Does the Immirzi Parameter Determine Black Hole Entropy in Loop Quantum Gravity?

[genneth]

marcus: I just read through that old Ashtekar review paper on isolated horizons; it seems that *all* LQG calculations of entropy uses isolated horizons? Ghosh+Perez even point out that their main contribution is to introduce "quantum hair" by counting punctures of the IH --- so I don't think the use of the isolated horizon is novel here. Certainly, for a classical Schwarzschild BH (the analogue of what Ghosh+Perez looks at) the event horizon should correspond to the IH (at least classically; the former is not well-defined with hbar > 0).

 

[marcus]

Certainly, for a classical Schwarzschild BH (the analogue of what Ghosh+Perez looks at) the event horizon should correspond to the IH (at least classically; the former is not well-defined with hbar > 0).

That is a good point! The earlier papers may not have used IH---they may have used EH. But the point is that the two should correspond.

So intuitively the difference must arise elsewhere---e.g. in their keeping track of the number of punctures.

Thanks for giving this a closer look!

"...it seems that *all* LQG calculations of entropy uses isolated horizons?..."

That's not how I remember it (I can check back to the 1996-1998 papers) but your point makes the issue seem unimportant.

 

[marcus]

I fished up the March 1996 paper of Rovelli, the earliest LQG BH entropy paper I know of, and the earliest that Ghosh Perez cite. It might be interesting to take a look. (I think this was before the concept of IH was defined, but you point out this should not matter.)

http://arxiv.org/abs/gr-qc/9603063
Black Hole Entropy from Loop Quantum Gravity
Carlo Rovelli
(Submitted on 30 Mar 1996)
We argue that the statistical entropy relevant for the thermal interactions of a black hole with its surroundings is (the logarithm of) the number of quantum microstates of the hole which are distinguishable from the hole's exterior, and which correspond to a given hole's macroscopic configuration. We compute this number explicitly from first principles, for a Schwarzschild black hole, using nonperturbative quantum gravity in the loop representation. We obtain a black hole entropy proportional to the area, as in the Bekenstein-Hawking formula.
5 pages

The concept of isolated horizon was, I think, introduced in this 1999 paper:

http://arXiv.org/abs/gr-qc/9905089
Isolated Horizons: the Classical Phase Space
A. Ashtekar, A. Corichi, K. Krasnov
(Submitted on 23 May 1999)
A Hamiltonian framework is introduced to encompass non-rotating (but possibly charged) black holes that are 'isolated'' near future time-like infinity or for a finite time interval. The underlying space-times need not admit a stationary Killing field even in a neighborhood of the horizon; rather, the physical assumption is that neither matter fields nor gravitational radiation fall across the portion of the horizon under consideration. A precise notion of non-rotating isolated horizons is formulated to capture these ideas. With these boundary conditions, the gravitational action fails to be differentiable unless a boundary term is added at the horizon. The required term turns out to be precisely the Chern-Simons action for the self-dual connection. The resulting symplectic structure also acquires, in addition to the usual volume piece, a surface term which is the Chern-Simons symplectic structure. We show that these modifications affect in subtle but important ways the standard discussion of constraints, gauge and dynamics. In companion papers, this framework serves as the point of departure for quantization, a statistical mechanical calculation of black hole entropy and a derivation of laws of black hole mechanics, generalized to isolated horizons. It may also have applications in classical general relativity, particularly in the investigation of of analytic issues that arise in the numerical studies of black hole collisions.
43 pages, 2 figures

I'm not sure (you may know) but the concept of IH may have been refined subsequently. And the definition of IH emended in later papers. I haven't followed it--this is just the earliest reference I can find.

 

[thesamer]

http://arxiv.org/abs/1111.0961"
Immirzi parameter and Noether charges in first order gravity
R. Durka
(Submitted on 3 Nov 2011)
The framework of SO(3,2) constrained BF theory applied to gravity makes it possible to generalize formulas for gravitational diffeomorphic Noether charges (mass, angular momentum, and entropy). It extends Wald's approach to the case of first order gravity with a negative cosmological constant, the Holst modification and the topological terms (Nieh-Yan, Euler, and Pontryagin). Topological invariants play essential role contributing to the boundary terms in the regularization scheme for the asymptotically AdS spacetimes, so that the differentiability of the action is automatically secured. Intriguingly, it turns out that the black hole thermodynamics does not depend on the Immirzi parameter for the AdS-Schwarzschild, AdS-Kerr, and topological black holes, whereas nontrivial modification appears for the AdS-Taub-NUT spacetime.

17 pages, to appear in The Proceedings of "Quantum Theory and Symmetries 7" Prague, Journal of Physics: Conference Series (JPCS)

 

[Demystifier]

A recent calculation suggests that the black-hole entropy in LQG does NOT depend on the Immirzi parameter:
http://xxx.lanl.gov/abs/1107.1320
More precisely, the entropy is A/4 (Bekenstein-Hawking, Immirzi independent) plus a topological Immirzi-dependent term which does not depend on the surface A.

This seems to be a very surprising result, but even more surprising is that Marcus hasn't already drawn our attention to this paper. :-)

I would like to add two info about this paper.

First, it is published in Phys. Rev. Lett.
http://prl.aps.org/abstract/PRL/v107/i24/e241301

Second, today a refutation of a critique of that paper appeared:
http://xxx.lanl.gov/abs/1204.4344

 

[marcus]

A recent calculation suggests that the black-hole entropy in LQG does NOT depend on the Immirzi parameter...

More precisely, the entropy is A/4 (Bekenstein-Hawking, Immirzi independent) plus a topological Immirzi-dependent term which does not depend on the surface A.

This seems to be a very surprising result, but even more surprising is that Marcus hasn't already drawn our attention to this paper. :-)

That was the Ghosh Perez paper and I did call attention to it as soon as it came out. But I did not succeed to raise much attention by calling, that time. :-D.

You might be interested to learn of a followup paper by Frodden Ghosh Perez, that came out in October 2011. I think people are closing in on the right answer--what Loop geometry should say about BH and BH entropy in particular. My hunch is that it is going to turn out that to first order there will be no dependence of S on Immirzi.

Maybe Ghosh Perez result is not right but my hunch is it is in the right direction. Just have to wait and see.
Here is the more recent Frodden Ghosh Perez paper:
http://arxiv.org/abs/1110.4055

 

[marcus]

A calculation posted on arxiv today indicates that in Loop gravity black hole entropy
S = A/4 in general to first order and as suggested earlier does not depend on the Immirzi parameter.

Some earlier posts on this thread from back around July 2011 discussed this possibility.

Today's paper represents the first time the coefficient 1/4 has been derived in general in any type of quantum gravity. (String theory results are for very special "extremal" black holes, not what one expects to find in nature.) So if confirmed, as I expect it will be, this is a landmark paper.

There are situations in Loop gravity when one may want the Immirzi to run with scale, so it's nice not to have it nailed down to one fixed specific value. Ted Jacobson already suggested the desirability of this back in the 2007 as I recall, in a paper about LQG black holes.

So this seems to be coming about. 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

Maybe there is no connection with Jacobson's earlier paper here, really. It just reminded me of it. Jacobson's paper hit my funnybone and I made one or two speculative comments about it when it came out:
https://www.physicsforums.com/showthread.php?t=178710
http://arxiv.org/abs/0707.4026
Renormalization and black hole entropy in Loop Quantum Gravity
Ted Jacobson
7 pages
(Submitted on 26 Jul 2007)

"Microscopic state counting for a black hole in Loop Quantum Gravity yields a result proportional to horizon area, and inversely proportional to Newton's constant and the Immirzi parameter. It is argued here that before this result can be compared to the Bekenstein-Hawking entropy of a macroscopic black hole, the scale dependence of both Newton's constant and the area must be accounted for. The two entropies could then agree for any value of the Immirzi parameter, if a certain renormalization property holds."

Jacobson's reference [15] is a Martin Reuter paper
[15] M. Reuter and J. M. Schwindt, “Scale-dependent metric and causal
structures in quantum Einstein gravity,” JHEP 0701, 049 (2007)
[arXiv:hep-th/0611294].

Ah! I see that Bianchi already made the connection and referred to Jacobson's 2007 paper in his conclusions section--as reference [20].

 

[atyy]

A calculation posted on arxiv today indicates that in Loop gravity black hole entropy
S = A/4 in general to first order and as suggested earlier does not depend on the Immirzi parameter.

Some earlier posts on this thread from back around July 2011 discussed this possibility.

Today's paper represents the first time the coefficient 1/4 has been derived in general in any type of quantum gravity. (String theory results are for very special "extremal" black holes, not what one expects to find in nature.) So if confirmed, as I expect it will be, this is a landmark paper.

There are situations in Loop gravity when one may want the Immirzi to run with scale, so it's nice not to have it nailed down to one fixed specific value. Ted Jacobson already suggested the desirability of this back in the 2007 as I recall, in a paper about LQG black holes.

So this seems to be coming about.


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

Maybe there is no connection with Jacobson's earlier paper here, really. It just reminded me of it. Jacobson's paper hit my funnybone and I made one or two speculative comments about it when it came out:
https://www.physicsforums.com/showthread.php?t=178710
http://arxiv.org/abs/0707.4026
Renormalization and black hole entropy in Loop Quantum Gravity
Ted Jacobson
7 pages
(Submitted on 26 Jul 2007)

"Microscopic state counting for a black hole in Loop Quantum Gravity yields a result proportional to horizon area, and inversely proportional to Newton's constant and the Immirzi parameter. It is argued here that before this result can be compared to the Bekenstein-Hawking entropy of a macroscopic black hole, the scale dependence of both Newton's constant and the area must be accounted for. The two entropies could then agree for any value of the Immirzi parameter, if a certain renormalization property holds."

Jacobson's reference [15] is a Martin Reuter paper
[15] M. Reuter and J. M. Schwindt, “Scale-dependent metric and causal
structures in quantum Einstein gravity,” JHEP 0701, 049 (2007)
[arXiv:hep-th/0611294].

Ah! I see that Bianchi already made the connection and referred to Jacobson's 2007 paper in his conclusions section--as reference [20].

From the section on p5, "partition function and spin foams", isn't this a semi-classical calculation?

 

[Demystifier]

In view of these new papers, what exactly is wrong with older papers which calculate entropy to be Immirzi-dependent?

 

[Physics Monkey]

I too would like to know if any more understanding is available.

I still have some elementary confusions/reservations. For example, looking at Eq. 2 in http://xxx.lanl.gov/pdf/1107.1320v3.pdf it looks to me like the authors have written S = S_{BH} + S_q with S_q almost defined so that S_{BH} is the right semiclassical answer. I don't doubt that the proposal has more content than this, but to the extent that N is proportional to A, then the full entropy is proportional to A and has IP dependence. Is their proposal that the semiclasssical answer gets the entropy wrong by an extensive amount? I also don't know how this connects up with the Bianchi work.

 

[Demystifier]

Perhaps we shall not have a good answer to that question until Rovelli writes a paper on it, because Rovelli seems to be the only guy in the LQG community able to write a paper truly understandable to a wider physics community.