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

fzero

In the latest Bianchi paper, which does not involve state counting, there is no difference. Since [itex]H \sim \sum_f K^z_f \sim \sum \gamma j_f >0[/itex] we can still identify it with the area.

If we try to discuss state counting, we should first note that the degenerate faces were always ignored in past calculations. There is a footnote on page 4 of the Rovelli-Wilson-Ewing (RW-E) paper that claims that they can be erased from the spin network in canonical LQG. We can still allow degenerate edges, which we need in order to glue [itex]k_f>0[/itex] and [itex]k_f<0[/itex] faces together.

The states that correspond to the BH entropy calculation can be determined from the usual prescription. We choose a triangulation [itex]\Delta[/itex] and then count the number of faces that pierce the surface of the horizon. The choice of orientation means that nondegenerate faces now come from two bins, so we have to sum over two species of spins.

Suppose that we denote [itex]k_f>0[/itex] faces by [itex]N^+_j[/itex] and [itex]k_f<0[/itex] faces by [itex]N^-_j[/itex]. If we also include [itex]N_0[/itex] degenerate faces, the number of states is now

[tex] W = \frac{ N!}{(N_0)!} \prod_j \frac{(2j+1)^{N^+_j+N^-_j}}{N^+_j!N^-_j!}.[/tex]

There are two constraints, namely

[tex] N = N_0 + \sum_j(N^+_j+N^-_j),[/tex]

[tex] 8\pi G\hbar \gamma \sum_j j(N^+_j+N^-_j) = A.[/tex]

There is no other constraint on [itex]N_0[/itex]. Having [itex]N^-\neq 0[/itex] means that we need to include degenerate edges, but only complete faces contribute to the state counting, not edges.

Let's first consider the case that [itex]N_0=0[/itex]. Taking the large N limit and then extremizing the entropy subject to the constraints leads to the occupation numbers

[tex] \frac{N^\pm_j}{N} = (2j+1) e^{-\mu j},[/tex]

where

[tex] N = \frac{A}{8\pi G\hbar \gamma\alpha},~~~\mu\sim 2.753,~~~\alpha\sim 0.4801.[/tex]

The entropy is

[tex] S = \frac{\mu A}{8\pi G\hbar \gamma},[/tex]

which results in

[tex]\gamma = \frac{\mu}{2\pi} \sim 0.4382.[/tex]

So we find the right entropy at a new value of the Immirzi parameter.

Now, if we were to allow degenerate faces ([itex]N_0\neq 0[/itex]), we don't have enough information to fix the occupation numbers. In this case,

[tex] \frac{N_0}{N} = \frac{1}{1+\sum_j(2j+1) e^{-\mu j}} , [/tex]

[tex] \frac{N_j}{N} = \frac{N_0}{N} (2j+1) e^{-\mu j}.[/tex]

The only constraint left is the area constraint and only the nondegenerate faces contribute to that. However, we have two unknowns, [itex]\mu[/itex] and [itex]N_0[/itex]. So we cannot compute the number of degenerate faces at this level of sophistication.

There is a physical explanation for this. Namely, it costs very little entropy to replace a pair of spin states (faces) with a degenerate face and a single higher spin face in such a way to keep the area fixed. The amount of entropy is much smaller than the leading term in the large N limit. We can actually use the number of states to determine the change in entropy if we replace a spin [itex]j_1[/itex] and [itex]j_2[/itex] state with a spin [itex]j_1+j_2[/itex] state and a degenerate face. It is

[tex] S (N_0+1) - S (N_0) = \ln \left[ \frac{2(j_1+j_2)+1}{(2j_1+1)(2j_2+1)} \frac{N_{j_1} N_{j_2}} {(N_0+1)( N_{j_1+j_2}+1)} \right].[/tex]

In the large N limit, we can use the occupation numbers solved for above to find

[tex] S (N_0+1) - S (N_0) \sim \mu ( j_1+j_2 - (j_1+j_2)) \sim 0.[/tex]

It might be useful to find a reference that explains why degenerate faces can be removed from the spin network.

atyy

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."

marcus

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.
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

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.

Not quite. Sen starts with the SU(2) CS result, [itex]-(3/2)\log A = -3\log a[/itex]. 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 [itex]\log a[/itex] 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

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.

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/201...py-in-lqg.html

MTd2

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

marcus

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

Whoa! Thanks! I missed that Baez comment. What I saw was his "tightening the noose" comment:
http://backreaction.blogspot.com/201...70896762383197

What you are quoting is a later comment by Baez that I didn't see until you pointed it out:
http://backreaction.blogspot.com/201...71751066857145

marcus

Now Aleksandar Mikovic has joined the discussion:
http://backreaction.blogspot.com/201...53172764855047

==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.
http://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/