Discussion of the Rovelli's paper on the black hole entropy in Loop Quantum Gravity In a recent discussion about black holes, space_cadet provided me with the following paper of Rovelli: Black Hole Entropy from Loop Quantum Gravity which claims to derive the Bekenstein-Hawking formula for the entropy of the black hole.
Parts of his derivation seem strange to me, so I hope someone will able to clarify them.
All of the computation hangs on the notion of distinguishable (by an outside observer) states. It's not clear to me how does one decide which of the states are distinguishable and which are not. Indeed, Rovelli mentions a different paper that assumes different condition and derives an incorrect formula. It seems to me that the concept of Rovelli's distinctness was arrived at either accidentally or a posteriori to derive the correct entropy formula.

Is the concept of distinguishable states discussed somewhere more carefully?

After this assumption is taken, the argument proceeds to count number of ordered partitions of a given number (representing the area of the black hole) and this can easily be seen exponential by combinatorial arguments, leading to the proportionality of the area and entropy.
But it turns out that the constant of proportionality is wrong (roughly 12 times smaller than the correct B-H constant). Rovelli says that this is because number of issues were not addressed. The correct computation of area would also need to take the effect of nodes intersecting the horizon. It's not clear to me that addressing this would not spoils the proportionality even further (instead of correcting it). 

Has a more proper derivation of the black hole entropy been carried out?

 A: [This was intended as a comment on Lubos' answer above, but grew too big to stay a comment.]
(@Lubos) It is well understood that the horizon is, by definition, a trapping surface. Consequently external observers can gain no information about anything that happens in the interior once the trapping surface is formed. This is not an understanding peculiar to LQG. That is in fact what makes the results of LQG more robust in the end.
You state that:

There is no calculation of an area-extensive entropy in LQG or any other discrete model of quantum gravity, for that matter.

An easy counterexample to that statement, for instance, is Srednicki's 1993 PRL "Entropy and Area" (which has 359 citations so far). This paper shows that this entropy-area relation is a very universal aspect of plain old quantum field theory with no inputs whatsoever from loops or strings. Also, the papers you cite (by Domagala, Lewandowski and Meissner) - while these fix an error in Rovelli's work they are not intended to negate the basic procedure of counting states associated with quanta of area, but to reinforce it. So you may hate or love that specific paper by Rovelli, but that does not change the validity of the rest of the vast amount of work done on this topic in LQG. For a comprehensive bibliography I suggested looking up the references in Ashtekar and Lewandowski's 2005 "LQG: Status Report" paper and by doing arXiv searches for papers by Alejandro Corichi and collaborators.
The fact that Black Hole entropy should be determined solely by counting the microscopic surface states of the horizon (and not those of the bulk interior) is something we know from Bekenstein and Hawking's work based on semiclassical QFT. Any microscopic theory, based on loops or strings or whatever, must ultimately yield the same results under coarse graining. LQG does this in a simple and natural way. The key lies in the notion of the area operator - which by itself is a construction natural to and shared by any theory of quantum geometry. Rovelli's paper is one the earliest (with Kirrill Krasnov, Baez and Ashtekar being among the other pioneers) which outlines the general notion. It is significant for these reasons.
Please allow me to stress that in no way am I trying to cast doubts on your (@Lubos') work with quasinormal modes and such. I have yet to properly understand that calculation and I also do not claim to have a universal understanding of all the work on black hole entropy from the loop perspective or otherwise. My hope is simply to refute the notion "that LQG actually doesn't work at all"! This statement is unfounded and far more evidence than simply noting the error in Rovelli's paper is needed to back up such claims. Needless to say there are errors in the early papers on quantum mechanics, general relativity and string theory. Do those mistakes imply that either one of these frameworks "doesn't work at all"?

Edit: There are some very recent papers which hopefully are big steps towards resolve the black hole entropy question in LQG, and should be of interest to some of the readers here - Detailed black hole state counting in loop quantum gravity (published in PRD) and Statistical description of the black hole degeneracy spectrum.
Edit (v2): There are some persistent misunderstandings as reflected in the comments about the nature of the Ashtekar formulation. Let me restate, as I mentioned below, that Ashtekar's variables are nothing more than a canonical transformation which lead to a simpler form of the ADM constraints. There are no assumptions about area quantization and such which go into the picture at this stage. Area and volume quantization is the outgrowth of natural considerations regarding quantum geometry. These were undertaken in the mid-90s, seven or eight years after Ashtekar's original papers. Perhaps the single best and most comprehensive reference for the Ashtekar variables and more generally the complete framework of canonical quantum gravity is Thomas Thiemann's habilitation thesis.
A: The distinction between distinguishable and indistinguishable microstates is the following. For an observer outside the BH, two microstates are distinguishable if they can affect the future evolution of the observer differently. Two microstates with a different geometry of the horizon are distinguishable. Instead, if the geometry differs only inside the horizon, there is no way the outside observer can be affected by the difference.   Why is this relevant for the entropy?  Because the entropy is a quantity that characterizes the heat exchanges with a system. These exchanges are determined by the number of different distinguishable microstates the system can be in, and not by the total number of states. If a system has a part which is completely isolated, including thermally, then its states are irrelevant for the thermodynamical behavior of the system. 
Does this mean that the entropy depends on which observer sees it? Yes of course, but this is well known. The entropy depends a lot on the observer; for instance it depends on the macroscopic quantities chosen to describe the system. A system has an entropy only after you specify how you are looking at it, namely which are the macroscopic quantities that you use to describe it. Then the entropy is determined by the number of states at those macroscopic parameters fixed. 
Yes, the story of BH entropy in Loop Gravity has much evolved since that paper of mine, and many more things have been understood.  I think that the BH counting in LQG is a success, but I also think that the problem is not resolved, and the situation is still perplexing. I am not convinced by the idea that the solution is just fixing a parameter to make it come out right.   If anybody is interested in what I think today about the black hole entropy calculations in LQG, the place to look is my very recent review http://fr.arxiv.org/abs/1012.4707, which is written for a large audience, and where I try to asses the state of the field, including the BH entropy problem.
A: Dear Marek, it has been showed that the paper by Rovelli was invalid for lots of reasons, including those related to yours.
First of all, as you hint, it is incorrect to treat the interior and exterior of the black hole asymmetrically because the location of the event horizon may only be determined a posteriori - after a star collapses. So there's no qualitative difference between the interior and the exterior.
It follows that in the "real LQG", there would also be an entropy coming from the interior which would be volume-extensive. No one has ever showed that this term is absent; the absence is just a wishful thinking, so the proportionality law to the surface is just a result of an omission.
However, even if one removes the interior by hand, Rovelli's paper was showed incorrect. The numerical constant turned out to be incorrect, and newer calculations showed that even with the assumption that the black hole entropy comes from the horizon - which could make the area-law for the entropy tautological - the actual calculable entropy is actually not proportional to the area at all. The corrections to Rovelli's paper - showing that his neglecting of the higher spins etc. were invalid - appeared e.g. in
http://arxiv.org/abs/gr-qc/0407051
http://arxiv.org/abs/gr-qc/0407052
If you're looking for papers that show that it suddenly makes sense, you will be disappointed. Quite on the contrary, it has been showed that none of the early dreams that LQG could produce the right black hole entropy works.  This is also particular self-evident in the case of the quasinormal modes that were hypothesized to know about the "right" unnatural value of the Immirzi parameter - a multiplicative discrepancy in the Rovelli-like calculations. 
I showed that for the Schwarzschild, the result really contained $\ln(3)/\sqrt{2}$ and similar right things, but we also showed with Andy Neitzke - and with many other people who followed - that the number extracted for other black holes is totally different and excludes the heuristic conjecture.
So today, it's known that the relationship supported by the same Immirzi parameter on "both sides" was actually wrong on both sides, not just one. There is no calculation of an area-extensive entropy in LQG or any other discrete model of quantum gravity, for that matter.
Best wishes
Lubos
A: By the way, the quantization of areas, as explained elsewhere, directly contradict special relativity. If you pick a near null surface in the Minkowski space, even though its coordinate differences may be macroscopic, its proper area can be arbitrarily small (but positive). This is implied by relativity because it is the Lorentz transform of a tiny spacelike (or mixed) area. In LQG, the proper area will be essentially the number of intersections of the area with the spin network - it can clearly never go to zero for near-null surfaces, implying a maximum violation of Lorentz symmetry. – Luboš Motl Jan 20 '11 at 9:27
that is related to?:
http://arxiv.org/pdf/gr-qc/0411101v1.pdf
...One such candidate is loop quantum gravity which leads to a discrete structure of the geometry of space. This discreteness can be expected to lead to small-scale corrections of dispersion relations, just as the atomic structure of matter modiﬁes continuum dispersion relations once the wave length becomes comparable to the lattice size. There have been several studies already which derive modiﬁed dispersion relations motivated from particular properties of loop quantum gravity...
...The diﬃculty lies in the fact that loop quantum gravity is very successful in
providing a completely non-perturbative and background independent quantization of general relativity which makes it harder to re-introduce a background such as Minkowski space over which a perturbation expansion could be performed...
