hep-th
Second Order Perturbative Calculation of Quasinormal Modes of
Schwarzschild Black Holes
Hsien-chung Kao
Department of Mathematics, University of Durham, Durham, DH1 3LE, UK. 1
Department of Physics, National Taiwan Normal University, Taipei, Taiwan 116.
hckao@phy.ntnu.edu.tw
Abstract
We analytically calculate to second order the correction to the asymptotic form of quasinormal
frequencies of four dimensional Schwarzschild black holes based on the monodromy analysis pro-
posed by Motl and Neitzke. Our results are in good agreement with those obtained from numerical
calculation.
1on leave from National Taiwan Normal University.
http://arxiv.org/abs/0704.0284v1
1 Introduction
Quasinormal modes (QNMs) were originally observed in considering the scattering or emission of
gravitational waves by Schwarzschild black holes [1]. It was found that a characteristic damped
oscillation, which only depends on the black hole mass, dominated the time evolution in a certain
period of time. Since then QNMs have been investigated extensively both analytically and numerically.
For a general review and classification, see Refs. [2, 3]. From numerical studies, an asymptotic formula
for quasinormal frequencies of Schwarzschild black holes was obtained [4]:
2GMωn ≈ 0.0874247+
i+O[n−1/2]. (1)
The real part in the above formula was later postulated to be 1
ln 3 [5] based on a discrete area
spectrum of quantum black holes proposed in Ref. [6]. This was confirmed later by Motl and Neitzke
[7]. The recent surge of interest in the QNMs derived from its possible application in determining
the Immirzi parameter in loop quantum gravity[8]. The numerical value ln 3 in the real part of the
asymptotic quasinormal frequencies in Schwarzschild black holes was at first taken as a hint that the
relevant gauge group in loop quantum gravity is SO(3) instead of the commonly believed SU(2).
However, as shown in Ref. [7], the value ln 3 is not universal and one should take the argument with
a grain of salt.
Another interesting application of QNMs was pointed out by Horowitz and Hubeny in their study
of a scalar field in the background of a Schwarzschild anti-de Sitter black hole [9]. According to
AdS/CFT correspondence, a large black hole in AdS spacetime corresponds to a thermal state in CFT
[10]. They argued the decay of the scalar field corresponds to the decay of a perturbation of this state.
In the BTZ black hole, a one-to-one correspondence was found between the QNMs in the bulk and
the poles of the retarded correlation function in the dual conformal field theory on the boundary [11].
The idea of dS/CFT correspondence has also been proposed and formulated [15]. Since there is a
cosmological horizon in de Sitter spacetime, QNMs may also be defined in principle. Similar studies of
QNMs have also been carried out in de Sitter spacetime trying to lent support for such correspondence
[16]. However, the situation there is more subtle and it seems QNMs only exist in odd dimensions [3].
Therefore, it is not clear whether such correspondence makes sense in even dimensions, and further
study is necessary.
2 Perturbative calculation of the asymptotic form of quasi-
normal frequencies
In Ref. [12], the author calculated the first order correction to the asymptotic form of quasinormal
frequencies of a Schwarzschild black hole using a WKB analysis. The result was extended to include
the scalar field case using the monodromy analysis developed by Motl and Neitzke [13]. The agreement
with numerical results is excellent. We will begin with a brief review of their method which made
systematic expansion more accessible. In a background spacetime described by a metric gµν , a massless
scalar Φ satisfies the following Klein-Gordon equation:
−g∂νΦ
= 0. (2)
For four dimensional Schwarzschild black holes, the metric is given by
ds2 = −f(r)dt2 + f(r)−1dr2 + r2dΩ2,
with f(r) = (1− r0
) and r0 = 2GM. Let
Φ(r, t,Ω) = r φ(r)Ylm(Ω) e
iωt. (3)
φ(r) now satisfies the following equation:
− f(r) d
+ V (r)φ = ω2φ, (4)
V (r) = (1 − r0
l(l + 1)
By a simple modification in the potential V (r) [2],
V (r) = (1− r0
l(l + 1)
(1− j2)r0
, (5)
the previous equation can also describes linearized perturbation of the metric or an electromagnetic
test fields. Here, j = 0, 1, 2 which is the spin of the relevant field. They can also be classified as
the tensor, vector, and scalar types of perturbation to the background Schwarzschild metric using the
master equations derived by Ishibashi and Kodama [14]. Introducing the tortoise coordinate:
x(r) = r + r0 ln(r/r0 − 1),
one obtain a Schrodinger-like equation
+ V [r(x)]
φ = ω2φ. (6)
Because of our convention in eq (3), QNMs are defined through the following out-going wave
boundary condition:
φ(x) ∼
eiωx as x → −∞ (horizon),
e−iωx as x → ∞ (spatial infinity),
assuming Reω > 0. Define
f(x) = eiωx φ ∼
e2iωx as x → −∞,
1 as x → ∞.
According to Ref. [7], the boundary condition at the horizon translates to the monodromy of f(x)
around it
M(r0) = e4πωr0 . (9)
The same monodromy can also be accounted for by those around r = 0 and r = ∞, and it has
been shown that only the former one is non-trivial. To find the monodromy around r = 0, one need
to introduce the complex coordinate variable
z = ω(x− iπr0) = ω[r + r0 ln(1− r/r0)], (10)
which is vanishing at the black hole singularity r = 0. In the limit |r/r0| ≪ 1, the potential can be
expanded as a series in
z/(ωr0):
V (z) = −ω
2(1− j2)
3l(l+ 1) + 1− j2
2(−ωr0)1/2z3/2
− 3l(l+ 1) + 1− j
2(−ωr0)3/2z1/2
+ . . . . (11)
Note that the third term in the above expression is of order (−ωr0)−3/2 and would not contribute
until we consider third order perturbation. To second order in perturbation theory, the wavefunction
can be expanded as
φ = φ(0) +
φ(1) +
φ(2) +O(ω−3/2). (12)
The zeroth, first and second order equations are given by
dφ(0)
1− j2
φ(0) = 0; (13)
dφ(1)
1− j2
φ(1) =
−ωr0 δV (z)φ(0); (14)
dφ(2)
1− j2
φ(2) =
−ωr0 δV (z)φ(1), (15)
respectively. Here,
δV (z) =
3l(l+ 1) + 1− j2
2(−ωr0)1/2z3/2
. (16)
Define φ
(z) to be the two linearly independent solutions to the zeroth order equation
(z) =
J±j/2(z). (17)
In the asymptotic region z ≫ 1
(z) ≈ cos[z − π(1 ± j)/4]. (18)
It has been shown by Musiri and Siopsis that φ
can be expressed in terms of φ
+ (z) = Cφ
+ (z)
dz1 δV (z1)φ
(z1)φ
+ (z1)− Cφ
dz1 δV (z1)φ
+ (z1)φ
+ (z1); (19)
(z) = Cφ
+ (z)
dz1 δV (z1)φ
(z1)φ
(z1)− Cφ(0)− (z)
dz1 δV (z1)φ
+ (z1)φ
(z1). (20)
where C =
−ωr0/ sin(πj/2) [13]. Similarly, φ(2)± can in turn be expressed in terms of φ
+ (z) = Cφ
+ (z)
dz2 δV (z2)φ
(z2)φ
+ (z2)− Cφ
dz2 δV (z2)φ
+ (z2)φ
+ (z2); (21)
(z) = Cφ
+ (z)
dz2 δV (z2)φ
(z2)φ
(z2)− Cφ(0)− (z)
dz2 δV (z2)φ
+ (z2)φ
(z2). (22)
In the limit, z → ∞,
(z) = c−± φ
+ (z)− c+± φ
(z); (23)
(z) = d−± φ
+ (z)− d+± φ
(z). (24)
Here,
c±± = C
dz1 δV (z1)φ
(z1)φ
(z1); (25)
d±± = C
dz1δV (z2) δV (z1)φ
+ (z2)φ
(z1)− φ(0)− (z2)φ
+ (z1)
(z1). (26)
Notice that φ
defined in eq (17) are in fact linearly dependent to each other when j is an even
integer. As a result, each of these coefficients is divergent by itself in these cases. It is reassuring to
see that all the divergent pieces cancel among themselves so that physically interested quantities do
have a smooth limit when j is an even integer. In zeroth order, the combination
φ(0)(z) = φ
+ (z)− e−iπ(j/2)φ
(z) ∼ e−iz (27)
in the asymptotic region z ≫ 1. This can be extended to second order
φ(z) =
+ (z) +
+ (z) +
+ (z)
− e−iπ(j/2)
1− ξ√
(z) +
(z) +
, (28)
by introducing two parameters ξ and ζ. Naturally, they are determined by the condition that the
coefficient of the eiz term is vanishing when z → ∞:
ξ = ξ+ + ξ−; (29)
ζ = −ξξ− + d++ eiπj/2 − d+− + d−− e−iπj/2 − d−+, (30)
where
ξ+ = c++ e
iπj/2 − c+−, ξ− = c−− e−iπj/2 − c−+. (31)
Substitute the above result back to eq (28), we have
φ(z) = i eiπ(1−j)/4 sin(πj/2) e−iz
1− ξ−√
ξ(ξm + c+−)− d−− e−iπj/2 + d−+
, (32)
where the identity c−+ = c+− has been used to simplify the expression.
When going around the black hole singularity by 3π, φ
and φ
both pick up an extra phase:
( e3iπz) = e3iπ(2±j)/2φ
(−z); (33)
( e3iπz) = e3iπ(3±j)/2φ
(−z). (34)
Consequently,
φ( e3iπz) = e3iπ(1+j)/2
+ (−z)− i
+ (−z)−
+ (−z)
− e−iπ(j/2)
1− ξ√
e3iπ(1−j)/2
(−z)− i 1√
(−z)− 1
. (35)
To second order,
φ( e3iπz) = −i eiπ(1−j)/4 sin(3πj/2) e−iz
(1 + ie3iπj)ξ+ + (1 + i)ξ−√
−ωr0(−1 + ei3πj)
−(1 + i)ξ ξ− + [(1 + ei3πj)(d++ eiπj/2 − d−+) + 2d−− e−iπj/2 − 2d+−]
−ωr0(−1 + ei3πj)
+ . . . , (36)
where the term eiz is not relevant for our calculation and has been neglected. Taking the ratio between
the coefficients of the term e−iz in eqs (36) and (32), we obtain the monodromy to second order:
M(r0) = −[1 + 2 cos(jπ)]
∆2c +∆2d
. (37)
Here,
(1 + ie3iπj)ξ+ + (i+ e
3iπj)ξ−
(−1 + ei3πj)
; (38)
∆2c = −(1− i)ξ+ ξ− − ξc+−; (39)
∆2d =
(1 + ei3πj)(d++ e
iπj/2 + d−− e
−iπj/2)− 2d+− − 2d−+ ei3πj
(−1 + ei3πj)
. (40)
The terms ∆2c and ∆2d depend on coefficients cµν and dµν , respectively. Although our expression for
∆1 here is different from that in Ref. [13] by a phase factor, our final result is identical to their.
Making use of the formula
I1(µ, ν) ≡
dz z−1/2Jµ(z)Jν(z) =
π/2Γ(1+2µ+2ν
Γ(3−2µ−2ν
)Γ(3+2µ−2ν
)Γ(3−2µ+2ν
, (41)
one can obtain explicitly
c++ =
3l2 + 3l+ 1− j2
) Γ(1−2j
) Γ(1+2j
) sin[
π(1−2j)
2 sin( j π
; (42)
c−− =
3l2 + 3l+ 1− j2
) Γ(1−2j
) Γ(1+2j
) sin[
π(1+2j)
2 sin( j π
; (43)
c+− =
3l2 + 3l+ 1− j2
) Γ(1−2j
) Γ(1+2j
) sin[
π(1−2j)
] sin[
π(1+2j)
2 sin( j π
. (44)
Note that
c−− = −c++(j → −j); c+− = −c−+(j → −j). (45)
These relation are also obeyed by dµν ’s, which can be used to reduce our work. With the above results,
we are ready to find ∆1 and ∆2c in eq (40):
∆1 = −
i(3l2 + 3l + 1− j2) Γ2(1
) cos(
) cos(jπ)
2π3/2[1 + 2 cos(jπ)]
; (46)
∆2c = −
(3l2 + 3l + 1− j2)2 Γ4(1
) Γ2(1−2j
) Γ2(1+2j
) cos(jπ)
1152π3
. (47)
The double integral
I2(µ2, ν2;µ1, ν1) ≡
dz1 z
1 Jµ2(z2)Jν2(z2)Jµ1(z1)Jν1 (z1) (48)
can be expressed in terms of the generalized hypergeometric functions, but the general formula is quite
complicated and not particularly illuminating. Therefore, we will just give the final result explicitly
for the coefficients d++ and d+−:
d++ = −
3l2 + 3l+ 1− j2
cot( jπ
) 5G4(
, 1+j
, 1−j
, 1, 2+j
, 2−j
576 sin2( j π
3l2 + 3l+ 1− j2
cot( j π
) cot(j π) Γ(1+2j
) Γ(1+2j
) Γ2(1+j
288 sin( jπ
1 + 2j
1 + j
1 + j
1 + 2j
2 + j
2 + j
5 + 2j
, 1 + j; 1)
3l2 + 3l+ 1− j2
) Γ2(1−2j
) Γ2(1+2j
) sin2[
π(1−2j)
] sin[
π(1+2j)
2π3 sin2( j π
; (49)
d+− = −
3l2 + 3l+ 1− j2
) 5G4(
, 1+j
, 1−j
, 1, 2+j
, 2−j
1152 sin3( j π
3l2 + 3l+ 1− j2
cot2( j π
) cot(j π) Γ(1−2j
) Γ(1−2j
) Γ2(1−j
1− 2j
1− 2j
5− 2j
, 1− j; 1)
3l2 + 3l+ 1− j2
) Γ2(1−2j
) Γ2(1+2j
) sin2[
π(1−2j)
] sin2[
π(1+2j)
2304 π3 sin2( j π
. (50)
Here, we have used the regularized generalized hypergeometric function 5G4(a1, a2, a3, a4, a5; b1, b2, b3, b4; z)
so that the pole structure of each term in these expressions are more explicit. It is related to the usual
generalized hypergeometric function by
5G4(a1, a2, a3, a4, a5; b1, b2, b3, b4; z) =
5F4(a1, a2, a3, a4, a5; b1, b2, b3, b4; z)
Γ(b1)Γ(b2)Γ(b3)Γ(b4)
. (51)
The other two coefficients can be obtained by relations analogous to those in eq (45)
d−− = −d++(j → −j); d−+ = −d+−(j → −j). (52)
On the face of it, each of the dµν ’s has a third order pole coming from terms involving the generalized
hypergeometric function when j is an even integer. On closer look, we see there are some cancelation
among the divergences and in the end all they have are just simple poles in such limit similar to the
cµν ’s. Another possible divergence arises in d−− when j = 1, which will again be canceled when we
calculate the monodromy.
It is now straightforward to obtain ∆2d by making use of the following two identities
−4π7/2 cos2(
) cot(jπ)Γ(
1 + 2j
1 + 2j
) Γ2(
1 + j
1 + 2j
1 + j
1 + j
1 + 2j
2 + j
2 + j
5 + 2j
, 1 + j; 1)
+4π7/2 cos2(
) cot(jπ)Γ(
1 − 2j
1− 2j
) Γ2(
1− 2j
1− 2j
5− 2j
, 1− j; 1)
−Γ4(1
) Γ2(
1 − 2j
) Γ2(
1 + 2j
) sin[
π(1− 2j)
] sin[
π(1 + 2j)
] = 0; (53)
−4π7/2 cos(jπ
1 + 2j
1 + 2j
) Γ2(
1 + j
1 + 2j
1 + j
1 + j
1 + 2j
2 + j
2 + j
5 + 2j
, 1 + j; 1)
−4π7/2 cos(
1− 2j
1 − 2j
) Γ2(
1 − j
1− 2j
1− 2j
5− 2j
, 1− j; 1);
+8π5 Γ(
) 5G4(
1 + j
2 + j
−Γ4(1
) Γ2(
1 − 2j
) Γ2(
1 + 2j
) cos(
)[1− cos(jπ)] = 0. (54)
Eventually, we achieve the following nice result
∆2d =
(3l2 + 3l+ 1− j2)2 Γ4(1
) Γ2(1−2j
) Γ2(1+2j
) cos(jπ)
1152π3[1 + 2 cos(jπ)]
, (55)
where all divergences have been canceled out.
Together with the result from eq (47), the asymptotic form of quasinormal frequencies of a four
dimensional Schwarzschild black hole is found to be
4πωr0 = (2n+ 1)πi+ ln[1 + 2 cos(jπ)]
i(3l2 + 3l + 1− j2) Γ2(1
) Γ(1−2j
) Γ(1+2j
) cos( jπ
) cos(jπ)
2π3/2[1 + 2 cos(jπ)]
(3l2 + 3l+ 1− j2)2 Γ4(1
) Γ2(1−2j
) Γ2(1+2j
) cos2(jπ)
576π3[1 + 2 cos(jπ)]2(−ωr0)
+O[(−ωr0)−3/2]. (56)
The physically interested cases are
≈ (2n+ 1)πi+ ln 3 + 1− i√
(l2 + l − 1)Γ4(1/4)
2π3/2
(l2 + l − 1)2Γ8(1/4)
2592π3
, for j = 2; (57)
≈ (2n+ 1)πi+ ln 3 + 1− i√
(l2 + l + 1/3)Γ4(1/4)
2π3/2
(l2 + l+ 1/3)2Γ8(1/4)
288π3
, for j = 0; (58)
≈ 2nπi+ i2π(l
2 + l)2
, for j = 1. (59)
A few comments are in order. First, all the second order corrections are purely imaginary. In particular,
when j = 2 (gravitational perturbation) the numerical coefficients of the i/n term (after divided by
4π) are 0.739, 3.58, 49.7 for l = 2, 3, 6, respectively. They are in good agreement with the known
numerical studies [4]. As for the real part, our result predicts vanishing correction. For j = 2, this is
again consistent with the numerical results in Ref. [4] for l = 2, 3. For l = 6 the numerical result is
0.263, which seems to be contradictory to ours. However, the numerical value for l = 6 has opposite
sign relative to those of l = 2, 3. This is peculiar, since in all other cases a given type of corrections
are always of the same sign irrespective of the specific value of angular momentum. Therefore, we
believe more study is needed to clarify whether there is really a discrepancy. As for the j = 1 case,
the numerical study in Ref. [17] suggests the leading correction is of the form b
. However, this
does not necessarily mean the two results are inconsistent. In fact, one can only extract the behavior
of the leading correction to the real part from their Fig. 2 and further numerical study is needed to
confirm or refute our prediction.
3 Conclusion
In sum, we have calculated to second order the correction to the asymptotic form of quasinormal
frequencies for Schwarzschild black holes in four dimensions. Most of our results are consistent with
the numerical ones when available. In cases where there seem to be contradiction, we think further
numerical studies are needed to clarify the situation. It would also be helpful if more detailed numerical
studies can be carried out for the j = 0 case so that more thorough comparisons are possible. It would
be interesting to generalize the method to other spacetime backgrounds [18]. Extension to higher order
is also desirable. It might enable us to find a quantitative prediction for the ”algebraically special”
frequencies in Schwarzschild black holes, where the quasinormal frequency is purely imaginary and it
increases with the fourth power of l [19, 4].
Acknowledgment
The author thanks Chong-Sun Chu for helpful discussions. The work is supported in part by the
National Science Council and the National Center for Theoretical Sciences, Taiwan.
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http://arxiv.org/abs/hep-th/0605128
	Introduction
	Perturbative calculation of the asymptotic form of quasinormal frequencies
	Conclusion

  We analytically calculate to second order the correction to the asymptotic
form of quasinormal frequencies of four dimensional Schwarzschild black holes
based on the monodromy analysis proposed by Motl and Neitzke. Our results are
in good agreement with those obtained from numerical calculation.

Introduction
Quasinormal modes (QNMs) were originally observed in considering the scattering or emission of
gravitational waves by Schwarzschild black holes [1]. It was found that a characteristic damped
oscillation, which only depends on the black hole mass, dominated the time evolution in a certain
period of time. Since then QNMs have been investigated extensively both analytically and numerically.
For a general review and classification, see Refs. [2, 3]. From numerical studies, an asymptotic formula
for quasinormal frequencies of Schwarzschild black holes was obtained [4]:
2GMωn ≈ 0.0874247+
i+O[n−1/2]. (1)
The real part in the above formula was later postulated to be 1
ln 3 [5] based on a discrete area
spectrum of quantum black holes proposed in Ref. [6]. This was confirmed later by Motl and Neitzke
[7]. The recent surge of interest in the QNMs derived from its possible application in determining
the Immirzi parameter in loop quantum gravity[8]. The numerical value ln 3 in the real part of the
asymptotic quasinormal frequencies in Schwarzschild black holes was at first taken as a hint that the
relevant gauge group in loop quantum gravity is SO(3) instead of the commonly believed SU(2).
However, as shown in Ref. [7], the value ln 3 is not universal and one should take the argument with
a grain of salt.
Another interesting application of QNMs was pointed out by Horowitz and Hubeny in their study
of a scalar field in the background of a Schwarzschild anti-de Sitter black hole [9]. According to
AdS/CFT correspondence, a large black hole in AdS spacetime corresponds to a thermal state in CFT
[10]. They argued the decay of the scalar field corresponds to the decay of a perturbation of this state.
In the BTZ black hole, a one-to-one correspondence was found between the QNMs in the bulk and
the poles of the retarded correlation function in the dual conformal field theory on the boundary [11].
The idea of dS/CFT correspondence has also been proposed and formulated [15]. Since there is a
cosmological horizon in de Sitter spacetime, QNMs may also be defined in principle. Similar studies of
QNMs have also been carried out in de Sitter spacetime trying to lent support for such correspondence
[16]. However, the situation there is more subtle and it seems QNMs only exist in odd dimensions [3].
Therefore, it is not clear whether such correspondence makes sense in even dimensions, and further
study is necessary.
2 Perturbative calculation of the asymptotic form of quasi-
normal frequencies
In Ref. [12], the author calculated the first order correction to the asymptotic form of quasinormal
frequencies of a Schwarzschild black hole using a WKB analysis. The result was extended to include
the scalar field case using the monodromy analysis developed by Motl and Neitzke [13]. The agreement
with numerical results is excellent. We will begin with a brief review of their method which made
systematic expansion more accessible. In a background spacetime described by a metric gµν , a massless
scalar Φ satisfies the following Klein-Gordon equation:
−g∂νΦ
= 0. (2)
For four dimensional Schwarzschild black holes, the metric is given by
ds2 = −f(r)dt2 + f(r)−1dr2 + r2dΩ2,
with f(r) = (1− r0
) and r0 = 2GM. Let
Φ(r, t,Ω) = r φ(r)Ylm(Ω) e
iωt. (3)
φ(r) now satisfies the following equation:
− f(r) d
+ V (r)φ = ω2φ, (4)
V (r) = (1 − r0
l(l + 1)
By a simple modification in the potential V (r) [2],
V (r) = (1− r0
l(l + 1)
(1− j2)r0
, (5)
the previous equation can also describes linearized perturbation of the metric or an electromagnetic
test fields. Here, j = 0, 1, 2 which is the spin of the relevant field. They can also be classified as
the tensor, vector, and scalar types of perturbation to the background Schwarzschild metric using the
master equations derived by Ishibashi and Kodama [14]. Introducing the tortoise coordinate:
x(r) = r + r0 ln(r/r0 − 1),
one obtain a Schrodinger-like equation
+ V [r(x)]
φ = ω2φ. (6)
Because of our convention in eq (3), QNMs are defined through the following out-going wave
boundary condition:
φ(x) ∼
eiωx as x → −∞ (horizon),
e−iωx as x → ∞ (spatial infinity),
assuming Reω > 0. Define
f(x) = eiωx φ ∼
e2iωx as x → −∞,
1 as x → ∞.
According to Ref. [7], the boundary condition at the horizon translates to the monodromy of f(x)
around it
M(r0) = e4πωr0 . (9)
The same monodromy can also be accounted for by those around r = 0 and r = ∞, and it has
been shown that only the former one is non-trivial. To find the monodromy around r = 0, one need
to introduce the complex coordinate variable
z = ω(x− iπr0) = ω[r + r0 ln(1− r/r0)], (10)
which is vanishing at the black hole singularity r = 0. In the limit |r/r0| ≪ 1, the potential can be
expanded as a series in
z/(ωr0):
V (z) = −ω
2(1− j2)
3l(l+ 1) + 1− j2
2(−ωr0)1/2z3/2
− 3l(l+ 1) + 1− j
2(−ωr0)3/2z1/2
+ . . . . (11)
Note that the third term in the above expression is of order (−ωr0)−3/2 and would not contribute
until we consider third order perturbation. To second order in perturbation theory, the wavefunction
can be expanded as
φ = φ(0) +
φ(1) +
φ(2) +O(ω−3/2). (12)
The zeroth, first and second order equations are given by
dφ(0)
1− j2
φ(0) = 0; (13)
dφ(1)
1− j2
φ(1) =
−ωr0 δV (z)φ(0); (14)
dφ(2)
1− j2
φ(2) =
−ωr0 δV (z)φ(1), (15)
respectively. Here,
δV (z) =
3l(l+ 1) + 1− j2
2(−ωr0)1/2z3/2
. (16)
Define φ
(z) to be the two linearly independent solutions to the zeroth order equation
(z) =
J±j/2(z). (17)
In the asymptotic region z ≫ 1
(z) ≈ cos[z − π(1 ± j)/4]. (18)
It has been shown by Musiri and Siopsis that φ
can be expressed in terms of φ
+ (z) = Cφ
+ (z)
dz1 δV (z1)φ
(z1)φ
+ (z1)− Cφ
dz1 δV (z1)φ
+ (z1)φ
+ (z1); (19)
(z) = Cφ
+ (z)
dz1 δV (z1)φ
(z1)φ
(z1)− Cφ(0)− (z)
dz1 δV (z1)φ
+ (z1)φ
(z1). (20)
where C =
−ωr0/ sin(πj/2) [13]. Similarly, φ(2)± can in turn be expressed in terms of φ
+ (z) = Cφ
+ (z)
dz2 δV (z2)φ
(z2)φ
+ (z2)− Cφ
dz2 δV (z2)φ
+ (z2)φ
+ (z2); (21)
(z) = Cφ
+ (z)
dz2 δV (z2)φ
(z2)φ
(z2)− Cφ(0)− (z)
dz2 δV (z2)φ
+ (z2)φ
(z2). (22)
In the limit, z → ∞,
(z) = c−± φ
+ (z)− c+± φ
(z); (23)
(z) = d−± φ
+ (z)− d+± φ
(z). (24)
Here,
c±± = C
dz1 δV (z1)φ
(z1)φ
(z1); (25)
d±± = C
dz1δV (z2) δV (z1)φ
+ (z2)φ
(z1)− φ(0)− (z2)φ
+ (z1)
(z1). (26)
Notice that φ
defined in eq (17) are in fact linearly dependent to each other when j is an even
integer. As a result, each of these coefficients is divergent by itself in these cases. It is reassuring to
see that all the divergent pieces cancel among themselves so that physically interested quantities do
have a smooth limit when j is an even integer. In zeroth order, the combination
φ(0)(z) = φ
+ (z)− e−iπ(j/2)φ
(z) ∼ e−iz (27)
in the asymptotic region z ≫ 1. This can be extended to second order
φ(z) =
+ (z) +
+ (z) +
+ (z)
− e−iπ(j/2)
1− ξ√
(z) +
(z) +
, (28)
by introducing two parameters ξ and ζ. Naturally, they are determined by the condition that the
coefficient of the eiz term is vanishing when z → ∞:
ξ = ξ+ + ξ−; (29)
ζ = −ξξ− + d++ eiπj/2 − d+− + d−− e−iπj/2 − d−+, (30)
where
ξ+ = c++ e
iπj/2 − c+−, ξ− = c−− e−iπj/2 − c−+. (31)
Substitute the above result back to eq (28), we have
φ(z) = i eiπ(1−j)/4 sin(πj/2) e−iz
1− ξ−√
ξ(ξm + c+−)− d−− e−iπj/2 + d−+
, (32)
where the identity c−+ = c+− has been used to simplify the expression.
When going around the black hole singularity by 3π, φ
and φ
both pick up an extra phase:
( e3iπz) = e3iπ(2±j)/2φ
(−z); (33)
( e3iπz) = e3iπ(3±j)/2φ
(−z). (34)
Consequently,
φ( e3iπz) = e3iπ(1+j)/2
+ (−z)− i
+ (−z)−
+ (−z)
− e−iπ(j/2)
1− ξ√
e3iπ(1−j)/2
(−z)− i 1√
(−z)− 1
. (35)
To second order,
φ( e3iπz) = −i eiπ(1−j)/4 sin(3πj/2) e−iz
(1 + ie3iπj)ξ+ + (1 + i)ξ−√
−ωr0(−1 + ei3πj)
−(1 + i)ξ ξ− + [(1 + ei3πj)(d++ eiπj/2 − d−+) + 2d−− e−iπj/2 − 2d+−]
−ωr0(−1 + ei3πj)
+ . . . , (36)
where the term eiz is not relevant for our calculation and has been neglected. Taking the ratio between
the coefficients of the term e−iz in eqs (36) and (32), we obtain the monodromy to second order:
M(r0) = −[1 + 2 cos(jπ)]
∆2c +∆2d
. (37)
Here,
(1 + ie3iπj)ξ+ + (i+ e
3iπj)ξ−
(−1 + ei3πj)
; (38)
∆2c = −(1− i)ξ+ ξ− − ξc+−; (39)
∆2d =
(1 + ei3πj)(d++ e
iπj/2 + d−− e
−iπj/2)− 2d+− − 2d−+ ei3πj
(−1 + ei3πj)
. (40)
The terms ∆2c and ∆2d depend on coefficients cµν and dµν , respectively. Although our expression for
∆1 here is different from that in Ref. [13] by a phase factor, our final result is identical to their.
Making use of the formula
I1(µ, ν) ≡
dz z−1/2Jµ(z)Jν(z) =
π/2Γ(1+2µ+2ν
Γ(3−2µ−2ν
)Γ(3+2µ−2ν
)Γ(3−2µ+2ν
, (41)
one can obtain explicitly
c++ =
3l2 + 3l+ 1− j2
) Γ(1−2j
) Γ(1+2j
) sin[
π(1−2j)
2 sin( j π
; (42)
c−− =
3l2 + 3l+ 1− j2
) Γ(1−2j
) Γ(1+2j
) sin[
π(1+2j)
2 sin( j π
; (43)
c+− =
3l2 + 3l+ 1− j2
) Γ(1−2j
) Γ(1+2j
) sin[
π(1−2j)
] sin[
π(1+2j)
2 sin( j π
. (44)
Note that
c−− = −c++(j → −j); c+− = −c−+(j → −j). (45)
These relation are also obeyed by dµν ’s, which can be used to reduce our work. With the above results,
we are ready to find ∆1 and ∆2c in eq (40):
∆1 = −
i(3l2 + 3l + 1− j2) Γ2(1
) cos(
) cos(jπ)
2π3/2[1 + 2 cos(jπ)]
; (46)
∆2c = −
(3l2 + 3l + 1− j2)2 Γ4(1
) Γ2(1−2j
) Γ2(1+2j
) cos(jπ)
1152π3
. (47)
The double integral
I2(µ2, ν2;µ1, ν1) ≡
dz1 z
1 Jµ2(z2)Jν2(z2)Jµ1(z1)Jν1 (z1) (48)
can be expressed in terms of the generalized hypergeometric functions, but the general formula is quite
complicated and not particularly illuminating. Therefore, we will just give the final result explicitly
for the coefficients d++ and d+−:
d++ = −
3l2 + 3l+ 1− j2
cot( jπ
) 5G4(
, 1+j
, 1−j
, 1, 2+j
, 2−j
576 sin2( j π
3l2 + 3l+ 1− j2
cot( j π
) cot(j π) Γ(1+2j
) Γ(1+2j
) Γ2(1+j
288 sin( jπ
1 + 2j
1 + j
1 + j
1 + 2j
2 + j
2 + j
5 + 2j
, 1 + j; 1)
3l2 + 3l+ 1− j2
) Γ2(1−2j
) Γ2(1+2j
) sin2[
π(1−2j)
] sin[
π(1+2j)
2π3 sin2( j π
; (49)
d+− = −
3l2 + 3l+ 1− j2
) 5G4(
, 1+j
, 1−j
, 1, 2+j
, 2−j
1152 sin3( j π
3l2 + 3l+ 1− j2
cot2( j π
) cot(j π) Γ(1−2j
) Γ(1−2j
) Γ2(1−j
1− 2j
1− 2j
5− 2j
, 1− j; 1)
3l2 + 3l+ 1− j2
) Γ2(1−2j
) Γ2(1+2j
) sin2[
π(1−2j)
] sin2[
π(1+2j)
2304 π3 sin2( j π
. (50)
Here, we have used the regularized generalized hypergeometric function 5G4(a1, a2, a3, a4, a5; b1, b2, b3, b4; z)
so that the pole structure of each term in these expressions are more explicit. It is related to the usual
generalized hypergeometric function by
5G4(a1, a2, a3, a4, a5; b1, b2, b3, b4; z) =
5F4(a1, a2, a3, a4, a5; b1, b2, b3, b4; z)
Γ(b1)Γ(b2)Γ(b3)Γ(b4)
. (51)
The other two coefficients can be obtained by relations analogous to those in eq (45)
d−− = −d++(j → −j); d−+ = −d+−(j → −j). (52)
On the face of it, each of the dµν ’s has a third order pole coming from terms involving the generalized
hypergeometric function when j is an even integer. On closer look, we see there are some cancelation
among the divergences and in the end all they have are just simple poles in such limit similar to the
cµν ’s. Another possible divergence arises in d−− when j = 1, which will again be canceled when we
calculate the monodromy.
It is now straightforward to obtain ∆2d by making use of the following two identities
−4π7/2 cos2(
) cot(jπ)Γ(
1 + 2j
1 + 2j
) Γ2(
1 + j
1 + 2j
1 + j
1 + j
1 + 2j
2 + j
2 + j
5 + 2j
, 1 + j; 1)
+4π7/2 cos2(
) cot(jπ)Γ(
1 − 2j
1− 2j
) Γ2(
1− 2j
1− 2j
5− 2j
, 1− j; 1)
−Γ4(1
) Γ2(
1 − 2j
) Γ2(
1 + 2j
) sin[
π(1− 2j)
] sin[
π(1 + 2j)
] = 0; (53)
−4π7/2 cos(jπ
1 + 2j
1 + 2j
) Γ2(
1 + j
1 + 2j
1 + j
1 + j
1 + 2j
2 + j
2 + j
5 + 2j
, 1 + j; 1)
−4π7/2 cos(
1− 2j
1 − 2j
) Γ2(
1 − j
1− 2j
1− 2j
5− 2j
, 1− j; 1);
+8π5 Γ(
) 5G4(
1 + j
2 + j
−Γ4(1
) Γ2(
1 − 2j
) Γ2(
1 + 2j
) cos(
)[1− cos(jπ)] = 0. (54)
Eventually, we achieve the following nice result
∆2d =
(3l2 + 3l+ 1− j2)2 Γ4(1
) Γ2(1−2j
) Γ2(1+2j
) cos(jπ)
1152π3[1 + 2 cos(jπ)]
, (55)
where all divergences have been canceled out.
Together with the result from eq (47), the asymptotic form of quasinormal frequencies of a four
dimensional Schwarzschild black hole is found to be
4πωr0 = (2n+ 1)πi+ ln[1 + 2 cos(jπ)]
i(3l2 + 3l + 1− j2) Γ2(1
) Γ(1−2j
) Γ(1+2j
) cos( jπ
) cos(jπ)
2π3/2[1 + 2 cos(jπ)]
(3l2 + 3l+ 1− j2)2 Γ4(1
) Γ2(1−2j
) Γ2(1+2j
) cos2(jπ)
576π3[1 + 2 cos(jπ)]2(−ωr0)
+O[(−ωr0)−3/2]. (56)
The physically interested cases are
≈ (2n+ 1)πi+ ln 3 + 1− i√
(l2 + l − 1)Γ4(1/4)
2π3/2
(l2 + l − 1)2Γ8(1/4)
2592π3
, for j = 2; (57)
≈ (2n+ 1)πi+ ln 3 + 1− i√
(l2 + l + 1/3)Γ4(1/4)
2π3/2
(l2 + l+ 1/3)2Γ8(1/4)
288π3
, for j = 0; (58)
≈ 2nπi+ i2π(l
2 + l)2
, for j = 1. (59)
A few comments are in order. First, all the second order corrections are purely imaginary. In particular,
when j = 2 (gravitational perturbation) the numerical coefficients of the i/n term (after divided by
4π) are 0.739, 3.58, 49.7 for l = 2, 3, 6, respectively. They are in good agreement with the known
numerical studies [4]. As for the real part, our result predicts vanishing correction. For j = 2, this is
again consistent with the numerical results in Ref. [4] for l = 2, 3. For l = 6 the numerical result is
0.263, which seems to be contradictory to ours. However, the numerical value for l = 6 has opposite
sign relative to those of l = 2, 3. This is peculiar, since in all other cases a given type of corrections
are always of the same sign irrespective of the specific value of angular momentum. Therefore, we
believe more study is needed to clarify whether there is really a discrepancy. As for the j = 1 case,
the numerical study in Ref. [17] suggests the leading correction is of the form b
. However, this
does not necessarily mean the two results are inconsistent. In fact, one can only extract the behavior
of the leading correction to the real part from their Fig. 2 and further numerical study is needed to
confirm or refute our prediction.
3 Conclusion
In sum, we have calculated to second order the correction to the asymptotic form of quasinormal
frequencies for Schwarzschild black holes in four dimensions. Most of our results are consistent with
the numerical ones when available. In cases where there seem to be contradiction, we think further
numerical studies are needed to clarify the situation. It would also be helpful if more detailed numerical
studies can be carried out for the j = 0 case so that more thorough comparisons are possible. It would
be interesting to generalize the method to other spacetime backgrounds [18]. Extension to higher order
is also desirable. It might enable us to find a quantitative prediction for the ”algebraically special”
frequencies in Schwarzschild black holes, where the quasinormal frequency is purely imaginary and it
increases with the fourth power of l [19, 4].
Acknowledgment
The author thanks Chong-Sun Chu for helpful discussions. The work is supported in part by the
National Science Council and the National Center for Theoretical Sciences, Taiwan.
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	Introduction
	Perturbative calculation of the asymptotic form of quasinormal frequencies
	Conclusion
