[McKay (OP)]

I am pretty sure the answer is "yes" - thats the whole idea with space elevators, isnt it? As long as you can keep applying a force away from the well, by pulling on a cable or just having super thrusters,  you should, eventually crawl out of the gravity well.

Now, to the real question - what about black holes? Their whole thing is that the escape velocity becomes >c  below the event horizon and, since nothing can travel >c, thats that... but what about slow crawling way I mentioned above?
To avoid spaghettification lets use a supermassive BH.

[Janus]

Escaping from a planetary gravity well is really all about energy.  You can give an object enough kinetic energy (accelerate to to surface escape velocity) or you can slowly lift it against gravity, as is the case with a space elevator.  The minimum energy needed is the same in both cases.  The advantage of the space elevator is that its energy source is planet bound, and the rocket has to carry its power source with it. The rocket doesn't only need to lift itself, but also its own fuel/reaction mass, which makes it less efficient than the space elevator.   In addition, the longer the rocket takes in making its climb, the less efficient it is.
Black holes are a different matter, as now you have to consider General Relativity, and in General Relativity, Gravity is the due to the curvature of Space-time.  Inside the event horizon, all world-lines lead to the center of the black hole.  A world-line is the path an object takes through space-time, and all such possible paths inside the event horizon end up at the center.  An object inside the event horizon has no "future" that doesn't result in it being at the center of the BH.

[jeffreyH]

Escape velocity is the velocity required to escape to infinity from a particular point in a gravity well. The value varies with change in radial distance from the source. If you wanted to move from earth to the moon you would not need to achieve escape velocity. If you want to move temporarily from the earth's surface to one foot above it you can simply jump. Just because at the event horizon you need to reach the speed of light to escape to infinity does not mean that you could not crawl away from the event horizon if you had enough energy. However the energy required at this point is infinite. This is because the amount of relativistic mass of an object is in theory infinite at the horizon.

[Bill S]

So, in theory you could crawl away, but in reality you couldn't, because you would be seriously overweight?

[evan_au]

If you want to move temporarily from the earth's surface to one foot above it you can simply jump. Just because at the event horizon you need to reach the speed of light to escape to infinity does not mean that you could not crawl away from the event horizon if you had enough energy.

In my primitive understanding, inside the event horizon, if you managed to jump, you would exceed the speed of light. And accelerating a massive object beyond the speed of light is impossible (so far as we know).

The geometry inside a black hole twists our usual notion of space and time, so even the concept of "away from the black hole singularity" has no meaning in the sense that we might use it outside the event horizon.

[jeffreyH]

So, in theory you could crawl away, but in reality you couldn't, because you would be seriously overweight?

In theory since there is a singularity at the horizon. Not the same one as at the centre. However there are no possible orbits between the photon sphere and the horizon. So that motion perpendicular to the field no longer holds true. It is a theoretically strange place. I have reservations about this viewpoint and in particular the Schwarzschild solution.

[jeffreyH]

If you want to move temporarily from the earth's surface to one foot above it you can simply jump. Just because at the event horizon you need to reach the speed of light to escape to infinity does not mean that you could not crawl away from the event horizon if you had enough energy.

In my primitive understanding, inside the event horizon, if you managed to jump, you would exceed the speed of light. And accelerating a massive object beyond the speed of light is impossible (so far as we know).

The geometry inside a black hole twists our usual notion of space and time, so even the concept of "away from the black hole singularity" has no meaning in the sense that we might use it outside the event horizon.

Inside the event horizon is a different kettle of fish!

[Bill S]

In theory since there is a singularity at the horizon.

Could you say a bit more about that "singularity", please?

[jeffreyH]

In theory since there is a singularity at the horizon.

Could you say a bit more about that "singularity", please?

It is about the nature of a mathematical singularity. This is a point at which a mathematical function breaks down. The relativistic version of gravitational potential or acceleration includes the radius to the event horizon. This is due to the Schwarzschild solution to the Einstein field equations. When a particles position is coincident with the horizon radius the mathematics are undefined.

https://en.m.wikipedia.org/wiki/Singularity_(mathematics)

[Bill S]

Thanks, Jeffrey. I'll not pretend to understand all the maths in the link, but I get the idea.

This seems to fit with Chris Baird's comments:

"In general, singularities are the non-physical mathematical result of a flawed physical theory. When scientists talk about black hole singularities, they are talking about the errors that appear in our current theories and not about objects that actually exist."

http://wtamu.edu/~cbaird/sq/2013/09/13/does-every-black-hole-contain-a-singularity/

[yor_on]

I suspect Janus expressed it succinctly. It's about the energy needed, and the energy needed for 'climbing out' should then be needed to be 'infinite', also presuming you find a way to define where/how to 'get out'. A singularity as a black hole represent a infinite 'border' between being outside versus inside.

And then it's also about how mass and 'forces', communicating your wish to climb will act. They will all have a geodesic pointing to a 'center'. They won't communicate any more.
=

Actually, thinking of it some more. That seems a pretty good argument for us not being inside some 'event horizon'. We, as in forces/mass interactions etc, do communicate.