Monday Quantum Gravity Seminar: Black hole entropy in LQG
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).