# General Relativity/Schwarzschild metric

Main article: Schwarzschild metric

The Schwarzschild metric can be put into the form

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

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

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

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

${\displaystyle r_{s}={\frac {2GM}{c^{2}}}}$

Note that as ${\displaystyle M\to 0}$ or ${\displaystyle r\rightarrow \infty }$ one recovers the Minkowski metric:
${\displaystyle ds^{2}=-c^{2}dt^{2}+dr^{2}+r^{2}d\Omega ^{2}.\,}$
Note that there are two singularities in the Schwarzschild metric: at r=0 and ${\displaystyle r=r_{s}={\frac {2GM}{c^{2}}}}$. 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.