# Looking at the Gregory-Laflamme 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 Gregory-Laflamme instability. Using numerical methods, we find the quasinormal modes and time-domain profiles of the black string perturbations in the stable sector and also show the appearance of the Gregory-Laflamme 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 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 Gregory-Laflamme 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:**In this work, we study the quasinormal modes (QNMs) of scalar field coupling to Einstein’s tensor in charged braneworld black hole. The shape of the potential function is illustrated and we find that lower coupling constant leads to more stable field. We then apply six-order WKB approximation to calculate the quasinormal frequencies (QNF) in weaker coupling field, and depict the dependence of the oscillation frequency on the coupling constant. Furthermore, we use finite difference method to shape the evolution of the coupling field and find that coupling field with lower multipole numbers l corresponds to stable field, while higher l tends to lead to instability when the coupling constant is larger than a threshold value. Finally the fitting curve of such threshold value is given numerically.International Journal of Theoretical Physics 01/2012; 51(8). · 1.19 Impact Factor - [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 six-order 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). · 1.19 Impact Factor - SourceAvailable from: arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**Generalized Kerr–NUT–de Sitter space–time is the most general space–time which admits a rank-2 closed conformal Killing–Yano tensor. It contains the higher-dimensional Kerr–de Sitter black holes with partially equal angular momenta. We study the separability of gravitational perturbations in the generalized Kerr–NUT–de Sitter space–time. We show that a certain type of tensor perturbations admits the separation of variables. The linearized perturbation equations for the Einstein condition are transformed into the ordinary differential equations of Fuchs type.International Journal of Modern Physics A 01/2012; 25(15). · 1.09 Impact Factor

Page 1

arXiv:0807.1897v2 [hep-th] 16 Jul 2008

Looking at the Gregory-Laflamme instability through quasi-normal modes

R. A. Konoplya,∗Keiju Murata,†and Jiro Soda‡

Department of Physics, Kyoto University, Kyoto 606-8501, Japan

A. Zhidenko§

Instituto de F´ ısica, Universidade de S˜ ao Paulo

C.P. 66318, 05315-970, S˜ ao Paulo-SP, 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 Gregory-Laflamme instability. Using numerical

methods, we find the quasinormal modes and time-domain profiles of the black string perturbations

in the stable sector and also show the appearance of the Gregory-Laflamme 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 Gregory-Laflamme 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 brane-world 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 Reissner-Nordstr¨ om-de Sitter black holes are

stable [1] in the Einstein gravity. On the contrary, black

holes in Gauss-Bonnet (GB) gravity are unstable for large

GB coupling for D = 5,6 [2], where D is the total number

of space-time dimensions. Kaluza-Klein black holes with

squashed horizon are stable against lowest zero mode

perturbations [3]. Unlike, Kaluza-Klein 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 brane-world

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.kyoto-u.ac.jp

‡Electronic address: jiro@tap.scphys.kyoto-u.ac.jp

§Electronic address: zhidenko@fma.if.usp.br

a black string in the full D-dimensional theory.

It is well known that such black strings suffer from the

so-called Gregory-Laflamme instability, which is the long

wavelength gravitational instability of the scalar type of

the metric perturbations [4], [5]. The Gregory-Laflamme

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 D-dimensional 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 so-called

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 D-dimensional

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 power-law

tails. The quasinormal modes and asymptotic tails are

very well studied for D-dimensional 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 D-dimensional 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 Gregory-Laflamme 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 Gundlah-Price-

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 space-time 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 z-direction is periodically identified by the relation

z = z + 2πR. We study perturbations of an (n + 1)-

spherically symmetric solution with the Killing vector in

z-direction. 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 non-vanishing 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 wave-like 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 Gundlach-Price-Pullin method (time

domain).

In time domain, we study the black string ringing using

a numerical characteristic integration method [28], that

uses the light-cone 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 time-domain

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 (quasi-resonance) 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 (t-d)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 well-known 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 time-domain 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(t-d)ω1(t-d)

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 time-domain

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 (quasi-resonance) 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: Time-domain profiles of black string perturbations

for k = 2.5 n = 2 (red, top), n = 3 (orange), n = 4 (green),

n = 5 (blue, bottom). Late-time 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 D-dimensional 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 higher-dimensional

Schwarzschild black holes [32]: we can see long-lived os-

cillations, which can be infinitely long lived modes called

the quasi-resonances [33]. The analytical explanation of

existence of the quasi-resonances 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 higher-dimensional

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

⋆

+ ω2|Ψ(r⋆)|2− V |Ψ(r⋆)|2

?

dr⋆.

Integration of the first term by parts gives

I = Ψ⋆(r⋆)dΨ(r⋆)

dr⋆

?????

∞

−∞

+

+

∞

?

−∞

?

ω2|Ψ(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 non-zero real part of the quasi-normal

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 I-III), 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: Time-domain 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 Gauss-Bonnet 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 power-law 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 Gregory-Laflamme

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 D-dimensional black strings for various D, that is for

the type of perturbations where the Gregory-Laflamme

instability forms in the long wavelength regime.

2) The time domain profiles indicate that the threshold

Let us stress

Page 6

6

FIG. 4: Time-domain 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). Quasi-normal 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 (non-oscillating), 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 Grant-in-Aid for Scientific Research No.

19 · 3715. J.S. was supported by the Japan-U.K. Re-

search Cooperative Program, Grant-in-Aid 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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