Open access peer-reviewed chapter

Semi-Analytic Techniques for Solving Quasi-Normal Modes

By Chun-Hung Chen, Hing-Tong Cho and Alan S. Cornell

Submitted: October 26th 2016Reviewed: February 24th 2017Published: June 7th 2017

DOI: 10.5772/68114

Downloaded: 593

Abstract

In this chapter, we discuss an approach to obtaining black hole quasi-normal modes known as the asymptotic iteration method, which was initially developed in mathematics as a new way to solve for eigenvalues in differential equations. Furthermore, we demonstrate that the asymptotic iteration method allows one to also solve for the radial quasi-normal modes on a variety of black hole spacetimes for a variety of perturbing fields. A specific example for Dirac fields in a general dimensional Schwarzschild black hole spacetime is given, as well as for spin-3/2 field quasi-normal modes.

Keywords

  • extra-dimensions
  • quasi-normal modes
  • quantum fields in curved space
  • supergravity
  • blackholes

1. Introduction

Quasi-normal modes (QNMs) are one of the most important theoretical results in modern cosmology, especially for studying the perturbations from various fields on black hole spacetimes. In this theory, the behaviour of a particle around a black hole is dominated by the radial equation, and the evolution of QNMs behaves like damped harmonic oscillators with specific frequencies. The frequencies are constructed by complex modes, where the real part is the actual frequency and the imaginary part represents the damping rate due to the gravitational emission. In lay terms, the QNMs are the characteristic sounds of the black hole.

With the recent ground breaking progress into the detection of gravitational wave data, where it is believed that the last part of the gravitational wave emission, called the ring down phase, is dominated by the QNMs, it is exciting that perturbation theory in curved spacetimes can now be possibly tested in a real experimental system, and that a large number of scientists from all over the world are involved in the data analysis. Related issues of the QNMs within the range of research into cosmology include studying the stability of black holes and probing the dimensionality of spacetime. It is, however, the observable gravitational wave data from the collisions of binary black hole systems, which indicates how the background spacetime will finally become a Kerr black hole spacetime through gravitational wave emission. Perturbation theory on a Kerr black hole spacetime still includes some difficulties in the higher dimensional cases, which will be a challenge for the theoretical community for some time to come.

Methods that are used to obtain QNMs can be both semi-analytic and numerical methods and were introduced by Cho et al. [1], the most famous of these is the WKB approximation methods [2]. Note that, the WKB approximation has been extended to sixth order [3] and is powerful in many cases, but like all methods have several limitations. A new method has been developed in recent years called the asymptotic iteration method (AIM), which is more efficient in some cases. This method was used to solve eigenvalue problems for the second-order homogeneous linear differential equations [4, 5] and also successfully used in calculating QNMs [6]. Reviewing this AIM and providing the tools “in detail” for studying QNMs in the higher dimensional spacetimes are the key focus of this chapter.

As such, this chapter is organised as follows: In the next section, we shall review the recent progress on perturbation theory in curved spacetimes. More precisely, we shall present a comparison of the spin-3/2 field in general dimensional Schwarzschild spacetimes with other spin fields, including the spherical harmonics and the radial equations. In Section 3, we shall review the AIM and present an exercise detailing how the QNMs of Dirac fields are obtained in general dimensional Schwarzschild spacetimes. Furthermore, we can also compare to spin-3/2 QNMs results. We shall conclude with a brief summary.

2. Perturbation theory in a general dimensional Schwarzschild spacetime

2.1. Eigenvalue problem on spheres

For perturbation theory in curved spacetimes, separability can always simplify the equations of motion and plays an important role. For the maximally symmetric spacetime cases, the eigenmodes on spheres allow us to separate the angular part for various spin fields and simplify the equations of motion from the general form into a “radial-time” presentation. For the case of bosonic fields, an earlier study by Rubin and Ordóñez presented a systematic study [7, 8] as well as in a later work by Higuchi [9]. For the case of fermionic fields, Camporesi and Higuchi presented the eigenmodes for spinor fields on arbitrary dimensional spheres [10], and in a recent work by the authors [11], the spinor-vector eigenmodes on arbitrary dimensional spheres were derived using a similar approach to Camporesi and Higuchi’s methods. In this section, we review the structure of these eigenmodes, especially for the case of spinor-vector fields, which shall be presented with the characteristics of both spinor and vector fields.

The metric of the N-sphere is given by

dΩN2=dθN2+sin2θNdΩN12,E1

where, in this metric, we restrict the sphere to radius r = 1. When we consider the eigenvalue problem in this spacetime, the bosonic field and fermionic field shall be studied with different operators. In the case of bosonic fields, the operator for the eigenvalue equation will be the Laplacian operator μμ, whereas, in the case of fermionic fields, it will be the Dirac operator γμμ, where γμ is the Dirac gamma matrices. In Table 1, we present the structure of the eigenmodes with various spin fields on the sphere and also the conditions on the specific mode, such as the transverse, traceless and symmetric conditions.

FieldsEigenfunctionEigenvalue
ScalarT(l)l(l+N1), l=0,1,2, .....
Spinorψ(j)±i(j+N12),j=1/2,3/2,5/2, .....
VectorLongitudinal eigenvector
Lμ(l)=μT(l)l(l+N1)+1,l=1,2,3, .....
Transverse eigenvector
Tμ(l),μTμ(l)=0l(l+N1)+(N1),l=1,2, .....
Spin.-Non-transverse-traceless eigenmode
Vectorψμ(j)=μψ(j)+a(±)γμψ(j)±ij2+(N1)j+14(N5)(N1)
Transverse and traceless eigenmode
ψμ(j)=(ψθN,ψθi),μψμ=γμψμ=0±i(j+N12),j=1/2,3/2…,N3
Sym.-gμνT(l)l(l+N1),l=0,1,2, .....
TensorLongitudinal-longitudinal (traceless) modes
LLμν(l)=2μνT(l)(2N)gμνααT(l)l(l+N1)+2N,l=2,3,4, .....
Longitudinal-transverse (traceless) modes
LTμν(l)=μTν(l)+νTμ(l)l(l+N1)+(N+2),l=2,3,4, .....
Transverse-traceless modes
Tμν(l),μTμν(l)=gμνTμν(l)=T[μν](l)=0l(l+N1)+2,l=2,3,4, .....

Table 1.

Structure of the eigenvalue problem for various fields on N-sphere.

Looking first at the longitudinal and non-transverse modes for bosonic fields, the longitudinal and non-transverse eigenfunctions for higher spins are the linear combination of the eigenfunctions for the lower spin one. For example, for the vector fields, the longitudinal eigenvector is the covariant derivative of a scalar eigenfunction. Furthermore, for the symmetric tensor fields, there are three types of non-transverse eigenfunctions. The first one is the metric element multiplied by a scalar eigenfunction, the second one is the longitudinal-longitudinal eigenfunction, which is the linear combination for longitudinal eigenvectors, and the last one is the longitudinal-transverse eigenfunction, which is the linear combination of transverse eigenvectors. Analogous to the non-transverse-traceless modes for fermionic fields, the non-transverse-traceless eigenspinor-vector is the linear combination of the eigenspinor. We note that there are two non-transverse-traceless eigenspinor-vectors, due to the a(+) and a(−) being different factors where

a(±)=i2(j+N12)±ij2+(N1)j+14(N5)(N1).E2

This indicates a special signature for spinor-vector harmonics that do not have the transverse-traceless eigenmodes for S2.

Next, we look at the transverse-traceless modes. For the bosonic fields, an unique way to construct this type of eigenfunction was suggested by Higuchi [9]. Analogous to the fermionic case, for the transverse-traceless eigenspinor-vector, ψμ=(ψθN,ψθi), ψθNbehaves like a spinor on N – 1 spheres, and ψθibehaves like a spinor-vector on N – 1 spheres. If we let ψθibe the linear combination of the non-transverse-traceless eigenspinor-vector on N – 1 spheres, ψθNhas to be non-zero to satisfy the transverse and traceless conditions. If we let ψθibe the linear combination of the transverse-traceless eigenspinor-vector on N – 1-spheres, ψθNhas to be zero, because the ψθialready satisfies the transverse and traceless condition.

On the other hand, we can take a look at the eigenvalue. For the bosonic fields, all of the eigenvalues contain a similar first term, which is the eigenvalue of the scalar fields; however, the starting value of the angular momentum quantum number l has to be considered case by case, as well as the second term for higher spin cases. For the fermionic cases, the eigenvalue of the non-transverse-traceless eigenspinor-vector has a very different value from the spinor eigenvalue, though the eigenvalue of the transverse and traceless eigenspinor-vector is exactly the same as the eigenspinor for the spin-1/2 field.

As a remark on this section, the eigenfunctions and eigenvalues on N-spheres are independent for bosonic and fermionic fields, even though they have a very similar style of structure, which indicates that the spherical harmonics for a bosonic field cannot be constructed by the spherical harmonics of a fermionic field, and vice versa. In this section, we presented a review of the eigenvalue problem for scalar, spinor, vector, spinor-vector, and symmetric tensor fields on spheres, where further details can be found in the papers referred to in this section.

2.2. Effective potentials

In perturbation theory with various fields in the Schwarzschild black hole spacetime, a radial equation (Schrödinger-like equation) will be derived from the equations of motion, which shall be the master equation of this study. In a general way, the studies of perturbation theory in maximally symmetric spacetimes are well established and include the (A)dS and Reissner-Nordström spacetimes, but not for the spin-3/2 fields yet. With our recent progress in the study of spin-3/2 fields, we may now do a comparison of various massless fields in a general dimensional Schwarzschild black hole spacetime in this section.

The metric of a general dimensional Schwarzschild spacetime is given by

dS2=f(r)dt2+f1(r)dr2+r2dΩD22,E3

where f(r)=12M/rD3, D is the dimensional factor and M is the mass of the black hole. The master equation can be obtained from the equations of motion by a change of coordinates to give the Schrödinger-like equation

d2dr*2Ψs+(ω2Vs)Ψs=0,E4

where the subscript s represents the “spin” and r* represents the “tortoise” coordinate, which can be defined as follows ddr*=f(r)ddr. The mathematical meaning of this coordinate is that a mapping of the location of the event horizon of the Schwarzschild black hole from r0 (where f(r0) = 0) is taken to minus infinity.

We shall first look at the four-dimensional cases, where, for the bosonic fields, the radial equation can be represented with the potential [12]

Vs=f[l(l+1)r2+(1s2)2Mr3],E5

where l is an integer, s = 0 represents the effective potential for scalar fields, s = 1 for the electromagnetic fields, and s = 2 represents the “vector-type” perturbation for gravitational fields (which is the Regge-Wheeler equation). The “scalar-type” perturbation for the gravitational field in the four-dimensional case is the Zerilli equation, with an effective potential

VZ,s=2=2fr3[9M3+3λ2Mr2+λ2(1+λ)r3+9M2λr(3M+λr)2],E6

where λ=(l1)(l+2)/2.

For the fermionic fields in the four-dimensional Schwarzschild case, the effective potential can be shown as follows [13, 14]:

Vs=±fdWsdr+Ws2,E7

where

Ws=12=f(j+12)r,j=12,32,….E8
Ws=32=f(j12)(j+12)(j+32)r[(j+12)2f],j=32,52,….E9

Note that the “±” represents two isospectral potentials, which were known as the supersymmetric partner potentials.

For the higher dimensional cases, there are more types of effective potentials, which can be presented in the four-dimensional case, which indicates that some special cases will not exist in the four-dimensional cases but will exist in the higher dimensional one. In the following sections, we shall discuss the higher dimensional effective potentials as presented in Table 2.

FieldsVs
Scalar [12]
Vs=0=f[l(l+D3)r2+D24(D4r2f+2fr)]
l=0,1,2, .....
Dirac [13]
Vs=1/2=±fdWs=1/2dr+Ws=1/22
where
Ws=1/2=f(j+D32r),j=1/2,3/2,5/2, .....
Electromagnetic [15]
Scalar-type perturbation
VS,s=1=f[l(l+D3)+(D2)(D4)4r2(3D8)(D4)M2rD1].
l=1,2,3, .....
Vector-type perturbation
VV,s=1=f[l(l+D3)+(D2)(D4)4r2D(D4)M2rD1].
l=1,2,3, .....
Rarita-Schwinger [11]
Related to the non-TT eigenmodes
VNTT,s=3/2=±fdWNTT,s=3/2dr+WNTT,s=3/22
where
WNTT,s=3/2=f(j+D32)r[(2D2)2(j+D32)21D4D2(2MrD3)(2D2)2(j+D32)2f],j=1/2,3/2,5/2….
Related to the TT eigenmodes
VTT,s=3/2=±fdWTT,s=3/2dr+WTT,s=3/22
where
WTT,s=3/2=f(j+D32r),j=1/2,3/2,5/2, ....
Gravitational [16, 17]
Scalar-type perturbation
VS,s=2=fH16r2[m+12N(N+1)(1f)]2,N=D2
H=N4(N+1)2(1f)3+N(N+1)[4(2N23N+4)m+N(N2)(N4)(N+1)](1f)212N[(N4)m+N(N+1)(N2)]m(1f)+16m3+4N(N+2)m2.m=l(l+N1)N,l=2,3,4.
Vector and tensor type perturbation
VV/T,s=2=fr2[l(l+D3)+(D2)(D4)4μV/T2(D2)2MrD3].
μV=3,lV=2,3,4…for vector-type perturbation.
μT=1,lT=1,2,3, ....for tensor-type perturbation.

Table 2.

The effective potential for various fields in the higher dimensional cases (D ≥ 5).

Starting with bosonic fields, we have one effective potential for scalar fields in the higher dimensional Schwarzschild spacetime, and it is necessary to satisfy Eq. (5) when D = 4 and s = 0. For the case of electromagnetic fields, there are two types of effective potentials, which are the “scalar-type perturbation potential” and the “vector-type perturbation potential”. It is accidental that both of these effective potentials satisfy Eq. (5) when D = 4 and s = 1, but it has been shown they have different behaviours in the higher dimensional cases. It is believed that these two types of effective potentials are strongly linked to two types of eigenvectors on spheres, which are longitudinal and transverse ones. For the case of gravitational fields, the “scalar-type perturbation potential” becomes the potential for the Zerilli equation, and the “vector type perturbation potential” becomes the potential for the Regge-Wheeler equation when D = 4 and s = 2. The “tensor-type perturbation potential” will be present when D ≥ 5 but absent in the four-dimensional case.

For the fermionic fields, the “±” still represents two isospectral supersymmetric partner potentials in the higher dimensional cases. We have one set of effective potentials for the Dirac field, which is strongly related to the eigenspinor on the sphere and reduces to Eq. (8) when D = 4. For the case of the Rarita-Schwinger field, the potentials related to the non-transverse-traceless eigenmodes are the leading equation both for the four-dimensional case and the higher dimensional one, which are strongly linked to the “non-transverse-traceless” eigenspinor-vector on spheres. Another effective potential for the Rarita-Schwinger fields is the one related to the transverse and traceless eigenmodes; however, this type of eigenspinor-vector was absent on the 2-sphere, which indicates that the potentials related to the transverse and traceless eigenmodes exist for the cases when D ≥ 5. We must note that, in this case, the effective potentials are exactly the same as the Dirac case.

Lastly, note that, most of the effective potentials in Table 2 are simple barrier like potentials. Nevertheless, some cases in the higher dimensions, or for the lowest energy state with j = 1/2, for the potentials related to the non-transverse-traceless eigenmodes exhibit special behaviours but not a simple barrier potential, which strongly suggests a link with the instabilities of the black hole [18] and warrants further study.

To summarise, we have in this section provided a brief review of the effective potentials, which play an important role in the perturbation theory of various spin fields in the general dimensional Schwarzschild black hole spacetimes.

3. QNM frequencies by AIM

3.1. AIM methods

The AIM is a well-established approach in solving the eigenvalue problem for the second-order differential equations, for example, Schrödinger-like equations. As mentioned in the previous section, the radial equations of the perturbation theory with various spin fields in general dimensional Schwarzschild spacetimes were presented as Schrödinger-like equations. The QNMs, which are the signature modes in the black hole perturbation theory, can be obtained naturally by using the AIM. In this subsection, we shall present a brief review of the AIMs, and in the next subsection, we shall present an example calculation, showing the methods used to obtain the quasi normal frequencies. We shall start with the second-order differential equation for the function χ(x)

χ=λ0(x)χ+s0(x)χ,E10

where χ=dχ/dx. The symmetric structure of the right-hand side of Eq. (10) leads to the method, where we differentiate on both sides of the equation we find that

χ=λ0χ+(λ0+s0)χ+s0χ,=(λ0+s0+λ02)χ+(s0+s0λ0)χλ1χ+s1χ.E11

Taking the second derivative of Eq. (10) we have

χ(4)=λ2χ+s2χ,E12

where

λ2=λ1+s1+λ0λ1;s2=s1+s0λ1.E13

Differentiating iteratively to the (n+1)thand the (n+2)thorder, we have

χ(n+1)=λn1χ+sn1χ;χ(n+2)=λnχ+snχ,E14

where

λn=λn1+sn1+λ0λn1;sn=sn1+s0λn1.E15

In the AIM, we suppose that for sufficiently large n, which represents the iterating number, the coefficients λn and sn will have the relation

snλn=sn+1λn+1=β(x),E16

where the general solution of Eq. (10) is

χ(x)=exp(xβ(x)dx)[C2+C1xexp(x(λ0(x)+2β(x))dx)dx].E17

C1 and C2 are constants determined by the normalisation, and the QNMs (or the energy eigenstates) can be obtained by the termination condition

snλn+1sn+1λn=0.E18

This is the basic idea for the AIM, where to appreciate the effectiveness of this methods, we refer the reader to Ciftci et al. [4, 5], which presented some studies for the constant coefficient, harmonic oscillator, and the energy eigenvalue problem for several well-known potentials. For the study of perturbation theory in curved spacetimes by the AIM, Cho et al. [1] present a review of the QNMs for the bosonic fields in the four-dimensional maximally symmetric and the Kerr black hole spacetimes. In the next subsection, as an example, we shall present how to obtain the QNMs for Dirac fields in the higher dimensional Schwarzschild black hole spacetimes by the AIM and compare these with other numerical semi-analytic results.

3.2. Example: How to obtain the QNMs for Dirac fields in general dimensional Schwarzschild black hole spacetimes by the AIM

The QNMs for Dirac particles in the higher dimensional Schwarzschild black hole spacetimes had been done in the earlier work by some of the authors [13] using the third-order WKB approximation but not the AIM. With recent progress in spin-3/2 fields [11], we find that these results greatly overlap with some of the spin-3/2 particles, which are represented by the relations in the radial equations of the transverse and traceless eigenmodes. In this section, as an example, we are going to show how to reproduce the results by the AIM.

In Table 2, the effective potential of the radial equation, Eq. (4), for the Dirac particle is as follows:

Vs=1/2=fdWs=1/2dr+Ws=1/22,Ws=1/2=f(j+D32r),j=12,32, .....E19

As we are going to reproduce the results in Ref. [13], a similar choose of f(r) will be

f(r)=1(rHr)D3,rHD3=8πMΓ((D1)/2)π(D1)/2(D2),M1,E20

where rH represents the location of the event horizon and M represents the mass of the black hole. By making a coordinate transformation

ξ2(r)=1rHr,E21

the radial equation becomes

[fdξdrddξ(fdξdrddξ)+ω2Vs=1/2]Ψs=1/2=0.E22

Simplifying Eq. (22), we have

[d2dξ2+(ff+ξξ)ddξ+ω2Vs=1/2f2ξ2]Ψs=1/2=0,E23

where

f=ddξf(ξ),ξ=ddrξ(r)|r=rH1ξ2,ξ=ddξξ.E24

Next, by setting the boundary behaviour of Ψs=1/2=α(ξ)χ(ξ)and together with Eq. (23), we have

d2dξχ=(ff+ξξ+2αα)ddξχ[ω2Vs=1/2fξ2+αα+(ff+ξξ)αα]χ.E25

This is the second-order differential equation, which is the same as Eq. (10) with

λ0=(ff+ξξ+2αα),s0=[ω2Vs=1/2fξ2+αα+(ff+ξξ)αα].E26

Note that the last parameter we have to define is the asymptotic behaviour function α(ξ), and we approach this by starting with the asymptotic behaviour of Ψs=1/2, which can be represented as an outgoing plane wave

Ψs=1/2eiωr*forr,Ψs=1/2eiωr*forr,E27

with r* being the tortoise coordinate, which has the relation with r

ddr*=f(r)ddr.E28

Solving Eq. (28) in the four-dimensional case, we have

r*=r+rHln(rHr1),E29

together with Eqs. (21) and (27), we have

Ψs=1/2eiωrH1ξ2(1ξ2)iωrHξ2iωrHforξ1,Ψs=1/2eiωrH1ξ2(1ξ2)iωrHξ2iωrHforξ0.E30

Collecting the leading terms, we can define α(ξ) in the four-dimensional case as follows:

α(ξ)=eiωrH1ξ2(1ξ2)iωrHξ2iωrH.E31

If we now consider the higher dimensional cases, Eq. (29) will become more complicated with

r*=r+a=1D3rHεD3ln(rrHε1),E32

where ε=eia2πD3. Eq. (27) in the higher dimensions can be presented in the general form

Ψs=1/2eiωrH1ξ2a=1D3(ε(1ξ2))iωrHεD3(1ε(1ξ2))iωrHεD3forξ1,Ψs=1/2eiωrH1ξ2a=1D3(ε(1ξ2))iωrHεD3(1ε(1ξ2))iωrHεD3forξ0.E33

Next, we shall consider how to define the asymptotic behaviour function α(ξ) in the higher dimensional cases, where in our experience, finding the dominant terms in Eq. (33) is helpful to the AIM calculation. When D = 5, Eq. (33) becomes

Ψs=1/2eiωrH1ξ2(ξ22)iωrH2(ξ2)iωrH2forξ1,Ψs=1/2eiωrH1ξ2(ξ22)iωrH2(ξ2)iωrH2forξ0.E34

By considering the dominant term for the boundary behaviour, a suitable choice of the asymptotic behaviour function is as follows:

α(D=5)(ξ)=eiωrH1ξ2(ξ2)iωrH2.E35

For D = 6, Eq. (33) becomes

Ψs=1/2eiωrH1ξ2(ei2π3(1ξ2))iωrHei2π33(1ei2π3(1ξ2))iωrHei2π33(ei4π3(1ξ2))iωrHei4π33(1ei4π3(1ξ2))iωrHei4π33(1ξ2)iωrH3(1ξ2)iωrH3forξ1,Ψs=1/2eiωrH1ξ2(ei2π3(1ξ2))iωrHei2π33(1ei2π3(1ξ2))iωrHei2π33(ei4π3(1ξ2))iωrHei4π33(1ei4π3(1ξ2))iωrHei4π33(1ξ2)iωrH3(1ξ2)iωrH3forξ0.E36

The dominant term for the boundary behaviour can be chosen as follows:

α(D=6)(ξ)=eiωrH1ξ2(ξ2)iωrH3(1ξ2)iωrH3.E37

For a similar discussion on the higher dimensional cases, we find that α(ξ) can be defined separately for the odd dimensional and even dimensional cases, where

α(ξ)=eiωrH1ξ2(ξ2)iωrHD3,Dodd;α(ξ)=eiωrH1ξ2(ξ2)iωrHD3(1ξ2)iωrHD3,Deven.E38

Substituting Eqs. (19), (20), (21) and (38) into Eq. (26), we find λ1 and s1 by the relation in Eq. (15). Next, by defining a suitable initial value of ω0, which represents the parameter ω in s0, with the termination condition Eq. (18), we can solve the parameter ω1, which represents the parameter ω in s1. This loop iterates, as with ω1 we can solve for ω2 in the next iteration and so on. For a sufficiently large number of iterations, the ωn-1 and ωn will become stable, and it will be the quasi normal frequency we are seeking.

The QNM results are obtained with the iteration number = 200 in Table 3, for the D = 5,6 dimensional Schwarzschild spacetimes, and in Table 4, for D = 7,8. The results using the third-order WKB methods are the same as presented in our previous work [13], where for completeness we also list the sixth-order WKB results.

5 Dimensions
lnThird-order WKBSixth-order WKBAIM
000.7247–0.3960 i0.7823–0.3635 i0.7252–0.3960 i
101.3158–0.3839 i1.3301–0.3852 i1.3163–0.3839 i
111.1490–1.2192 i1.1800–1.2021 i1.1495–1.2192 i
201.8754–0.3838 i1.8801–0.3844 i1.8802–0.3840 i
211.7541–1.1818 i1.7672–1.1791 i1.7674–1.1779 i
221.5588–2.0318 i1.5620–2.0517 i1.5593–2.0318 i
302.4251–0.3839 i2.4271–0.3840 i2.4256–0.3839 i
312.3322–1.1686 i2.3382–1.1674 i2.3327–1.1686 i
322.1704–1.9891 i2.1691–1.9980 i2.1709–1.9891 i
331.9611–2.8411 i1.9385–2.9063 i1.9616–2.8411 i
402.9716–0.3839 i2.9726–0.3839 i2.9721–0.3839 i
412.8963–1.1624 i2.8994–1.1618 i2.8968–1.1624 i
422.7596–1.9668 i2.7574–1.9711 i2.7601–1.9668 i
432.5773–2.7984 i2.5569–2.8330 i2.5778–2.7984 i
442.3583–3.6512 i2.3126–3.7665 i2.3588–3.6512 i
503.5166–0.3838 i3.5171–0.3838 i3.5171–0.3838 i
513.4533–1.1591 i3.4550–1.1588 i3.4538–1.1591 i
523.3353–1.9536 i3.3333–1.956 i3.3358–1.9536 i
533.1741–2.7711 i3.1580–2.7908 i3.1746–2.7711 i
542.9785–3.6089 i2.9383–3.6779 i2.9790–3.6089 i
552.7525–4.4625 i2.6857–4.6302 i2.7530–4.4625 i
6 Dimensions
001.2806–0.6391 i1.4364–0.5821 i1.2811–0.6391 i
102.1006–0.6276 i2.1350–0.6423 i2.1011–0.6276 i
111.7071–2.0417 i1.8139–1.9981 i1.7076–2.0417 i
202.8671–0.6308 i2.8797–0.6354 i2.8676–0.6308 i
212.5777–1.962 i2.6235–1.9629 i2.5782–1.962 i
222.1056–3.4196 i2.1297–3.4716 i2.1061–3.4196 i
303.6132–0.6321 i3.6194–0.6329 i3.6196–0.6325 i
313.3917–1.9342 i3.4143–1.9318 i3.4180–1.9342 i
323.0002–3.3204 i3.0081–3.3404 i3.0007–3.3204 i
332.4883–4.7849 i2.4206–4.9568 i2.4888–4.7849 i
404.3518–0.6324 i4.3550–0.6325 i4.3523–0.6324 i
414.1724–1.9214 i4.1845–1.9192 i4.1729–1.9214 i
423.8423–3.2694 i3.8437–3.2771 i3.8428–3.2694 i
433.3973–4.6831 i3.3411–4.7697 i3.3978–4.6831 i
442.8578–6.1516 i2.6999–6.4718 i2.8583–6.1516 i
505.0872–0.6325 i5.0890–0.6324 i5.0877–0.6325 i
514.9359–1.9143 i4.9428–1.9128 i4.9364–1.9143 i
524.6513–3.2399 i4.6503–3.2435 i4.6518–3.2399 i
534.2586–4.6195 i4.2147–4.6677 i4.2591–4.6195 i
543.7773–6.0494 i3.6482–6.2378 i3.7778–6.0494 i
553.2168–7.5218 i2.9732–8.0104 i3.2173–7.5218 i

Table 3.

Low-lying (nl, with l=j1/2) spin-1/2 field QNM frequencies using the WKB methods and the AIM with D = 5,6.

7 Dimensions
lnThird-order WKBSixth-order WKBAIM
001.7861–0.8090 i2.0640–0.7502 i1.7866–0.8090 i
102.7344–0.8066 i2.7827–0.8558 i2.7349–0.8066 i
112.0521–2.6832 i2.2892–2.6106 i2.0526–2.6832 i
203.6130–0.8166 i3.6327–0.8322 i3.6135–0.8166 i
213.1092–2.5590 i3.2053–2.5943 i3.1097–2.5590 i
222.2671–4.5340 i2.3344–4.6355 i2.2676–4.5340 i
304.4610–0.8206 i4.4730–0.8233 i4.4615–0.8206 i
314.0779–2.5187 i4.1298–2.5218 i4.0784–2.5187 i
323.3809–4.3622 i3.4158–4.4006 i3.3814–4.3622 i
332.4624–6.363 i2.3163–6.6549 i2.4629–6.363 i
405.2968–0.8218 i5.3035–0.8220 i5.3033–0.8220 i
414.9878–2.5009 i5.0176–2.4966 i4.9883–2.5009 i
424.4049–4.2771 i4.4224–4.2810 i4.4054–4.2772 i
433.6066–6.1782 i3.4954–6.3058 i3.6071–6.1782 i
442.6393–8.1950 i2.2624–8.7432 i2.6398–8.1950 i
506.1273–0.8221 i6.1310–0.8219 i6.1309–0.8221 i
515.8671–2.4910 i5.8847–2.4870 i5.8676–2.4910 i
525.3673–4.2295 i5.3756–4.2263 i5.3678–4.2295 i
534.6654–6.0660 i4.5819–6.1252 i4.6659–6.0660 i
543.8002–8.0044 i3.5047–8.3081 i3.8007–8.0044 i
552.7964–10.035 i2.1874–10.910 i2.7969–10.0348i
8 Dimensions
002.2437–0.9238 i2.6486–0.8930 i2.2442–0.9238 i
103.2663–0.9359 i3.3105–1.0469 i3.2668–0.9359 i
112.2397–3.1796 i2.6722–3.0659 i2.2402–3.1796 i
204.2077–0.9556 i4.2279–0.9928 i4.2082–0.9556 i
213.4491–3.0091 i3.6051–3.1428 i3.4496–3.0091 i
222.1453–5.4463 i2.2891–5.5740 i2.1458–5.4463 i
305.1097–0.9641 i5.1285–0.9704 i5.1102–0.9642 i
314.5378–2.9607 i4.6284–2.9895 i4.5383–2.9607 i
323.4576–5.1805 i3.5481–5.2651 i3.4581–5.1805 i
332.0383–7.6877 i1.7944–8.0198 i2.0388–7.6877 i
405.9947–0.9669 i6.0062–0.9669 i5.9952–0.9669 i
415.5365–2.9415 i5.5943–2.9365 i5.5370–2.9415 i
424.6421–5.0549 i4.6967–5.0480 i4.6426–5.0549 i
433.4017–7.3877 i3.2284–7.5066 i3.4022–7.3877 i
441.9199–9.9347 i1.2118–10.564 i1.9204–9.9347 i
506.8720–0.9677 i6.8785–0.9669 i6.8772–0.9659 i
516.4875–2.9309 i6.5231–2.9224 i6.4880–2.9309 i
525.7272–4.9883 i5.7608–4.9630 i5.7277–4.9883 i
534.6390–7.2092 i4.5109–7.2241 i4.6395–7.2092 i
543.3023–9.6186 i2.7516–9.9294 i3.3028–9.6186 i
551.7759–12.195 i0.5680–13.282 i1.7764–12.1949 i

Table 4.

Low-lying (nl, with l=j1/2) spin-1/2 field QNM frequencies using the WKB methods and the AIM with D = 7,8.

4. A remark on the spin-3/2 field QNMs

In this section, we are going to present a discussion of our recent work [11] on the spin-3/2 fields in general dimensional Schwarzschild spacetimes. Note that, this section will not go into as great a detail as the previous one but shall just list some improvements made in light of recent considerations. Starting with the radial equation related to the “non-TT eigenmodes”, the radial equation can still be represented as the Schrödinger like one, Eq. (4), where

VNTT,s=3/2=±fdWNTT,s=3/2dr+WNTT,s=3/22,WNTT,s=3/2=f(j+D32)r[(2D2)2(j+D32)21D4D2(2Mr)D3(2D2)2(j+D32)2f],E39

and

f(r)=1(2Mr)D3.E40

Because the general form of the radial equation is the same as the Dirac case, Eqs. (22)–(38) are still sufficient in this case, but the effective potential will be Eq. (39), not Eq. (19). What we need to consider here is that the asymptotic behaviour function is simpler than what we had used previously and successfully generates better results for the QNMs.

In Table 5, for some lower modes in the seven-dimensional spacetime, the new AIM results (with the current choice of boundary behaviour function, Eq. (38)) are better than the previous results for the first few modes. This is why we believe Eq. (38) is a suitable choice of the boundary behaviour function for the AIM methods in the higher dimensional spacetimes.

7 Dimensions
lnThird-order WKBSixth-order WKBAIM (earlier)AIM (new)
000.7725–0.2978 i0.7530–0.3037 i0.7008–0.3036 i0.7535–0.3036 i
101.1441–0.2893 i1.1415–0.2831 i1.1231–0.2976 i1.142–0.2831 i
110.9465–0.9065 i0.9267–0.8783 i0.9266–0.8782 i0.9271–0.8782 i
8 Dimensions
441.0674–3.5290 i0.7606–3.5381 i1.0674–3.5290 i1.0679–3.5291 i
541.5711–3.4654 i1.3755–3.4609 i1.5711–3.4653 i1.5716–3.4654 i
551.0321–4.3822 i0.5837–4.5299 i1.0321–4.3822 i1.0326–4.3822 i

Table 5.

Selected QNM frequencies of low-lying (nl, with l=j3/2) for spin-3/2 field related to the non-TT eigenmodes by using the WKB methods and the AIM.

5. Summary

In this chapter, we presented a brief review of recent progress on perturbation theory with various fields in curved spacetimes, especially for a systematic comparison to the fermionic and bosonic fields. Generally, the first step in this topic is the obtaining of the radial equations as discussed in Section 2. There are then various methods, but no unique approach for considering a specific field on a specific spacetime, where we strongly suggest the reader follow the references provided to develop a step-by-step approach. On the other hand, the process for obtaining the radial equations always relates to the separability of a spacetime, this being the main reason that we still have difficulties for perturbation theory in the higher dimensional Kerr spacetimes. As such we mentioned in the introduction section that for the gravitational wave experiments, the QNMs in the higher dimensional Kerr spacetimes are definitely an interesting next stage of study for this area, and it shall be interesting seeing further progress in this direction.

In Section 3, we presented the AIM, giving an in detail example to study the perturbation theory in the higher dimensional spacetimes and also presented a remark for our recent work on the spin-3/2 fields. Since there are several numerical methods for obtaining the QNMs, we believe that the AIM is a straightforward way to study the QNMs due to its simple mathematical structure.

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Chun-Hung Chen, Hing-Tong Cho and Alan S. Cornell (June 7th 2017). Semi-Analytic Techniques for Solving Quasi-Normal Modes, Trends in Modern Cosmology, Abraao Jesse Capistrano de Souza, IntechOpen, DOI: 10.5772/68114. Available from:

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