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

[marcus]

I mentioned that I'm beginning to see this paper in the context of a small revolution in Loop gravity.
A number of young researchers are posting Loop BH papers which break from the earlier work (1990s) and often find the entropy independed of Immirzi to first order. (Besides Bianchi, some names are Ghosh, Perez, Engle, Noui, Pranzetti, Durka. And there are groundbreaking Loop BH papers by Modesto, Premont-Schwarz, Hossenfelder. I'm probably forgetting some. )
So part of understanding Bianchi's paper, for me, is catching up on the context of other recent Loop BH papers. Here is one that came out earlier this month. You can see there is significant conceptual overlap.

http://arxiv.org/abs/1204.0702
Radiation from quantum weakly dynamical horizons in LQG
Daniele Pranzetti
(Submitted on 3 Apr 2012)
Using the recent thermodynamical study of isolated horizons by Ghosh and Perez, we provide a statistical mechanical analysis of isolated horizons near equilibrium in the grand canonical ensemble. By matching the description of the dynamical phase in terms of weakly dynamical horizons with this local statistical framework, we introduce a notion of temperature in terms of the local surface gravity. This provides further support to the recovering of the semiclassical area law just by means of thermodynamical considerations. Moreover, it allows us to study the radiation process generated by the LQG dynamics near the horizon, providing a quantum gravity description of the horizon evaporation. For large black holes, the spectrum we derive presents a discrete structure which could be potentially observable and might be preserved even after the inclusion of all the relevant transition lines.
Comments: 9 pages, 2 figures

 

[fzero]

He shows that for all pure states of the quantum Rindler horizon it is identically true that
∂A/4 = ∂E/T
The argument that this extends by linearity to superpositions---to mixed states of the quantum Rindler horizon, and large assemblies thereof---is not made explicitly. But a relevant observation is made immediately after equation (20) on page 4:

"Notice that the entropy density is independent of the acceleration a, or equivalently from the distance from the horizon."​

This opens the way to our concluding that ∂A/4 = ∂E/T applies as well to mixed states and collections thereof.

I went through the statistical computation of the entropy in Bianchi's polymer model using the energy constraint. I was wrong when I said that there would be a rescaling of the Lagrange multiplier by a factor of \gamma. The important point is that the energy constraint is not merely equivalent to the area constraint, they are in fact exactly the same. From equation (9) of the present paper (let's call it B12 for Bianchi-2012),

E = \sum_f \hbar\gamma j_f a = \frac{a}{8\pi G} \sum_f A_f = \frac{aA_H}{8\pi G}.

We can rewrite this constraint as

\sum_f j_f = \frac{A_H}{8\pi \gamma G\hbar}.

This is the same as the area constraint used in equation (16) of the polymer paper (B10 for Bianchi-2010), except, as previously discussed, here we are using a slightly different basis where the eigenvalues of |\vec{L}| are j rather than \sqrt{j(j+1)}. You can check that the distribution of states depends only on the degeneracy and not on the precise eigenvalue, so the rest of equ (16) is unchanged.

I'd already gone through all of the math in that section of the paper before realizing that there weren't any numerical differences between the constraints, so I might as well report on the result. The derivation of the entropy only differs in the numerical value of the constants derived there. This is the effect of the different eigenvalue spectrum. For example, the occupation numbers at equilibrium satisfy

p_j \equiv \frac{N_j^*}{N^*} \approx (2j+1) e^{-\mu^* j}.

Imposing the normalization requirement

\sum_j p_j =1

can be done by approximating the sums by an integral. I find that \mu^* is the solution to

\int_0^\infty dx(x+2) e^{-\mu^* (x+1)/2} =1.

This leads to the equation

4(\mu^*+1) e^{-\mu^*/2} = (\mu^*)^2,

which has a numerical solution at

\mu^* \sim 2.086.

This is a little bit different from the value obtained in B10, but in a reasonable neighborhood given the similarity of the normalization constraints.

Similarly, the constant

\alpha^* = \sum_j j p_j = \frac{1}{4} \int_0^\infty dx(x+1)(x+2) e^{-\mu^* (x+1)/2}
= \frac{e^{-\mu^*/2} }{(\mu^*)^3} ( (\mu^*)^2 + 2 \mu^* + 4) \sim 0.486.

The leading term in the entropy is once again

S = \frac{\kappa}{4G\hbar} \frac{\mu^*}{2\pi\gamma} A_H,

so that we require

\gamma =\frac{\mu^*}{2\pi} \sim 0.322.

Presumably the difference between this value of the Immirzi parameter and earlier results is due to the difference in the spacing between the area eigenvalues \sqrt{j(j+1)} vs j. We are effectively using slightly different scales to quantize the area operator.

 

[marcus]

Compliments on giving these papers a close reading, and thanks for sharing what you are finding out.
It occurs to me that Pranzetti's April 2012 paper may actually be to your liking. At least I hope so! Canonical ensemble etc.
==quote page 3 middle of second column==
Namely, if we assume a stationary near-horizon geometry and we use the Unruh temperature βκ = 2π/hbar for our local accelerated observer O, the entropy expression (9) gives exactly S = A/(4l_p²), at the leading order.
==endquote==
http://arxiv.org/abs/1204.0702
here kappa is the local surface gravity, so beta is the Unruh temperature. As one sees, no dependence on Immirzi.
The derivation may conform with your standards. Not sure, but it might.

==quote Pranzetti beginning of page 2==
In this letter, we want to investigate further and more in detail the analogy between a quantum horizon with its punctures and a gas of particles by introducing the main ingredients for a grand canonical ensemble analysis. The basic idea is to regard the bulk and the horizon as forming together an isolated system. The two subsystems are considered separately in thermal equilibrium; then, at some point, a weakly dynamical phase takes place and they interact with each other. This local interaction allows for the possibility of exchange of energy and particles between the two. After such a change of thermodynamic state has taken place, the two subsystems go back to a situation of thermal equilibrium. This picture will be made more precise in the following, where we will concentrate only on the spherically symmetric case. However, let us at this point clarify the framework we are working in: no background structure is introduced at any point, we will work in the quantum gravity regime; no matter is going to be coupled to gravity; the radiation spectrum we will derive is related entirely to emission of quanta of the gravitational field due to dynamical processes described by the LQG approach...
==endquote==

==quote page 2 start of "Entropy" section==
Entropy. Let us now first concentrate on the derivation of the entropy of the gas of punctures (see [18] for the original microcanonical derivation and [19] for a recent review). By working in the grand canonical ensemble— which represents the physically most suitable framework to describe the horizon+bulk system—, it can be shown how the Bekenstein-Hawking semiclassical entropy can be recovered only through thermodynamical considerations. Moreover, the description of the radiation process in the second part of the paper justifies the interpretation of the local notion of surface gravity introduced above as a temperature, which is a fundamental ingredient to recover the semiclassical area law (see below). In this sense, the result of the second part of the paper puts on more solid ground the recent derivation of [16]. This section simply presents a more detailed derivation of the IH entropy in the grand canonical ensemble already performed in [16]. The original part of the paper is contained in the next section.
The grand canonical partition function for the gas of punctures is given by...
===endquote===

Pranzetti is at the Max Planck Institute for Gravitation in a little place called Golm outside Berlin. I wonder whose group he's in. Bianca Dittrich or Dan Oriti probably. I checked. Oriti's group. Pranzetti was a Marseille PhD student before that.

 

[fzero]

Compliments on giving these papers a close reading, and thanks for sharing what you are finding out.
It occurs to me that Pranzetti's April 2012 paper may actually be to your liking. At least I hope so! Canonical ensemble etc.
==quote page 3 second column==
Namely, if we assume a stationary near-horizon geometry and we use the Unruh temperature βκ = 2π/hbar for our local accelerated observer O, the entropy expression (9) gives exactly S = A/(4l_p²), at the leading order.
==endquote==
http://arxiv.org/abs/1204.0702
here kappa is the local surface gravity, so beta is the Unruh temperature.
The derivation may conform with your standards. Not sure, but it might.

The calculation he's referring to in that particular paragraph is a semiclassical one, so it's on par with the one in B12 and Hawking's original one. Someone should probably be able to explain why the semiclassical results seem somewhat universal. I have a feeling that it's just that adding quantum bits of area are the same as what Bekenstein and Hawking were doing back in the 70s. As long as you cook up the right relationship between the energy of the state (being added) and the corresponding area, you will find the same result when you use the Clausius formula.

This is not to say that you don't learn something from these approaches. But don't be confused that the semiclassical computations are as exciting as a quantum treatment that accounts for the right microstates.

The rest of the Pranzetti paper seems to be about developing the grand canonical ensemble for the spin microstates. This is interesting, and something like it is needed to properly treat radiation (he addresses this of course). However, as you can see from his equation (9), the grand canonical formalism is a specific type of correction to the leading term computed by the microcanonical ensemble. So it's probably not immediately relevant to the specific issues I've been discussing.

Addendum: I saw your last edit and believe that the above comments continue to apply. The "thermodynamic" calculation is the semiclassical one. The corrections from the grand canonical ensemble do not affect the leading order entropy computed in the microcanonical formalism at large occupation numbers. The corrections represent quantum corrections due to particle exchange between the horizon and the exterior, so represent subleading corrections. These are likely of smaller order than terms that we've already dropped in the N_j\gg 1 limit.

 

[marcus]

Thanks for the comment on Pranzetti's paper! Not sure exactly what you mean by "semiclassical" though.

You are right that the derivation of S = A/4 (without Immirzi dependence) happens in the first 3 pages.

Indeed the grand canonical ensemble of quantum states of horizon's geometry is set up in the first few equations on page 2, where he gives the "canonical partition function" of quantum states, e.g. equation (3).

There is no hint of "QFT on a curved spacetime" there. Punctures are simply where spin network states go through the horizon. Their edges carry area quantum numbers j.
A class of spin network states is specified by {s_j} in his notation. There s_j is the number of spin network edges with spin label j, which pass thru the horizon.
So the approach is fully background independent. There is no prior geometry. All the geometry is in the spin networks which are quantum states of geometry. Standard in Loop gravity.So he sets up to derive S = A/4 with those equations (2) (3) (4) ...(9)

The fully quantum conclusion is equation (9) where you see the quantum corrections as well as the leading term which has the 1/4 coefficient.

Then to recover the Bek Hawk. result he of course takes a limit so that the quantum corrections go away. That is equation (10) in the next paragraph.

But already before that in equation (9), which is not semiclassical, you see there is no dependence on Immirzi.

Anyway that is how I read it. Do you see equations (2 - 9) as in any way semiclassical?
For me they come entirely within ordinary spin network Loop gravity.

 

[fzero]

I was a bit too fast to comment on the Pranzetti paper. This is for two reasons: 1. I was swayed by his own reference to a semiclassical analysis to fix \beta \bar{\kappa} = 2\pi/\hbar, and 2. My own thermodynamics is quite rusty.

It appears to me that Pranzetti's calculation in the grand canonical ensemble (GCE) is equivalent (there is a question of the role of the large N limit that I comment on below) to the calculation Bianchi made in the microcanonical ensemble (MCE) with the area/energy constraint. In both cases you find the correct leading term in the entropy, but for a fixed value of the Immirzi parameter. In the Bianchi calculation, the fixing of the Immirzi parameter is explicit. In the Pranzetti calculation it is hidden, but he refers to it below his equation (11). I will discuss this more in a bit.

Now, using the MCE is rather transparent. We're aiming to count microstates and not worry about dynamical processes like emission or absorption, so we can fix the number of quanta and energy. To ensure that we're using the correct mixed state, we impose the area constraint and extremize the entropy.

In the GCE, we allow the number of quanta and energy to fluctuate but we use a heat bath to fix the temperature of the system state. We end up getting the same answer for the entropy as before when we fix the average energy to be the appropriate multiple of the energy. The setup is a bit unphysical for a real BH, but for our purposes we can always imagine feeding the right amount of matter into balance the radiation coming out. It also seems to me to be nicer to impose the area constraint dynamically, rather than by hand, but this is more opinion than a serious objection.

However, what the GCE also seems to do for us is let us avoid the large N limit. For Bianchi, the large N limit was not just important to allow us to use the Stirling approximation, but it was also important in obtaining a manageable form for the number of states with the same occupation numbers (\Omega in B10). Pranzetti's use of the GCE seems to remove the need for us to take this limit, at the expense of an extra free parameter, the chemical potential. This leads us naturally to the new conundrum surrounding the Immirzi parameter.

We will work with an area operator that is a mix of the ones that Pranzetti and Bianchi use:

\hat{H} | \{ s_j\}\rangle = \hbar \bar{\kappa} \gamma \sum_j s_j j | \{ s_j\}\rangle .

I am using Pranzetti's notation, but I have set \hbar G = \ell_p^2 for convenience, as well as chosen the Schwinger-type basis of Bianchi to simplify some calculations later.

Now, it could be that I don't understand the state space well enough and there is some inequivalence between the Bianchi and Pranzetti pictures. This would presumably address the large N questions above. Somehow the difference would go away in the large N limit, explaining why they agree. In any case, I will keep going under the assumption that I understand the mechanics of the states, if not their complete motivation.

It will be convenient to define a parameter

\nu = \hbar \beta \bar{\kappa} \gamma.

If we set \beta \bar{\kappa} = 2\pi/\hbar, then we can write the Immrizi parameter as

\gamma = \frac{\nu}{2\pi}.

Now, we can write all thermodynamic quantities in terms of the function

f(\nu) = \sum_j (2j+1) e^{-\nu j} \longrightarrow \frac{4}{\nu^2} (\nu+1) e^{-\nu/2} ~\mathrm{as}~N\rightarrow \infty.

This is the same expression that turned up in the result for the occupation numbers in B10, so I've included the value for the sum that we find in the large N limit.

The relevant equation from Pranzetti is (10), which we write as

f(\nu) = \frac{\bar{N}}{\bar{N}+1} e^{-\beta \mu} .

In general, this is a transcendental equation that determines \nu (equivalently \gamma) in terms of \mu and \bar{N}. In the large \bar{N} limit, the explicit \bar{N} dependence drops out. If we did not wish to take the limit, we could just eliminate \bar{N} using the energy constraint, which is

\frac{\bar{\kappa}}{8\pi G} A = \bar{E} = - \hbar \bar{\kappa} \gamma (\bar{N}+1) \frac{d}{d\nu} \log f.

Now the chemical potential represents the cost in energy to take a particle from the heat bath and place it into the system (the isolated horizon or BH). Physically, we might think that this is related to the surface gravity of the horizon. Perhaps we might think that it is zero, since this is precisely the notion of energy that is ambiguous in a gravitational system. It might be possible to address this by thinking more carefully about how we have to define the GCE. I will probably think a bit more about it, but in any case, the Immrizi parameter now depends implicitly on the chemical potential.

If we set the chemical potential to zero and take the large \bar{N} limit, then we will recover the same numerical solution as in the modified Bianchi calculation

\nu \sim 2.086,~~~\gamma \sim 0.322.

For other values of \mu, we will obtain some other value of \gamma.

This is either strange or expected. On the one hand, it might seem strange that we need to change the quantum of area to accommodate a change in chemical potential. On the other hand, we might think that whatever change we made to the system represents some sort of change in the natural energy scale of the problem and the Immirzi parameter must run.

Anyway, the upshot of all of this is that B10 and Pranzetti look like correct computations for the state spaces they are using. Their results seem to agree qualitatively and quantitatively in the common regime of validity. I still think that B12 is pulling a bit of a fast one by using a pure state, but I understand it to be correct as a semiclassical computation, rather than a purely quantum one.

 

[marcus]

Thanks for the comment on Pranzetti's paper! Not sure exactly what you mean by "semiclassical" though.
...
...
But already before that in equation (9), which is not semiclassical, you see there is no dependence on Immirzi.
...

I was a bit hasty myself (we had company yesterday evening ). What I meant and should have said is that already in equation (9), which is not semiclassical, you can see there is no dependence on Immirzi in the leading term, which is the area term. The other two terms can be considered corrections, that appear in the full quantum version of the entropy equation, namely equation (9).

I would expect there to be some dependence on Immirzi in the second term. Perhaps both, I haven't thought much about it. But it is the leading term that is the area term, where you see the proportionality of the entropy with the area, and that is where you don't get dependence on Immirzi. I should have made that clearer.

BTW just for clarification the μ that appears in the second term of (9) is called the chemical potential. Doubtless familiar to you, Fzero, but others might be reading. The N-bar that appears in the second term is the average number of punctures. I write it Ñ to avoid having to resort to LaTex.

I really like this Pranzetti paper! For convenience here's the link:
http://arxiv.org/abs/1204.0702
The second term in equation (9) is quite interesting. I think I first saw it in the Ghosh Perez paper last year, but I'm not sure. It is -μβÑ. The β, as he says, can be interpreted as a "generic temperature β for the preferred local observer O hovering outside the horizon at proper distance l, as result..." It seems to be this second term which you are scrutinizing in your above post #56. More power to you .

 

[fzero]

The second term in equation (9) is quite interesting. I think I first saw it in the Ghosh Perez paper last year, but I'm not sure. It is -μβÑ. The β, as he says, can be interpreted as a "generic temperature β for the preferred local observer O hovering outside the horizon at proper distance l, as result..." It seems to be this second term which you are scrutinizing in your above post #56. More power to you .

As it turns out, the leading term proportional to area comes entirely from the \beta \bar{E} term. For the subleading terms we find

-\mu \beta \bar{N} =-\bar{N} \log z = \bar{N} \log f + \bar{N} \log\frac{\bar{N}+1}{\bar{N}},

\log \mathscr{Z}=\log(\bar{N}+1).

So we can write

S = \frac{A}{4G\hbar} + \bar{N} \log f + \log \frac{(\bar{N}+1)^{\bar{N}+1}}{\bar{N}^\bar{N}}.

Now, remember that

(\log f)' = c \frac{A}{\bar{N}+1}

by the energy constraint. We can't explicitly integrate this because \bar{N} is (defined transcendentally as) a function of \nu.

It is interesting to try to compare this to the large N result in B10:

S = \frac{A}{4G\hbar} - \frac{3}{2} \log \frac{A}{G\hbar}.

Naively, it doesn't appear possible to reproduce the \log A correction, since \log f\rightarrow 0in the \mu=0, large \bar{N} limit. Perhaps there is some subtlety in taking these limits.

 

[marcus]

I don't think people are entirely sure what the correction terms are. They all seem to agree that the MAIN term for the entropy is A/4.
And that obviously does not depend on the Immirzi.

And from what i see they expect the correction terms, which arise when you do a full quantum treatment of the BH entropy, to involve the Immirzi.

Bianchi says as much in his recent paper. And if I remember right this is explictly the case both with Ghosh Perez and with Pranzetti.

So this is the picture that is emerging more or less across the board with these young researchers' work on Loop BH.

Supposing they are right and the work is born out, then I think this allows for the Immirzi to run. Run with what? With scale? with energy density?

Then the size of the BH would affect the slight corrections in the formula by which the entropy was calculated.

Maybe the "bare" UV value of Immirzi is, say, 0.274. And she runs to zero when you go to larger and larger scale.

Just speculating
========================

I should copy Pranzetti's equation (9) since that seems to be the main equation we are discussing.

...
BTW just for clarification the μ that appears in the second term of (9) is called the chemical potential. ... The N-bar that appears in the second term is the average number of punctures. I write it Ñ to avoid having to resort to LaTex.

I really like this Pranzetti paper! For convenience here's the link:
http://arxiv.org/abs/1204.0702
The second term in equation (9) is quite interesting. I think I first saw it in the Ghosh Perez paper last year, but I'm not sure. It is -μβÑ. The β, as he says, can be interpreted as a "generic temperature β for the preferred local observer O hovering outside the horizon at proper distance l, as result..." ...

===quote http://arxiv.org/abs/1204.0702 page 3 equation (9)===

S = (βκ/8πG)A -μβÑ + log curlyZ
==endquote==

curlyZ is defined in equation (2) as a function of the local observer's temperature β and is the grand canonical partition function for the gas of punctures. These are the links of spin networks sticking out thru the BH horizon.

The term (βκ/8πG) turns out to be 1/4, in the appropriate units, with G = hbar = c = 1.
This is when β is seen to be the Unruh temperature associated with the acceleration which the observer must maintain in order to continue hovering at a fixed distance above the horizon, which has surface gravity κ.

I have no particular reason to copy in equation (9) at this point. We have been discussing it for the past I-don't-know-how-many posts. But I just wanted to finally write it in for completeness, for the record so to speak.

 

[fzero]

I don't think people are entirely sure what the correction terms are. They all seem to agree that the MAIN term for the entropy is A/4.
And that obviously does not depend on the Immirzi.

You can't quite say that the leading contribution to the entropy doesn't depend on the Immirzi parameter. Either it does, or a new parameter is introduced into the model that serves to fix the Immirzi parameter mid-calculation. In particular, we have roughly 4 different modern methods of state counting that have appeared in the literature:

1. SU(2) Chern-Simons with level k\rightarrow \infty, microcanonical ensemble: entropy depends on \gamma as

S = \frac{c_1}{\gamma} \frac{A}{4G}.

2. Microcanonical ensemble with area/energy constraint: entropy depends on \gamma as

S = \frac{c_2}{\gamma} \frac{A}{4G},

where c_2 is set at a critical value via the constraint. Value agrees with approach 1 above.

3. SU(2) CS with finite level, MCE: entropy does not depend explicitly on \gamma, but after extremizing \gamma depends on the level (see, for example, the discussion around fig 6 in http://arxiv.org/abs/1103.2723v1). Value quickly converges to k=\infty result for k\geq 4, so consistent with 1 and 2.

4. Grand canonical ensemble with area/energy constraint, chemical potential \mu introduced: entropy does not depend on \gamma.
Thermodynamics, including energy constraint, fix \gamma in terms of \mu. In \mu =0, large A limit, recover value of \gamma consistent with approaches 1 and 2.

And from what i see they expect the correction terms, which arise when you do a full quantum treatment of the BH entropy, to involve the Immirzi.

Bianchi says as much in his recent paper. And if I remember right this is explictly the case both with Ghosh Perez and with Pranzetti.

Engle et al, in 1103.2723 linked above, claim that

"As we have just seen, k does modify the leading but does not modify the subleading corrections of the entropy. In that sense, the logarithmic corrections seems to be universal and independent of the Immirzi parameter"

But I think they're making the mistake of forgetting that their critical exponent \alpha is defined in terms of \gamma. Essentially the ratio of \alpha and \gamma must take a critical value when computing the entropy.

Supposing they are right and the work is born out, then I think this allows for the Immirzi to run. Run with what? With scale? with energy density?

Then the size of the BH would affect the slight corrections in the formula by which the entropy was calculated.

Maybe the "bare" UV value of Immirzi is, say, 0.274. And she runs to zero when you go to larger and larger scale.

Running of the Immirzi parameter already has consequences at lowest orders in the approaches outlined below. Extending the observations of Larsen and Wilzcek and Jacobsen, the space of running parameters is (G,\gamma, \mu, k, \ldots). The fact that critical parameters appear in most approaches also gives another effective quantity that will be a function of scale. I certainly don't know enough to speculate on the consequences.

 

[marcus]

Let's suppose Pranzetti's equation is right. Here's his equation (9)
===quote http://arxiv.org/abs/1204.0702 page 3 equation (9)===

S = (βκ/8πG)A -μβÑ + log curlyZ
==endquote==

Here is the first term,
(βκ/8πG)A

Are you saying that this term depends on the Immirzi?

 

[marcus]

I don't see how you could possibly be saying that and i don't see anything in your posts that implies it. So a simple "no" answer would suffice.

Just to be clear, I'd like to be sure of that. So that I know we both agree that the first term in Pranzetti's eqn (9) does not depend on the Immirzi.

And in that case we can look at the other two terms, try to estimate their size etc, if you are so inclined. But first let's be sure we understand each other about the leading term.

 

[fzero]

I don't see how you could possibly be saying that and i don't see anything in your posts that implies it. So a simple "no" answer would suffice.

Just to be clear, I'd like to be sure of that. So that I know we both agree that the first term in Pranzetti's eqn (9) does not depend on the Immirzi.

The leading term doesn't depend on \gamma in the following sense. It gives

S = \frac{A}{4G},

where A is a macroscopic parameter. In terms of the microscopic parameters,

A = F(\gamma,\mu).

So A is the macroscopic area at a specific value of \gamma, much the same way that the microcanonical result c A/\gamma is the area at a specific \gamma.

I've been looking at whether or not there's some way to derive an expression for the entropy that makes sense without appealing to the area constraint. I haven't found anything useful so far.

And in that case we can look at the other two terms, try to estimate their size etc, if you are so inclined. But first let's be sure we understand each other about the leading term.

I wrote down expressions for the other two terms in post #58. They are also implicitly functions of \gamma, but they vanish in the large N limit.

I worked out what was confusing me about the \log A term. What had happened was B10 partially reproduces the "quantum" corrections from the CS theory (they at least agree at large N). These corrections have been ignored in Pranzetti, so there's no point in looking for them.

 

[marcus]

The leading term doesn't depend on \gamma in the following sense. It gives

S = \frac{A}{4G},

where A is a macroscopic parameter. In terms of the microscopic parameters,

A = F(\gamma,\mu)...

I'm not sure you understand. In the Loop papers I've seen where γ → 0 all the areas remain constant. Spin network labels are increased precisely in accordance with this requirement. So jγ = const. Having gamma, the Immirzi parameter, run does not necessarily introduce any variation in the area. That holds for any area, not only for the areas of BH horizons.

So I would say your first statement is right. The leading term coefficient has no Immirzi dependence.
S = A/4
But your second statement A = F(gamma, mu) does not connect with how I've seen things done in Loop gravity.

I think it's pretty clear that the leading term in (9) need not change as gamma runs, as, for example, γ → 0. It would be interesting, though, to learn something about the dependence of the other two terms, and their sizes relative to the leading term.

Various papers by Bianchi, Magliaro, Perini exemplify this so-called "double scaling limit" it makes sense to keep the overall region of space the same size as you vary parameters. I suspect that the proven usefulness of this type of limit is one of the motivations here: i.e. reasons for interest in the new work giving Immirzi parameter greater freedom.

 

[fzero]

I'm not sure you understand. In the Loop papers I've seen where γ → 0 all the areas remain constant. Spin network labels are increased precisely in accordance with this requirement. So jγ = const. Having gamma, the Immirzi parameter, run does not necessarily introduce any variation in the area. That holds for any area, not only for the areas of BH horizons.

So I would say your first statement is right. The leading term coefficient has no Immirzi dependence.
S = A/4
But your second statement A = F(gamma, mu) does not connect with how I've seen things done in Loop gravity.

I can be more specific using the expressions in post #56. There are some missing factors in those expressions, so let me give some more detail here and clear up the mistakes.

We start with

\log \mathscr{Z} = - \log ( 1 - z \sum_j (2j+1) e^{-\beta E_j} ), ~~z = e^{\beta\mu}.

We use the Schwinger basis and G\hbar =\ell_p^2, then

E_j = \bar{\kappa} \hbar \gamma j .

Using \beta\bar{\kappa} = 2\pi/\hbar, we can write

\beta E_j = 2\pi \gamma j.

The ensemble energy is

\bar{E} = - \frac{\partial}{\partial \beta} \log \mathscr{Z} = \frac{z\sum_j (2j+1)E_j e^{-\beta E_j}}{1 - z \sum_j (2j+1) e^{-\beta E_j}}.

Let's get a neater expression from this by noting that

\sum_j (2j+1) E_j e^{-\beta E_j} = -\frac{ \gamma f&#039;(\gamma) }{\beta}<br /> ,

where

f(\gamma) = \sum_j (2j+1) e^{-2\pi\gamma j}.

Now the energy constraint is

\frac{\bar{\kappa} A}{8\pi G} = \bar{E} =-\frac{1}{\beta} \frac{\gamma f&#039;(\gamma)}{1-z f(\gamma)},

so we can write

A =- 4G\hbar \frac{\gamma f&#039;(\gamma)}{1-z f(\gamma)}.

The right-hand side of this expression is what we mean by F(\gamma,\mu). The area A is a fixed input, so it is a transcendental equation that relates \gamma and \mu.

We can also note immediately that the leading contribution to the entropy is

S =- \frac{\gamma f&#039;(\gamma)}{1-z f(\gamma)} +\cdots.

In terms of microscopic quantities, this looks \gamma dependent, but the area constraint sets it to a macroscopic constant.

I think it's pretty clear that the leading term in (9) need not change as gamma runs, as, for example, γ → 0. It would be interesting, though, to learn something about the dependence of the other two terms, and their sizes relative to the leading term.

Various papers by Bianchi, Magliaro, Perini exemplify this so-called "double scaling limit" it makes sense to keep the overall region of space the same size as you vary parameters. I suspect that the proven usefulness of this type of limit is one of the motivations here: i.e. reasons for interest in the new work giving Immirzi parameter greater freedom.

As for the other terms, there are a variety of ways to express them using the expressions

\bar{N} = \frac{zf}{1-zf}, ~~~ zf = \frac{\bar{N}}{\bar{N}+1}.

In particular

S = \beta \bar{E} - \beta \mu \bar{N} + \log\mathscr{Z} ,
=- \frac{\gamma f&#039;(\gamma)}{1-z f(\gamma)} -\beta\mu \frac{zf}{1-zf} - \log (1-zf),
=-(\bar{N}+1) \gamma f&#039; - \beta\mu \bar{N} + \log(\bar{N}+1).

To try to examine these terms, it's useful to work at large \bar{N}, for which

e^{-\beta\mu} \approx f(\gamma) \approx \frac{2\pi\gamma + 1}{\pi^2\gamma^2} e^{-\pi\gamma}.

One thing to note about this expression is that there doesn't seem to be any limiting value of \mu for which \gamma\rightarrow 0. In any case, we can use this to write

S\approx -\bar{N}\gamma f&#039; +\bar{N} \log f + \log\bar{N}.

The first two terms are roughly of the same order for \gamma = O(1). The relation between \gamma and \mu is too unwieldy to do much analytically, but maybe some rough numerics could prove insightful.

Edit: Actually, when \mu =0, f\approx 1, so \log f\approx 0. From the 1st term, it turns out that

\bar{N} \approx 0.4227 \frac{A}{4G},

so the 3rd term goes like \log A. However, as I mentioned earlier, there are other corrections to the partition function that have not been taken into account that contribute to the log.

 

[marcus]

Since we're on a new page I should probably recap what the main topic is. Haven't done that for a while.

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

==quote first paragraph==
There is strong theoretical evidence that Black Holes have a finite thermodynamic entropy equal to one quarter the area A of the horizon [1]. Providing a microscopic derivation of the Bekenstein-Hawking entropy

S_BH = A/(4G hbar)

is a major task for a candidate theory of quantum gravity. Loop Gravity [2] has been shown to provide a geometric explanation of the finiteness of the entropy and of the proportionality to the area of the horizon [3]. The microstates are quantum geometries of the horizon [4]. 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. This is achieved in this letter.
==endquote==

Over the past year or so there have been several Loop gravity papers by various authors (Ghosh, Perez, Pranzetti, Frodden, Engle, Noui...) supporting this general conclusion. If it is sustained (and I think Bianchi's treatment of it will be, possibly among others) this will constitute a landmark. AFAIK no other approach to Quantum Gravity has achieved such a result at the equivalent level of generality. In stringy context the 1/4 prefactor was derived only for highly special extreme cases not expected to be observed in nature.
So it would be natural if Bianchi's paper were to occasion an incredulous outcry from some quarters. We'll have to see if that happens. Anyway the story isn't over, Bianchi and Wieland have a followup paper in the works. Others I mentioned (or forgot to mention) may have as well.

A nice choice of units is made in this paper. c = k_B = 1, so that at all times one sees the dependence on G and hbar and can immediately see what the effect of varying them would be. IOW time is measured in meters and temperature is measured in joules. In such units the Planck area is Ghbar so A/Ghbar, as a ratio of areas, is dimensionless (a unitless number) and also, since k_B=1, entropy, which might otherwise be expressed as energy/temperature, turns out to be dimensionless. So the above equation is simply an equality of pure numbers.

 

[marcus]

Additional perspective on the significance of the LQG entropy result can be gleaned from this excerpt at the close of Bianchi's conclusion section.
==quote conclusions, page 5==
The result obtained directly addresses some of the difficulties found in the original Loop Gravity derivation of Black-Hole entropy where the area-ensemble is used [3] and the Immirzi parameter shows up as an ambiguity in the expression of the entropy [20]. Introducing the notion of horizon energy in the quantum theory, we find that the entropy of large black holes is independent from the Immirzi parameter. Quantum gravity corrections to the entropy and the temperature of small black holes are expected to depend on the Immirzi parameter.
==endquote==

 

[marcus]

Now that we have the "Discrete Symmetries" paper (May 2012 Rov+Wilson-Ewing) a natural question to ask about Bianchi's entropy paper is what if any changes would follow from changing over to the proposed S' action?

In "Discrete Symmetries" RWE consider the effect of time and parity reversal on the conventional Holst action S[e,ω] that has so far been the basis of covariant LQG, i.e. of spinfoam dynamic geometry.

They propose two alternative actions, since these are closely related I will just consider one (S') for simplicity. You can look up the other (S") in their paper if you wish.

The classical basis for spinfoam QG is the Holst action. A 4D manifold M equipped with internal Minkowski space M at each point together with a tetrad e (one-forms valued in M) and a connection ω. The conventional Holst action is:
S[e,ω]=∫e^IΛe^JΛ(∗ + 1/γ) F_{I J}

Here the ∗ denotes the Hodge dual. Rovelli and Wilson-Ewing propose a new action S' that uses the signum of det e: s = sign(det e) defined to be zero if det e = 0 and otherwise ±1.

S'[e,ω]=∫e^IΛe^JΛ(s ∗ + 1/γ) F_{I J}

===========
http://arxiv.org/abs/1205.0733
Discrete Symmetries in Covariant LQG
Carlo Rovelli, Edward Wilson-Ewing
(Submitted on 3 May 2012)
We study time-reversal and parity ---on the physical manifold and in internal space--- in covariant loop gravity. We consider a minor modification of the Holst action which makes it transform coherently under such transformations. The classical theory is not affected but the quantum theory is slightly different. In particular, the simplicity constraints are slightly modified and this restricts orientation flips in a spinfoam to occur only across degenerate regions, thus reducing the sources of potential divergences.
8 pages

So what if any effect would the modified simplicity constraints have on the BH entropy results of Bianchi and others? Here is the modified simplicity constraint for S':
K+sγL=0
This seems to conflict with the idea in the Bianchi paper of a γ-simple representation for which K-γL=0
Section IV "Quantum Theory" starting on page 3 of the RWE paper is specifically about this kind of question:

"Let us now study the effect of using the modified simplicity condition on the quantum theory. We refer the readers to [1, 9, 11, 12] for the general construction. In the quantum theory, π_f^IJ is promoted to a quantum operator which is identified as the generator of SL(2, C) over a suitable space formed by SL(2,C) unitary representations. K_f and L_f are then the generators of boosts and rotations respectively...
...Therefore the key effect of the introduction of the sign s is that the quantum theory now includes both positive and negative k representations..."
This seems very interesting--I'm just now trying to understand it.

 

[fzero]

Now that we have the "Discrete Symmetries" paper (May 2012 Rov+Wilson-Ewing) a natural question to ask about Bianchi's entropy paper is what if any changes would follow from changing over to the proposed S' action?

...

"Let us now study the effect of using the modified simplicity condition on the quantum theory. We refer the readers to [1, 9, 11, 12] for the general construction. In the quantum theory, π_f^IJ is promoted to a quantum operator which is identified as the generator of SL(2, C) over a suitable space formed by SL(2,C) unitary representations. K_f and L_f are then the generators of boosts and rotations respectively...
...Therefore the key effect of the introduction of the sign s is that the quantum theory now includes both positive and negative k representations..."
This seems very interesting--I'm just now trying to understand it.

In the latest Bianchi paper, which does not involve state counting, there is no difference. Since H \sim \sum_f K^z_f \sim \sum \gamma j_f &gt;0 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 k_f&gt;0 and k_f&lt;0 faces together.

The states that correspond to the BH entropy calculation can be determined from the usual prescription. We choose a triangulation \Delta 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 k_f&gt;0 faces by N^+_j and k_f&lt;0 faces by N^-_j. If we also include N_0 degenerate faces, the number of states is now

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

There are two constraints, namely

N = N_0 + \sum_j(N^+_j+N^-_j),

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

There is no other constraint on N_0. Having N^-\neq 0 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 N_0=0. Taking the large N limit and then extremizing the entropy subject to the constraints leads to the occupation numbers

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

where

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

The entropy is

S = \frac{\mu A}{8\pi G\hbar \gamma},

which results in

\gamma = \frac{\mu}{2\pi} \sim 0.4382.

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

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

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

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

The only constraint left is the area constraint and only the nondegenerate faces contribute to that. However, we have two unknowns, \mu and N_0. 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 j_1 and j_2 state with a spin j_1+j_2 state and a degenerate face. It is

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].

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

S (N_0+1) - S (N_0) \sim \mu ( j_1+j_2 - (j_1+j_2)) \sim 0.

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]

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!