Analytic treatment of the excited instability spectra of the magnetically charged SU(2) Reissner-Nordström black holes

The magnetically charged SU(2) Reissner-Nordström black-hole solutions of the coupled nonlinear Einstein-Yang-Mills field equations are known to be characterized by infinite spectra of unstable (imaginary) resonances {ωn(r+, r−)}n = 0n = ∞ (here r± are the black-hole horizon radii). Based on direct numerical computations of the black-hole instability spectra, it has recently been observed that the excited instability eigenvalues of the magnetically charged black holes exhibit a simple universal behavior. In particular, it was shown that the numerically computed instability eigenvalues of the magnetically charged black holes are characterized by the small frequency universal relation ωn(r+ − r−) = λn, where {λn} are dimensionless constants which are independent of the black-hole parameters. In the present paper we study analytically the instability spectra of the magnetically charged SU(2) Reissner-Nordström black holes. In particular, we provide a rigorous analytical proof for the numerically-suggested universal behavior ωn(r+ − r−) = λn in the small frequency ωnr+ ≪ (r+ − r−)/r+ regime. Interestingly, it is shown that the excited black-√hole resonances are characterized by the simple universal relation ωn + 1/ωn = e− 2π/3. Finally, we confirm our analytical results for the black-hole instability spectra with numerical computations.

(1.1) 1 As discussed in [14], the SU(2) Reissner-Nordström black-hole spacetime is only an approximate intermediate attractor of the nonlinear gravitational collapse of the Yang-Mills field because it is characterized by an infinite family of exponentially growing (unstable) perturbation modes.
2 See [15] for an excellent review on the black-hole critical phenomena in nonlinear gravitational collapse. 3 It is worth noting that this physically interesting fact refers to type I and Type III nonlinear critical behaviors in gravitational collapse, see [14,[16][17][18][19] for details. 4 Here the quantity |p − p * | provides a measure in the physical parameter space for the distance of the initial field data from the threshold (critical) solution of the nonlinear Einstein-Yang-Mills theory [15].

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It is interesting to note that the critical exponents in the scaling behavior (1.1), which characterizes the nonlinear near-critical gravitational collapse of the Yang-Mills field, are directly related to the imaginary eigenvalues which characterize the instability spectrum of the corresponding magnetically charged Reissner-Nordström black-hole spacetime [14]: It is therefore physically interesting to investigate the characteristic instability (imaginary) resonance spectra {ω n (r + , r − )} n=∞ n=0 5 of these magnetically charged black-hole solutions of the coupled nonlinear Einstein-Yang-Mills equations.
In his important numerical work, Rinne [14] has recently determined numerically the first three imaginary (unstable) resonant frequencies which characterize the magnetically charged SU(2) Reissner-Nordström black-hole spacetimes. 6 Subsequently, in [20] we have analyzed the detailed numerical data provided by Rinne [14] and revealed the intriguing fact that, to a good degree of accuracy, the numerically computed [14] excited instability eigenvalues of the magnetically charged SU(2) Reissner-Nordström black holes are characterized by the remarkably simple universal behavior where {λ n } are dimensionless constants which seem to be independent of the black-hole parameters.
The main goal of the present paper is to determine analytically the characteristic instability spectra of the magnetically charged SU(2) Reissner-Nordström black-hole solutions of the coupled Einstein-Yang-Mills theory. In particular, in this paper we shall provide a rigorous analytical proof for the validity of the numerically suggested [14,20] universal behavior (1.3) which characterizes the excited instability spectra of the SU(2) Reissner-Nordström black-hole spacetimes.

Description of the system
The SU(2) Reissner-Nordström black-hole spacetime of mass M and unit magnetic charge is characterized by the spherically-symmetric line element [6] where the radially dependent mass function m = m(r) is given by 7 Here r± are the black-hole horizon radii [see eq. (2.3) below]. 6 As noted above, the SU(2) Reissner-Nordström black-hole solutions of the coupled nonlinear Einstein-Yang-Mills field equations are characterized by infinite spectra {ωn(r+, r−)} n=∞ n=0 of imaginary (unstable) resonant frequencies [10,11]. Reference [14] has provided, for the first time, detailed numerical results for the first three resonant frequencies which quantify the instability growth rates of these magnetically charged black-hole spacetimes. 7 We shall use natural units in which G = c = = 1.

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The radii of the black-hole (outer and inner) horizons are given by As shown in [21], linearized perturbation modes ξ(r)e −iωt 8 of the magnetically charged black-hole spacetime are governed by the wave equation where the radial coordinate x = x(r) is defined by the differential relation 9 The effective radial potential which governs the Schrödinger-like wave equation (2.4) is given by [21] U It is worth emphasizing the fact that the radial function U (x) in eq. (2.4), which determines the spatial behavior of the black-hole perturbation modes, has the form of an effective binding potential. In particular, it is a negative definite function of the radial coordinate x and it vanishes asymptotically at the two boundaries x → ±∞ of the magnetically charged black-hole spacetime. As shown in [21], well-behaved perturbation modes of the black-hole spacetime are characterized by spatially bounded (exponentially decaying) radial eigenfunctions at the two asymptotic boundaries: where ω = i|ω|. The radial differential equation (2.4), supplemented by the physically motivated boundary conditions (2.7) and (2.8) [21], determine the discrete family {ω n (r + , r − )} n=∞ n=0 of unstable (ℑω > 0) resonances which characterize the SU(2) Reissner-Nordström black-hole spacetimes [9][10][11]. Interestingly, below we shall show explicitly that the characteristic resonance spectrum of the magnetically charged black holes can be studied analytically in the regime |ω n |r + ≪ 1 of small imaginary resonant frequencies. In particular, we shall derive a remarkably compact analytical formula [see eq. (4.3) below] for the excited instability eigenvalues which characterize the SU(2) Reissner-Nordström black-hole solutions of the coupled Einstein-Yang-Mills theory.

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3 The characteristic resonance condition The recent numerical results of Rinne [14] reveal that the excited instability resonances {ω n } n=∞ n=1 of the magnetically charged SU(2) Reissner-Nordström black-hole spacetimes are characterized by the property As we shall now show, the Schrödinger-like differential equation (2.4), which governs the dynamics of the black-hole perturbation modes, is amenable to an analytical treatment in the regime (3.1) of small resonant frequencies.
It was pointed out in [12,13] that, in the M ≫ 1 regime (the regime of weaklymagnetized SU(2) Reissner-Nordström black holes), the Schrödinger-like perturbation equation (2.4) can be transformed using the well-known Chandrasekhar transformations [22] to the physically equivalent Teukolsky-like radial equation [23]: where the complex number plays the role of an effective spherical harmonic index (see [12,13] for details), and [23] ∆(r; M ≫ 1) = r 2 − 2M r . The mathematical Chandrasekhar transformations [22] can also be used in the case of generic magnetically charged SU(2) Reissner-Nordström black holes, 10 in which case the generalized expression for the radial function ∆(r) in the Teukolsky-like radial perturbation equation (3.2) is given by [23] ∆(r; M ) = r 2 − 2M r + 1 .

(3.5)
It is convenient to use the dimensionless physical variables [24][25][26] in terms of which the radial differential equation (3.2) reads

14)
11 That is, the mathematical solution which respects the physically motivated near-horizon boundary condition (2.7). 12 Note that in the near-horizon region kz ≪ 1 one can neglect the first two terms inside the square brackets in eq. (3.7) [24]. 13 That is, the mathematical solution which respects the physically motivated asymptotic boundary condition (2.8) at spatial infinity.
14 Note that in the asymptotic region z ≫ ̟ + 1 one can neglect the last two terms inside the square brackets in eq. (3.7) [24]. 15 It is worth noting that the overlap radial region ̟ ≪ z ≪ 1/k exists in the regime |ω|r+ ≪ 1 of small resonant frequencies [see eqs.

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(3.15) A spatially bounded (normalizable) radial eigenfunction which satisfies the physically motivated boundary condition (2.8) at spatial infinity [21] is characterized by the asymptotic relation ψ(z → ∞) → 0. Thus, the coefficient ψ 2 of the exploding exponent in the asymptotic expression (3.13) should vanish, yielding the characteristic resonance equation [see eq. (3.15)] for the instability eigenvalues of the SU(2) Reissner-Nordström black-hole spacetimes. Taking cognizance of eq. (3.3), one can write the resonance equation (3.16) for the instability spectra of the magnetically charged black holes in the form 16 .  In this small frequency regime the resonance equation (3.17) can be approximated by which yields the characteristic infinite spectrum 18,19 .
It is worth emphasizing again that the analytically derived formula (4.3) for the characteristic instability spectra of the magnetically charged SU(2) Reissner-Nordström black holes is valid in the small frequency regime [see (3.6) and (4.1)] This inequality implies that, for a given value of the black-hole dimensionless temperature (r + − r − )/r + , the analytical formula (4.3) describes an infinite family of unstable (imaginary) black-hole resonances in the regime It is interesting to note that the instability spectra (4.3) of the magnetically charged SU(2) Reissner-Nordström black-hole spacetimes have the simple generic form ω n × (r + − r − ) = constant n ≡ λ n [see eq. (1.3)]. We have therefore provided here an analytical proof for the numerically-observed [14,20] universal behavior (1.3) of the black-hole excited instability eigenvalues.

Numerical confirmation
It is of considerable physical interest to verify the validity of the analytically derived formula (4.3) for the excited instability eigenvalues of the SU(2) Reissner-Nordström black holes. The corresponding instability eigenvalues of the magnetically charged black holes were recently computed numerically in the very interesting work of Rinne [14]. In table 1 we present the dimensionless ratio ω ana 2 /ω num 2 , where {ω ana 2 (r + )} are the analytically calculated excited instability eigenvalues of the SU(2) Reissner-Nordström black holes as given by the analytical formula (4.3) and {ω num 2 (r + )} are the numerically computed [14] instability eigenvalues of the magnetically charged black holes. From the data presented in table 1 one finds a fairly good agreement between the analytically derived formula (4.3) for the excited instability spectra of the SU(2) Reissner-Nordström black holes and the corresponding numerically computed black-hole instability eigenvalues.

JHEP03(2017)072 6 Summary
The magnetically charged SU(2) Reissner-Nordström black-hole spacetimes describe a family of unstable solutions of the coupled nonlinear Einstein-Yang-Mills field equations [7][8][9][12][13][14]. In particular, these magnetically charged black holes are known to be characterized by infinite spectra of imaginary (unstable) resonant frequencies {ω n (r + , r − )} n=∞ n=0 . Based on direct numerical computations of the black-hole instability spectra [14], it has recently been pointed out [20] that the excited instability eigenvalues of the magnetically charged SU(2) Reissner-Nordström black holes are described, to a very good degree of accuracy, by the simple universal relation (1.3).
In the present paper we have studied analytically the characteristic instability spectra of the magnetically charged SU(2) Reissner-Nordström black-hole spacetimes in the small frequency regime. In particular, we have provided a simple analytical proof for the numerically-observed [14,20] Finally, it is interesting to note that one finds from (4.3) the simple dimensionless ratio ω n+1 ω n = e −2π/ √ 3 (6.3) for the characteristic excited instability eigenvalues of the magnetically charged SU(2) Reissner-Nordström black holes. It is worth emphasizing the fact that the relation (6.3), which characterizes the instability resonance spectra of the magnetically charged blackhole spacetimes, is universal in the sense that it is independent of the physical parameters (masses and magnetic charges) of the SU(2) Reissner-Nordström black holes.