Discussion of the Rovelli's paper on the black hole entropy in Loop Quantum Gravity

Question

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?

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Marek

Comments

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.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user user346

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@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 :-)

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Marek

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I know that Ashtekar has published something more recently than that Rovelli paper. Maybe later tonight, I'll go and look it up.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Jerry Schirmer

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One HAS to mention this: youtube.com/watch?v=FMSmJCKaaC0

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Sklivvz

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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.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Matt Reece

Answer 1

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.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Carlo Rovelli

Comments

"... 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.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user user346

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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 :)

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user user346

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ok, space_cadet, you convinced me. i have edited away all personal considerations.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Carlo Rovelli

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Thanks @Carlo. There is indeed a great deal of misinformation on LQG on this site. Hopefully your arrival should change that for the better!

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user user346

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No, you haven't! There's still a bunch of nasty comments against Lubos Motl in your answer.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Dimensio1n0

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@Bernhard: It's funny to see the summary "Removed hate speech" for a removal of greetings .

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Dimensio1n0

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@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. :)

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Bernhard

Answer 2

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

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Luboš Motl

Comments

@space_cadet: as for my origin: almost correct, I am from Slovakia ;-) But I definitely don't follow anyone. If by follow you don't mean respecting an answer of an established physicist who has moreover backed up his argument with papers :-)

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Marek

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@Marek we are very far from arriving at such a destination where "there's probably nothing more to talk about on this site" about LQG. If you feel that way, again that's totally 100% perfecto cool. I'm quite sure that physics.SE is broad enough in scope and capacity to allow discussions on opposing lines of thought and by practitioners in different ("opposing"? - I don't like that word) camps to continue in parallel without the risk that everything will be consumed in a flame war. That is the essence of a democracy, right?

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user user346

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@space_cadet: sorry, I didn't want to insult you or anything. But you have to acknowledge that science is about invalidating invalid hypotheses and it just wouldn't be cool to discuss any run-of-the-mill incorrect theory on physics.SE. Now, I am not saying that LQG is one of these (although it does appear to be based on the information I've yet seen; feel free to prove me wrong) but its status is definitely open for a discussion and if (again, if) it can be shown that it's not a physical theory, it's all over. There's no democracy in science.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Marek

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@Marek we have been waiting 25 years now for String Theory to be validating to the level of rigor that would put it on the same standing as QM or GR. It is instructive to look back at history and realize that there were long periods of time in string theory's early development when its feasibility as a consistent fundamental theory was considered about as likely as the ones you afford LQG at present. Should we give up Strings because of these past failures? Ok, I'm being a bit facetious there. But you get my point, I hope. In any case, I rest my case.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user user346

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@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? :-)

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Marek

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@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 ;)

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user user346

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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.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Luboš Motl

Answer 3

[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"?

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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.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user user346

Comments

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.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Luboš Motl

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@Lubos you may or may not agree that LQG provides a natural description of the degrees of freedom of quantum geometry. You also may or may not agree that the counting of black hole microstates in LQG is done in the spirit of Einstein's dictum "... as simple as possible but no simpler." You are entitled to your opinion. However you cannot, of your own accord, elevate that opinion to the level of "fact" any more than I, or anybody else, is entitled to do the same with any other theory they disagree with or find fault in. In other words let us agree to disagree and leave it at that. Cheers.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user user346

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"there are errors in the early papers on...", sure and then there are later papers which correct them. I'd like to see those later papers for LQG. Going to look at that Status Report. However, what you talk about in this answer/comment seems like non-sense. Why should any theory of QG have an area operator? And why should the entropy arise because of the states on the boundary? It's perfectly well imaginable that those states live inside the hole just that there's less of them (for some reason). You are not allowed to draw that conclusion from B-H law. That's circular logic.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Marek

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#1. A theory of quantum gravity must also necessarily be a theory of quantum geometry - if you disagree please describe why. I feel I'm in the corner of conventional wisdom on this one. And in order to "quantize" geometry it is obvious we will need quantized versions of geometric operators such as area and volume. #2. Actually that is exactly the conclusion that you are lead to if you believe in the Bekenstein-Hawking arguments leading up to the area law and the laws of black hole thermodynamics. But I think I'll have to elaborate on this point in an edit.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user user346

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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.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Matt Reece

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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.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Matt Reece

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Hi @Matt. If we want to speak of quantum geometry then wouldn't the natural place to start be to try to think of operator analogs of geometric properties such as area and volume? And yes, in trying to measure Planck sized geometric observables we would run into new conceptual difficulties. But we ran into such difficulties when QM taught us that $x$ and $p$ (or $E$ and $t$) cannot be measured simultaneously and yet we have come around to understanding how classical behavior (where both $x$ and $p$ are properly localized as in a coherent state for eg.) emerges from a quantum picture.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user user346

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And interestingly enough there is, IMHO, in string theory geometric operators are naturally present - the String action is given simply as $\int d\tau d\sigma \sqrt{h} $ i.e. the integral over the area form of the worldsheet. Now this is action describes the dynamics of a *classical* string. In a quantum theory, $\sqrt{h}$ would have an operator representation. I have more to say on this, but the bottom line is that again (IMHO) strings and loops have far more in common than the nature of the present debate would seem to suggest.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user user346

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@space_cadet: unfortunately I don't know enough about string theory to refute your statements but the pure fact that google doesn't bring up anything with regard to area operator in string theory seems convincing enough that the notion is completely useless (if not impossible to define in the first place). Also the reasons presented by Matt are very valid: if there is no way to measure something (at least in principle), it has no physical meaning. Could you point out some way how to realize LQG area measurements?

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Marek

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The pure fact that google doesn't bring up anything in regards to statement X in string theory, means zilch. String theory is a nice subject but to define it as the Universal Oracle is going a bit far. As for area quantization, it is a very well-defined concept. Its simplest occurrence is in the quantum hall effect - which as Susskind, Polychronakos and others have shown - can be cast into the language of matrix models. Ref1, Ref2.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user user346

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Dear space_cadet, by the way, concerning the claim that one can't see inside the trapping surface. I completely agree, this follows from general arguments in quantum gravity. But one of the problems is that this general fact on quantum gravity contradicts loop quantum gravity. It's because the fact implies that the entropy inside can't be high - but LQG does predict volume-extensive entropy unless this term is removed "by hand". This is fudging with the results, an attempt to hide the contradiction between gravity and LQG. If you care about spirits, LQG is too simple.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Luboš Motl

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And by the way, quantum gravity also implies that the area of a surface - with a Planckian resolution - cannot be a good operator. This can also be seen by the fact that there can't exist any operational procedure to measure it. If all areas (including those of measuring "sticks") are roughly quantized in units of the Planck area, no tool can measure other areas with a better precision. That's why it's very important that a good theory deals with physical quantities such as energy and momentum of scattering particles. Good theories (ST) do so automatically - teach us the right observables.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Luboš Motl

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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.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Luboš Motl

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@Lubos I am not aware of any such volume-extensive entropy contribution from LQG. I do know of logarithmic corrections to the area law but that is not what you mention, and those contributions also depend only on the area. If you could provide a reference which makes this claim or provide other pointers that would be helpful. Or perhaps you could cast this into a question, with greater detail than can be put in a comment, and we could hash it out there.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user user346

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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.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Luboš Motl

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@lubos - 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 - this is simply not true. The Ashtekar variables are nothing more and nothing less than a canonical transformation on the phase space on general relativity. Ashtekar's formulation is a classical theory completely equivalent to general relativity. An analogy - transforming to complex variables $z = q+ip$ makes the SHO easier to quantize. But it doesn't change any of the underlying physics.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user user346

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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!

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Carlo Rovelli

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@CarloRovelli: This is not a reasonable response to Lubos's cogent criticism. The point here is that the area law is nothing like the traced-out entropy of a free field, it is nothing like boundary degrees of freedom in additional to bulk, it is boundary degrees of freedom replacing bulk, and this holographic counting is not reproduced in loop quantum gravity. The entropy of classical fluctuations of a gravity field (or other fields) near a black hole is divergent, and it is double-counting to look at both boundary and bulk. This makes serious reservations about discrete Regge anything.

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user Ron Maimon

Answer 4

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...

This post imported from StackExchange Physics at 2014-04-01 17:35 (UCT), posted by SE-user oswaldosalcedo