Slowly decaying resonances of charged massive scalar fields in the Reissner-Nordstr\"om black-hole spacetime

We determine the characteristic timescales associated with the linearized relaxation dynamics of the composed Reissner-Nordstr\"om-black-hole-charged-massive-scalar-field system. To that end, the quasinormal resonant frequencies $\{\omega_n(\mu,q,M,Q)\}_{n=0}^{n=\infty}$ which characterize the dynamics of a charged scalar field of mass $\mu$ and charge coupling constant $q$ in the charged Reissner-Nordstr\"om black-hole spacetime of mass $M$ and electric charge $Q$ are determined {\it analytically} in the eikonal regime $1\ll M\mu<qQ$. Interestingly, we find that, for a given value of the dimensionless black-hole electric charge $Q/M$, the imaginary part of the resonant oscillation frequency is a monotonically {\it decreasing} function of the dimensionless ratio $\mu/q$. In particular, it is shown that the quasinormal resonance spectrum is characterized by the asymptotic behavior $\Im\omega\to0$ in the limiting case $M\mu\to qQ$. This intriguing finding implies that the composed Reissner-Nordstr\"om-black-hole-charged-massive-scalar-field system is characterized by extremely long relaxation times $\tau_{\text{relax}}\equiv 1/\Im\omega\to\infty$ in the $M\mu/qQ\to 1^-$ limit.


I. INTRODUCTION
Wheeler's 'no-hair' conjecture [1,2] presents a simple physical picture according to which asymptotically flat black holes in Einstein's theory of gravitation cannot support static matter fields outside their horizons. Interestingly, various no-hair theorems [1][2][3][4][5][6][7][8] provide strong support for the general validity of Wheeler's famous conjecture. It is therefore expected that fundamental matter fields that propagate in a static black-hole spacetime would eventually be absorbed by the black hole or be scattered away to infinity. For rotating black-hole spacetimes, a third possibility also exists [9,10]: thanks to the intriguing physical mechanism of superradiant scattering of bosonic fields in spinning black-hole spacetimes, these black holes can support stationary (rather than static) linearized bound-state massive scalar configurations in their external regions [9]. Moreover, as explicitly shown in [10], rotating black holes can also support genuine bosonic hair (that is, non-linear stationary bosonic field configurations) in their external regions. These rotating hairy black-hole-bosonic-field configurations [9,10] provide explicit counterexamples to the no-hair conjecture in asymptotically flat non-static spacetimes.
It is important to stress the fact that the elegant no-hair theorems [1][2][3][4][5][6][7][8], which are valid for asymptotically flat static black-hole configurations, say nothing about the timescale associated with the dynamical process of black-hole hair shedding [11]. This characteristic relaxation timescale, τ relax , will be the main focus of the present study.
The dynamics of fundamental matter and radiation fields in black-hole spacetimes are characterized by damped quasinormal oscillations of the form e −iωt [12][13][14]. These exponentially decaying oscillations are characterized by complex quasinormal resonant frequencies {ω n } n=∞ n=0 whose values depend on the physical parameters (such as mass, charge, angular momentum, and intrinsic spin) of the composed black-hole-field system. In accord with Wheeler's no-hair conjecture [1,2], these characteristic damped oscillations reflect the gradual decay of the fields in the external regions of the black-hole spacetimes. In particular, the characteristic timescale associated with the relaxation dynamics of an external field in a black-hole spacetime is determined by the imaginary part of the fundamental (least damped) quasinormal resonant frequency which characterizes the composed black-hole-field system: The main goal of the present paper is to determine the characteristic relaxation timescales, τ relax , associated with the relaxation dynamics of charged massive scalar fields in the charged Reissner-Nordström (RN) black-hole spacetime.
To that end, we shall explore below the quasinormal resonance spectrum which characterizes the linearized relaxation dynamics [15,16] of the composed RN-black-hole-charged-massive-scalar-field system. As we shall show below, the characteristic quasinormal resonances of this composed black-hole-field system can be studied analytically in the eikonal regime [17] where {q, µ} are respectively the charge coupling constant and proper mass of the field, and {M, Q} are respectively the mass and electric charge of the RN black hole. In particular, below we shall reveal the interesting fact that this composed RN-black-hole-charged-massive-scalar-field system is characterized by extremely long dynamical relaxation times, τ relax ≡ 1/ℑω 0 → ∞, in the limiting case M µ/qQ → 1 − [18,19].

II. DESCRIPTION OF THE SYSTEM
We shall analyze the quasinormal resonance spectrum which characterizes the linearized relaxation dynamics of a scalar field Ψ of mass µ and charge coupling constant q [20] in the spacetime of a Reissner-Nordström black hole of mass M and electric charged Q [21]. The curved black-hole spacetime is described by the line element [22] where The zeros of the radial function f (r), determine the horizon radii of the charged RN black hole. The familiar Klein-Gordon wave equation [23][24][25][26][27][28][29] [ determines the linearized dynamics of the charged massive scalar field in the curved RN black-hole spacetime, where A ν = −δ 0 ν Q/r is the electromagnetic potential of the charged black hole. Substituting the field decomposition [30] into the Klein-Gordon wave equation (6), and using the black-hole metric function (4), one obtains [23][24][25] two ordinary differential equations of the confluent Heun type [31,32] for the eigenfunctions R(r) and S(θ) which respectively describe the radial and angular behaviors of the charged massive scalar field in the curved black-hole spacetime. The ordinary differential equation which determines the spatial behavior of the radial eigenfunction R(r) is given by [23][24][25] where ∆ = r 2 f (r), and Here K l = l(l + 1) (with l ≥ |m|) is the characteristic eigenvalue of the angular eigenfunction S lm (θ) [31][32][33].
Defining the "tortoise" radial coordinate y by the differential relation and using the new radial eigenfunction one can transform the radial equation (8) into the more familiar form of a Schrödinger-like ordinary differential equation, where the effective radial potential in (12) is given by with H(r; M, Q, µ, l) = µ 2 + l(l + 1) The quasinormal resonant frequencies {ω n (M, Q, µ, q, l)} n=∞ n=0 , which characterize the linearized relaxation dynamics of the charged scalar field in the charged black-hole spacetime, are determined by imposing on the Schrödinger-like wave equation (12) the physically motivated boundary conditions of purely ingoing waves at the black-hole horizon and purely outgoing waves at spatial infinity [34]. That is, In the next section we shall study analytically the quasinormal resonance spectrum {ω n (M, Q, µ, q, l)} n=∞ n=0 which characterizes the relaxation dynamics of the composed Reissner-Nordström-black-hole-charged-massive-scalar-field system in the eikonal large-mass regime (2).

III. THE QUASINORMAL RESONANCE SPECTRUM OF THE COMPOSED REISSNER-NORDSTRÖM-BLACK-HOLE-CHARGED-MASSIVE-SCALAR-FIELD SYSTEM
In the present section we shall perform a WKB analysis in order to determine the complex resonant frequencies which characterize the composed Reissner-Nordström-black-hole-charged-massive-scalar-field system in the large-mass regime In the eikonal large-mass regime (16), the radial potential (13), which characterizes the dynamics of the charged massive scalar field in the charged RN black-hole spacetime, can be approximated by This radial potential has the form of an effective potential barrier whose maximum r 0 is located at As we shall now show, the fundamental complex resonances associated with the effective scattering potential (13) can be determined analytically in the large-mass regime (16) using standard WKB methods [35][36][37][38]. In particular, as shown in [35,36], the WKB resonance condition which characterizes the complex scattering resonances (the quasinormal frequencies) of the Schrödinger-like radial equation (12) in the eikonal large-frequency regime is given by where the various derivatives V (k) 0 ≡ d k V /dy k (with k ≥ 0) that appear in the WKB resonance equation (19) are evaluated at the maximum point y = y 0 (r 0 ) [see Eq. (18)] which characterizes the effective scattering potential V (y).
Substituting Eqs. (17) and (18) into the WKB resonance equation (19) and using the differential relation (10), one finds the characteristic resonance condition for the quasinormal resonant frequencies of the composed Reissner-Nordström-black-hole-charged-massive-scalar-field system. As we shall now show, the rather complicated resonance equation (20) can be solved analytically in the regime [39] ω R ≫ ω I .
This strong inequality, which characterizes the fundamental quasinormal frequencies of the composed black-hole-field system in the eikonal regime (2) [see Eqs. (25) and (29) below], enables one to decouple the real and imaginary parts of the WKB resonance equation (20). In particular, one finds for the real part of the resonance equation (20), and for the imaginary part of the resonance equation (20). Substituting Eq. (18) into Eq. (22), one finds and Note that the relations (24) and (25) are valid in the regimeμ <Q ≤ 1 [40], which corresponds to Interestingly, one finds from (25) that, for a given value of the dimensionless black-hole electric chargeQ, the real part of the resonant oscillation frequency, ω R , is a monotonically increasing function of the dimensionless ratioμ. In particular, one finds the limiting behaviors [see Eq. (25)] Substituting Eqs. (24) and (25) into Eq. (23), one finds for the imaginary parts of the quasinormal resonances which characterize the composed RN-black-hole-chargedmassive-scalar-field system in the regime (2). Interestingly, one finds from (29) that, for a given value of the dimensionless black-hole electric chargeQ, the imaginary part of the resonant oscillation frequency, ω I , is a monotonically decreasing function of the dimensionless ratioμ. In particular, one finds the limiting behaviors [see Eq. (29)] {M ω I → 1 −Q 2 (1 + 1 −Q 2 ) 2 · (n + 1/2) forμ/Q → 0} and Note that the expression (29) for the imaginary parts of the RN-black-hole-charged-massive-scalar-field resonances can be written in the compact form [see Eqs. (5) and (24)] where is the Bekenstein-Hawking temperature of the RN black hole [41][42][43].

IV. THE REGIME OF VALIDITY OF THE WKB APPROXIMATION
It is important to emphasize that our WKB results (25) and (29) for the real and imaginary parts of the quasinormal resonant frequencies which characterize the relaxation dynamics of the charged massive scalar fields in the charged RN black-hole spacetime were derived under the assumption that higher-order correction terms that appear in the large-frequency WKB approximation can be neglected. In particular, as explicitly shown in [35][36][37][38], an extension of the WKB approximation to include higher-order derivatives of the effective scattering potential yields the correction term on the r.h.s of the resonance equation (19). Thus, the WKB resonance condition (19) is valid provided Substituting Eqs. (10), (17), (24), and (25) into Eq. (34), one finds Λ(n) = 1 which implies that the WKB resonance condition that we have used, Eq. (19), is valid in the large coupling (eikonal) regime [see Eq. (34)] [44,45] qQ 1 −μ 2 ≫ n + 1 2 .

V. SUMMARY AND DISCUSSION
We have determined the characteristic timescales associated with the relaxation dynamics of the composed Reissner-Nordström-black-hole-charged-massive-scalar-field system. To that end, the quasinormal resonance spectrum {ω n (µ, q, M, Q)} n=∞ n=0 which characterizes the dynamics of a linearized charged scalar field of mass µ and charge coupling constant q in the charged RN black-hole spacetime of mass M and electric charge Q was studied analytically in the eikonal regime 1 ≪ M µ < qQ [see Eq. (2)]. In particular, we have derived the analytical expression [see Eqs. (25) and (29)] M ω(M, Q, µ, q; n) = qQ · for the quasinormal resonant frequencies which characterize the composed RN-black-hole-charged-massive-scalar-field system in the eikonal regime (2) [46,47].
The characteristic timescale τ relax ≡ 1/ℑω 0 associated with the linearized relaxation dynamics of the composed RNblack-hole-charged-massive-scalar-field system is determined by its fundamental (least damped) quasinormal resonant frequency. In particular, from (37) one finds the expression for the characteristic relaxation time of the composed black-hole-field system. Interestingly, one finds from (38) that the composed RN-black-hole-charged-massive-scalar-field system is characterized by extremely long relaxation times in the limiting caseμ/Q → 1 − : Thus, although Reissner-Nordström black holes cannot support static matter fields outside their horizons [3,15,16,48,49], we conclude that these black-hole spacetimes may host extremely long-lived (exponentially decaying with long relaxation times, τ relax ≫ M ) charged massive scalar fields in their external regions.