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Based on the well known fact that the quasinormal frequencies are the poles of the frequency domain Green’s function we describe a method that allow s us to calculate exactly the quasinormal frequencies of the Klein-Gordon field moving in the three-dimensional rotating Bañados-Teitelboim-Zanelli black hole. These quasinormal frequencies are already published and widely explored in several applications, but we use this example to expound the proposed method of computation. We think that the described procedure can be useful to calculate exactly the quasinormal frequencies of classical fields propagating in other backgrounds. Furthermore, we compare with previous results and discuss some related facts.

Several dynamical aspects of the black hole perturbations are encoded in their quasinormal modes (QNM), namely in the solutions to their equations of motion that are purely ingoing near the event horizon. Also, in the far region they satisfy a boundary condition determined by the asymptotic structure of the spacetime, for example, for asymptotically flat black holes, it is imposed that the perturbation is purely outgoing in the far region, whereas for asymptotically anti-de Sitter black holes a Dirichlet type boundary condition is commonly imposed in the far region [

Motivated by the AdS-CFT correspondence, the QNF spectrum of asymptotically anti-de Sitter spacetimes has been studied, because they determine the decay times of the dual quantum field theories living at the boundary [

Among the asymptotically anti-de Sitter solutions, the three-dimensional Bañados-Teitelboim-Zanelli black hole (BTZ black hole in what follows) [

As is well known, in the coordinates ( t , r , ϕ ) the metric of the three-dimensional rotating BTZ black hole is^{1} [

d s 2 = − f ( r ) d t 2 + f ( r ) − 1 d r 2 + r 2 ( d ϕ − J 2 r 2 d t ) 2 , (1)

where the function f is equal to

f ( r ) = − M + r 2 l 2 + J 2 4 r 2 , (2)

the factor l 2 is related to the negative cosmological constant by Λ = − 1 / l 2 and we point out that the mass M and the angular momentum J of the black hole are determined by

M = r + 2 + r − 2 l 2 , J = 2 r + r − l , (3)

with r ± denoting the outer and inner radii of the black hole [

^{1}As in Ref. [

In analogy to Refs. [

1) The field is ingoing near the outer horizon.

2) The field goes to zero as r → ∞ .

For the cases in which an exact solution of the radial equation is available, the usual method for computing exactly the QNF is to impose the boundary conditions on the radial function and then we calculate exactly the QNF from the conditions on the radial solutions [^{2} As far as we know this method is not used previously in the exact computation of the QNF for asymptotically anti-de Sitter black holes [

For the Schrödinger type equation [

d 2 Φ d x 2 + ω 2 Φ − V Φ = 0 , (4)

where Φ is related to the radial function of the field, ω denotes the frequency and V is the effective potential that is characteristic of the field under study, it is well known that its frequency domain Green’s function is equal to [

G ˜ ( x 1 , x 2 , ω ) = ( Φ + ( ω , x 1 ) Φ ∞ ( ω , x 2 ) W x ( ω ) , x 1 < x 2 ; Φ + ( ω , x 2 ) Φ ∞ ( ω , x 1 ) W x ( ω ) , x 2 < x 1 , (5)

where Φ + is the solution of the Schrödinger type equation satisfying the boundary condition near the horizon of the black hole. In a similar way Φ ∞ fulfills the boundary condition as r → ∞ , and

W x ( Φ ∞ , Φ + ) = W x ( ω ) = Φ ∞ d Φ + d x − Φ + d Φ ∞ d x , (6)

^{2}For the exact calculation of the QNF spectrum for the Klein-Gordon field moving in the static BTZ black hole see Ref. [

^{3}In the rest of this work we follow closely the notation of Ref. [

is the Wronskian of these solutions. In the rest of this section, for the Klein-Gordon field we determine the Wronskian (6) of the solutions satisfying the boundary conditions at the event horizon and at the asymptotic region. Taking as a basis the expression (5) for the frequency domain Green’s function we calculate exactly the corresponding QNF by finding the zeros of its Wronskian, that is, the poles of the frequency domain Green’s function.

The Klein-Gordon equation takes the form^{3}

( □ 2 − μ 2 l 2 ) Ψ = ( 1 | g | ∂ μ ( | g | g μ ν ∂ ν ) − μ 2 l 2 ) Ψ = 0, (7)

where □ 2 is the d’Alembertian operator, μ is related to the mass of the field, g μ ν denotes the inverse metric, g its determinant and Ψ denotes the Klein-Gordon field. As shown in Ref. [

Ψ = R ( r ) e − i ω t e i m ϕ , (8)

where m is the angular eigenvalue, then the Klein-Gordon equation simplifies to the following differential equation for the radial function R

d d r ( f d R d r ) + 1 r f d R d r + ( ω 2 f − J ω m f r 2 − ( r 2 / l 2 − M ) m 2 f r 2 ) R = 0. (9)

We notice that the previous expression can be transformed to a Schrödinger type equation of the form (4) as follows

d 2 R ^ d x 2 + ω 2 R ^ − { J ω m r 2 + ( r 2 / l 2 − M ) m 2 r 2 − f g d d r ( f d g d r ) − f g d g d r } R ^ = 0 , (10)

with

f d d r = d d x , g = 1 r , R = g R ^ . (11)

In Equation (10) the expression in curly braces is the effective potential of the Klein-Gordon field in the rotating BTZ black hole.

In the variable z defined by

z = r 2 − r + 2 r 2 − r − 2 , (12)

we find that the radial Equation (9) simplifies to [

z ( 1 − z ) d 2 R d z 2 + ( 1 − z ) d R d z + ( A z + B + C 1 − z ) R = 0 , (13)

where

A = l 4 4 ( r + 2 − r − 2 ) 2 ( ω r + − m l r − ) 2 , C = − μ 2 4 , B = − l 4 4 ( r + 2 − r − 2 ) 2 ( ω r − − m l r + ) 2 . (14)

Taking the radial function R in the form

R = z α ( 1 − z ) β F , (15)

we obtain that the function F must be a solution to the hypergeometric differential equation [

z ( 1 − z ) d 2 F d z 2 + [ c − ( a + b + 1 ) z ] d F d z − a b F = 0 , (16)

when the quantities α and β are solutions to the algebraic equations

α 2 + A = 0 , β 2 − β + C = 0 , (17)

and the parameters of the hypergeometric equation are equal to

a = α + β + i − B , b = α + β − i − B , c = 2 α + 1. (18)

As in Ref. [

Hence, around z = 0 the solutions of the radial equation take the form

R = z α ( 1 − z ) β ( C 1 F 2 1 ( a , b ; c ; z ) + C 2 z 1 − c F 2 1 ( a − c + 1 , b − c + 1 ; 2 − c ; z ) ) , (19)

where C 1 , C 2 are constants and 2 F 1 ( a , b ; c ; z ) denotes the hypergeometric function [

R + = C 1 z α ( 1 − z ) β F 2 1 ( a , b ; c ; z ) . (20)

Instead of following the usual way to calculate exactly the QNF [

u = 1 − z . (21)

Using this variable we find that the function F of the Formula (15) must be solution of a hypergeometric differential Equation (16) with parameters [

a ˜ = a , b ˜ = b , c ˜ = 1 + a + b − c , (22)

and therefore the radial function takes the form

R = ( 1 − u ) α u β ( K 1 F 2 1 ( a , b ; a + b − c + 1 ; u ) + K 2 u c − a − b F 2 1 ( c − a , c − b ; c − a − b + 1 ; u ) , (23)

where K 1 and K 2 are constants. Analyzing its behavior as r → ∞ we obtain that the radial function fulfilling the boundary condition b) at the asymptotic region is

R ∞ = K 2 ( 1 − u ) α u β u c − a − b F 2 1 ( c − a , c − b ; c − a − b + 1 ; u ) , (24)

since the term proportional to K 1 diverges as u → 0 .

Considering Equation (11) we obtain that for the BTZ black hole Φ + = r R + and Φ ∞ = r R ∞ and hence we get that their Wronskian is equal to

W x ( Φ ∞ , Φ + ) = r W x ( R ∞ , R + ) . (25)

Owing to the hypergeometric differential Equation (16) is of second order, we know that among any three solutions there is a linear relation [

F 2 1 ( a , b ; c ; z ) = Γ ( c ) Γ ( c − a − b ) Γ ( c − a ) Γ ( c − b ) F 2 1 ( a , b ; a + b − c + 1 ; 1 − z ) + Γ ( c ) Γ ( c − a − b ) Γ ( a ) Γ ( b ) ( 1 − z ) c − a − b F 2 1 ( c − a , c − b ; c − a − b + 1 ; 1 − z ) . (26)

From the Kummer property of the hypergeometric function and considering that the Wronskian of the solutions to the hypergeometric equation is [

W z ( F 2 1 ( a , b ; a + b − c + 1 ; 1 − z ) , ( 1 − z ) c − a − b F 2 1 ( c − a , c − b ; c − a − b + 1 ; 1 − z ) ) = ( a + b − c ) z − c ( 1 − z ) c − a − b − 1 , (27)

we find that the Wronskian of the radial solutions Φ + and Φ ∞ is equal to

W x ( Φ ∞ , Φ + ) = − 2 l 2 K 2 C 1 ( r + 2 − r − 2 ) ( a + b − c ) Γ ( c ) Γ ( c − a − b ) Γ ( c − a ) Γ ( c − b ) . (28)

We notice that W x ( Φ ∞ , Φ + ) does not depend on the coordinate x (or r), as we expect [

c − a = − n , c − b = − n , n = 0 , 1 , 2 , 3 , ⋯ , (29)

and therefore the QNF of the Klein-Gordon field in the rotating BTZ black hole are equal to [

ω = m l − 2 i l 2 ( r + − r − ) ( n + 1 2 + 1 + μ 2 2 ) ,

ω = − m l − 2 i l 2 ( r + + r − ) ( n + 1 2 + 1 + μ 2 2 ) , (30)

that is, the proposed method produces the same QNF that the procedure used in Ref. [

The QNF of the rotating BTZ black hole have been useful in several applications [

It is convenient to mention that for the Klein-Gordon field propagating in the rotating BTZ black hole, the proposed procedure yields the same values for the QNF as the usual method of Refs. [

As far as we are aware this method is not previously used in the exact determination of the QNF for black holes, and we believe that this procedure can be a useful tool in the study of spacetime perturbations. Doubtless the applicability of the method that we expound in the present work to the determination of the QNM for other gravitational systems deserves further research.

This work was supported by CONACYT México, SNI México, COFAA-IPN, EDI-IPN, and the Research Project IPN SIP-20181408.

López-Ortega, A. and Mata-Pacheco, D. (2018) BTZ Quasinormal Frequencies as Poles of Green’s Function. Journal of Applied Mathematics and Physics, 6, 1170-1178. https://doi.org/10.4236/jamp.2018.66099