On the branching of the quasinormal resonances of near-extremal Kerr black holes

Abstract

It has recently been shown by Yang et al. (Phys Rev D 87:041502(R), 2013a; Phys Rev D 88:044047, 2013b) that rotating Kerr black holes are characterized by two distinct sets of quasinormal resonances. These two families of quasinormal resonances display qualitatively different asymptotic behaviors in the extremal (\(a/M\rightarrow 1\)) black-hole limit: the zero-damping modes are characterized by relaxation times which tend to infinity in the extremal black-hole limit (\(\mathfrak {I}\omega \rightarrow 0\) as \(a/M\rightarrow 1\)), whereas the damped modes (DMs) are characterized by non-zero damping rates (\(\mathfrak {I}\omega \rightarrow \) finite-values as \(a/M\rightarrow 1\)). In this paper we refute the claim made by Yang et al. that co-rotating DMs of near-extremal black holes are restricted to the limited range \(0\le \mu \lesssim \mu _{\text {c}}\approx 0.74\), where \(\mu \equiv m/l\) is the dimensionless ratio between the azimuthal harmonic index m and the spheroidal harmonic index l of the perturbation mode. In particular, we use an analytical formula originally derived by Detweiler in order to prove the existence of DMs (damped quasinormal resonances which are characterized by finite\(\mathfrak {I}\omega \) values in the \(a/M\rightarrow 1\) limit) of near-extremal black holes in the \(\mu >\mu _{\text {c}}\) regime, the regime which was claimed by Yang et al. not to contain DMs. We show that these co-rotating DMs (in the regime \(\mu >\mu _{\text {c}}\)) are expected to characterize the resonance spectra of rapidly rotating (near-extremal) black holes with \(a/M\gtrsim 1-10^{-9}\).