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Dipartimento di Fisica e Astronomia, Alma Mater Università di Bologna, via Irnerio 46, 40126 Bologna, Italy


Istituto Nazionale di Fisica Nucleare I.N.F.N., Sezione di Bologna, viale Berti Pichat 6-2, 40127 Bologna, Italy


Institute of Space Science, Atomistilor 409, 077125 Magurele, Ilfov, Romania


Author to whom correspondence should be addressed.

Academic Editors: Remo Garattini and Kevin H. Knuth

Abstract We review some features of Bose–Einstein condensate BEC models of black holes obtained by means of the horizon wave function formalism. We consider the Klein–Gordon equation for a toy graviton field coupled to a static matter current in a spherically-symmetric setup. 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 a continuous occupation number. An attractive self-interaction is needed for bound states to form, the case in which one finds 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 horizon wave function formalism 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, resulting in agreement with the semiclassical calculations and which does not hold for a single very massive particle. The spectrum of these systems has two components: 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 modeled with a Planckian distribution at the expected Hawking temperature. Assuming the main effect of the internal scatterings is the Hawking radiation, 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 = Nm and Entropy 2015, 17 6894 a Planckian distribution for E > M at the same Hawking temperature. This can be used to compute the partition function and to find the usual area law for the entropy, with a logarithmic correction related to the Hawking component. The backreaction of modes with ω > m is also shown to reduce the Hawking flux. The above corrections suggest that for black holes in this quantum state, the evaporation properly stops for a vanishing mass. View Full-Text

Keywords: black holes; horizon wave function; Hawking radiation; generalized uncertainty principle black holes; horizon wave function; Hawking radiation; generalized uncertainty principle

Author: Roberto Casadio 1,2,* , Andrea Giugno 1,2, Octavian Micu 3 and Alessio Orlandi 1,2



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