Int J Theor Phys (2012) 51:2702–2713DOI 10.1007/s10773-012-1172-0

Quasinormal Modes in Double-Charge de Sitter BlackHole

Ying Zhou

Received: 4 February 2012 / Accepted: 5 April 2012 / Published online: 29 April 2012© Springer Science+Business Media, LLC 2012

Abstract The black hole, as a hot topic to be regarded as a normally research to become astrong evidence for its existence, made more and more people get involved in its research. Tocalculating the quasinormal modes for massless scalar field and Maxwell’s field in double-charge de Sitter black hole by using WKB approximation method, there is a fact that thespeed of weakening electromagnetic perturbation will be reduced. The quasinormal modesin black hole mainly depends on angular quantum number l when its real part is in lower-frequency circumstances. At the same time, imaginary part mainly depends on the overtonenumber n. When the black hole carries the same electronic quantity, the more the electroniccharges have, the smaller the real part and imaginary part of quasinormal modes will be.

Keywords Quasinormal modes · WKB approximation · Scalar field · Maxwell’s field

1 Introduction

While the perturbation of black hole is studied, a rule has been found that the evolutionalprocedure of the black hole in space time can be divided into three steps [1]. The first stepis initial outbust period,which can’t be regarded as a study target because it is highly relatedwith its initial pulse. The second step is the period of quasinormal oscillation, which isthe Medium-term attenuation behavior of perturbation field. Its frequency is plural form.Meanwhile, the frequency and resistance of oscillation are not depended on the originalperturbation but on the parameters such as quality, charge, angular momentum, etc. Thethird step is the period of the inverse power-law decaying tale. In this period, the situation ofperturbation decay does not refer to exponential decay decreasing any more. The researchon quasinormal modes contributes to a better understanding of the black hole, because eachblack hole has its own particular spectrum of oscillation frequency. When the gravitationalwave is detected before long, quasinormal modes will become a strong evidence of the blackhole existence. Therefore the research on quasinormal modes has a great directive function

Y. Zhou (�)Institute of Theoretical Physics, China West Normal University, Nanchong 637002, Chinae-mail: 55732961@qq.com

Int J Theor Phys (2012) 51:2702–2713 2703

to the astronomic observation. The research showed that the quasinormal modes of Adsblack hole space time can be explained well in conformal field theory, which is consideredas ADS/CFT correspondence probe. That’s the reason why a large amount of work hasbeen done to calculate quasinormal modes of Ads black holes [2–9]. Some research evendiscusses the correspondence of DS/CFT by trying to study on the relation of quasinormalmodes of s space time and conformal field theory that plays an essential role in research onde Sitter black hole quasinormal modes.

It is obvious that the main step to calculate the black hole quasinormal modes is tosolve partial differential equation. Numerical calculating is involved because of the greatdifficulties of analytic solution. Nevertheless, the limitation of numerical calculating sta-bility and computer function makes quasinormal modes calculating a really hard work. In1975, Chandrasekhar and Detweiler [10] successfully calculated the quasinormal modes inSchwarzschild black hole space time. After that, a lot of calculating methods have been im-proved and calculating precision developed. So plenty of methods have been developed tocalculate quasinormal modes in black hole from then on. Here are some dominating ways:(1) Poshl-Teller Approach Method [11]; (2) Continuous Fraction Method [12]; (3) FiniteDifference Method [13]; (4) WKB Approximation [14–17]; (5) Single Value Method [18]etc. WAB approximation was applied in this dissertation to calculate the quasinormal modesfor massless scalar field and electromagnetic field in double-charge de Sitter black hole.

2 Charged Rotating Black Hole in Gauged Supergravity

Establishing Supergravity Theory needs the existence of graviton and supersymmetry.Graviton was firstly mentioned by Pauli and Fields, also by whom the relativistic field equa-tions with massless Spin-2 was found, which suggests the existence of quantum gravity.The graviton has not been found by far but it was thought that graviton produces gravityin substance exchanging as photons exchanging produces electromagnetic force in electro-magnetic interaction. Supersymmetry is an abstract symmetry between bosons and fermions.Some scholars think every particle has a supersymmetry particle partner at present. However,the interaction between particle partner and substance is too feeble to observe. Supersym-metric geometry is developed as a basis of Gravitational Geometry Theory, which results ina Supergravity Theory including Einstein General Relativity.

Z.-W. Chong, M. Cvetic, H. Lu, C.N. Pope did a research on Kerr–Newman–AdS blackhole and concluded in a solution with pairwise-equal charge [19–21];

ds2 = − ρ2 − 2mr

r1r2 + a2 cos2 θ

(dt − a sin2 θ(r2 − 2mr − r1r2)dφ

ρ2 − 2mr

)2

+ (r1r2 + a2 cos2 θ

)(dr2

�+ dθ2 + � sin2 θdφ

ρ2 − 2mr

2)(1)

After that, a black hole solution with double charges in gauged supergravity was conjecturedand the correctness of which was proved. Charged rotating black hole in gauged supergravityprovides a new background of the further study on Ads/CFT. Here is the metric form:

ds2 = −�r

w

(dt2 − a sin2 θdφ

) + w

(dr2

�r+ dθ2

�θ

)

+ �θ sin2 θ

w

[adt − (

r21 r2

2 + a2)dφ

]2(2)

2704 Int J Theor Phys (2012) 51:2702–2713

where:

�r = r2 + a2 − 2mr − g2r1r2

(r1r2 + a2

), �θ = 1 − g2a2 cos2 θ

w = r1r2 + a2 cos2 θ r1 = r + 2ms2i si = sinh δi i = 1,2

(3)

According to the equation, this black hole is a static spherically symmetric black hole whilea = 0. The double-charge de Sitter black hole calculating in this thesis is the charged rotatingblack hole in gauged supergravity when a = 0. Here is the metric below:

ds2 = −�r

wdt2 + w

�rdr2 + wdθ2 + sin2 θ

wr2

1 r22 dφ2 (4)

where:

�r = r2 − 2mr − g2r21 r2

2 w = r1r2 r1 = r + 2ms21

si = sinh δi i = 1,2(5)

The value of the metric determinant is

g = −w2 sin2 θ (6)

The non-zero versus contravariant metrics:

g00 = − w

�r; g11 = �r

w; g22 = 1

w; g33 = 1

w sin2 θ(7)

3 Massless Scalar Field Perturbation Equation and WKB Numerical Resultsof Double-Charged de Sitter Black Holes

Perturbations of the massless scalar field which governed by the Klein–Gordon equation is:

1√−g

∂

∂xμ

(√−ggμν ∂Φ

∂xν

)= 0 (8)

Substitute Eq. (7) into Eq. (8)[− w2

�r

∂2

∂t2+ ∂

∂r�r

∂

∂r+ 1

sin θ

∂

∂θsin θ

∂

∂θ+ 1

sin2 θ

∂2

∂ϕ2

]Φ = 0 (9)

The wave function Φ can be separated as:

Φ(t, r, θ,ϕ) = e−iωtw− 12 R(r, t)Y (θ,ϕ) (10)

The radial equation can be obtained as below:{∂

∂r�r

∂

∂r+

[w2

�rω2 − l(l + 1)

]}w− 1

2 R = 0 (11)

where ω, l are the particle energy and angular quantum number, and Y is the SphericalHarmonics. Because radial spread due to radiation, so we are only interested in the radialequation.

Define tortoise coordinate: dr∗ = r1r2�r

dr , and multiply w− 12 both sides of the equation,

then we can get the new radial equation:

∂2Φ

∂x2+ (

ω2 − V)Φ = 0 (12)

Int J Theor Phys (2012) 51:2702–2713 2705

Fig. 1 Variation of the v formassless scalar field with l

Fig. 2 Variation of the v formassless scalar field with g

With the effective potential:

V = �r

(r1r2)2

[l(l + 1) + (r1r2)

12∂[ r1+r2

2 (r1r2)− 3

2 ]∂r

](13)

From Eq. (13), we can find that the angel quantum number l, g, m and S2i affect the value of

potential function, but has nothing to do with ω. According to the expressions of the effectivepotential, the connection of effective potential along with l, g, and S2

i can be analyzed as inFigs. 1–4.

From Fig. 1, it is easy to find that if the angel quantum number increases, the maximumvalue of the effective potential will increase and r corresponded with maximum in the sametime. When the angular quantum number tends to be infinite, the maximum of effectivepotential and the corresponding r trend to be a fixed value.

From Fig. 2, it is easy to find the way how the effective potential is varied by g. Anincreasing g will decrease the maximum of the effective potential and will increase the r

which is corresponded with maximum of effective potential.And Fig. 3 suggests the way of effective potential variation following electronic charges.

Here the two electronic charges are regarded the same. When the electronic charges increase,the maximum of effective potential will decrease and the corresponding r will increase inthe same time.

Figure 4 suggests the way of effective potential variation following a definite charge.Here one of the charges is considered as 0. When the electronic charges increase, the max-imum of effective potential will decrease and the corresponding r will increase in the sametime. The maximum of potential function increases compared with Fig. 3.

2706 Int J Theor Phys (2012) 51:2702–2713

Fig. 3 Variation of the v formassless scalar field with S2

i

Fig. 4 Variation of the v formassless scalar field with S2

1

WKB approximation is a semi-classical approximation method to solution Schrödingerequation proposed by Gregor Wentzel [22], Hendrik Anthony Kramers [23], and Leon Bril-louin [24, 25] in 1926. In quantum mechanics, WKB approximation is used to deal with theparticle barrier penetration problem [26–28]. The effective potential in perturbation field ofblack holes is equivalent to a potential barrier in quantum mechanical. The form of pertur-bation equation is similar to Schrödinger equation. WKB approximation method, therefore,is used to estimate quasinormal modes of black holes. The perturbation equation of blackholes will finally form a second-order differential equation like (12).

In Eq. (12), x is a “tortoise coordinate” r∗, which ranges from the horizon to infinity.The boundary conditions for the quasinormal modes are usually defined as following: atr∗ = +∞ the waves are purely outgoing, while in the black hole horizon r∗ = −∞ thewaves are purely ingoing. This is determined by black hole’s special physical system. Theresult shows that only the discrete complex frequency satisfies such a boundary condition.

B.F. Schutz and C.M. Will [14] firstly developed the WKB approximation method tocalculate the quasinormal frequencies for black holes but there were still a large numberof errors. Afterward, S. Iyer and C.M. Will [15] develop this way to a third order, whichincrease the precision in large scale. And R.A. Konoplya [17] was extended to sixth order,which further decreased the errors. The form of the sixth order result is:

iω2 − V0√−2V ′′

0

− Λ2 − Λ3 − Λ4 − Λ5 − Λ6 = n + 1

2(14)

According to the formula, V0 is the maximum value of the potential function V and V ′′0 is the

second derivative of V0. The correction terms Λ2,Λ3 can be found in [26]. The correction

Int J Theor Phys (2012) 51:2702–2713 2707

Table 1 The quasinormalfrequencies of massless scalarfield: g = 0.01, s2

1 = s22 = 1

l n Third order Sixth order

1 0 0.119034 −0.0315253i 0.119644 −0.0315310i

2 0 0.197629 −0.0313298i 0.197776 −0.0313371i

3 0 0.276230 −0.0312764i 0.276286 −0.0312787i

3 1 0.272703 −0.0943025i 0.272882 −0.0942742i

3 2 0.266237 −0.1584610i 0.266370 −0.1585250i

4 0 0.354884 −0.0312537i 0.354911 −0.0312546i

4 1 0.352146 −0.0940436i 0.352233 −0.0940330i

4 2 0.346964 −0.1575690i 0.347022 −0.1575970i

4 3 0.339769 −0.2220280i 0.339551 −0.2223960i

Table 2 The quasinormalfrequencies of massless scalarfield: l = 2; n = 0; s2

1 = s22 = 0.3

G Third order Sixth order

0 0.342606 −0.0615363i 0.342847 −0.0615348i

0.01 0.341614 −0.0613926i 0.341853 −0.0613911i

0.02 0.338624 −0.0609573i 0.338859 −0.0609555i

0.04 0.326451 −0.0591456i 0.326669 −0.0591432i

0.06 0.305323 −0.0558591i 0.305511 −0.0558568i

0.08 0.273620 −0.0506268i 0.273763 −0.0506254i

0.10 0.227789 −0.0425669i 0.227875 −0.0425640i

0.12 0.157481 −0.0295478i 0.157519 −0.0295417i

Table 3 The quasinormalfrequencies of massless scalarfield: g = 0.01, l = 2, n = 0,s21 = s2

2

S2i

Third order Sixth order

0.1 0.425192 −0.0816964i 0.425524 −0.0816756i

0.2 0.379211 −0.0702861i 0.379486 −0.0702770i

0.3 0.341614 −0.0613926i 0.341853 −0.0613911i

0.4 0.310373 −0.0543167i 0.310588 −0.0543198i

0.5 0.284050 −0.0485855i 0.284035 −0.0486276i

0.6 0.261603 −0.0438701i 0.261788 −0.0438772i

0.7 0.242259 −0.0399363i 0.242433 −0.0399440i

0.8 0.225432 −0.0366140i 0.225596 −0.0366218i

1.0 0.197629 −0.0313298i 0.197776 −0.0313371i

terms Λ4,Λ5,Λ6 can be found in [28]. At low frequency n < l, WKB approximation whichcan gain more accurate solutions. This formula shows that when the ω and the effectivepotential is in dependent, we can calculate the quasinormal modes frequency. We can fix thevalue of l, g, m and S2

i , find the value of r at which V attain maximum, then substitute r

into formula.Now we will calculated some quasinormal mode frequencies for massless scalar pertur-

bations in double-charge de Sitter black hole. The results are presented in Tables 1, 2, 3and 4.

2708 Int J Theor Phys (2012) 51:2702–2713

Table 4 The quasinormalfrequencies of massless scalarfield: l = 2; n = 0; g = 0.01

S21 S2

2 Third order Sixth order

0.1 0 0.452654 −0.0887767i 0.453026 −0.0887466i

0.2 0 0.426744 −0.0820827i 0.427069 −0.082058i

0.6 0 0.349632 −0.0631374i 0.349831 −0.063121i

0.8 0 0.321594 −0.0566168i 0.321754 −0.0566016i

1.0 0 0.298126 −0.0513135i 0.298255 −0.0512989i

1.2 0 0.278146 −0.0469116i 0.278251 −0.0468975i

2.0 0 0.220709 −0.0348603i 0.220756 −0.0348473i

4 Electromagnetic Field Perturbation Equation and WKB Numerical Resultsof Double-Charge de Sitter Black Hole

Maxwell’s equations is

1√−g

∂

∂xμ

(√−gFμν) = Jμ (15)

where Jμ is four-dimensional current density. Electromagnetic field which in this paperdiscussed is the electromagnetic field, so Jμ = 0 ◦ Fμν is the contravariant electromagnetictensor while Fμν is versus covariant electromagnetic tensor in the background space-time ofblack holes Fμν = Aν,μ − Aμ,ν . The following electromagnetic potential is selected:

At = Ar = Aθ = 0, Aφ = ψ(r) sin(ωt) sin θdPl(cos θ)

dθ(16)

Substitute Eq. (16) into expression of Fμν , the nonzero contravariant electromagnetic tensorscan be obtained:

F03 = ∂A3

∂x0− ∂x0

∂A3= ∂ψ(r, t)

∂tsin θ

dPl(cos θ)

dθ

F13 = ∂A3

∂x1− ∂x1

∂A3= ∂ψ(r, t)

∂rsin θ

dPl(cos θ)

dθ(17)

F23 = ∂A3

∂x2− ∂x2

∂A3= ψ(r, t) cos θ

dPl(cos θ)

dθ+ ψ(r, t) sin θ

d2Pl(cos θ)

dθ

As the electromagnetic tensor is antisymmetric tensor, Fμν = −Fνμ so:

F30 = −∂ψ(r, t)

∂tsin θ

dPl(cos θ)

dθ

F31 = −∂ψ(r, t)

∂rsin θ

dPl(cos θ)

dθ(18)

F32 = −ψ(r, t) cos θdPl(cos θ)

dθ− ψ(r, t) sin θ

d2Pl(cos θ)

dθ

Because the covariant versus contravariant electromagnetic tensors have the following rela-tionship: Fμν = gμμgννFμν .

The nonzero versus contravariant electromagnetic tensors are:

Int J Theor Phys (2012) 51:2702–2713 2709

F 30 = 1

�r sin θ

∂ψ(r, t)

∂t

dPl(cos θ)

dθ

F 31 = − �r

w2 sin θ

∂ψ(r, t)

∂r

dPl(cos θ)

dθ(19)

F 32 = − 1

w2 sin2 θψ(r, t)

[cos θ

dPl(cos θ)

dθ+ sin θ

d2Pl(cos θ)

dθ

]

Substitute Eq. (19) into Maxwell’s equations:

1√−g

[∂

∂t

(√−gF 30) + ∂

∂t

(√−gF 31) + ∂

∂t

(√−gF 32)] = 0 (20)

Substitute F 30,F 31,F 32 into (20):

1

�r sin θ

dPl(cos θ)

dθ

∂2ψ(r, t)

∂t2+

{− 1

w sin θ

dPl(cos θ)

dθ

∂

∂r

[�r

w

∂ψ(r, t)

∂t

]}

+{

ψ(r, t)

w2 sin θ

(csc2 θ

dPl(cos θ)

dθ− ctg θ

d2Pl(cos θ)

dθ2− d3Pl(cos θ)

dθ3

]}= 0 (21)

This is defined according to Legendre equation:

csc2 θdPl(cos θ)

dθ− ctg θ

d2Pl(cos θ)

dθ2− d3Pl(cos θ)

dθ3= l(l + 1)

dPl(cos θ)

dθ(22)

multiply w− 12 on both sides of the equation, which can be rewritten as:

−∂2ψ(r, t)

∂t2+ �r

w

∂

∂r

[�r

w

∂ψ(r, t)

∂t

]− �r

wl(l + 1)ψ(r, t) = 0 (23)

Define tortoise coordinate: ddr∗ = �r

wddr

and ψ(r, t) = ψ(r) sin(ωt)

Expression Eq. (23) can be rewritten as:

∂2R

∂r2∗+

[ω2 − �r

w2l(l + 1)

]R = 0 (24)

The effective potential function of electromagnetic perturbation in the double-charge deSitter black hole can be written as follows:

V = �r

w2l(l + 1) (25)

Equation (25) suggests that the angel quantum number l, g, m and S2i affect the value of

potential function but has nothing to do with ω. According to formula (25), the connectionof effective potential along with l, g, and S2

i can be analyzed as in Figs. 5–8.The variation condition of comparative electromagnetic and effective potential of scalar

field presents a fact that when the parameters are the same, the effective potential maximumof electromagnetic perturbation is smaller than the effective potential maximum of scalarperturbation along with the increasing of the related r .

The definite values are chosen to deal with WKB approximation calculation in terms ofdouble-charged de Sitter black holes’ electromagnetic perturbation. The numerical resultsof quasinormal modes calculated are listed in Tables 5, 6, 7 and 8.

2710 Int J Theor Phys (2012) 51:2702–2713

Fig. 5 Variation of the v forelectromagnetic field with l

Fig. 6 Variation of the v forelectromagnetic field with g

Fig. 7 Variation of the v forelectromagnetic field with S2

i

5 Discussion

The real parts of black hole’s space-time quasinormal modes determine the oscillation fre-quency while the imaginary parts determine the rate at which each mode is damped as aresult of the emission of radiation. By using the six order WKB approximation to calculatethe quasinormal modes of double-charge de Sitter black hole, the following conclusions canbe get.

When the values of g, n, S2i keep invariant, how the influence of quasinormal modes

varying by a changing l has been investigated. Figures 1 and 5 show that with the increasingof l, the quasinormal modes of scalar field perturbation with two charges increase and the

Int J Theor Phys (2012) 51:2702–2713 2711

Fig. 8 Variation of the v forelectromagnetic field with S2

1

Table 5 The quasinormalfrequencies of electromagneticfield: g = 0.01, s2

1 = s22 = 1

l n Third order Sixth order

1 0 0.104460 −0.0300952i 0.105253 −0.0302315i

2 0 0.189132 −0.0308264i 0.189295 −0.0308387i

3 0 0.270193 −0.0310199i 0.270252 −0.0310228i

3 1 0.266584 −0.0935524i 0.266775 −0.0935242i

3 2 0.259982 −0.157258i 0.260137 −0.157328i

4 0 0.350197 −0.0310985i 0.350225 −0.0310995i

4 1 0.347422 −0.0935848i 0.347513 −0.0935742i

4 2 0.342174 −0.156823i 0.342237 −0.156852i

4 3 0.334894 −0.221013i 0.334683 −0.221392i

Table 6 The quasinormalfrequencies of electromagneticfield: l = 2; n = 0; S2

i= 0.3

g Third order Sixth order

0 0.325955 −0.0606064i 0.326221 −0.0606069i

0.01 0.325090 −0.0604449i 0.325354 −0.0604454i

0.02 0.322480 −0.0599581i 0.322740 −0.0599583i

0.04 0.311811 −0.0579697i 0.312052 −0.0579689i

0.06 0.293119 −0.0544929i 0.293329 −0.0544909i

0.08 0.264631 −0.0492051i 0.264798 −0.0492023i

0.10 0.222482 −0.041392i 0.222593 −0.0413895i

0.12 0.155710 −0.0290057i 0.155756 −0.0290036i

imaginary parts decrease while the real part and imaginary part of quasinormal modes ofelectromagnetic perturbation both increase. Among which, the real part varies faster whilethe imaginary part varies slower. When the values of g, l, S2

i keep invariant, how the influ-ence of quasinormal modes varying by a changing modulus n has been investigated again.With the increasing of modulus n, both the real parts of perturbation of two-charged deSitter black holes’ scalar field and electromagnetic perturbation decrease while both imag-inary parts increase. The smaller the modulus n is, the smaller the imaginary part will bewith slower attenuation and longer attenuation cycle. At the same time, if only n has beenchanged, the real parts increase very slowly while the imaginary parts increase more quickly.In summary, the real parts of the black hole’s quasinormal modes in lower frequency mainly

2712 Int J Theor Phys (2012) 51:2702–2713

Table 7 The quasinormalfrequencies of electromagneticfield: g = 0.01; l = 2; n = 0

S2i

Third order Sixth order

0.1 0.403246 −0.0803217i 0.403608 −0.0802953i

0.2 0.360317 −0.0691642i 0.360619 −0.0691545i

0.3 0.32509 −0.0604449i 0.325354 −0.0604454i

0.4 0.295731 −0.0534929i 0.29597 −0.0534995i

0.5 0.270934 −0.0478526i 0.270911 −0.0479055i

0.6 0.249744 −0.0432059i 0.249948 −0.0432179i

0.7 0.23145 −0.0393257i 0.231642 −0.0393385i

0.8 0.215512 −0.0360461i 0.215694 −0.0360591i

1.0 0.189132 −0.0308264i 0.189295 −0.0308387i

Table 8 The quasinormalfrequencies of electromagneticfield: l = 2; n = 0; g = 0.01

S21 S2

2 Third order Sixth order

0.1 0 0.428809 −0.0872326i 0.429212 −0.0871927i

0.2 0 0.404702 −0.0807103i 0.405057 −0.0806794i

0.6 0 0.332681 −0.0622194i 0.332905 −0.0622026i

0.8 0 0.306392 −0.0558432i 0.306574 −0.0558283i

1.0 0 0.284346 −0.050652i 0.284495 −0.0506381i

1.2 0 0.265546 −0.0463394i 0.265669 −0.046326i

2.0 0 0.211343 −0.0345111i 0.211402 −0.0344986i

depend on angular quantum number l while imaginary part depend on modulus quantumnumber n.

Figures 2 and 6 reveal that when the values of n, l, S2i keep invariant, with an increasing g,

the real and imaginary parts of quasinormal modes both decrease.Figures 3 and 7 reveal that when the values of g, n, l keep invariant, with the same and

increasing charges of black hole, the real and imaginary parts of quasinormal modes bothdecrease.

The comparison between Figs. 3 with 4 and Figs. 7 with 8 reveal that two electroniccharges are the same as x in Figs. 3 and 7 and are different as 2x and 0 in Figs. 4 and 8.The quasinormal modes in two situations are not equal. That is to say, the carried chargesare equal but the number of charges and quasinormal modes are not equal. The more theelectronic charges are, the smaller the real and imaginary parts of quasinormal modes willbe.

A large amount of work is to be done in this area such as following.

1. This thesis deals to calculate the situation in supergravity ranges when electronic-chargedrotating black hole a = 0. This situation can be also called at static state. Calculating inthe situation that when a is not 0 can be done in further research.

2. This thesis only calculate in the condition of massless scalar field and Maxwell field ofquotient perturbation. The gravitation fields and Dirac fields need further understood.The quasinormal mode, as an effective instrument, requires much further study in thisarea to be helpful to know more and further of the essence of black holes.

Acknowledgements Great thanks are due to S.Q. Wu for invaluable discussions. This work is supportedby the National Natural Science Foundation of China under grant number 11178018.

Int J Theor Phys (2012) 51:2702–2713 2713

References

1. Vishveshawara, C.V.: Nature 227, 936 (1970)2. Chan, J.S.F., Mann, R.B.: Phys. Rev. D 59, 064025 (1999)3. Horowitz, G.T., Hubeny, V.E.: Phys. Rev. D 62, 024027 (2000)4. Cardoso, V., Lemos, J.P.S.: Phys. Rev. D 63, 124015 (2001)5. Wang, B., Lin, C.Y., Molina, C.: Phys. Rev. D 70, 064025 (2004)6. Zhu, J.M., Wang, B., Abdalla, E.: Phys. Rev. D 63, 124004 (2001)7. Winsanlcy, E.: Phys. Rev. D 64, 104010 (2001)8. Konoplya, R.A.: Phys. Rev. D 66, 044009 (2002)9. Berti, E., Kokkotas, K.D.: Phys. Rev. D 67, 064020 (2003)

10. Chandrasekhar, S., Detweiler, S.: Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 343, 289 (1975)11. Ferrari, V., Mashhoon, B.: Phys. Rev. D 30, 295 (1984)12. Leaver, E.: Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 402, 285 (1985)13. Gundlach, C., Price, H.R., Pullin, J.: Phys. Rev. D 49, 883 (1994)14. Schutz, B.F., Will, C.M.: Astron. J. 291, L33 (1985)15. Iyer, S., Will, C.M.: Phys. Rev. D, Part. Fields 35, 3621 (1987)16. Kokkotas, K.D., Schutz, B.F.: Phys. Rev. D 37, 3378 (1988)17. Konoplya, R.A.: Phys. Rev. D 68, 024018 (2003)18. Motl, L., Neitzke, A.: Theor. Math. Phys. 7, 307 (2003)19. Chong, Z.-W., Cvetic, M., Lu, H., Pope, C.N.: Nucl. Phys. B 717, 246 (2005)20. Duff, M.J., Liu, J.T.: Physica B, Condens. Matter 554, 237 (1999). hep-th/990114921. Wu, S.-Q.: Phys. Lett. B 705, 383 (2011)22. Wenzel, G.: Z. Phys. 38, 518 (1926)23. Kramers, H.M.: Z. Phys. 39, 828 (1926)24. Brillouin, L.: Compes Rendus 183, 24 (1926)25. Brillouin, L.: J. Phys. Radium 7, 353 (1926)26. Yang, S.Z., Lin, K.: Phys. Mech. & Astron. 40, 507 (2010)27. Lin, K., Yang, S.Z.: Acta Phys. Sin. 59, 2223 (2010)28. Jiang, Q.-Q., Wu, S.-Q.: Phys. Lett. B 635, 151 (2006)