Monday, May 14, 2012

A note on the black hole entropy in LQG

If you know anything about Loop Quantum Gravity, you know that people working on it suffer from an inferiority complex because, when counting black hole microstates, they only get the Bekenstein-Hawking entropy of black holes up to a factor. This factor, known as the Barbero-Immirzi parameter, enters the theory through the quantization condition and then has to be fixed to match the correct semi-classical result.

Now there has been a recent paper on the arXiv by Eugenio Bianchi
which addresses the issue, or at least that's what I thought when I first read the abstract. Bianchi derives the black hole entropy in a spin-foam formalism and finds the usual Bekenstein-Hawking entropy - without any additional factors.

I've been scratching my head over this paper for a while now. The purpose of this blogpost is twofold: First to draw your attention to what is potentially an important contribution in the field, check. And second, I want to offer you my interpretation of that finding, and hope some reader who knows more about LQG than I will correct me when I'm wrong.

The Bekenstein-Hawking entropy is not a quantum gravitational result. One finds that black holes have a temperature by considering a quantum field (usually a massless scalar field but that doesn't matter) in the classical background geometry of a black hole. If one has the mass of the black hole, one can identify it with the total energy and integrate dE = TdS to get the entropy. The validity of this argument breaks down at the Planck scale, but that's not the regime of interest here. One can also abuse the Unruh effect to argue that black holes have a temperature, same result.

If one has some candidate theory for quantum gravity, one can ideally go and compute the microstates of a black hole. In LQG, areas and volumes are quantized in multiples of the Barbero-Immirzi parameter. Even without knowing the details, this leads one to expect that the number of possible microstates depends on that parameter. Thus, the number of microstates will generically not reproduce the Bekenstein-Hawking entropy, unless the parameter is chosen suitably. Now what I would conclude at this point is that the Bekenstein-Hawking entropy is not a measure for the microstates of the black holes. Alas, most of my colleagues seem to believe it is, especially the string theorists, and there's then the origin of the loop quantized inferiority complex.

So, with that avant-propos, why does Bianchi get a result different from the previous LQG results, a result that reproduces the Bekenstein-Hawking entropy?

Well, it looks to me like that's because he doesn't count the black hole microstates to begin with. He considers an observer in the black hole background with a two-level detector and finds the temperature, then S=E/T, and no Barbero-Immirzi parameter ever appears because it's a kinetic effect that has nothing to do with the quantization of areas and volumes. I am greatly simplifying and omitting many details, but that is what it looks like to me.

It is good to see this can be done by constructing the worldline of the observer in the spin-network and express the acceleration and so on in the proper kinetic formalism; that is an interesting calculation in its own right. But does that solve the problem with the black hole entropy in LQG?

In my opinion, it doesn't. In fact, it only manifests the problem further. Now not only is the microstate counting inconsistent with the Bekenstein-Hawking entropy unless a free parameter of the model is fixed appropriately, but the kinetic result is inconsistent with the microstate counting within the same theory.

Truth be said, this paper has created more questions for me than it has answered. I am wondering now for example, what really is the observer fundamentally? It ought to be described by quantum fields. But these quantum fields have a quantization prescription. And that quantization prescription, not having anything to do with gravity, doesn't have an additional parameter in it. That after all is why the Bekenstein-Hawking result is reproduced, because it doesn't have anything to do with the quantization of gravity. But the fields interact with the geometry, so how can they have a different quantization prescription?

If somebody can point me into the direction of a helpful reference or a bottle of ibuprofen, please dump in the comments.

27 comments:

Mitchell said...

Long discussion of the paper here: http://www.physicsforums.com/showthread.php?t=599812

Bee said...

Thanks. Ah. Well, at least I'm not the only one scratching her head over that ;o)

Nidnus Rep said...

Hi Bee,

What's your opinion about the recent Ashoke Sen paper claiming contradictions with LQG results?

Thanks for a nice blog,

P

Arun said...

Hi Bee,

Now what I would conclude at this point is that the Bekenstein-Hawking entropy is not a measure for the microstates of the black holes.

Interesting thought - so the string theory results only establish that B-H entropy is only sometimes a measure of microstates of a black hole? I was under the impression that stringy results were rather generic.

Thanks!
-Arun

Bee said...

Hi Nidnus,

I haven't read it. Best,

B.

Bee said...

Dear Arun,

What do you mean with "generic"? They're generic for what under which assumptions? The result is generic in that any approach to qg that properly reproduces the semi-classical limit better finds the correct BH entropy by an argument similar to Hawking's. I don't see why the result that the microstate counting leads to the same result is generic to whatever approach to qg you try. It seems non-trivial to me, as it depends on the way the gravitational degrees of freedom are organized (or desorganized in that case). Best,

B.

Uncle Al said...

Planck domain observation self-gravitates to a black hole. As local gravitation implodes, local space goes elliptic - local volume explodes. Perhaps infinities balance to a finite event horizon with no core singularity. Gravitation need not be quantized because it isn’t.

John Baez said...
This comment has been removed by the author.
John Baez said...

...it only manifests the problem further. Now not only is the microstate counting inconsistent with the Bekenstein-Hawking entropy unless a free parameter of the model is fixed appropriately, but the kinetic result is inconsistent with the microstate counting within the same theory."

That sounds like tremendous progress, then! If a theory suffers from some sort of internal problem unless a parameter is tuned to a specific value, that's a kind of wedge one can use to understand the theory better - and either improve it, or destroy it completely.

This has always been true of the Immirzi parameter problem in loop quantum gravity, but you're making it sound that Eugenio Bianchi has tightened the noose in a potentially very useful way.

Arun said...

So, is it the idea that the B-H entropy is a property of the classical geometry of the black hole and has is at best indirectly connected with quantum microstates?

Bee said...

Hi John,

Yes, I see what you mean. You have a point there. Best,

B.

Zephir said...

/**.if somebody can point me into the direction of a helpful reference or a bottle of ibuprofen, please dump in the comments...*/
I'm not dealing the heparotoxic drugs, but I'd like to point to Black hole complementarity, which essentially makes the black hole entropy incalculable in unique way (the Mathur's fuzzball concept accounts to this problem partially too). In particular, the Bekenstein-Hawking's model accounts to entropy of black hole from extrinsic perspective, whereas the LQG model are applying the intrinsic perspective - well, at least partially.

Bee said...

Hi Arun,

The BH entropy is a property of that part of the field quanta that go to asymptotic infinity. It has a priori nothing to do neither with the geometric degrees of freedom (it does of course have something to do with the causal structure), nor with the matter that formed the black hole. Leaving aside the question what happens to the matter, making a connection between the quantum geometric degrees of freedom and the process of evaporation induced by the BH temperature is thus nontrivial. Best,

B.

Plato Hagel said...
This comment has been removed by the author.
Plato Hagel said...

Hi Bee,

What is conformal field theory explaining then?

Best,

Bee said...

Hi Plato,

Sorry, but I don't understand the question. Best,

B.

Plato Hagel said...

Hi Bee,

Below Planck scale there is no geometry that explains what is happening inside the interior of the blackhole, yet, we may say there is a description process that allows us to see "the thing"(Lee Smolin)yet there is specifics(Baez and Motl) why the paper linked article does not work? Why does it not work exactly?

Best

Plato Hagel said...

Just an update for today here

While one tries to solidify a chance for implementing a process and examination can linked article do better then what already exists?

Best,

Arun said...

So, Bee, I suspect you are not tremendously impressed by the various string theory calculations of black hole entropy and the leading order corrections to that.

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.

Bee said...

Hi Plato,

Well, it does work. Essentially, the purpose of my post was to explain why it works. Best,

B.

Bee said...

Hi Arun,

That is correct. Best,

B.

A. Mikovic said...

Bianchi obtains the entropy not by counting the microstates, but by deriving the temperature of the horizon. He derives this temperature by idetifying 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.

Bee said...

Hi Aleksandar,

"Bianchi obtains the entropy not by counting the microstates..."

That's what I wrote, no? Best,

B.

Plato Hagel said...

Hi Bee,

Ya okay it works...John sees new hope(?)....what phenomenology in implementation can you apply to this case? That is the true test isn't it?

Best,

Plato Hagel said...

Perhaps something in the Crab Nebula that Fermi has picked up? :)

Quantum Geometry said...

I think, to have a sound explanation of what Bekenstein argued to explain that S proportional to A ,one needs to have a microscopic theory of the geometric degrees of freedom. However, to make it thermal one needs input from semiclassical gravity and that can not fixed from the microscopic theory itself. So, entropy being lack of information, one has to have a description of underlying microscopic degrees of freedom which are not accessible in the classical theory. To explain Bekenstein-Hawking entropy to be what it is, one needs both non-perturbative quantum gravity and semi-classical gravity. Otherwise, entropy and area of a black hole will continue to be analogous but not equivalent. This is all my personal opinion.

However, I do not support the fact that Bianchi has calculated black hole entropy from LQG without fixing the `notorious' parameter.