Numerical evidence for universality in the excited instability spectrum of magnetically charged Reissner–Nordström black holes

Abstract

It is well known that the SU(2) Reissner–Nordström 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.

1 Introduction

The familiar U(1) Reissner–Nordström spacetime is well known to describe a stable black-hole solution of the coupled Einstein–Maxwell equations [1, 2] and the coupled Einstein–Maxwell-scalar equations [3–5]. Yasskin [6] 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 [7–10]. In fact, it was proved in [11, 12] 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 [13] has revealed that these unstable SU(2) Reissner–Nordström black-hole spacetimes play the role of approximate¹ codimension-two intermediate attractors (that is, nonlinear critical solutions [14]) in the dynamical gravitational collapse of the Yang–Mills field.² In particular, this interesting numerical study [13] 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 law³

$$\begin{aligned} \tau =\text {const}-\gamma \ln |p-p^*|. \end{aligned}$$

(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 [13]:

$$\begin{aligned} \gamma =1/\omega _{\text {instability}}. \end{aligned}$$

(2)

It is therefore of physical interest to explore the instability spectrum $\{\omega _n\}_{n=0}^{n=\infty }$ of the SU(2) Reissner–Nordström black holes. Indeed, Rinne [13] has recently computed numerically the characteristic unstable eigenvalues of these magnetically charged black-hole solutions of the Einstein–Yang–Mills theory.⁴

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.

2 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 [6]

$$\begin{aligned} \mathrm{d}s^2= & {} -\Bigg (1-{{2m}\over {r}}\Bigg )\mathrm{d}t^2\nonumber \\&+\Bigg (1-{{2m}\over {r}}\Bigg )^{-1}\mathrm{d}r^2+r^2(\mathrm{d}\theta ^2+\sin ^2\theta \mathrm{d}\phi ^2), \end{aligned}$$

(3)

where the mass function $m=m(r)$ is given by⁵

$$\begin{aligned} m(r)=M-{{1}\over {2r}}. \end{aligned}$$

(4)

The black-hole temperature is given by

$$\begin{aligned} T_{\text {BH}}={{r_+-r_-}\over {4\pi r^2_+}}, \end{aligned}$$

(5)

where

$$\begin{aligned} r_{\pm }=M\pm \sqrt{M^2-1}\ \end{aligned}$$

(6)

are the (outer and inner) horizons of the black hole.

Linearized perturbations $\xi (r) \mathrm{e}^{-i\omega t}$ ⁶ of the magnetically charged black-hole spacetime are governed by the Schrödinger-like wave equation [15]

$$\begin{aligned} \Bigg \{{{\mathrm{d}^2}\over {\mathrm{d}x^2}}+\omega ^2+{1 \over {r^2}}\Bigg [1-{{2m(r)} \over r}\Bigg ] \Bigg \}\xi =0, \end{aligned}$$

(7)

where the “tortoise” radial coordinate $x$ is defined by the relation⁷

$$\begin{aligned} \mathrm{d}x/\mathrm{d}r=[1-2m(r)/r]^{-1}. \end{aligned}$$

(8)

Well-behaved (spatially bounded) perturbation modes are characterized by the boundary conditions

$$\begin{aligned} \xi (x\rightarrow -\infty )\sim \mathrm{e}^{|\omega |x}\rightarrow 0\ \end{aligned}$$

(9)

and

$$\begin{aligned} \xi (x\rightarrow \infty )\sim xe^{-|\omega |x}\rightarrow 0, \end{aligned}$$

(10)

where $\omega =i|\omega |$. As shown in [9, 11, 12], these boundary conditions single out a discrete set of unstable ($\mathfrak {I}\omega >0$) black-hole eigenvalues $\{\omega _n(r_+)\}_{n=0}^{n=\infty }$.

3 Numerical evidence for universality in the excited instability spectrum

Most recently, Rinne [13] 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 black-hole instability spectrum.

In Table 1 we present the first excited instability eigenvalues $\{\omega _1(r_+)\}$ of the magnetically charged SU(2) Reissner–Nordström black holes. In particular, we display the dimensionless ratio $\omega _1(r_+)/\pi T_{\text {BH}}$, where the black-hole temperature $T_{\text {BH}}$ is given by (5). We also display the ratio between the dimensionless quantity $\omega _1(r_+)\times (r_+-r_-)$ for generic SU(2) Reissner–Nordström black holes and the corresponding quantity $\omega _1(r_+=10)\times (10-1/10)$ for the weakly magnetized Reissner–Nordström black hole with $r_+=10$.⁸ Remarkably, the numerical data presented in Table 1 reveals that the black-hole instability eigenvalues in the regime $\omega _1(r_+)/T_{\text {BH}}\ll 1$ are characterized, to a good degree of accuracy, by the universal relation⁹

$$\begin{aligned} \omega _1(r_+-r_-)=\lambda _1; \quad \lambda _1={\text {constant}}. \end{aligned}$$

(11)

In order to support this intriguing finding, we display in Table 2 the second excited instability eigenvalues $\{\omega _2(r_+)\}$ of the SU(2) Reissner–Nordström black holes. Remarkably, the numerical data presented in Table 2 provide compelling evidence for the validity of the suggested universal behavior of the black-hole instability eigenvalues in the regime $\omega _2(r_+)/T_{\text {BH}}\ll 1$. In particular, one finds¹⁰

$$\begin{aligned} \omega _2(r_+-r_-)=\lambda _2; \quad \lambda _2={\text {constant}}. \end{aligned}$$

(12)

Table 1 The instability eigenvalues of SU(2) Reissner–Nordström black holes. The data shown refers to the first excited eigenvalues $\{\omega _1(r_+)\}$ of these magnetically charged black holes. We display the dimensionless ratio $\omega _1(r_+)/\pi T_{\text {BH}}$, where $T_{\text {BH}}$ is the black-hole temperature. Also shown is the ratio between the dimensionless quantity $\omega _1(r_+)\times (r_+-r_-)$ for generic SU(2) Reissner–Nordström black holes and the corresponding quantity $\omega _1(r_+=10)\times (10-1/10)$ for the weakly magnetized Reissner–Nordström black hole with $r_+=10$ (see footnote 8). One finds that the instability eigenvalues in the regime $\omega _1(r_+)/\pi T_{\text {BH}}\lesssim 0.1$ are characterized, to a good degree of accuracy, by the universal relation $\omega _1(r_+-r_-)={\text {constant}}$

Table 2 The instability eigenvalues of SU(2) Reissner–Nordström black holes. The data shown refers to the second excited eigenvalues $\{\omega _2(r_+)\}$ of these magnetically charged black holes. We display the dimensionless ratio $\omega _2(r_+)/\pi T_{\text {BH}}$, where $T_{\text {BH}}$ is the black-hole temperature. Also shown is the ratio between the dimensionless quantity $\omega _2(r_+)\times (r_+-r_-)$ for generic SU(2) Reissner–Nordström black holes and the corresponding quantity $\omega _2(r_+=10)\times (10-1/10)$ for the weakly magnetized Reissner–Nordström black hole with $r_+=10$ (see footnote 8). One finds that the instability eigenvalues in the regime $\omega _2(r_+)/T_{\text {BH}}\ll 1$ are characterized, to a good degree of accuracy, by the universal relation $\omega _2(r_+-r_-)={\text {constant}}$

4 Summary

The U(1) Reissner–Nordström black holes are known to be stable within the framework of the coupled Einstein–Maxwell theory [1–5]. This stability property of the black holes manifests itself in the form of an infinite spectrum of damped quasi-normal resonances [16, 17]. 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 [7–10]. This instability property of the magnetically charged black holes manifests itself in the form of an infinite spectrum of exponentially growing black-hole resonances [11, 12]. 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

$$\begin{aligned} \omega _n(r_+-r_-)={\text {constant}}_n \quad \text {for}\quad \omega _n\ll T_{\text {BH}}. \end{aligned}$$

(13)

We believe that it would be highly interesting to find an analytical explanation for this numerically suggested universal behavior.