Quasinormal spectrum and quantization of charged black holes

Abstract

Black-hole quasinormal modes have been the subject of much recent attention, with the hope that these oscillation frequencies may shed some light on the elusive theory of quantum gravity. We study analytically the asymptotic quasinormal spectrum of a charged scalar field in the (charged) ReissnerNordström spacetime. We find an analytic expression for these black-hole resonances in terms of the black-hole physical parameters: its Bekenstein-Hawking temperature TBH , and its electric potential Φ. We discuss the applicability of the results in the context of black-hole quantization. In particular, we show that according to Bohr's correspondence principle, the asymptotic resonance corresponds to a fundamental area unit ∆A = 4h ln 2.Everything in our past experience in physics tell us that general relativity and quantum theory are approximations, special limits of a single, universal theory. However, despite the flurry of research in this field we still lack a complete theory of quantum gravity. In many respects the black hole plays the same role in gravitation that the atom played in the nascent of quantum mechanics [1]. It is therefore believed that black holes may play a major role in our attempts to shed light on the nature of a quantum theory of gravity.The quantization of black holes was proposed long ago by Bekenstein [2,3], based on the remarkable observation that the horizon area of a non-extremal black hole behaves as a classical adiabatic invariant. In the spirit of the Ehrenfest principle [4] -any classical adiabatic invariant corresponds to a quantum entity with a discrete spectrum, and based on the idea of a minimal increase in black-hole surface area [2], Bekenstein conjectured that the horizon area of a quantum black hole should have a discrete spectrum of the formwhere γ is a dimensionless constant, andis the Planck length (we use units in which G = c =h = 1 henceforth). This type of area quantization has since been reproduced based on various other considerations (see e.g., [5] for a detailed list of references). In order to determine the value of the coefficient γ, Mukhanov and Bekenstein [6][7][8] have suggested, in the spirit of the Boltzmann-Einstein formula in statistical physics, to relate g n ≡ exp[S BH (n)] to the number of the black hole microstates that correspond to a particular external macro-state, where S BH is the black-hole entropy. In other words, g n is the degeneracy of the nth area eigenvalue. Now, the thermodynamic relation between black-hole surface area and entropy, S BH = A/4h, can be met with the requirement that g n has to be an integer for every n only whenwhere k is some natural number.Identifying the specific value of k requires further input. This information may emerge by applying Bohr's correspondence principle to the (discrete) quasinormal mode (QNM) spectrum of black holes [9]. Gravitational waves emitted by a perturbed black hole are dominated by this 'quasinormal ringing', damped oscillations with a discrete spectrum (see e.g., [10] for a detailed review). At late...

Charged reflecting stars supporting charged massive scalar field configurations

“…for the static charged massive scalar field into the characteristic Klein-Gordon wave equation [50][51][52][53][54][55][56][57] …”

Spectroscopy of a Reissner–Nordström black hole via an action variable

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Quantum-gravity fluctuations and the black-hole temperature

“…It is well known that a Schwarzschild black hole is characterized by a discrete spectrum of gravitational resonances [8][9][10] with the fundamental asymptotic frequency [7,11] …”