## Schwarzchild Radius

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.