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The no hair theorem says that black holes rapidly converge to a state that is completely described just by their mass, spin and charge. Black hole thermodynamics says that the black hole entropy is proportional to the surface area of the event horizon. As I understand it in information theoretic terms the black hole entropy is 1 bit per planck unit of area, which should then be the amount of information required to fully describe a black hole. These two statements seem to be incompatible. It seems that the thermodynamics says that if we fix the macroscopic no hair parameters there still remain all the surface bits to be fixed. Is there any real conflict here or is somehow illusory? Is it a quantum vs classical issue? Does the entropic information not count in some sense? If not why, can 2 equal mass Schwartzchild black holes be distinguished by their "surface information"? Is it that the surface information is just not observable because the surface is considered in the black hole? Or is this apparent discrepancy part of the black hole information paradox?

There is a very similar question with a detailed answer here, but I believe I am actually asking something different. The question and excellent answer there focus on the objects as they fall into a black hole. I am more interested in how to think about the total information in a steady state black hole and the current status of the no hair theorem wrt black hole entropy and the information paradox.

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First of all, this is a very good question. Classically black holes have no hair, and so are specified by a handful of charges (mass, angular momentum, etc). Quantum mechanically, they act like thermodynamic systems. They have temperature and a large entropy, and in fact all the laws of thermodynamics have black hole analogues.

To explain away the apparent tension between black holes apparently storing very little, and also very much information, consider a less exotic system--a volume of fluid. The fluid is specified by state variables such as pressure, temperature, volume, etc. Of course, the actual system consists of a bunch of vibrating and colliding particles, but the thermodynamic properties are described by just a few parameters. A given microstate in a thermodynamic ensemble corresponds to all the constituent particles having definite positions and velocities.

Black hole entropy should be thought of in the same way. The microstate of a black hole is harder to identify, and has been the source of a tremendous amount of work in the last few decades. However the modern view is that black holes are quantum mechanical systems comprised of degrees of freedom which scale with the horizon area. For a given ensemble (specified by temperature, pressure, etc), there are $e^S$ microstates.

This view received fantastic support in the work of Vafa and Strominger http://arxiv.org/abs/hep-th/9601029, who were able to identify the microstates of black holes in string theory, and moreover were able to show that the counting worked out--that the entropy of black holes does indeed scale with area (and not volume). This result also helped to convince many theorists that string theory was a strong candidate theory of everything.

Also, there is a lot of modern work on the microstates. The name this work goes under is the "Fuzzball program" if you're interested in learning more about it.

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  • $\begingroup$ "This result also helped to convince many theorists that string theory was a strong candidate theory of everything." I wish this was a joke. How is information entropy ascribed to a mathematical model of extreme conditions we have no direct experience with any sign of advance in understanding Nature? For such advance there have to be experiments with surprising results and some theory successfully explaining them in a coherent way. $\endgroup$ – Ján Lalinský Feb 3 '15 at 2:48
  • $\begingroup$ It's true that the Strominger-Vafa paper calculates the entropy for a very unphysical system--a 10 dimensional black hole (the internal space isn't even compactified to a small radius that would render it invisible) for a theory that certainly does not describe our universe. But the fact that a theory of quantum gravity could account for the Bekenstein-Hawking entropy is a tremendous achievement. In fact this calculation was a precursor to the AdS/CFT correspondence, one of the deepest and most interesting advances since Einstein/QFT IMHO. $\endgroup$ – Surgical Commander Feb 3 '15 at 2:53

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