[knzhou]

He proved this in the context of classical general relativity. Hawking radiation is a (semiclassical) quantum gravity effect which is remarkable for being able to break this classical result.

The identification of area with entropy doesn't mean that entropy decreases as a black hole evaporates, because the emitted particles carry the entropy.

[kovrik]

But mass decreases which implies that event horizon shrinks, right?

Also, I thought never-increasing entropy was only applicable to closed systems. In that sense, when particles leave black hole through Hawking radiation, total entropy of the Universe remains the same. Am I missing something?

[pdonis]

> mass decreases which implies that event horizon shrinks, right?

Yes. So the entropy of the hole by itself decreases. But the entropy of the whole system, including the hole and the Hawking radiation it emits, increases.

> when particles leave black hole through Hawking radiation, total entropy of the Universe remains the same

We have no way of evaluating the total entropy of the universe. The concept itself might not make any sense, since we have no way of making measurements or running experiments on the universe from the outside.