YITP-11-97, WITS-CTP-83, OU-HET-735/2011

A New Approach to Black Hole Quasinormal Modes:

A Review of the Asymptotic Iteration Method
H. T. Cho,1, A. S. Cornell,2, Jason Doukas,3, T. -R. Huang,1 and Wade Naylor4,
1

Department of Physics, Tamkang University, Tamsui, Taipei, Taiwan, Republic of China

2

National Institute for Theoretical Physics; School of Physics,

University of the Witwatersrand, Wits 2050, South Africa

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan
4

International College & Department of Physics,

Osaka University, Toyonaka, Osaka 560-0043, Japan

arXiv:1111.5024v1 [gr-qc] 21 Nov 2011

(Dated: 18th November, 2011)

We discuss an approach to obtaining black hole quasinormal modes (QNMs) using the asymptotic
iteration method (AIM), initially developed to solve second order ordinary differential equations.
We introduce the standard version of this method and present an improvement more suitable for
numerical implementation. We demonstrate that the AIM can be used to find radial QNMs for
Schwarzschild, Reissner-Nordstr
om (RN) and Kerr black holes in a unified way. An advantage
of the AIM over the standard continued fraction method (CFM) is that for differential equations
with more than three regular singular points Gaussian eliminations are not required. However, the
convergence of the AIM depends on the location of the radial or angular position, choosing the
best such position in general remains an open problem. This review presents for the first time the
spin 0, 1/2 & 2 QNMs of a Kerr black hole and the gravitational and electromagnetic QNMs of the
RN black hole calculated via the AIM, and confirms results previously obtained using the CFM.
We also presents some new results comparing the AIM to the WKB method. Finally we emphasize
that the AIM is well suited to higher dimensional generalizations and we give an example of doubly
rotating black holes.

Keywords: asymptotic iteration method, quasinormal modes, extra dimensions

Email:
Email:
Email:
Email:

htcho@mail.tku.edu.tw
alan.cornell@wits.ac.za
jasonad@yukawa.kyoto-u.ac.jp
naylor@phys.sci.osaka-u.ac.jp

2
CONTENTS

I. Introduction
II. The Asymptotic Iteration Method
A. The Method
B. The Improved Method
C. Two Simple Examples
1. The Harmonic Oscillator
2. The Poschl-Teller Potential

3
4
4
5
6
6
7

III. Schwarzschild (A)dS Black Holes

A. The Schwarschild Asymptotically Flat Case
B. The de-Sitter Case
C. The Spin-Zero Anti-de-Sitter Case

8
8
9
11

IV. Reissner-Nordstr
om Black Holes

14

V. Kerr Black Holes

A. The Spin-Zero Case
B. The Spin-Half Case
C. The Spin-Two Case
VI. Doubly Rotating Kerr (A)dS Black Holes
A. Radial Quasi-Normal Modes

16
17
20
23
25
27

VII. Summary and Outlook

28

Acknowledgments

29

A. Angular Eigenvalues for Spin-Weighted Spheroidal Harmonics

29

B. Higher Dimensional Scalar Spheroidal Harmonics with two Rotation Parameters

30

References

32

3
I.

INTRODUCTION

The study of quasinormal modes (QNMs) of black holes (black holes) is an old and well established subject, where
the various frequencies are indicative of both the parameters of the black hole and the type of emissions possible.
Initially the calculation of these frequencies was done in a purely numerical way, which requires selecting a value
for the complex frequency, integrating the differential equation, and checking whether the boundary conditions are
satisfied. Note that in the following we shall use the definition that QNMs are defined as solutions of the perturbed
field equations with boundary conditions:
(
eix
x
(x)
,
(1.1)
ix
e
x
for an eit time dependence (which corresponds to ingoing waves at the horizon and outgoing waves at infinity).
Also note the boundary condition as x does not apply to asymptotically anti-de Sitter spacetimes, where
instead something like a Dirichlet boundary condition is imposed, for example see Ref. [1]. Since those conditions
are not satisfied in general, the complex frequency plane must be surveyed for discrete values that lead to QNMs.
This technique is time consuming and cumbersome, making it difficult to systematically survey the QNMs for a wide
range of parameter values. Following early work by Vishveshwara [2], Chandrasekhar and Detweiler [3] pioneered this
method for studying QNMs.
In order to improve on this, a few semi-analytic analyses were also attempted. In one approach, employed by
Mashoon et al. [4], the potential barrier in the effective one-dimensional Schrodinger equation is replaced by a
parameterized analytic potential barrier function for which simple exact solutions are known. The overall shape
approximates that of the true black hole barrier, and the parameters of the barrier function are adjusted to fit the
height and curvature of the true barrier at the peak. The resulting estimates for the QNM frequencies have been
applied to the Schwarzschild, Reissner-Nordstr
om and Kerr black holes, with agreement within a few percent with the
numerical results of Chandrasekhar and Detweiler in the Schwarzschild case [3], and with Gunter [5] in the ReissnerNordstr
om case. However, as this method relies upon a specialized barrier function, there is no systematic way to
estimate the errors or to improve the accuracy.
The method by Leaver [6], which is a hybrid of the analytic and the numerical, successfully generates QNM
frequencies by making use of an analytic infinite-series representation of the solutions, together with a numerical
solution of an equation for the QNM frequencies which involves, typically by applying a Frobenius series solution
approach, the use of continued fractions. This technique is known as the continued fraction method (CFM).
Historically, another commonly applied technique is the WKB approximation [7]. Even though it is based on an
approximation, this approach is powerful as the WKB approximation is known in many cases to be more accurate,
and can be carried to higher orders, either as a means to improve accuracy or as a means to estimate the errors
explicitly. Also it allows a more systematic study of QNMs than has been possible using outright numerical methods.
The WKB approximation has since been extended to sixth-order [8].
However, all of these approaches have their limitations, where in recent years a new method has been developed
which can be more efficient in some cases, called the asymptotic iteration method (AIM). Previously this method
was used to solve eigenvalue problems [9] as a semi-analytic technique for solving second-order homogeneous linear
differential equations. It has also been successfully shown by some of t he current authors that the AIM is an efficient
and accurate technique for calculating QNMs [10].
As such, we will review the AIM as applied to a variety of black hole spacetimes, making (where possible) comparisons with the results calculated by the WKB method and the CFM a la Leaver [6]. Therefore, the structure of this
paper shall be: In Sec. II we shall review the AIM and the improved method of Ciftci et al. [9] (also see Ref. [11]),
along with a discussion of how the QNM boundary conditions are ensured. Applications to simple concrete examples,

4
such as the harmonic oscillator and the Poschl-Teller potential are also provided. In Sec. III the case of Schwarzschild
(A)dS black holes shall be discussed, developing the integer and half-spin equations. In Sec. IV a review of how
the QNMs of the Reissner-Nordstr
om black holes shall be made, with several frequencies calculated in the AIM and
compared with previous results. Sec. V will review the application of the AIM to Kerr black holes for spin 0, 1/2, 2
fields. Sec. VI will give discuss the spin-zero QNMs for doubly rotating black holes. We then summarize and conclude
in Sec. VII.

II.

THE ASYMPTOTIC ITERATION METHOD

A.

The Method

To begin we shall now review the idea behind the AIM, where we first consider the homogeneous linear second-order
differential equation for the function (x),
00 = 0 (x)0 + s0 (x) ,

(2.1)

where 0 (x) and s0 (x) are functions in C (a, b). In order to find a general solution to this equation, we rely on the
symmetric structure of the right-hand of Eq. (2.1) [9]. If we differentiate Eq. (2.1) with respect to x, we find that
000 = 1 (x)0 + s1 (x) ,
where
1 = 00 + s0 + (0 )2 and s1 = s00 + s0 0 .
Taking the second derivative of Eq.(2.1) we get
0000 = 2 (x)0 + s2 (x) ,
where
2 = 01 + s1 + 0 1

and

s1 = s00 + s0 0 .

Iteratively, for the (n + 1)th and the (n + 2)th derivatives, n = 1, 2, ..., we have
(n+1) = n1 (x)0 + sn1 (x) ,

(2.2)

and thus bringing us to the crucial observation in the AIM is that differentiating the above equation n times with
respect to x, leaves a symmetric form for the right hand side:
(n+2) = n (x)0 + sn (x) ,

(2.3)

where
n (x) = 0n1 (x) + sn1 (x) + 0 (x)n1 (x)

and

sn (x) = s0n1 (x) + s0 (x)n1 (x) .

(2.4)

For sufficiently large n the asymptotic aspect of the method is introduced, that is:
sn1 (x)
sn (x)
=
(x) ,
n (x)
n1 (x)

(2.5)

where the QNMs are obtained from the quantization condition

n = sn n1 sn1 n = 0 ,

(2.6)

5
which is equivalent to imposing a termination to the number of iterations [11]. From the ratio of the (n + 1)th and
the (n + 2)th derivatives, we have


sn
0

(n+2)
n
n
d


.
ln((n+1) ) = (n+1) =
(2.7)
dx

0
n1 + sn1
n1

From our asymptotic limit, this reduces to

d
n
ln((n+1) ) =
,
dx
n1

(2.8)

which yields
(n+1)

Z

(x) = C1 exp

n (x0 )
dx0
n1 (x0 )

Z
= C1 n1 exp

x
0

( + 0 )dx

(2.9)

where C1 is the integration constant and the right-hand side of Eq. (2.4) and the definition of (x) have been used.
Substituting this into Eq. (2.2), we obtain the first-order differential equation
Z x

0 + = C1 exp
( + 0 )dx0 ,
(2.10)
which leads to the general solution
 Z
(x) = exp

(x0 )dx0

(Z

C2 + C1

x0

exp

)
[0 (x00 ) + 2(x00 )] dx00

!
dx0

(2.11)

The integration constants, C1 and C2 , can be determined by an appropriate choice of normalisation. Note, that for
the generation of exact solutions C1 = 0.

B.

The Improved Method

Ciftci et al. [9] were among the first to note that an unappealing feature of the recursion relations in Eqs. (2.4) is
that at each iteration one must take the derivative of the s and terms of the previous iteration. This can slow the
numerical implementation of the AIM down considerably and also lead to problems with numerical precision.
To circumvent these issues we developed an improved version of the AIM which bypasses the need to take derivatives
at each step [10]. This greatly improves both the accuracy and speed of the method. We expand the n and sn in a
Taylor series around the point at which the AIM is performed, :
n () =
sn () =

X
i=0

cin (x )i ,

(2.12)

din (x )i ,

(2.13)

i=0

where the cin and din are the ith Taylor coefficients of n () and sn () respectively. Substituting these expressions
into Eqs.(2.4) leads to a set of recursion relations for the coefficients:
i
cin = (i + 1)ci+1
n1 + dn1 +

i
X

ck0 cik
n1 ,

(2.14)

k=0

din = (i + 1)di+1
n1 +

i
X
k=0

dk0 cik
n1 .

(2.15)

6
In terms of these coefficients the quantization condition Eq.(2.6) can be re-expressed as
d0n c0n1 d0n1 c0n = 0 ,

(2.16)

and thus we have reduced the AIM into a set of recursion relations which no longer require derivative operators.
Observing that the right hand side of Eqs. (2.14) and (2.15) involve terms of order at most n 1, one can recurse
these equations until only ci0 and di0 terms remain (that is, the coefficients of 0 and s0 only). However, for large
numbers of iterations, due to the large number of terms, such expressions become impractical to compute. We avert
this combinatorial problem by beginning at the n = 0 stage and calculating the n + 1 coefficients sequentially until
the desired number of recursions is reached. Since the quantisation condition only requires the i = 0 term, at each
iteration n we only need to determine coefficients with i < N n, where N is the maximum number of iterations to
be performed. The QNMs that we calculate in this paper will be determined using this improved AIM.

C.

Two Simple Examples

1.

The Harmonic Oscillator

In order to understand the effectiveness of the AIM, it is appropriate to apply this method to a simple concrete
problem: The harmonic oscillator potential in one dimension,


d2
(2.17)
2 + x2 = E .
dx
When |x| approaches infinity, the wave function must approach zero. Asymptotically the function decays like a
Gaussian distribution, in which case we can write
(x) = ex

/2

f (x) ,

(2.18)

where f (x) is the new wave function. Substituting Eq. (2.18) into Eq. (2.17) then re-arranging the equation and
dividing by a common factor, one can obtain
df
d2 f
= 2x
+ (1 E)f .
(2.19)
dx2
dx
We recognise this as Hermites equation. For convenience we let 1 E = 2j, such that in our case 0 = 2x and
s0 = 2j. We define
n = n sn1 n1 sn ,

for

n = 1, 2, 3, . . .

(2.20)

Thus using Eqs.(2.6) one can find that

n = 2n+1

n
Y

(j i) ,

(2.21)

i=0

and the termination condition Eq.(2.4) can be written as n = 0. Hence j must be a non-negative integer, which
means
Ek = 2k + 1 ,

for

k = 0, 1, 2, . . .

(2.22)

and this is the exact spectrum for such a potential. Moreover, the wave function (x) can also be derived in this
method.
We should point out that in this case the termination condition, n = 0, is dependent only on the new eigenvalue j
for a given iteration number n, and this is the reason why we can obtain an exact eigenvalue. However, for the black
hole cases in the subsequent sections, the termination condition depends also on x, and therefore one can only obtain
approximate eigenvalues by terminating the procedure after n iterations.

7
2.

The Poschl-Teller Potential

To conclude this section we will also demonstrate that the AIM can be applied to the case of QNMs, which have
unbounded (scattering) like potentials, by recalling that we can find QNMs for Scarf II (upside-down Poschl-Tellerlike) potentials [12]. This is based on observations made by one of the current authors [13] relating QNMs from
quasi-exactly solvable models. Indeed bound state Poschl-Teller potentials have been used for QNM approximations
previously by inverting black hole potentials [4]. However, the AIM does not require any inversion of the black hole
potential as we shall show.
Starting with the potential term
V (x) =

1
sech2 x ,
2

(2.23)

and the Schr

odinger equation, we obtain:


1
d2
2
2
+

sech
x
=0.
dx2
2

(2.24)

As we shall also see in the following sections, it is more convenient to transform our consideration to a finite domain.
Hence, we shall use the transformation y = tanh x, which leads to

 



1
2 d
2
2 d
2
(1 y )
1y
(1 y )
+
=0,
dy
dy
2




2y
2
1
d2
d

=0,
(2.25)
dy 2
1 y 2 dy
(1 y 2 )2
2(1 y 2 )
where 1 < y < 1. The QNM boundary conditions in Eq. (1.1) can then be implemented as follows. As y 1 we
shall have eix (1 y)i/2 . Hence our boundary condition eix (1 y)i/2 . Likewise, as
y 1 we have eix (1 + y)i/2 and the boundary condition eix (1 + y)i/2 . As such we
can take the boundary conditions into account by writing
= (1 y)i/2 (1 + y)i/2 ,

(2.26)

2y(1 i) d 1 2i 2 2
d2
+
,
=
2
dy
1 y 2 dy
2(1 y 2 )

(2.27)

and therefore have

where
2y(1 i)
,
1 y2
1 2i 2 2
.
s0 =
2(1 y 2 )

0 =

(2.28)
(2.29)

Following the AIM procedure, that is, taking n = 0 successively for n = 1, 2, , one can obtain exact eigenvalues:


1
1
n = i n +
.
(2.30)
2
2
This exact QNM spectrum is the same as the one in Ref. [13] obtained through algebraic means.
The reader might wonder about approximate results for cases where Poschl-Teller approximations can be used, such
as Schwarzschild and SdS backgrounds, e.g., see Refs. [1, 4]. In fact when the black hole potential can be modeled
by a Scarf like potential the AIM can be used to find the eigenvalues exactly [12] and hence the QNMs numerically.
We demonstrate this in the next section.

8
III.

SCHWARZSCHILD (A)DS BLACK HOLES

We shall now begin the core focus of this paper, the study of black hole QNMs using the AIM. Recall that the
perturbations of the Schwarzschild black holes are described by the Regge-Wheeler [14] and Zerilli [15] equations, and
the perturbations of Kerr black holes are described by the Teukolsky equations [16]. The perturbation equations for
Reissner-Nordstr
om black holes were also derived by Zerilli [17], and by Moncrief [1820]. Their radial perturbation
equations all have the one-dimensional Schr
odinger-like form with an effective potential.
Therefore, we shall commence in the coming subsections by describing the radial perturbation equations of
0
0
Schwarzschild black holes first, where our perturbed metric shall be g = g
+ h , and where g
is spherically
symmetric. As such it is natural to introduce a mode decomposition to h . Typically we write
lm (t, r, , ) =

eit ul (r)
Ylm (, ) ,
r

where Ylm (, ) are the standard spherical harmonics. The function ul (r, t) then solves the wave equation
 2

d
2

V
(r)
ul (r) = 0 ,
l
dx2

(3.1)

(3.2)

where x, defined by dx = dr/f (r) are the so-called tortoise coordinates, and V (x) is a master potential of the form
[21]



2M
(4 s2 )
`(` + 1)
2
+
(1

s
)

.
(3.3)
V (r) = f (r)
r2
r3
6
In this section
f (r) = 1

2M

r2
r
3

(3.4)

with cosmological constant . Here s = 0, 1, 2 denotes the spin of the perturbation: scalar, electromagnetic and
gravitational, (for half-integer spin see Refs. [15, 22, 23] and Sec. V B).

A.

The Schwarschild Asymptotically Flat Case

To explain the AIM we shall start with the simplest case of the radial component of a perturbation of the
Schwarzschild metric outside the event horizon [15]. For an asymptotically flat Schwarzschild solution ( = 0)
f (r) = 1

2M
,
r

(3.5)

where from dx = dr/f (r) we have

x(r) = r + 2M ln

 r

1 ,
2M

(3.6)

for the tortoise coordinate, x.

Note that for the Schwarzschild background the maximum of this potential, in terms of r, is given by [24]
r0 =

h

1/2 i
3M
1
14
`(` + 1) (1 s2 ) + `2 (` + 1)2 + `(` + 1)(1 s2 ) + (1 s2 )2
.
2 `(` + 1)
9

(3.7)

The choice of coordinates is somewhat arbitrary and in the next section (for SdS) we will see how an alternative
choice leads to a simpler solution. Firstly, consider the change of variable:
=1

2M
,
r

(3.8)

9
with 0 < 1. In terms of , our radial equation then becomes


1 3 d
`(` + 1)
1 s2
4M 2 2
d2
+

+ 2
=0.
d 2
(1 ) d
(1 )4
(1 )2
(1 )

(3.9)

To accommodate the out-going wave boundary condition eix = ei(r+2M ln(r/2M 1)) as (x, r) in terms
of (which is the limit 1) and the regular singularity at the event horizon ( 0), we define
() = 2iM (1 )2iM e

2iM
1

() ,

(3.10)

where the Coulomb power law is included in the asymptotic behaviour (cf. Ref. [6] Eq. (5)). The radial equation
then takes the form:
00 = 0 ()0 + s0 () ,

(3.11)

where
4M i(2 2 4 + 1) (1 3)(1 )
,
(1 )2
16M 2 2 ( 2) 8M i(1 ) + `(` + 1) + (1 s2 )(1 )
s0 () =
.
(1 )2

0 () =

(3.12)
(3.13)

Note that primes of denote derivatives with respect to .

Using these expressions we have tabulated several QNM frequencies and compared them to the WKB method of
Ref. [24] and the CFM of Ref. [6] in Table I. For completeness in Table I we have also included results from an
approximate semi-analytic 3rd order WKB method [24]. More accurate semi-analytic results with better agreement
to Leavers method can be obtained by extending the WKB method to 6th order [8] and indeed in Sec. V we use this
to compare with the AIM for results where the CFM has not been used.
It might also be worth mentioning that a different semi-analytic perturbative approach has recently been discussed
by Dolan and Ottewill [25], which has the added benefit of easily being extended to any order in a perturbative
scheme.

B.

The de-Sitter Case

We have presented the QNMs for Schwarzchild gravitational perturbations in Table I, however, to further justify
the use of this method, it is instructive to consider some more general cases. As such, we shall now consider the
Schwarzschild de Sitter (SdS) case, where we have the same WKB-like wave equation and potential as in the radial
equation earlier, though now
f (r) = 1

2M
r2
,
r
3

(3.14)

where > 0 is the cosmological constant. Interestingly the choice of coordinates we use here leads to a simpler AIM
solution, because there is no Coulomb power law tail; however, in the limit = 0 we recover the Schwarzschild results.
Note that although it is possible to find an expression for the maximum of the potential in the radial equation, for
the SdS case, it is the solution of a cubic equation, which for brevity we refrain from presenting here. In our AIM
code we use a numerical routine to find the root to make the code more general.
In the SdS case it is more convenient to change coordinates to = 1/r [1], which leads to the following master
equation (cf. Eq. (3.3))



`(` + 1) + (1 s2 ) 2M (4 s2 ) 62
d2 p0 d 2
= 0 ,
+
+

(3.15)
d 2
p d
p2
p

10
TABLE I. QNMs to 4 decimal places for gravitational perturbations (s = 2) where the fifth column is taken from Ref. [24].
Note that the imaginary part of the n = 0, ` = 2 result in [24] has been corrected to agree with Ref. [6]. Note also that if the
number of iterations in the AIM is increased, to say about 50, then we find agreement with Ref. [6] accurate to 6 significant
figures.
`

Leaver

AIM (after 15 iterations)

W KB

0.3737 - 0.0896 i[*]

0.3737 - 0.0896 i

0.3732 - 0.0892 i

(<0.01%)(<0.01%)

(-0.13%)(0.44%)[*]

0.3467 - 0.2739 i

2
3
3

0
1

0.3011 - 0.4783 i
0.2515 - 0.7051 i
0.5994 - 0.0927 i
0.5826 - 0.2813 i

0.5517 - 0.4791 i

0.5120 - 0.6903 i

0.4702 - 0.9156 i

0.4314 - 1.152 i

0
1
2
3
4

0.8092 - 0.0942 i
0.7966 - 0.2843 i
0.7727 - 0.4799 i
0.7398 - 0.6839 i
0.7015 - 0.8982 i

0.3467 - 0.2739 i

0.3460 - 0.2749 i

(<0.01%)(<0.01%)

(-0.20%)(-0.36%)

0.3012 - 0.4785 i

0.3029 - 0.4711 i

(0.03%)(-0.04%)

(0.60%)(1.5%)

0.2523 - 0.7023 i

0.2475 - 0.6703 i

(0.32%)(0.40%)

(-1.6%)(4.6%)

0.5994 - 0.0927 i

0.5993 - 0.0927 i

(<0.01%)(<0.01%)

(-0.02%)(0.0%)

0.5826 - 0.2813 i

0.5824 - 0.2814 i

(<0.01%)(<0.01%)

(-0.03%)(-0.04%)

0.5517 - 0.4791 i

0.5532 - 0.4767 i

(<0.01%)(<0.01%)

(0.27%)(0.50%)

0.5120 - 0.6905 i

0.5157 - 0.6774 i

(<0.01%)(-0.03%)

(0.72%)(1.9%)

0.4715 - 0.9156 i

0.4711 - 0.8815 i

(0.28%)(<0.01%)

(0.19%)(3.7%)

0.4360 - 1.147 i

0.4189 - 1.088 i

(1.07%)(0.43%)

(-2.9%)(5.6%)

0.8092 - 0.0942 i

0.8091 - 0.0942 i

(<0.01%)(<0.01%)

(-0.01%)(0.0%)

0.7966 - 0.2843 i

0.7965 - 0.2844 i

(<0.01%)(<0.01%)

(-0.01%)(-0.04%)

0.7727 - 0.4799 i

0.7736 - 0.4790 i

(<0.01%)(<0.01%)

(0.12%)(0.19%)

0.7398 - 0.6839 i

0.7433 - 0.6783 i

(<0.01%)(<0.01%)

(0.47%)(0.82%)

0.7014 - 0.8985 i

0.7072 - 0.8813 i

(-0.01%)(-0.03%)

(0.81%)(1.9%)

where we have defined

p = 2 2M 3 /3

p0 = 2(1 3M ) .

(3.16)

It may be worth mentioning that for SdS we can express [1]:

i

eix = ( 1 ) 21 ( 2 ) 22 ( 3 ) 23

(3.17)

in terms of the roots of f (r), where 1 is the event horizon and 2 is the cosmological horizon (and n is the surface
gravity at each n ). This is useful for choosing the appropriate scaling behaviour for QNM boundary conditions.
Based on the above equation an appropriate choice for QNMs is to scale out the divergent behaviour at the

11
cosmological horizon:1
() = eix u() ,

(3.18)





u=0,
pu00 + (p0 2i)u0 `(` + 1) + (1 s2 ) 2M (4 s2 ) 2
6

(3.19)

which implies

in terms of . Furthermore, based on the scaling in Eq. (3.18), the correct QNM condition at the horizon 1 implies
i

u(x) = ( 1 ) 1 (x) ,

(3.20)

where
1 =


1
1 df 
= M 12
,

2 dr rr1
3 1

(3.21)

with 1 = 1/r1 , and r1 is the smallest real solution of f (r) = 0, implying p = 0. The differential equation then takes
the standard AIM form:
00 = 0 ()0 + s0 () ,

(3.22)

where


1 0
2i
p
2i ,
(3.23)
p
1 ( 1 )




 i


1
i
i
0
`(` + 1) + (1 s2 ) 2M (4 s2 ) 2 +
. (3.24)
s0 () =
+
1
+
(p

2i)
p
6
1 ( 1 )2 1
1 ( 1 )

0 () =

Using these equations, we present in Table II results for SdS with s = 2.

Identical results were generated by the AIM and CFM, both after 50 iterations. Though results are presented for
n = 1, 2, 3, ` = 2, 3 modes only, the AIM is robust enough to be applied to any other case; where like for the = 0
case, agreement with other methods in more extreme parameter choices would only require further iterations.
As far as we are aware, only Ref. [26] (who used a semi-analytic WKB approach) has presented tables for general
spin fields for the SdS case. We have also compared our results to those in [26] for the s = 0, 1 cases and find identical
results (to a given accuracy in the WKB method).
It may be worth mentioning that a set of three-term recurrence relations was derived in Ref. [10] for the CFM, valid
for electromagnetic and gravitational perturbations (s = 1, 2), while for s = 0 this reduces to a five-term recurrence
relation. However, for the AIM we can treat the s = 0, 1, 2 perturbations on an equal footing, see Ref. [10] for more
details. Typically n Gaussian Elimination steps are required to reduce an n + 3 recurrence to a 3-term continued
fraction, e.g., for Reissner-N
ordtrom see Ref. [27] and for higher-dimensional Schwarzschild backgrounds see Ref.
[28]. However, all that is necessary in the AIM is to factor out the correct asymptotic behaviour at the horizon(s)
and infinity (we showed this for higher-dimensional scalar spheroids in Ref. [29]).

C.

The Spin-Zero Anti-de-Sitter Case

There are various approaches to finding QNMs for the SAdS case (an eloquent discussion is given in the appendix
of Ref. [30]). One approach is that of Horowitz and Hubeny [31], which uses a series solution chosen to satisfy the

Note that this is opposite to the case presented in Ref. [1], where they define the QNMs as solutions with boundary conditions
(x) eix as x , for eit time dependence.

12
TABLE II. QNMs to 6 significant figures for Schwarzschild de Sitter gravitational perturbations (s = 2) for ` = 2 and ` = 3
modes. We only present results for the AIM method, because the results are identical to those of the CFM after a given number
of iterations (in this case 50 iterations for both methods). The n = 1, 2 modes can be compared with the results in [26] for
s = 2.
(` = 2)

n=1

n=2

n=3

0.373672 - 0.0889623 i

0.346711 - 0.273915 i

0.301050 - 0.478281 i

0.02

0.338391 - 0.0817564 i

0.318759 - 0.249197 i

0.282732 - 0.429484 i

0.04

0.298895 - 0.0732967 i

0.285841 - 0.221724 i

0.259992 - 0.377092 i

0.06

0.253289 - 0.0630425 i

0.245742 - 0.189791 i

0.230076 - 0.319157 i

0.08

0.197482 - 0.0498773 i

0.194115 - 0.149787 i

0.187120 - 0.250257 i

0.09

0.162610 - 0.0413665 i

0.160789 - 0.124152 i

0.157042 - 0.207117 i

0.10

0.117916 - 0.0302105 i

0.117243 - 0.0906409 i

0.115876 - 0.151102 i

0.11

0.0372699 - 0.00961565 i

0.0372493 - 0.0288470 i

0.0372081 - 0.0480784 i

(` = 3)

n=1

n=2

n=3

0.599443 - 0.0927030 i

0.582644 - 0.281298 i

0.551685 - 0.479093 i

0.02

0.543115 - 0.0844957 i

0.530744 - 0.255363 i

0.507015 - 0.432059 i

0.04

0.480058 - 0.0751464 i

0.471658 - 0.226395 i

0.455011 - 0.380773 i

0.06

0.407175 - 0.0641396 i

0.402171 - 0.192807 i

0.392053 - 0.322769 i

0.08

0.317805 - 0.0503821 i

0.315495 - 0.151249 i

0.310803 - 0.252450 i

0.09

0.261843 - 0.0416439 i

0.260572 - 0.124969 i

0.257998 - 0.208412 i

0.10

0.189994 - 0.0303145 i

0.189517 - 0.0909507 i

0.188555 - 0.151609 i

0.11

0.0600915 - 0.00961888 i

0.0600766 - 0.0288567 i

0.0600469 - 0.0480945 i

SAdS QNM boundary conditions. This method can easily be applied to all perturbations (s = 0, 1, 2). The other
approach is to use the Frobenius method of Leaver [6], but instead of developing a continued fraction the series must
satisfy a boundary condition at infinity, such as Dirichlet [1].
The AIM does not seem easy to apply to metrics where there is an asymptotically anti-de Sitter background,
because for general spin, s, the potential at infinity is a constant and hence would include a combination of ingoing
and outgoing waves, leading to a sinusoidal dependence [32]. However, for the scalar spin zero (s = 0) case, the
potential actually blows up at infinity and is effectively a bound state problem. In this case the AIM can easily be
applied as we show below.
Let us consider the scalar wave equation in SAdS spacetime, where = 3/R2 , and R is the AdS radius. The
master equation takes the same form as for the graviational case, except that the potential becomes

V =

2
1 + r2
r



2
+2
r3

2(r 1)(r2 + r + 2)(r3 + 1)

.
r4


=

(3.25)

Here for simplicity we have taken the AdS radius R = 1, the mass of the black hole M = 1, and the angular momentum
number l = 0. Hence the horizon radius r+ = 1. Thus, with this choice we can compare with the data in Table 3.2
on page 37 of Ref. [33] (see Table III below).
To implement the AIM we first look at the asymptotic behavior of . As r r+ = 1, the potential V goes to zero.
In addition,

i[
4 ln(r1)]

i/4

(r 1)


i/4
1
1
.
r

(3.26)

13
For QNMs we choose the out-going (into the black hole) boundary condition. That is,
e

ix

i/4

1
.
1
r

(3.27)

On the other extreme of our space, r , the potential goes to infinity. This is a crucial difference from the case
of gravitational
perturbations. In that case, the potential goes to a constant. However, in the scalar case, as r ,

2+1/2
(1/r)
and to implement the Dirichlet boundary condition, we take
  21 +2
1

.
r

(3.28)

For the AIM one possible choice of variables is

=1

1
,
r

(3.29)

and we see that to accommodate the asymptotic behaviour of the wavefunction we should take
= i/4 (1 )

2+ 12

(3.30)

Finally, after some work we find the scalar perturbation equation is

00 = 0 ()0 + s0 () ,

(3.31)

where

iq + 2[4 + 2(9 + 4 2) (21 + 10 2) 2 + 4(2 + 2) 3 ]

,
0 =
2q
(3.32)


1
s0 =
4i[9 + 8 2 2(7 + 5 2) + (6 + 4 2) 2 ]q + 2 (1 + )2 (40 + 41 20 2 + 4 3 )
2
16q


4[4 5 + 2 2 ][8(3 + 2 2) + 8(10 + 7 2) (91 + 64 2) 2 + (34 + 24 2) 3 ] ,

(3.33)
and q = (4 + 9 7 2 + 2 3 ). Using the AIM we find the results presented in Table III below.
TABLE III. Comparison of the first few QNMs to 6 significant figures for Schwarzschild anti de Sitter scalar perturbations
(s = 0) for ` = 0 modes with r+ = 1. The second column corresponds to data [33] using the Horowitz and Hubeny (HH)
method [31], while the third column is for the AIM using 70 iterations. [*] Note the mismatch for the real part of the n = 3
mode in [33]; we have confirmed this using the Mathematica notebook provided in Ref. [21].
n

HH method

AIM

2.7982 - 2.6712 i

2.79823 -2.67121 i

4.75849 - 5.03757 i

4.75850 -5.03757 i

6.71927 - 7.39449 i

6.71931 -7.39450 i

8.68223[*] - 9.74852 i

8.68233 -9.74854 i

10.6467 - 12.1012 i

10.6469 -12.1013 i

12.6121 - 14.4533 i

12.6125 -14.4533 i

14.5782 - 16.8049 i

14.5788 -16.8050 i

16.5449 - 19.1562 i

16.5457 -19.1563 i

14
IV.

REISSNER-NORDSTROM
BLACK HOLES

The procedure for obtaining the quasinormal frequencies of Reissner-Nordstrom black holes in four-dimensional
spacetime is similar to that of our earlier cases. Starting with the Reissner-Nordstrom metric
ds2 = f (r)dt2 +

dr2
+ r2 d2 + r2 sin2 d2 ,
f (r)

(4.1)



2
where f (r) = 1 1r + Q
and |Q| 12 is the charge of the black hole. If we consider perturbations exterior to
2
r
the event horizon, the perturbation equations of the Reissner-Nordstrom (charged and non-rotating) geometry can be
separated into two pairs of Schr
odinger-like equations, which describe the even- and odd-parity oscillations respectively
[1720]. They are given by
 2

d
()
()
2
Vi
Zi = 0 ,
(4.2)
dx2
where (+) corresponds to even- and () to odd-parity modes:


4Q2

()
,
Vi (r) = 5 Ar qj +
r
r



d
(+)
()
,
Vi (r) = Vi (r) + 2qj
dx r2 [(l 1)(l + 2)r + qj ]

(4.3)
(4.4)

for i = j = 1, 2 (i 6= j) where
dr

=
,
dx
r
= r2 r + Q2 (r r+ )(r r ) ,
A = l(l + 1) ,
i
p
1h
q1 =
3 + 9 + 16Q2 (l 1)(l + 2) ,
2
i
p
1h
q2 =
3 9 + 16Q2 (l 1)(l + 2) ,
2

(4.5)
(4.6)
(4.7)
(4.8)
(4.9)

and = i. Here is the frequency, l the angular momentum parameter and r and r+ the radii of the inner and
outer (event) horizons of the black hole respectively. Note that r+ = 1 and r = 0 at the Schwarzschild limit (Q = 0);
r+ = r = 21 at the extremal limit (Q = 12 ). Here the tortoise coordinate is given by
Z
x=

2
r+
r2
r2
dr = r +
ln(r r+ )
ln(r r ) ,

r+ r
r+ r

(4.10)

which ranges from at the event horizon to + at spatial infinity.

The QNMs of the Reissner-Nordstr
om black holes are ordinarily accompanied by the emission of both electromagnetic and gravitational radiation, except at the Schwarzschild limit [27, 34]. Eq. (4.2) corresponds to purely
()
()
gravitational perturbations for the radial wave functions Z2 and purely electromagnetic perturbations for Z1 at
(+)
the Schwarzschild limit. Chandrasekhar [35] has shown that the solution Zi for the even-parity oscillations and
()
Zi for the odd-parity oscillations have the relationship
(
)
()
2qj2
dZ
(+)
()
[A(A 2) 2qj ] Zi = A(A 2) + 3
Zi + 2qj i ,
(4.11)
r [(A 2)r + qj ]
dx
so one can just consider solutions for a specific parity, as in the Schwarzschild case, to understand the property of the
()
(+)
black hole. Since the formalism of the effective potential Vi
in the odd-parity equation is much simpler than Vi
in the even-parity equation, it is customary to compute the QNMs for the odd-parity modes.

15
Note that the mass M of the Reissner-Nordstrom black hole has been scaled to 2M = 1, so its quasinormal
frequencies are uniquely determined by the charge Q, the angular momentum l, and the overtone number n of the
mode.
The following procedure is similar to that in Sec. III. At first we change r to the variable x in Eq. (4.2) with
odd-parity mode. From Eq. (4.5) we have
d
d
= 2
,
dx
r dr

(4.12)

and
d2
=
dx2

r2



r 2Q2
r3

2

d2
.
dr2

(4.13)

i
d () h 2
()
()
Zi + Vi
Zi = 0 .
dr

(4.14)

d
+
dr

r2

Substituting Eq. (4.13) into Eq. (4.2) for the odd-parity mode, we get


r2

2

d2 ()
Z
+
dr2 i

r2



r 2Q2
r3

Considering the QNM boundary conditions in the Reissner-Nordstrom case

(
eix ; x
()
Zi
,
eix ; x +
()

and incorporating this into the radial wave function Zi

()

Zi

, we have a form involving the asymptotic behaviour [27]

1 r

= er r1 (r r )

(4.15)

2
r+
+ r

2
r+

(r r+ ) r+ r Zi (r) .

(4.16)

Differentiating Eq. (4.16) one and two times with respect to r, we have
2

()

Zi,r

r+
r+
d ()
1 r r
+
(r r ) r+ r (
Zi = er r1 (r r )
+
Zi ,r + Z Zi ) ,
dr

(4.17)

and
()
Zi,rr

r 1

=e

1 r

(r r )

2
r+
+ r

(r r+ )

2
r+
r+ r





Zi ,rr + 2Z Zi ,r + 2Z + Z,r Zi ,

(4.18)

where Z is defined by
Z =

2
2
r+
1 (1 )(r+ r ) r+
+
+
.
r
(r+ r )(r r )
(r+ r )(r r+ )

(4.19)

Substituting Eqs. (4.17) and (4.18) into Eq. (4.14), we obtain

"
 2
 2   
#

r 2Q2
Zi ,rr + 2Z
+
Zi ,r
r2
r2
r2
r3
( 
)
 

2
i
h

r 2Q2
()
2
2
+
Z + Z,r +
Z + Vi
Zi = 0 .
r2
r2
r3
For the same reason as in Sec. III, here we change the variable r to by the definition = 1
from 0 at the event horizon to 1 at spatial infinity. Thus we have
d
(1 )2 d
=
,
dr
r+ d

(4.20)
r+
r ,

which ranges

(4.21)

16
and
(1 )4 d2
(1 )3 d
d2
=
2
.
2
2
2
2
dr
r+ d
r+
d

(4.22)

Substituting Eqs. (4.21) and (4.22) into Eq. (4.20), and rewriting the equation in the AIM form, we obtain
Zi , = Zi ()Zi , + sZi ()Zi ,

(4.23)

where
2
r+ 2Q2 (1 )
2r+ Z

2
1
(1 )
(1 )2
h
i
()
6
2 2
2 + Vi
r+

r+
Z
r+ Z 
r+ Z,
2

,
sZi () =
r

2Q
(1

+
2
8
4
4
(1 )
(1 )
(1 )
(1 )2


2
(1 )(r+ r ) r+
(1 )
1
r+ (1 )
Z =
+
+
,
r+
(r+ r ) [r+ r (1 )]
(r+ r )


4Q2 (1 )
(1 )5 Ar+
()

q
+
Vi
,
=
j
5
r+
1
r+
r+ [r+ r (1 )]
.
=
(1 )2

Zi () =

(4.24)

(4.25)
(4.26)
(4.27)
(4.28)

The numerical results to four decimal places are presented in Table IV, V and VI. They are compared with Leaver
and W KB from [27] and [34] respectively. The quasinormal frequencies appear as complex conjugate pairs in ; we
list only the ones with Im() > 0. Note that we arrange as (Im(), Re()). In Table VI the quasinormal frequencies
obtained by the WKB method are not available. It is apparent that the quasinormal frequencies obtained by the AIM
are very accurate except for n = 3 in the extremal case Q = 21 in Tables IV and VI.
The QNMs of l = 2 and i = 2 in Table IV reduce to the purely gravitational QNMs in the Schwarzschild case at
Q = 0, while the QNMs of l = 2 and i = 1 in Table V reduce to the purely electromagnetic QNMs at Q = 0.

V.

KERR BLACK HOLES

A rotating black hole carrying angular momentum is described by the Kerr metric (in Boyer-Lindquist coordinates)
as




a2 r sin2
r  2 2ar sin2
dtd + dr2 d2 + r2 + a2 +
sin2 d ,
ds2 = 1
dt

(5.1)

= r2 + a2 2M r (r r )(r r+ ) ,

(5.2)

= r2 + a2 cos2 ,

(5.3)

with

and where a is the Kerr rotation parameter with 0 a M , where we have included a general black hole mass, M .
The horizons r and r+ are again the inner and the outer (event) horizons respectively. Teukolsky [16] showed that
the perturbation equations in the Kerr geometry are separable, where the separated equations for the angular wave
function s Slm () and the radial wave function R(r) are given by


(m + su)2
2
2 2 2
[(1 u )S,u ],u + a u 2asu + s + s Alm
(5.4)
s Slm = 0 ,
1 u2
R,rr + (s + 1)(2r 1)R,r + K(r)R = 0 ,
(5.5)

17
TABLE IV. Reissner-Nordstr
om quasinormal frequency parameter values ( = i) for the fundamental (n = 0) and two
lowest overtones for l = 2 and i = 2.
n

Leaver

(0.7473,-0.1779)

0.2

0.4

AIM

(0.7569,-0.1788)
(0.8024,-0.1793)

0.495

(0.8586,-0.1685)

(0.6934,-0.5478)

0.2

(0.7035,-0.5503)

0.4

(0.7538,-0.5499)

0.495

(0.8070,-0.5140)

0.2

0.4

0.495

(0.6021,-0.9566)
(0.6129,-0.9599)
(0.6703,-0.9531)
(0.7078,-0.8872)

W KB

(0.7473,-0.1779)

(0.7463,-0.1784)

(<0.01%)(<0.01%)

(-0.13%)(-0.28%)

(0.7569,-0.1788)

(0.7558,-0.1793)

(<0.01%)(<0.01%)

(-0.15%)(-0.28%)

(0.8024,-0.1793)

(0.8011,-0.1797)

(<0.01%)(<0.01%)

(-0.16%)(-0.22%)

(0.8586,-0.1685)

(0.8566,-0.1706)

(<0.01%)(<0.01%)

(-0.23%)(-1.25%)

(0.6934,-0.5478)

(0.6920,-0.5478)

(<0.01%)(<0.01%)

(-0.20%)(-0.37%)

(0.7035,-0.5502)

(0.7020,-0.5522)

(<0.01%)(0.02%)

(-0.21%)(-0.36%)

(0.7538,-0.5499)

(0.7510,-0.5525)

(<0.01%)(<0.01%)

(-0.37%)(-0.47%)

(0.8067,-0.5164)

(0.8068,-0.5287)

(-0.04%)(0.47%)

(0.01%)(-2.21%)

(0.6021,-0.9566)

(0.6059,-0.9421)

(<0.01%)(<0.01%)

(0.63%)(1.52%)

(0.6128,-0.9599)

(0.6164,-0.9458)

(0.02%)(<0.01%)

(0.57%)(1.47%)

(0.6703,-0.9531)

(0.6717,-0.9455)

(<0.01%)(<0.01%)

(0.21%)(0.80%)

(0.8350,-0.8347)

(0.7344,-0.9135)

(17.97%)(5.92%)

(2.66%)(-2.96%)

where the function

K(r) =

2


o
1 n 2
r + a2 2 2amr + a2 m2 + is am (2r 1) r2 a2
+ 2isr a2 2 s Alm .

(5.6)

In the above u = cos , s is the spin weight, s Alm is the spin-weighted separation constant for the angular equation,
and m is another angular momentum parameter. For completeness the evaluation of the separation constant s Alm
using the AIM is discussed in Appendix A.
In order to use the AIM we need to solve for the angular solution in the radial equation. However, for nonzero s
the effective potential of the radial equation is in general complex. A straight forward application of the AIM does
not give the correct answer. In fact a similar problem occurs in both numerical [36] and WKB [7] methods. For this
reason we shall look at each of the spin cases (0, 12 , 2) separately in the following subsections.

A.

The Spin-Zero Case

Because the AIM works better on a compact domain, we define a new variable y = 1 rr+ , which ranges from 0 at
the event horizon (r = r+ ) to 1 at spatial infinity. It is then necessary to incorporate the boundary conditions, which

18
TABLE V. Reissner-Nordstr
om quasinormal frequency parameter values ( = i) for the fundamental (n = 1) and two lowest
overtones for l = 2 and i = 1.
n

Leaver

(0.9152,-0.1900)

0
0

0.2

AIM

(0.9599,-0.1929)

0.4

(1.1403,-0.1984)

0.495

(1.3855,-0.1773)

(0.8731,-0.5814)

0.2

(0.9200,-0.5894)

0.4

(1.1100,-0.6021)

0.495

(1.3573,-0.5350)

2
2
2
2

(0.8024,-1.0032)

0.2

(0.8530,-1.0143)

0.4

(1.0582,-1.0263)

0.495

(1.3019,-0.9024)

W KB

(0.9152,-0.1900)

(0.9143,-0.1901)

(<0.01%)(<0.01%)

(-0.10%)(-0.05%)

(0.9599,-0.1929)

(0.9590,-0.1930)

(<0.01%)(<0.01%)

(-0.09%)(-0.05%)

(1.1403,-0.1981)

(1.1395,-0.1980)

(<0.01%)(0.15%)

(-0.07%)(-0.20%)

(1.3855,-0.1773)

(1.3850,-0.1783)

(<0.01%)(<0.01%)

(-0.04%)(-0.56%)

(0.8731,-0.5814)

(0.8717,-0.5819)

(<0.01%)(<0.01%)

(-0.16%)(-0.09%)

(0.9200,-0.5894)

(0.9186,-0.5897)

(<0.01%)(<0.01%)

(-0.15%)(-0.05%)

(1.1100,-0.6021)

(1.1081,-0.6014)

(<0.01%)(<0.01%)

(-0.17%)(0.12%)

(1.3573,-0.5350)

(1.3579,-0.5423)

(<0.01%)(<0.01%)

(0.04%)(-1.36%)

(0.8024,-1.0032)

(0.8046,-0.9917)

(<0.01%)(<0.01%)

(0.27%)(1.15%)

(0.8530,-1.0143)

(0.8548,-1.0037)

(<0.01%)(<0.01%)

(0.21%)(1.05%)

(1.0582,-1.0263)

(1.0568,-1.0181)

(<0.01%)(<0.01%)

(-0.13%)(0.80%)

(1.3019,-0.9024)

(1.3141,-0.9222)

(<0.01%)(<0.01%)

(0.94%)(-2.19%)

expressed in the new compact domain is


(y) =

r
1
(1 y)
r+

i

r+

y + (1 y)r+ ei 1y (y) .

(5.7)

By making the change of coordinates and change of function, Eq. (5.7) takes the form
(y) = 0 (y) + s0 (y) ,

(5.8)

1 dg
1 df

,
g dy f dy

(5.9)

where
0 = 2
and
s0 =

1 d2 g
1 df
1 dg
1

2 V |r=r+ (1y)1 .
g dy 2
f dy
g dy f 2

(5.10)

In the above we have defined


f=


dy 
,
r2 + a2 dr r=r+ (1y)1

(5.11)

19
TABLE VI. Reissner-Nordstr
om quasinormal frequency parameter values ( = i) for the fundamental (n = 1) and two
lowest overtones for l = 1 and i = 1.
n

Leaver

(0.4965,-0.1850)

AIM
(0.4965,-0.1850)
(<0.01%)(<0.01%)

0.2

(0.5238,-0.1883)

(0.5238,-0.1883)
(<0.01%)(<0.01%)

0.4

(0.6470,-0.1965)

(0.6470,-0.1965)
(<0.01%)(<0.01%)

0.495

(0.8428,-0.1742)

(0.4290,-0.5873)

0.2

(0.4598,-0.5953)

0.4

(0.5980,-0.6107)

0.495

(0.7979,-0.5293)

(0.8428,-0.1742)
(<0.01%)(<0.01%)
(0.4292,-0.5873)
(<0.01%)(0.05%)
(0.4598,-0.5953)
(<0.01%)(<0.01%)
(0.5980,-0.6107)
(<0.01%)(<0.01%)
(0.7978,-0.5280)
(0.01%)(0.25%)

(0.3496,-1.0504)

(0.3495,-1.0504)
(0.03%)(<0.01%)

0.2

(0.3832,-1.0596)

(0.3832,-1.0596)
(<0.01%)(<0.01%)

0.4

(0.5340,-1.0660)

(0.5340,-1.0659)
(<0.01%)(<0.01%)

0.495

(0.7104,-0.9055)

(0.6248,-1.0574)
(-12.05%)(-16.78%)

and

i
1
r
g = (1 y)2i 1
(1 y)
y i+ eir+ (1y) ,
r+

(5.12)

where = r2 + a2 2M r,
=

1
[(r2 + a2 ) + am] ,
r+ r

(5.13)

and a is again our rotation parameter. Eq. (5.8) is now in the correct form to use the AIM for QNM frequency
calculations.
As presented in Tables VII and VIII are the QNM frequencies for the scalar perturbations of the Kerr black hole
with the two extreme (minimum and maximum) values of the angular momentum per unit mass, i.e, a = 0.00 and
a = 0.80. m was set to 0, while l was given values of 0, 1 and 2 and n varied accordingly.
Included in Table VII are the numerically determined QNM frequencies published by Leaver in 1985 [6]. The
percentages bracketed under each QNM frequency via the AIM are the percentage differences between the calculated
value and the numerical value published by Leaver. With the exceptions of the QNM frequencies for l = 0, n = 0 and
l = 2, n = 2, the AIM values correspond to the CFM up to four decimal places and even those anomalies differ by
less than 0.30%. Proving, at least in this case, the AIM is a precise semi-analytical technique.

20
In Table VIII, all three values were calculated for this work, even though published values are available for the third
order WKB(J), at least graphically. Since the WKB(J) is a generally accepted semi-analytical technique for QNM
frequency calculations, the percentages below the AIM values are the differences to the sixth order WKB(J) values.
Only in the case of l = 0, n = 0 does the AIM QNM frequency significantly differ from the sixth order WKB(J) value.
Note that in an upcoming work, further values will be presented for values of a = 0.20, a = 0.40 and a = 0.60, with
the same variations of l and n with M = 1 [37].
TABLE VII. The QNM Frequencies for the Scalar Perturbations of the Kerr Black Hole, with a = 0.00, i.e., the Schwarzschild
limit (M = 1).
l

Numerical

Third Order WKB(J)

Sixth Order WKB(J)

AIM

0.1105 - 0.1049i

0.1046 - 0.1001i

0.1105 - 0.1008i

0.1103 - 0.1046i

(-5.34% , 9.82%)

(<0.01% , -2.91%)

(-0.18% , -0.29%)

0.2929 - 0.0977i

0.2911 - 0.0989i

0.2929 - 0.0978i

0.2929 - 0.0977i

(-0.61% , 1.23%)

(<0.01% , 0.10%)

(<0.01% , <0.01%)

0.2645 - 0.3063i

0.2622 - 0.3074i

0.2645 - 0.3065i

0.2645 - 0.3063i

(-0.87% , 0.36%)

(<0.01% , 0.07%)

(<0.01% , <0.01%)

0.4836 - 0.0968i

0.4832 - 0.0968i

0.4836 - 0.0968i

0.4836 - 0.0968i

(-0.08% , <0.01%)

(<0.01% , <0.01%)

(<0.01% , <0.01%)

0.4639 - 0.2956i

0.4632 - 0.2958i

0.4638 - 0.2956i

0.4639 - 0.2956i

(-0.15% , 0.07%)

(-0.02% , <0.01%)

(<0.01% , <0.01%)

0.4305 - 0.5086i

0.4317 - 0.5034i

0.4304 - 0.5087i

0.4306 - 0.5086i

(0.28% , -1.02%)

(-0.02% , 0.02%)

(0.02% , <0.01%)

TABLE VIII. The QNM Frequencies for the Scalar Perturbations of the Kerr Black Hole, with a = 0.80 (M = 1).
l

Third Order WKB(J)

Sixth Order WKB(J)

AIM

0.1005 - 0.1007i

0.1211 - 0.0897i

0.1141 - 0.0939i

0.3029 - 0.0891i

0.3053 - 0.0893i

0.2758 - 0.2779i

0.2821 - 0.2755i

0.5035 - 0.0885i

0.5041 - 0.0886i

0.4866 - 0.2693i

0.4885 - 0.2690i

0.4585 - 0.4570i

0.4607 - 0.4581i

(-5.78% , 4.68%)
0.3052 - 0.0892i
(-0.03% , -0.11%)
0.2817 - 0.2756i
(-0.14% , 0.04%)
2

0.5041 - 0.0886i
(<0.01% , <0.01%)
0.4885 - 0.2689i
(<0.01% , -0.04%)

B.

The Spin-Half Case

For the spin-1/2 case we would like to know how the AIM can be used to derive an appropriate form of the Dirac
equation in this spacetime background using the basis set up by four null vectors which are the basis of the NewmanPenrose formalism, for further details see Ref. [37]. That is, in the Kerr background we adopt the following vectors

21
as the null tetrad:
lj =
nj =
mj =
lj =
nj =
mj =

1
(, 2 , 0, a sin2 ) ,

1
(, 2 , 0, a sin2 ) ,
22
1
(ia sin , 0, 2 , i(r2 + a2 ) sin ) ,
2

1 2
(r + a2 , , 0, a) ,

1
(r2 + a2 , , 0, a) ,
2

1
1
(ia sin , 0, 1, i
),
sin
2

where m
j and m
j are nothing but complex conjugates of mj and mj respectively.
It is clear that the basis vectors basically become derivative operators when these are applied as tangent vectors to
the function ei(t+m) . Therefore we can write
1
1
~l = D = D0 , ~n = D = D , m
~ = =
~ = = L0 , m
L0 ,
22 0
2

(5.14)

where,
rM
K
+ 2n
,

K
rM
Dn = r i + 2n
,

Ln = + Q + n cot ,
Dn = r + i

Ln = Q + n cot ,
and K = (r2 + a2 ) + am with Q = a sin + m csc .
The spin coefficients can be written as a combination of basis vectors in the Newman-Penrose formalism which are
now expressed in terms of the elements of different components of the Kerr metric. So by combining these different
components of basis vectors in a suitable manner we get the spin coefficients as
=====0.
1
cot
ia sin
,=
,=
,

2 2

2(
)2
ia sin

rM
=
, = 2 , =+
,
2
2

2
2
2
= .

(5.15)

(5.16)

Using the above definitions and results and choosing f1 = F1 , g2 = G2 , f2 = F2 and g1 = G1 (where F1,2 and G1,2
are the pair of spinors), the Dirac equation is reduced to
1
D0 f1 + L 12 f2 = 0 ,
2

D 1 f2 2L 1 f1 = 0 ,
2

1
D0 g2 L1 f1 = 0 ,
2 2

D 1 g1 + 2L 12 g2 = 0 .
2

(5.17)

22
We separate the Dirac equation into radial and angular parts by choosing,
f1 (r, ) = R 12 (r)S 21 () , f2 (r, ) = R 12 (r)S 21 () ,
g1 (r, ) = R 12 (r)S 12 () , g2 (r, ) = R 21 (r)S 12 () .
Replacing these fj and gj (j = 1, 2) and using as the separation constant, we get,
L 12 S 12 = S 12 ,

(5.18)

L1 S 12 = S 21 ,
2

2 D0 R 12 = 2 R 12 ,

1
2

1
D0 2 R 12

(5.19)

= R 21 ,

where 2 2 R 12 is redefined as R 12 .
Eqs. (5.18) and (5.19) are the angular and radial Dirac equation respectively, in a coupled form with the separation
constant [35]. Decoupling Eq. (5.18) gives the eigenvalue/angular equation for spin half particles as
h
i
L 12 L1 + 2 S 21 = 0 ,
(5.20)
2

and S 12 satisfies the adjoint equation (obtained by replacing by ). Decoupling Eq. (5.19) then gives the radial
equation for spin half particles as
h
i
(5.21)
D1 D0 2 R 21 = 0 ,
2

1
2

and R 21 satisfies the complex-conjugate equation. Furthermore, note that unlike the case of a scalar particle, a
spin half particle is not capable of extracting energy from a rotating black hole, that is, there is no Penrose Process
(superradiance) equivalent scenario [37].
Returning now to the AIM, recall that is works better on a compact domain, as such we define a new variable
2
y = 1 rr+ , which ranges from 0 at the event horizon (r = r+ ) to 1 at spatial infinity. It is then necessary to
incorporate the boundary conditions, which expressed in the new compact domain is

(y) =

 21 i
r
1
r
i +
2
1
(1 y )
(y 2 ) 2 + (1 y 2 )r+ e 1y2 (y) .
r+

(5.22)

By making the change of coordinates and change of function, our equation takes the form
(y) = 0 (y) + s0 (y) ,

(5.23)

1 df
1 dg

,
g dy f dy

(5.24)

where as in Sub-Sec.V A we have

0 = 2
and
s0 =

1 d2 g
1 df
1 dg
1

2 2 V |r=r+ (1y2 )1 .
2
g dy
f dy
g dy f

(5.25)

As presented in Table IX and Table X the QNM frequencies for the spin half perturbations of the Kerr black hole
with the two extreme values of the angular momentum per unit mass, i.e. a = 0.00 and a = 0.80, m was set to 0,
while l was given values of 0, 1 and 2 and n varied accordingly.
Included in Table IX are the numerically determined QNM frequencies published by Jing et al. [38]. Even though
the WKB method has been used to calculate the Schwarzschild limit QNM frequencies before [23], the sixth order

23
WKB values and AIM values are novel to this work and shall be explored more fully in Ref. [37]. The percentages
bracketed under each QNM frequency, is the percentage difference between the calculated value and the numerical
value published by Jing et al. [38].
As expected, since there are additional correction terms, the sixth order WKB QNM frequencies are closer to the
numerical values than the third order WKB values. While the AIM does not prove as accurate in its calculation of
the spin half QNM frequencies as it did with the scalar values, none of the differences between the AIM values and
the numerical values exceed 0.30%, except for when l = 2, n = 2.
Similarly in Table X are the numerically determine QNM frequencies published by Jing et al. [38]. Both the third
and sixth order WKB values along with the AIM values are novel to this work and shall also be explored more fully
in Ref. [37]. The percentages bracketed under each QNM frequency, are the percentage differences between the
calculated value and the numerical value published by Jing, at least for l = 0 and l = 1. For l = 2, the AIM values
are compared to the sixth order WKB values. As already noted, since there are additional correction terms, the sixth
order WKB QNM frequencies are closer to the numerical values than the third order WKB values. The AIM does
not appear to be as precise in calculating the QNM frequencies for spin half perturbations of the Kerr black hole as
it was for the scalar perturbations. As in the previous subsection, additional tables and plots of these Kerr processes
shall constitute a future work [37], and shall be explored further there.
TABLE IX. The QNM Frequencies for the Spin Half Perturbations of the Kerr black hole, with a = 0.00, i.e., the Schwarzschild
limit (M = 1).
l

Numerical

0.1830 - 0.0970i

0
1

0
1
2

0.3800 - 0.0964i
0.3558 - 0.2975i
0.5741 - 0.0963i
0.5570 - 0.2927i
0.5266 - 0.4997i

Third Order WKB

Sixth Order WKB

AIM

0.1765 - 0.1001i

0.1827 - 0.0949i

0.1830 - 0.0969i

(-3.55% , 3.20%)

(-0.16% , -2.16%)

(<0.01% , -0.10%)

0.3786 - 0.0965i

0.3801 - 0.0964i

0.3800 - 0.0964i

(-0.37% , 0.10%)

(0.03% , <0.01%)

(<0.01% , <0.01%)

0.3536 - 0.2987i

0.3559 - 0.2973i

0.3568 - 0.2976i

(-0.62% , 0.40%)

(0.03% , -0.07%)

(0.28% , 0.03%)

0.5737 - 0.0963i

0.5741 - 0.0963i

0.5741 - 0.0963i

(-0.07% , <0.01%)

(<0.01% , <0.01%)

(<0.01% , <0.01%)

0.5562 - 0.2930i

0.5570 - 0.2927i

0.5573 - 0.2928i

(-0.14% , 0.10%)

(<0.01% , <0.01%)

(0.05% , 0.03%)

0.5273 - 0.4972i

0.5265 - 0.4997i

0.5189 - 0.5213i

(0.13% , -0.50%)

(-0.02% , <0.01%)

(-1.46% , 4.32%)

C.

The Spin-Two Case

As we have mentioned earlier, the radial equation for nonzero spin s is in general complex. In fact, it does not even
reduce back to the Regge-Wheeler and Zerilli equations when the rotation parameter a 0. Detweiler [36] has found
a way to overcome this problem, where he defined a new function




s/2
2
2 1/2
s+1 dR
X=
r +a
(r)R + (r)
.
(5.26)
dr
If the functions (r) and (r) are required to satisfy
2 0 s+1 + 0 s+1 2 2s+1 K = constant ,

(5.27)

24
TABLE X. The QNM Frequencies for the Spin Half Perturbations of the Kerr black hole, with a = 0.80 (M = 1).
l

Numerical

Third Order WKB(J)

Sixth Order WKB(J)

AIM

0.1932 - 0.0891i

0.1883 - 0.0896i

0.1914 - 0.0865i

0.1920 - 0.0872i

(-2.54% , 0.56%)

(-0.93% , -2.92%)

(-0.62% , -2.13%)

0
1

0.3993 - 0.0893i
0.3789 - 0.2728i

0.3956 - 0.0881i

0.3967 - 0.0880i

0.3965 - 0.0880i

(-0.93% , -1.34%)

(-0.65% , -1.46%)

(-0.70% , -1.46%)

0.3751 - 0.2701i

0.3777 - 0.2687i

(-1.00% , -0.99%)

(-0.32% , -1.50%)

0.5984 - 0.0881i

0.5987 - 0.0881i

0.5987 - 0.0882i
(<0.01% , 0.11%)

0.5844 - 0.2669i

0.5855 - 0.2667i

0.5847 - 0.2644i
(-0.14% , -0.86%)

0.5600 - 0.4512i

0.5609 - 0.4517i

then it can be shown that the radial equation in Eq. (5.5) becomes
d2 X
VX =0,
dx2

(5.28)

dG
,
dx

(5.29)

where
V =

U
(r2

2
a2 )

+ G2 +

+
r
s(2r 1)
,
+
G=
2
2
2
2 (r + a ) (r + a2 )2

0
20 + 0 s+1
U =K+
,
s
r+
r
x=r+
ln (r r+ )
ln (r r ) .
r+ r
r+ r

(5.30)
(5.31)
(5.32)

As Detweiler has indicated, it is possible to choose the functions (r) and (r) so that the resulting effective potential
V (r) is real and has the form
(
)
f (f + 2) b [ (g 0 g0 )] [g b (g 0 g0 )]
2
4 +
V =
2
2
g + b

(r2 + a2 )
2 (g + b) (g b)
"
#2
#
"
ram

d
ram
Y2
+

(5.33)
2
2 ,
r2 + a2 dr (r2 + a2 )2
(r2 + a2 )
(r2 + a2 )
where
g = f 2 + 3(r2 + a2 ) 3r2 ,
am
= r2 + a2
,

n
h
i
o1/2
am 
= 9 2f a2
(5f + 6) 12a2 + 2bf (f + 2)
,



am
b = 3 a2
,

2
2
Y = am (r + a ) ,

(5.36)

f = A + a2 2 2am .

(5.39)

(5.34)
(5.35)

(5.37)
(5.38)

25
When the Kerr rotation parameter a approaches zero, the potential V in Eq. (5.33) coincides with the Regge-Wheeler
potential for negative , and coincides with the Zerilli potential for positive . Here we choose to be negative, where
the choice of the sign in Eq. (5.37) is determined by the sign of m [7, 36].
The QNM boundary conditions for X are
(
eikx ; x
X
,
(5.40)
eix ; x
where
k=

am
.
r+

(5.41)

Hence, we write
X=e

ir

i
r+ r

(r r+ )r+
(r r )r

 r

ik
+ r

G .

Substituting this into Eq. (5.28), we have

"



2 #

G r 2 a 2
r2 + a2
r 2 a2
2

G,r + G + G,r +
V G = 0 ,
G,rr + 2G +
(r2 + a2 )
(r2 + a2 )

(5.42)

(5.43)

where
G = i +

ikr
i

.
r(r+ r )

(5.44)

As before we define the variable = 1 r+ /r which has a compact domain 0 < < 1. Eq. (5.43) can then be written
in the AIM form
G, = G ()G, + sG ()G ,

(5.45)

where


2
a2 (1 )2
r+
2
1 r+

G =
2G +
,
2 + a2 (1 )2
1
(1 )2
r+
(
)
2
2
2
2
+ a2 (1 )2
a2 (1 )2
r+
r+
G r+
r+
2
sG =
V

G, .
2 + a2 (1 )2 G
(1 )4
(1 )2
r+
(1 )2

(5.46)
(5.47)

The results for the gravitational (spin-two) case are presented in Tables XI and XII. In general the error of the
separation constant is smaller than that of the quasinormal frequencies. As for the quasinormal frequencies the error
in the Kerr case is larger than that of either the Schwarzschild or the Reissner-Nordstrom cases, where this is due
to our consideration of the angular and the radial equations simultaneously. The number of iterations that can be
performed in the code is relatively small, much like the number of continued fractions in the CFM is typically smaller
due to the coupling between radial and angular equations.

VI.

DOUBLY ROTATING KERR (A)DS BLACK HOLES

Rotating black holes in higher dimensions were first discussed in the seminal paper by Myers and Perry [39]. One
of the unexpected results to come from this work was that some families of solutions were shown to have event
horizons for arbitrarily large values of their rotation parameters. The stability of such black holes is certainly in
question [40, 41], but no direct proof of instability has been provided. Another new feature of the Myers-Perry (MP)

26
TABLE XI. Spin-2 angular separation constants and Kerr gravitational quasinormal frequencies for the fundamental mode
corresponding to l = 2 and m = 0 compared with the CFM [6] (M = 1/2).
a

ALeaver

AAIM

Leaver

AIM

(4.0000, 0.0000)

(4.0000, 0.0000)

(0.7473, -0.1779)

(0.7413, -0.1780)

0.1

(3.9972, 0.0014)

0.2

(3.9886, 0.0056)

0.3

(3.9730, 0.0126)

0.4

(3.9480, 0.0223)

0.45

(3.9304, 0.0276)

(<0.01%)(<0.01%)
(3.9973, 0.0014)

(-0.80%)(-0.06%)
(0.7502, -0.1774)

(<0.01%)(<0.01%)
(3.9887, 0.0056)

(-0.77% , -0.06%)
(0.7594, -0.1757)

(<0.01%)(<0.01%)
(3.9733, 0.0126)

(0.7715, -0.1722)
(-0.59%)(-0.12%)

(0.8038, -0.1643)

(0.8025, -0.1639)

(0.8240, -0.1570)

(0.8250, -0.1591)

(<0.01%)(-0.45%)
(3.9303, 0.0280)

(0.7540, -0.1763)
(-0.71%)(-0.03%)

(0.7761, -0.1720)

(<0.01%)(<0.01%)
(3.9482, 0.0222)

(0.7444, -0.1775)

(-0.16%)(0.24%)

(<0.01%)(1.45%)

(0.12%)(-1.34%)

TABLE XII. Spin-2 angular separation constants and Kerr gravitational quasinormal frequencies for the fundamental mode
corresponding to l = 2 and m = 1 compared with the CFM [6] (M = 1/2).
a

ALeaver

AAIM

Leaver

AIM

(4.0000, 0.0000)

(4.0000, 0.0000)

(0.7473, -0.1779)

(0.7413, -0.1780)

0.1

(3.8932, 0.0252)

(<0.01%)(<0.01%)
(3.8937, 0.0250)

(-0.80%)(-0.06%)
(0.7765, -0.1770)

(-0.01%)(-0.89%)
0.2

(3.7676, 0.0532)

(3.7681, 0.0526)

(-0.51% , 0.86%)
(0.8160, -0.1745)

(0.01%)(-1.12%)
0.3

(3.6125, 0.0835)

(3.6123, 0.0826)

(3.4023, 0.1122)

(3.4011, 0.1110)

(0.8719, -0.1693)

(3.2535, 0.1195)

(3.2491, 0.1173)
(-0.03%)(-1.82%)

(0.8722, -0.1674)
(0.03%)(1.00%)

(0.9605, -0.1559)

(-0.03%)(-1.00%)
0.45

(0.8143, -0.1726)
(-0.02%)(1.09%)

(<0.01%)(-0.99%)
0.4

(0.7726, -0.1755)

(0.9620, -0.1543)
(0.16%)(1.05%)

(1.0326, -0.1396)

(1.0376, -0.1369)
(0.48%)(1.97%)

solutions was that they in general have b D1

2 c spin parameters, making them somewhat more complex than the four
dimensional Kerr solution. The first asymptotically non-flat five-dimensional MP metric was given in [42]. Subsequent
generalizations to arbitrary dimensions was done in [43], and finally the most general Kerr-(A)dS-NUT metric was
found by Chen, L
u and Pope [44] (for more details see some of the other reviews in this special issue).
In this section we review how the AIM can be used to solve the D 6 two-rotation scalar perturbation equations
(for more details on the metric and resulting separation see [45]). The scalar field master equations are found to be
[45]:

  2

1 d
dRr
(r + a21 )2 (r2 + a22 )2 2 a21 a22 j(j + D 7)
2
0 = D6
rD6 r
+

b
r

b
(6.1)
1
2 Rr ,
r
dr
dr
r
r2
" 
# 
 D6

D6
ai
d
yi
dRi
(a21 yi2 )2 (a22 yi2 )2 2
a21 a22 j(j + D 7)
2
0=
yi

yi +
+ b1 y i b2 R i ,
yi
dyi
ai
dyi
y i
yi2
(6.2)

27
where
r = (1 + g 2 r2 )(r2 + a21 )(r2 + a22 ) 2M r7D ,
yi = (1

g 2 yi2 )(a21

yi2 )(a22

yi2 )

(6.3)
(6.4)

the radial and angular frequencies are defined by:




m2 a2
m1 a1
+ 2
,

r = (1 + g r )
r2 + a21
r + a22


m2 a2
m1 a1
+ 2
,

yi = (1 g 2 yi2 )
a21 yi2
a2 yi2
2 2

(6.5)
(6.6)

and i = 1, 2. In the above g is the curvature of the spacetime and a1 , a2 are the two rotation parameters and for later
reference we define  = a2 /a1 .
Doubly rotating black holes are more complicated than simply rotating black holes (cf. Ref. [46]), because two
rotation planes lead to two coupled spheroids which are also needed for the solution of the radial equation.

A.

Radial Quasi-Normal Modes

For simplicity we will consider the flat case, setting g = 0, which leads to easier QNM boundary conditions (cf.
Schwarzschild to Schwarzschild-dS). These satisfy the boundary condition that there are only waves ingoing at the
black hole horizon and outgoing waves at asymptotic infinity.
As we have shown with the previous examples, it is easier to work on a compact domain and define the variable
x = 1/r, so that infinity is mapped to zero and the outer horizon stays at xh = 1/rh = 1. The domain of x, therefore,
will be [0, 1]. Thus the QNM boundary condition is translated into the statement that the waves move leftward at
x = 0 and rightward at x = 1. We again choose the AIM point in the middle of the domain, i.e., at x = 1/2.
In terms of x the radial equation (6.1) becomes:

  2

d
(x + a21 )2 (x2 + a22 )2 2
b1
dR
0 = xD4
x8D x
+

x a21 a22 j(j + D 7)x2 2 b2 R ,

(6.7)
dx
dx
x
x
where x (x) r (r = 1/x) and x (x) r (r = 1/x).
After performing some asymptotic analysis, we find that for the solutions to satisfy the QNM boundary conditions
we must have:
R (1 x)ih h x(D2)/2 eix /x y(x) ,

(6.8)

h x (x = 1) ,
(1 + a21 )(1 + a22 )
.
h
0x (x = 1)

(6.9)

where

(6.10)

We then substitute this ansatz into Eq. (6.7) and rewrite into the AIM form:
y 00 = 0 y 0 + s0 y .

(6.11)

This final step was performed in Mathematica and then the resulting expressions for 0 and s0 were fed into the AIM
routine. The method we use to find the QNMs proceeds in a fashion similar to that used in Ref. [47, 48] except we
use the AIM instead of the CFM.
First we set the number of AIM iterations in both the eigenvalue and QNM calculations to sixteen. We start with
the Schwarzschild values (b1 , b2 , ), i.e., at the point (a1 , a2 ) 0 and then increment a1 and a2 by some small value.

28

Re @D
0.90
0.85
0.80
0.75
0.70
0.65

Im @D
-1

-0.5

0.5

-0.2
-0.3
-0.4
-0.5

a1

=1
.8
.6
.4
.2
0

-1

-0.5

0.5

0
.2
.4
.6
.8
=1

a1

FIG. 1. An example of the D = 6 fundamental (j, m1 , m2 , n1 , n2 ) = (0, 0, 0, 0, 0) QNM. On the left is a plot of the imaginary
part and on the right plot of the real part.

We take the initial eigenvalues (b1 , b2 ), insert them into the radial equation (6.8) then use the AIM to find the new
QNM that is closest to using the Mathematica routine FindRoot.
We then take this new value of omega, 0 , insert it into the two angular equations (at the same value of a1 and a2 )
then solve using the AIM, searching closest to the previous b1 and b2 values. Thereby obtaining the new eigenvalues
b01 , b02 . We then repeat this process with the new ( 0 , b01 , b02 ) as the starting point until the results converge and we
have achieved four decimal places of accuracy. When this occurs we increment a1 and a2 again and repeat the process.
In this way, we are able to find the QNMs and eigenvalues along lines passing approximately through the origin (i.e.,
starting from the near Scwharzschild values) in the (a1 , a2 ) parameter space.
As an example, we have plotted various values of  = a2 /a1 (= 0, 0.2, 0.4, 0.6, 0.8, 1) against a1 and used an
interpolating function to interpolate between these values as shown in Fig. 1 (for further details see [45]).

VII.

SUMMARY AND OUTLOOK

In this review we have shown that the AIM can be used to calculate the radial QNMs of a variety of black hole
spacetimes. In particular, we have used it to calculate perturbations of Schwarzschild (in asymptotically flat, de Sitter
and anti-DeSitter), RN and Kerr (for spin 0, 1 and 2 perturbations) black holes in four dimensions. We argued that
the method will be of use in studies of extra dimensional black holes and gave an explicit example of this in the case
of the doubly rotating Myers Perry black hole.
In summary, we have demonstrated how the AIM can also be applied to radial QNMs and not just to spheroidal
eigenvalue problems [11, 29]. Given the fact that the AIM can be used in both the radial and angular wave equations
[29] we expect no problems in obtaining QNMs for Kerr-dS black holes in four and higher dimensions. Note that
this was only recently accomplished via the CFM in Ref. [49] using Heuns equation [50] to reduce the problem to a
3-term recurrence relation. In higher dimensions a similar method was used for simply rotating Kerr-AdS black holes
[46].
It remains to be seen if the AIM can be tailored to handle asymptotic QNMs (see Ref. [51] for an adapted version
of the CFM). However, given the close relationship between the AIM and the exact WKB approach [52] it might be
possible to adapt the AIM to find asymptotic QNMs [5356] numerically or even semi-analytically.

29
TABLE XIII. Selected spin two eigenvalues, 2 Alm , obtained from the AIM for a Kerr black hole with varying values of c = a
and m = 1. The same number of iterations in the AIM, nA and the number of recursions in the CFM, nC for c = 0.1, 0.8 at a
working precision of 15 digit precision, where results are presented to 10 s.f.
l

c = 0.1 (nA =nC =15, )

c = 0.8 (nA =35, nC =70)

c = 10i (nA =80, nC =130 CFM)

c = 10 (nA =75, nC =145)

-0.1391483511

-1.462479552

(12.44128209, 0.8956143162)

-101.8949078

5.929826236

5.247141863

(32.31138608, 1.302608040)

-63.74900642

13.95640426

13.45636668

(51.27922784, 1.946041848)

-30.35607486

23.96944247

23.54163307

(69.25750923, 3.012877154)

-4.557015739

35.97681567

35.58524928

(85.86796852, 4.990079008)

-2.555206382

9.98139515

49.61077286

(99.20081385, 6.801617108)

12.32203552

We hope that the AIM might be of some topical use, for example, in the angular spheroids/QNMs needed in the
phenomenology of Hawking radiation from spinning higher-dimensional black holes, for a recent review see [57]. We
have recently used a combination of all the techniques discussed in this work to evaluate the angular eigenvalues,
Akjm , for real c = a, which are needed for the tensor graviton emission rates on a simply rotating Kerr-de Sitter
black hole background in (n + 4)-dimensions [58] (also see Ref. [59]) and it might also be interesting to find QNMs of
doubly rotating Kerr-(A)dS black holes (for Kerr asymptotically flat see [45]). Finally, attempting to solve the QNMs
for all spins on the Schwarschild-AdS background via the AIM also seems an interesting problem.
As such we hope to have provided the reader with enough technical details, and to have addressed some of the
possible questions to allow him/her to pursue the study of QNMs with the AIM.2

ACKNOWLEDGMENTS

HTC was supported in part by the National Science Council of the Republic of China under the Grant NSC 992112-M-032-003-MY3, and the National Science Centre for Theoretical Sciences. The work of JD was supported by
the Japan Society for the Promotion of Science (JSPS), under fellowship no. P09749. WN would like to thank the
Particle Physics Theory Group, Osaka University for computing resources.

Appendix A: Angular Eigenvalues for Spin-Weighted Spheroidal Harmonics

As mentioned in Sec. V C, aside from radial QNMs the AIM can also be applied to various kinds of spin-weighted
spheroidal harmonics, s Slm (), e.g. see [48]. Therefore in this appendix we briefly compare the AIM with the CFM
for the four-dimensional spin-weighted spheroids.
With the regular boundary conditions, the angular wave function can be written as [6]
1

S = eau (1 + u) 2 |ms| (1 u) 2 |m+s| A (u) .

(A1)

Putting this back into Eq.(5.4) and rewriting the equation in AIM form, we have
A,uu = A (u)A,u + sA (u)A ,

(A2)

Source code and other information for some of the cases presented here can be found on the AIM link at http://www-het.phys.sci.
osaka-u.ac.jp/~naylor/.

30
where
2u
2N ,
1 u2




(m + su)2
1
2 2 2
+ 2asu + 2uN a u + s + s Alm N 2 N,u ,
sA (u) =
1 u2
1 u2
|m s|
|m + s|
N = a +

.
2(1 + u) 2(1 u)

A (u) =

(A3)
(A4)
(A5)

These are the relevant equations for calculating the eigenvalues of the spheroidal harmonics in the four-dimensional
case. It was noticed in Ref. [11] that the AIM converges fastest at the maximum of the potential, when the AIM
is written in WKB form. In four dimensions this occurs at x = 0 and is true for general spin-s as we have verified.
Note that for higher dimensional generalizations it is not easy to explicitly find a maximum [29]. It may be worth
mentioning that for the case where c = 0 an exact analytic solution of the above equations leads to [48]:
s Alm

= l(+1) s(s + 1) .

(A6)

The AIM recovers the c 0 results directly while the CFM appears to take the limit c  1 only (at least within our
numerical procedure of using NSOLVE within Mathematica).
For the purposes of consistency we have calculated (see Table XIII) the 2 Al1 eigenvalues for the lowest l = 2, . . . , 7
modes to 10 significant figures and have also compared this with the results of the CFM. In both the AIM and
CFM larger l modes requires more iterations/recursions to achieve convergence in a given l eigenvalue to the required
precision. Care should be taken when comparing the number of iterations in the AIM with that of the number of
recursions in the CFM, because one iteration of the AIM is not equivalent to one iteration of the CFM. In fact
although we typically need to iterate the improved AIM on average a lesser number of times, the CFM the is typically
faster for smaller values of c. However, for larger values of c both methods can be faster or slower on a case by case
basis.
The results of the first few l eigenvalues for different values of c = a, with m = 1, are presented in Table XIII. As
far as we are aware this is the first time tables of spin-2 spheroids have been presented using the AIM. Further results
are presented in Sec. V C along with the radial QNMs for the spin two perturbations of the Kerr black hole in Table
XII.

Appendix B: Higher Dimensional Scalar Spheroidal Harmonics with two Rotation Parameters

The two Eqs. (6.2) are in fact the two-rotation generalization of the higher dimensional spheroidal harmonics
(HSHs) studied in [47]. In this case, the existence of two rotation parameters leads to a system of two coupled second
order ODEs3 . In general, one would expect that the generalizations of the HSHs to b D1
2 c rotation parameters would
lead to even larger systems of equations. While these systems would also be useful generally in studies of MP black
holes, here we will only focus on the two rotation case.
The angular equations can be written in the Sturm-Liouville form (assuming momentarily that and b2 are real):
w(i )Ri (i ) =

d
di


p(i )


d
Ri (i ) + q(i )Ri (i )
di

For the moment we are considering to be an independent parameter.

(B1)

31

b2 a1 2
6

b1
10.0
9.5
9.0
8.5
8.0
7.5

5
4
=1

=0.01
=0.1
=0.5

=1

=0.5
=0.1

=0.01

1
0.2

0.4

0.6

0.8

1.0

a1
0.0

0.2

0.4

0.6

0.8

1.0

a1

FIG. 2. (Color Online) D = 6, g = 0, (j, m1 , m2 , n1 , n2 ) = (0, 1, 1, 0, 0). A plot of the eigenvalues for various choices of
 a2 /a1 . Note that the dependence on a1 has been scaled into the other quantities.

b2 a1 2
3.6
3.4
3.2
3.0
2.8
2.6
2.4

b1
11
10
9
g a1 =0
8

g a1 =0.5 i

g a1 =0.5
0.2

0.4

0.6

0.8

a1

1.0

g a1 =0.5 i
g a1 =0
g a1 =0.5

0.2

0.4

0.6

0.8

1.0

a1

FIG. 3. (Color Online) D = 6,  a2 /a1 = 1/2, (j, m1 , m2 , n1 , n2 ) = (0, 1, 1, 0, 0). A plot of the eigenvalues for ga1 = 0.5i, 0, 0.5,
corresponding to deSitter, flat, and anti-deSitter spacetimes respectively. Note that the dependence on a1 has been scaled into
the other quantities.
(D5)/2

with the weight function w1 (i ) = 41 i

, the eigenvalue = b1 , and

(D5)/2

p(i ) = i

i ,


1 (D7)/2 (a21 i )2 (a22 i )2 2
a2 a2 j(j + D 7)
q(i ) = i

i + 1 2
b2 ,
4
i
i

(B2)
(B3)

where i and
i are defined in the obvious way under the change of coordinates. Since w() > 0 we can define the
two norms:
Z a21
(D5)/2
N1 (R1 )
1
|R1 |2 d1 ,
(B4)
a22

a22

N2 (R2 )

(D5)/2

|R2 |2 d2 .

(B5)

For further details see Ref. [45].

The regular solutions are found to be:
R1 (1 a22 )
j/2

|m2 |
2

R2 2 (a22 2 )

(a21 1 )
|m2 |
2

2 ;

|m1 |
2

1 ;

1 (a22 , a21 ),

2 (0, a22 ).

(B6)
(B7)

32
Now for a given value of we can determine b1 and b2 simply by performing the improved AIM [10] on both of
the angular equations separately. This will result in two equations in the two unknowns b1 , b2 which we can then
solve using a numerical routine such as the built-in Mathematica functions NSolve or FindRoot. More specifically we
rewrite Eqs. (6.2) using (B6) and (B7) and transform them into AIM form:
d1
d2 1
= 01
+ s01 1 ,
2
d1
d1
d2
d2 2
= 02
+ s02 2 .
2
d2
d2

(B8)
(B9)

The AIM requires that a special point be taken about which the 0i and s0i coefficients are expanded. As was shown
in [29] different choices of this point can worsen or improve the speed of the convergence. In the absence of a clear
selection criterion we simply choose this point conveniently in the middle of the domains:
01 =

a21 + a22
,
2

02 =

a22
.
2

(B10)

Some results are plotted in Figs. 2, 3 above. This method can now be used in the radial QNM equation in Sec.
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