Is the recently claimed solution to black hole information paradox published on PRL valid?

Question

In a recent paper on PRL, http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.111301, the authors claimed the black hole radiation is a pure state thus there is no information problems. The conclusion is based on solving a Schrodinger equation of the radiation.

While, in my mind, to support their main conclusion, the solution of Schrodinger equation is unnecessary: The essential problem is, can the radiation be exactly described by some action and Hamiltonian within current quantum theory? If so, the radiation must be always a pure state with the pure-state-assumption of the initial condition, just a primary property of all Schrodinger equations. Hence my question is, is it valid? 

 My major is not on this but I'm trying to understand the research on black hole information paradox. I feel, a little messy.

Answer 1

@Yu-XiangZhang That problem is that you are using a widespread but not a self-consistent logic. You can't say that since the process is described by Schrodinger equation, then it must be automatically unitary.  Note that Hawking never solved the full problem, he just noticed that since the radiation coming from the horizon is thermal, then something will go wrong when trying to describe the collapse within quantum mechanics. You have to find a way to treat the gravitational collapse in the context of quantum mechanics and check if something goes wrong. This paper actually shows that noting goes wrong as far as an asymptotic observer is concerned. But note that the paper does not claim that this solves the paradox in its entirety. You have to check what happens with an observer who sees the horizon and singularity.  

Comments

You can't say that since the process is described by Schrodinger equation, then it must be automatically unitary.

This is quite counterintuitive to me, could you elaborate?

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@Jia  This is why it is called a paradox. The clam of the paradox is that if you try to treat gravitational collapse in the context of the standard quantum mechanics (e.g. as described by the Schrodinger equation) something will go wrong, i.e. the process will not be unitary. Hawking called for modification of quantum mechanics to accommodate formation and evaporation of a black hole.   Again, it is important to note that he did not solve the problem of  formation and evaporation of a black hole. He worked in an asymptotic limit when the black hole is already formed, noticed thermal radiation from the horizon and concluded that since there is a disconnected region in space-time (beyond the horizon), and only featureless thermal radiation is coming out of it, then unitarity must be violated in some way. That is why it is very important to solve a full time dependent problem and see what is really going on (i.e. whether Schrodinger equation really  leads to non-unitarity or not).

Otherwise, you could say the following: Hawking in his original paper tried to solve for the evolution of the scalar field in the curved background of a black hole. He used the formalism of the field theory. Since the Klein-Gordon equation in the field theory formalism is unitary (preserves probabilities etc.), the there can't be any paradox whatsoever. Of course, that is not what Hawking claimed. He claimed that when you apply the formalism to this gravitational system, it looks like unitarity breaks down.   

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OK, I see. So I guess this belongs to the kind of issue that can't be understood without going into technical details. 

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Yes.

Answer 2

It is an interesting calculation, I haven't checked the steps and Please use Arxiv links when available

First thing to notice is the Infalling shell is treated in Classical GR, and Scalar field theory is quantized in this Gravitational Background. So it is not a full calculation Involving quantum gravity.

So this wont constitute as a resolution to Black hole information Paradox. Ideally even the infalling matter must be treated Quantum Mechanically.

Assuming the calculation is correct, Trace(Pho^2) = 1 Implying that the Scalar field evolved in a unitary manner is an interesting one. But I don't know what would happen if one were to change the co-ordinate system.

I have Little expertise on this subject so many thing I would have said might be naive, Any comment/corrections will be appreciated.

Comments

I mean that the calculation, no matter correct or not, is irrelevant. Because when you assume the initial state is pure and the evolution follows some action or Schrodinger equation, the result must be pure, and thus "unitary" and no information problems, Trace(\(\rho^2\))=1 is not surprising at all (if the observer could observe the entire quantum field, not like the Unruh effect where only one Rindler wedge is accessible ). But I don't know whether it is valid to make such assumptions,  and why the paradox is so thorny?

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@Yu-XiangZhang, What about the entanglement between the infalling matter and scalar field?

If they are interacting gravitationally they must entangle right?

@Yu-XiangZhang I was skeptical too. It would be good If someone else commented on the paper.

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@Prathyush, I edited your comment to @ user Yu-Xiang Zhang. If I'm not mistaken, user won't automatically be notified of your further comments even if he/she commented under your answer. 

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@Prathyush, I think so. However, the entanglement is absent in the PRL paper, which says radiation from the collapsing object is always a pure state. Pure states cannot be entangled with other systems. So I don't believe their result.