For simple setups, where the radiation field deviates not too far from thermodynamic equilibrium (< 10 %), corrections to the Planckian thermal emission spectrum can be calculated (and measured) from the statistical operator method, using the heat flux and the photon flux as additional constraints in the entropy maximazation procedure:


Now I am courious if similar considerations could be applied to the situation of an evaporating black hole. Can the information, orignally stored in an evaporating black hole, in principle be reconstracted from analyzing it`s nonequilibrium emission spectrum (obtained from including a sufficient larger number of additional relevant parameters as constraints in the statistical operator method)?


I`m pondering if one could describe an evaporating black hole applying nonequilibrium statistical mechanics such that the global mean temperature is not sufficient to describe its thermodynamic state and additional (microscopic) degrees of freedom (which are important to store the information of infalling objects too) must be taken into account. My question then is: Are these deviations from thermodynamic equilibrium translated to deviations of the spectrum of the Hawking radiation from Planck s law? And if so, can this in principle be used to reconstract the information stored in the microscopic degrees of freedom of the black hole?


2 Answers 2


The deviations from thermal equilibrium for a black hole are negligible. The outgoing radiation still encodes the exact state. These are not contradictory statements, because a thermal state has a maximum number of microstates which can produce it, so that the microstate doesn't have to leave a macroscopic imprint.

I heard a classical example of this from Sidney Coleman in the early 1990s. Consider heating a lump of coal at absolute zero with a perfectly coherent laser until it glows red hot, emitting photons until it cools down to absolute zero again. The incoming photons are in a pure state, the outgoing photons are thermal. In principle, all the information about the incoming photons are contained in the outgoing photons, but it can be shown that the reduced density matrix of even 50% of all the photons is essentially perfectly random and thermal. You need to look at nearly all the photons to get the density matrix to reveal the substructure.

Any way of experimentally demonstrating the nonthermalness of a black hole will require an impossible measurment of an observable which combines nearly all the photons. The only reasonable way out is to consider a nearly extremal black hole absorbing a tiny amount of matter, and reemitting it. Within string theory, such a process is not thermal at all but is described (in model black holes) by a brane-matter interaction whose output is reasonably calculable. Gubser and others performed this calculation for D-branes scattering gravitons in string theory in 1995, and this method of coherent model black hole incoming/outgoing scattering was one of the major inputs leading to AdS/CFT.

  • $\begingroup$ Thanks a log @Ron Maimon, the last (for me a little bit too commpact) paragraph contains a lot of useful information about nonthermal black holes (I was some kind of up to I think). Maybe I`ll come up with follow up comments or questions about it later... $\endgroup$
    – Dilaton
    Commented Dec 18, 2011 at 11:06
  • 2
    $\begingroup$ @Ron Maimon Can you point to a reference where I can read Coleman's calculations. It would be very nice to see you posting again. $\endgroup$
    – Prathyush
    Commented Mar 25, 2013 at 20:41
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    $\begingroup$ @Prathyush: Coleman didn't publish this, he just explained it to me pedagogically. It's a reference to Page's results, which can be read by googling Page time. $\endgroup$
    – Ron Maimon
    Commented Mar 26, 2013 at 1:03

Take all the outgoing Hawking radiation, evolve it backward in time, then evolve it forward again along an infalling trajectory. Sure, go ahead. Capture all the outgoing Hawking radiation with all the quantum coherence preserved intact including all the relative phases, feed it into a quantum computer which doesn't make a single error and evolve it backward exactly. If you can fight the second law of thermodynamics this way, you can do it.

  • $\begingroup$ Thanks @Ursecker for this answer, however my question did not aim at evolving things backward against the second law of thermodynamics. $\endgroup$
    – Dilaton
    Commented Dec 18, 2011 at 9:41
  • $\begingroup$ @Uresecker: also, extracting the wavefunction of the outgoing photons violates no-cloning. You would need a physical reverser which can coherently reverse the photon's motion. $\endgroup$
    – Ron Maimon
    Commented Dec 18, 2011 at 10:13

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