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I am aware of Hawking radiation. Which means that a black hole can actually spew out matter. Would it be possible for it to spew out enough to destabilise it, and "de-collapse" it?
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A black hole emits Hawking radiation, which is energy, and therefore mass. The mass is taken away from the black hole, so it gets smaller over time. The rate at which it gets smaller
increases the smaller it gets, so its kind of like a runaway reaction. Nobody knows if the black hole ever disappears entirely. It eventually gets small enough that quantum gravity is required to describe what's going on.
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As far as I know hawking radiation works as such: (correct me if I'm wrong)
Near the "threshold of no return" of a black hole a matter/antimatter particle pair comes into existence. Normally these particles would instantly annihilate each other, but in this circumstance, one of the particles is caught by the gravity of the black hole, but the other manages to escape. Which of the particles escapes? The particle or anti particle? And why? Surely both are under the influence of the massive gravity?
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This has been the popular line of how Hawking radiation works for over 20 years, but the usual calculations have nothing to do with particle/antiparticle pairs. There's no good intuitive picture for them actually.
Then a couple of years ago someone decided to see if this cartoon picture actually made any sense in reality. Amazingly, it all worked out. This is a valid way of looking at it. (just thought I'd clarify the history)
Anyways, both the particle and antiparticle have an equal probability to escape.
As for your other questions, the answers are mostly that nobody knows. First of all, things are really messed up and nonintuitive in standard classical relativity. Almost every notion from classical physics is thrown out the window. Its therefore very difficult to define what entropy and the 2nd law even mean except in the most idealized situations.
Add to that that entropy is best defined (not the only way though) by comparing microstates and macrostates. Nobody has much of an idea what microstates are in curved spacetime because there's no quantum gravity theory.
Also, singularities are problems with the classical theory. Understanding them will require a quantum description.