The gravitational potential energy of an object of mass
\[m\]
a distance \[r\]
from centre of a spherical mass \[M\]
is \[GRAVITATIONAL \: POTENTIAL \: ENERGY=GPE=- \frac{GMm}{r}\]
.The body is given a speed is
\[v_{ESCAPE}\]
, just sufficient to escape the gravitational field, and have zero energy. Then
\[TOTAL \: ENERGY=KINETIC \: ENERGY + GPE=\frac{1}{2}mv^2_{ESCAPE}- \frac{GMm}{r^2} =0 \]
.Rearranging gives
\[v_{ESCAPE}= \sqrt{\frac{2GM}{r}}\]
.Suppose now that the particle is a photon. A photon is massless, but the same formula applies, with
\[v_{ESCAPE}=c\]
, the speed of light.
\[c=\sqrt{\frac{2GM}{r}}\]
.
Rearranging again for \[r\]
gives \[r=\frac{2GM}{c^2}\]
.This is an important concept. A mass
\[M\]
confined to a radius \[r=\frac{2GM}{c^2}\]
or less will allow no light to escape. We call this expression the equation for the 'Schwarzchild Radius'. Any mass contained within it's Scawrchzchild radius is a black hole.