Onset of superradiant instabilities in the composed Kerr-black-hole-mirror bomb

It was first pointed out by Press and Teukolsky that a system composed of a spinning Kerr black hole surrounded by a reflecting mirror may develop instabilities. The physical mechanism responsible for the development of these exponentially growing instabilities is the superradiant amplification of bosonic fields confined between the black hole and the mirror. A remarkable feature of this composed black-hole-mirror-field system is the existence of a critical mirror radius, $r^{\text{stat}}_{\text{m}}$, which supports {\it stationary} (marginally-stable) field configurations. This critical (`stationary') mirror radius marks the boundary between stable and unstable black-hole-mirror-field configurations: composed systems whose confining mirror is situated in the region $r_{\text{m}}<r^{\text{stat}}_{\text{m}}$ are stable (that is, all modes of the confined field decay in time), whereas composed systems whose confining mirror is situated in the region $r_{\text{m}}>r^{\text{stat}}_{\text{m}}$ are unstable (that is, there are confined field modes which grow exponentially over time). In the present paper we explore this critical (marginally-stable) boundary between stable and explosive black-hole-mirror-field configurations. It is shown that the innermost ({\it smallest}) radius of the confining mirror which allows the extraction of rotational energy from a spinning Kerr black hole approaches the black-hole horizon radius in the extremal limit of rapidly-rotating black holes. We find, in particular, that this critical mirror radius (which marks the onset of superradiant instabilities in the composed system) scales linearly with the black-hole temperature.


I. INTRODUCTION
One of the most intriguing phenomenon in black-hole physics is the superradiant scattering of bosonic fields by spinning black holes: it was first pointed out by Zel'dovich [1] that a co-rotating integer-spin wave field of frequency ω interacting with a spinning Kerr black hole can be amplified (gain energy) if the composed system is in the superradiant regime [2] ω < ω c ≡ mΩ H .
Here m is the azimuthal harmonic index of the incident wave field, and is the angular velocity of the spinning Kerr black hole [3]. The amplification of the scattered bosonic fields in the superradiant regime (1) is accompanied by a decrease in the rotational energy and angular momentum of the central spinning black hole [4][5][6][7][8].
Soon after Zel'dovich's discovery [1] of the superradiance phenomenon in black-hole physics, it was realized by Press and Teukolsky [9] that the coupled black-hole-field system may develop exponentially growing instabilities. This unstable system, known as the black-hole bomb, is composed of three ingredients [9]: (1) a spinning black hole whose rotational energy serves as the energy source of the composed system, (2) a co-rotating bosonic cloud which orbits the central black hole and interacts with it to extract its rotational energy, and (3) a reflecting mirror which surrounds the central black hole and prevents the amplified bosonic field from radiating its energy to infinity [10,11].
A remarkable feature of this composed physical system is the existence, for each given set (ā, l, m) of the black hole and field parameters [12], of a critical (minimum) mirror radius which marks the boundary between stable and unstable black-hole-mirror-field configurations. The critical ('stationary' [13]) mirror radius, r stat m (ā, l, m), corresponds to stationary confined field configurations which are characterized by the critical superradiant frequency (1).
The composed black-hole-mirror-scalar-field system was studied in the important work by Cardoso et. al. [14,15]. In particular, it was shown in [14] that, for a given set of parameters (ā, l, m), black-hole-mirror-field systems whose mirror radii lie in the regime r m < r stat m (ā, l, m) are stable (the confined scalar mode decays in time) whereas black-hole-mirror-field systems whose mirror radii lie in the regime r m > r stat m (ā, l, m) are unstable (the confined scalar mode grows exponentially over time).
The numerical results presented in [14] indicate that the critical ('stationary' [13]) mirror radius, r stat m (ā, l, m), has the following three important features: • For a confined field mode of given harmonic indexes (l, m) [see Eq. (10) below], the stationary mirror radius is a decreasing function of the black-hole angular momentum a. In other words, rapidly-rotating Kerr black holes are characterized by stationary mirror radii which are smaller (closer to the black-hole horizon) than the corresponding stationary mirror radii of slowly-rotating black holes.
• For a given value of the spheroidal harmonic index l of the confined field, the stationary mirror radius decreases with increasing values of the azimuthal harmonic index m.
The equatorial m = l mode is therefore characterized by the innermost (smallest) stationary mirror radius among confined field modes which share the same spheroidal harmonic index l.
• For confined equatorial m = l modes, the stationary mirror radius decreases with increasing values of the spheroidal harmonic index l.
From these characteristics of the stationary mirror radius, one concludes that the asymptotic radius provides the innermost location of the confining mirror. The physical significance of this critical mirror radius, r * m , lies in the fact that this is the innermost (smallest) radius of the confining mirror which allows the extraction of rotational energy from spinning Kerr black holes. Below we shall explore the physical properties of this critical mirror radius. Cardoso et. al. [14] also provided an analytic treatment of the black-hole-mirror-field system in the small frequency regime aω ≪ 1. Substituting the critical (marginal) superradiant frequency ω c = mΩ H [see Eq. (1)] into equation (30) of [14], one finds that the stationary mirror radius r stat m (ā, l, m) is given as a solution of the characteristic equation where J l+1/2 (x) is the Bessel function. It should be emphasized that the characteristic equation (4) for the 'stationary' (critical) radii of the mirror is valid in the regime aω ≪ 1 considered in [14]. Thus, the characteristic equation (4) can only determine the stationary mirror radii of slowly-rotating black holes in the regime mā ≪ 1.
One of the goals of the present study is to obtain an analogous characteristic equation for the stationary mirror radii of rapidly-rotating (ā ≃ 1) black holes. As discussed above, the numerical results presented in [14] indicate that these near-extremal black holes are characterized by stationary mirror radii which are closer to the black-hole horizon than the corresponding stationary mirror radii (4) of slowly-rotating black holes. Below we shall confirm this expectation analytically.
In addition, as we shall show below, our characteristic equation for the critical ('stationary' [13]) mirror radii of rapidly-rotating black holes [see Eq. (21) below] is valid for arbitrarily large values of the harmonic indexes (l, m) of the confined field [16]. This fact will allow us to address the following interesting question regarding the nature of this critical mirror radius: what is the asymptotic behavior of the stationary mirror radius in the eikonal l ≫ 1 limit?
As discussed above, the asymptotic mirror radius, r * m ≡ r stat m (ā → 1, l = m → ∞), corresponds to the innermost location of the confining mirror which allows the extraction of rotational energy from spinning Kerr black holes. In addition, for generic confined field configurations [17], this critical mirror radius marks the boundary between stable and explosive black-hole-mirror-field configurations. One of the goals of the present study is to determine this fundamental (asymptotic) mirror radius r * m .

II. DESCRIPTION OF THE SYSTEM
The explored physical system is composed of a spinning Kerr black hole of mass M and angular momentum Ma linearly coupled to a massless scalar field Ψ. In the Boyer-Lindquist coordinate system (t, r, θ, φ) the black-hole spacetime geometry is described by the line-element [18,19] ds 2 = − 1 − 2Mr ρ 2 dt 2 − 4Mar sin 2 θ ρ 2 dtdφ + ρ 2 ∆ dr 2 + ρ 2 dθ 2 + r 2 + a 2 + 2Ma 2 r sin 2 θ ρ 2 sin 2 θdφ 2 , (5) where ∆ ≡ r 2 − 2Mr + a 2 and ρ 2 ≡ r 2 + a 2 cos 2 θ. The zeroes of ∆ determine the radii of the black-hole (event and inner) horizons: As discussed above, the critical ('stationary' [13]) mirror radius which characterizes the composed black-hole-mirror-field system is a decreasing function of the black-hole rotation parameterā. Thus, rapidly-rotating black holes are expected to be characterized by the smallest (innermost) stationary mirror radii. In the present study we shall analyze the physical properties of the black-hole-mirror-field system in this physically interesting regime of rapidly-rotating Kerr black holes withā The dynamics of the scalar field Ψ in the curved geometry is determined by the Klein- which, in the rotating Kerr spacetime (5) becomes [20,21] ( It is convenient to decompose the scalar field Ψ in the form in which case one finds [20] that the radial R lm and angular S lm wave functions are determined by two coupled ordinary differential equations of the confluent Heun type [22][23][24][25], see Eqs. (11) and (13) below.
It is worth emphasizing that the sign of ℑω in (10)  The angular functions S lm (θ; aω) are known as the spheroidal harmonic functions. These functions are solutions of the angular differential equation [22,23,25,26] in the interval θ ∈ [0, π]. Regular eigenfunctions [27] exist for a discrete set {A lm (aω)} of angular eigenvalues which are labeled by the two integers m and l ≥ |m|. These angular eigenvalues can be expanded in the form [28][29][30] A lm (aω) = l(l where the expansion coefficients {c k (l, m)} are given in [28][29][30].
The radial functions R lm (r; a, ω) satisfy the ordinary differential equation [22] ∆ where K ≡ (r 2 + a 2 )ω − ma. Note that the angular differential equation (11) and radial differential equation (13)  We shall be interested in solutions of the radial wave equation (13) with the physical requirement (boundary condition) of purely ingoing waves (as measured by a comoving observer) crossing the black-hole horizon [22]. As shown in [22], this boundary condition corresponds to the behavior of the radial eigenfunction in the vicinity of the black-hole horizon, where the "tortoise" radial coordinate y is defined by dy = [(r 2 + a 2 )/∆]dr. In addition, following [7,14] we shall assume that the scalar field vanishes at the location r m of the confining mirror: For the analysis of the radial Teukolsky equation (13), it is convenient to define new dimensionless variables in terms of which Eq. (13) becomes where V ≡ K 2 /r 2 + x(x + τ ) − (a 2 ω 2 − 2maω + A) and K = r 2 + ωx 2 + r + kx + r + ̟τ /2.

III. MARGINALLY STABLE BLACK-HOLE-MIRROR-FIELD CONFIGURA-TIONS
We shall now analyze the stationary (ℑω = 0) resonances of the composed system.
As discussed above, these stationary (marginally-stable) black-hole-mirror-field configurations are characterized by the critical frequency (1) for superradiant scattering in the blackhole spacetime. In particular, in this section we shall find the discrete set of mirror radii, {r stat m (ā, l, m; n)}, which support stationary confined field configurations. (Here n = 1, 2, 3, ... is the resonance parameter).
We consider a rapidly-rotating Kerr black hole [31] surrounded by a reflecting mirror which is placed in the vicinity of the black-hole horizon. In particular, we shall assume the following inequalities: In the near-horizon region x ≪ 1 the radial equation is given by (17) The physical radial solution obeying the ingoing boundary condition (14) at the black-hole horizon with the critical (marginallystable) superradiant frequency ̟ = 0 is given by [22,30] R where 2 F 1 (a, b; c; z) is the hypergeometric function [30], and We shall henceforth consider the case of real δ [32,33]. The mirror-like boundary condition (15)] now reads the stationary mirror radii is valid in the complementary regime x m ≫ 1, or equivalently in the regime of slowly-rotating black holes with mā ≪ 1.
One important conclusion which can immediately be drawn from the resonance condition (21) is the fact that, for rapidly-rotating black holes, the critical mirror radius scales linearly with the black-hole temperature [34]: x stat m ∝ τ .
This implies that the stationary mirror radius x stat m is a decreasing function of the black-hole rotation parameterā (an increasing function of the black-hole dimensionless temperature τ ).
As we shall now show, for small and moderate values of the field harmonic indexes (l, m), the stationary mirror radii x stat m (ā, l, m; n) are characterized by the relation in which case the resonance condition (21) can be solved analytically. In the regime (23) one can use the large-z asymptotic behavior of the hypergeometric function 2 F 1 (a, b; c; z) [30] to approximate the resonance condition (21) by which yields x stat It is easy to verify that, for real values of the angular eigenvalue δ, the characteristic equation (25) corresponds to real values of the stationary mirror radii [15]: taking the logarithm of both sides of (25), one obtains the characteristic equation [35] 2δ ln for the discrete set {x stat m (ā, l, m; n)} of stationary mirror radii, where n = 1, 2, 3, ... is the resonance parameter of the mode. Inspection of the resonance condition (26)  and are therefore purely real numbers (see Eq. 6.1.23 of [30]). The fourth term on the r.h.s of (26) is obviously a purely real number. This implies that the r.h.s of (26) is a purely real number. One therefore concludes that, for real values of the angular eigenvalue δ, the dimensionless mirror radii obtained from the resonance condition (26) are purely real numbers. These stationary [13] mirror radii are given by In Table I we display the discrete radii of the reflecting mirror corresponding to the stationary confined equatorial l = m = 2 mode as obtained from the analytical formula (27) [36]. We also display the corresponding mirror radii as obtained from a direct numerical solution [37] of the characteristic resonance equation (21). One finds a remarkably good agreement between the exact mirror radii [as obtained numerically from the resonance condition (21)] and the approximated mirror radii [as obtained from the analytical formula (27)].    I: Stationary resonances of the composed black-hole-mirror-field system. We display the scaled mirror radii, x stat m (l = m = 2; n)/τ , corresponding to the stationary confined mode with l = m = 2, as obtained from a direct numerical solution of the characteristic resonance equation (21). We also display the corresponding mirror radii as obtained from the analytical formula (27).
One finds a remarkably good agreement between the exact (numerically computed) mirror radii and the approximated (analytically calculated) mirror radii.
In Table II we display the discrete set of mirror radii, {x stat m (l = m; n)/τ }, corresponding to composed black-hole-mirror configurations with stationary confined equatorial (l = m) field modes [38][39][40]. The data presented in Table II correspond to a direct numerical solution [37] of the characteristic resonance condition (21). It is worth recalling that, for a confined pure field (that is, a confined mode characterized by a given value of the azimuthal harmonic index m), the smallest stationary mirror radius, x stat m (ā, m; n = 1), marks the boundary between stable and unstable black-hole-mirror-field configurations for that particular pure mode.
From Table II one learns that the stationary mirror radius, x stat m (ā, m; n = 1), is a decreasing function of the azimuthal harmonic index m. This is a generic feature of the composed black-hole-mirror-field system: in Table III we display the values of the stationary mirror radii in the asymptotic m ≫ 1 regime [41,42]. One finds that the data presented in Table   III is described extremely well by the simple asymptotic formula: What we find most interesting is the fact that the coefficient α in (28) has a finite asymptotic value. This fact suggests that the composed Kerr-black-hole-mirror-field system is characterized by a finite asymptotic limit of the critical ('stationary') mirror radius: (3)]. This feature of the spinning black-hole-mirror-field bomb should be contrasted with the corresponding case of the charged black-hole-mirror-field bomb [6,7]. In the charged case one finds [7] that the critical ('stationary') mirror radius x stat m can be made arbitrarily small (that is, the mirror can be placed arbitrarily close to the black-hole horizon) in the asymptotic [43] qQ ≫ 1 regime [7]: x stat m (qQ ≫ 1)   (20)].
One finds that the critical (stationary) radius of the mirror, x stat m (m; n), is a decreasing function of the azimuthal harmonic index m.
In order to support our findings, according to which the composed Kerr-black-hole-mirrorfield system is characterized by a finite asymptotic value of the critical mirror radius, we   III: Stationary resonances of the composed black-hole-mirror-field system. We display the dimensionless mirror radii, x stat m (m; n = 1)/τ , corresponding to stationary confined modes in the asymptotic l = m ≫ 1 regime. One finds that the critical (stationary) radius of the mirror, x stat m (m; n = 1), decreases monotonically to an asymptotic finite value [see Eq. (29)] in the eikonal l ≫ 1 limit.
shall prove in the next section that composed systems whose mirror radii lie in the regime x m /τ ≪ 1 cannot support stationary confined field configurations.

IV. NO STATIONARY CONFINED FIELD CONFIGURATIONS IN THE
We have seen that the (scaled) critical mirror radius, x stat m (m)/τ , decreases monotonically with increasing values of the azimuthal harmonic index m. This fact naturally gives rise to the following question: Can the ratio x stat m (m)/τ be made arbitrarily small in the asymptotic eikonal regime m → ∞? One can also formulate this question in a more practical form: Is it possible to extract the black-hole rotational energy by placing a confining mirror arbitrarily close to its horizon [44]?
Our analysis in Sec. III provides compelling evidence that the answer to the above questions is "no". In particular, our results suggest that the Kerr-black-hole-mirror-field system is characterized by a finite asymptotic value of the critical ('stationary' [13]) mirror (29)]. In order to support this finding, we shall prove in this section that composed black-hole-mirror systems whose mirror radii lie in the regime x m /τ ≪ 1 cannot support stationary confined field configurations [45].
To that end, it proves useful to write the radial Teukolsky equation in the form of a Schrödinger-like wave equation [46]: where ψ = (r 2 + a 2 ) 1/2 R and the "tortoise" radial coordinate y was defined in Sec. II. The potential V in Eq. (31) is given by [46] where and In the near-horizon region [47] x ≪ τ , one finds [48] y ≃ r 2 which implies ∆ ≃ (r + − r − ) 2 e τ y/2M and where In the eikonal regime, l, m ≫ 1, one can use the relation [42] in order to find [see Eq. (34)] From (40) one concludes that for all values of the dimensionless ratio µ (note that µ ≤ 1).
Taking cognizance of Eqs. (31), (37), and (38), one obtains the radial wave equation for stationary field configurations (with ω = mΩ H ). Defining the radial wave equation (42) can be written in the form Using Eq. (9.1.54) of [30], one finds that the physical solution to the radial equation (44) is described by the Bessel function of the first kind: This radial solution can also be written [see Eqs. (36) and (43)] in the form [49] ψ Taking cognizance of Eq. (41), one finds that the argument of the Bessel function in (46) is purely imaginary. It is well known that the Bessel function J 0 (ix) with x ∈ R has no zeroes [that is, ψ(x m ) = 0 for x m ∈ R]. One therefore concludes that stationary solutions (with ω = mΩ H ) of the wave field are not compatible with the mirror-like boundary condition (15) for confining mirrors in the region x m ≪ τ [see Eq. (35)].
The proof presented in this section supports our previous conclusion that, the composed Kerr-black-hole-mirror-field system is characterized by a finite asymptotic value [see Eq.
(29)] of the dimensionless ratio x * m /τ , where x * m is the critical mirror radius.

V. SUMMARY AND DISCUSSION
In this paper, we have used analytical tools in order to study the stationary (marginallystable) resonances of the composed black-hole-mirror-field system. These resonances are fundamental to the physics of confined bosonic fields in black-hole spacetimes. In particular, these resonances mark the onset of superradiant instabilities in the black-hole bomb mechanism of Press and Teukolsky [9].
We have derived the characteristic resonance condition (21) for the marginally-stable (stationary) black-hole-mirror-field configurations. In particular, it was shown that, for rapidly-rotating black holes, the stationary resonances of the system are described by the simple zeroes of the hypergeometric function.
The characteristic resonance condition (21) determines the discrete set of mirror radii, {x stat m (ā, l, m; n)}, which support stationary confined field configurations in the black-hole spacetime. One nice feature of this resonance condition lies in the fact that it immediately reveals that the critical ('stationary' [13]) mirror radii scale linearly with the black-hole temperature [34], see Eq. (22). This fact implies that x stat m (ā, l, m; n) is a decreasing function of the black-hole rotation parameterā -the larger the black-hole spin, the closer to the black-hole horizon the confining mirror can be placed.
It was shown that the stationary mirror radius, x stat m (m), decreases monotonically with increasing values of the azimuthal harmonic index m of the confined field mode. In particular, our results provide compelling evidence that the composed Kerr-black-hole-mirrorfield system is characterized by a finite asymptotic value of the critical mirror radius: x stat m (m ≫ 1)/τ ≃ 0.36 [50].
The physical significance of the asymptotic stationary mirror radius, x * m ≡ x stat m (ā → 1, l = m → ∞), lies in the fact that it is the innermost location of the confining mirror which allows the extraction of rotational energy from spinning black holes. This implies that, for generic confined field configurations [17], this critical mirror radius marks the onset of superradiant instabilities in the composed system: composed black-hole-mirror-field systems whose mirror radii lie in the regime x m < x * m are stable (that is, all modes of the confined field decay in time), whereas composed black-hole-mirror-field systems whose mirror radii lie in the regime x m > x * m are unstable (that is, there are confined field modes which grow exponentially over time).
sented in [14]. In addition, as we shall discuss below, another notable advantageous of our analysis lies in the fact that, the characteristic resonance condition (21) can easily be solved for asymptotically large values of the harmonic indexes (l, m). Below we shall be interested in this asymptotic (eikonal) regime.
[38] One of our goals is to determine the smallest mirror radius r * m [see Eq. (3)] which allows the extraction of rotational energy from spinning Kerr black holes. (This critical mirror radius also marks the boundary between stable and unstable black-hole-mirror-field configurations.) As discussed above, the numerical results of [14] indicate that, for a given value of the spheroidal harmonic index l of the confined field, the equatorial m = l mode is characterized by the innermost location of the stationary mirror radius. We therefore focus our attention on these equatorial modes.
[39] It is worth emphasizing again that the mirror-like boundary condition (21) is valid for mirror radii in the x m ≪ 1 regime [see Eq. (18)]. Solving the resonance condition (21) for small and moderate values of the spheroidal harmonic index l, one finds x stat m (ā, m; n = 1)/τ 1, see Table II. Thus, the stationary mirror radii presented in Table II are valid in the regime of rapidly-rotating (near-extremal) black holes with τ x stat m (ā, m; n = 1) ≪ 1.
[43] Here Q and q are respectively the electric charge of the Reissner-Nordström black hole and the charge coupling constant of the confined field [7].
[44] The answer to this question may one day be of practical importance: In order to save construction materials, one would like to build the confining mirror as close as possible to the black-hole horizon.
[45] The hypergeometric function 2 F 1 (a, b; c; z) is characterized by the property 2 F 1 (a, b; c; z) → 1 for a · b · z/c ≪ 1 [30]. Taking cognizance of the characteristic resonance condition (21), one realizes that there are no confined stationary field configurations in the regime m 2 x m /τ ≪ 1