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When I first heard about the black hole information paradox, I thought it had no content. At the time, papers about it had been written for numerous years and they keep on coming. Now that the press got wind of Hawking's latest one, I thought I should ask about it:

The information paradox relies on the no-hair conjecture. However, all its proofs I'm aware of rely on the fact that we end up with a stationary black hole. So once we introduce Hawking radiation, the theorem evaporates right besides its subject.

Basically, we're wondering why a theorem we have proven for the stationary case does not hold for the non-stationary case. That seems hardly surprising to me, but I may have missed something obvious.

On a related note, I always found the no-hair theorem somewhat suspicious because it means after formation of the black hole, we end up with a result stronger than Gauss's law, whereas before formation of the black hole, the generalizations of Gauss's law to relativistic gravity are (again, as far as I know) generally weaker.


To illustrate the argument from a different point of view, let me describe the thermodynamic information paradox:

First, lets start with the no-hair theorem, which states that isolated systems will tend towards a stationary equilibrium state, uniquely described by just a few parameters.

While going forward, instead of looking at completely isolated systems, we now allow interaction via absoption and emission of radiation.

The asumption is that because the system has no hairs, no matter the incoming radiation, the outgoing radiation will obey the totally probabilistic thermal laws.

Let's also assume that we're going to reach $T=0$ after all energy has been radiated.

This is, as far as I can tell, a pretty close analogy to the black hole paradox, and has a simple resolution: Physical equilibrium states fluctuate and thus have hairs. In fact, thermal radiation alone will disrupt equilibrium, and just assuming that it doesn't leads to nonsense.

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  • $\begingroup$ It doesn't rely on no-hair theorems. No matter what the black hole is, as long as it evaporates completely, it takes a pure state to a mixed state. May be a better name would be "black hole loss of unitarity paradox". $\endgroup$ – MBN Jan 28 '14 at 10:23
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    $\begingroup$ @MBN: I may have missed something obvious, but I don't think that's it: after all, we only end up with a mixed state because we trace out irrevocably lost degrees of freedom - but they are only irrevocably lost if there are no hairs $\endgroup$ – Christoph Jan 28 '14 at 10:43
  • $\begingroup$ Well, most people still believe that it is actually possible to cross the horizon effectively, so you shouldn't expect that them to understand that there is a foundamental difference between a static phenomenon (eternal black hole) with a dynamical one (real black hole if I can say). This said, I find your heuristic quite convincing, and I would also add that another good heuristic toward quantum gravity would be to forbid any kind of matter to cross the horizon. That is, horizon is just as fictive as it is for Rindler's observer. $\endgroup$ – sure Mar 26 '15 at 15:33
  • $\begingroup$ @MBN Mathur (2009) says that evaporation of a black hole with hair could conceivably evade Hawking's theorem and end in a pure state. $\endgroup$ – tparker Feb 9 '17 at 21:39
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I'm kind of in your boat. Hawking radiation violates almost all of the energy conditions, and a stacked set of apparent horizons is two-way transversible when their area decreases with time. I see no reason why the typical assumptions like cosmic censorship should apply. And if cosmic censorship is gone, and the black hole is two-way transversible, then why is it a problem to have the degrees of freedom live inside of the black hole?

I've asked this question of professors at conferences and never gotten satisfactory answers.

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I substantially agree with your analysis. I usually paraphrase the information paradox in this way: "show me a solution of the gravity equation (Einstein or beyond) that is consistent with unitarity and the physical attributes of a black hole". By attributes I mean that from far away it appears black, characterized by the asymptotic charges (Mass, Electric-Magnetic Charges, Angular Momentum) and has the correct range of Masses and other astrophysical characteristics. We know that usual gravitational solutions are bad to describe a real "physical" blackhole-like object, but we don't know what to consider instead.

In the classical picture of the black hole, the horizon is an information-free area. That is why is not possible to retrieve information of what is inside. Now the crucial point is that if you try to do a small perturbation of the the known black hole solution or to give some small structure at the horizon, you are not able to solve the problem. Basically every perturbation either falls beyond the horizon or is emitted. The only way to preserve something at the horizon without drastic changes is having something massless, see the soft graviton proposal of Hawking-Perry-Strominger, even though this last proposal has many obscure points, and still does not solve the problem of the central singularity.

It appears that to solve the problem you need a change of order 1 at the horizon, in such a way to have structure and therefore no loss of information. The fuzzball proposal solves all these problems in string theory by explicitly giving you a non singular solution made of intersecting D-branes that has structure even at the horizon, in which hawking radiation is nothing more that closed string emissions. From far away is nearly impossible to tell the difference between a black hole and a fuzzball, but near the horizon the difference is order 1. For completeness, the current issue with this proposal is that we are only able to work in supergravity, while realistically we will need the full string theory, that now is not under control.

Other approaches (ER=EPR conjecture) postulates non local interactions between inside and outside, and falls into the broader framework of looking at gravity as an emergent phenomenon from something more fundamental, perhaps entanglement in some quantum theory.

In every case it appears that a solution to the "paradox" will require a strong departure from the classical black hole picture. That's why is called (a bit misleadingly) information paradox.

The literature on the subject is enormous, I can suggest Mathur (2009) as an introduction.

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