### Monday Quantum Gravity Seminar: Black hole entropy in LQG

Today instead of our usual QG group meeting we had a proper seminar, as Jacobo Díaz Polo from Universidad de Valencia gave a talk on black hole entropy calculations in Loop Quantum Gravity. Kirill Krasnov, who was one of the pioneers of this research, was a sitting in the audience as a memeber of our group but was also willing to give a hand at answering questions, so it was a unique opportunity to learn the real state of affairs here. Warning: I am not an expert of any kind in LQG, knowing only what I have gathered from reading Rovelli's textbook (but not doing most of the math) and hearing quite a few talks on the subject, specially at the Loops '05 conference. So there may be some details (or perhaps more than that) which I get wrong. However this post is not a "popularization" one like the series on special relativity; nonphysicists are unlikely to gather what this is about from what follows.

The question to answer for a BH entropy calculation in LQG is: In how many ways consistent with a given total area can the spin networks states (which describe spatial geometry in loop quantum gravity) puncture the sphere representing the horizon of the black hole? Each puncture has a two quantum spin numbers, j and m, with possible values of j being 1/2, 1, 3/2.... and possible values of m for each j going from -j to j. The logaritm of the total number of states gives the entropy of the black hole, which should agree with the semiclassical Bekenstein-Hawking entropy (one quarter of the area in Planck units) . The diffeomorphism constraint, which represents invariance of GR under spatial diffeomorphisms, is implemented via the requirement that states with the same number of punctures in the horizon are considered equivalent. The Hamiltonian constraint, which should encode the dynamics of the theory, is not implemented at all. I raised my hand at this point of the seminar and asked why; I thought the answer would be "because the black hole is stationary" or something like that, and I wanted to inquire further on how is the "stationarity" defined at the quantum level. But the answer Kirill gave was the Hamiltonian constraint describes the geometry of the bulk (as being a solution to the quantized version of the Einstein equations, I imagine) and we are looking only at the geometry of the horizon surface. We are assuming therefore that of all the states that we can imagine puncturing this surface, "most" of them will be also be extendable to solutions of the Hamiltonian constraint on the bulk.

When we get to the actual counting of states, Jacobo said he would give a lower bound and an upper bound to the total number of states. The lower bound comes from assuming that all the j numbers are 1/2, the lowest possible spin. I couldn't follow exactly how the upper bound was found, it had something to do with how the density of states grows with the area... but the point is that the upper bound gives exactly the same quantity as the lower bound, and therefore this quantity is the number of states. Its logaritm is the entropy and is equal to: ln(2)/(4 Pi Sqrt(3) g) times the classical area of the black hole in Planck units. Here g is the Immirizi parameter, usually represented by a gamma, and Sqrt(3) means square root of 3. (I should get one of those nice math display programs that some other physics bloggers have if I'm going to write posts like this one!) Comparison with the Bekenstein-Hawking entropy fixes the Immirizi paramenter as ln(2)/(Pi Sqrt(3)).

This is the result Kirill and his collaborators had found in their original paper, but since then their result has been submitted to criticism. My supervisor Jorma Louko made a question about this, how does this result contrast with the different one found in this paper. Between Kirill and Jacobo they explained that different definitions can be given of "number of states that give a same total area". The calculation made originally by Kirill et. al. assumed one of those definitions; under it, the 1/2 value for the spins dominates over all others for large values of the area, and therefore it is justified to consider only states with spin 1/2, which is essentially what Jacobo had presented in the talk. Under other definition this is no longer true, higher spins must be taken into account, and the result for the entropy (and consequently the fixing of the Immirizi parameter) are different; g turns out now to be defined by an ugly trascendental equation.

Appearently there is no agreement on which is the correct counting method. I had to leave at this point because I had to get to a class where I am demostrating (answering questions of undergraduates on their classwork), which was a pity because although the talk had ended already I would have liked to go on discussing these matters. I would specially like to know if the uncertainity on the proper counting procedure may be linked to the lack of knowledge on how to impose the Hamiltonian constraint, and the consequent decision to ignore it. (I remember now overhearing John Baez say something that could be interpreted in this direction in a conversation on the subject at the Loops '05 conference).

The question to answer for a BH entropy calculation in LQG is: In how many ways consistent with a given total area can the spin networks states (which describe spatial geometry in loop quantum gravity) puncture the sphere representing the horizon of the black hole? Each puncture has a two quantum spin numbers, j and m, with possible values of j being 1/2, 1, 3/2.... and possible values of m for each j going from -j to j. The logaritm of the total number of states gives the entropy of the black hole, which should agree with the semiclassical Bekenstein-Hawking entropy (one quarter of the area in Planck units) . The diffeomorphism constraint, which represents invariance of GR under spatial diffeomorphisms, is implemented via the requirement that states with the same number of punctures in the horizon are considered equivalent. The Hamiltonian constraint, which should encode the dynamics of the theory, is not implemented at all. I raised my hand at this point of the seminar and asked why; I thought the answer would be "because the black hole is stationary" or something like that, and I wanted to inquire further on how is the "stationarity" defined at the quantum level. But the answer Kirill gave was the Hamiltonian constraint describes the geometry of the bulk (as being a solution to the quantized version of the Einstein equations, I imagine) and we are looking only at the geometry of the horizon surface. We are assuming therefore that of all the states that we can imagine puncturing this surface, "most" of them will be also be extendable to solutions of the Hamiltonian constraint on the bulk.

When we get to the actual counting of states, Jacobo said he would give a lower bound and an upper bound to the total number of states. The lower bound comes from assuming that all the j numbers are 1/2, the lowest possible spin. I couldn't follow exactly how the upper bound was found, it had something to do with how the density of states grows with the area... but the point is that the upper bound gives exactly the same quantity as the lower bound, and therefore this quantity is the number of states. Its logaritm is the entropy and is equal to: ln(2)/(4 Pi Sqrt(3) g) times the classical area of the black hole in Planck units. Here g is the Immirizi parameter, usually represented by a gamma, and Sqrt(3) means square root of 3. (I should get one of those nice math display programs that some other physics bloggers have if I'm going to write posts like this one!) Comparison with the Bekenstein-Hawking entropy fixes the Immirizi paramenter as ln(2)/(Pi Sqrt(3)).

This is the result Kirill and his collaborators had found in their original paper, but since then their result has been submitted to criticism. My supervisor Jorma Louko made a question about this, how does this result contrast with the different one found in this paper. Between Kirill and Jacobo they explained that different definitions can be given of "number of states that give a same total area". The calculation made originally by Kirill et. al. assumed one of those definitions; under it, the 1/2 value for the spins dominates over all others for large values of the area, and therefore it is justified to consider only states with spin 1/2, which is essentially what Jacobo had presented in the talk. Under other definition this is no longer true, higher spins must be taken into account, and the result for the entropy (and consequently the fixing of the Immirizi parameter) are different; g turns out now to be defined by an ugly trascendental equation.

Appearently there is no agreement on which is the correct counting method. I had to leave at this point because I had to get to a class where I am demostrating (answering questions of undergraduates on their classwork), which was a pity because although the talk had ended already I would have liked to go on discussing these matters. I would specially like to know if the uncertainity on the proper counting procedure may be linked to the lack of knowledge on how to impose the Hamiltonian constraint, and the consequent decision to ignore it. (I remember now overhearing John Baez say something that could be interpreted in this direction in a conversation on the subject at the Loops '05 conference).

## 2 Comments:

The following are some comments which hopefully will make your (good) posting even better.

You say: The diffeomorphism constraint, which represents invariance of GR under spatial diffeomorphisms, is implemented via the requirement that states with the same number of punctures in the horizon are considered equivalent.

There is more to say on this. Because the punctures are labelled by spins j, not only the total number of punctures matters as the "diffeomorphism invariant" data, but also the labels, the spins. In other words, what matters in the counting is a list

{j_1, ..., j_p}

where p is the total number of punctures. The order of spins in the list is not important, i.e. lists that only differ by a permutation are considered the same data.

You say: Between Kirill and Jacobo they explained that different definitions can be given of "number of states that give a same total area".

To be more precise, different definitions can be given of (surface) quantum states that has to be counted when computing the entropy. In one possible definition the surface states are specified by their (j,m) labels. One then counts lists

{j_1, ..., j_p}

that are unordered (as explained above). Each such set of labels has multiplicity

Prod (2j+1)

which in addition has to be corrected by the condition that

Sum m = 0 (mod k)

where k is the level of the associated CS theory (roughly the horizon area in Planck units).

This (quite reasonable) counting problem is what was dealt with in the original

paper.

However, as a detailed analysis of the quantization presented in paper revealed, an extra label appears in the description of the surface states. Thus, in addition to the quantum numbers (j,m), one has to introduce the label "a" at each puncture. This label refers to the data of U(1) CS theory on the surface, and specifies the holonomy of the U(1) connection around the puncture. The "a" data must be compatible with the (j,m) data in the sense that

a = - 2m (mod k)

at every puncture and that moreover

Sum a = 0 (mod k)

The question is how much of the data (j,m,a) should be considered as specifying the BH state, and therefore counted to get the BH entropy. The fact that there are any ambiguities as to this was not appreciated when the paper paper was written. The counting presented in this paper still counted the states as specified by the lists (j,m) such that an "admissible" set of a's exists. The issue was revisited in paper which showed that there is another possibility. Namely, instead of counting the sets (j,m) one may decide that it is the a's that specify the surface state, and count sets of a's such that there is a set of compatible (j,m). As was shown in this paper, this leads to a different counting. This was presented as pointing out "a mistake" made in ABK paper. I would like to emphasize, however, that Domagala-Lewandowski counting is simply a counting of different sets with a different result. It has very little to do with the counting in ABK paper. Ironically, the Domogala-Lewandowski counting is very similar to the "old" entropy counting in loop quantum gravity that was proposed in work, see formula (9) in this paper. This "old" counting was later discarded in favor of the "new" one in paper for the reason that the sets of j's counted were treated as distinguishable. It is quite ironic that the entropy calculations in the framework of loop quantum gravity are now back to where they started many years ago.

To summarize, there are several possibilities as to what should be counted to get the BH entropy. A good discussion as to what the other possibilities are is presented in

paper by Polychronakos. As is clear from this discussion, the issue is far from being settled and more work is needed in this direction.

By Kirill Krasnov, at 5:32 PM, February 21, 2006

Thanks for commenting, Kirill. That made things clearer. By the way, I can't click on the links you provide, so unless you haven't that problem (and the fault is my PC's), could you please give the links again?

Just some random questions as they come to my mind now. Assuming as hypothesis that LQG is a correct theory, the Immirizi parameter must have some definite value. Therefore there is (at most!) one way of counting the surface states that gives agreement with the Bekenstein-Hawking entropy. What kind of principle would single

thatcounting method as the one that gives the semiclassical entropy? Do you expect this principle to be buried in the details of the semiclassical limit of the full quantum theory (e.g., there are many correct ways to define the number of states of a surface at the quantum level, but if one could take the semiclassical limit correctly one would see that of all the possible "entropies" definable as "log number of states" there is only one which has the thermodynamical properties discovered by Hawking, regulating the BH's exchange of heat, etc.)? Or do you think there is a "principled" argument which one could discover a priori without knowing how to take the semiclassical limit and which singles out one of the countings as the "correct" one?Another question, are they any ideas around that might provide an independent fixing of the value of gammma? I vaguely remember reading about some, but I can't remember where and I might be mistaken.

By Alejandro, at 6:29 PM, February 21, 2006

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