# Resonance spectra of caged black holes

## Abstract

Recent numerical studies of the coupled Einstein–Klein–Gordon system in a cavity have provided compelling evidence that *confined* scalar fields generically collapse to form black holes. Motivated by this intriguing discovery, we here use analytical tools in order to study the characteristic resonance spectra of the confined fields. These discrete resonant frequencies are expected to dominate the late-time dynamics of the coupled black-hole-field-cage system. We consider caged Reissner–Nordström black holes whose confining mirrors are placed in the near-horizon region \(x_{\text {m}}\equiv (r_{\text {m}}-r_+)/r_+\ll \tau \equiv (r_+-r_-)/r_+\) (here \(r_{\text {m}}\) is the radius of the confining mirror and \(r_{\pm }\) are the radii of the black-hole horizons). We obtain a simple analytical expression for the fundamental quasinormal resonances of the coupled black-hole-field-cage system: \(\omega _n=-i2\pi T_{\text {BH}} \cdot n\left[ 1+O(x^n_{\text {m}}/\tau ^n)\right] \), where \(T_{\text {BH}}\) is the temperature of the caged black hole and \(n=1,2,3,...\) is the resonance parameter.

### Keywords

Black Hole Scalar Field Recent Numerical Study Charged Scalar Field Cage Black Hole## 1 Introduction

Caged black holes^{1} have a long and broad history in general relativity. These composed objects were extensively studied in the context of black-hole thermodynamics [1, 2, 3, 4, 5, 6, 7, 8]. In addition, the physics of caged black holes was studied with relation to the black-hole bomb mechanism of Press and Teukolsky [9, 10, 11, 12, 13, 14, 15, 16, 17].

Recently there is a renewed interest in the physics of caged black holes. This renewed interest stems from the important work of Bizoń and Rostworowski [18] who revealed that asymptotically anti-de Sitter (AdS) spacetimes are nonlinearly unstable. In particular, it was shown in [18] that the dynamics of massless, spherically symmetric scalar fields in asymptotically AdS spacetimes generically leads to the formation of Schwarzschild–AdS black holes.

It is well known that the AdS spacetime can be regarded as having an infinite potential wall at asymptotic infinity^{2} [19, 20, 21, 22, 23]. One therefore expects the dynamics of *confined* scalar fields^{3} to display a qualitatively similar behavior to the one observed in [18]. In an elegant work, Maliborski [24] (see also [25, 26]) has recently confirmed this physically motivated expectation. In particular, the recent numerical study by Okawa et al. [25] provides compelling evidence that spherically symmetric confined scalar fields generically collapse to form caged black holes.

The late-time dynamics of perturbation fields in a black-hole spacetime^{4} is characterized by quasinormal ringing, damped oscillations which reflect the dissipation of energy from the black-hole exterior region (see [27, 28, 29] for excellent reviews and detailed lists of references). The observation of these characteristic complex resonances may allow one to determine the physical parameters of the newly born black hole.

^{5}The recent interest [24, 25, 26] in the dynamics and formation of caged black holes makes it highly important to study their characteristic resonance spectra. As we shall show below, the resonant frequencies of caged black holes can be determined

*analytically*in the regime

^{6}Here \(r_{\text {m}}\) is the radius of the confining cage (mirror) and \(r_{\pm }\) are the radii of the black-hole horizons [see Eq. (4) below].

## 2 Description of the system

^{7}

^{8}

## 3 Boundary conditions

- (1)The Dirichlet-type boundary condition implies$$\begin{aligned} R(r=r_{\text {m}})=0. \end{aligned}$$(11)
- (2)The Neumann-type boundary condition implies$$\begin{aligned} {{\mathrm{d}R}\over {\mathrm{d}r}}(r=r_{\text {m}})=0. \end{aligned}$$(12)

## 4 The resonance conditions

The boundary conditions (11) and (12) single out two discrete families of complex resonant frequencies \(\{\omega (M{,}Q{,}r_{\text {m}}{,}l{;}n)\}\) ^{9} which characterize the late-time dynamics of the composed black-hole-field-cavity system (these characteristic resonances are also known as “boxed quasinormal frequencies” [10, 11]). The main goal of the present paper is to determine these characteristic resonances *analytically*.

^{10}

^{11}

^{12}

^{13}

## 5 The discrete resonance spectra of caged black holes

Taking cognizance of the near-horizon condition (24), one realizes that the r.h.s. of the resonance conditions (25) and (29) are small quantities. This observation follows from the fact that, for damped modes with \(\mathfrak {I}\varpi <0\) [\(\mathfrak {R}(i\varpi )>0\)], one has \(z^{i\varpi }_{\text {m}}\ll 1\) in the regime (24). We can therefore use an iteration scheme in order to solve the resonance conditions (25) and (29).

^{14}

## 6 Summary and discussion

Recent numerical studies of the Einstein–Klein–Gordon system in a cavity [25, 26] have provided compelling evidence that confined scalar fields generically collapse to form (caged) black holes. Motivated by these intriguing studies, we have explored here the late-time dynamics\({}^{4}\) of these confined fields in the background of caged black holes.

*analytically*for caged black holes whose confining mirrors are placed in the vicinity of the black-hole horizon [that is, in the regime \(x_{\text {m}}\ll \tau \); see Eq. (24)]. Remarkably, the resonant frequencies of these caged black holes can be expressed in terms of the Bekenstein–Hawking

*temperature*\(T_{\text {BH}}\) of the black-hole:

^{15}

^{16}

^{17}scale linearly with the black-hole temperature):

- (1)
- (2)
Near-extremal asymptotically flat Reissner–Nordström black holes coupled to charged scalar fields [43].

- (3)
Asymptotically flat charged black holes coupled to charged scalar fields in the highly charged regime \(qQ\gg 1\)

^{18}[44, 45]. - (4)
The “subtracted” black-hole geometries studied in [46].

- (5)
Asymptotically AdS black holes in the regime \(r_+\gg R\)

^{19}[47]. This last example, together with our result (34) for the resonant frequencies of caged black holes, provide an elegant demonstration of the analogy, already discussed in the Introduction, between asymptotically AdS black-hole spacetimes and caged black-hole spacetimes.

^{20}at the black-hole horizon. The fact that the black-hole spacetimes mentioned above [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47] share this same property (namely, they are characterized by a linear scaling of their resonances with the black-hole temperature) suggests that the dynamics of perturbation fields in these black-hole spacetimes are mainly determined by the near-horizon properties of these geometries.

## Footnotes

- 1.
We us e the term “Caged black holes” to describe black holes which are confined within finite-volume cavities.

- 2.
- 3.
That is, scalar fields which are confined within finite-volume cavities.

- 4.
That is, the dynamics of the fields well after the formation of the black-hole horizon.

- 5.
It is worth emphasizing again that caged black holes may serve as a simple toy-model for the physically more realistic AdS black holes.

- 6.
We use the term “Tightly caged black holes” to reflect the fact that the boundary of the confining cavity is placed in the vicinity of the black-hole horizon: \(r_{\text {m}}-r_+\ll r_+-r_-\).

- 7.
We use natural units in which \(G=c=\hbar =1\).

- 8.
We shall henceforth omit the indices \(l\) and \(m\) for brevity.

- 9.
The integer \(n\) is the resonance parameter.

- 10.
Note that the near-horizon region (1) corresponds to \(x\le x_{\text {m}}\ll \tau \le 1\). This also implies [see Eq. (14)] \(y\rightarrow -\infty \) (and thus \(\mathrm{e}^{\tau y/r_+}\rightarrow 0\)) in the region (1).

- 11.
Here we have used the relation \((\lambda +2M/r-2Q^2/r^2)/r^2=(\lambda +\tau )/r^2_+[1+O(x)]\) in the near-horizon region (1).

- 12.
Here we have used the relation (15) in the near-horizon region (1).

- 13.
See equations 9.1.10 and 9.1.11 of [36] for the sub-leading correction terms.

- 14.
Here we have used the relation \(\tan (x+n\pi )=\tan (x)\). In addition, we have used the relation \(\tan (x)=x+O(x^3)\) in the \(x\ll 1\) regime [see Eq. (24)].

- 15.
- 16.
That is, the resonant frequencies scale linearly with the black-hole temperature \(T_{\text {BH}}\).

- 17.
It is worth emphasizing that, the characteristic relaxation time of generic field perturbations is determined by the (reciprocal of the) imaginary part of the fundamental \((n=1)\) resonance: \(\tau _{}=1/\mathfrak {I}{\omega _1}\).

- 18.
Here \(q\) is the charge coupling constant of the field.

- 19.
Here \(R\) is the AdS radius.

- 20.
Note that the surface gravity is proportional to the black-hole temperature.

## Notes

### Acknowledgments

This research is supported by the Carmel Science Foundation. I thank Yael Oren, Arbel M. Ongo and Ayelet B. Lata for helpful discussions.

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