Abstract
Inspired by the recent conjecture that black holes are condensates of gravitons, we investigate a simple model for the black hole degrees of freedom that is consistent both from the point of view of Quantum mechanics and of General Relativity. Since the two perspectives should “converge” into a unified picture for small, Planck size, objects, we expect our construction to be a useful step for understanding the physics of microscopic, quantum black holes. In particular, we show that a harmonically trapped condensate gives rise to two horizons, whereas the extremal case (corresponding to a remnant with vanishing Hawking temperature) is not contained in the spectrum.
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Notes
- 1.
We shall use units such that \(c=1\), \(\hbar =\ell _\mathrm{p}\,m_\mathrm{p}\) and the Newton constant \(G_\mathrm{N}=\ell _\mathrm{p}/m_\mathrm{p}\).
- 2.
This is nothing but the Newton oscillator, which would correspond to a homogenous BEC distribution in the Newtonian approximation.
- 3.
Note we have already integrated out the angular coordinates.
- 4.
The squared length \(\theta \) should not be confused with one of the angular coordinates of the previous expressions. Also, note \(\rho \) has already been integrated over the angles.
- 5.
Note that for vanishing impact parameter, one does not expect any emission of classical gravitational waves.
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Acknowledgments
This work is supported in part by the European Cooperation in Science and Technology (COST) action MP0905 “Black Holes in a Violent Universe”.
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Orlandi, A., Casadio, R. (2016). Quantum Harmonic Black Holes. In: Nicolini, P., Kaminski, M., Mureika, J., Bleicher, M. (eds) 1st Karl Schwarzschild Meeting on Gravitational Physics. Springer Proceedings in Physics, vol 170. Springer, Cham. https://doi.org/10.1007/978-3-319-20046-0_32
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