We study analytically the fundamental resonances of near-extremal, slowly
rotating Kerr-Newman black holes. We find a simple analytic expression for
these black-hole quasinormal frequencies in terms of the black-hole physical
parameters: omega=m Omega-2i pi T(l+1+n), where T and Omega are the temperature
and angular velocity of the black hole. The mode parameters l and m are the
spheroidal harmonic index and the azimuthal harmonic index of a co-rotating
mode, respectively. This analytical formula is valid in the regime Im omega <<
Re omega <<1/M, where M is the black-hole mass.Comment: 4 page

Abramowitz

Berti

Chandrasekhar

Davis

de la Cruz

Detweiler

Flammer

Keshet

Leaver

Morse

Nollert

Page

Press

Regge

Ronveaux

Ruffini

Shahar Hod

Teukolsky

Vishveshwara

Physics Letters B

English

arXiv

AbstractWe study analytically the fundamental resonances of near-extremal, slowly rotating Kerr–Newman black holes. We find a simple analytic expression for these black-hole quasinormal frequencies in terms of the black-hole physical parameters: ω=mΩ−2iπTBH(l+1+n), where TBH and Ω are the temperature and angular velocity of the black hole. The mode parameters l and m are the spheroidal harmonic index and the azimuthal harmonic index of a co-rotating mode, respectively. This analytical formula is valid in the regime ℑω≪ℜω≪M−1, where M is the black-hole mass

Hod, Shahar

Elsevier - Publisher Connector 

Physics Letters B 666 (2008) 483–485Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Quasinormal resonances of near-extremal Kerr–Newman black holes
Shahar Hod a,b,∗
a The Ruppin Academic Center, Emeq Hefer 40250, Israel
b The Hadassah Institute, Jerusalem 91010, Israel
a r t i c l e i n f o a b s t r a c t
Article history:
Received 5 June 2008
Accepted 1 August 2008
Available online 6 August 2008
Editor: B. Grinstein
We study analytically the fundamental resonances of near-extremal, slowly rotating Kerr–Newman black
holes. We ﬁnd a simple analytic expression for these black-hole quasinormal frequencies in terms of the
black-hole physical parameters: ω =mΩ − 2iπ TBH(l+ 1+n), where TBH and Ω are the temperature and
angular velocity of the black hole. The mode parameters l and m are the spheroidal harmonic index and
the azimuthal harmonic index of a co-rotating mode, respectively. This analytical formula is valid in the
regime ω  ω  M−1, where M is the black-hole mass.
© 2008 Elsevier B.V. All rights reserved.The statement that black holes have no hair was introduced by
Wheeler [1] in the early 1970’s. The various no-hair theorems state
that the external ﬁeld of a dynamically formed black hole (or a
perturbed black hole) relaxes to a Kerr–Newman spacetime, char-
acterized solely by three parameters: the black-hole mass, charge,
and angular momentum. This implies that perturbation ﬁelds left
outside the black hole would either be radiated away to inﬁnity, or
be swallowed by the black hole.
This relaxation phase in the dynamics of perturbed black holes
is characterized by ‘quasinormal ringing’, damped oscillations with
a discrete spectrum (see, e.g., [2] for a detailed review). At late
times, all perturbations are radiated away in a manner reminis-
cent of the last pure dying tones of a ringing bell [3–6]. Being
the characteristic ‘sound’ of the black hole itself, these free oscilla-
tions are of great importance from the astrophysical point of view.
They allow a direct way of identifying the spacetime parameters
(the mass, charge, and angular momentum of the black hole). This
fact has motivated a ﬂurry of research during the last four decades
aiming to compute the quasinormal mode (QNM) spectrum of var-
ious types of black-hole spacetimes [2].
The dynamics of black-hole perturbations is governed by the
Regge–Wheeler equation [7] in the case of a spherically symmet-
ric Schwarzschild black hole, and by the Teukolsky equation [8]
for rotating Kerr–Newman spacetimes. The black hole QNMs corre-
spond to solutions of the wave equations with the physical bound-
ary conditions of purely outgoing waves at spatial inﬁnity and
purely ingoing waves crossing the event horizon [9]. Such bound-
ary conditions single out a discrete set of black-hole resonances
{ωn} (assuming a time dependence of the form e−iωt ). (In analogy
* Correspondence address: The Ruppin Academic Center, Emeq Hefer 40250, Is-
rael.
E-mail address: shaharhod@gmail.com.0370-2693/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.physletb.2008.08.002with standard scattering theory, the QNMs can be regarded as the
scattering resonances of the black-hole spacetime. They thus cor-
respond to poles of the transmission and reﬂection amplitudes of
a standard scattering problem in a black-hole spacetime.)
Since the perturbation ﬁeld can fall into the black hole or ra-
diate to inﬁnity, the perturbation decays and the corresponding
QNM frequencies are complex. It turns out that there exist an inﬁ-
nite number of quasinormal modes, characterizing oscillations with
decreasing relaxation times (increasing imaginary part) [10]. The
mode with the smallest imaginary part (known as the fundamen-
tal mode) determines the characteristic dynamical timescale τ for
generic perturbations to decay.
In this work we determine analytically the fundamental (least-
damped) resonant frequencies of rotating Kerr–Newman black
holes. (For a recent progress in the study of the highly-damped res-
onances, see [11,12].) The spectrum of quasinormal resonances can
be studied analytically in the slow rotation, near-extremal limit
(M2 − Q 2 − a2)1/2  a  M , where M , Q , and a and the mass,
charge, and angular momentum per unit mass of the black hole,
respectively. In order to determine the black-hole resonances we
shall analyze the scattering of massless scalar and neutrino waves
in the Kerr–Newman spacetime [13].1 The dynamics of a pertur-
bation ﬁeld Ψ in the Kerr–Newman spacetime is governed by the
Teukolsky equation [8]. One may decompose the ﬁeld as (we use
natural units in which G = c = h¯ = 1)
Ψslm(t, r, θ,φ) = eimφ Sslm(θ;aω)ψslm(r;aω)e−iωt , (1)
where (t, r, θ,φ) are the Boyer–Lindquist coordinates, ω is the
(conserved) frequency of the mode, l is the spheroidal harmonic
1 It is worth mentioning that all attempts to decouple the gravitational and elec-
tromagnetic perturbations of the Kerr–Newman spacetime have failed so far, see,
e.g., [13].
484 S. Hod / Physics Letters B 666 (2008) 483–485index, and m is the azimuthal harmonic index with −l  m  l.
The parameter s is called the spin weight of the ﬁeld, and is
given by s = ± 12 for massless neutrino perturbations, and s = 0 for
scalar perturbations. (We shall henceforth omit the indices s, l,m
for brevity.) With the decomposition (1), ψ and S obey radial and
angular equations, both of conﬂuent Heun type [14,15], coupled by
a separation constant A(aω).
The angular functions S(θ;aω) are the spin-weighted spher-
oidal harmonics which are solutions of the angular equation [8,15]
1
sin θ
∂
∂θ
(
sin θ
∂ S
∂θ
)
+
[
a2ω2 cos2 θ − 2aωs cos θ
− (m + s cos θ)
2
sin2 θ
+ s + A
]
S = 0. (2)
The angular functions are required to be regular at the poles
θ = 0 and θ = π . These boundary conditions pick out a dis-
crete set of eigenvalues labeled by an integer l. In the aω  1
limit these angular functions become the familiar spin-weighted
spherical harmonics with the corresponding angular eigenvalues
A = l(l + 1) − s(s + 1) + O (aω).2
The radial Teukolsky equation is given by
−s d
dr
(
s+1 dψ
dr
)
+
[
K 2 − 2is(r − M)K

− a2ω2 + 2maω − A + 4isωr
]
ψ = 0, (3)
where  ≡ r2−2Mr+Q 2+a2 and K ≡ (r2+a2)ω−am. The zeroes
of , r± = M ± (M2 − Q 2 − a2)1/2, are the black hole (event and
inner) horizons.
For the scattering problem one should impose physical bound-
ary conditions of purely ingoing waves at the black-hole horizon
and a mixture of both ingoing and outgoing waves at inﬁnity
(these correspond to incident and scattered waves, respectively).
That is,
ψ ∼
{
e−iωy +R(ω)eiωy as r → ∞ (y → ∞),
T (ω)e−i(ω−mΩ)y as r → r+ (y → −∞),
(4)
where the “tortoise” radial coordinate y is deﬁned by dy = [(r2 +
a2)/]dr. Here Ω ≡ a
r2++a2
is the angular velocity of the black-
hole horizon. The coeﬃcients T (ω) and R(ω) are the transmission
and reﬂection amplitudes for a wave incident from inﬁnity. The
discrete black-hole resonances are the poles of these transmission
and reﬂection amplitudes. (The pole structure reﬂects the fact that
the QNMs correspond to purely outgoing waves at spatial inﬁnity.)
These resonances determine the ringdown response of a black hole
to outside perturbations.
The transmission and reﬂection amplitudes satisfy the usual
probability conservation equation |T (ω)|2 + |R(ω)|2 = 1. The cal-
culation of these scattering amplitudes in the low frequency limit,
ω  ω  M−1, is a common practice in the physics of black
holes, see, e.g., [13,16,17] and references therein. Deﬁne
x ≡ r − r+
r+ − r− ,  ≡
ω −mΩ
4π TBH
, k ≡ ω(r+ − r−), (5)
where TBH = (r+−r−)4π(r2++a2) is the Bekenstein–Hawking temperature of
the black hole. Then a solution of Eq. (3) obeying the ingoing
boundary conditions at the horizon (r → r+ , kx  1) is given by
[18,19]
2 In the small aω limit we shall take A = l(l + 1) − s(s + 1) in Eq. (3), where l is
nearly an integer with a small correction of order O (aω).ψ = x−s−i (x+ 1)−s+i 2F1(−l − s, l − s + 1;1− s − 2i ;−x),
(6)
where 2F1(a,b; c; z) is the hypergeometric function. In the asymp-
totic (r 
 M , x 
 | | + 1) limit one ﬁnds the solution [18,19]
ψ = C1e−ikxxl−s1F1(l − s + 1;2l + 2;2ikx)
+ C2e−ikxx−l−s−11F1(−l − s;−2l;2ikx), (7)
where 1F1(a; c; z) is the conﬂuent hypergeometric function. The
coeﬃcients C1 and C2 can be determined by matching the two
solutions in the overlap region | | + 1 x  1/k. This yields
C1 = (2l + 1)(1− s − 2i)
(l − s + 1)(l + 1− 2i), (8)
and
C2 = (−2l − 1)(1− s − 2i)
(−l − s)(−l − 2i) . (9)
Finally, the asymptotic form of the conﬂuent hypergeometric func-
tions [18,19] can be used to write the solution in the form given
by Eq. (4). After some algebra one ﬁnds
∣∣T (ω)∣∣2 = 
{[
(l − s)!(l + s)!
(2l)!(2l + 1)!
]2
(l + 1− 2i)
(−l − 2i) (2ik)
2l+1
}
, (10)
for the transmission probability.
The quasinormal frequencies are the scattering resonances
of the black-hole spacetime. They thus correspond to poles of
the transmission and reﬂection amplitudes. Taking cognizance of
Eq. (10) and using the well-known pole structure of the Gamma
functions [19], one ﬁnds the resonance condition l+1−2i = −n,
where n 0 is a non-negative integer. This yields a simple formula
for the black-hole resonances:
ω =mΩ − 2iπ TBH(l + 1+ n), (11)
in the near-extremal limit. It is worth emphasizing again that this
formula is valid in the ω  ω  M−1 regime. This requires
(M2 − Q 2 − a2)1/2  a  M and m > 0.3
In summary, we have studied analytically the quasinormal
mode spectrum of near-extremal, slowly rotating Kerr–Newman
black holes. It was shown that the fundamental resonances can
be expressed in terms of the black-hole physical parameters: the
temperature TBH, and the horizon angular velocity Ω .
The fundamental resonances are expected to dominate the re-
laxation dynamics of a perturbed black-hole spacetime. Taking cog-
nizance of Eq. (11), one realizes that in the near-extremal limit
ω approaches zero linearly with the black-hole temperature TBH
for all modes co-rotating with the black hole (i.e., modes having
m > 0). We therefore conclude that the characteristic relaxation
timescale τ ∼ 1/ω of the black hole is of the order of O (T−1BH ).4
3 Taking cognizance of Eq. (10), one ﬁnds that the total reﬂection modes (TRM),
which are characterized by the condition T (ω) = 0, are given by the requirement
1/(−l − 2i) = 0, that is −l − 2i = −n. We recall that l, as deﬁned from the
relation A = l(l + 1) − s(s + 1) [where the separation constants {A} themselves are
obtained from Eq. (2)], is nearly an integer with a small correction of order O (aω).
This implies that each quasinormal frequency is separated from a nearby total re-
ﬂection frequency by a small term of order O (aω). Thus, in order to determine the
QNMs in numerical calculations (and distinguish them from the TRMs), one would
have to use numerical schemes of very high precision.
4 We note that a spherically symmetric Schwarzschild black hole has only one
time/length scale—its horizon radius, r+ (or equivalently, its mass M). One therefore
expects to ﬁnd τ ∼ r+ (and ω ∼ r−1+ ) on dimensional grounds. On the other hand,
Kerr–Newman black holes have an additional lengthscale—the black-hole inverse
temperature T−1BH . Here we have established that the relevant relaxation timescale
of the perturbed black hole is determined by its inverse temperature, T−1BH , and
not by its horizon radius r+ . We emphasize that T−1BH is much larger than r+ in
S. Hod / Physics Letters B 666 (2008) 483–485 485It is worth emphasizing that this result, τ ∼ T−1BH , is in accord
with the spirit of the recently proposed universal relaxation bound
[20,21].
Acknowledgements
This research is supported by the Meltzer Science Foundation.
I thank Liran Shimshi, Yael Oren and Clovis Hoppman for helpful
discussions.
References
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Quasinormal resonances of near-extremal Kerr–Newman black holes

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Quasinormal resonances of near-extremal Kerr-Newman black holes

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