Bianchi's entropy result-what to ask, what to learn from it

[marcus]

Motl points to an interesting paper by Sen: "we apply Euclidean gravity to compute logarithmic corrections to the entropy of various non-extremal black holes in different dimensions ... For Schwarzschild black holes in four space-time dimensions the macroscopic result seems to disagree with the existing result in loop quantum gravity."

The gist of what I had to say in the other thread was threefold
A. it's completely speculative what the best QG formula for BH entropy is. I wouldn't guess or bet unless forced to. We don't know that any particular approach even has the right degrees of freedom to describe a BH quantum geometrically. That includes Sen with the "Euclidean" approach. And of course Nature has the last word.

B. It doesn't matter much, but just "for the record" Sen does not accurately reflect what I think are the prevailing ideas of the log term among Loop researchers. He seems off by a factor of 2. It looks on first sight like a factor of 4, but half of that is a difference in notation.

C. If I were forced to bet, I'd guess Bianchi (and others who find the area-term coefficient to be 1/4 independent of Immirzi) are moving in the right direction. I expect followup papers to appear and it would be naive to assume that they will use the same methodology. Insights and methods don't stand still so one cannot predict the future course of research.

My post #2 from the other thread says pretty much where I stand.

Nice to have the connections drawn and links laid out. Thanks! I'll add a possibly useful reference. Here is a review paper:
http://arXiv.org/abs/1101.3660
Detailed black hole state counting in loop quantum gravity
Ivan Agullo, J. Fernando Barbero G., Enrique F. Borja, Jacobo Diaz-Polo, Eduardo J. S. Villaseñor
(Submitted on 19 Jan 2011)
We give a complete and detailed description of the computation of black hole entropy in loop quantum gravity by employing the most recently introduced number-theoretic and combinatorial methods. The use of these techniques allows us to perform a detailed analysis of the precise structure of the entropy spectrum for small black holes, showing some relevant features that were not discernible in previous computations. The ability to manipulate and understand the spectrum up to the level of detail that we describe in the paper is a crucial step towards obtaining the behavior of entropy in the asymptotic (large horizon area) regime.

This review paper is what Sen does not square with. Agullo et all have a table on page 30 which shows the currently prevailing Loop BH log terms. With A standing for area they are predominantly - 0.5 log A.

On the other hand Sen says that in the Loop context the log term is -log A. IOW off by a factor of two. I suppose he is depending mostly on older or marginal sources. What he actually says is let a be the linear scale of the BH, in other words essentially sqrt(A) then the Loop term is -2log(a). This amounts to the same thing as -log(A).
It's of little if any consequence. For clarity/completeness, I'll include the rest of my comment:
==quote post #2==
These authors have a different log term (see table on page 30) from what Ashoke Sen refers to as characterizing the Loop BH entropy.
They say -(1/2)log a and he says (on page 28) -2log a.
Superficially different at least--perhaps reconcilable but I don't see how.
I'm not sure any of that will hold over the long term--still too much technical disagreement.

As I guess you are well aware, the question of black hole entropy is not settled in LQG.
Even in the pre-2012 work, where the authors think that they must specify a value of the Immirzi parameter in order to recover Bek.Hawk semiclassical, they use different enough methods so that some get γ=0.237 and others get γ=0.274.
Again see the table on page 30 of the Agullo et al paper. http://arXiv.org/abs/1101.3660 Crisp summary of differences.
And then Bianchi posted a paper last month (April 2012) which finds the entropy to be quite different from either group. Basically proportional to area with coefficient 1/4 without fixing the value of Immirzi at all!

If I had to bet, I'd guess that Bianchi is closer to being right---that the BH entropy relation does not require fixing a particular value of Immirzi (a radical innovation in context of earlier work). And Bianchi has not yet worked out the quantum corrections, or any way not posted. His paper does not specifically mention a log term at all. So we'll just have to wait and see if there is a log term and if so what it is.
==endquote==

 

[fzero]

This review paper is what Sen does not square with. Agullo et all have a table on page 30 which shows the currently prevailing Loop BH log terms. With A standing for area they are predominantly - 0.5 log A.

The 1/2 vs 3/2 here depends on whether you use U(1) or SU(2) Chern-Simons theory. Sen addresses this in the comments above his (4.4). Page 27 of the Agullo et al review discusses the pros and cons of the SU(2) theory.

On the other hand Sen says that in the Loop context the log term is -log A. IOW off by a factor of two. I suppose he is depending mostly on older or marginal sources. What he actually says is let a be the linear scale of the BH, in other words essentially sqrt(A) then the Loop term is -2log(a). This amounts to the same thing as -log(A).
It's of little if any consequence.

Not quite. Sen starts with the SU(2) CS result, $-(3/2)\log A = -3\log a$. Then in point #1 starting on page 27, he explains that this is the entropy corresponding to counting states/unit area. However, he wants to compare to his result, which counted the number of states per unit mass interval. He argues that you need to add $\log a$ to the LQG result.

He also shows that the logarithmic term actually vanishes in the U(1) CS theory after converting to his measure. However this is consistent with completely averaging the SU(2) result over spins.

 

[marcus]

Hmmm, so Sen calculates -(3/2)log A, in effect. If I remember right, some of the Loop papers also calculated the log term to be -(3/2)log A. (Recent example by Romesh Kaul http://arxiv.org/abs/1201.6102 ) Nice to see agreement between what Sen *thinks* prevailing Loop results are and what they actually are, at least in that case. I still don't see him attributing -(1/2)logA, though, which I think is more typical.

As I believe I indicated earlier, my attitude towards this business is agnostic. I'm not convinced that humans have lit on the right way yet to calculate BH entropy (in quantum geometry, not the classical approximation).

I don't believe you can make assumptions about what methods creative researchers in an active field are going to use next, in following up the latest papers we have. It's difficult to guess the future of research (almost by definition.)

On the other hand I'm very glad to see that you are so interested and knowledgeable about BH entropy. I benefit from some of your explanations and I expect others do as well.

 

[fzero]

Hmmm, so Sen calculates -(3/2)log A, in effect. If I remember right, some of the Loop papers also calculated the log term to be -(3/2)log A. (Recent example by Romesh Kaul http://arxiv.org/abs/1201.6102 ) Nice to see agreement between what Sen *thinks* prevailing Loop results are and what they actually are, at least in that case. I still don't see him attributing -(1/2)logA, though, which I think is more typical.

I took a quick look at the Kaul paper above. I think that he's saying the following. The U(1) CS theory is obtained from the SU(2) theory by a partial gauge fixing. The papers that derive the -1/2 coefficient do not apply the constraint on counting due to the gauge fixing. Once this constraint is applied, the -3/2 coefficient is obtained. This is explained in Kaul's section 3.2.

As I believe I indicated earlier, my attitude towards this business is agnostic. I'm not convinced that humans have lit on the right way yet to calculate BH entropy (in quantum geometry, not the classical approximation).

I don't believe you can make assumptions about what methods creative researchers in an active field are going to use next, in following up the latest papers we have. It's difficult to guess the future of research (almost by definition.)

The important point here is that the definition of the LQG observables for the BH problem were conjectured more than 15 years ago. Since then, the brute force and more creative methods of computing the entropy have been in agreement, or at least discrepancies have been understood (like the 3/2 vs 1/2 result above). Any creative method of getting some new answer would either have to expose an error in earlier work or start from different assumptions for extremely well-motivated reasons.

Once the observables are defined here, the counting problem is technically complicated, but not otherwise mysterious. There is essentially no room to obtain some other answer without changing the definition of the observables. There's plenty of room for creativity there, but there will still be constraints coming from LQG foundations.

 

[marcus]

Bee Hossenfelder comments on Eugenio Bianchi's recent paper:
http://backreaction.blogspot.com/2012/05/note-on-black-hole-entropy-in-lqg.html

 

[MTd2]

Bee Hossenfelder comments on Eugenio Bianchi's recent paper:
http://backreaction.blogspot.com/2012/05/note-on-black-hole-entropy-in-lqg.html

It's worth taking a look at John Baez' comments in that blog entry. It seems he liked he paper.

 

[marcus]

It's worth taking a look at John Baez' comments in that blog entry. It seems he liked he paper.

Let's be clear: Baez contemplates the possibility that the Loop gravity program could self-destruct by discovering unresolvable contradictions. He welcomes Bianchi's paper in part because it could lead to progress by "tightening the noose" of internal contradiction. In scientific theory both positive and negative results constitute progress.

At this point, as I see it, we cannot say if the tension among these different ways of computing the entropy will be resolved or not, and what effect this will have. It's definitely exciting.

I note that Baez did not mention that several previous papers by other authors came to similar conclusions to Bianchi---that the coefficient of area is simply 1/4 and independent of Immirzi. I don't know why he made no reference, even in passing, to the other research.

BTW a new paper just appeared on arxiv that joins this "Immirzi-independence" chorus. (It could be wrong of course!):

http://arxiv.org/abs/1205.3487
A New Term in the Microcanonical Entropy of Quantum Isolated Horizon
Abhishek Majhi
(Submitted on 15 May 2012)
The quantum geometric framework for Isolated Horizon has led to the Bekenstein-Hawking area law and the quantum logarithmic correction for the black hole entropy. The point to be noted here is that all the results have been derived in a model independent way and completely from within the quantum geometric framework where the quantum degrees of freedom are described by the states of the SU(2) Chern Simons theory on the Isolated Horizon. Here we show that a completely new term independent of the area of the Isolated Horizon appears in the microcanonical entropy. It has a coeffcient which is a function of the Barbero Immirzi parameter.
4 pages

According to Majhi, the dependence of entropy on Immirzi splits into two parts. There is the linear area part A/4 which does NOT depend, and then there is this N term involving number of spin-network links passing thru horizon which DOES depend. The coefficient of that term is a function of Immirzi, as you can see from the abstract. Majhi had an earlier paper that as far as I can see said roughly the same thing, which he cites. And of course there is the log area term.

No idea if this is helpful. Paper just came out. Anyway, exciting times for Loop.

 

[MTd2]

Not really:

http://www.blogger.com/profile/11573268162105600948

"John Baez said...
Actually, now that I look at them, I see Bianchi's calculations are based on a quite different theory than the old loop quantum gravity black hole entropy calculations. It's using a Lorentz group spin foam model, not an SU(2) formulation of loop quantum gravity; the area operator does not involve sqrt(j(j+1)), he's not quantizing a phase space of classical solutions with isolated horizones, etc. etc. So, there's not really any possibility of an 'inconsistency'. Instead, there's the possibility that the new theory is better than the old one."

 

[marcus]

Not really:

http://www.blogger.com/profile/11573268162105600948

"John Baez said...
Actually, now that I look at them, I see Bianchi's calculations are based on a quite different theory than the old loop quantum gravity black hole entropy calculations. It's using a Lorentz group spin foam model, not an SU(2) formulation of loop quantum gravity; the area operator does not involve sqrt(j(j+1)), he's not quantizing a phase space of classical solutions with isolated horizones, etc. etc. So, there's not really any possibility of an 'inconsistency'. Instead, there's the possibility that the new theory is better than the old one."

Whoa! Thanks! I missed that Baez comment. What I saw was his "tightening the noose" comment:
http://backreaction.blogspot.com/20...howComment=1337048785509#c4372570896762383197

What you are quoting is a later comment by Baez that I didn't see until you pointed it out:
http://backreaction.blogspot.com/20...howComment=1337127782679#c4303871751066857145

 

[marcus]

Now Aleksandar Mikovic has joined the discussion:
http://backreaction.blogspot.com/20...howComment=1337247199848#c4265253172764855047

==quote==
...Bianchi obtains the entropy not by counting the microstates, but by deriving the temperature of the horizon. He derives this temperature by identifying an operator which can be considered as an energy of the horizon and by using a 2-state thermometer. He uses the EPRL formalism, and there areas of triangles are gamma times the spin, so that gamma disappears inside the area.

The fact that gamma does not appear in classical quantities like areas and and entropy in EPRL spin foam model is consistent with the result for the effective action for EPRL derived by myself and M. Vojinovic: the classical limit is the Regge action, which is independent of gamma, since it depends on triangle areas and the deficit angles, see arXiv:1104.1384, Effective action and semiclassical limit of spin foam models, by A. Mikovic and M. Vojinovic, Class. Quant. Grav. 28, 225004 (2011). However, the quantum corrections to the effective action will depend on gamma, and hence the quantum corrections to the entropy will be gamma dependent...
==endquote==

For various reasons it seems to me possible that Eugenio Bianchi did not make a mistake! IOW that there is no inconsistency between the version of Loop gravity used and the conclusion that the entropy of a fixed area BH does not depend strongly (linearly) on the Immirzi.

Here is the paper that Mikovic refers to in his comment:
http://arxiv.org/abs/1104.1384
Effective action and semiclassical limit of spin foam models
A. Mikovic, M. Vojinovic
(Submitted on 7 Apr 2011)
We define an effective action for spin foam models of quantum gravity by adapting the background field method from quantum field theory. We show that the Regge action is the leading term in the semi-classical expansion of the spin foam effective action if the vertex amplitude has the large-spin asymptotics which is proportional to an exponential function of the vertex Regge action. In the case of the known three-dimensional and four-dimensional spin foam models this amounts to modifying the vertex amplitude such that the exponential asymptotics is obtained. In particular, we show that the ELPR/FK model vertex amplitude can be modified such that the new model is finite and has the Einstein-Hilbert action as its classical limit. We also calculate the first-order and some of the second-order quantum corrections in the semi-classical expansion of the effective action.
15 pages

 

[marcus]

We might learn a bit more about this Immirzi-free BH entropy result in about 10 days from now, if EB chooses to say something about it when he gives the Perimeter Institute Colloquium talk on 30 May.
http://pirsa.org/12050053

 

[marcus]

Since we're on a new page I'll give a link to the paper which is the main focus of discussion here:

http://arxiv.org/abs/1204.5122
Entropy of Non-Extremal Black Holes from Loop Gravity
Eugenio Bianchi
(Submitted on 23 Apr 2012)
We compute the entropy of non-extremal black holes using the quantum dynamics of Loop Gravity. The horizon entropy is finite, scales linearly with the area A, and reproduces the Bekenstein-Hawking expression S = A/4 with the one-fourth coefficient for all values of the Immirzi parameter. The near-horizon geometry of a non-extremal black hole - as seen by a stationary observer - is described by a Rindler horizon. We introduce the notion of a quantum Rindler horizon in the framework of Loop Gravity. The system is described by a quantum surface and the dynamics is generated by the boost Hamiltonion of Lorentzian Spinfoams. We show that the expectation value of the boost Hamiltonian reproduces the local horizon energy of Frodden, Ghosh and Perez. We study the coupling of the geometry of the quantum horizon to a two-level system and show that it thermalizes to the local Unruh temperature. The derived values of the energy and the temperature allow one to compute the thermodynamic entropy of the quantum horizon. The relation with the Spinfoam partition function is discussed.
6 pages, 1 figure

 

[marcus]

Two more related just came out. Smolin builds directly on the work of Bianchi this thread is about, plus the FGP paper Bianchi cites, and on a remarkable 1995 paper of Ted Jacobson where he shows that the Einstein equation arises as a "collective" thermodynamic effect of a swarm of unspecified degrees of freedom.

Here is the first of two new papers by Bianchi and Wieland on this subject. There is another still in progress.

http://arxiv.org/abs/1205.5325
Horizon energy as the boost boundary term in general relativity and loop gravity
Eugenio Bianchi, Wolfgang Wieland
(Submitted on 24 May 2012)
We show that the near-horizon energy introduced by Frodden, Ghosh and Perez arises from the action for general relativity as a horizon boundary term. Spin foam variables are used in the analysis. The result provides a derivation of the horizon boost Hamiltonian introduced by one of us to define the dynamics of the horizon degrees of freedom, and shows that loop gravity provides a realization of the horizon Schrodinger equation proposed by Carlip and Teitelboim.
3 pages, 1 figure

Here's Smolin's new one:

http://arxiv.org/abs/1205.5529
General relativity as the equation of state of spin foam
Lee Smolin
(Submitted on 24 May 2012)
Building on recent significant results of Frodden, Ghosh and Perez (FGP) and Bianchi, I present a quantum version of Jacobson's argument that the Einstein equations emerge as the equation of state of a quantum gravitational system. I give three criteria a quantum theory of gravity must satisfy if it is to allow Jacobson's argument to be run. I then show that the results of FGP and Bianchi provide evidence that loop quantum gravity satisfies two of these criteria and argue that the third should also be satisfied in loop quantum gravity. I also show that the energy defined by FGP is the canonical energy associated with the boundary term of the Holst action.
9 pages, 3 figures

What Smolin's argument tends to show is that the underlying degrees of freedom (which Jacobson left unspecified, and of which the thermodynamic equation of state is the classic Einstein GR equation) are specifically those of spinfoam QG set out, as Smolin indicates, in the Zakopane lectures. The paper seems to tie several strands of development together in a neat fashion.

 

[marcus]

Eugenio just posted the title and abstract of his Perimeter Colloquium talk to be given Wednesday afternoon at 2PM.

http://pirsa.org/12050053/
Black Hole Entropy from Loop Quantum Gravity
Speaker(s): Eugenio Bianchi
Abstract: There is strong theoretical evidence that black holes have a finite thermodynamic entropy equal to one quarter the area A of the horizon. Providing a microscopic derivation of the entropy of the horizon is a major task for a candidate theory of quantum gravity. Loop quantum gravity has been shown to provide a geometric explanation of the finiteness of the entropy and of the proportionality to the area of the horizon. The microstates are quantum geometries of the horizon. What has been missing until recently is the identification of the near-horizon quantum dynamics and a derivation of the universal form of the Bekenstein-Hawking entropy with its 1/4 prefactor. I report recent progress in this direction. In particular, I discuss the covariant spin foam dynamics and and show that the entropy of the quantum horizon reproduces the Bekenstein-Hawking entropy S=A/4 with the proper one-fourth coefficient for all values of the Immirzi parameter.
Date: 30/05/2012 - 2:00 pm

One thing to note is that Eugenio's 24 May http://arxiv.org/abs/1205.5325 already cites Smolin's 24 May http://arxiv.org/abs/1205.5529 General relativity as equation of state of spin foam.
So when he says that in the Colloquium talk he's going to report recent progress it could mean there will be some discussion of both the papers that were just posted.
I've started a thread on the related Smolin paper "GR=EoS of SF" in case anyone would like to comment.
https://www.physicsforums.com/showthread.php?t=608890

 

[marcus]

Eugenio should be starting his Colloquium talk about now. It's an interesting issue. Will the coefficient of area in Loop BH entropy turn out to be independent of γ (as he and several others have found)? My guess is that it will and that EB is on the right track.

From the talk's abstract:
"In particular, I discuss the covariant spin foam dynamics and and show that the entropy of the quantum horizon reproduces the Bekenstein-Hawking entropy S=A/4 with the proper one-fourth coefficient for all values of the Immirzi parameter."

As Bianchi points out at the conclusion of his April paper, correction terms would still be expected to depend on γ. http://arxiv.org/abs/1204.5122
The video was put online by around 5 PM Eastern time, less than two hours after the conclusion of the talk.
Just watched it. Perfect talk. Good questions from audience and thoroughly interesting Q&A discussion for about 20 minutes after, so the whole video lasts about 67 minutes. X-G Wen asked several questions. Beginning around minute 60 there was even some discussion of what can be learned from the earlier LQG derivation, and where the erroneous step occurred. Comment by Lee about that.

http://pirsa.org/12050053/

 

[fzero]

Beginning around minute 60 there was even some discussion of what can be learned from the earlier LQG derivation, and where the erroneous step occurred. Comment by Lee about that.

I finally had a chance to listen to some of the talk. Smolin claims that the state counting was wrong because the area operator doesn't commute with the boost Hamiltonian. But we have already deduced (back on page 1 of this thread) that these operators do in fact commute on the microstates that are used to build the horizon. The key ingredient needed to see this is the simplicity constraint. So the discussion in the question period hasn't in fact shed any light on the discrepancy.

 

[marcus]

So the discussion in the question period hasn't in fact shed any light on the discrepancy.

Sounds like neither explanation of the discrepancy did anything for you. Glad you finally had time to listen to the talk. So?

 

[marcus]

I finally had a chance to listen to some of the talk...

I hope someone (perhaps you?) has 30 minutes so they listen from minute 35 to minute 65.

It is Bianchi himself who explains the discrepancy of the earlier results right around minute 62! This is before the discrepancy issue is even raised explicitly! He begins to talk about state counting and says "what should we expect" but is interrupted. Smolin's comment is so brief that it doesn't count as explanation, it basically just says the earlier calculations were wrong. He doesn't take time to adequately spell out his reasoning.

Bianchi drew the key distinction between counting intrinsic and extrinsic states of geometry already (if I remember right) before the question was raised. Then later around minute 63 someone from the audience (is it Razvan Gurau?) raises the issue and at minute 65 Bianchi has to repeat what he said before, with emphasis.

At minute 65 says that the earlier counting was correct! and in fact ROBUST--but it was counting intrinsic states of geometry. That is not what is relevant for the observer who is hovering outside. Entropy depends on who sees it. That, I think, is the real explanation

This is partly work in progress by Bianchi. He is developing the quantum statistical mechanics version of his derivation which so far has been quantum thermodynamical. We won't know for sure until we see a paper but here is what I think he is saying: The observer is in space outside and lives his worldline in spacetime outside. So what matters are the states of EMBEDDED geometry. You have to count the states of the horizon as it is embedded in spacetime.

The whole thing can be made independent of any particular observer (Bianchi has done this with his previous results so I would expect that also here) but first one must be sure one is dealing with the full states of the horizon, the extrinsic geometry, not just the internal business of how many and what shapes of facets comprise it.

 

[marcus]

It's becoming increasingly clear that Bianchi's is a landmark result, which changes the Loop picture significantly.

Next year, at the main biennial conference Loops 2013, we can expect a lot of papers along the lines set out here, in the paper
http://arxiv.org/abs/1204.5122
and in the hour-long colloquium talk+QA
http://pirsa.org/12050053/

Next year the Loops conference will be held at Perimeter Institute in Canada. My guess, since he's at PI, is that Eugenio Bianchi is one of the organizers. It's going to be really interesting to see how the field is shaping up by looking at details of the Loops 2013 program as it comes out.

 

[fzero]

I listened to the rest of the talk, and to the answers to Wen and Gurau's questions a couple of times.

Bianchi drew the key distinction between counting intrinsic and extrinsic states of geometry already (if I remember right) before the question was raised. Then later around minute 63 someone from the audience (is it Razvan Gurau?) raises the issue and at minute 65 Bianchi has to repeat what he said before, with emphasis.

At minute 65 says that the earlier counting was correct! and in fact ROBUST--but it was counting intrinsic states of geometry. That is not what is relevant for the observer who is hovering outside. Entropy depends on who sees it. That, I think, is the real explanation

The intrinsic states on the horizon are precisely what Rovelli and others have argued are relevant for the outside observer. Aren't they the same states $|j\rangle$ that Bianchi is using? His $\delta S$ is precisely the change in entropy in which an extrinsic state attaches to the horizon, after which it is an intrinsic state.

This is partly work in progress by Bianchi. He is developing the quantum statistical mechanics version of his derivation which so far has been quantum thermodynamical. We won't know for sure until we see a paper but here is what I think he is saying: The observer is in space outside and lives his worldline in spacetime outside. So what matters are the states of EMBEDDED geometry. You have to count the states of the horizon as it is embedded in spacetime.

Aren't these the states $|j\rangle$ that were supposed to be associated with edges of tetrahedra that compose the horizon?

The whole thing can be made independent of any particular observer (Bianchi has done this with his previous results so I would expect that also here) but first one must be sure one is dealing with the full states of the horizon, the extrinsic geometry, not just the internal business of how many and what shapes of facets comprise it.

Do you have some more illuminating definition of what he's calling intrinsic and extrinsic geometry? It looks like the state $|\Omega\rangle$ that he uses in his density matrix is presumably the state composed of the "intrinsic" degrees of freedom forming the horizon. So how would the statistical mechanics compute some other degrees of freedom?

 

[marcus]

I listened to the rest of the talk, and to the answers to Wen and Gurau's questions a couple of times...
...
Do you have some more illuminating definition of what he's calling intrinsic and extrinsic geometry? It looks like the state $|\Omega\rangle$ that he uses in his density matrix is presumably the state composed of the "intrinsic" degrees of freedom forming the horizon. So how would the statistical mechanics compute some other degrees of freedom?

I'm glad you got to hear the rest of the talk, including the Q&A towards the end around minute 60. You have a really good question to write email to Bianchi about.
I'm sure he would appreciate interest from physics colleagues and would be happy to clarify the distinction.

Relevant links in case anyone else is reading the thread:
http://arxiv.org/abs/1204.5122
and in the hour-long colloquium talk+QA
http://pirsa.org/12050053/ [video]

Physicsmonkey already earlier in this thread had a question that he wrote to Bianchi about and quickly got a reply. It was back near the start of thread, I forget exactly where.

Yeah, it was on the second page of the thread, here:

I reached out to bianchi for clarification about his area formula. In the interest of keeping his privacy, I will just summarize the main points of his brief reply that are apparently common knowledge.

In short, both $\sqrt{j(j+1)}$ and $j$ are acceptable area operators (they differ by an operator ordering ambiguity that vanishes as $\hbar \rightarrow 0$ (which I guess here means something like $j \rightarrow \infty$ as fzero and others suggested).

The two criteria for an area operator are apparently 1) that its eigenvalues go to j in the large j limit and 2) that its eigenvalue vanish for j=0.

More systematically, bianchi is using a Schwinger oscillator type representation where we have two operators $a_i$ and the spins are $\vec{J} = \frac{1}{2} a^+ \vec{\sigma} a$. The total spin of the representation can be read off from the total number $N = a_1^+ a_1 + a_2^+ a_2 = 2j$. On the other hand, you can work out $J^2$ for yourself to find $J^2 = \frac{1}{4}( N^2 + 2N)$ which one easily verifies gives $J^2 = j(j+1)$. Thus by $|\vec{L}|$ bianchi appears to mean $N/2$.

It is again interesting to see this kind of representation appearing in a useful way since it is quite important in condensed matter.

Among other things this PhMo post reminds me of the nice point of courtesy that one does not quote someone's email without first asking permission, but one can paraphrase points which are treated as common knowledge. It seems like the right way for someone at advanced academic level to get clarification. I hope, if you write Bianchi about this you will share the main points of his reply with us as PhMo did.

I should probably not interject my own perception of this as it might only cause confusion but, that said, I would like to comment.
Entropy can only be defined with an implied/explicit observer. I believe the idea of an HORIZON is also observer-dependent. If one generalizes and gets away from designating a particular observer, the mathematical language will nevertheless indicate a class or family of observers which share the horizon.
Bianchi develops the Loop BH entropy in a way that seems to me clearly aware of the observer at each stage, although he eventually is able to generalize and cancel out dependence on any particular class or family.

This is in contrast to how I remember the Loop treatment of BH entropy back in the 1990s. I could well be wrong--not having checked back and reviewed those earlier LQG papers. But as I recall it was not so clear, with them, where the observer was and what he was looking at and measuring.

The analysis, as I recall, was done more in a conceptual vacuum. So one was looking at states only of the BH horizon ("intrinsic") without any surrounding geometric or dynamically interacting ("extrinsic") context.
I think Bianchi is going deeper, imagining more, including more in his analysis. I like the fact that he has an actual quantum THERMOMETER with which the observer a little ways outside the horizon can measure the temperature. Stylistically I like the concrete detail in the Colloquium slide where the coffee mug falls in and a new FACET of the quantum state of the horizon is created. The whole treatment AFAICS is deeper, more concrete, more interactive than what I remember from the 1990s papers.

But this is just my personal take. To get a satisfactory answer to your question about the precise meaning of the intrinsic/extrinsic distinction I would guess requires an email to Bianchi. Unless Physicsmonkey or the likes thereof care to explain.

 

[marcus]

It's nice that one of us at PF exchanged an email with Bianchi and got a point in the paper clarified.

I reached out to bianchi for clarification about his area formula. In the interest of keeping his privacy, I will just summarize the main points of his brief reply that are apparently common knowledge.

In short, both $\sqrt{j(j+1)}$ and $j$ are acceptable area operators (they differ by an operator ordering ambiguity that vanishes as $\hbar \rightarrow 0$ (which I guess here means something like $j \rightarrow \infty$ as fzero and others suggested).
...
...
It is again interesting to see this kind of representation appearing in a useful way since it is quite important in condensed matter.

I see there are signs of fairly wide interest in Bianchi's result. He is scheduled to give two talks next week at a big international conference in Stockholm--the Marcel Grossmann triennial meeting (over 1000 participants have registered for this year's MG13). Eugenio has a 30 minute time slot in the parallel session QG1 (Tuesday 3 July) and another 30 minute in the Thursday session QG4. I just learned the titles of his two tallks and found the abstracts.

http://ntsrvg9-5.icra.it/mg13/FMPro...s&talk_accept=yes&-max=50&-recid=42004&-find=
Session QG4 - Loop quantum gravity cosmology and black holes
Speaker: Bianchi, Eugenio
Entropy of Non-Extremal Black Holes from Loop Gravity
Abstract: We compute the thermodynamic entropy of non-extremal black holes using the quantum dynamics of Loop Gravity. The horizon entropy is finite, scales linearly with the area A, and reproduces the Bekenstein-Hawking expression S = A/4 with the one-fourth coefficient for all values of the Immirzi parameter. The near-horizon geometry of a non-extremal black hole - as seen by a stationary observer - is described by a Rindler horizon. We introduce the notion of a quantum Rindler horizon in the framework of Loop Gravity. The system is described by a quantum surface and the dynamics is generated by the boost Hamiltonion of Lorentzian Spinfoams. We show that the expectation value of the boost Hamiltonian reproduces the local horizon energy of Frodden, Ghosh and Perez. We study the coupling of the geometry of the quantum horizon to a two-level system and show that it thermalizes to the local Unruh temperature. The derived values of the energy and the temperature allow one to compute the thermodynamic entropy of the quantum horizon. The relation with the Spinfoam partition function is discussed.
Talk view--------------------------

http://ntsrvg9-5.icra.it/mg13/FMPro...s&talk_accept=yes&-max=50&-recid=42199&-find=
Session QG1 - Loop Quantum Gravity, Quantum Geometry, Spin Foams
Speaker: Bianchi, Eugenio
Horizons in spin foam gravity
Abstract:Spin foams provide a formulation of loop quantum gravity in which local Lorentz invariance is a manifest symmetry of quantum space-time. I review progress in determining horizon boundary conditions in this approach, and discuss the thermal properties of the quantum horizon.
Talk view: [No link, I suppose that some of the talks will be viewable next week, and this field will be filled in for some of them.]

For an overview of the parallel sessions including links to specific ones, see:
http://www.icra.it/mg/mg13/parallel_sessions.htm
There are 4 specifically Loop sessions each about 4:30 long--each making time for 8 thirty-minute talks and a coffee break. Or more if some talks are limited to 20 minutes.
QG1 A and B ("Loop Quantum Gravity, Quantum Geometry, Spin Foams") chaired by Lewandowski
QG4 A and B ("Loop quantum gravity cosmology and black holes") chaired by Pullin and Singh
Plus there are two more related sessions on devising tests of QG not limited to Loop.
QG2 A and B ("Quantum Gravity Phenomenology") chaired by Amelino-Camelia

 

[marcus]

It would be interesting to see a PERTURBATIVE confirmation of Bianchi's result. A uniformly accelerating observer in Minkowski space has a Rindler horizon (beyond which stuff can't affect him, is out of causal touch with him).

So one can have gravitons as perturbations of Minkowski geometry and look at entropy in that situation.

I should look at Bianchi's ILQGS talk again. He just recently gave a seminar talk, which is online.
Slides: http://relativity.phys.lsu.edu/ilqgs/bianchi101612.pdf
Audio: http://relativity.phys.lsu.edu/ilqgs/bianchi101612.wav
Since this talk was in October, there is sure to be new stuff compared with the May 2012 paper we started this discussion thread with.

 

[francesca]

It would be interesting to see a PERTURBATIVE confirmation of Bianchi's result.

Here it is: http://arxiv.org/abs/1211.0522

Horizon entanglement entropy and universality of the graviton coupling
Eugenio Bianchi
(Submitted on 2 Nov 2012)
We compute the low-energy variation of the horizon entanglement entropy for matter fields and gravitons in Minkowski space. While the entropy is divergent, the variation under a perturbation of the vacuum state is finite and proportional to the energy flux through the Rindler horizon. Due to the universal coupling of gravitons to the energy-momentum tensor, the variation of the entanglement entropy is universal and equal to the change in area of the event horizon divided by 4 times Newton's constant - independently from the number and type of matter fields. The physical mechanism presented provides an explanation of the microscopic origin of the Bekenstein-Hawking entropy in terms of entanglement entropy.
Comments: 7 pages

 

[marcus]

... gravitons in Minkowski space... The physical mechanism presented provides an explanation of the microscopic origin of the Bekenstein-Hawking entropy in terms of entanglement entropy.

Yes! It's significant that the analysis is done in flat 4D space. It does not require a black hole event horizon! No largescale curved geometry is needed. The explanation is quantum field theoretical.

It independently confirms the Bekenstein-Hawking entropy S=A/4 and so gives a reasonable QFT suggestion for where that entropy comes from.

This might be a step towards understanding what the microscopic degrees of freedom are that underlie quantum spacetime and from which GR arises at large scale. Thanks, Francesca!