Quasinormal Modes of Kerr Black Holes in Four and Higher Dimensions

0.16 MB·English

by  Hsien-chung Kao

Quasinormal Modes of Kerr Black Holes in Four and Higher Dimensions
Hsien-chung Kao1 and Dan Tomino2
Department of Physics, National Taiwan Normal University, Taipei, Taiwan 116.
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We analytically calculate to leading order the asymptotic form of quasinormal frequencies of
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Kerr black holes in four, five and seven dimensions. All the relevant quantities can be explicitly
q
- expressed in terms of elliptical integrals. In four dimensions, we confirm the results obtained by
r
g Keshest and Hod bycomparing theanalytic results tothe numerical ones.
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[email protected]
[email protected]
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1 Introduction
Perturbation of black holes are known to reveal characteristic damped oscillation modes which domi-
nate the time evolution in certain intermediate period of time [1]. Since the frequencies are complex,
they arecalledquasinormalmodes(QNMs). Forageneralreviewandclassification,seeRefs. [2,3,4].
They depend only on the fundamental parameters of the black holes, such as mass, angular momen-
tum and charge. For a Schwarzschild black hole, an asymptotic formula for high overtones has been
obtained using the monodromy matching method [5]:
1
ω = ln3+(2n 1)πi (1)
n
8πM −
(cid:26) (cid:27)
in the units G=c=h¯ =1. The analytic value ln3 in the real part of the above formula was used to
arguethattherelevantgaugegroupinloopquantumgravityshouldbeSO(3)[6]. Althoughthisresult
turns out to be not universal [5], one still expects QNMs to play an important role in understanding
black holes and quantum gravity.
The monodromy method has been generalized so that first order correction to the asymptotic
form of quasinormal frequencies can also be calculated [7]. In the case of Schwarzschild black holes,
comparisonswithnumericalresultshavebeenmadeandtheagreementis excellent[8,9]. InRef. [10],
thecalculationisfurtherextendedtothesecondorder. Theagreementisnotasgoodasthefirstorder
case. In fact, there seems to be sizable discrepancy in higher angular momentum case (l = 6). More
study is needed to clarify the situation.
Although high overtones of QNMs in spherically symmetric black holes have been extensively
studied, they are not as well studied in the case of rotating black holes. In Ref. [11], convergent
numerical results have been presented. Making use of the monodromy analysis used in Refs. [5, 12],
Keshet and Hod obtain the first analytic results that are in excellent agreement with the numerical
ones[13]. TherearetwodistinctpropertiesofrotatingblackholesthatmakesitshighlydampedQNMs
quite differentfromthatofsphericallysymmetricones. First,there aremorethanone turning points.
Second, there is a term linear in the frequency ω, which contributes to the real part in leading order.
In this paper, we will extend their method to find the asymptotic formula of quasinormal frequency
for higher dimensional Kerr black holes.
2 Calculation of the asymptotic form of quasinormal frequen-
cies
Consider a massless scaler field Φ in the background of a D-dimensional Kerr black hole. Using the
Boyer-Linquist coordinate, we can write the metric as
ds2 = ∆ dt asinθ2dφ 2+ρ2 dr2 +dθ2 + sin2θ adt (r2+a2)dφ 2+r2cos2θdΩ2 . (2)
−ρ2 − ∆ ρ2 − D−4
(cid:20) (cid:21)
(cid:2) (cid:3) (cid:2) (cid:3)
Here,
ρ2 =r2+a2cos2θ, (3)
∆=r2 µr5 D +a2. (4)
−
−
2
dΩ2 is the metric onS /Z , with Ω =(ψ ,...,ψ ) and0 ψ π for all i. The location
D−4 D−4 2 D−4 1 D−4 ≤ i ≤
of an event horizon is determined by ∆ =0. For D = 4, there are both the outer and inner horizons
r = µ/2 µ2/4 a2. For D 5, there is only one horizon. The surface r = r is also a Killing
+
± ± − ≥
horizon. The corresponding Killing vector is ξ = ∂ +Ω ∂ . Here, Ω = a is the angular
p t H φ H r2+a2
+
(D 3)r2+(D 5)a2
velocity on the horizon. From ξ, one can find surface gravity κ = − + − . The ADM mass
2r+(r+2+a2)
is given by M = (D−21)6AπD−2µ, with An = 2πΓ((nn++11)/)2 the area of an n-dimensional unit sphere. The
2
BdSek=en2sκπte(idnM-Ha−wΩkiHndgJe)n,toronpeythisengifivnend bJy=SA=D−412A(8rπ+2D+−a22()ra+2. +a2)r+D−4 [14]. Making use of the identity
It’s well known that Φ satisfies the Klein-Gordon equation:
1
∂ gµν√ g∂ Φ =0. (5)
µ ν
√ g −
−
(cid:8) (cid:9)
Let
Φ(r,t,θ,φ,Ω )=R˜ (r)S (u)Y (Ω )e iωt+imφ, (6)
D−4 lmL lmL L,M1,...,MD−5 D−4 −
where u=cos2θ and Y (Ω ) the spherical harmonics. R (r) and S (u) satisfy
L,M1,...,MD−5 D−4 lmL lmL
1 d dR (r)
rD−4∆ lmL
rD 4dr dr
− (cid:20) (cid:21)
ω2[(r2+a2)2 a2∆] ω[2mµa]+a2[m2 L(L+D 5)∆/r2]
+ − − − − Λ R (x)=0, (7)
rD 5∆ − lmL lmL
(cid:26) − (cid:27)
d2S (u) 1 dS (u)
lmL lmL
u(1 u) + [(D 3) (D 1)u]
− du2 2 − − − du
1 m L(L+D 5)
+ ω2a2u − +Λ S (u)=0, (8)
lmL lmL
4 − 1 u − u
(cid:20) − (cid:21)
respectively. Λ is the separation constant. Define T (u) = u L/2(1 u) m/2S (u) and the
lmL lmL − − lmL
−
angular equation becomes
d2T (u) D 3 D 1 dT (u)
lmL lmL
u(1 u) + ( − +L) ( − +m+L)u
− du2 2 − 2 du
(cid:20) (cid:21)
1
+ ω2a2u (m+L)(m+L+D 3)+Λ T (u)=0. (9)
lmL lmL
4 − −
(cid:2) (cid:3)
When a 0,the solutionto the aboveequationbecomes the hypergeometricfunction F (A,B;C,u)
2 1
→
with
2
1 D 3 D 3
A= m+L+ − − +Λ , (10)
lmL
2 2 −s 2 
(cid:18) (cid:19)
 
2
1 D 3 D 3
B = m+L+ − + − +Λ , (11)
lmL
2 2 s 2 
(cid:18) (cid:19)
 D 3 
C =L+ − . (12)
2
In such limit, T(0) (u) together with eimφ and Y (Ω ) form the spherical harmonics on
lmL L,M1,...,MD−5 D−4
SD 2, and therefore Λ =l(l+D 3). As a result, A= 1(m+L l)= p, where p specifies the
− lmL − 2 − −
number of zeros of T(0) (u) for u in the interval [0,1].
lmL
3
It is straightforward to generalize Flammer’s method to arbitrary dimension D [15]. In the limit
ω i ,
→− ∞
Λ =Λ c+Λ +O(c 1), (13)
lmL 0 1 −
1
T (u)=Z (x)+ Z (x)+O(c 2), (14)
lmL 0 1 −
c
where c=iωa and x=cu. To leading order in c 1, Z (x) satisfies
− 0
1 1
xZ (x)+ (D 3+2L)Z (x) (x Λ )Z (x)=0. (15)
0′′ 2 − 0′ − 4 − 0 0
The solution is given by Z (x) = e x/2L(L) (x), where L(n)(x) is the generalized Laguerre
0 − 14(Λ0−D+3−2L) p
polynomials with p zeros. For Z (x) to have the same number of zeros as T(0) (u), we find
0 lmL
Λ =[2(l m)+(D 3)]. (16)
0
−| | −
This isconsistentwiththe resultinthe D =4 case[11]. LetR (r)=[rD 4∆] 1/2R˜ (r), andthe
lmL − − lmL
radial equation becomes
d2 ω2q (r)+ωq (r)+q (r)
+ 0 1 2 R˜ (r)=0. (17)
dr2 ∆2 lmL
(cid:20) (cid:21)
Here,
q (r)=(r2+a2)2 a2∆=r4+a2r2+µa2r5 D; (18)
0 −
−
q1(r)=m[ 2µar5−D]+Λ0[ ia∆]; (19)
− −
a2∆
q (r)=m2[a2]+L(L+D 5)[− ]+Λ [ ∆]
2 − r2 1 −
(D 2)(D 4)∆2(r) 4(D 5)a2∆ [2a2 (D 3)µr5 D]2
−
− − − − − − − . (20)
− 4r2
Similar to Ref.[13], we define z rV(r )dr with V(r) = ∆ 1(r)[q (r)+ω 1q (r)]1/2. The radial
′ ′ − 0 − 1
≡
equation becomes
R
d2
ω2 V (r) Rˆ (r)=0, (21)
−dz2 − − 1 lmL
(cid:26) (cid:27)
where Rˆ =V1/2R˜ and
q (r) V (r) 3[V (r)]2
2 ′′ ′
V (r)= + . (22)
1 −∆2(r)V2(r) 2V3(r) − 4V4(r)
It has been shown using the monodromy matching method that the condition for quasinormal modes
is given by
1
2iω Vdr =2πi n+ . (23)
2
ZCt,o (cid:18) (cid:19)
Here, C is a contour in the complex plane that runs from t to t lying beyond the outer horizon
t,o 1 2
r =r . t ,t are the transition points in the right half complex plane determined by the condition:
+ 1 2
q (t)+ω 1q (t)=0. (24)
0 − 1
4
ImHrL
t
2
r r
- +
ReHrL
t
1
Figure 1: The Stokes lines (dashed) and anti-Stokes lines (solid) emanting from the turning points t
1
and t in complex r-plane for D = 4 a = 0.5 are shown. r and r are the inner and outer horizon
2 +
−
radii, repectively.
In the limit ω i ,
→− ∞
q (r) q (r)
0 1
2ω Vdr 2ω dr+ dr. (25)
→ ∆ ∆ q (r)
ZCt,o ZCr,o p ZCr,o 0
Making use of the explicit expression for q (r) and following the cponvention in Ref. [13], we have
1
q (r)
0
δ = 2i dr; (26)
0
− ∆
ZCr,o p
µar5 D
−
δ =2i dr; (27)
m
∆ q (r)
ZCr,o 0
a
δ =2i p dr. (28)
Λ
2 q (r)
ZCr,o 0
The sign convention is oppposite to that in Ref. [13]. It is chosen to make δ ,δ and δ all positive.
0 m Λ
The QNM quantization condition in eq (23) becomes
ωδ +mδ +iΛ δ = iπ(2n+1). (29)
0 m 0 Λ
−
To leading order in c 1,
−
ω = i(nδˆ+φˆ) mωˆ, (30)
− −
5
where δˆ=2π/δ ,φˆ=(Λ δ +π)/δ ,ωˆ =δ /δ . So far,ourresultshavebeencompletelygeneraland
0 0 Λ 0 m 0
hold for arbitrary dimensions. It has been pointed out that the integrals for δ ,δ can be expressed
0 m
in terms of elliptical functions for D = 4 [13]. From eqs (26) to (28), it is easy to see that similar
situation also occurs for D =5 and 7. Therefore, we will only focus on these cases hereafter.
Since no explicit expressions for the D = 4 case are given in Ref. [13], we think it is appropriate
to present them here. The explicit form of q (r) is given by
0
q (r)=r4+a2r2+µa2r. (31)
0
There are four transition points. To leading order in c 1, they are r =0, r = 2u , and r =u iv ,
− 1 1 1
− ±
with
a(λ1/3 λ 1/3)
−
u = − , (32)
1
2√3
a(λ1/3+λ 1/3)
−
v = , (33)
1
2
3√3µ 27µ2
λ= + 1+ . (34)
2a 4a2
r
For convenience, define
u1+iv1 r(r+2u )[(r u )2+v2]
f (r ,u ,v )= 1 − 1 1 dr; (35)
0 0 1 1
(r r )
Zu1−iv1 p − 0
u1+iv1 1
f (r ,u ,v )= dr. (36)
m 0 1 1
Zu1−iv1 (r−r0) r(r+2u1)[(r−u1)2+v12]
Using mathematica, we find p
4iu v i(r u )(9u2+v2) 4iu v
f (r ,u ,v )=ir 3u2+v2+2iu v E 1 1 0− 1 1 1 K 1 1
0 0 1 1 0q 1 1 1 1 (cid:20)3u21+v12+2iu1v1(cid:21)− 3u21+v12+2iu1v1 (cid:20)3u21+v12+2iu1v1(cid:21)
i2r (3u +iv )(r u +iv ) 2iv (r +2u ) 4iu v
0 1 1 0 1 1 1 0p 1 1 1
− Π − ,
− 3u21+v12+2iu1v1 (cid:20)(r0−u1−iv1)(3u1−iv1) 3u21+v12+2iu1v1(cid:21)
i(3u +iv )(3u2 v2 2r2) 2iv 4iu v
1p 1 1− 1 − 0 Π − 1 , 1 1 , (37)
− 3u21+v12+2iu1v1 (cid:20)3u1−iv1 3u21+v12+2iu1v1(cid:21)
and p
2
f (r ,u ,v )=
m 0 1 1
r 3u2+v2 2iu v
0 1 1 − 1 1
p 3u2+v2 2iu v 3u2+v2+2iu v 3u2+v2+2iu v
×(F "sin−1 s3u121+v112−+2iu11v11 !,3u211+v112−2iu11v11#−K(cid:20)3u121+v112−2iu11v11(cid:21))
4u
1
−r (r +2u ) 3u2+v2 2iu v
0 0 1 1 1 − 1 1
r (3pu iv ) 3u2+v2 2iu v 3u2+v2+2iu v
×(Π"(r0+02u11)(−u1−1iv1),sin−1 s3u121+v112−+2iu11v11 !,3u121+v112−2iu11v11#
r (3u iv ) 3u2+v2+2iu v
Π 0 1− 1 , 1 1 1 1 . (38)
− (cid:20)(r0+2u1)(u1−iv1) 3u21+v12−2iu1v1(cid:21))
6
Here, E(m),E(ϕ,m),K(m),F(ϕ,m),Π(n,m) and Π(n,ϕ,m) are the elliptical integrals:
π
2
E(m)= [1 msin2θ]1/2dθ, (39)
−
Z0
ϕ
E(ϕ,m)= [1 msin2θ]1/2dθ, (40)
−
Z0
π
2
K(m)= [1 msin2θ]−1/2dθ, (41)
−
Z0
ϕ
F(ϕ,m)= [1 msin2θ] 1/2dθ, (42)
−
−
Z0
π
2
Π(n,m)= [1 nsin2θ]−1[1 msin2θ]−1/2dθ, (43)
− −
Z0
ϕ
Π(n,ϕ,m)= [1 nsin2θ] 1[1 msin2θ] 1/2dθ. (44)
− −
− −
Z0
Note that because of the pole at r =r and branch cuts, there is ambiguity in eqs (37) and (38). To
0
findthe correctanalyticexpression,wecomparetheresultwiththe numericalintegration. After some
trial and error, we find
f (r ,u ,v ) f (r ,u ,v ) i2πµr
δ0 =−2i 0 + 1 r1 −r0 − 1 1 + r r+ θ(3u1r+−3u21−v12) , (45)
(cid:26) +− − +− − (cid:27)
where θ(x) is the step function. The term with a step function is introduced to compensate the
discontinuity caused by the term Π (r+−−u21iv−1i(vr1+)(+32uu11−)iv1),3u21+4vi12u+1v21iu1v1 . The magnitude of jump is
given by the residue of q1/2/∆ at r =h r . Similarly, i
0 +
µa[r f (r ,u ,v ) r f (r ,u ,v )]
+ m + 1 1 m 1 1
δm = 2i − − − − . (46)
− r r
(cid:26) +− − (cid:27)
Note thathere there is no discontinuity andwe do notneedto introduce any termwith stepfunction.
Comparisons to the numerical results from Ref. [11] are given in Fig.2 and 3.
`
ΜΩ(cid:144)2
0.07
0.06
0.05
0.04
0.03
0.02
0.01
2a(cid:144)Μ
0.2 0.4 0.6 0.8 1
Figure 2: Comparison between the analytic and numerical results for the real part of the highly
damped QNM frequency ωˆ(a).
7
`
Μ∆(cid:144)2
0.26
0.258
0.256
0.254
0.252
2a(cid:144)Μ
0.2 0.4 0.6 0.8 1
Figure 3: Comparison between the analytic and numerical results for the level spacing δˆ.
In the extremal limit, a µ/2. As a result, λ √7+√3,u µ, and v √7µ. Using
→ → 2 1 → 4 1 → 4
L’Hospital’s rule, we find
i 1 i√7 7+i5√7 7+i5√7
δ = − 2(3 i√7)E 3(1 i√7)K
0
p 2 ( − (cid:20) 16 (cid:21)− − (cid:20) 16 (cid:21)
7+i√7 7+i5√7 7 i3√7 7+i5√7
2(3+i√7)Π , +2(5 i√7)Π − , 24.52, (47)
− (cid:20) 4 16 (cid:21) − (cid:20) 8 16 (cid:21))≈
i 5+i√7 7+i5√7 7+i5√7
δ = − (5+i√7)E 8K 1.27. (48)
m
p 8 ( (cid:20) 16 (cid:21)− (cid:20) 16 (cid:21))≈
The limit is smooth in constrast to the Reissner-Nordstrom case as pointed out in Ref. [13]. In the
Schwarzschild limit a 0, even though the asymptotic QNMs are not continuous the level spacing
→
does goes to the Schwarzschildresult δˆ=(D 3)/(2µ) smoothly.
−
For D =5,
q (r)=r4+a2r2+µa2. (49)
0
There are four transition points: r = u iv , with
1 1
± ±
õ 1
u =a , (50)
1
2a − 4
r
õ 1
v =a + , (51)
1
2a 4
r
µ
r =a 1. (52)
+ a2 −
r
Here, we have
4πµ u +iv (u iv )2 (u iv )2
δ0 = 2r +2i(u1+iv1) E sin−1 u1 iv1 ,(u1+−iv1)2 −E (u1+−iv1)2
(cid:18) +(cid:19) (cid:26) (cid:20) (cid:18) 1− 1(cid:19) 1 1 (cid:21) (cid:20) 1 1 (cid:21)(cid:27)
2i r+2 −(u1−iv1)2 F sin−1 u1+iv1 ,(u1−iv1)2 K (u1−iv1)2
− (u +iv ) u iv (u +iv )2 − (u +iv )2
(cid:2) 1 1 (cid:3)(cid:26) (cid:20) (cid:18) 1− 1(cid:19) 1 1 (cid:21) (cid:20) 1 1 (cid:21)(cid:27)
2iµ2 (u iv )2 u +iv (u iv )2 (u iv )2 (u iv )2
+r2(u +iv ) Π 1−r2 1 ,sin−1 u1 iv1 ,(u1+−iv1)2 −Π 1−r2 1 ,(u1−+iv1)2 . (53)
+ 1 1 (cid:26) (cid:20) + (cid:18) 1− 1(cid:19) 1 1 (cid:21) (cid:20) + 1 1 (cid:21)(cid:27)
8
4πa
δ = −
m
2r
(cid:18) + (cid:19)
−2i√µ(ur21−iv1) Π (u1−r2iv1)2,sin−1 uu1+iivv1 ,((uu1+−iivv1))22 −Π (u1−r2iv1)2,((uu1−+iivv1))22 . (54)
+ (cid:26) (cid:20) + (cid:18) 1− 1(cid:19) 1 1 (cid:21) (cid:20) + 1 1 (cid:21)(cid:27)
The results are shown in Fig. 4 and 5. In contrst to the D = 4 case, there is a critical value
a 0.678µ1/2. When a > a , ωˆ changes sign. The physical meaning is not clear at the moment. It
c c
≈
might be related to the ”algebraically special” frequencies [16].
Μ1(cid:144)2Ω`
0.2
0.1
a(cid:144)Μ1(cid:144)2
0.2 0.4 0.6 0.8 1
-0.1
-0.2
Figure4: TheanalyticresultfortherealpartofthehighlydampedQNMfrequencyωˆ(a)whenD =5.
`
Μ1(cid:144)2∆
1.2
1
0.8
0.6
0.4
0.2
a(cid:144)Μ1(cid:144)2
0.2 0.4 0.6 0.8 1
Figure 5: Analytic result for level spacing δˆwhen D =5.
For D =7,
q0(r)=r4+a2r2+µa2r−2. (55)
There are six transition points: r = u iv and iw . It is convenient to introduce the variable
1 1 1
± ± ±
9
y =r2. Consequently,
u1+iv1 r r6+a2r4+µa2 u2+iv2 (y+w )[(y u )2+v2]
dr = 2 − 2 2 dy, (56)
r4+a2r2 µ 2(y y )(y y )
Zu1−iv1 p − Zu2−iv2 p − + − −
u1+iv1 µar
− dr
Zu1−iv1 (r4+a2r2−µ) r6+a2r4+µa2
u2+iv2 p µa
= − dy. (57)
Zu2−iv2 2(y−y+)(y−y−) (y+w2)[(y−u2)2+v22]
Here, p
a2
u = (λ1/3+λ 1/3 2), (58)
2 −
6 −
√3a2
v = (λ1/3 λ 1/3), (59)
2 −
6 −
a2
w = (λ1/3+λ 1/3+1), (60)
2 −
3
27µ µ 27µ2
λ=1+ +3√3 + , (61)
2a4 a4 4a8
r
a2 a4
y = − µ+ . (62)
± 2 ±r 4
Similarly, define
u2+iv2 (y+w )[(y u )2+v2]
f (y ,u ,v ,w )= 2 − 2 2 dy, (63)
0 0 2 2 2
y y
Zu2−iv2 p − 0
u2+iv2 1
f (y ,u ,v ,w )= dy. (64)
m 0 2 2 2
Zu2−iv2 (y−y0) (y+w2)[(y−u2)2+v22]
We find p
2
f (y ,u ,v ,w )= √u +iv +w (2u w y )
0 0 2 2 2 2 2 2 2 2 0
3 − −
u +iv +w u iv +w u iv +w
E sin−1 2 2 2 , 2− 2 2 E 2− 2 2
×( " ru2−iv2+w2 ! u2+iv2+w2#− (cid:20)u2+ıv2+w2(cid:21))
2 1
+ [3y2 3(u iv w )y +(2v2 2iu v +iv w 3u w )]
3√u +iv +w 0− 2− 2− 2 0 2 − 2 2 2 2− 2 2
2 2 2
u +iv +w u iv +w u iv +w
F sin−1 2 2 2 , 2− 2 2 K 2− 2 2
×( " ru2−iv2+w2 ! u2+iv2+w2#− (cid:20)u2+ıv2+w2(cid:21))
1
2 [(y u )2+v2]
− √u +iv +w 0− 2 2
2 2 2
u iv +w u +iv +w u iv +w
Π 2− 2 2,sin−1 2 2 2 , 2− 2 2
×( " w2+y0 ru2−iv2+w2 ! u2+iv2+w2#
u iv +w u iv +w
2 2 2 2 2 2
Π − , − , (65)
− (cid:20) w2+y0 u2+ıv2+w2(cid:21))
10