Spectroscopy of Rotating Linear Dilaton Black Holes From Boxed Quasinormal Modes

TL;DR

This paper analytically derives the resonance spectra of confined scalar fields around rotating linear dilaton black holes and uses these results to explore their entropy and area quantization.

Contribution

It provides an analytical calculation of boxed quasinormal modes for rotating linear dilaton black holes, linking these modes to entropy and area spectra.

Findings

Derived boxed quasinormal mode frequencies analytically.

Connected quasinormal modes to black hole entropy and area spectra.

Validated the use of Maggiore's method for this black hole geometry.

Abstract

Numerical studies of the coupled Einstein-Klein-Gordon system have recently revealed that confined scalar fields generically collapse to form caged black holes. In the light of this finding, we analytically study the characteristic resonance spectra of the confined scalar fields in rotating linear dilaton black hole geometry. Confining mirrors (cage) are assumed to be placed in the near-horizon region of a caged rotating linear dilaton black hole $\frac{r_{m}-r_{2}}{r_{2}}\ll 1$ ($r_{m}$ is the radius of the cage and $r_{2}$ represents the event horizon). The radial part of the Klein-Gordon equation is written as a Schr\"{o}dinger-like wave equation, which reduces to a Bessel differential equation around the event horizon. Using analytical tools and proper boundary conditions, we obtain the boxed-quasinormal mode frequencies of the caged rotating linear dilaton black hole. Finally, we employ Maggiore's method, which evaluates the transition frequency in the adiabatic invariant quantity from the highly damped quasinormal modes, in order to investigate the entropy/area spectra of the rotating linear dilaton black hole.

Equations

$$A_n=\epsilon n\hslash (n=1,2,3,....),$$

$$I_{adb}=\int \frac{dE}{\Delta \omega },$$

$$\begin{gathered} S=\frac{1}{16\pi }\int d^{4}x\sqrt{\left|g\right|}\{R-\frac{1}{2}{e}^{4\varphi }{\partial }_{\mu }\aleph {\partial }^{\mu }\aleph -2{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi \\ -\aleph F_{\mu \nu }{\tilde{F}}^{\mu \nu }-e^{-2\varphi }{F}_{\mu \nu }{F}^{\mu \nu }\}, \end{gathered}$$

$$d{s}^2=-f(r)d{t}^2+\frac{d{r}^2}{f(r)}+h(r)\left[d{\theta }^2+{\sin }^2\theta {\left(d\phi -\frac{a}{h(r)}dt\right)}^2\right],$$

$$h(r)=r{r}_0,$$

$$f(r)=\frac{Z}{h(r)},$$

$$r_1$$

$$r_2$$

$$e^{-2\varphi }=\frac{h(r)}{s(r)},$$

$$\aleph =-\frac{r_{0}a\cos \theta }{s(r)},$$

$${\mathcal{A}}_{em}=\frac{1}{\sqrt{2}}(e^{2\varphi }dt+a{\sin }^2\theta d\phi ).$$

$$T_H$$

$$=\frac{\hslash \left(r_2-r_1\right)}{4\pi r_2{r}_0}.$$

$$S_{BH}=\frac{A_H}{4\hslash }=\frac{\pi r_2{r}_0}{\hslash }.$$

$${\Omega }_H=-{\frac{g_{tt}}{g_{t\phi }}|}_{r=r_2}=\frac{a}{r_2{r}_0}.$$

$$d{M}_{QL}=T_{H}d{S}_{BH}+{\Omega }_{H}dJ.$$

$${\partial }_k(\sqrt{-g}{\partial }^k\Phi )=0,k=\mathrm{0...3},$$

$$\Phi =\frac{\rho (r)}{\sqrt{r}}{e}^{-i\omega t}{Y}_l^m(\theta ,\phi ),Re(\omega )>0,$$

$$Z{\partial }_{rr}\rho (r)+(\frac{r^2-r_1{r}_2}{r}){\partial }_r\rho (r)+[\frac{{\left(\omega r{r}_0-ma\right)}^2}{Z}-l(l+1)+\frac{(3r_1-r)r_2-r{r}_1-r^2}{4r^2}]\rho (r)=0,$$

$$\left(-{\partial }_{r^{\ast }{r}^{\ast }}+V\right)\rho (r^{\ast })={\omega }^2\rho (r^{\ast }),$$

$$V=f(r)\left\{\frac{1}{h(r)}\left[\frac{r^2+2Mr-3a^2}{4r^2}+l(l+1)\right]-\frac{m\tilde{\Omega }}{f(r)}(m\tilde{\Omega }-2\omega )\right\},$$

$$\tilde{\Omega }=\frac{a}{r{r}_0}.$$

$$r^{\ast }=\int \frac{dr}{f(r)},$$

$$r^{\ast }=\frac{r_0}{r_2-r_1}\ln \left[\frac{(\frac{r}{r_2}-1{)}^{r_2}}{(r-r_1{)}^{r_1}}\right].$$

$$\underset{r\to r_2}{\lim }{r}^{\ast }=-\infty \text{ and }\underset{r\to \infty }{\lim }{r}^{\ast }=\infty .$$

$$y=\frac{r-r_2}{r_2},$$

$$f(r)\to f(y)=\frac{1}{r_0}\left(r_{2}y-\frac{y}{1+y}{r}_1\right),$$

$$f_{NH}(y)\cong \tau y+O(y^2),$$

$$\tau =\frac{r_2-r_1}{r_0}.$$

$$r^{\ast }\cong \int \frac{r_{2}dy}{f_{NH}(y)}\cong \frac{r_2}{\tau }\ln y=\frac{1}{2\kappa }\ln y,$$

$$y=e^{2\kappa r^{\ast }}.$$

Full text

Rotating Linear Dilaton Black Hole, Boxed Quasinormal Modes, Quantization, Caged Black Hole, Zerilli Equation, Bessel Functions

I Introduction

Quantum mechanics (QM) and general relativity (GR) fail when confronted with each other’s principles, and are therefore limited in their ability to describe the Universe. Therefore, one must unify the theories to make them compatible with one another. The resulting theory is expected to describe the behavior of the Universe, from leptons and quarks to galaxies. This is the quantum gravity theory (QGT) Rovelli . The onset of the QGT dates back to the 1970s when Hawking Hawking74 ; Hawking75 and Bekenstein Bek72 ; Bek73 ; Bek74 ; Bek74n2 ; Bek75 amalgamated GR and QM by using the black hole (BH) as a theoretical arena. However, at present, QGT is under construction and not completed.

In the spirit of Ehrenfest’s adiabatic hypothesis Ehren ; Ehren2 , Bekenstein Bek74n2 conjectured that the surface area of a BH can only take on a discrete spectrum of values:

[TABLE]

where $\varepsilon$ is a dimensionless constant. One can easily deduce from Eq. (1) that the minimum change in the horizon area corresponds to $\Delta A_{\min}=A_{n}-A_{n-1}=\varepsilon\hbar$. Namely, the BH horizon is formed by patches of equal area Bek74n2 . Based on Bekenstein’s conjecture, many different attempts have been made in the literature to derive the quantum spectrum of BHs, and consequently different spectra with different $\varepsilon$ have been obtained. The best-known values of $\varepsilon$ are $8\pi$ Bek74n2 ; Magg and $4lnp$ where $p=2,3,...$ Mukhon1 ; Mukhon2 ; HodPRL ; Bek2015 . Among the methods applied, one of the most important contributions was made by Hod HodPRL , who suggested that $\varepsilon$ could be determined by utilizing the quasinormal mode (QNM) frequencies BertiCQG . QNMs are the characteristic sounds of BHs and neutron stars Nollert . Hod considered the real part of the asymptotic QNM as a transition frequency (the smallest energy a BH can emit) in the semiclassical limit, and in sequel showed that its quantum emission gives rise to a change in the mass of BH, which is related to $A_{n}$. In this way, he found that $\varepsilon=4\ln 3$ (for a Schwarzschild BH) HodPRL . Later on, Kunstatter Kunst managed to quantize the area of the Schwarzschild BH with the help of the adiabatic invariance:

[TABLE]

where $I_{adb}$ is the adiabatic invariant quantity and $\Delta\omega$ denotes the transition frequency between the two neighboring levels of a BH possessing total energy $dE=T_{H}dS_{BH}$ in which $T_{H}$ and $S_{BH}$ stand for the Hawking temperature and the Bekenstein-Hawking entropy, respectively. The result obtained was nothing but Hod’s result HodPRL : the reproduction of $\varepsilon=4\ln 3$.

In 2007, Maggiore Magg postulated that a BH can be viewed as a damped harmonic oscillator whose physically relevant frequency is identical to the complex QNM frequencies having both real and imaginary parts. Using the fact that for the highly excited QNMs the imaginary part is dominant over the real part, he activated Kunstatter’s formula (2). According to the Bohr-Sommerfeld quantization rule (BSQR) $I_{adb}\simeq n\hbar$ when the quantum number $n\rightarrow\infty$. Considering this fact, Maggiore proved that for the Schwarzschild BH the area spectrum is exactly equal to Bekenstein’s original result Bek74n2 . Following the seminal study of Maggiore, the area spectra of numerous BHs have been thoroughly investigated (see for instance MMA ; MMB ; MM1 ; MM2 ; MM2b ; MM2c ; MM3 ; MM4 ; MM5 ; MM6 ; MM6b ; MM7 ; MM8 ; MM9 ).

In this paper, we study the spectroscopy of a rotating linear dilaton black hole (RLDBH) Clem2003 , which is a solution to the Einstein-Maxwell-dilaton-axion (EMDA) theory Leygnac . To this end, we use Maggiore’s method (MM) Magg . However while doing this computation, instead of the ordinary QNMs (or vibrational modes) we shall consider the boxed-quasinormal modes (BQNMs) (also known as quasi-bound states) Bqnm1 . These modes reveal when the considered BH is caged Okaw . Namely, RLDBH is assumed to be confined in a cavity with finite-volume. In this setup, confinement can be achieved by placing an artificial spherical reflecting surface at some distance $r_{m}$. This configuration is akin to the “perfect mirror” (used by Press and Teukolsky Press ) to perform a BH bomb. However, for studying the resonant frequencies of the caged BHs, one can follow the recent studies of Bqnm1 ; Bqnm2 in which $r_{m}$ was chosen to be close to the near-horizon (NH) region epjc15 . The propagating scalar fields $\Psi$ are imposed to vanish at $r_{m}$. This requirement is fulfilled by using two boundary conditions: the Dirichlet boundary condition (DBC) ($\left.\Psi(r)\right|_{r=r_{m}}=0$) and the Neumann boundary condition (NBC) ($\left.\frac{d\Psi(r)}{dr}\right|_{r=r_{m}}=0$). In addition to this, one should use the fact that QNMs are the pure ingoing waves at the event horizon. These three conditions guide us to find the resonance condition, which yields the BQNMs after some manipulations. Here, we will use the transition frequency of the BQNMs in Eq. (2) to derive the RLDBH’s spectroscopy (area/entropy spectra). It is worth mentioning that the spectroscopy of RLDBH has recently been studied by Sakalli MM9 via the standard QNMs, and the present paper aims to give support to that study.

This paper is arranged as follows. In Sec. 2, we briefly overview the RLDBH metric, analyze the Klein-Gordon equation (KGE) for a massless scalar field, and derive the Zerilli equation (ZE) CS in this geometry. In Sec. 3, it is shown that the ZE reduces to a Bessel differential equation around the NH. Next, we impose the required boundary conditions for computing the complex BQNMs of the caged RLDBH. Then, applying MM we obtain the spectroscopy of RLDBH. Section 4 concludes . (Throughout the paper we use units in which fundamental constants are $c=k_{B}=G=1$).

II RLDBH and Separation of Massless KGE

The action of EMDA theory is given by Leygnac

[TABLE]

where $\phi$ and $\aleph$ denote the dilaton and axion (pseudoscalar coupled to an Abelian vector field $\mathcal{A}_{em}$) fields, respectively. $R$ denotes the Ricci scalar, $F_{\mu\upsilon}$ is the well-known Maxwell two-form associated with a $U(1)$ subgroup of $E_{8}\times E_{8}$ (or Spin$(32)/Z_{2}$), and $\widetilde{F}^{\mu\nu}$ stands for the dual of $F_{\mu\upsilon}$. RLDBH spacetime is described by Clem2003 ; Leygnac

[TABLE]

where

[TABLE]

[TABLE]

are the metric functions. The constant parameter $r_{0}$ seen in Eq. (5) is associated with the background electric charge $Q:$ $r_{0}=\sqrt{2}Q$. In Eq. (6), $Z=(r-r_{2})(r-r_{1})$, where $r_{1}$ and $r_{2}$ denote the inner and outer (event) horizons, respectively, derived from the condition of $f(r)=0$. Explicit form of these radii are

[TABLE]

where the constant $M$ is related to mass, and $a$ is the rotation parameter modulating the angular momentum: $J=\frac{ar_{0}}{2}$. For a BH solution, $M\geq a$. The non-asymptotically flat geometry of the RLDBH entails the computation of the quasilocal mass $M_{QL}$ BYform . As a result, one finds that $M=2M_{QL}$. Furthermore, the background fields are given by

[TABLE]

[TABLE]

where $s(r)=r^{2}+a^{2}\cos^{2}\theta$. The electromagnetic four-vector potential reads

[TABLE]

The Hawking temperature of the RLDBH is provided by the surface gravity $\kappa$ Wald  as

[TABLE]

As can be seen from Eq. (11), in the static case ($a=0$, i.e., $r_{1}=0$) this BH emits radiation with a constant (mass-independent) temperature, which corresponds to the isothermal process PasaIz ; AstrSak leading to the uninformed Hawking radiation AlSak . If $A_{H}$ stands for the area of the event horizon, then the RLDBH’s entropy is given by

[TABLE]

Angular velocity of the RLDBH:

[TABLE]

Thus, the first law of thermodynamics can be validated for the RLDBH via

[TABLE]

Meanwhile, one may examine why Eq. (14) does not contain electric charge $Q$. This is because here $Q$ is nothing but a background charge of fixed value Clem2003 .

Utilizing the massless KGE:

[TABLE]

with the following ansatz for the scalar field $\Phi$

[TABLE]

where $Y_{l}^{m}(\theta,\varphi)$ denotes the spheroidal harmonics with the eigenvalue $-l(l+1)$ Damora , we get

[TABLE]

From Eq. (17), one can obtain the ZE CS as follows

[TABLE]

where the effective potential or the so-called Zerilli potential is given by

[TABLE]

by which

[TABLE]

The tortoise coordinate $r^{\ast}$ is defined as

[TABLE]

which yields

[TABLE]

The asymptotic limits of $r^{\ast}$ become

[TABLE]

III BQNM Frequencies of Caged RLDBH and Spectroscopy Computation

In this section, we aim to read the BQNM frequencies of the caged RLDBH. These resonant frequencies can be analytically determined when the confining mirrors are placed in the vicinity of the event horizon. To this end, we will impose the Hod’s boundary condition that nothing is supposed to come out of the event horizon of the caged BH and to survive at the cage Bqnm1 .

Setting

[TABLE]

the metric function $f(r)$ transforms into

[TABLE]

which has the following NH behavior:

[TABLE]

wherein

[TABLE]

Thus, the tortoise coordinate (21) around the NH behaves as

[TABLE]

so that we have

[TABLE]

Substituting Eq. (24) into Eq. (19), one finds the NH Zerilli potential as follows

[TABLE]

where the parameters of Eq. (30) are given by

[TABLE]

[TABLE]

in which

[TABLE]

Here $\widetilde{\omega}$ connotes the wave frequency measured by the observer rotating with the horizon Clem2003 . Therefore, the ZE (18) around the NH becomes

[TABLE]

One can find that the general solution of Eq. (34) is given by

[TABLE]

where $u=2i\sqrt{\eta}e^{\kappa r^{\ast}}$ and $D_{1},_{2}$ are constants. $J_{-i\overline{\omega}}(u)$ and $Y_{-i\overline{\omega}}(u)$ denote the Bessel functions AS of the first and second kinds, respectively. The parameters of the Bessel functions are given by

[TABLE]

[TABLE]

Using the following limiting forms (when $z\rightarrow 0$) of the Bessel functions AS ; OR

[TABLE]

[TABLE]

we get the NH ($e^{\kappa r^{\ast}}\ll 1$) behavior of Eq. (35):

[TABLE]

One can deduce that $D_{1}$ and $D_{2}$ are related to the amplitude of the ingoing and outgoing waves at NH, respectively. Now, one can impose the condition of having QNMs, which stipulates purely ingoing waves at the horizon. Hence, we pick $D_{1}\neq 0$ and set $D_{2}=0$. Thus, the physical solution of Eq. (35) is given by

[TABLE]

or

[TABLE]

Following Bqnm1 ; Bqnm2 ; Okaw , we consider the DBC at the surface $y\equiv y_{m}=\frac{r_{m}-r_{2}}{r_{2}}$ of the confining cage:

[TABLE]

Using the following relation AS

[TABLE]

we can rewrite the condition (43) as

[TABLE]

which is called the resonance condition Bqnm1 ; Bqnm2 . According to Hod Bqnm1 , the boundary of the confining cavity leading to the characteristic resonance spectra is close to the NH ($r_{m}\approx r_{2}$). So

[TABLE]

With the aid of Eqs. (38) and (39), we find the resonance condition (45) approximates to

[TABLE]

Finally, one should also impose the NBC Bqnm1 on the solution (42):

[TABLE]

which admits the following expression AS

[TABLE]

Recalling Eq. (44), one can reproduce the following expression

[TABLE]

Substituting Eq. (49) into Eq. (50), we derive the resonance condition for the NBC:

[TABLE]

From Eqs. (38) and (39), we get

[TABLE]

Thus, we can rewrite the resonance condition (51) as

[TABLE]

According to Eq. (46), the damped modes with ${Im}(\overline{\omega})<0$ lead to $\left(l_{m}\right)^{i\overline{\omega}}\ll 1$. So, it is clear that both resonance conditions (47) and (53) take small quantities.

We now use an iteration scheme to obtain the BQNMs of the caged RLDBH. For both DBC and NBC, the $0^{th}$ order resonance condition is given by Bqnm1 :

[TABLE]

which results in

[TABLE]

The $1^{st}$ order resonance condition is obtained after substituting Eq. (55) into r.h.s of Eqs. (47) and (53). Thus, we have

[TABLE]

which is equivalent to

[TABLE]

where plus (minus) stands for the NBC (DBC). In the $u\ll 1$ regime, we have

[TABLE]

Using this for Eq. (57), we find the characteristic resonance spectra as follows

[TABLE]

which yields

[TABLE]

Putting this into Eqs. (37) and (33) gives the BQNMs of the RLDBH

[TABLE]

For the highly damped modes, Eq. (61) becomes

[TABLE]

which is in accordance with the recent study of Sakalli MM9 in which the quantization of the RLDBH was studied with the standard QNMs. According to MM, the transition frequency now reads

[TABLE]

From Eqs. (2) and (63), we now have

[TABLE]

Acting upon the BSQR ($I_{adb}=\hbar n$), we obtain the entropy spectrum as

[TABLE]

which also yields the area spectrum as follows

[TABLE]

Thus, the minimum area spacing of the RLDBH becomes

[TABLE]

The above results support Bekenstein’s conjecture Bek74n2 . Furthermore, we see that the spectroscopy of the RLDBH is independent from its characteristic parameters, and its area spectrum is evenly spaced.

IV Conclusion

Motivated by recent studies Bqnm1 ; Bqnm2 which have shown that the BQNMs occur when a BH is caged by a perfect mirror (artificial) located in the NH region, we have studied the entropy/area spectra of RLDBHs. To this end, we first analyzed the KGE for a massless scalar field in this geometry. Using the separability feature of the KGE, we derived the ZE (18) in the background of the RLDBH, and read its effective potential (19). It has been shown that NH-ZE (34) has a solution in terms of the Bessel functions (35). Thus, by using the boundary conditions of QNM (only ingoing waves survive at the NH), DBC, and NBC, we obtained the BQNMs (61) analytically. Then, the transition frequency obtained from the highly damped BQNMs was substituted in Eq. (2) to compute the quantum entropy/area spectra of RLDBH. The resulting area spectrum is equidistant and fully consistent with Bekenstein’s conjecture Bek74n2 . Finally, our results support the study of Sakalli MM9 , which proves that RLDBH entropy/area spectra are equally spaced from ordinary QNMs.

Acknowledgments

We would like to thank Gérard Clément, Shahar Hod, and Mustafa Halilsoy for helpful discussions.