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Claus Kiefer, "Quantum gravity", 3rd ed., page 220/221, says in chapter 7 "Quantization of black holes":

"A theory of quantum gravity should give a definite answer to the question of whether unitarity (with respect to an outside observer) is preserved or not."

I am not able to see the problem here. According to the basic characteristics of a black hole, an outside observer can never observe what is happening inside the event horizon, and for everything happening outside the event horizon there is no unitarity issue.

So is there an error in the cited phrase or am I missing something? What exactly should a theory of quantum gravity provide?

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    $\begingroup$ In the case of quantum mechanics near the horizont, the Schrödinger equation considered in the Schwarzschild metric becomes a heat equation when crossing the horizont. Therefore, unitarity is not conserved by this approach. $\endgroup$ – kaffeeauf Feb 5 '17 at 18:18
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    $\begingroup$ See the black hole information paradox. $\endgroup$ – Qmechanic Feb 5 '17 at 18:36
  • $\begingroup$ @kaffeeauf: cosmic censorship prohibits (at least de facto) the application of current concepts such as the Schrödinger equation inside or at the event horizon. In contrast, from the point of view of the reference frame of an outside observer, infalling particles get never lost because they never reach the event horizon. $\endgroup$ – Moonraker Feb 5 '17 at 19:43
  • $\begingroup$ B@Moonraker. Cosmic censorship only says there are no naked singularities. Equations of QFT hold inside the horizon, at least a little inside, because gravity is far from needing quantization at those gravitational fields near the horizon for macroscopic black holes. The time is a separate issue and makes no difference. See the wiki article referenced by QMechanic about the information paradox for the black hole and unitarity, and possible solutions $\endgroup$ – Bob Bee Feb 6 '17 at 3:38
  • $\begingroup$ Ps/QFT holds in GR backgrounds if you do what's expected, make it covariant. Hawkings did it for Black Hole radiation and there's books and papers on other uses of it. $\endgroup$ – Bob Bee Feb 6 '17 at 4:15
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Semi-classical gravity does infer a situation which complicates the subject of unitarity within black hole physics. The evolution of two states after forming the black holes are identical, leading to a mixed state obtained through integrating the thermal Hawking radiation states. It leads to the information paradox.

The problem with this is that the final states are identical - we cannot recover the initial state of the evolution just by knowing the final state, even in principle. This contradicts unitarity evolution in quantum mechanics.

Reference: "Black Holes, Information, and Hilbert Space for Quantum Gravity" (2012).

In principle unitarity preserves the ability to recover the initial state if we know the final state by applying the inverse of the time evolution $e^{+iHt}$.

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The quote is referring to the information loss paradox.

The paradox arises when considering quantum fields in a curved black hole background. Hawking has shown long ago that black holes (where black stems for zero emitting objects) actually becomes radiating objects with a black (!) body spectrum. Moreover this radiation make them slowly evaporate, therefore unitarity of quantum mechanics and evaporating black hole solutions seems to be mutually incompatible, even though they are solid predictions of the respective theories.

A quantum theory of gravity should:

  1. Provide a mechanism to describe unitary black hole evaporation (or the contrary, even though few people (no one?) are willing to give up unitarity)

  2. Get rid of the central singularity

  3. Give a microscopic interpretation of the entropy formula for black holes

  4. Give a gravitational descriptions of these microstates

String theory has achieved 1) and 3) 20 years ago. Some approaches (the fuzzball proposal for instance) are working on solving even 2) and 4). In some particular (but physically unrealistic) systems the problem has been completely solved (D1D5 system).

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  • $\begingroup$ I don't see any answer to my question: If the outside observer is taken as the reference (as Kiefer does), how can the the unitarity problem (or also the information loss paradox) exist with respect to the point of view of an outside observer? $\endgroup$ – Moonraker Feb 9 '17 at 16:53
  • $\begingroup$ The external observer will see the black hole evaporate completely and a pure state becoming a mixed state. All the initial mass of the black hole will be re-emitted, but not the information of what has fallen inside. $\endgroup$ – Rexcirus Feb 9 '17 at 22:44
  • $\begingroup$ From your comments I guess that you are considering an eternal black hole. But that's a black hole that doesn't evaporate, and this is not realistic, since is not taking into account quantum mechanics. Google the Penrose diagram for an evaporating black hole and you will see that the causal structure is strongly different. $\endgroup$ – Rexcirus Feb 9 '17 at 22:51
  • $\begingroup$ Can you explain what you mean of getting rid of the singularity, in a way that it explains whether a black hole follows unitary evolution? I didn't understand this one. $\endgroup$ – Gareth Meredith Apr 10 at 2:50
  • $\begingroup$ It's not an hard requirement to explain unitarity. But, many approaches to solve the information paradox emphasise that the central singularity is a simplified description of the physics, as a point like electric charge is only an approximate description of a quantum field. Also by standard GR theorems, singularity --> horizon. $\endgroup$ – Rexcirus Apr 10 at 3:53

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