A mystery of black-hole gravitational resonances

More than three decades ago, Detweiler provided an analytical formula for the gravitational resonant frequencies of rapidly-rotating Kerr black holes. In the present work we shall discuss an important discrepancy between the famous {\it analytical} prediction of Detweiler and the recent {\it numerical} results of Zimmerman et. al. In addition, we shall refute the claim that recently appeared in the physics literature that the Detweiler-Teukolsky-Press resonance equation for the characteristic gravitational eigenfrequencies of rapidly-rotating Kerr black holes is not valid in the regime of damped quasinormal resonances with $\Im\omega/T_{\text{BH}}\gg1$ (here $\omega$ and $T_{\text{BH}}$ are respectively the characteristic quasinormal resonant frequency of the Kerr black hole and its Bekenstein-Hawking temperature). The main goal of the present paper is to highlight and expose this important {\it black-hole quasinormal mystery} (that is, the intriguing discrepancy between the analytical and numerical results regarding the gravitational quasinormal resonance spectra of rapidly-rotating Kerr black holes).


I. THE BLACK-HOLE QUASINORMAL MYSTERY
One of the earliest and most influential works on the quasinormal resonance spectra of black holes is Detweiler's "Black holes and gravitational waves. III. The resonant frequencies of rotating holes" [1]. In this famous paper [2], he derived the characteristic resonance for the quasinormal frequencies of near-extremal (rapidly-rotating) Kerr black holes. Here [3,4] τ ≡ 8πMT BH ; ̟ ≡ M(ω − mΩ H ) ;ω ≡ 2ωr + , where are the Bekenstein-Hawking temperature and the angular velocity of the rotating Kerr black hole, respectively. The parameters {s, m} are the spin-weight and azimuthal harmonic index of the field mode [5], and δ is closely related to the angular-eigenvalue of the angular Teukolsky equation [5,6].
Detweiler's resonance equation (1) is based on the earlier analyzes of Teukolsky and Press [5] and Starobinsky and Churilov [7] who studied the scattering of massless spin-s perturbation fields in the rotating Kerr black-hole spacetime in the double limit a/M → 1 (T BH → 0) and ω → mΩ H . These limits correspond to [see Eq. (2)] The rather complicated resonance equation (1) can be solved analytically in two distinct regimes: (1) In his original analysis [1], Detweiler studied the regime and obtained the black-hole eigenfrequencies where the integer n is the resonance parameter of the mode, and (2) On the other hand, in [8] (see also [9]) we have analyzed the regime and obtained the characteristic relation [8] ℑ̟ n = −2πT BH (n + 1 2 + ℑδ)  [8,9]. On the other hand, in their numerical study, Yang. et. al. [10] have found no trace of the black-hole quasinormal resonances (6) predicted by Detweiler. This discrepancy between the analytical prediction (6) of [1] and the numerical results of [10] is the essence of the black-hole quasinormal mystery.
Most recently, Zimmerman et. al. [11] have claimed that the discrepancy between Detweiler's analytical prediction (6) and their numerical results [10,11] stems from the fact that his resonance equation (1) is not valid in the regime (5). In particular, they have claimed that the standard matching procedure used in [5,7] to match the near-horizon of the Teukolsky radial equation with the far-region x ≫ max(̟, τ ) solution [see equation (A5) of [5]] of the Teukolsky radial equation is invalid in the regime (5) studied by Detweiler.
Since Detweiler's analysis is based on the matching procedure of [5,7], Zimmerman et. al.
have claimed that Detweiler's analysis is also invalid in the regime (5).

II. THE MYSTERY IS STILL UNSOLVED
In this paper we would like to point out that the assertion made in Ref. [11] is actually erroneous. In particular, we shall show below that the matching procedure of [5,7,12] is valid in the overlap region In their matching procedure, Teukolsky and Press [5] (see also [7]) use the identity [see for the near-horizon hypergeometric function (11) [The notation (δ → −δ) in (13) means "replace δ by −δ in the preceding term."]. In order to perform the matching procedure, Teukolsky and Press [5] (see also [7]) take the limit for the hypergeometric functions that appear in the expression (13) of the radial eigenfunction R.
In Ref. [11] Zimmerman et. al. have recently claimed that the limit (14) used in [5,7,12] is invalid in the regime (5) studied by Detweiler. To support their claim, they plot (see Fig.   3 of [11]) the hypergeometric functions of (13) in the regime Not surprisingly, Zimmerman et. al. found that, in the regime (15), the asymptotic behavior (14) used in the matching procedure of [5,7,12] is not valid. They then concluded that the matching procedure of [5] is not valid in the regime (5) studied by Detweiler.
However, here we would like to stress the fact that the analytical arguments of Zimmer- a · b · z c ≪ 1 .
Taking cognizance of the arguments (a, b, c, z) of the hypergeometric functions in (13), one realizes that, for moderate values of the field azimuthal harmonic index m, the asymptotic behavior (14) assumed in the matching procedure of [5,7,12] is valid in the regime [see Eqs. (13) and (17)] In particular, one finds from (18) that, in the regime ̟/τ ≫ 1 [see (5)] studied by Detweiler, the asymptotic behavior (14) used in [5,7,12] is valid for This is certainly not the regime plotted in Fig. 3 Fig. 3 of [11]) are simply irrelevant for the discussion about the validity of the matching procedure used in [5,7]! Furthermore, taking cognizance of (10) and (18), one concludes that the matching procedure of [5,7] is valid in the overlap region It is worth emphasizing again that Detweiler's analysis is based on the matching procedure of [5,7]. As such, his resonance equation (1) for the characteristic eigen-frequencies of rapidly-rotating Kerr black holes is expected to be valid in the regime (20) [and not in the regime (15) considered in Fig. 3 of [11]].
In Table I we present the hypergeometric functions (13) used in the matching procedure of [5,7]. We display the values of these functions for the field mode l = m = s = 2 with n = 2, 3, 4, 5 and x = 0.1 [see (10)]. This mode is characterized by the angular eigenvalue δ = 2.051 [15], which yields the Detweiler resonance spectrum [see Eqs. (6) and (7)] Inspection of the data presented in Table I reveals that, for resonant modes with n ≥ 3, the hypergeometric functions 2 F 1 that appear in the expression (13) for the radial eigenfunction R are described extremely well by the asymptotic behavior (14) as originally assumed in [5,7,12]. Our results therefore support the validity of the matching procedure used by Teukolsky and Press [5] and by Starobinsky and Churilov [7].   (13) used in the matching procedure of [5,7] for the field mode l = m = s = 2 with x = 0.1. The first row corresponds to the hypergeometric (13), whereas the second row corresponds to the hypergeometric function 2 F 1 (−iω + s + 1/2 + iδ, −iω + 1/2 + iδ + 4i̟/τ ; 1 + 2iδ; −τ /x) in (13). One finds that, for resonant modes with n ≥ 3, these functions are described extremely well by the asymptotic behavior (14).

III. SUMMARY
Currently, we face a mystery which can be summarized as follows: (1) Contrary to the claim made in [11], Detweiler's resonance equation (1) is valid in the regime (5).
(2) In his original analytical study, Detweiler has shown that the resonance equation (1) predicts the existence of the near-extremal Kerr black-hole quasinormal resonances (6) in the regime (5). The main goal of the present work was to highlight the discrepancy between the analytical prediction (6) of Detweiler and the numerical results of Zimmerman et. al. We would like to emphasize that currently we have no solution to this black-hole quasinormal mystery. We believe, however, that it is important to highlight this open problem. We hope, in particular, that the present work would encourage researchers in the fields of black-hole physics and general relativity to further explore this interesting physical problem. Hopefully, future studies of the black-hole quasinormal spectrum will shed light on this intriguing mystery.