克雷奇曼标量

本书暂定概念：尚未进入正式概念库。下面是本书给出的定义原文。

定义 · 6.1 施瓦西解

最简单的标量当然就是里奇标量。但对爱因斯坦方程的任意真空解，它必然有 $R=0$，因此无助于判断奇点的性质。$R_{\mu \nu }{R}^{\mu \nu }$ 也是如此。正因如此，最简单的曲率判据是克雷奇曼标量(Kretschmann scalar)，即 $R^{\mu \nu \rho \sigma }{R}_{\mu \nu \rho \sigma }$。对施瓦西解，它由下式给出

$$R^{\mu \nu \rho \sigma }{R}_{\mu \nu \rho \sigma }=\frac{48G^2{M}^2}{r^6}\text{\hspace{1em}(6.3)}$$

英文原文

The simplest scalar is, of course, the Ricci scalar. But this is necessarily $R=0$ for any vacuum solution to the Einstein equation so it is not helpful in detecting the nature of singularities. The same is true for $R_{\mu \nu }{R}^{\mu \nu }$ . For this reason, the simplest curvature diagnostic is the Kretschmann scalar, $R^{\mu \nu \rho \sigma }{R}_{\mu \nu \rho \sigma }$ . For the Schwarzschild solution it is given by

$$R^{\mu \nu \rho \sigma }{R}_{\mu \nu \rho \sigma }=\frac{48G^2{M}^2}{r^6}\text{\hspace{1em}(6.3)}$$

出处：6.1 施瓦西解 · node a-9cfdb4e6c735fc394fd8f802 · 源节点 SHA-256 4ee28b85442af78d…