# What is the escape velocity of a Black Hole?

The escape velocity of Earth is $v=\sqrt{\frac {2GM}{R}}$, 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 $c$.

But I want to know the escape velocity of black holes. Is it much more than light speed?

• The escape velocity formula is $v_e=\sqrt{\frac {2GM}{R}}$
– voix
Aug 10 '12 at 21:05
• You might be interested in my response to this related question, as it points out that escape velocities from black holes are not quite the same as Newtonian escape velocities.
– user10851
Aug 29 '12 at 4:42

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.

• 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. Aug 11 '12 at 12:36
• @AlanSE - Yes, for a point-like object.
– voix
Aug 11 '12 at 20:24
• I don't buy your derivation. The energy of a particle is not given by the second formula you've written since it neglects the gravitational field, you are considering a free particle. The correct energy is $E=\gamma mc^2 \sqrt{1-2GM/R}$, which leads to $v_e=\sqrt{2GM/R - (2GM/R)^2}$. Aug 15 '17 at 21:29
• After coming back to this question, your answer is valid for a Schwarzschild BH when the escaping particle travels radially and you measuring the speed from the starting point. The formula I wrote assumes you measure the distance from a point at infinity, which is not the same since speed is coordinate dependent. Sep 14 '19 at 9:59

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 $c$.

• Where the "surface" is the event horizon. There isn't necessarily any material surface there. Aug 10 '12 at 21:38
• Yes, that means event horizon. Aug 10 '12 at 21:38
• But if the escape velocity is $c$, why can't light escape? 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. Aug 11 '12 at 22:20