Looking at the GregoryLaflamme instability through quasinormal modes
ABSTRACT We study evolution of gravitational perturbations of black strings. It is well known that for all wave numbers less than some threshold value, the black string is unstable against the scalar type of gravitational perturbations, which is named the GregoryLaflamme instability. Using numerical methods, we find the quasinormal modes and timedomain profiles of the black string perturbations in the stable sector and also show the appearance of the GregoryLaflamme instability in the time domain. The dependence of the black string quasinormal spectrum and latetime tails on such parameters as the wave vector and the number of extra dimensions is discussed. There is numerical evidence that at the threshold point of instability, the static solution of the wave equation is dominant. For wave numbers slightly larger than the threshold value, in the region of stability, we see tiny oscillations with very small damping rate. While, for wave numbers slightly smaller than the threshold value, in the region of the GregoryLaflamme instability, we observe tiny oscillations with very small growth rate. We also find the level crossing of imaginary part of quasinormal modes between the fundamental mode and the first overtone mode, which accounts for the peculiar time domain profiles.

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ABSTRACT: We study the stability of $D \geq 7$ asymptotically flat black holes rotating in a single twoplane against tensortype gravitational perturbations. The extensive search of quasinormal modes for these black holes did not indicate any presence of growing modes, implying the stability of simply rotating MyersPerry black holes against tensortype perturbations. Comment: 9 pages, 10 figures, RevTeXPhysical review D: Particles and fields 04/2009;  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, the quasinormal modes (QNMs) of electromagnetic field perturbation to asymptotic safe (AS) black hole are discussed. Through sixorder WKB approach we investigate the effects of quantum correction to the quasinormal modes (QNMs) numerically. Meanwhile by means of finite difference method, the evolutions of such perturbation to the safe black hole are figured out with corresponding parameters. It is found that the stability of black hole remains although the decay frequency and damping speed of oscillations are respectively increased and lowered by the quantum correction to classic Schwarzschild black hole.International Journal of Theoretical Physics 52(5). · 1.09 Impact Factor  SourceAvailable from: Alexander Zhidenko[Show abstract] [Hide abstract]
ABSTRACT: Perturbations of black holes, initially considered in the context of possible observations of astrophysical effects, have been studied for the past ten years in string theory, braneworld models and quantum gravity. Through the famous gauge/gravity duality, proper oscillations of perturbed black holes, called quasinormal modes (QNMs), allow for the description of the hydrodynamic regime in the dual finite temperature field theory at strong coupling, which can be used to predict the behavior of quarkgluon plasmas in the nonperturbative regime. On the other hand, the braneworld scenarios assume the existence of extra dimensions in nature, so that multidimensional black holes can be formed in a laboratory experiment. All this stimulated active research in the field of perturbations of higherdimensional black holes and branes during recent years. In this review recent achievements on various aspects of black hole perturbations are discussed such as decoupling of variables in the perturbation equations, quasinormal modes (with special emphasis on various numerical and analytical methods of calculations), latetime tails, gravitational stability, AdS/CFT interpretation of quasinormal modes, and holographic superconductors. We also touch on stateoftheart observational possibilities for detecting quasinormal modes of black holes.Review of Modern Physics 02/2011; 83(3). · 44.98 Impact Factor
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arXiv:0807.1897v2 [hepth] 16 Jul 2008
Looking at the GregoryLaflamme instability through quasinormal modes
R. A. Konoplya,∗Keiju Murata,†and Jiro Soda‡
Department of Physics, Kyoto University, Kyoto 6068501, Japan
A. Zhidenko§
Instituto de F´ ısica, Universidade de S˜ ao Paulo
C.P. 66318, 05315970, S˜ ao PauloSP, Brazil
We study evolution of gravitational perturbations of black strings. It is well known that for
all wavenumber less than some threshold value, the black string is unstable against scalar type
of gravitational perturbations, which is named the GregoryLaflamme instability. Using numerical
methods, we find the quasinormal modes and timedomain profiles of the black string perturbations
in the stable sector and also show the appearance of the GregoryLaflamme instability in the time
domain. The dependence of the black string quasinormal spectrum and late time tails on such
parameters as the wave vector and the number of extra dimensions is discussed. There is a numer
ical evidence that in the threshold point of instability the static solution of the wave equation is
dominant. For wavenumbers slightly larger than the threshold value, in the region of stability, we
see tiny oscillations with very small damping rate. While, for wavenumbers slightly smaller than
the threshold value, in the region of the GregoryLaflamme instability, we observe tiny oscillations
with very small growth rate. We also find the level crossing of imaginary part of quasinormal modes
between the fundamental mode and the first overtone mode, which accounts for the peculiar time
domain profiles.
PACS numbers: 04.30.Nk,04.50.+h
I. INTRODUCTION
Unlike four dimensional Einstein gravity, which allows
existence of black holes, higher dimensional theories, such
as the braneworld scenarios and string theory, allow ex
istence of a number of ”black” objects: higher dimen
sional black holes, black strings and branes, black rings
and saturns and others. In higher than four dimensions
we lack the uniqueness theorem, so that stability may
be the criteria which will select physical solutions among
this variety of solutions. Up to now, we know that higher
dimensional ReissnerNordstr¨ omde Sitter black holes are
stable [1] in the Einstein gravity. On the contrary, black
holes in GaussBonnet (GB) gravity are unstable for large
GB coupling for D = 5,6 [2], where D is the total number
of spacetime dimensions. KaluzaKlein black holes with
squashed horizon are stable against lowest zero mode
perturbations [3]. Unlike, KaluzaKlein black holes, the
black string metric is a solution of the Einstein equations
in five or higher dimensional gravity that has a factor
ized form consisting of the Tangherlini black hole and an
extra flat dimension [6]. According to the braneworld
scenarios, if the matter localised on the brane undergoes
gravitational collapse, a black hole with the horizon ex
tended to the transverse extra direction will form. This
object looks like a black hole on the brane, but is, in fact,
∗Electronic address: konoplya˙roma@yahoo.com
†Electronic address: murata@tap.scphys.kyotou.ac.jp
‡Electronic address: jiro@tap.scphys.kyotou.ac.jp
§Electronic address: zhidenko@fma.if.usp.br
a black string in the full Ddimensional theory.
It is well known that such black strings suffer from the
socalled GregoryLaflamme instability, which is the long
wavelength gravitational instability of the scalar type of
the metric perturbations [4], [5]. The GregoryLaflamme
instability has been intensively studied during the recent
decade [6] and the threshold values of the wave vector k
at which the instability appears are known [7]. In the
present paper we are aimed at studying the evolution
of linear perturbations of Ddimensional black strings
in time and frequency domains. This task is motivated
mainly by the two reasons: first to realize what happens
on the edge of instability of black strings and how the
perturbations will develop in time. Second, the socalled
quasinormal modes of a stable black string might be an
observational characteristic for the future Large Hadron
Collider Experiments, if such objects as black strings ex
ist.
The latter needs some more explanation. In this re
search, we shall show that, if a stable Ddimensional
black string is gravitationally perturbed, it will undergo
damped oscillations, called quasinormal ringing, similar
to that of a black hole [8]. At asymptotically late time,
this quasinormal ringing goes over into the powerlaw
tails. The quasinormal modes and asymptotic tails are
very well studied for Ddimensional black holes [10] and
for black holes localised on the brane [11]. We find here
that the quasinormal ringing of black strings has a num
ber of differences from that of Ddimensional black holes
[10], especially when approaching the edge of instabil
ity. In particular, we find a numerical evidence that in
the threshold point of instability, the static solution of
the wave equation is dominant. In the region of the sta
Page 2
2
bility, for k slightly larger than the threshold value, we
see modes with tiny oscillation frequencies and damping
rates. In the region of the GregoryLaflamme instability,
for k slightly smaller than threshold value, we observe
tiny oscillations and instability growth. We also find the
level crossing of imaginary part of quasinormal modes
between the fundamental mode and the first overtone
mode.
The paper is organized as follows. Sec II gives ba
sic formulas for the black string perturbations and the
wave equation for the scalar type of gravitational per
turbations. Sec III analyses quasinormal modes of black
strings in frequency domain with the help of the Frobe
nius method, and in time domain by the GundlahPrice
Pullin method. We also discuss features of the Gregory
Laflamme instability using the obtained results. The fi
nal section is devoted to the conclusion.
II.BASIC FORMULAE
In this section, we shall briefly review the results of
the paper [7], where the wave equation for the scalar
type of gravitational perturbations were obtained. This
wave equation will be our starting point for numerical
investigation.
For the static string in D = n + 4 spacetime dimen
sions, the background metric can be written as
ds2= gµνdxµdxν= −f(r)dt2+dr2
f(r)+ r2dΩ2
n+1+ dz2,
(1)
where
f(r) = 1 −
?r+
r
?n
,
and dΩ2
properties of black string have been extensively studied
in recent years and we refer a reader to the papers [12]
[27] for more detailed information on black strings.
The zdirection is periodically identified by the relation
z = z + 2πR. We study perturbations of an (n + 1)
spherically symmetric solution with the Killing vector in
zdirection. Therefore, we can write perturbations in the
form
n+1is the metric on a unit (n+1)sphere. Various
δgµν= eikzaµν(t,r),k =m
R,
m ∈ Z.
The perturbed vacuum Einstein equations have the form
δRµν= 0 (2)
The perturbations can be reduced to the form, where the
only nonvanishing components of aµν are
att= ht,arr= hr,azz= hz,
atr=˙hv,azr= −ikhv.
The linearized Einstein equations give a set of cou
pled equations determining the four radial profiles above.
However, we may eliminate hv, hr and ht from these
equations in order to produce a single second order equa
tion for hz:
¨hz= f(r)2h′′
z+ p(r)h′
z+ q(r)hz,(3)
where
p(r) =
f(r)2
r
?
1 +
n
f(r)−
4(2 + n)k2r2
2k2r2+ n(n + 1)(r+/r)n
?
,
q(r) = −k2f(r)2k2r2− n(n + 3)(r+/r)n
2k2r2+ n(n + 1)(r+/r)n.
Defining
hz(t,r) =
r−(n−1)/2
2k2r2+ n(n + 1)(r+/r)nΨ(t,r),
we can reduce the equation (3) to the wavelike equation
?∂2
dr
f(r)is the tortoise coordinate. Here, the
effective potential V (r) is given by
∂t2−∂2
∂r2
⋆
+ V (r)
?
Ψ = 0, (4)
where dr⋆ =
V (r) =f(r)
4r2
U(r)
(2k2r2+ n(n + 1)(r+/r)n)2,
where
U(r) = 16k6r6+ 4k4r4(n + 5)(3f(r) − 2n + 3nf(r)) −
−4k2r2n(n+1)?n(n+5) + f(r)(2n2+7n+9)??r+
−n2(n + 1)3(f(r) − 2n + nf(r))
The above effective potential does not vanish at asymp
totic infinity but has an effective ”mass” term, containing
k, at the spatial infinity.
r
?n
−
?r+
r
?2n
.
III.EVOLUTION OF PERTURBATIONS
ANALYZED WITH THE FROBENIUS METHOD
AND TIME DOMAIN INTEGRATION
TECHNIQUE
First of all, let us briefly describe the two methods
which we used here: the Frobenius method (frequency
domain) and the GundlachPricePullin method (time
domain).
In time domain, we study the black string ringing using
a numerical characteristic integration method [28], that
uses the lightcone variables u = t−r⋆and v = t+r⋆. In
the characteristic initial value problem, initial data are
specified on the two null surfaces u = u0 and v = v0.
The discretization scheme we used, is
Ψ(N) = Ψ(W) + Ψ(E) − Ψ(S) −
−∆2V (W)Ψ(W) + V (E)Ψ(E)
(5)
8
+ O(∆4) ,
Page 3
3
TABLE I: Fundamental mode (ω0) found by the timedomain
integration, first (ω1) and second (ω2) overtones of spherically
symmetric black string perturbations (n = 1) found by the
Frobenius method. As k grows, the first overtone decreases
its imaginary part, becoming the fundamental mode, reaching
purely real frequency (quasiresonance) at k ≈ 0.94 and then
disappearing. The second overtone, as well as the fundamen
tal purely imaginary mode, increases its damping rate with
k.
kω0 (td)ω1 (Frob.)
0.84 +0.011i 0.802 − 0.0173i
0.85 +0.008i 0.810 − 0.0157i
0.86 +0.005i 0.818 − 0.0140i
0.87 +0.002i 0.827 − 0.0124i
0.88 −0.001i 0.835 − 0.0107i
0.89 −0.004i 0.844 − 0.0090i
0.90 −0.007i 0.852 − 0.0074i
0.91 −0.011i 0.861 − 0.0058i
0.92 −0.014i 0.870 − 0.0042i
0.93 −0.017i 0.879 − 0.0025i
0.94 −0.021i 0.888 − 0.0007i
kω2 (Frob.)
0.8 0.361 − 0.632i
0.9 0.373 − 0.679i
1.0 0.378 − 0.724i
1.1 0.374 − 0.767i
1.2 0.364 − 0.806i
1.3 0.347 − 0.841i
1.4 0.324 − 0.870i
1.5 0.294 − 0.894i
where we have used the following definitions for the
points: N = (u+∆,v+∆), W = (u+∆,v), E = (u,v+∆)
and S = (u,v).
In frequency domain we used the wellknown Frobe
nius method [29]. In order to study the QN spectrum
in frequency domain, we have separated time and radial
coordinates in (3)
hz(t,r) = e−iωthω(r).
Here hω(r) satisfies the quasinormal mode boundary con
ditions, which are purely ingoing wave at the event hori
zon and purely outgoing wave at the spatial infinity.
Thus, the appropriate Frobenius series are
hω(r) =
?
1 −r+
r
?−iωr+/n
ei√ω2−k2rr(n+3+α)/2y(r),
(6)
where α =i(2ω2− k2)r+
√ω2− k2
for n = 1 and α = 0 for n > 1.
It is crucial that y(r) must be regular at the event horizon
and at the spatial infinity and can be expanded as
y(r) =
∞
?
i=0
ai
?
1 −r+
r
?i
.
Substituting (6) into (3), we have found that the coeffi
cients aisatisfy a (3n + 5)term recurrence relation. We
found the coefficients of the recurrence relation, and then
we obtained the equation with the infinite continued frac
tion, which is algebraic equation with respect to the QN
frequency ω. Numerical solutions of this algebraic equa
tion give us the QN spectrum.
Now let us discuss the obtained results for QN modes
and time domain profiles. The Frobenius method for the
considered cumbersome potential gives rise to a num
ber of technical difficulties: first, the convergence of the
TABLE II: Dominant QNMs of spherically symmetric black
string perturbations (n = 2) found with the timedomain and
the Frobenius method. For k ≥ 1.4 the ω0 mode is not domi
nating anymore and is difficult for detection by the time do
main integration.
kω0(td)ω1(td)
1.3 −0.012i 1.418 − 0.141i 1.396 − 0.118i
1.4 −0.047i 1.491 − 0.099i 1.483 − 0.107i
1.5n/a1.584 − 0.095i 1.570 − 0.096i
1.6n/a1.674 − 0.088i 1.659 − 0.086i
1.7 n/a1.763 − 0.081i
1.8n/a1.852 − 0.077i
1.9n/a1.936 − 0.054i
2.0n/a2.032 − 0.046i
ω1 (Frobenius)
n/a
n/a
n/a
n/a
TABLE III: Dominant QNMs of spherically symmetric black
string perturbations (n = 3) found with the timedomain
method. For k ≥ 2 the ω1 mode is a dominating one.
kω0
1.6 −0.007i 1.869 − 0.214i
1.7 −0.043i 1.930 − 0.211i
1.8 −0.082i 2.013 − 0.212i
1.9 −0.130i 2.106 − 0.185i
2.0 −0.250i 2.223 − 0.179i
2.1n/a2.309 − 0.172i
2.2n/a2.400 − 0.167i
2.3n/a2.472 − 0.161i
ω1
Frobenius series is rather slow. Second, when searching
for the solutions of the algebraic equation in the region
close to the threshold of instability, one needs very good
initial guess for ω to ”fall” into the minimum of the con
tinued fraction equation. This can be easily understood.
As we shall see from the time domain integration, the
dominant solution in the threshold point corresponds to
some static solution with ω = 0, so that nearby funda
mental mode has tiny real and imaginary parts. The
Frobenius method naturally requires slow convergence
and excellent initial guess for ω for such small ω.
In tables I, we have listed fundamental mode ω0, first
ω1and second ω2overtone of spherically symmetric black
string perturbations in the case of n = 1. First, we notice
the level crossing of imaginary part of quasinormal modes
between the fundamental mode and the first overtone
mode. This level crossing is peculiar to the black strings.
As k grows, the first overtone ω1decreases its imaginary
part, becoming the fundamental mode, reaching purely
real frequency (quasiresonance) at k ≈ 0.94 and then
disappearing. The second overtone, as well as the fun
damental purely imaginary mode, increases its damping
rate with k.
In table II, we listed dominant QNMs of spherically
symmetric black string perturbations in the case of n = 2.
We can see that the fundamental mode near the threshold
of instability has no real part. Yet, higher overtones have
detectable real part.
In table III, we listed dominant QNMs of spherically
symmetric black string perturbations in the case of n =
Page 4
4
0.2 0.40.60.8 1.0
k
0.2
0.4
0.6
0.8
1.0
Re?w?
0.20.4 0.60.81.0
k
?1.2
?1.0
?0.8
?0.6
?0.4
?0.2
Im?w?
FIG. 1: Real and imaginary part of first two overtones for n = 2 black string perturbations as a function of k. The first overtone
ω1 (blue) approaches ω = k (red line). The second overtone ω2 (green) becomes pure imaginary in stable region (k ∼ 1).
FIG. 2: Timedomain profiles of black string perturbations
for k = 2.5 n = 2 (red, top), n = 3 (orange), n = 4 (green),
n = 5 (blue, bottom). Latetime decay of perturbations for
n ≥ 3 is ∝ t−(n+6)/2. For lower n the law of decay is different:
∝ t−0.93for n = 1 and ∝ t−1.2for n = 2.
3. We can see the level crossing of imaginary part of
quasinormal modes between the fundamental mode and
the first overtone mode.
The real part of the second mode (ω1in tables I, III)
asymptotes to k at large k, while the imaginary part
monotonically decreases when k is increasing. The third
mode does not asymptotes to k, but has monotonically
decreasing real and imaginary parts as can be seen from
Fig. 1. Because of the risk to ”fall” into another over
tone, in order to obtain the higher overtones, we had to
start from the Ddimensional black holes with k = 0,
for which the QN frequencies are known [9], and then to
”move” towards higher k in the Frobenius method with
a very small step (see Fig. 1).
An essential advantage of the time domain method in
comparison with the Frobenius method is that we do not
have any decreasing of the convergence or loss of accuracy
when approaching the point of instability. Therefore our
time domain method is more complete than the frequency
one, at least for the dominant mode, which can always
be extracted from the time domain picture. In Fig. 2,
one can see examples of time domain profiles for various
n and a fixed k. There one can see that for n ≥ 3 the
intermediate late time asymptotic is power law like
Ψ ∝ t−(n+6)/2,n ≥ 3,
while for other n, the asymptotics are
Ψ ∝ t−0.93, forn = 1,
Ψ ∝ t−1.2
forn = 2.
Let us note that this asymptotic apparently should be
considered as intermediate. They are expected to go over
into other power law ones at very late times, as it takes
place for massive fields in general [30], [31], [33], [34].
Let us note, that k plays the role of the effective
mass.At asymptotically late time we observe power
law damped tails, which have oscillation frequency equal
to k, resembling asymptotical behavior of massive fields
near Schwarzschild black holes.
behavior is qualitatively similar to that of the funda
mental mode for massive fields of higherdimensional
Schwarzschild black holes [32]: we can see longlived os
cillations, which can be infinitely long lived modes called
the quasiresonances [33]. The analytical explanation of
existence of the quasiresonances was found in [34].
The first overtone’s
• For D = 5 (n = 1), as k grows, the imaginary part
of the first overtone quickly decreases and vanishes
for some threshold value of k, while its real part
stays smaller than the threshold value (see Table
I). After the threshold value of k is reached, the
first overtone “disappears”.
• For D ≥ 6 (n ≥ 2), the imaginary part of the first
overtone becomes small for large k, while the real
part asymptotically approaches k (see Fig. 1).
Page 5
5
Even though the first overtone of the spherically sym
metric black strings behaves similarly to the funda
mental mode of massive fields near higherdimensional
Schwarzschild black holes, the other modes have com
pletely different behavior. The fundamental mode of a
black string perturbation is purely imaginary. It grows
for small values of k, leading to instability of the black
string. This behavior is common for the unstable modes.
Indeed, let us multiply the equation (4) by the complex
conjugated function Ψ⋆and assume that the dependance
on time is Ψ(t,r⋆) = e−iωtΨ(r⋆). Let us study the inte
gral of the obtained equation
I =
∞
?
−∞
?
Ψ⋆(r⋆)d2Ψ(r⋆)
dr2
⋆
+ ω2Ψ(r⋆)2− V Ψ(r⋆)2
?
dr⋆.
Integration of the first term by parts gives
I = Ψ⋆(r⋆)dΨ(r⋆)
dr⋆
?????
∞
−∞
+
+
∞
?
−∞
?
ω2Ψ(r⋆)2− V Ψ(r⋆)2−
????
dΨ(r⋆)
dr⋆
????
2?
dr⋆= 0.
Taking into account the boundary conditions (6), we find
that the imaginary part of the integral is
Im(I) = Re(
?
ω2− k2)Ψ(∞)2+ Re(ω)Ψ(−∞)2+
∞
?
−∞
Since the sign of Re(√ω2− k2) coincides with the sign
of Re(ω), the nonzero real part of the quasinormal
frequency implies that the imaginary part is negative.
Therefore, the unstable modes (Im(ω) > 0) must have
zero real part. In other words, unstable modes cannot be
oscillating.
In the stability region, the fundamental mode is also
purely imaginary, but damped mode, whose damping
rate grows quickly with k. Because of its quick growth,
this mode cannot be considered as the fundamental one
for large k. In fact, for larger k (see Tables IIII), the
first overtone turns out to be the fundamental mode (the
mode with the largest lifetime). The real part of the
second overtone decreases, as k grows, and reaches zero
for some value of k, while the imaginary part remains
negative.
Let us now look at the Fig.3. At moderately large
values of k, sufficiently far from instability, the profile
has the same form as that for massive fields, yet, when
approaching the instability point the real oscillation fre
quency (Re(ω)) and the decay rate (Im(ω) < 0) decrease
considerably. After crossing the instability point we ob
serve, that starting from some tiny values, Im(ω) > 0 are
slowly increasing (while Re(ω) is still zero for the funda
mental mode and tiny oscillations, observed in the time
+2Re(ω)Im(ω)
Ψ(r⋆)2dr⋆= 0.
FIG. 3: Timedomain profiles of black string perturbations
for n = 1 k = 0.84 (magenta, top), k = 0.87 (red), k =
0.88 (orange), k = 0.9 (green), k = 1.1 (blue, bottom). We
can see two concurrent modes: for large k the oscillating one
dominates , near the critical value of k the dominant mode
does not oscillate (looks like exponential tail), for unstable
values of k the dominant mode grows. The plot is logarithmic,
so that straight lines correspond to an exponential decay.
domain, come from the next decayed mode). Therefore
we conclude, that the there is some static solution ω = 0
of the wave equation (4), which shows itself exactly in
the threshold point of instability. We would say that this
picture of instability is natural, if the instability develops
on the fundamental mode. However, if instability occurs
at higher multipoles ℓ, as it takes place for instance in
the GaussBonnet theory [2], the picture of instability is
quite different: growing modes appear only after rather
long period of decaying oscillations (see [2]). Note also,
that here we confirmed the threshold values of k found
in [7] with a very good accuracy by the time domain in
tegration (see for instance Fig. 3 for n = 1). Thus the
threshold values are: k = 0.876 for n = 1, k = 1.269 for
n = 2, k = 1.581 for n = 3, k = 1.849 for n = 4.
Finally, in Fig. 4, we can see the region of the profiles
where the period of the quasinormal ringing goes over
into the powerlaw tail behavior. Close to the critical
point, there exists a period where the oscillation ceases.
This is because the pure damping mode becomes the fun
damental mode near the critical point.
IV.CONCLUSION
We have numerically studied the GregoryLaflamme
instability through quasinormal modes.
three main results obtained here:
1) We have found the quasinormal modes and late
time tails for scalar type of gravitational perturbations
of Ddimensional black strings for various D, that is for
the type of perturbations where the GregoryLaflamme
instability forms in the long wavelength regime.
2) The time domain profiles indicate that the threshold
Let us stress
Page 6
6
FIG. 4: Timedomain profiles of black string perturbations for n = 4 k = 2.2 (red, left top), k = 2.3 (orange, right top), k = 2.4
(green, left bottom), k = 2.5 (blue, right bottom). Quasinormal ringing and tails have the same frequency of oscillation which
is close to k. One can see a period where the oscillation ceases close to the critical point.
instability value of k corresponds to dominance of some
static solution ω = 0.
3) Near the instability point (in k) the fundamental
mode is pure imaginary (nonoscillating), and, as k is in
creasing, the lifetime of the second mode is increasing,
so that at some moderate k both modes are dominating
at the late time of the ringing. At larger k, the domi
nance goes over to the second (oscillating) mode, as to
the longer lived one.
Our research could be improved in a number of ways.
First of all, one could compute QNMs for higher multi
pole numbers, starting from the effective potential de
rived in [35] and also for other types of gravitational
perturbation. Though vector and tensor types of per
turbations do not contain instabilities, such investigation
would give us complete data on QNMs and evolution of
gravitational perturbations.
The main limitation of our analysis is that we cannot
say what happen with unstable black string some time
since the moment of initial perturbations: the pertur
bations will grow and become large, so that the linear
approximation will not be valid anymore. However, it is
beyond the scope of this paper.
Acknowledgments
R. A. K. was supported by the Japan Society for the
Promotion of Science (JSPS), Japan.
ported by JSPS GrantinAid for Scientific Research No.
19 · 3715. J.S. was supported by the JapanU.K. Re
search Cooperative Program, GrantinAid for Scientific
Research Fund of the Ministry of Education, Science and
Culture of Japan No.18540262 and No.17340075. A. Z.
was supported by Funda¸ c˜ ao de Amparo ` a Pesquisa do
Estado de S˜ ao Paulo (FAPESP), Brazil.
K.M. was sup
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