Black Hole Horizons and Thermodynamics: A Quantum Approach

Abstract

We focus on quantization of the metric of a black hole restricted to the Killing horizon with universal radius r0. After imposing spherical symmetry and after restriction to the Killing horizon, the metric is quantized employing the chiral currents formalism. Two "components of the metric" are indeed quantized: The former behaves as an affine scalar field under changes of coordinates, the latter is instead a proper scalar field. The action of the symplectic group on both fields is realized in terms of certain horizon diffeomorphisms. Depending on the choice of the vacuum state, such a representation is unitary. If the reference state of the scalar field is a coherent state rather than a vacuum, spontaneous breaking of conformal symmetry arises and the state contains a Bose-Einstein condensate. In this case the order parameter fixes the actual size of the black hole with respect to r0. Both the constructed state together with the one associated with the affine scalar are thermal states (KMS) with respect to Schwarzschild Killing time when restricted to half horizon. The value of the order parameter fixes the temperature at the Hawking value as well. As a result, it is found that the quantum energy and entropy densities coincide with the black hole mass and entropy, provided the universal parameter r0 is suitably chosen, not depending on the size of the actual black hole in particular.