Gravitational potential relative a black hole

Question

If I understand correctly, the escape velocity is $c$ at the event horizon of a black hole. This would seem to mean that an object falling into the black hole from an infinite distance would attain velocity $c$ - regardless of the size of the black hole.

I presume the above reasoning is flawed. For one thing, to an external observer an object slows down due to time dilation and never enters the event horizon at all, never mind at light speed.

What I want to know is, since the Newtonian approach doesn't work, what is the gravitational potential of an infinitely distant object with respect to a black hole, and how does it vary with the black hole's size? Since nothing escapes the event horizon I'm only asking about the energy that can be extracted outside of the horizon.

Answer 1

The gravitational field points in the negative r direction, so when we integrate it over r to get potential, we obtain a negative number. Adding potential energy means making the negative number closer to zero.

For distances far enough from the event horizon to use classical physics,

$$V=-\frac{GMm}r$$

Gravitational potential energy is always negative, with $V=0$ at $r=\infty$ for all $M$ and $m$.

For finite r, then, gravitational potential energy varies directly with the mass of the black hole.

Close to the event horizon, we'll need to use physics that I don't understand that employ tensor mathematics that I definitely don't understand, so maybe somebody can chime in with that aspect of it.

---

You wrote,

to an external observer an object slows down due to time dilation and never enters the event horizon at all

This is not correct. The external observer sees the falling object accelerate normally and vanish into the black hole. It is the internal observer for whom time is slower.