Numerical evidence for universality in the excited instability spectrum of magnetically charged Reissner-Nordstr\"om black holes

It is well-known that the SU(2) Reissner-Nordstr\"om black-hole solutions of the Einstein-Yang-Mills theory are characterized by an infinite set of unstable (imaginary) eigenvalues $\{\omega_n(T_{\text{BH}})\}_{n=0}^{n=\infty}$ (here $T_{\text{BH}}$ is the black-hole temperature). In this paper we analyze the excited instability spectrum of these magnetically charged black holes. The numerical results suggest the existence of a universal behavior for these black-hole excited eigenvalues. In particular, we show that unstable eigenvalues in the regime $\omega_n\ll T_{\text{BH}}$ are characterized, to a very good degree of accuracy, by the simple universal relation $\omega_n(r_+-r_-)={\text{constant}}$, where $r_{\pm}$ are the horizon radii of the black hole.


I. INTRODUCTION
The familiar U(1) Reissner-Nordström spacetime is known to describe a stable black-hole solution of the coupled Einstein-Maxwell equations [1] and the coupled Einstein-Maxwell-scalar equations [2]. Yasskin [3] has proved that the Einstein-Yang-Mills theory also admits an explicit black-hole solution which is described by the magnetically charged SU(2) Reissner-Nordström spacetime. However, the SU(2) Reissner-Nordström black-hole solution of the coupled Einstein-Yang-Mills equations is known to be unstable [4][5][6]. In fact, it was proved in [7] that the magnetically charged Reissner-Nordström black-hole spacetime is characterized by an infinite family of unstable (growing in time) perturbation modes.
The recent numerical work of Rinne [8] has revealed that these unstable SU(2) Reissner-Nordström black-hole spacetimes play the role of approximate [9] codimension-two intermediate attractors (that is, nonlinear critical solutions [10]) in the dynamical gravitational collapse of the Yang-Mills field [11]. In particular, this interesting numerical study [8] has explicitly demonstrated that, during a near-critical evolution of the Yang-Mills field, the time spent in the vicinity of an unstable SU(2) Reissner-Nordström black-hole solution is characterized by the critical scaling law [12] τ = const − γ ln |p − p * | . (1) Interestingly, the critical exponents of the scaling law (1) are directly related to the characteristic instability eigenvalues of the corresponding SU(2) Reissner-Nordström black holes [8]: It is therefore of physical interest to explore the instability spectrum {ω n } n=∞ n=0 of the SU(2) Reissner-Nordström black holes. Indeed, Rinne [8] has recently computed numerically the characteristic unstable eigenvalues of these magnetically charged black-hole solutions of the Einstein-Yang-Mills theory [13].
In the present paper we shall analyze these numerically computed black-hole eigenvalues in an attempt to identify a possible hidden pattern which characterizes the black-hole instability spectrum. As we shall show below, the numerical results indeed suggest the existence of a universal behavior for these black-hole unstable eigenvalues.

II. DESCRIPTION OF THE SYSTEM
The Reissner-Nordström black-hole solution of the Einstein-Yang-Mills theory with unit magnetic charge is described by the line element [3] where the mass function m = m(r) is given by [14] The black-hole temperature is given by where are the (outer and inner) horizons of the black hole. Linearized perturbations ξ(r)e −iωt [15] of the magnetically charged black-hole spacetime are governed by the Schrödinger-like wave equation [16] where the "tortoise" radial coordinate x is defined by the relation [17] dx Well-behaved (spatially bounded) perturbation modes are characterized by the boundary conditions and where ω = i|ω|. As shown in [5,7], these boundary conditions single out a discrete set of unstable (ℑω > 0) black-hole eigenvalues {ω n (r + )} n=∞ n=0 .

III. NUMERICAL EVIDENCE FOR UNIVERSALITY IN THE EXCITED INSTABILITY SPECTRUM
Most recently, Rinne [8] computed numerically the first three instability eigenvalues which characterize the SU(2) Reissner-Nordström black-hole solutions of the coupled Einstein-Yang-Mills equations. We have examined these numerically computed eigenvalues in an attempt to reveal a possible hidden pattern which characterizes the blackhole instability spectrum.

IV. SUMMARY
The U(1) Reissner-Nordström black holes are known to be stable within the framework of the coupled Einstein-Maxwell theory [1,2]. This stability property of the black holes manifests itself in the form of an infinite spectrum of damped quasi-normal resonances [21]. To the best of our knowledge, for generic U(1) Reissner-Nordström black holes, there is no simple universal formula which describes the infinite family of these damped black-hole quasi-normal resonances.
On the other hand, the SU(2) Reissner-Nordström black holes are known to be unstable within the framework of the coupled Einstein-Yang-Mills theory [4][5][6]. This instability property of the magnetically charged black holes manifests itself in the form of an infinite spectrum of exponentially growing black-hole resonances [7]. In this paper we have provided compelling numerical evidence that the infinite family of these unstable black-hole resonances can be described, to a very good degree of accuracy, by the simple universal formula ω n (r + − r − ) = constant n for ω n ≪ T BH .
We believe that it would be highly interesting to find an analytical explanation for this numerically suggested universal behavior.