Thermal BEC black holes

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We review some features of BEC models of black holes obtained by means of the HWF formalism. We consider the KG equation for a toy graviton field coupled to a static matter current in spherical symmetry. The classical field reproduces the Newtonian potential generated by the matter source, while the corresponding quantum state is given by a coherent superposition of scalar modes with continuous occupation number. An attractive self-interaction is needed for bound states to form, so that (approximately) one mode is allowed, and the system of N bosons can be self-confined in a volume of the size of the Schwarzschild radius. The HWF is then used to show that the radius of such a system corresponds to a proper horizon. The uncertainty in the size of the horizon is related to the typical energy of Hawking modes: it decreases with the increasing of the black hole mass (larger number of gravitons), in agreement with semiclassical calculations and different from a single very massive particle. The spectrum contains a discrete ground state of energy m (the bosons forming the black hole), and a continuous spectrum with energy ω> m (representing the Hawking radiation and modelled with a Planckian distribution at the expected Hawking temperature). The N-particle state can be collectively described by a single-particle wave-function given by a superposition of a total ground state with energy M = N m and a Planckian distribution for E > M at the same Hawking temperature. The partition function is then found to yield the usual area law for the entropy, with a logarithmic correction related with the Hawking component. The backreaction of modes with ω> m is also shown to reduce the Hawking flux and the evaporation properly stops for vanishing mass.