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Entropy 2015, 17(10), 6893-6924; doi:10.3390/e17106893

Thermal BEC Black Holes

1
Dipartimento di Fisica e Astronomia, Alma Mater Università di Bologna, via Irnerio 46, 40126 Bologna, Italy
2
Istituto Nazionale di Fisica Nucleare (I.N.F.N.), Sezione di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy
3
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
Received: 9 September 2015 / Revised: 6 October 2015 / Accepted: 9 October 2015 / Published: 15 October 2015
(This article belongs to the Special Issue Entropy in Quantum Gravity and Quantum Cosmology)
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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
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Casadio, R.; Giugno, A.; Micu, O.; Orlandi, A. Thermal BEC Black Holes. Entropy 2015, 17, 6893-6924.

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