The Schwarzschild Radius

For a body of mass  
  on the Earth's surface, the gravitational potential energy is  
\[GPE=- \frac{GM_{EARTH}m}{R_{EARTH}}\]
This means that in order for that body to just escape the Earth's gravitational influence it must be given an amount of kinetic energy  
  sufficient to cancel out the negative gravitational potential; energy.
\[\frac{1}{2}mv^2 = \frac{GM_{EARTH}m}{R_{EARTH}} \rightarrow v^2=\frac{2GM_{EARTH}}{R_{EARTH}} \]
The Schwarzschild radius is the answer to this question: Suppose a mass is on the surface of a spherically uniform non rotating body of mass  
. What would the radius of the mass  
  have to be for the escape velocity to equal the speed of light  
This is really a question in General Relativity, but surprising the result is exactly (1), that is  
\[c^2=\frac{2GM}{R_{SCWARZSCHILD}} \]
We can rearrange this to give  
Of course, nothing can travel faster than light, not even light, so anything on the surface of such a body will stay there forever. The body is called a black hole.

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