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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-
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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) ,
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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
1.6 n/a1.674 − 0.088i 1.659 − 0.086i
1.7 n/a1.763 − 0.081i
1.8 n/a1.852 − 0.077i
1.9 n/a1.936 − 0.054i
2.0n/a 2.032 − 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
2.1 n/a2.309 − 0.172i
2.2 n/a2.400 − 0.167i
2.3 n/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 =
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0.20.40.60.81.0
k
0.2
0.4
0.6
0.8
1.0
Re?w?
0.2 0.40.60.81.0
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).
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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
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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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