Thermal BEC black holes

TL;DR

This paper models black holes as Bose-Einstein condensates of gravitons using the HWF formalism, showing how quantum features relate to classical horizons, Hawking radiation, and entropy corrections.

Contribution

It introduces a BEC model of black holes with a quantum superposition of scalar modes, linking horizon size uncertainty to Hawking modes and deriving the area law with corrections.

Findings

Horizon radius matches the Schwarzschild radius in the BEC model.

Hawking radiation spectrum is Planckian at the Hawking temperature.

Entropy follows the area law with a logarithmic correction.

Abstract

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 $\omega > m$ (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 $\omega > m$ is also shown to reduce the Hawking flux and the evaporation properly stops for vanishing mass.