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I just watched a BBC Horizon episode where they talked about the Hawking Paradox. They mentioned a controversy about information being lost but I couldn't get my head around this.

Black hole thermodynamics gives us the formula $$S ~=~ \frac{A k c^3 }{4 h G}.$$

And we also have Einstein's famous $E = m c^2$, which mean that mass can be turned into energy, right? Hence information is either lost or it is preserved, but now in energy-form instead of mass-form.

I can't understand why radiation from black holes would be any different than an atomic bomb for example, where mass is also turned into energy?

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This issue is much more sophisticated than you think. However, since there are real physicists here, I won't dare attempting at answering the question (maybe next year!) So, let's wait for someone to clear the fog of confusion here. Meanwhile, I suggest you read this article: iopscience.iop.org/1742-6596/171/1/012009 – stupidity May 29 '12 at 20:16
Thanks, I read through the pdf, but since I'm a mere mortal with basic physic education, it was quite hard to understand :) – Joakim May 30 '12 at 19:36

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Radiation normally contains subtle correlations. For all practical purposes you can't use it, but it's there. Hawking radiation is, according to the theory, perfectly thermal and does not contain any more information than the temperature itself. The problem is that then the process of black hole evaporation is not reversible, in principle. Unlike all other processes that we know of (which might be irreversible in practice, but are reversible in principle). That irreversibility (which implies non-unitarity) is incompatible with quantum mechanics. That is the problem in a nutshell. It is really a conflict between quantum mechanics and semi-classical general relativity.

There are many more things to be said but I get the impression you haven't done a lot of reading about this and details would be rather pointless. I suggest you browse around for a bit with that starting point in mind.

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See WIMP's answer to physics.stackexchange.com/questions/29175. Your example of an atom bomb is in principle time reversible (though obviously not in practice), but Hawking radiation is not time reversible even in principle. – John Rennie May 30 '12 at 9:05
You are quite right, I've never heard about the paradox until yesterday (my profession is computer science, not physics). Hmm, can you think of the thermal radiation as a radio wave that you could tune in to? And in ordinary thermal radiation you can hear the music of past events but in the radiation from black holes it is total silence - and that is the problem? – Joakim May 30 '12 at 20:02
No, it's got nothing to do with finding the right frequency. You have correlations. Think about mixing dough. If you mix it, you're not, theoretically, destroying information. If you'd carefully take note of the exact location of every single atom (of egg, flour, milk, etc) you could tell what the dough looked like before mixing, because they're oh-so-exactly matched together to get egg flour and milk back in time. There is no such fine matching in black hole radiation. It's just dough that you don't know where it came from, if you will. Maybe not the best analogy. – WIMP May 31 '12 at 6:19
I appreciate that you're trying to explain this to me! It seems like I have some reading to do before fully understanding this though :) One more thing, if you have the patience to help me. What is the reason to the missing correlations? A friend of mine talked about particles and antiparticles appearing at the event horizon and the antiparticle falling in meanwhile the particle was left behind - does that have anything to do with it? – Joakim May 31 '12 at 22:18
Yes, it's got something to do with that. Your friend was talking about a popular (though slightly incorrect) explanation for Hawking radiation. One way to understand the missing correlations in the radiation is that: The Hawking radiation (which you can picture as pairs of particles and antiparticles created at the horizon, and one being captured) is produced at the horizon and has nothing to do with the matter that collapsed. The part that escapes and goes to infinity is random (thermal), and knows nothing about the collapsed matter. Thus, the information of the collapsed matter is lost. – WIMP Jun 1 '12 at 8:11
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