Spin networks describing BH have a central node with label taken from an intertwiner Hilbertspace of high dimension (corresp. to hole entropy). Horizon area corresponds to the network links that pass through the horizon. The horizon Hilbertspace turns out to have the same high dimension (again corresponding to the hole entropy.) It was found in 2009 that Chern Simons theory enters usefully here---triggering publication of a followup paper by Krasnov and Rovelli.(adsbygoogle = window.adsbygoogle || []).push({});

http://arxiv.org/abs/0905.3168

Black hole entropy and SU(2) Chern-Simons theory

Jonathan Engle, Karim Noui, Alejandro Perez

4 pagesPhysical Review Letters, additional detail in longer http://arxiv.org/abs/1006.0634

(Submitted on 19 May 2009)

"Black holes in equilibrium can be defined locally in terms of the so-called isolated horizon boundary condition given on a null surface representing the event horizon. We show that this boundary condition can be treated in a manifestly SU(2) invariant manner. Upon quantization, state counting is expressed in terms of the dimension of Chern-Simons Hilbert spaces on a sphere with marked points. Moreover, the counting can be mapped to counting the number of SU(2) intertwiners compatible with the spins that label the defects. The resulting BH entropy is proportional to a_{H}with logarithmic corrections Δ S=-3/2 log a_{H}. Our treatment from first principles completely settles previous controversies concerning the counting of states."

http://arxiv.org/abs/0905.4916

Black holes in full quantum gravity

Kirill Krasnov, Carlo Rovelli

5 pagesClassical and Quantum Gravity

(Submitted on 29 May 2009)

"Quantum black holes have been studied extensively in quantum gravity and string theory, using various semiclassical or background dependent approaches. We explore the possibility of studying black holes in the full non-perturbative quantum theory, without recurring to semiclassical considerations, and in the context of loop quantum gravity. We propose a definition of a quantum black hole as the collection of the quantum degrees of freedom that do not influence observables at infinity. From this definition, it follows thatfor an observer at infinity a black hole is described by an SU(2) intertwining operator. The dimension of the Hilbert space of such intertwiners grows exponentially with the horizon area. These considerations shed some light on the physical nature of the microstates contributing to the black hole entropy. In particular, it can be seen that the microstates being counted for the entropy have the interpretation of describing different horizon shapes. The space of black hole microstates described here is related to the one arrived at recently by Engle, Noui and Perez, and sometime ago by Smolin, but obtained here directly within the full quantum theory."

Then in June 2010 a longer piece by Engle et al further developing the idea:

http://arxiv.org/abs/1006.0634

Black hole entropy from an SU(2)-invariant formulation of Type I isolated horizons

Jonathan Engle, Karim Noui, Alejandro Perez, Daniele Pranzetti

30 pages, 1 figure

(Submitted on 3 Jun 2010)

"A detailed analysis of the spherically symmetric isolated horizon system is performed in terms of the connection formulation of general relativity. The system is shown to admit a manifestly SU(2) invariant formulation where the (effective) horizon degrees of freedom are described by anSU(2) Chern-Simonstheory. This leads to a more transparent description of the quantum theory in the context of loop quantum gravity and modifications of the form of the horizon entropy."

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# Chern-Simons, LQG black hole entropy, and central intertwiner

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