---
abstract: 'We study analytically the Klein-Gordon wave equation for stationary massive scalar fields linearly coupled to spinning Kerr black holes.  In particular, using the WKB approximation, we derive a compact formula for the discrete spectrum of scalar field masses which characterize the stationary composed Kerr-black-hole-massive-scalar-field configurations in the large-coupling regime $M\mu\gg1$ (here $M$ and $\mu$ are respectively the mass of the central black hole and the proper mass of the scalar field).  We confirm our analytically derived formula for the Kerr-scalar-field mass spectrum with numerical data that recently appeared in the literature.' 
author:
- Shahar Hod
title: 'Spinning Kerr black holes with stationary massive scalar clouds: The large-coupling regime'
---

Introduction
============

The complex interactions between fundamental matter fields and general relativistic black holes have been the focus of intense research efforts during the last five decades (see [@Chan; @Zel; @PressTeu2; @Bekc; @Hodkr; @Herkr; @Notemas; @CarDias] and references therein).  Especially interesting are the physical properties of astrophysically realistic spinning black holes coupled to bosonic matter fields.  As explicitly demonstrated recently [@Hodkr; @Herkr], these integer-spin fields can form non-trivial spatially regular [*stationary*]{} matter configurations in rotating curved black-hole spacetimes. 

In particular, recent analytical [@Hodkr] and numerical [@Herkr] studies of the stationary sector of the coupled Einstein-Klein-Gordon field equations have revealed the intriguing fact that, for a given spinning black hole of mass $M$ and angular momentum per unit mass $a$, there exists a [*discrete*]{} spectrum of resonant field masses, $\{M\mu(a/M,m;n)\}^{n=\infty}_{n=0}$ [@Noteunit; @Notedim; @Notemn; @Notesmp], which characterize a family of stationary spatially regular scalar configurations that can be supported in the external region of the corresponding rotating black-hole spacetime. 

Interestingly, these stationary bound-state bosonic matter configurations are characterized by proper azimuthal frequencies which are in resonance with (that is, are multiple integers of) the angular frequency of the supporting central black hole [@Hodkr; @Herkr; @Notecrit]: $$\label{Eq1}
\omega_{\text{field}}=\omega_{\text{c}}\equiv
m\Omega_{\text{H}}\  ,$$ where the integer $m$ is the azimuthal harmonic index of the scalar mode and [@Chan; @Kerr; @Noterp] $$\label{Eq2}
\Omega_{\text{H}}={{a}\over{r^2_++a^2}}\$$ is the angular velocity which characterizes the outer horizon of the spinning Kerr black hole. 

For a scalar field of proper mass $\mu$, the resonant field frequency (\[Eq1\]) which, for a given value of the azimuthal harmonic index $m$, characterizes a stationary bound-state scalar configuration in the Kerr black-hole spacetime, is also bounded from above by the inequality [@Hodkr; @Herkr] $$\label{Eq3}
\omega^2_{\text{field}}<\mu^2\  .$$ The characteristic relation (\[Eq3\]) guarantees that the massive scalar configuration is spatially bounded to the black hole.  In particular, low frequency stationary scalar modes in the regime (\[Eq3\]) are captured inside an effective potential well which characterizes the composed black-hole-massive-scalar-field system \[see Eq. ( \[Eq18\]) below\].  Thus, matter configurations made of massive scalar fields in the regime (\[Eq3\]) cannot radiate their energies to asymptotic infinity [@Hodkr; @Herkr; @Notediv]. 

The discrete spectrum of resonant field masses $\{M\mu(a/M,m)\}$, which characterizes the stationary bound-state scalar configurations in the spinning Kerr black-hole spacetime, has been computed [*numerically*]{} in [@Herkr].  For rapidly-rotating (near-extremal) black holes in the regime $a/M\to 1$, the characteristic mass spectrum of the composed black-hole-linearized-massive-scalar-field configurations has been studied [*analytically*]{} in [@Hodkr].  To the best of our knowledge, the characteristic resonant spectrum $\{M\mu(a/M,m)\}$ of the stationary bound-state massive scalar configurations has not been determined analytically thus far for [*generic*]{} [@Notegen] values $a/M\in(0,1)$ of the dimensionless rotation parameter which characterizes the central supporting black holes. 

The main goal of the present paper is to derive a compact [*analytical*]{} formula for the discrete spectrum $\{M\mu(a/M,m)\}$ of field masses which characterize the stationary bound-state scalar configurations in [*generic*]{} [@Notegen] Kerr black-hole spacetimes.  To this end, we shall study analytically the Klein-Gordon wave equation for stationary massive scalar fields linearly coupled to rotating Kerr black-hole spacetimes.  In particular, using the WKB approximation, we shall derive below a compact analytical formula for the discrete spectrum $\{M\mu(a/M,m)\}$ of scalar field masses which characterize the stationary bound-state Kerr-black-hole-massive-scalar-field configurations in the eikonal large-mass $M\mu\gg1$ regime.  Most importantly, this analytically derived formula \[see Eq. ( \[Eq27\]) below\] for the scalar resonance spectrum would be valid in the [*entire*]{} physical range $a/M\in(0,1)$ of the black-hole rotation parameter. 

Description of the system
=========================

We shall analyze the properties of a physical system which is composed of a stationary scalar field $\Psi$ of mass $\mu$ and proper frequency $\omega_{\text{field}}=m\Omega_{\text{H}}$ \[see Eqs. ( \[Eq1\]) and (\[Eq2\])\] which is linearly coupled to a central rotating Kerr black hole of mass $M$ and angular momentum per unit mass $a$. Using the Boyer-Lindquist coordinates $(t,r,\theta,\phi)$, one can describe the spinning black-hole spacetime by the line element [@Chan; @Kerr] $$\begin{aligned}
\label{Eq4}
ds^2=-{{\Delta}\over{\rho^2}}(dt-a\sin^2\theta
d\phi)^2+{{\rho^2}\over{\Delta}}dr^2+\rho^2
d\theta^2+{{\sin^2\theta}\over{\rho^2}}\big[a
dt-(r^2+a^2)d\phi\big]^2\  ,\end{aligned}$$ where $$\label{Eq5}
\Delta\equiv r^2-2Mr+a^2\ \ \ ; \ \ \ \rho^2\equiv
r^2+a^2\cos^2\theta\  .$$ The zeros of $\Delta$, $$\label{Eq6}
r_{\pm}=M\pm\sqrt{M^2-a^2}\  ,$$ determine the horizon radii of the Kerr black-hole spacetime. 

The spatial and temporal properties of a linearized massive scalar field in the curved black-hole spacetime are determined by the familiar Klein-Gordon wave equation $$\label{Eq7}
(\nabla^{\nu}\nabla_{\nu}-\mu^2)\Psi=0\  .$$ Substituting the metric components of the Kerr black-hole line element (\[Eq4\]) into the Klein-Gordon wave equation (\[Eq7\]) and using the scalar field decomposition [@Noteanz] $$\label{Eq8}
\Psi(t,r;\omega,\theta,\phi)=\sum_{l,m}e^{im\phi}{S_{lm}}(\theta;m,a\sqrt{\mu^2-\omega^2})
{R_{lm}}(r;M,a,\mu,\omega)e^{-i\omega t}\  ,$$ one obtains the characteristic ordinary differential equation (known as the spheroidal angular equation) [@Stro; @Heun; @Fiz1; @Teuk; @Abram; @Hodasy; @Barma; @Hodpp] $$\begin{aligned}
\label{Eq9}
{1\over {\sin\theta}}{{d}\over{d\theta}}\Big(\sin\theta {{d
S_{lm}}\over{d\theta}}\Big)
+\Big[K_{lm}+a^2(\mu^2-\omega^2)\sin^2\theta-{{m^2}\over{\sin^2\theta}}\Big]S_{lm}=0\\end{aligned}$$ for angular part $S_{lm}$ of the scalar eigenfunction $\Psi$ [@Notespd].  Spheroidal harmonic eigenfunctions $\{S_{lm}\}$ which are regular at the angular poles $\theta=0$ and $\theta=\pi$ are known to be characterized by a discrete set $\{K_{lm}\}$ of angular eigenvalues [@Stro; @Heun; @Fiz1; @Teuk; @Abram; @Hodasy; @Barma; @Hodpp].  In particular, it has recently been proved that, in the eikonal ($|m|\gg1$) regime, the angular eigenvalues are given by the asymptotic large-$m$ analytical formula [@Hodpp; @Notebpo; @Noteklm] $$\label{Eq10}
K_{lm}=m^2-a^2(\mu^2-\omega^2)+[2(l-|m|)+1]\sqrt{m^2+a^2(\mu^2-\omega^2)}+O(1)\  .$$

Substituting the scalar field decomposition (\[Eq8\]) into (\[Eq7\]), and using the line element (\[Eq4\]) which characterizes the spinning Kerr black-hole spacetime, one finds the characteristic ordinary differential equation [@Teuk; @Stro] $$\label{Eq11}
\Delta{{d} \over{dr}}\Big(\Delta{{d R_{lm}
}\over{dr}}\Big)+\Big\{[\omega(r^2+a^2)-ma]^2
+\Delta[2ma\omega-\mu^2(r^2+a^2)-K_{lm}]\Big\}R_{lm}=0\$$ for the radial part $R_{lm}$ of the scalar eigenfunction $\Psi$, where the angular eigenvalues $\{K_{lm}\}$ that explicitly appear in the radial equation (\[Eq11\]) should be determined from the spheroidal differential equation (\[Eq9\]) \[see, in particular, the asymptotic angular spectrum (\[Eq10\]) [@Hodpp]\].  Physically acceptable solutions of the scalar radial equation (\[Eq11\]) are characterized by purely ingoing waves (as measured by a comoving observer) at the black-hole outer horizon [@Notemas; @Hodkr; @Herkr]: $$\label{Eq12}
R_{lm} \sim e^{-i(\omega-m\Omega_{\text{H}})y}\ \ \ \text{ for }\ \
\ r\rightarrow r_+\ \ (y\rightarrow -\infty)\  ,$$ where the radial coordinate $y$, to be used below, is defined by the differential relation $dy=(r^2/\Delta)dr$ \[see Eq. ( \[Eq15\]) below\].  In addition, the spatially regular bound-state (finite-mass) configurations of the massive scalar fields in the black-hole spacetime are characterized by asymptotically decaying radial eigenfunctions at spatial infinity [@Notemas; @Hodkr; @Herkr]: $$\label{Eq13}
R_{lm} \sim {1\over r}e^{-\sqrt{\mu^2-\omega^2}y}\ \ \ \text{ for }\
\ \ r\rightarrow\infty\ \ (y\rightarrow \infty)\  .$$

The physically motivated boundary conditions (\[Eq12\]) and (\[Eq13\]) of the radial scalar equation (\[Eq11\]), together with the resonance condition (\[Eq1\]), determine the discrete spectrum $\{M\mu(m,a/M;n)\}$ of scalar field masses which characterize the stationary composed Kerr-black-hole-linearized-massive-scalar-field configurations.  In the next section we shall use [*analytical*]{} techniques in order to determine the characteristic Kerr-scalar-field resonance spectrum in the large mass ($M\mu\gg1$) regime. 

The resonance mass spectrum of the composed Kerr-black-hole-massive-scalar-field configurations
===============================================================================================

In the present section we shall derive a compact analytical formula for the discrete spectrum $\{M\mu(m,a/M;n)\}$ of scalar field masses which characterize the composed stationary Kerr-black-hole-massive-scalar-field configurations in the eikonal large-mass regime.  The trick is to transform the radial differential equation (\[Eq11\]) of the scalar field into the familiar form of a Schrödinger-like radial differential equation and then to perform a standard WKB analysis. 

Defining the new radial eigenfunction $$\label{Eq14}
\psi=rR\$$ and using the coordinate transformation [@Notemap] $$\label{Eq15}
dy={{r^2}\over{\Delta}}dr\  ,$$ one obtains from (\[Eq11\]) the Schrödinger-like ordinary differential equation $$\label{Eq16}
{{d^2\psi}\over{dy^2}}-V(y)\psi=0\  ,$$ with the effective radial potential $$\label{Eq17}
V=V(r;\omega,M,a,\mu,l,m)={{2\Delta}\over{r^6}}(Mr-a^2)+{{\Delta}\over{r^4}}
[K_{lm}-2ma\omega+\mu^2(r^2+a^2)]-{{1}\over{r^4}}[\omega(r^2+a^2)-ma]^2\
.$$

Substituting into (\[Eq17\]) the resonant oscillation frequency (\[Eq1\]), which characterizes the stationary massive scalar field configurations in the spinning Kerr black-hole spacetime, and using the asymptotic large-$m$ analytical formula (\[Eq10\]) for the characteristic eigenvalues of the angular differential equation (\[Eq9\]), one finds the expression $$\label{Eq18}
V(r)=m^2\cdot \Big\{{{r-r_+}\over{r^3(r^2_++a^2)^2}}\big[\beta
a^2r^2-a^2(2M+\beta
r_-)r+2Mr^3_+\big]+{{1}\over{m}}\cdot{{\Delta}\over{r^4}}(2k+1)\sqrt{1+\beta{{a^4}\over{(r^2_++a^2)^2}}}+O(m^{-2})\Big\}\$$ for the effective radial potential which characterizes the composed stationary Kerr-black-hole-massive-scalar-field configurations in the eikonal $m\gg1$ regime [@Notelarmu], where $k\equiv l-m\ll m$ [@Noteklm].  Here we have used the dimensionless physical parameter $\beta$ which relates the proper mass of the stationary scalar field to its resonant oscillation frequency [@Notebet]: $$\label{Eq19}
\mu^2=(1+\beta)\cdot\omega^2_{\text{c}}\  .$$

As we shall now show, the characteristic Schrödinger-like differential equation (\[Eq16\]) is amenable to a standard WKB analysis [@WKB1; @WKB2; @WKB3; @Will; @Hodalm].  In particular, one finds that the effective radial potential (\[Eq18\]) of the stationary composed Kerr-massive-scalar-field configurations is characterized by the presence of a [*binding*]{} potential well outside the black-hole horizon [@Hodbmu].  As shown in [@WKB1; @WKB2; @Hodalm], the WKB resonance condition which characterizes the bound-state resonances of the standard Schrödinger-like ordinary differential equation (\[Eq16\]) in the eikonal large-$m$ regime is given by $$\label{Eq20}
{V_{\text{min}}\over{\sqrt{2V^{''}_{\text{min}}}}}=-(n+{1\over 2})\ \ \ ; \ \ \ n=0,1,2,...\   ,$$ where a prime denotes a spatial derivative of the effective radial potential (\[Eq18\]) with respect to the radial coordinate $y$ \[see Eq. ( \[Eq15\])\] of the Schrödinger-like differential equation.  The subscript “min" in (\[Eq20\]) means that the effective radial potential and its spatial derivatives are evaluated at the radial coordinate $r^{\text{min}}$ which characterizes the minimum point of the effective binding potential (\[Eq18\]) [@Notemin]. 

The WKB resonance equation (\[Eq20\]) with the (rather cumbersome) radial potential (\[Eq18\]) can be solved analytically using an iteration scheme (that is, using an expansion in inverse powers of $m$).  Keeping only the leading order $O(m^2)$ term of the effective radial potential (\[Eq18\]), one finds from (\[Eq20\]) the leading-order solution $$\label{Eq21}
r^{\text{min}}_0=r_+\cdot {{\gamma[1-\sqrt{1-\gamma^2}]}\over{2-\gamma^2-2\sqrt{1-\gamma^2}}}\$$ with $$\label{Eq22}
\beta_0={{(1+\gamma)(2-\gamma^2-2\sqrt{1-\gamma^2})}\over{\gamma^3}}\  ,$$ where $$\label{Eq23}
\gamma\equiv {{r_-}\over{r_+}}\$$ is the dimensionless ratio between the horizon radii \[see Eq. ( \[Eq6\])\] of the rotating Kerr black-hole spacetime. 

Next, substituting $$\label{Eq24}
r^{\text{min}}=r^{\text{min}}_0\cdot\big[1+{{r_1}\over{m}}+O(m^{-2})\big]\$$ and $$\label{Eq25}
\beta=\beta_0\cdot\big[1+{{\beta_1}\over{m}}+O(m^{-2})\big]\  ,$$ and keeping terms of orders $O(m^2)$ and $O(m)$ in the effective radial potential (\[Eq18\]), one finds from (\[Eq20\]) after some lengthy algebra [@Noter1] $$\label{Eq26}
\beta_1=-{{(2-\gamma^2-2\sqrt{1-\gamma^2})\sqrt{1+{{1}\over{\gamma}}}}\over{(1-\sqrt{1-\gamma^2})^2}}\cdot
\Big[\sqrt{-1+\gamma+\sqrt{1-\gamma^2}}\cdot(1+2n)+\sqrt{2+\gamma-2\sqrt{1-\gamma^2}}\cdot(1+2k)\Big]\  .$$

Finally, taking cognizance of Eqs. ( \[Eq1\]), (\[Eq2\]), (\[Eq19\]), (\[Eq22\]), (\[Eq23\]), (\[Eq25\]), and (\[Eq26\]), one finds the analytical formula $$\label{Eq27}
M\mu=m\cdot [\Gamma_0+{{\Gamma_1}\over{m}}+O(m^{-2})]\$$ for the discrete spectrum of scalar field masses which characterize the stationary composed Kerr-black-hole-massive-scalar-field configurations in the eikonal large-mass ($M\mu\gg1$) regime, where $$\label{Eq28}
\Gamma_0={{\sqrt{2(1+\gamma)(1-\sqrt{1-\gamma^2})-\gamma^2}}\over{2\gamma}}\$$ and [@Noteg1] $$\label{Eq29}
\Gamma_1=-{{(1+{{1}\over{\gamma}})^{3/2}(2-\gamma^2-2\sqrt{1-\gamma^2})^2\Big[\sqrt{-1+\gamma+\sqrt{1-\gamma^2}}\cdot(1+2n)+\sqrt{2+\gamma-2\sqrt{1-\gamma^2}}\cdot(1+2k)\Big]}\over{4(1-\sqrt{1-\gamma^2})^2\sqrt{2(1+\gamma)(1-\sqrt{1-\gamma^2})-\gamma^2}}}\  .$$

Numerical confirmation
======================

It is of physical interest to test the accuracy of the analytically derived formula (\[Eq27\]) for the discrete mass spectrum of the composed stationary Kerr-black-hole-massive-scalar-field configurations.  The corresponding scalar field masses of the stationary configurations were computed numerically in the very interesting work of Herdeiro and Radu [@Herkr].  In Table \[Table1\] we display the dimensionless ratio ${{\mu_{\text{numerical}}}/{\mu_{\text{wkb}}}}$ for the fundamental ($n=0$) resonant mode with $l=m=10$, where $\{\mu_{\text{numerical}}(a/M)\}$ are the [*numerically*]{} computed [@Herkr] field masses of the stationary scalar configurations and $\{\mu_{\text{wkb}}(a/M)\}$ are the [*analytically*]{} derived masses of the stationary bound-state scalar fields as given by the WKB expression (\[Eq27\]).  From the data presented in Table \[Table1\] one finds that the agreement between the numerical data [@Herkr] and the analytically derived formula (\[Eq27\]) for the resonant masses of the stationary black-hole-scalar-field configurations is generally better than $1\%$ [@Notenex]. 

     $s\equiv {{a}/{M}}$      ${{\mu(l=m=10)}\over{\mu_{\text{wkb}}}}$  
  ------------------------- ---------------------------------------------
             0.1                              0.99946   
             0.2                              0.99993   
             0.3                              1.00009   
             0.4                              1.00017   
             0.5                              0.99974   
             0.6                              1.00037   
             0.7                              1.00076   
             0.8                              1.00134   
             0.9                              1.00266   
            0.95                              1.00399   
            0.99                              1.00750   
            0.999                             1.01219   

  : Stationary massive scalar clouds linearly coupled to spinning Kerr black holes.  We display the dimensionless ratio ${{\mu_{\text{numerical}}}/{\mu_{\text{wkb}}}}$, where $\{\mu_{\text{numerical}}(a/M)\}$ are the [*numerically*]{} computed [@Herkr] field masses of the stationary scalar fields and $\{\mu_{\text{wkb}}(a/M)\}$ are the [*analytically*]{} derived scalar field masses as given by the WKB expression (\[Eq27\]).  The data presented are for the fundamental ($n=0$) resonant mode with $l=m=10$. One finds that the agreement between the numerical data and the analytical formula (\[Eq27\]) is generally better than $1\%$.[]{data-label="Table1"}

Summary
=======

The intriguing physical mechanism of rotational superradiance [@Zel; @PressTeu2] allows spinning Kerr black holes to support stationary bound-state matter configurations made of massive scalar fields in their exterior regions.  The physical properties of these stationary composed black-hole-field configurations have been studied extensively in recent years [@Hodkr; @Herkr].  Interestingly, it has been shown [@Hodkr; @Herkr] that, for a given set $\{M,a,m\}$ of the black-hole-field physical parameters, there exists a [*discrete*]{} spectrum $\{M\mu(a/M,m;n)\}$ of field masses which characterize the composed stationary Kerr-black-hole-massive-scalar-field configurations.  In particular, for near-extremal (rapidly-rotating) Kerr black holes, the characteristic discrete mass spectrum of the composed black-hole-linearized-massive-scalar-field configurations has been studied [*analytically*]{} in [@Hodkr]. 

In the present paper we have studied analytically the Klein-Gordon wave equation for stationary massive scalar fields linearly coupled to generic [@Notegen] spinning Kerr black holes.  In particular, using the WKB approximation, we have derived an explicit analytical formula \[see Eqs. ( \[Eq27\])-(\[Eq29\])\] for the discrete spectrum of scalar field masses which characterize the stationary composed Kerr-black-hole-massive-scalar-field configurations in the eikonal large-mass $M\mu\gg1$ regime.  Interestingly, the [*analytically*]{} derived formula (\[Eq27\]) for the discrete mass spectrum of the composed Kerr-scalar-field configurations was shown to agree remarkably well with direct [*numerical*]{} computations [@Herkr] of the corresponding black-hole-massive-scalar-field resonances. 

Finally, it is worth noting that, to the best of our knowledge, the analytically derived expression (\[Eq27\]), which is valid in the entire physical range $a/M\in(0,1)$ of the black-hole rotation parameter [@Noteacc], provides the first analytical formula for the discrete mass spectrum of the stationary bound-state linearized massive scalar field configurations in the spacetimes of [*generic*]{} [@Notegen] spinning Kerr black holes. 

[**ACKNOWLEDGMENTS**]{}

This research is supported by the Carmel Science Foundation.  I would like to thank C. A. R. Herdeiro and E. Radu for sharing with me their numerical data.  I thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B. Tea for stimulating discussions. 

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We shall use natural units in which $G=c=\hbar=1$.

Note that the proper mass $\mu$, which characterizes the scalar field mode, stands for $\mu/\hbar$. Hence, this physical parameter of the matter field has the dimensions of $($length$)^{-1}$.

Here the dimensionless integer parameters $m$ and $n$ are respectively the azimuthal harmonic index and the resonance parameter of the stationary bound-state massive scalar field mode \[see Eqs. ( \[Eq8\]) and (\[Eq20\]) below\]. 

We shall assume $a>0$ and $m>0$ for the black-hole-field physical parameters without loss of generality. 

It is worth noting that, for a given value of the azimuthal harmonic index $m$ of the scalar mode, the characteristic proper frequency $\omega_{\text{field}}=m\Omega_{\text{H}}$ of the stationary field configuration coincides with the threshold (critical) frequency $\omega_{\text{c}}$ for the superradiant scattering phenomenon of bosonic fields in the spinning Kerr black-hole spacetime [@Zel; @PressTeu2]. 

Here $r_+$ is the radius of the outer black-hole horizon \[see Eq. ( \[Eq6\]) below\]. 

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It is worth noting that the [*stationary*]{} bound-state scalar configurations, which are characterized by the resonant oscillation frequencies (\[Eq1\]), divide the parameter space of the composed black-hole-scalar-field system into two physically distinct regimes [@Zel; @PressTeu2; @Bekc; @Hodkr; @Herkr; @Notemas; @CarDias].  In particular, for a given value of the azimuthal harmonic index $m$, massive bosonic field configurations in the regime $\omega_{\text{field}}<\omega_{\text{c}}$ are known to develop exponentially growing superradiant instabilities \[provided that they also conform to the inequality (\[Eq3\]) which characterizes the bound-state configurations in the black-hole spacetime\], whereas field configurations in the regime $\omega_{\text{field}}>\omega_{\text{c}}$ are known to decay in time [@Zel; @PressTeu2; @Bekc; @Hodkr; @Herkr; @Notemas; @CarDias]. 

We use here the term ‘generic Kerr black holes’ to describe spinning black holes that belong to the [*entire*]{} physical range $a/M\in(0,1)$ of the rotation parameter (That is, spinning black holes which are not necessarily near-extremal ones). 

Here the integer $l\geq|m|$ is the spheroidal harmonic index of the bound-state scalar field mode. 

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The angular eigenfunctions $\{S_{lm}\}$ are the familiar spheroidal harmonics. 

As explicitly proved in [@Hodpp], the large-$m$ asymptotic formula (\[Eq10\]) is valid in the regime $-a^2(\mu^2-\omega^2)<m^2$. Taking cognizance of the inequality $\omega^2<\mu^2$ which characterizes the bound-state massive scalar resonances in the black-hole spacetime \[see (\[Eq3\])\], one finds that the requirement $-a^2(\mu^2-\omega^2)<m^2$ is trivially satisfied. 

Note that, as explicitly proved in [@Hodpp], the large-$m$ analytical formula (\[Eq10\]) is valid in the $l-|m|\ll \sqrt{m^2+a^2(\mu^2-\omega^2)}$ regime. 

It is worth noting that the radial range $r\in [r_+,\infty]$ is mapped into the radial range $y\in [-\infty,+\infty]$ by the radial differential relation (\[Eq15\]). 

Note that the eikonal large-$m$ regime also corresponds to the large-mass $M\mu\gg1$ regime \[see Eq. ( \[Eq27\]) below\]. 

Note that the characteristic inequality (\[Eq3\]) for the bound-state resonances of the massive scalar fields in the black-hole spacetime implies that $\beta>0$ . 

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That is, the radial coordinate $r^{\text{min}}$ is defined by the relation $V^{'}[r=r^{\text{min}}(y)]=0$ with $V^{''}[r=r^{\text{min}}(y)]>0$.

The explicit expression of the dimensionless expansion coefficient $r_1$ is rather cumbersome and is not important for our purposes. 

Note that $\Gamma_1<0$, which implies $M\mu<m\Gamma_0$ \[see Eq. ( \[Eq27\])\] in the asymptotic $m\to\infty$ regime [@Hodbmu]. 

It is worth mentioning that, for the particular case of rapidly-rotating ([*near-extremal*]{}) black holes, one can use an alternative mass formula for the stationary bound-state scalar fields (see [@Hodkr] for details). 

It is worth emphasizing again that, as we explicitly shown in this paper (see the data presented in Table \[Table1\]), the analytical formula (\[Eq27\]) is not only valid in the entire physical regime $a/M\in(0,1)$ of the supporting black hole, but it also agrees remarkably well with the exact (numerically computed [@Herkr]) proper masses which characterize the stationary bound-state scalar fields in the spinning Kerr black-hole spacetime.