入射爱丁顿-芬克尔斯坦坐标

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

这就是用入射爱丁顿-芬克尔斯坦坐标(ingoing Eddington-Finkelstein coordinates)写出的施瓦西黑洞。我们看到，$d{r}^2$ 项现在已消失，因此在 $r=2GM$ 处度规中不再有奇点。然而，$d{v}^2$ 项在 $r=2GM$ 处为零，而且在 $r<2GM$ 时还会变号。你也许会担心，这意味着坐标在那里仍然失效，甚至意味着当我们穿过视界时度规的号差会发生改变。为了消除这种担心，我们只需计算度规行列式

$$\det (g)=\det \left(\begin{array}{cccc}-(1-\frac{2GM}{r})& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& r^2& 0\\ 0& 0& 0& r^2{\sin }^2\theta \end{array}\right)=-r^4{\sin }^2\theta$$

英文原文

This is the Schwarzschild black hole in ingoing Eddington-Finkelstein coordinates. We see that the $d{r}^2$ terms have now disappeared, and so there is no singularity in the metric at $r=2GM$ . However, the $d{v}^2$ term vanishes at $r=2GM$ and, moreover, flips sign for $r<2GM$ . You might worry that this means that the coordinates still go bad there, or even that the signature of the metric changes as we cross the horizon. To allay such worries, we need only compute the determinant of the metric

$$\det (g)=\det \left(\begin{array}{cccc}-(1-\frac{2GM}{r})& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& r^2& 0\\ 0& 0& 0& r^2{\sin }^2\theta \end{array}\right)=-r^4{\sin }^2\theta$$

出处：6.1 施瓦西解 · node a-eb00429f4c3215f9ed570528 · 源节点 SHA-256 140b828d38b63a95…