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There is a big concern about loosing information inside a black hole. As far as I understand, the black hole is characterized classically by its mass, angular momentum and charge. Semi-classically, thermal radiation is added.

Lot's of effort is paid to somehow restore the amount of information that crossed the horizon of a black hole, e.g. holographically on the surface of the event horizon. Why is it not allowed to simply loose the information? Is it related to the reversibility of fundamental laws of nature?

If I burn the book, all the information that was stored in the special arrangement of paper and ink atoms are lost as well. Of course, the atoms still exist, but the content of the book that they were storing is lost.

What is the difference in the quality of information loss between the burning and the crossing of the event horizon?

Could the black hole just simply be a cosmic fire that burns everything irreversibly into thermal radiation?

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I'll answer based on limited reading in this area of physics.

As I understand it, the background is the view that general relativity and quantum physics both espouse a unitary approach. That is, the evolution of the physical systems (quantum fields plus spacetime) is unitary. In Hawking radiation there is a non-unitary element apparently "creeping in", but it depends on exactly how the calculation is done. If we allow a black hole to completely evaporate then, arguably, the overall evolution "ought not" to be non-unitary. On this argument all the information that went into the black hole during its lifetime should reappear in the physical phenomena, a bit like burning a book (where the information goes into the smoke and ashes). The information would be in the Hawking radiation.

I put the words "ought not" in inverted commas in order to signal that this is not an area where we can claim sufficient confidence to say for sure that it has to be unitary. After all, general relativity suggests a singularity inside the black hole horizon; if we suppose there is not really a singularity then that's all very well but then we don't know what there would be instead.

But let's propose that the overall evolution is unitary. Can we make that plausible?

The difficulty is in making this intuition look convincing. One of the problems is that you can imagine something like a book or a living thing falling in (lots of information) or something of the same mass but little structure (little information) and there is no particular reason why the Hawking radiation immediately after it fell in would be very different in the two cases. Also, for a large black hole the Hawking radiation is much too weak to account for information of stuff falling in right there and then. So to get the information back at evaporation there has to be a way for it to be stored somehow. As I understand it, attempts so far to give an account of the whole process have not looked convincing enough to gain widespread assent. It is more a case of different people placing bets on whether a feasible model will be found sometime.

But as I say, I am not an expert on this, just reporting as one who is familiar with the background. If there is recent news then I hope another person will answer and provide it.

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    $\begingroup$ "here is no particular reason why the Hawking radiation immediately after it fell in would be very different in the two cases" - In your frame of reference as an external observer of the Hawking radiation, nothing ever crosses the horizon. You cannot see a result of somethng that never happens in your world. $\endgroup$
    – safesphere
    Commented Nov 14, 2020 at 7:33
  • $\begingroup$ @safesphere your comment is not relevant in this context. The infinite time to cross is an artifact of a specific set of coordinates only for a spherically symmetric vacuum solution. The information question does not assume spherical symmetry and Hawking radiation has nonzero stress energy. So the spacetime is neither spherically symmetric nor vacuum. $\endgroup$
    – Dale
    Commented Nov 16, 2020 at 13:58
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Let's first consider how information is put in the book: Some nice clean paper is tarnished by ink.

There exists a only a very small probability that the ink leaves the paper and goes back to the ink canister. But the laws of physics allow it by being time-symmetric.

Surely a recycling pant can take ink out of paper. But said plant must use energy and it must cool itself, IOW it must tarnish the energy.

Well, the waste heat will then contain the information that was in the book. Waste heats produced when processing different books are different from each other.

Now, a black hole can be used as a cooling device in a recycling plant. Then later empty space can be used as a cooling device that cools the black hole.

Does a black hole clean the waste heat, by removing the information, like the recycling plant cleans the paper? Well, it shouldn't if it obeys the same laws as the plant.

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    $\begingroup$ To make the point more clear: What if, instead of burning, I annihilated the book with anti-matter, such that I would be left with a big number of photons. These high-energetic photons, if they met, wouldn't necessarily create the same particles and anti-particles that they had been created from. They could in principle create neutrino and anti-neutrino pairs. This would be completely non-deterministic. I don't understand, how any information could be retrieved, even in principle. $\endgroup$
    – p6majo
    Commented Nov 14, 2020 at 13:29
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    $\begingroup$ @p6majo Don't different books produce different explosions when annihilated? You just study very closely the explosion, then you find out what exploded. $\endgroup$
    – stuffu
    Commented Nov 14, 2020 at 13:42
  • $\begingroup$ Ok, I start to see the point. If two different books of equal mass but different ink distributions (different words, different pieces of information) were annihilated, there would be a slightly different outcome in the final distribution of photons. Whereas in the case of a black hole the increase in mass would be the same in each case, so both final states would be equivalent. From an outside observer, it will be hard to tell, since it takes infinitely long. However, in the frames of reference of the books it would happen in a finite amount of time. $\endgroup$
    – p6majo
    Commented Nov 14, 2020 at 14:28

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