# General Relativity/Schwarzschild metric

Main article: Schwarzschild metric

The Schwarzschild metric can be put into the form

$ds^{2} = -c^2 \left(1-\frac{2GM}{c^2 r} \right) dt^2 + \left(1-\frac{2GM}{c^2 r}\right)^{-1}dr^2+ r^2 d\Omega^2$,

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

$d\Omega^2 = d\theta^2+\sin^2\theta d\phi^2\,$

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

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

Note that as $M\to 0$ or $r \rightarrow\infty$ one recovers the Minkowski metric:
$ds^{2} = -c^2dt^2 + dr^2 + r^2 d\Omega^2.\,$
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.