Schwarzschild solution - Black Holes

This is the simplest solution to the Einstein’s equations. It is a spherically symmetric vacuum solution. It describes the spacetime outside compact gravitating bodies.

The metric can be written as

$d{s}^2=-e^{2\alpha }d{t}^2+e^{2\beta }d{r}^2+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)$

Note: A metric with a timelikeKilling-Vector is stationary. If in addition there are no dtdx cross terms, the metric is static.

Birkhoff’s Theorem

Now, after solving Einstein’s equations, we obtain the Schwarzschild metric (with G=1) $d{s}^2=-(1-\frac{2M}{r})d{t}^2+(1-\frac{2M}{r}{)}^{-1}d{r}^2+r^{2}d{\Omega }^2$

• Geodesics - Schwarzschild
• Kruskal Coordinates