Now there has been a recent paper on the arXiv by Eugenio Bianchi
Entropy of Non-Extremal Black Holes from Loop Gravity
By Eugenio Bianchi
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.