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

It has recently been shown by Yang. et. al. [Phys. Rev. D {\bf 87}, 041502(R) (2013)] 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 ($a/M\to 1$) black-hole limit: The zero-damping modes (ZDMs) are characterized by relaxation times which tend to infinity in the extremal black-hole limit ($\Im\omega\to 0$ as $a/M\to 1$), whereas the damped modes (DMs) are characterized by non-zero damping rates ($\Im\omega\to$ finite-values as $a/M\to 1$). 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 $0\leq \mu\lesssim\mu_{\text{c}}\approx 0.74$, 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 $\Im\omega$ values in the $a/M\to 1$ 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 damped modes. 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 $a/M\gtrsim 1-10^{-9}$.


I. INTRODUCTION
Perturbed black holes display a unique pattern of damped oscillations, known as quasinormal resonances, which characterize the relaxation phase of the black-hole spacetime. The spectrum of quasinormal resonances reflects the physical parameters (such as mass, charge, and angular momentum) of the black-hole spacetime.
The complex quasinormal resonances correspond to perturbation fields which propagate in the black-hole spacetime with the physically motivated boundary conditions of purely outgoing waves at spatial infinity and purely ingoing waves crossing the black-hole horizon [1]. These boundary conditions single out a discrete set, {ω QNM (n; m, l)} n=∞ n=0 , of complex black-hole resonances for each perturbation mode (here m and l are the azimuthal harmonic index and the spheroidal harmonic index of the wave field, respectively).
In a very interesting paper, Yang et. al. [2] have recently studied numerically the quasinormal spectrum of nearextremal (rapidly-rotating) Kerr black holes. The authors of [2] have reached the remarkable conclusion that these rapidly-rotating black holes are characterized by two qualitatively distinct sets of quasinormal resonances: • Zero-damping modes (ZDMs), which are characterized by the asymptotic property [3] and • Damped modes (DMs), which are characterized by the asymptotic property Here is the dimensionless Bekenstein-Hawking temperature of the black hole [4], where r ± ≡ M ± (M 2 − a 2 ) 1/2 are the black-hole (event and inner) horizons. This dimensionless temperature approaches zero in the extremal a → M (r − → r + ) limit of rapidly-rotating black holes.

II. THE ERRONEOUS CLAIM MADE IN [2] AND DETWEILER'S DAMPED RESONANCES
It has been asserted in Ref. [2] that the ZDMs (1) exist for all co-rotating modes (m ≥ 0) [3], whereas the DMs (2) exist for counter-rotating modes (m < 0) and for co-rotating modes in the limited range Here is the dimensionless ratio between the azimuthal harmonic index m and the spheroidal harmonic index l of the perturbation mode. The critical ratio, µ c , is given by µ c = 15− √ 193 2 ≃ 0.74 in the eikonal limit [2,5]. This critical value of the dimensionless ratio µ marks the boundary between perturbations modes (those with µ < µ c ) which are characterized by imaginary values of the angular-eigenvalue δ [6,7] and perturbations modes (those with µ > µ c ) which are characterized by real values of the angular-eigenvalue δ [6,7].
In this Comment we would like to point out that the assertion made in Ref. [2], according to which co-rotating DMs exist only in the limited range 0 ≤ µ < ∼ µ c [see Eqs. (2) and (4)], is actually erroneous. In particular, we shall show that co-rotating DMs of near-extremal black holes [see Eq. (10) below] actually exist in the entire range In fact, Detweiler [8] has obtained an analytic expression for co-rotating DMs of near-extremal black holes which is valid in the regime µ > µ c [9]: where Ω H ≡ a/2M r + is the angular-velocity of the black-hole horizon, and the integer n is the resonance parameter of the mode. Here we have used the definitions [8] re iθ ≡ Γ(2iδ) Γ(−2iδ) It is worth emphasizing again that the expression (7), originally derived in [8], describes DMs in the µ > µ c (δ 2 > 0) regime, the regime which was claimed in [2] not to contain DMs.

III. THE SOURCE OF THE ERRONEOUS CLAIM MADE IN [2]
It is important to understand the reason for the failure of Yang et. al. [2] to observe the DMs (7) of [8] in the regime µ > µ c [10]. In order to understand the null result of [2] in finding numerically the DMs (7), one should examine the regime of validity of the analyzes presented in [7] and [8].
A careful check of these analyzes reveals that the expression (7) for the black-hole DMs [8] is valid in the regime where the dimensionless coordinate x ≡ (r−r + )/r + belongs to an overlapping region in which two different expressions for the radial Teukolsky wave function (hypergeometric and confluent hypergeometric functions) can be matched, see [7,8] for details. Taking cognizance of the inequalities in (9), one realizes that the expression (7) for co-rotating DMs with µ > µ c is only valid in the regime of near-extremal (rapidly-rotating) black holes. In particular, since each inequality sign in (9) roughly corresponds to an order-of-magnitude difference between two variables (that is, τ /̟ < ∼ 10 −1 , ̟/x < ∼ 10 −1 , and x < ∼ 10 −1 ), the expression (7) for the black-hole DMs [8] is not expected to be valid outside the regime [11] τ < ∼ 10 −4 .
The inequality (10) corresponds to rapidly-rotating black holes with [see Eq.
It is worth noting that the numerical analysis presented in [2] did not explore the deep near-extremal regime (11) of the rotating Kerr black holes [12]. As a consequence, the co-rotating DMs (7) in the regime µ > µ c have not been observed in the numerical study of [2]. This simple fact has probably led Yang et. al. [2] to the erroneous conclusion that co-rotating DMs are restricted to the limited range 0 ≤ µ < ∼ µ c .
In this Comment we have refuted the claim made in Ref. [2] that co-rotating DMs of near-extremal black holes are restricted to the limited range 0 ≤ µ < ∼ µ c [see Eqs. (2) and (4)]. In particular, we have pointed out that the analytical expression (7), originally derived in [8], describes DMs in the µ > µ c regime [9], the regime which was claimed in [2] not to contain DMs.
Most importantly, we have emphasized the fact that the analytical expression (7) for the black-hole DMs is not expected to be valid outside the deep near-extremal regime (11) of rapidly-rotating black holes.
Finally, it is worth emphasizing that rapidly-rotating black holes in the regime (11) are probably of no astrophysical relevance [13]. However, these near-extremal black holes are very important from the point of view of quantum field theory. In particular, these black holes play a key role in the conjectured relation between the quantum states of near-extremal black holes and the corresponding quantum states of a two-dimensional conformal field theory [14][15][16][17].