Renormalization and black hole entropy in Loop Quantum Gravity

Microscopic state counting for a black hole in Loop Quantum Gravity yields a result proportional to horizon area, and inversely proportional to Newton’s constant and the Immirzi parameter. It is argued here that before this result can be compared to the BekensteinHawking entropy of a macroscopic black hole, the scale dependence of both Newton’s constant and the area must be accounted for. The two entropies could then agree for any value of the Immirzi parameter, if a certain renormalization property holds. The number of microscopic states of a black hole has been computed in Loop Quantum Gravity (LQG), in the state space of spin networks. The result for the entropy of a black hole with horizon area A is SLQG = b γ A ~G , (1) where b is a numerical constant and γ is the Immirzi parameter. These calculations have a long and continuing history (see for example [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] and for reviews [11, 12, 13]), including some controversy over the correct evaluation of the number of states. The results differ only in the value of b however (unless states related by surface diffeomorphisms are identified, as has discussed for example in [14]). In addition to the case of spherically symmetric, static black holes, the result (1) has been shown to hold, with the same value of b, in the presence of scalar, Maxwell, and Yang-Mills fields [15] E-mail: [email protected]