Quasinormal frequencies of the dimensionally reduced BTZ black hole

Abstract

We calculate numerically the quasinormal frequencies of the Klein-Gordon and Dirac fields moving in the two-dimensional dimensionally reduced BTZ black hole. Our work extends results previously published on the damped oscillations of this black hole. First, we compute the quasinormal frequencies of the minimally coupled Klein-Gordon field for a range of the dimensionally reduced BTZ black hole physical parameters that is not previously analyzed. Furthermore we determine, for the first time, the quasinormal frequencies of the Dirac field propagating in the dimensionally reduced BTZ black hole. For the Dirac field we use the Horowitz-Hubeny method and the asymptotic iteration method, while for the Klein-Gordon field the extension of the previous results is based on the asymptotic iteration method. Finally we discuss our main results.

Appendices

Appendix A Asymptotic iteration method

An useful method to find the eigenvalues of linear second order differential equations is the asymptotic iteration method [31,32,33]. This method works with linear second order differential equations of the form

$$\begin{aligned} y''= \lambda _0 (x)y'+s_0(x)y, \end{aligned}$$

(A1)

where the prime denotes differentiation with respect to the independent variable. Taking advantage of the symmetrical structure of Eq. (A1) we find that its first derivative takes a similar mathematical form

$$\begin{aligned} y'''= \lambda _1 (x)y'+s_1(x)y, \end{aligned}$$

(A2)

with

$$\begin{aligned} \lambda _1 =\lambda _0' + s_0 + \lambda _0^2, \quad \text {and} \quad s_1=s_0'+s_0 \lambda _0. \end{aligned}$$

(A3)

In an analogous way we find that the $(n+2)$-th derivative of the function y is equal to [31,32,33]

$$\begin{aligned} y^{(n+2)}= \lambda _{n} (x)y'+s_{n}(x)y, \end{aligned}$$

(A4)

where

$$\begin{aligned} \lambda _n =\lambda _{n-1}' + s_{n-1} + \lambda _0 \lambda _{n-1}, \quad \text {and} \quad s_n=s_{n-1}'+s_0 \lambda _{n-1}. \end{aligned}$$

(A5)

The asymptotic aspect of the method imposes that for sufficiently large n the functions $\lambda _n$ and $s_n$ satisfy [31,32,33]

$$\begin{aligned} \frac{s_n}{\lambda _n} =\frac{s_{n-1}}{\lambda _{n-1}} = \alpha \end{aligned}$$

(A6)

or in an equivalent form

$$\begin{aligned} s_n \lambda _{n-1} - s_{n-1} \lambda _n = 0. \end{aligned}$$

(A7)

Solving this equation, usually known as discretization condition, we can find the QNFs of the black hole under study [32, 33].

Nevertheless, the computation of the recurrence relations (A5) requires many resources [31,32,33]. Owing this fact, in Ref. [32] is proposed an improved version of the AIM. In this version of the method we expand the functions $\lambda _n$ and $s_n$ around a convenient point $\xi$, that is,

$$\begin{aligned} \lambda _n (\xi )= \sum ^\infty _{i=0} c^i _n (x- \xi )^i, \qquad \qquad s _n (\xi )= \sum ^\infty _{i=0} d^i _n (x- \xi )^i, \end{aligned}$$

(A8)

to find that the recurrence relations (A5) imply that the coefficients $c^i _n$ and $d^i _n$ satisfy

$$\begin{aligned} c^i _n&= (i+1)c^{i+1} _{n-1} + d^i _{n-1} + \sum ^i _{k=0} c^k _0 c^{i-k} _{n-1}, \nonumber \\ d^i _n&= (i+1) d^{i+1} _{n-1} + \sum ^i _{k=0} d^k _0 c^{i-k} _{n-1}. \end{aligned}$$

(A9)

Furthermore, we get that the discretization condition (A7) takes the form

$$\begin{aligned} d^0 _n c^0 _{n-1} - d^0 _{n-1} c^0 _n =0. \end{aligned}$$

(A10)

Solving this equation for different values of n, we numerically obtain the QNFs of the studied black hole [32, 33]. In this work we use this improved formulation of the AIM to find the QNFs of the Klein-Gordon and Dirac fields propagating in the DRBTZ black hole.

Appendix B Horowitz-Hubeny method for the Dirac field

For the Klein-Gordon field moving in the DRBTZ black hole, in Ref. [22] are calculated its QNFs taking as a basis the Horowitz-Hubeny method. Nevertheless a restriction on the values of the horizons radii was imposed, as we previously commented. For the Dirac field we can use the Horowitz-Hubeny method to compute its QNFs in the DRBTZ black hole. In what follows we describe the steps to transform the radial Eqs. (31) into a convenient form to use the Horowitz-Hubeny method [6].

First, to fulfill the boundary condition near the event horizon, we propose that the solutions of Eqs. (31) take the form

$$\begin{aligned} R_s(r)=e^{-i(\omega \pm \frac{i \kappa }{2})r_*} U_s(r), \end{aligned}$$

(B11)

to find that the functions $U_s$ must satisfy the differential equations

$$\begin{aligned} f\frac{d^2U_s}{dr^2}+\Big (\frac{df}{dr}-2i\omega \pm \kappa \Big )\frac{dU_s}{dr}+\frac{1}{f}\Big (\frac{\kappa ^2}{4}\mp i\omega \kappa -V_s \Big )U_s=0. \end{aligned}$$

(B12)

To have an independent variable that changes in a finite interval, following to Horowitz-Hubeny [6], we make the change of variable

$$\begin{aligned} x=\frac{1}{r}, \end{aligned}$$

(B13)

to get that the radial Eqs. (B12) transform into

$$\begin{aligned} S_s(x)\frac{d^2U_s}{dx^2}+\frac{t_s(x)}{x-x_+}\frac{dU_s}{dx}+ \frac{u_s(x)}{(x-x_+)^2}U_s=0, \end{aligned}$$

(B14)

where the functions $t_s(x)$, $u_s(x)$, and $S_s(x)$ are equal to

$$\begin{aligned} t_s(x)&= 2x(x^2-x_+^2)(x^2-x_-^2)^2(x + x_+) + 2x^5(x + x_+)(x^2 - x_-^2) \nonumber \\&\quad -2 x x_+^2 x_-^2 (x + x_+)(x^2 - x_-^2) + 2i\omega x^2(x+x_+)(x^2-x_-^2)x_+^2 x_-^2 \nonumber \\&\quad \mp x^2x_+(x_-^2-x_+^2)(x+x_+)(x^2-x_-^2), \nonumber \\ u_s(x)&= \frac{1}{4}x^2 x_+^2(x_+^2-x_-^2)^2 \mp i \omega x^2x_+^3x_-^2(x_-^2-x_+^2) \nonumber \\&\quad \pm i \omega xx_+^2x_-^2(x_+^2x_-^2-x^4) + \frac{5}{4}x^8 - \frac{3}{2} x^6(x_+^2+x_-^2) + \frac{5}{2}x^4x_+^2x_-^2 \nonumber \\&\quad - \frac{1}{2} x^2 x_+^2 x_-^2 (x_+^2 + x_-^2) + \frac{1}{4} x_+^4x_-^4-m^2(x^2-x_+^2)(x^2-x_-^2)x_+^2x_-^2,\nonumber \\ S_s(x)&= x^2(x+x_+)^2(x^2-x_-^2)^2. \end{aligned}$$

(B15)

In the previous equations we define $x_+= 1/r_+$ and $x_-=1/r_-$.

As in Ref. [6] we propose that the solutions to Eqs. (B14) take the form

$$\begin{aligned} U_s=(x-x_+)^{\nu _s} \sum _{k=0}^{\infty } a_{k,s} (\omega )(x-x_+)^k, \end{aligned}$$

(B16)

but to fulfill the boundary condition of the QNMs near the event horizon we must take $\nu _s=0$ [6]. Substituting these simplified forms of the functions $U_s$ into Eq. (B14), we find that the coefficients $a_{k,s}$ must satisfy the recurrence relations

$$\begin{aligned} a_{k,s}=-\frac{1}{k(k-1)S_{0,s}+k t_{0,s}} \sum _{n=0}^{k-1} a_{n,s} (n(n-1)S_{k-n,s}+n t_{k-n,s}+u_{k-n,s}), \end{aligned}$$

(B17)

where $S_{0,s}= S_s (x_+)$, $t_{0,s} = t_s(x_+)$, the coefficients $u_{k,s}$ are given by

$$\begin{aligned} u_s(x)= \sum _{k=0}^{\infty } u_{k,s} (x-x_+)^k, \end{aligned}$$

(B18)

and similar definitions are valid for the coefficients $S_{k,s}$ and $t_{k,s}$.

At the asymptotic region, the boundary condition of the QNMs imposes that as $x \rightarrow 0$ ($r \rightarrow \infty$) the functions $U_s$ go to zero, that is,

$$\begin{aligned} U_s(x=0)=\sum _{k=0}^{\infty } a_{k,s} (\omega )(-x_+)^k=0 \end{aligned}$$

(B19)

and the QNFs of the Dirac field can be calculated by finding the roots of the previous equation when we replace the infinite sum by a finite sum [6].

As for the Klein-Gordon field moving in the DRBTZ black hole, the Horowitz-Hubeny method allows us to calculate the QNFs of the Dirac field when $r_+ > 2r_-$. The reason is that the series (B16) has a radius of convergence as large as the distance to the nearest singular point. Since the radius of convergence must include the point $x=0$, the distance from the expansion point at $x_+$ to the other singularity at $x_-$ must be larger than the distance between $x_+$ and 0, that is, $x_--x_+ > x_+$, or equivalently $r_+ > 2r_-$.