LQG and black holes -- back to basics

[naima]

Hi PF

I am reading an old Rovelli paper http://arxiv.org/abs/gr-qc/9603063
Rovelli computes the entropy of a black hole.
He says two things.
a) A state is described with an equivalence class of s-knots.
b) the entropy of a black hole can be computed by counting the number of the intersection of the horizon and the s-knot (an equivalence class)
So when he writes :
Let i = 1...n label such intersections
What is this n? what is this i labelling?

 

[marcus]

Hi Naima, you are right that it is an old paper! I believe it was preliminary with some gaps still to fill in. As I recall this idea for calculating the entropy was not developed, because about the same time the Ashtekar et al paper came out, with what seemed a more complete analysis. In a later paper with Krasnov, Rovelli spoke of having "discarded" this 1996 approach of his in favor of the one in Ashtekar et al. But it could still be interesting.

It is a combinatorial approach where one one considers all the partitions of the area number. the number n is the number of terms in the partition.

Imagine that each leg of a spin network is labeled with an number that can represent an element of area of any surface that leg passes through. The total area is the sum of all those areas of all the legs that "puncture" the surface. It could be a sum of many small terms (large n) or a sum of fewer larger terms (small n).
The number n can range within wide bounds (for a reasonable sized area).
http://arxiv.org/abs/gr-qc/9603063
At the moment I cannot recall how he connects that to the entropy. But you asked about the number n, and that I can say is the number of summands in a given partition of the area quantity A.

 

[naima]

Thank you Marcus.

As a S-knod is an equivalence class of spin networks does it mean that two such networks have the same number of legs?
Could you give a more recent paper which gives the LQG description of the Bekenstein area result ?

 

[marcus]

...As a S-knod is an equivalence class of spin networks does it mean that two such networks have the same number of legs?...

Yes, same number of legs. A diffeomorphism can change the embedding, but it would not change the topology of the graph.
I will try to hunt up some recent papers about LQG treatment of BH.
I will look for papers by Alejandro Perez
also by Gambini and Pullin
also by Eugenio Bianchi
If someone else happens to read this and can think of other LQG authors who have studied BH entropy, please tell us! I enjoy watching the developments in LQG but I certainly do not qualify as expert and cannot be sure to think of every relevant paper. but let's do the ARXIV.ORG search :D

Here is an invited review article, to be a chapter in a book edited by Ashtekar and Pullin
http://arxiv.org/abs/1501.02963
Quantum Geometry and Black Holes
J. Fernando Barbero G., Alejandro Perez
(Submitted on 13 Jan 2015)
We present an overall picture of the advances in the description of black hole physics from the perspective of loop quantum gravity. After an introduction that discusses the main conceptual issues we present some details about the classical and quantum geometry of isolated horizons and their quantum geometry and then use this scheme to give a natural definition of the entropy of black holes. The entropy computations can be neatly expressed in the form of combinatorial problems solvable with the help of methods based on number theory and the use of generating functions. The recovery of the Bekenstein-Hawking law and corrections to it is explained in some detail. After this, due attention is paid to the discussion of semiclassical issues. An important point in this respect is the proper interpretation of the horizon area as the energy that should appear in the statistical-mechanical treatment of the black hole model presented here. The chapter ends with a comparison between the microscopic and semiclassical approaches to the computation of the entropy and discusses a number of issues regarding the relation between entanglement and statistical entropy and the possibility of comparing the subdominant (logarithmic) corrections to the entropy obtained with the help of the Euclidean path integral with the ones obtained in the present framework.
39 pages. Contribution to appear in the World Scientific series "100 Years of General Relativity" edited by A. Ashtekar and J. Pullin

Some additional links:
http://arxiv.org/abs/1309.4563
Statistics, holography, and black hole entropy in loop quantum gravity
Amit Ghosh, Karim Noui, Alejandro Perez

http://arxiv.org/abs/1405.7287
Statistical and entanglement entropy for black holes in quantum geometry
Alejandro Perez

http://arxiv.org/abs/1412.5851
Black holes as gases of punctures with a chemical potential: Bose-Einstein condensation and logarithmic corrections to the entropy
Olivier Asin, Jibril Ben Achour, Marc Geiller, Karim Noui, Alejandro Perez

Naima, my personal opinion is that there is no one unique official way, in LQG, to derive the Bekenstein-Hawking area law for BH entropy. Maybe if you want a "most representative" derivation you could find it in that review article by Perez. However what I also like is that the law is derived in several different ways, sometimes stimulating of different points of view
http://arxiv.org/abs/1211.0522
Black hole entropy from graviton entanglement
Eugenio Bianchi
(Submitted on 2 Nov 2012 (v1), last revised 7 Jan 2013)
We argue that the entropy of a black hole is due to the entanglement of matter fields and gravitons across the horizon. While the entanglement entropy of the vacuum is divergent because of UV correlations, we show that low-energy perturbations of the vacuum result in a finite change in the entanglement entropy. The change is proportional to the energy flux through the horizon, and equals 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 phenomenon is local in nature and applies both to black hole horizons and to cosmological horizons, thus providing a microscopic derivation of the Bekenstein-Hawking area law. The physical mechanism presented relies on the universal coupling of gravitons to the energy-momentum tensor, i.e. on the equivalence principle.
4 pages

http://arxiv.org/abs/1204.5122
Entropy of Non-Extremal Black Holes from Loop Gravity
Eugenio Bianchi