Quasi-bound states of massive scalar fields in the Kerr black-hole spacetime: Beyond the hydrogenic approximation

Rotating black holes can support quasi-stationary (unstable) bound-state resonances of massive scalar fields in their exterior regions. These spatially regular scalar configurations are characterized by instability timescales which are much longer than the timescale $M$ set by the geometric size (mass) of the central black hole. It is well-known that, in the small-mass limit $\alpha\equiv M\mu\ll1$ (here $\mu$ is the mass of the scalar field), these quasi-stationary scalar resonances are characterized by the familiar hydrogenic oscillation spectrum: $\omega_{\text{R}}/\mu=1-\alpha^2/2{\bar n}^2_0$, where the integer $\bar n_0(l,n;\alpha\to0)=l+n+1$ is the principal quantum number of the bound-state resonance (here the integers $l=1,2,3,...$ and $n=0,1,2,...$ are the spheroidal harmonic index and the resonance parameter of the field mode, respectively). As it depends only on the principal resonance parameter $\bar n_0$, this small-mass ($\alpha\ll1$) hydrogenic spectrum is obviously degenerate. In this paper we go beyond the small-mass approximation and analyze the quasi-stationary bound-state resonances of massive scalar fields in rapidly-spinning Kerr black-hole spacetimes in the regime $\alpha=O(1)$. In particular, we derive the non-hydrogenic (and, in general, non-degenerate) resonance oscillation spectrum ${{\omega_{\text{R}}}/{\mu}}=\sqrt{1-(\alpha/{\bar n})^2}$, where $\bar n(l,n;\alpha)=\sqrt{(l+1/2)^2-2m\alpha+2\alpha^2}+1/2+n$ is the generalized principal quantum number of the quasi-stationary resonances. This analytically derived formula for the characteristic oscillation frequencies of the composed black-hole-massive-scalar-field system is shown to agree with direct numerical computations of the quasi-stationary bound-state resonances.


I. INTRODUCTION
Recent analytical [1] and numerical [2] studies of the coupled Einstein-scalar equations have revealed that rotating black holes can support stationary spatially regular configurations of massive scalar fields in their exterior regions. These bound-state resonances of the composed black-hole-scalar-field system owe their existence to the well-known phenomenon of superradiant scattering [3,4] of integer-spin (bosonic) fields in rotating black-hole spacetimes.
The stationary black-hole-scalar-field configurations [1,2] mark the boundary between stable and unstable boundstate resonances of the composed system. In particular, these stationary scalar field configurations are characterized by azimuthal frequencies ω field which are in resonance with the black-hole angular velocity Ω H [5]: where m = 1, 2, 3, ... is the azimuthal quantum number of the field mode. Bound-state field configurations in the superradiant regime ω field < mΩ H are known to be unstable (that is, grow in time), whereas bound-state field configurations in the regime ω field > mΩ H are known to be stable (that is, decay in time) [3,4]. The bound-state scalar resonances of the rotating Kerr black-hole spacetime are characterized by at least two different time scales: (1) the typical oscillation period τ oscillation ≡ 2π/ω R ∼ 1/µ of the bound-state massive scalar configuration (here µ is the mass of the scalar field [6]), and (2) the instability growth time scale τ instability ≡ 1/ω I associated with the superradiance phenomenon. Former studies [7][8][9][10][11] of the Einstein-massive-scalar-field system have revealed that these two time scales are well separated. In particular, it was shown [7][8][9][10][11] that the composed system is characterized by the relation or equivalently The strong inequality (2) implies that the bound-state massive scalar configurations may be regarded as the quasistationary resonances of the composed system. As shown in [12,13], the physical significance of the characteristic black-hole-scalar-field oscillation frequencies {ω R (n)} n=∞ n=0 [14] lies in the fact that the corresponding quasi-stationary scalar resonances dominate the dynamics of massive scalar fields in curved black-hole spacetimes. In particular, recent numerical simulations [12,13] of the dynamics of massive scalar fields in the Kerr black-hole spacetime have demonstrated explicitly that these quasistationary bound-state resonances dominate the characteristic Fourier power spectra P (ω) of the composed blackhole-massive-scalar-field system [15].

II. THE SMALL-MASS HYDROGENIC SPECTRUM
As shown by Detweiler [7], the massive scalar resonances can be calculated analytically in the small-mass regime M µ ≪ 1. In particular, one finds [7] that the quasi-stationary bound-state scalar resonances are characterized by the familiar hydrogenic spectrum where the integern 0 (l, n; α → 0) = l + 1 + n is the principal quantum number of the quasi bound-state resonances. Here the integer l ≥ |m| is the spheroidal harmonic index of the field mode and n = 0, 1, 2, ... is the resonance parameter. It is worth emphasizing that the hydrogenic spectrum (4) depends only on the principal resonance parameter (quantum number)n 0 = l + 1 + n. This small-mass oscillation spectrum is therefore degenerate. That is, two different modes, (l, n) and (l ′ , n ′ ) with l + n = l ′ + n ′ , are characterized by the same resonant frequency: ω R (l, n) = ω R (l ′ , n ′ ) for l + n = l ′ + n ′ .
To the best of our knowledge, the oscillation frequency spectrum which characterizes the quasi-stationary boundstate resonances of massive scalar fields in the rotating Kerr black-hole spacetime has not been studied analytically beyond the hydrogenic regime (4) of small (α ≪ 1) field masses. The main goal of the present paper is to analyze the oscillation spectrum of the composed black-hole-massive-scalar-field system in the α = O(1) regime. To that end, we shall use the resonance equation [see Eq. (11) below] derived in [10] for the bound-state resonances of massive scalar fields in rapidly-rotating (near-extremal) Kerr black-hole spacetimes. As we shall show below, this resonance equation can be solved analytically to yield the characteristic oscillation spectrum {ω R (n)} n=∞ n=0 of the quasi-stationary bound-state scalar resonances in the regime α 1.

III. DESCRIPTION OF THE SYSTEM
We consider a scalar field Ψ of mass µ linearly coupled to a rapidly-rotating (near-extremal) Kerr black hole of mass M and dimensionless angular momentum a/M → 1 − . The dynamics of the scalar field in the black-hole spacetime is governed by the Klein-Gordon (Teukolsky) wave equation Substituting the field decomposition [16][17][18] into the wave equation (6), one finds [19,20] that the radial function R and the angular function S obey two ordinary differential equations of the confluent Heun type [21,22]. The angular eigenfunctions, known as the spheroidal harmonics, are determined by the angular Teukolsky equation The regularity requirements of these functions at the two boundaries θ = 0 and θ = π single out a discrete set of angular eigenvalues {A lm } [see Eq. (14) below] labeled by the integers l and m. The radial Teukolsky equation is given by [19,20,24] where ∆ ≡ (r − r + )(r − r − ) [25]. Note that the angular Teukolsky equation (8) and the radial Teukolsky equation (9) are coupled by the angular eigenvalues {A lm }.
The quasi-stationary bound-state resonances of the massive scalar fields in the black-hole spacetime are characterized by the boundary conditions of purely ingoing waves at the black-hole horizon (as measured by a comoving observer) and a spatially decaying (bounded) radial eigenfunction at asymptotic infinity [7][8][9]26]: where Ω H is the angular velocity of the black-hole horizon [see Eq. (12) below]. The boundary conditions (10) imposed on the radial eigenfunctions single out a discrete set of eigenfrequencies {ω n (a/M, l, m, α)} n=∞ n=0 which characterize the quasi-stationary bound-state resonances of the massive scalar fields in the Kerr black-hole spacetime [7][8][9].

IV. THE CHARACTERISTIC RESONANCE EQUATION AND ITS REGIME OF VALIDITY
Solving analytically the radial Klein-Gordon (Teukolsky) equation (9) in two different asymptotic regions and using a standard matching procedure for these two radial solutions in their common overlap region [see Eq. (15) below], we have derived in [10] the characteristic resonance equation for the bound-state resonances of the composed Kerr-black-hole-massive-scalar-field system. Here is the angular velocity of the black-hole horizon, and where {A lm } are the angular eigenvalues which couple the radial Teukolsky equation (9) to the angular (spheroidal) equation (8). These angular eigenvalues can be expanded in the form [23] A lm = l(l + 1) where the expansion coefficients {c k } are given in [23]. Before proceeding, it should be emphasized that the validity of the resonance equation (11) is restricted to the regime where τ ≡ (r + − r − )/r + ≪ 1 is the dimensionless temperature of the rapidly-rotating (near-extremal) Kerr black hole, and the dimensionless coordinate x o ≡ (r o − r + )/r + belongs to the overlap region in which the two different solutions of the radial Teukolsky equation (hypergeometric and confluent hypergeometric radial wave functions) can be matched together, see [10,27] for details. The inequalities in (15) imply that the resonance condition (11) should be valid in the regime [28]

V. THE QUASI-STATIONARY BOUND-STATE RESONANCES OF THE COMPOSED BLACK-HOLE-MASSIVE-SCALAR-FIELD SYSTEM
As we shall now show, the resonance condition (11) can be solved analytically in the physical regime (16). In particular, in the present section we shall derive a (remarkably simple) analytical formula for the discrete spectrum of oscillation frequencies, {ω R (l, m, α; n)} n=∞ n=0 , which characterize the quasi-stationary bound-state resonances of the composed Kerr-black-hole-massive-scalar-field system.
Our analytical approach is based on the fact that the right-hand-side of the resonance equation (11) is small in the regime (16) with β ∈ R [29]. The resonance condition can therefore be approximated by the simple zeroth-order equation As we shall now show, this zeroth-order resonance condition can be solved analytically to yield the real oscillation frequencies which characterize the bound-state scalar resonances. We first use the well-known pole structure of the Gamma functions [23] in order to write the resonance equation (17) in the form [10] where the integer n = 0, 1, 2, ... is the resonance parameter of the field mode.
Finally, taking cognizance of the relation (19), one finds the discrete spectrum of oscillation frequencies which characterize the quasi-stationary bound-state resonances of the composed black-hole-massive-scalar-field system. Here is the generalized (finite-mass) spheroidal harmonic index. Note that ℓ → l in the small mass α ≪ 1 limit, in which case one recovers from (24) the well-known hydrogenic spectrum (4) of [7]. It is worth noting that, in general, the parameter ℓ(α) is not an integer. This implies that, for generic values of the dimensionless mass parameter α, the non-hydrogenic oscillation spectrum (24) is not degenerate [31].

VI. NUMERICAL CONFIRMATION
It is of physical interest to test the accuracy of the analytically derived formula (24) for the characteristic oscillation frequencies ω ana R (n)/µ of the quasi-stationary massive scalar configurations. The quasi bound-state resonances can be computed using standard numerical techniques, see [9,12] for details. In Table I   (n) of [12]. One finds a good agreement between the analytical formula (24) and the numerical data of [12] .
analytically derived oscillation frequencies (24) and the numerically computed resonances [12]. The data presented is for the fundamental l = m = 1 mode with a/M = 0.99 [32] and α = 0.42 [33,34]. One finds a good agreement between the analytically calculated oscillation frequencies (24) and the numerically computed resonances of [12].
In order to compare the accuracy of the newly derived analytical formula (24) with the accuracy of the familiar hydrogenic (small-mass) spectrum (4), we display in Table II the physical quantity ǫ(n) [see Eq. (19)] which provides a quantitative measure for the deviation of the resonant oscillation frequency ω R (n) from the field mass parameter µ. In particular, we present the dimensionless ratios ǫ ana /ǫ num and ǫ ana-hydro /ǫ num , where ǫ ana (n) is given by the analytical formula (24), ǫ ana-hydro (n) is defined from the hydrogenic spectrum (4), and ǫ num (n) is obtained from the numerically computed resonances of [12]. One finds that, in general, the newly derived formula (24) performs better than the hydrogenic formula (4) [35]. We display the dimensionless ratios ǫ ana /ǫ num and ǫ ana-hydro /ǫ num , where ǫ ana (n) is given by the analytical formula (24), ǫ ana-hydro (n) is defined from the hydrogenic spectrum (4), and ǫ num (n) is obtained from the numerically computed resonances of [12]. One finds that, in general, the newly derived analytical formula (24) performs better than the hydrogenic formula (4) [35].

VII. SUMMARY
In summary, we have studied the resonance spectrum of quasi-stationary massive scalar configurations linearly coupled to a near-extremal (rapidly-rotating) Kerr black-hole spacetime. In particular, we have derived a compact analytical expression [see Eq. (24)] for the characteristic oscillation frequencies ω ana R (n)/µ of the bound-state massive scalar fields. It was shown that the analytically derived formula (24) agrees with direct numerical computations [12] of the black-hole-scalar-field resonances.
It is well known that the characteristic hydrogenic spectrum (4) in the small-mass α ≪ 1 limit is highly degenerate -it depends only on the principal resonance parametern 0 ≡ l + 1 + n [36] [Thus, according to (4), two different modes which are characterized by the integer parameters (l, n) and (l ′ , n ′ ) with l + n = l ′ + n ′ share the same resonant frequency ω R (n 0 ) in the α → 0 limit]. On the other hand, the newly derived resonance spectrum (24), which is valid in the α = O(1) regime, is no longer degenerate. That is, for generic values of the dimensionless mass parameter α, two quasi-stationary modes with different sets of the integer parameters (l, n) are characterized, according to (24), by different oscillation frequencies [37].
Finally, it is worth emphasizing again that the physical significance of the characteristic oscillation frequencies (24) lies in the fact that these quasi-stationary (long-lived) resonances dominate the dynamics [and, in particular, dominate the characteristic Fourier power spectra P (ω) [12]] of the massive scalar fields in the black-hole spacetime.

ACKNOWLEDGMENTS
This research is supported by the Carmel Science Foundation. I would like to thank Yael Oren, Arbel M. Ongo,