The charged black-hole bomb: A lower bound on the charge-to-mass ratio of the explosive scalar field

The well-known superradiant amplification mechanism allows a charged scalar field of proper mass $\mu$ and electric charge $q$ to extract the Coulomb energy of a charged Reissner-Nordstr\"om black hole. The rate of energy extraction can grow exponentially in time if the system is placed inside a reflecting cavity which prevents the charged scalar field from escaping to infinity. This composed black-hole-charged-scalar-field-mirror system is known as the {\it charged black-hole bomb}. Previous numerical studies of this composed physical system have shown that, in the linearized regime, the inequality $q/\mu>1$ provides a necessary condition for the development of the superradiant instability. In the present paper we use analytical techniques to study the instability properties of the charged black-hole bomb in the regime of linearized scalar fields. In particular, we prove that the lower bound ${{q}\over{\mu}}>\sqrt{{{r_{\text{m}}/r_--1}\over{r_{\text{m}}/r_+-1}}}$ provides a necessary condition for the development of the superradiant instability in this composed physical system (here $r_{\pm}$ are the horizon radii of the charged Reissner-Nordstr\"om black hole and $r_{\text{m}}$ is the radius of the confining mirror). This {\it analytically} derived lower bound on the superradiant instability regime of the composed black-hole-charged-scalar-field-mirror system is shown to agree with direct {\it numerical} computations of the instability spectrum.


I. INTRODUCTION
Kerr black holes may contain large amounts of rotational energies which can be released by bosonic fields that scatter off these spinning black holes. In this physical process, which is known as superradiant scattering [1][2][3], an incident bosonic field whose proper frequency lies in the superradiant regime [1][2][3][4] 0 < ω < mΩ H (1) can extract the rotational energy and angular momentum of the spinning Kerr black hole (Here m is the azimuthal harmonic index of the incident bosonic field and Ω H is the angular velocity of the black-hole horizon). The rate of energy extraction from the black hole can grow exponentially in time if the bosonic field is prevented from escaping to infinity. The required confinement mechanism can be provided either by a reflecting mirror which surrounds the black hole [2,5] or, for a massive bosonic field, by the mutual gravitational attraction between the central black hole and the extracting field [6,7].
It should be emphasized that not all bosonic modes trigger the black-hole superradiant instability. In particular, it was proved in [8] that the inequality provides a necessary condition for the development of the superradiant instability in the composed Kerr-black-holemassive-scalar-field system, where µ is the proper mass of the exploding scalar field. As pointed out by Bekenstein [9], an analogous superradiant amplification of bosonic fields may occur when a charged field scatters off a charged black hole. In particular, a charged scalar field whose proper frequency lies in the superradiant regime [9] 0 < ω < qΦ H can extract the Coulomb energy and electric charge of a charged Reissner-Nordström (RN) black hole (Here q is the charge coupling constant of the incident scalar field and Φ H is the electric potential of the charged black hole). Interestingly, it was proved in [10] that, contrary to the spinning (Kerr) case, in the charged (RN) case the gravitational attraction between the black hole and the massive charged scalar field cannot provide the confinement mechanism which is required in order to trigger the black-hole superradiant instability. The charged black-hole bomb must therefore include a reflecting mirror which surrounds the black hole and prevents the amplified charged bosonic field from escaping to infinity [11][12][13].
In a very interesting work, Degollado et. al. [11] have used numerical techniques to study the instability properties of the composed RN-black-hole-charged-scalar-field-mirror system. In particular, it was found in [11] that, in the linearized regime [14], the inequality provides a necessary condition for the development of the superradiant instabilities in this charged system. The main goal of the present paper is to explore the superradiant instability regime of the composed RN-linearizedcharged-scalar-field-mirror system (the charged black-hole bomb) using analytical techniques. In particular, below we shall provide an analytical explanation for the characteristic inequality (4) observed numerically in the interesting study of Degollado et. al. [11]. Moreover, in this paper we shall derive a stronger lower bound [see Eq. (45) below] on the dimensionless charge-to-mass ratio which characterizes the explosive charged massive scalar fields.

II. DESCRIPTION OF THE SYSTEM
We shall study the dynamics of a charged massive scalar field Ψ linearly coupled to a non-extremal charged RN black hole of mass M and electric charge Q. The charged RN black-hole spacetime is described by the line element [15,16] where the metric function f (r) is given by The zeros of f (r), determine the horizon radii of the charged RN black hole [15]. The dynamics of a test scalar field Ψ of proper mass µ and charge coupling constant q [17,18] in the background of the RN black-hole spacetime is governed by the Klein-Gordon wave equation [19][20][21] where A ν = −δ 0 ν Q/r is the electromagnetic potential of the charged black hole. One can decouple the radial and angular parts of the scalar field Ψ and express it in the form where ω, l, and m are respectively the conserved frequency of the field mode and its angular harmonic indices [22,23]. It is worth noting that the sign of ℑω in (9) determines the (in)stability properties of the scalar field mode: stable modes (that is, modes decaying in time) are characterized by ℑω < 0, whereas unstable modes (that is, modes growing exponentially in time) are characterized by ℑω > 0. Stationary modes with ℑω = 0 mark the boundary between stable and unstable solutions of the Klein-Gordon wave equation (8). These marginally stable field modes are characterized by the critical (marginal) frequency [see Eq. (3)] for the superradiant scattering phenomenon [9]. Substituting the scalar field decomposition (9) into the Klein-Gordon wave equation (8) and using the line element (5) of the RN black-hole spacetime, one finds that the radial function R(r) is determined by the characteristic radial equation [19][20][21] where ∆ ≡ r 2 f (r) (12) and Here K l = l(l + 1) (where m and l ≥ |m| are integers) are the characteristic eigenvalues of the angular function S(θ) [19][20][21]. The characteristic equation (11) for the radial eigenfunction R(r) should be supplemented by the physical boundary condition of purely ingoing waves at the black-hole horizon [5][6][7]11]: where the radial coordinate y is determined by the relation dy = dr/f (r) [see Eq. (18) below]. For field modes in the superradiant regime (3), the near-horizon boundary condition (14) corresponds to an outgoing flux of Coulomb energy and electric charge from the charged RN black hole [9][10][11][12][13]. In addition, the reflecting mirror which surrounds the composed black-hole-field system dictates the boundary condition [5, [11][12][13] R(r = r m ) = 0 (15) for the confined scalar field, where r m is the radial location of the mirror.

III. THE EFFECTIVE RADIAL POTENTIAL OF THE COMPOSED RN-BLACK-HOLE-CHARGED-MASSIVE-SCALAR-FIELD SYSTEM
The radial equation (11), together with the boundary conditions (14) and (15), determine a discrete family of complex field resonances {ω n (r m )} [5,11,12,24]. As mentioned above, Degollado et. al. [11] have performed a very interesting numerical study of these characteristic resonances of the composed RN-black-hole-charged-scalarfield-mirror system. In particular, Degollado et. al. [11] have found numerically that unstable (exploding) charged field modes are characterized by the property q > µ [see Eq. (4)]. The main goal of the present paper is to provide an analytical explanation for this (numerically observed) characteristic inequality. Moreover, below we shall derive a stronger lower bound on the dimensionless charge-to-mass ratio of these explosive (unstable) charged massive scalar fields.
In order to analyze the physical properties of the composed RN-black-hole-charged-scalar-field-mirror system, we shall first express the radial equation (11) for the charged massive scalar fields in the form of a Schrödinger-like wave equation. To this end, it proves useful to define the new radial function in terms of which the radial equation (11) can be expressed in the form where the radial coordinate y is defined by the relation The effective radial potential in (17) is given by where In the next section we shall analyze the near-horizon properties of the effective radial potential V (r) that appears in the Schrödinger-like wave equation (17) for the charged massive scalar fields in the charged RN black-hole spacetime. We shall then use these properties in order to study the near-horizon spatial behavior of the radial eigenfunction ψ which characterizes the charged massive scalar fields.

IV. THE NEAR-HORIZON BEHAVIOR OF THE CHARGED SCALAR EIGENFUNCTIONS
Our main goal is to explore the onset of superradiant instabilities in the composed RN-black-hole-charged-scalarfield-mirror system. Thus, we shall henceforth analyze the behavior of the marginally stable (stationary) charged field modes (10) which mark the boundary of the superradiant instability regime [25]. In particular, in this section we shall study the near-horizon spatial behavior of the radial eigenfunction ψ which characterizes the stationary (marginally stable) resonances of the charged scalar fields in the charged RN black-hole spacetime. Specifically, we shall prove below that this characteristic function is a positive [26], increasing, and convex function in the near-horizon r−r+ r+−r− ≪ 1 region of the RN black-hole spacetime.
To that end, we shall first define the dimensionless variables and study the near-horizon x ≪ τ [27] behavior of the effective radial potential (19). Substituting the characteristic resonant frequency (10) of the marginally stable charged scalar fields into the expression (19) of the effective radial potential, one finds in the near-horizon region where [see Eq. (20)] Remembering that 1 − Q 2 /r 2 + > 0 , one finds the characteristic inequality for the massive charged scalar fields. Equations (22) and (25) imply that in the near-horizon region (23). Integrating the relation (18) in the near-horizon region, one finds which implies [28] x = e τ y/r+ [1 + O(e τ y/r+ )].
That is, Taking cognizance of the characteristic near-horizon spatial behavior (34) of the radial eigenfunction ψ [32], together with the boundary condition (15) which is dictated by the presence of the reflecting mirror, one concludes that the radial eigenfunction ψ must have (at least) one maximum point, x = x max , between the black-hole horizon [where ψ is a positive and increasing function, see (34)] and the reflecting mirror [where ψ vanishes, see (15)]. We note, in particular, that the radial eigenfunction ψ is characterized by the relations at the maximum point x = x max .

V. THE SUPERRADIANT INSTABILITY REGIME OF THE CHARGED BLACK-HOLE BOMB
It the previous section we have proved that the radial eigenfunction ψ, which characterizes the confined charged scalar fields in the charged RN black-hole spacetime, must have (at least) one maximum point, r = r max , between the black-hole horizon and the reflecting mirror. That is, Taking cognizance of Eqs. (17) and (35), one finds that the effective radial potential is characterized by the relation at this maximum point. We shall now use this characteristic inequality in order to derive a generic bound on the superradiant instability regime of the charged black-hole bomb. Substituting the characteristic resonant frequency (10) of the marginally stable charged scalar fields into the expression (19) of the effective radial potential, one finds the relation which yields the inequality [see (37)] Using the inequality r max < r m [see Eq. (36)], one finds and l(l + 1) In addition, for charged RN black holes the expression 2M/r 3 − 2Q 2 /r 4 is a concave function whose maximum is located at r = 4Q 2 /3M . One can therefore write [33] 2M From Eqs. (20), (41), and (42), one finds the lower bound Substituting the inequalities (40) and (43) into (39), one can write the lower bound on the dimensionless quantity qQ in terms of the physical parameters {r ± , r m } of the black hole and its confining mirror: It is worth emphasizing that the analytically derived lower bound (44) provides a necessary condition for the development of the superradiant instabilities in the composed RN-black-hole-charged-scalar-field-mirror system [34,35].

VI. NUMERICAL CONFIRMATION
We shall now verify the validity of the analytically derived lower bound (44) on the superradiant instability regime of the charged black-hole bomb. The instability spectrum of this composed RN-black-hole-charged-massive-scalar-fieldmirror system was investigated numerically in [11]. In Table I we display the dimensionless ratio (qQ) stat /(qQ) bound , where (qQ) stat is the numerically computed [11] value of the quantity qQ which corresponds to the stationary (marginally stable) charged scalar configurations [36], and (qQ) bound is the analytically derived lower bound on the superradiant instability regime given by Eq. (44). One finds from Table I that the charged black-hole bomb is characterized by the relation (qQ) stat /(qQ) bound > 1, in agreement with the analytically derived lower bound (44).  The superradiant instability regime of the composed RN-black-hole-charged-scalar-field-mirror system (the charged black-hole bomb). We display the dimensionless ratio (qQ) stat /(qQ) bound , where (qQ) stat is the numerically computed [11] value of the quantity qQ which corresponds to the stationary (marginally stable) charged scalar configurations [36], and (qQ) bound is the analytically derived lower bound on the superradiant instability regime given by Eq. (44). The data presented is for the case M µ = 0.3, M q = 0.36, and l = 1. One finds that the superradiant instability regime of the charged black-hole bomb is characterized by the relation (qQ) stat /(qQ) bound > 1, in agreement with the analytically derived lower bound (44).

VII. SUMMARY AND DISCUSSION
We have studied analytically the superradiant instability regime of the charged black-hole bomb. This physical system is composed of a charged massive scalar field which, on the one hand, extracts the Coulomb energy of a charged Reissner-Nordström black hole and, on the other hand, is prevented from escaping to infinity by a reflecting mirror which surrounds the black hole. We have proved that in order for the superradiant instability to develop in this composed charged black-hole bomb, the dimensionless quantity qQ of the black-hole-field system must be bounded from below as in (44).
In a very interesting study, Degollado et. al. [11] have used numerical techniques to study the instability spectrum of the charged black-hole bomb. In particular, it was found in [11] that the inequality q/µ > 1 [see Eq. (4)] provides a necessary condition for the development of the superradiant instability in this composed system. We can now provide an analytical explanation for this numerically observed [11] necessary condition: Using the relation Q 2 = r + r − , one finds from (44) on the dimensionless charge-to-mass ratio of the scalar fields in the explosive (unstable) regime of the charged blackhole bomb [38,39]. It is worth emphasizing that this lower bound provides a necessary condition for the development of the superradiant instabilities in the composed RN-black-hole-charged-scalar-field-mirror system [40]. Thus far, we have treated the composed charged black-hole bomb at the classical level. It should be emphasized, however, that the well known Schwinger quantum pair-production mechanism [41][42][43][44] restricts the physical parameters of the composed RN-black-hole-charged-massive-scalar-field system. In particular, this vacuum polarization effect sets the upper bound [41][42][43][44][45] E + ≪ E c ≡ µ 2 /qh on the strength of the black-hole electric field (here E + = Q/r 2 + is the electric field at the horizon of the charged RN black hole). The quantum production of charged particle/anti-particle pairs in the charged black-hole spacetime (the Schwinger discharge of the RN black hole) therefore sets the upper bound qQ ≪ µ 2 r 2 + on the physical parameters of the composed black-hole-field system. Taking cognizance of Eq. (44) [46] one finds that, in the superradiant explosive regime, the dimensionless quantity qQ is restricted by the two inequalities µr + < qQ ≪ µ 2 r 2 + .
The two inequalities in (46) imply that, in physically acceptable situations [47], the explosive charged massive scalar fields must be characterized by the strong inequalities 1 ≪ µr + < qQ .
It is worth noting that the physical restriction µr + ≫ 1 [see (47)] imposed by the quantum Schwinger pair-production mechanism (the vacuum polarization effect) implies that, in physically acceptable situations [47], the lower bound (44) is well approximated by the lower bound (45) [48]. Finally, it is worth emphasizing again that in this study we have treated the charged massive scalar fields at the linear level. Our analytical results are therefore expected to be valid in the early stages of the development of the superradiant instability (that is, in the ignition stage of the black-hole bomb), when the external charged scalar fields are weak and can still be regarded as perturbation fields on the background of the charged RN black-hole spacetime. As we demonstrated explicitly in this paper, the main advantage of this perturbative (linearized) approach stems from the fact that the physical properties of the composed RN-black-hole-charged-massive-scalar-field-mirror system (the charged black-hole bomb) can be explored analytically in the linear regime. It should be emphasized, however, that the late-time (non-linear) development of the superradiant instability can only be tackled with numerical techniques, as recently done in the interesting numerical work of Sanchis-Gual et. al. [49].