标题： 修正爱因斯坦方程 显示英文标题

标题： Fixing Einstein's equations

摘要： 爱因斯坦的广义相对论方程，当被视为演化初始数据的动力系统时，存在一个严重缺陷：无法证明其适定性（除非在特殊坐标中）。也就是说，它们不能产生对初始数据光滑依赖的唯一解。为弥补这一缺陷，最近广泛兴趣在于将爱因斯坦理论重新表述为双曲型微分方程组。原始理论的物理和几何内容保持不变，但动力演化变得合理。在这里，我们提出了一种新的双曲型表述，涉及$g_{ij}$、$K_{ij}$和$\bGam_{kij}$，它与爱因斯坦原始方程的空间加时间（“3+1”）形式非常接近。事实上，其组成部分的熟悉性使得这种表述的存在更加出人意料。这是目前我们所知的最经济的一阶对称双曲型表述，它对于所有（非物质）变量只有物理特征速度，即零或光速。该系统澄清了爱因斯坦原始方程与爱因斯坦-里奇和弗里泰利-雷拉双曲型表述之间的关系，并建立了与其他双曲型表述的联系。 显示英文摘要

摘要： 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.