# What is the escape velocity of a Black Hole?

The escape velocity of Earth is $v=\sqrt{\frac {GM}{R}}$, where $M$ is the mass of the Earth and $R$ it's radius (approximating it as a sphere), and is much less than light speed $c$.

What is the escape velocity of a Black hole? Is it much more than light speed?

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The escape velocity formula is $v_e=\sqrt{\frac {2GM}{R}}$ – voix Aug 10 '12 at 21:05
– Chris White Aug 29 '12 at 4:42

## 2 Answers

In General relativity, energy formula of a body thrown straight up to the infinity is

$\large {E=\frac{mc^2}{\sqrt{1-R_S/R}}}$

As we know relativistic energy formula is

$\large {E=\frac{mc^2}{\sqrt{1-v^2/c^2}}}$

So

$\large {\frac{mc^2}{\sqrt{1-v_e^2/c^2}}=\frac{mc^2}{\sqrt{1-R_S/R}}}$

hence escape velocity equation in General relativity is

${\large {v_e^2=c^2\frac{R_S}{R}}}$

where $R_S=2GM/c^2$ - Schwarzschild radius of a black hole, and $R>R_S$

It's easy to derive that

${\large {v_e=c\sqrt{\frac{R_S}{R}}}=\sqrt{\frac {2GM}{R}}}$

So escape velocity formula in General relativity and Newton gravity is the same.

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So the escape velocity at the event horizon would go to the speed of light as the ratio in the radical goes to 1?  Am I reading that right?  That would still quite simply imply infinite energy, so it seems consistent. – AlanSE Aug 11 '12 at 12:36
@AlanSE - Yes, for a point-like object. – voix Aug 11 '12 at 20:24

The escape velocity from the surface (i.e., the event horizon) of a Black Hole is exactly $c$, the speed of light.

Actually the very prediction of the existence of black holes was based on the idea that there could be objects with escape velocity equal to $c$.

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Where the "surface" is the event horizon. There isn't necessarily any material surface there. – Keith Thompson Aug 10 '12 at 21:38
Yes, that means event horizon. – Anixx Aug 10 '12 at 21:38
But if the escape velocity is $c$, why can't light escape? – seriousdev Aug 11 '12 at 14:18
It can if it comes from a place just above the horizon. But it looses much of its energy and becomes redshifted. – Anixx Aug 11 '12 at 22:20
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