Fermion Quasi-normal modes of the Kerr Black-Hole

0.43 MB·English

by  W. A. Carlson

WITS-CTP-86
Fermion Quasi-normal modes of the Kerr Black-Hole
W. A. Carlson,1,∗ A. S. Cornell,1,† and B. Jordan1,‡
1National Institute for Theoretical Physics; School of Physics,
University of the Witwatersrand, Wits 2050, South Africa
(Dated: 15th January 2012)
In this paper we study the fermion quasi-normal modes of a 4-dimensional rotating black-hole
using the WKB(J) (to third and sixth order) and the AIM semi-analytic methods in the massless
Dirac fermion sector. These semi-analytic approximations are computed in a pedagogical manner
with comparisons made to the numerical values of the quasi-normal mode frequencies presented in
the literature. It was found that The WKB(J) method and AIM show good agreement with direct
numerical solutions for low values of the overtone number n and angular quantum number l.
PACSnumbers: 04.70.Dy;03.65.Pm;04.25.-g;04.70.-s
Keywords: black-hole,quasi-normalmodes,fermion
2
1
0
2
I. INTRODUCTION

n
a
J For a number of years black-hole (BH) quasi-normal modes (QNMs) have been of great interest to both gravi-
tational theorists and gravitational wave experimentalists. QNMs have been studied in the contexts of ringing of
7
2 astrophysical BHs where QNMs are relevant to gravitational wave astronomy [1–3], loop gravity interpretations [4],
lower dimensional BHs [5], and in the AdS/CFT correspondence [6] where BHs are thermal states in the field theory

] andtheirdecayscorrespondtoadecayofaperturbationofthisstate[7]. Assuch,knowledgeofQNMswillprovidean
c
understanding of the stability of BHs against external perturbations; where more recently the interest in the Hawk-
q
- ing radiation and QNMs of BHs has stemmed from the idea that mini-TeV scale BHs might be created in particle
r accelerators such as the Large Hadron Collider (LHC) [8–12]. QNM oscillations have been found in perturbation
g
[ calculations of particles falling into Schwarzschild [13] and Kerr BHs [14] and in the collapse of a star to form a black
hole [15].
2
Let us first recall that QNMs are the damped, resonant, non-radial perturbations of BHs which are characterized
v
byaspectrumofdiscrete,complexfrequencies,withrealpartdeterminingtheoscillationfrequency,andanimaginary
7
part that determines the rate at which each mode is damped as a result of an emission. For a given kind of physical
6
2 perturbation the complex frequencies are uniquely determined by the mass and angular momentum of the BH, the
3 angularharmonicindices(l,m)ofthedeformation,andthedegreeoftheharmonicofthemoden. Ashasbeenshown
. in many contexts, the fundamental equations describing the perturbations of BHs can be reduced to a single second-
1
0 order ordinary differential equation similar to the one-dimensional Schr¨odinger equation for a particle encountering
2 a potential barrier. However, the nature of the potential precludes an exact, closed-form solution in terms of known
1 functions. That is, our equations for wave propagation reduce to
:
v
Xi d2ψ +(cid:0)ω2−V(x)(cid:1)ψ =0, (1.1)
dx2
r
a where x is a spatial co-ordinate. The BH event-horizon is mapped to −∞ in the radial co-ordinate, and the potential
V(x) is a function of the spatial co-ordinate only. Note that we have proceeded in the usual way by assuming a time
dependence, Ψ(t,x)=eiωtψ(x).
This separation is not restrictive since once we have a solution for Eq.(1.1), a general time dependent solution can
be given as a continuous Fourier transform of such a solution, where the form of Eq.(1.1) parallels a normal mode
analysis. We shall now restrict ourselves to asymptotically flat space-times such that the potential V(x) is positive
and satisfies
V(x)→0, x→±∞. (1.2)
∗Email: [email protected]
†Email: [email protected]
‡Email: [email protected]
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Note that there exist no normal mode expansions for such systems since such potentials do not allow bound states.
From the form of the potential in Eq.(1.2) we have that near to the boundaries ±∞ the solutions behave as plane
waves,
ψ(x)∼e±iωx, x→±∞. (1.3)
Here in-going at the horizon means entering into the BH, whilst the QNM frequencies and the associated wave-
functions ψ (which are the solutions to Eq.(1.1)) are the QNMs specified by the boundary conditions of Eq.(1.3).
ThestudyofBHperturbationswasinitiatedbyReggeandWheeler[16]andextendedbyZerilli[17]. Followingthe
work by Vishveshwara [18], Chandrasekhar and Detweiler [19] pioneered numerical methods for the study of QNMs.
More recently, semi-analytic methods have proven useful in the study of BH perturbations with the WKB(J) method
of Schutz, Iyer and Will [20, 21], and more recently the Asymptotic Iteration Method (AIM) of Ciftci, Hall and Saad
[22], which is a valuable tool for computing solutions to general second order ordinary differential equations.
Inthisnoteweextendtosixthorderinthesemi-analyticapproximationtheworksofSeidelandIyer[23],Konoplya
[24], Iyer and Will [21], and following the works of Refs. [25, 26], we extend to the fermion sector the Kerr BH
QNMs. Here we shall compare the results of direct numerical, WKB(J) approximation and the AIM computation of
QNMs of fermionic BH excitations in the Kerr space-time. Note that we shall assign c = (cid:126) = G = 1 and M = 1
for the calculated QNM frequencies in this work, where this note is organised as follows: In Sec.II we shall give a
brief overview ofthe Newman-Penrose (NP)formalism [27] andits application tothe QNMsof massless spin-1 fields.
2
Sec.III and IV contain reviews of the semi-analytic methods employed here, whilst Sec.V contains the numerical
results of the QNMs for spin-1 field in a Kerr space-time, followed by a short discussion of these results.
2
II. NEWMAN PENROSE FORMALISM
Inordertodevelopamasterequationasdiscussedintheprevioussection, forafermion, weshallintroducetheNP
formalism [27]. Let (M,g) be a pseudo-Riemannian manifold with metric g of dimension n, and denote the tangent
space at p∈M by T M. A natural basis for T M is the set of co-ordinate partial derivatives e ≡∂ , although we
p p µ µ
may choose a basis as we wish. Therefore, we choose one which is orthonormal with the signature of the metric g on
M, that is
g(e ,e )=η ,
a b ab
whereη isablockdiagonalmatrixintheNPformalism. Avectormaybeexpressedasalinearcombinationofbasis
ab
vectors such that
e =eae , (2.1)
µ µ a
and the operators eµ are n×n invertable matrices which satisfy,
a
eaeν =δν ,
µ a µ
where Latin letters index tetrad co-ordinates and Greek letters index manifold co-ordinates. The operator ea, now
µ
called a vielbein, is a projection operator from the co-ordinate basis to the non-co-ordinate basis. The NP formalism
is a special case for the vielbein, or tetrad, formalism in which tensors defined on a space-time are projected on to a
complete basis vector set at each point p. This basis set is the set of complex null vectors l,n,m and m¯, of which l
and n are real and m and m¯ are complex conjugate. By the null condition of this basis vector set, it is clear that
l·l=n·n=m·m=m¯ ·m¯ =0.
In addition, these basis vectors satisfy the orthogonality relations,
l·m=l·m¯ =n·m=n·m¯ =0,
as well as the normalisation conditions l·n=1 and m·m¯ =−1. The corresponding fundamental matrix η is
ab
0 1 0 0 
1 0 0 0
η =ηab =  ,
ab 0 0 0 −1
0 0 −1 0
3
to which there corresponds a basis, e =l, e =n, e =m and e =m¯. When used to define directional derivatives
1 2 3 4
we assign to each basis vector a special symbol, e =D, e =D∗, e =δ, and e =δ∗.
1 2 3 4
Having developed the NP formalism, we begin our analysis by considering the Dirac equation in the null tetrad
basis,
(−iγµ∇ +m)Ψ=0, (2.2)
µ
where ∇ is the covariant derivative taking the form
µ
i
∇ =∂ − ω a η γcb . (2.3)
µ µ 4 µ b ca
The spin connection ω a satisfies the vielbein ansatz,
µ b
∂ ea+ω a eb −Γσ ea =0, (2.4)
µ ν µ b ν µν σ
and γab =γaγb−γbγa with γa being the ath Dirac matrix. Note that the spin connection ω a may be rewritten in
µ b
terms of the affine connection as
ω a =eaeλΓν −eλ∂ ea . (2.5)
µ b ν b µλ b µ λ
The spin connection becomes the affine metric connection under the following co-ordinate transformation,
Γλ =eλ(cid:0)∂ ea+ω a eb(cid:1) . (2.6)
µν a µ ν µ b ν
We shall focus our attention on the massless limit of Eq.(2.2),
γµ∇ Ψ=0, (2.7)
µ
rewriting Ψ in terms of Weyl spinnors ψ and ψ˜, that is,
(cid:18) (cid:19)
ψ
Ψ= , (2.8)
ψ˜
and substituting Eq.(2.8) into Eq.(2.7) yields the dynamical Weyl spinor equations,
σµ ∇ ψa =0, and σ˜µ ∇˜ ψ˜b˙ =0, (2.9)
ab˙ µ ab˙ µ
inthechiralrepresentation. σa andσ˜a˙ arePaulimatrices,and∇ and∇˜µarethethecovariantderivativesofψa and
µ
ψ˜a˙,respectively. The Pauli matrices take on new forms in the vielbein formalism for curved space-time, ρµ = eµσa,
a
ρ =eaσ , ρ˜µ =eµσ˜a and ρ˜ =eaσ˜ , where eµ are the vielbein basis elements and σµ are Pauli matrices. Explicitly
µ µ a a µ µ a a
we find
ρi =(cid:18) lµ mµ (cid:19) and ρ ab˙ =(cid:18) nµ −mµ (cid:19) . (2.10)
ab˙ m¯µ nµ i −m¯ l
µ µ
Following the approach of Ref. [28], the covariant derivative of Eqs.(2.7) and (2.9) may be written as
ρi ∇ ψa =0, and ρ˜i ∇˜ ψ˜b˙ =0. (2.11)
ab˙ µ ab˙ µ
Consider now the 4-dimensional Kerr background space-time with line element
(cid:18) 2Mr(cid:19) (cid:18)4Mrasin2θ(cid:19)
ds2 = − 1− dt2− dtdθ
ρ2 ρ2
+(cid:18)ρ2(cid:19)dr2+(cid:0)ρ2(cid:1)dθ2+(cid:18)r2+a2+ 2Mrasin2θ(cid:19)sin2θdφ2 , (2.12)
∆ ρ2
where ρ2 =r2+a2cos2θ, ∆=r2+a2−2Mr and a= J ∈[0;1) is the rotation parameter of the BH. It is useful to
M
define ρ¯=r+iacosθ and ρ¯∗ =r−iacosθ such that ρ2 =ρ¯ρ¯∗. The basis of the NP formalism as defined on the Kerr
geometry as
l = 1 (cid:0)∆,−ρ2,0,−a∆sin2θ(cid:1) , n = 1 (cid:0)∆,ρ2,0,−a∆sin2θ(cid:1) ,
j ∆ j 2ρ2
m = √1 (cid:0)−iasinθ,0,−ρ2,−i(cid:0)r2+a2(cid:1)sinθ(cid:1) , m¯ =m∗ ,
j j j
2ρ¯
4
and
lj = 1 (cid:0)r2+a2,∆,0,a(cid:1) , nj = 1 (cid:0)r2+a2,−∆,0,a(cid:1) ,
∆ 2ρ2
(cid:18) (cid:19)
mj = √1 iasinθ,0,1, i , m¯j =mj∗ .
2ρ¯ sinθ
We may now define the Ricci rotation coefficients (spin coefficients in that which follows) with respect to the curved
space-time,
κ=ω , ρ=ω , (cid:15)= 1(ω +ω ) ,
311 314 2 211 341
σ =ω , µ=ω , γ = 1(ω +ω ) ,
313 243 2 212 324 (2.13)
λ=ω , τ =ω , α= 1(ω +ω ) ,
244 312 2 213 344
ν =ω , π =ω , β = 1(ω +ω ) .
242 241 2 213 343
Recallthatthecomplexconjugateofanyspincoefficientisobtainedbyreplacingthenumericalindex3withthevalue
4 and vice versa [28]. The resulting equations are
(D+(cid:15)−ρ˜)ψ0+(δ∗+π−α)ψ1 = 0,
(D∗+µ−γ)ψ1+(δ+β−τ)ψ0 = 0,
(D+(cid:15)∗−ρ˜∗)ψ˜0˙ +(δ+π∗−α∗)ψ˜1˙ = 0,
(D∗+µ∗−γ∗)ψ˜1˙ +(δ∗+β∗−τ∗)ψ˜0˙ = 0.
It is clear that basis vectors are mapped to derivative operators when applied to the tangent space as tangent
vectors to the wave-functions ψ(t,r,θ,φ)=ei(ωt+mφ)ψ(r,θ). Therefore, we write
∆
l=D =D , n=D∗ =− D† ,
0 2ρ2 0
1 1
m=δ = √ L† , m¯ =δ∗ = √ L ,
2ρ¯ 0 2ρ¯∗ 0
where
K r−M K r−M
D =∂ +i +2n , D† =∂r−i +2n ,
n r ∆ ∆ n ∆ ∆
n inasinθ n inasinθ
L =∂ +Q+ − , L† =∂ −Q+ + ,
n θ tanθ ρ¯ n θ tanθ ρ¯∗
with K =(cid:0)r2+a2(cid:1)ω+am, and Q=aωsinθ+ m .
sinθ
The spin coefficients are linear combinations of basis vectors in the NP formalism. We may therefore express
the collection of spin connections in terms metric tensor elements. We shall choose the representation of the spin
connections so as to simplify the algebra. A suitable choice for this assignment of elements is
1 iasinθ
ρ˜=− , β = √ 1 , π = √ ,
ρ¯∗ 2 2ρ¯∗tanθ 2(ρ¯∗)2
iasinθ r−M
τ =− √ , µ=− ∆ , γ =µ+ ,
2ρ2 2ρ¯∗ρ2 2ρ2
α=π−β∗ ,
which results in,
κ=σ =λ=ν =(cid:15)=0.
Now let f = ρ¯∗ψ0, f = ψ1, g = ψ˜1˙, g = −ρ¯ψ˜0˙. Following these assignments, it can be shown [28] that the Dirac
1 2 1 2
equations reduce to
1
D f + √ L f = 0, (2.14a)
0 1 1 2
2 2
√
∆D†f − 2L†f = 0, (2.14b)
1 2 1 1
2 2
1
D g − √ L†g = 0, (2.14c)
0 2 2 12 1
√
∆D†g + 2L g = 0. (2.14d)
1 1 1 2
2 2
5
Using Eq.(2.14), we separate the Dirac equation into radial and angular parts by making the ansatz:
f (r,θ) = R (r)S , (2.15a)
1 −1 −1
2 2
f (r,θ) = R (r)S , (2.15b)
2 1 1
2 2
g (r,θ) = R (r)S , (2.15c)
1 1 −1
2 2
g (r,θ) = R (r)S . (2.15d)
2 −1 1
2 2
Substituting Eq.(2.15) into Eq.(2.14) yields,
L S = −λS , (2.16a)
1 1 −1
2 2 2
L† S = λS , (2.16b)
1 −1 1
2 2 2
and
√ √
∆D R(cid:48) = λ ∆R , (2.17a)
0 −1 1
√ √ 2 2
∆D† ∆R = λR(cid:48) , (2.17b)
0 −1 −1
2 2
√
where R(cid:48) = 2R .
−12 12
Decoupling Eq.(2.16) yields,
(cid:104) (cid:105)
L L +λ2 S =0, (2.18)
±1 ±1 ∓1
2 2 2
where S is the adjoint of S under the replacement of θ by π−θ. Similarly, decoupling Eq.(2.17) yields,
1 −1
2 2
(cid:104) (cid:105)
∆D†D R =0, (2.19)
1 0 −1
2 2
√
and ∆R satisfies the complex conjugate relation. Once decoupled, Eqs.(2.18) and (2.19) are eigenvalue equations
1
2
with known eigenvalues λ and unknown eigen-functions R and S .
±1 ±1
2 2
ForthepurposeofcomputingQNMsweshallfocusourinterestontheradialequationforthespin-1 field. Wemap
2
Eq.(2.19) to the tortoise co-ordinate x,
(cid:18) (cid:19)
dr ∆
= , (2.20)
dx K¯
K
where K¯ = eliminates the co-ordinate singularity at the horizon. After this co-ordinate transformation, operators
ω
D and D† simplify to,
0 0
K¯ (cid:18) d (cid:19)
D = +iω , (2.21a)
0 ∆ dx
K¯ (cid:18) d (cid:19)
andD† = −iω . (2.21b)
0 ∆ dx
√
The assignments P = ∆R and P = R , in conjunction with Eq.(2.21), lead to the new form of the radial
1 1 −1 −1
2 2 2 2
equations
√
(cid:18) (cid:19)
d ∆
−iω P = λ P , (2.22a)
dx 21 K¯ −21
√
(cid:18) (cid:19)
d ∆
+iω P = λ P . (2.22b)
dx −12 K¯ 12
If we define,
Z =P ±P , (2.23)
± 1 −1
2 2
6
and combine Eqs.(2.22a) and (2.22b) we obtain
(cid:32) √ (cid:33)
d ∆
−λ Z = iωZ ,
dx K¯ + −
(cid:32) √ (cid:33)
d ∆
+λ Z = iωZ ,
dx K¯ − +
which may be separated to yield
(cid:18)d2Z±(cid:19)(cid:0)ω2−V (x)(cid:1)Z =0, (2.24)
dx2 ± ±
with
(cid:32)√ (cid:33)
∆ d ∆
V (x)=λ2 ±λ . (2.25)
± K¯2 dx K¯
Eq.(2.24) is now in the form of Eq.(1.1) with a potential given by Eq.(2.25). Eq.(2.24) is the QNM master equation
for a spin-1 field in a Kerr space-time. In the following two sections we shall review two semi-analytic approaches to
2
solving these equations for an ultimate comparison with the direct numerical results of Refs. [29, 30].
III. WKB(J) METHOD
We shall begin with the WKB(J) approximation of Schutz, Iyer and Will [20, 21] which is a general method for
generatingapproximatesolutionstoordinarylinearsecondorderdifferentialequationsandhasbeenusedextensivelyin
variousBHcases(seeRef. [24]andreferencestherein). Forfurtherdiscussionofthismethod,seeRefs. [21,23,24,31].
We may use the WKB(J) method to generate approximate solutions to differential equations of the form
(cid:18)d2Ψ(cid:19)
+Q(x)Ψ=0, (3.1)
dx2
forasmoothfunction, atO(1), Q(x)=ω2−f(x)withω ∈C. NowassumethatΨhasamodeexpansionoftheform,
Ψ(x,(cid:15))=Ae(cid:80)∞n=0iSn(x)(cid:15)n−1 , (3.2)
where A ∈ C and (cid:15) (cid:28) 1 tracks the order of the expansion. By substituting Eq.(3.2) into Eq.(3.1), and equating
powers of (cid:15), yields
(cid:90) x
(cid:112)
S (x)=±i dt Q(t), (3.3)
0
and
1
S (x)=− lnQ(x), (3.4)
1 4
where the two choices in sign in Eq.(3.3) correspond to either incoming or outgoing waves at ±∞. When x → +∞,
Q(x) → ω2 such that S → +iωx for the outgoing wave to infinity and S → −iωx for the incoming wave from
0 0
infinity. Similarly, for x → −∞, S → +iωx for the wave incoming from x → −∞, while S → −ωx to a wave
0 0
outgoing to x→−∞, see Fig.1. Designating these four solutions,
ΨI ∼e+ωx , ΨI ∼e−ωx , x→+∞
+ − (3.5)
ΨIII ∼e+ωx , ΨIII ∼e−ωx , x→−∞
− −
The general solutions to regions I and III are
Ψ∼ZINΨ−+ZOUTΨI , region I,
I I I + (3.6)
Ψ∼ZINΨ− +ZOUTΨI , region III,
III II III +
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RegionI RegionIII
RegionII Q(x)
FIG. 1: The reduced potential barrier as used in the WKB(J) method.
where Z−,Z+,Z− ,Z+ ∈C.
I I III III
The amplitudes at +∞ are related to amplitudes at −∞ through the linear relation
(cid:18)Z+ (cid:19) (cid:18)S S (cid:19) (cid:18) ZI (cid:19)
III = 11 12 · + . (3.7)
Z− S S ZI
III 21 22 −
WematchthesolutionsinregionsIandIIIwithasolutioninregionIIthroughthetwoturningpointsofQ(x)=0. If
the turningpoints are closelyspaced(−Q(x) (cid:28)Q(±∞)), then thesolution in regionII can be well approximated
max
by a Taylor series expansion,
1 (cid:16) (cid:17)
Q(x)=Q(x )+ Q(cid:48)(cid:48)(x )(x−x )2+O (x−x )3 , (3.8)
0 2 0 0 0
where x is the location of the maximum of the function Q(x) and the prime notation signifies differentiation with
0
respect to x. We shall assign Q =Q(x ) in what follows. Region II corresponds to
0 0
(cid:115)
−2Q
|x−x0|< Q(cid:48)(cid:48)0 ≈(cid:15)12 . (3.9)
0
We introduce new functions
1
k = Q(cid:48)(cid:48) ,
2 0
t = (4k)41ei4π (x−x0) , (3.10)
1 −iQ
and ν+ = 0 . (3.11)
(cid:112)
2 2Q(cid:48)(cid:48)
0
Rewriting Eq.(3.1) as
(cid:18)d2Ψ(cid:19) (cid:18) 1 1 (cid:19)
+ ν+ − t2 Ψ=0, (3.12)
dt2 2 4
which has, is terms of parabolic cylinder functions F (t), the general solution
ν
Ψ(t)=AF +BF (it). (3.13)
ν ν−1
It can be shown that the large |t| asymptotics of this solution give an S-matrix,
(cid:18)Z+ (cid:19)  eiπν iR2eiπν(2π)12  (cid:18) ZI (cid:19)
ZII−IIII = R−2(2π)12 Γ−(νe+iπ1ν) · Z−+I , (3.14)
Γ(−ν)
where Γ is the Gamma function and R is a general exponential function depending on the nature of the potential
Q. When expanding to higher orders, the S matrix has the same general form as shown in Eq.(3.14), with modified
8
expressionsfortheoff-diagonalelements. NotethatforaBHnowavesarereflectedbythehorizon,suchthatZIII =0,
−
and due to the QNM boundary conditions there are no waves incoming from infinity, ZI = 0. These conditions are
−
satisfied in Eq.(3.7) only if Γ(−ν)=0. Consequently, ν ∈Z, and
1 −iQ
α=ν+ = 0 , n=0,1,3,... . (3.15)
(cid:112)
2 2Q(cid:48)(cid:48)
0
The above equation gives the quantisation condition which labels QNMs by an overtone number n = −ν. Since the
coefficientsintheS-matrixthatconnectamplitudesnearthehorizonwiththoseatinfinitydependonlyonα(related
to the overtone number n), we need only expand Eq.(3.2) to higher orders in (cid:15) to obtain the WKB formula to higher
orders. The result has the form
iQ 1
0 −Λ −Λ −Λ −Λ −Λ =n+ , (3.16)
(cid:112)Q(cid:48)(cid:48) 2 3 4 5 6 2
0
where the correction terms Λ ,Λ ,Λ can be found in Ref. [31] and the included URL.
4 5 6
IV. ASYMPTOTIC ITERATION METHOD
Our second semi-analytic technique is the AIM, which produces an exact solution to an ordinary second order
partial differential equation under the assumption that the coefficients in the Taylor expansion of the differential
equation form an asymptotic series. To see this, consider the homogeneous linear second-order differential equation
for the function χ(x),
χ(cid:48)(cid:48) =λ (x)χ(cid:48)+s (x)χ, , (4.1)
0 0
where λ (x) and s (x) are functions in C (a,b). In order to find a general solution to this equation we rely on the
0 0 ∞
symmetric structure of the right-hand side of Eq.(4.1) [22]. If we differentiate Eq.(4.1) with respect to x, we find that
χ(cid:48)(cid:48)(cid:48) =λ (x)χ(cid:48)+s (x)χ,
1 1
where λ =λ(cid:48) +s +(λ )2 and s =s(cid:48) +s λ . Taking the second derivative of Eq.(4.1) yields
1 0 0 0 1 0 0 0
χ(cid:48)(cid:48)(cid:48)(cid:48) =λ (x)χ(cid:48)+s (x)χ,
2 2
where λ =λ(cid:48) +s +λ λ , and s =s(cid:48) +s λ . The (n+1)th and the (n+2)th derivatives, n=1,2,..., are found
2 1 1 0 1 1 0 0 0
iteratively,
χ(n+1) =λ (x)χ(cid:48)+s (x)χ. (4.2)
n−1 n−1
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:
χ(n+2) =λ (x)χ(cid:48)+s (x)χ, (4.3)
n n
where,
λ (x)=λ(cid:48) (x)+s (x)+λ (x)λ (x), (4.4a)
n n−1 n−1 0 n−1
and
s (x)=s(cid:48) (x)+s (x)λ (x). (4.4b)
n n−1 0 n−1
For sufficiently large n
s (x) s (x)
n = n−1 ≡β(x), (4.5)
λ (x) λ (x)
n n−1
where from the ratio of the (n+1)th and the (n+2)th derivatives, we have
(cid:16) (cid:17)
d lnχ(n+1) = χ(n+2) = λn χ(cid:48)+ λsnnχ = d lnχ(n+1) = λn . (4.6)
(cid:16) (cid:17)
dx χ(n+1) λ χ(cid:48)+ sn−1χ dx λn−1
n−1 λn−1
9
This then yields,
(cid:18)(cid:90) x λ (x(cid:48)) (cid:19) (cid:18)(cid:90) x (cid:19)
χ(n+1)(x)=C exp dx(cid:48) n =C λ exp dx(cid:48)(β+λ ) , (4.7)
1 λ (x(cid:48)) 1 n−1 0
n−1
where C is the integration constant and the right-hand side of Eq.(4.4) and the definition of β(x) have been used.
1
Substituting this into Eq.(4.2), we obtain the first-order differential equation
(cid:18)(cid:90) x (cid:19)
χ(cid:48)+βχ=C exp dx(cid:48)(β+λ ) . (4.8)
1 0
The general solution can be determined as [22]
(cid:20) (cid:90) x (cid:21)(cid:32) (cid:90) x (cid:90) x(cid:48) (cid:33)
χ(x)=exp − dx(cid:48)β(x(cid:48)) C +C dx(cid:48)exp dx(cid:48)(cid:48)[λ (x(cid:48)(cid:48))+2β(x(cid:48)(cid:48))] . (4.9)
2 1 0
The integration constants, C and C , can be determined by an appropriate choice of normalisation. Note that for
1 2
the generation of exact solutions C =0. Eq.(4.5) gives the sequence termination condition,
1
s λ −s λ =0, (4.10)
n n−1 n−1 n
which is also the QNM “quantisation condition” as given in Ref. [32]. Note that the sequence termination condition
is true only at a particular value n since there is no guarantee that the sequence λ is truly asymptotic.
n
Improvements to the efficiency of the AIM have been made which eliminates the need to take derivatives at each
step in the iteration process, thereby reducing the computational overhead while also improving the overall accuracy
ofthemethod. ThisisdonebyreducingtheAIMtoasetofrecursionrelationswhichnolongerinvolvedifferentiation.
For more details on this improved AIM see Ref. [25].
V. RESULTS AND CONCLUDING REMARKS
Tables I, II, III, IV and V constitute the main results of this paper, and contain the numerically computed QNM
frequencies for varying values of the BH rotation parameter a = J , along with the semi-analytic WKB(J) (to third
M
andsixthorder)andAIMresults. Ineachcase, andwherenumericalresultswereavailable, theywerecomparedwith
the semi-analytic solutions, these numerical results were taken from Refs. [29, 30]. At this point we re-iterate that
we have set c=G=1 and M =1. Note that in the absence of numerical results, QNM frequencies calculated using
the sixth order WKB(J) method were compared to the values from the AIM.
Unlike earlier cases, such as the case of scalar perturbations, previously published numerical values exist for the
QNM frequencies that result from spin-half perturbations of a Kerr BH [29, 30], at least for the angular numbers of
l = 0 and l = 1. In comparing these numerical results, the accuracy of the WKB(J) method was worst (relative to
the pure numerical evaluation) for parameter values (a,l,n)=(0,0,0), with approximate errors of (3.55%,3.20%) at
thirdorderand(0.16%,2.16%)atsixthorder,andshowedsteadyimprovementinaccuracywithincreasinglandn,to
(0.13%,0.50%) at third order and (0.02%,<0.01%) at sixth order. However, the AIM method experienced degraded
accuracywithincreasinglandnata=0,rangingfrom(<0.01%,0.10%)atl=n=0to(1.46%,4.32%)atl=n=2.
However, the accuracy of the AIM was at its highest when a was at its lowest, n was at its lowest and l was at its
highest. Note that this for an iterative of only 15, though with this iterative depth the AIM was still significantly
slower that the sixth order WKB(J) method.
For the QNM frequencies when l = 2 and l = 3, the AIM values were then compared to the sixth order WKB(J)
method values. In these cases the accuracy of the AIM, as compared to the sixth order WKB(J) method was at its
highest when n was at its lowest and the l values were higher, such as (a,l,n)=(0,3,0), (0.20,2,0), (0.40,2or 3,0),
(0.60,2or 3,0) and (0.80,3,0). Once again, numerical processors did evaluate significantly faster for the sixth order
WKB(J) method than for the AIM.
In conclusion, we have, using the NP formalism, developed the Dirac equation in a Kerr background in a peda-
gogical manner. Using this result we have applied two semi-analytic techniques to calculate the QNM frequencies for
comparisonwithavailablenumericalresults[29,30]findingthatthecomputationaloverheadfortheWKB(J)method
is lower at low n than for the AIM. As such the WKB(J) method executes considerably more quickly than the AIM.
10
TABLE I: The QNM Frequencies for the Spin-1 Perturbation of the Kerr BH with rotation parameter a = 0.00, which is the
2
Schwarzschild limit.
l n Numerical Third Order WKB Sixth Order WKB AIM
0 0 0.1830 - 0.0970i 0.1765 - 0.1001i 0.1827 - 0.0949i 0.1830 - 0.0969i
(-3.55% , 3.20%) (-0.16% , -2.16%) (<0.01% , -0.10%)
1 0 0.3800 - 0.0964i 0.3786 - 0.0965i 0.3801 - 0.0964i 0.3800 - 0.0964i
(-0.37% , 0.10%) (0.03% , <0.01%) (<0.01% , <0.01%)
1 0.3558 - 0.2975i 0.3536 - 0.2987i 0.3559 - 0.2973i 0.3568 - 0.2976i
(-0.62% , 0.40%) (0.03% , -0.07%) (0.28% , 0.03%)
2 0 0.5741 - 0.0963i 0.5737 - 0.0963i 0.5741 - 0.0963i 0.5741 - 0.0963i
(-0.07% , <0.01%) (<0.01% , <0.01%) (<0.01% , <0.01%)
1 0.5570 - 0.2927i 0.5562 - 0.2930i 0.5570 - 0.2927i 0.5573 - 0.2928i
(-0.14% , 0.10%) (<0.01% , <0.01%) (0.05% , 0.03%)
2 0.5266 - 0.4997i 0.5273 - 0.4972i 0.5265 - 0.4997i 0.5189 - 0.5213i
(0.13% , -0.50%) (-0.02% , <0.01%) (-1.46% , 4.32%)
3 0 0.7672 - 0.0963i 0.7674 - 0.0963i 0.7674 - 0.0963i
(<0.01% , <0.01%)
1 0.7540 - 0.2910i 0.7543 - 0.2910i 0.7544 - 0.2910i
(0.01% , <0.01%)
2 0.7305 - 0.4909i 0.7298 - 0.4919i 0.7267 - 0.4928i
(-0.42% , 0.18%)
3 0.6999 - 0.6957i 0.6967 - 0.7023i
Acknowledgments
BJ would like to thank Roman Konoplya for his advice and useful discussions during the production of this work.
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