General Relativity/Schwarzschild metric

<General Relativity

Main article: Schwarzschild metric

The Schwarzschild metric can be put into the form

$d{s}^2=-c^2(1-\frac{2GM}{c^{2}r})d{t}^2+{(1-\frac{2GM}{c^{2}r})}^{-1}d{r}^2+r^{2}d{\mathrm{\Omega }}^2$,

where $G$ is the gravitational constant, $M$ is interpreted as the mass of the gravitating object, and

$d{\mathrm{\Omega }}^2=d{\theta }^2+{\sin }^2\theta d{\varphi }^2$

is the standard metric on the 2-sphere. The constant

$r_s=\frac{2GM}{c^2}$

is called the Schwarzschild radius.

Note that as $M\to 0$ or $r\to \mathrm{\infty }$ one recovers the Minkowski metric:

$d{s}^2=-c^{2}d{t}^2+d{r}^2+r^{2}d{\mathrm{\Omega }}^2.$

Intuitively, this means that around small or far away from any gravitating bodies we expect space to be nearly flat. Metrics with this property are called asymptotically flat.

Note that there are two singularities in the Schwarzschild metric: at r=0 and $r=r_s=\frac{2GM}{c^2}$. It can be shown that while the latter singularity can be transformed away with a change of metric, the former is not. In other words, r=0 is a bonafide singularity in the metric.