（特殊坐标除外）。也就是说，它们不会产生平稳依赖于初始数据的独特解决方案。为了弥补这一缺陷，最近人们对将爱因斯坦的理论重新表述为双曲微分方程组产生了广泛的兴趣。原始理论的物理和几何内容保持不变，但动力学演化变得合理。在这里，我们以 $g_{ij}$、$K_{ij}$ 和 $\bGam_{kij}$ 形式提出了一个新的双曲公式，它非常接近爱因斯坦原始方程的空间加时间（“3+1”）形式。事实上，其组成部分的熟悉性使得这一表述的存在更加出乎意料。这是我们目前所知的最经济的一阶可对称双曲线公式，对于所有（非物质）变量只有物理特征速度，要么为零，要么光速。该系统阐明了爱因斯坦的原始方程与广义相对论的爱因斯坦-里奇和弗里特利-鲁拉双曲公式之间的关系，并建立了与其他双曲公式的联系。
Einstein's equations for general relativity, when viewed as a dynamical system for evolving initial data, have a serious flaw: they cannot be proven to be well-posed (except in special coordinates). That is, they do not produce unique solutions that depend smoothly on the initial data. To remedy this failing, there has been widespread interest recently in reformulating Einstein's theory as a hyperbolic system of differential equations. The physical and geometrical content of the original theory remain unchanged, but dynamical evolution is made sound. Here we present a new hyperbolic formulation in terms of $g_{ij}$, $K_{ij}$, and $\bGam_{kij}$ that is strikingly close to the space-plus-time (``3+1'') form of Einstein's original equations. Indeed, the familiarity of its constituents make the existence of this formulation all the more unexpected. This is the most economical first-order symmetrizable hyperbolic formulation presently known to us that has only physical characteristic speeds, either zero or the speed of light, for all (non-matter) variables. This system clarifies the relationships between Einstein's original equations and the Einstein-Ricci and Frittelli-Reula hyperbolic formulations of general relativity and establishes links to other hyperbolic formulations.