Asymptotic quasinormal modes of Reissner-Nordström and Kerr black holes

Abstract

According to a recent proposal, the so-called Barbero-Immirzi parameter of loop quantum gravity can be fixed, using Bohr's correspondence principle, from a knowledge of highly damped black hole oscillation frequencies. Such frequencies are rather difficult to compute, even for Schwarzschild black holes. However, it is now quite likely that they may provide a fundamental link between classical general relativity and quantum theories of gravity. Here we carry out the first numerical computation of very highly damped quasinormal modes (QNM's) for charged and rotating black holes. In the Reissner-Nordstr\"om case QNM frequencies and damping times show an oscillatory behavior as a function of charge. The oscillations become faster as the mode order increases. At fixed mode order, QNM's describe spirals in the complex plane as the charge is increased, tending towards a well defined limit as the hole becomes extremal. Kerr QNM's have a similar oscillatory behavior when the angular index $m=0.$ For $l=m=2$ the real part of Kerr QNM frequencies tends to $2\ensuremath{\Omega},$ $\ensuremath{\Omega}$ being the angular velocity of the black hole horizon, while the asymptotic spacing of the imaginary parts is given by $2\ensuremath{\pi}{T}_{H}.$