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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?

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  • $\begingroup$ About the proportionality constant, keep in mind that the Rovelli paper is 14 years old, when LQG was still in its infancy. Anyway, I'll get back to you in greater detail in an answer. $\endgroup$ – user346 Jan 7 '11 at 15:44
  • $\begingroup$ @space_cadet: oh, I didn't notice the date, thanks for pointing that out. So I guess all of the problems have been sorted out already and I am looking forward to reading newer papers on the topic :-) $\endgroup$ – Marek Jan 7 '11 at 17:17
  • $\begingroup$ I know that Ashtekar has published something more recently than that Rovelli paper. Maybe later tonight, I'll go and look it up. $\endgroup$ – Jerry Schirmer Jan 7 '11 at 20:46
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    $\begingroup$ One HAS to mention this: youtube.com/watch?v=FMSmJCKaaC0 $\endgroup$ – Sklivvz Jan 15 '11 at 23:09
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    $\begingroup$ Since this old question has popped back up on the front page, it may be worth mentioning that Ashoke Sen has decisively shown that loop quantum gravity is inconsistent with general relativity, based on black hole entropy calculations, in arxiv.org/abs/arXiv:1205.0971. $\endgroup$ – Matt Reece Sep 10 '12 at 0:29
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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

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    $\begingroup$ Thank you again Luboš! Indeed, it was my suspicion that LQG actually doesn't work at all for a long time but that was just based on other people's (you included) reports, so I wanted to get acquainted with LQG myself so that I could see the contradictions first-hand. I am sorry to state that even for a theoretical physics student it hasn't taken more than few hours (true time devoted to reading few basic papers and thinking) to arrive at that conclusion. $\endgroup$ – Marek Jan 14 '11 at 14:16
  • $\begingroup$ @Lubos you have a follower! @Marek aren't you also from the Czech republic, not that would bias your opinion in any way, I'm sure :) Anyhow your feelings seem to have shifted radically from what I saw reflected in your earlier questions on LQG and from your reactions to some of my answers. Maybe someday you'll feel less regret over having spent a few hours learning LQG. Fingers crossed ;) $\endgroup$ – user346 Jan 14 '11 at 18:30
  • $\begingroup$ Thanks, space_cadet! Your positive words are appreciated. ;-) By the way, judging by the name Marek which looks purely Czech - our version of Marc - I would also say that Marek is my countrymate but I honestly don't know. There may be another nation who spells it the same way. $\endgroup$ – Luboš Motl Jan 14 '11 at 18:57
  • $\begingroup$ @space_cadet: sure, things have changed precisely because I learned things I hadn't know before (and I don't regret that at all). Actually, I still like LQG approach from the mathematical point of view, the connection with spin-networks etc. There seem to be some interesting ideas. But if it can be shown (as seems to be the case) that LQG is not a physical theory then there's probably nothing more to talk about on this site, right? $\endgroup$ – Marek Jan 14 '11 at 19:18
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    $\begingroup$ @space_cadet: I don't think I follow. Early forms of string theory (like 26d bosonic string) were wrong and are dead now (if you don't count the revival in the form of heterotic strings). If you suggest LQG is in the same state today as strings were back then then I'd be happy to agree: it seems not to be a viable physical theory currently. Maybe some modification of it will be one day but not right now. Is that what you are saying? :-) $\endgroup$ – Marek Jan 14 '11 at 21:06
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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.

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  • $\begingroup$ "... the sky opened up and a chorus of angels appeared from the heaven" Welcome to Physics.SE @Carlo :) Some things are best heard from the horse's mouth, so to speak. $\endgroup$ – user346 Jan 27 '11 at 4:09
  • $\begingroup$ on a more serious note, with all due respect I would suggest an edit to remove your somewhat more personal comments about @Lubos. While, morally, you are entitled to defend your work in the strongest terms possible, I think such personal opinions are not needed to support your answer :) $\endgroup$ – user346 Jan 27 '11 at 4:13
  • $\begingroup$ ok, space_cadet, you convinced me. i have edited away all personal considerations. $\endgroup$ – Carlo Rovelli Jan 28 '11 at 14:55
  • $\begingroup$ Thanks @Carlo. There is indeed a great deal of misinformation on LQG on this site. Hopefully your arrival should change that for the better! $\endgroup$ – user346 Jan 28 '11 at 16:28
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    $\begingroup$ @Dimension10 Haha, this is how it works if you press "improve" after approving a suggested edit and you're too lazy to fill anything new in. :) $\endgroup$ – Bernhard Aug 10 '13 at 6:08
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[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.

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    $\begingroup$ Dear space_cadet, I wasn't making any controversial statement. Your "counterexample" is not a counterexample because I only spoke about discrete models of quantum gravity and there is nothing discrete whatsoever about Mark Srednicki's paper. It's a standard massless field. ... I don't know what you mean by "reinforcing a procedure etc.". My statement was merely that the result of the procedure disagrees with the value required by gravity. This fact may be obscured but it is a fact: LQG doesn't work. Science is not a business "it doesn't matter". Falsification in science kills a conjecture. $\endgroup$ – Luboš Motl Jan 14 '11 at 19:02
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    $\begingroup$ It is not at all obvious that to "quantize" geometry you need quantized versions of area and volume. One reason to have doubts about that notion is that you should try to define things in an operational and gauge-invariant way. It's not clear to me how I would measure tiny areas or volumes of order Planck size. It sounds like a suspiciously local question, and for quite general reasons one should have doubts about sharp definitions of extremely local quantities in theories of quantum gravity. $\endgroup$ – Matt Reece Jan 14 '11 at 22:43
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    $\begingroup$ And as a general rule of thumb, when making a statement along the lines of "any theory of quantum gravity must do X," it is useful to ask yourself "does string theory do X?" I'm not aware of any stringy version of an "area operator," and for the reasons alluded to in my last comment I doubt that one exists. $\endgroup$ – Matt Reece Jan 14 '11 at 22:45
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    $\begingroup$ LQG was "man-made" - the very Ashtekar field redefinition, trying to argue that a bulk SU(2) gauge field "is the same thing" as a bulk gravity, was derived from the assumption that the areas should be quantized - which they're not. It's a wrong initial guess, a lethal bomb in the very pillars of LQG that can be identified as the culprit of all the contradictions between LQG and gravity. In proper science, like string theory, one makes many fewer arbitrary assumptions - the careful analyses of the theory teach us the answers. $\endgroup$ – Luboš Motl Jan 20 '11 at 9:24
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    $\begingroup$ Lubos is obviosuly wrong in saying that "the Ashtekar field redefinition was derived from the assumption that the areas should be quantized". The Ashtekar field definition was made in 1986, almost ten years earlier anybody even thought about area quantization (1994)!! Maybe Lubos thinks that Ashtekar reads the future! $\endgroup$ – Carlo Rovelli Jan 29 '11 at 6:22
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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...

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