Theoretically, can a black hole collapse until it gets smaller than its Schwarzschild radius?

So, the concept is this:

When a supergiant star collapses, its nucleus becomes a black hole. So, what if that black hole collapses again?

I don't know if it is possible, but what if it happens?

Answer

No. And maybe, but probably not.

General Relativity tells us that time passes more slowly in an intense gravitational field (from the point of view of someone in a less intense region of the field). This has been verified experimentally on a small scale.

Imagine you're near (but not too near) a black hole, in communication with a probe that's dropping into it. As you observe an analog clock on the probe, you'll see its second hand moving more and more slowly as the probe approaches the event horizon, coming closer and closer to a dead stop.

You'll never actually see the probe reach the event horizon; time on the probe, as seen from an outside vantage point, will come arbitrarily close to the moment it reaches it, but it will never actually get there.

The same thing happens to any matter falling into a black hole. It will quickly reach a point where you can no longer see it, but it will never quite reach the actual event horizon.

Inside the event horizon, time passes infinitely slowly (again, this is all from an outside point of view). So once the event horizon forms during the initial collapse that forms the black hole, nothing further happens inside it. Everything inside the event horizon is frozen in time, and cannot collapse any further.

On the other hand, from the point of view of the probe itself (say, if you foolishly volunteered to ride along), local time continues to pass at its normal rate of 1 second per second. You'll pass through the event horizon, and if the black hole is massive enough for the tidal stress at that point to be manageable, you might not even notice. You might see events, including further collapse, continue to occur after you're inside. But if you look back as you're falling, you'll see time in the outside universe pass more and more quickly, and just as you cross the event horizon you'll see eternity pass in a finite amount of time. So from the point of view of a hypothetical observer who's fallen into a black hole, yes, the body that formed the black hole can continue to collapse -- but any such observer might as well be outside the universe, since there's no way to communicate with them.

All this is based on relativity, ignoring quantum mechanics. Current theory (see Hawking radiation) says that quantum effects cause black holes to evaporate. All information about anything that fell into the black hole is lost (i.e., this isn't a way for you to escape after you've fallen in), but all the mass/energy will eventually come out as random radiation. This happens in a finite amount of time from the point of view of an outside observer -- which means that if you're riding the probe, the black hole will evaporate just as you're crossing the event horizon. (You will not survive the experience.)

Disclaimer: I am not an astrophysicist. I hope that one will come along soon and explain how I've gotten this wrong.