My understanding is that light can not escape from within a black hole (within the event horizon). I've also heard that information cannot propagate faster than the speed of light. It would seem to me that the gravitational attraction caused by a black hole carries information about the amount of mass within the black hole. So, how does this information escape? Looking at it from a particle point of view: do the gravitons (should they exist) travel faster than the photons?
General Relativity – How Does Gravity Escape a Black Hole?
black-holesgeneral-relativitygravityspeed-of-light
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This is a great question, because it's a subtle variation on the usual question about spaghettification and supermassive black holes, and shows somewhat deeper thinking.
So let's assume the black hole is supermassive -- or more specifically that you are really tiny compared to the black hole -- so that we can ignore tidal effects. Tidal effects are the difference in gravitational "force" on two different parts of an object. In this case, I mean the difference between the acceleration of your feet and your head. Your feet are slightly closer to the center of the black hole, so they will experience a slightly greater acceleration than your head. You would feel this as a slight tug on your feet. The bigger the hole or the farther you are from it, the smaller the difference will be. At some point, it will be so small that it's "in the noise" and you don't even notice it. We're assuming that.
If your head were somehow stuck just outside the horizon,† you would be right. I don't think anyone would claim you wouldn't feel anything if your head were attached to a rocket keeping you out, while your feet dangled inside the black hole. :) But those aren't tidal effects; they're acceleration effects.
On the other hand, if you are falling into the supermassive black hole (even if you jumped off this crazy rocket just an instant earlier), things are very different. Your head and feet are being "accelerated" at basically the same rate (relative to some stationary coordinate system, let's say) because you are so small compared to the black hole. So your head is moving at roughly the same speed as your feet, which means that the signal doesn't have to actually move outward relative to these stationary coordinates (it can't). Instead, it just needs to move inward more slowly than your head. And that's entirely allowed everywhere, even well inside the black hole.
You'll typically see this sort of thing represented by a graph of the light cones. And inside the horizon, those light cones "tip over" towards the singularity. This means that even light pointed outward can't actually move outward; the outward-pointing light ray will still be moving toward the singularity. But your head (and your feet) are moving toward the singularity faster, so your head enters into the light cone of your feet. Which means that relative to your head light can still move outward, as can a nerve impulse. Basically, think of two light rays given off by your feet: one directed toward the singularity, and the other directed away from it. You'll probably believe that they have different speeds. The speed of your feet is somewhere between those two, as is the speed of your head.
So all that needs to happen is for your head to enter the future light cone of your feet before your head hits the singularity. Not a problem, since the black hole is so large and you've still got a while to go. Now, you might be concerned that your feet will hit the singularity before your head gets that first signal, which would seem weird. But then you remember that the concept of simultaneity is relative. Your head and feet are in the same reference frame -- at least far from the singularity -- so they experience things at basically the same rate, and nearly the same time as judged in their own reference frame.
† Just as a side note, you should try to distinguish between an event horizon and an apparent horizon. Technically, you're talking about the latter, which is the local surface where light rays that are directed outward can't actually move outward. An event (or absolute) horizon, on the other hand, has nothing to do with local effects -- at least not directly. You can only know if something is an event horizon if you know the entire future history of the universe. Unfortunately, the term "event horizon" is thrown around in popular descriptions of black holes when it shouldn't be. They happen to be the same for certain special black holes, but they really are different concepts, and the right way to think about a horizon is different in the two cases. I just use the term "horizon", and anyone who knows the difference will figure it out. A good (and accurate) popular reference for all such things is Thorne's "Black holes and time warps". The standard technical reference is Hawking & Ellis's "The large-scale structure of space-time".
Best Answer
There are some good answers here already but I hope this is a nice short summary:
Electromagnetic radiation cannot escape a black hole, because it travels at the speed of light. Similarly, gravitational radiation cannot escape a black hole either, because it too travels at the speed of light. If gravitational radiation could escape, you could theoretically use it to send a signal from the inside of the black hole to the outside, which is forbidden.
A black hole, however, can have an electric charge, which means there is an electric field around it. This is not a paradox because a static electric field is different from electromagnetic radiation. Similarly, a black hole has a mass, so it has a gravitational field around it. This is not a paradox either because a gravitational field is different from gravitational radiation.
You say the gravitational field carries information about the amount of mass (actually energy) inside, but that does not give a way for someone inside to send a signal to the outside, because to do so they would have to create or destroy energy, which is impossible. Thus there is no paradox.