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International Journal of Modern Physics DVol. 21, No. 6 (2012) 1250054 (8 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S021827181250054X

QUASINORMAL MODES AND STABILITY ANALYSISFOR FOUR-DIMENSIONAL LIFSHITZ BLACK HOLE

P. A. GONZALEZ

Escuela de Ingenierıa Civil en Obras Civiles,Facultad de Ciencias Fısicas y Matematicas,

Universidad Central de Chile,Avenida Santa Isabel 1186, Santiago, Chile

Universidad Diego Portales, Casilla 298-V, Santiago, Chilepgonzalezm@ucentral.cl

JOEL SAAVEDRA

Instituto de Fısica,Pontificia Universidad Catolica de Valparaıso,

Casilla 4950, Valparaıso, Chilejoel.saavedra@ucv.cl

YERKO VASQUEZ

Departamento de Ciencias Fısicas, Facultad de Ingenierıa,Ciencias y Administracion, Universidad de La Frontera,

Avenida Francisco Salazar 01145, Casilla 54-D, Temuco, Chileyvasquez@ufro.cl

Received 9 May 2012Revised 23 May 2012Accepted 24 May 2012Published 30 June 2012

We study the Lifshitz black hole in four dimensions with dynamical exponent z = 2and we calculate analytically the quasinormal modes of scalar perturbations. Thesequasinormal modes allow to study the stability of the Lifshitz black hole and we haveobtained that Lifshitz black hole is stable.

Keywords: Quasinormal modes; stability analysis; Lifshitz black hole.

1. Introduction

Quasinormal modes (QNMs) of stars and black holes have a long history, from theoriginal works1–3 to the modern point of view given in the literature.4–6 Amongthe main characteristic of QNMs, an interesting feature is their connection to ther-mal conformal field theories. According to the AdS/CFT correspondence,7 classical

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P. A. Gonzalez, J. Saavedra and Y. Vasquez

gravity backgrounds in AdS space are dual to conformal field theories at the bound-ary (for a review see Ref. 8). Using this principle it was established that the relax-ation time of a thermal state of the conformal theory at the boundary is proportionalto the inverse of imaginary part of QNMs of the dual gravity background.9 There-fore, the knowledge of QNMs spectrum determines how fast a thermal state in theboundary theory will reach thermal equilibrium. On the other hand, Bekenstein10

was the first who proposed the idea that in quantum gravity the area of blackhole horizon is quantized leading to a discrete spectrum which is evenly spaced.About this topic, QNMs allows to study the quantum area spectrum of the blackhole horizon. In those lines an interesting proposal was made by Hod11 who conjec-tured that the asymptotic QNMs frequency is related to the quantized black holearea. The black hole spectrum can be obtained imposing the Bohr–Sommerfeldquantization condition to an adiabatic invariant quantity involving the energy E

and the vibrational frequency ω(E).12 Identifying ω(E) with the real part ωR ofQNMs, the Hod’s conjecture leads to an expression of the quantized black holearea, which however is not universal for all black hole backgrounds. Furthermore itwas argued,13 that in the large damping limit the identification of ω(E) with theimaginary part of QNMs could lead to the Bekenstein universal bound.10 Therefore,the analytic computation of QNMs for black holes geometry is an important taskin order to connect different key ideas about black hole physics and gravitationaltheories. Recently another consequence of AdS/CFT correspondence7 has been theapplication, beyond high energy physics to another areas of physics, for examplequantum chromodynamics and condensed matter. In this sense, Lifshitz spacetimeshave received great attention from the condensed matter point of view i.e. thesearching of gravity duals of Lifshitz fixed points. From quantum field theory pointof view, there are many scale invariant theories which are of interest in studyingsuch critical points. These kind of theories exhibit the anisotropic scale invariancet → λzt, x → λx, with z �= 1, where z is the relative scale dimension of time andspace and they are of particular interest in the studies of critical exponent theoryand phase transition. Lifshitz spacetimes, described by the metric

ds2 = −l2(r−2zdt2 + r−2dr2 + r−2dx 2), (1)

where x represents a D − 2 spatial vector and l denotes the length scale in thegeometry.

In this work we will consider a matter distribution outside the horizon of oneLifshitz black hole, we will focus our study in the analytic solution found in Ref. 14.The action is

S =12

∫d4x(R − 2Λ) −

∫d4x

(e−2φ

4F 2 +

m2A

2A2 + (e−2φ − 1)

), (2)

where, R is the Ricci scalar, Λ is the cosmological constant, A is a gauge field,F is the field strength and m2

A = 2z. The black hole solution of this system that

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Quasinormal Modes and Stability Analysis

asymptotically approaches to the Lifshitz spacetime is given by

φ = −12

log(

1 +r2

r2H

),

A =f(r)r2

dt,

ds2 = −f(r)dt2

r2z+dx 2

r2+

dr2

f(r)r2,

(3)

with

f(r) = 1 − r2

r2H. (4)

For z = 2, the metric of Lifshitz black hole is

ds2 = −f(r)r4

dt2 +1

f(r)r2dr2 +

1r2dx 2. (5)

In order to compute QNMs, we perturbed this geometry with a scalar field andalso we are assuming that the back reaction is vanished over this geometry. Then wejust need to solve the Klein–Gordon equation of scalar field for the geometry underconsideration. Therefore, we obtain the analytic solution of the quasinormal modeswhich are characterized by a spectrum that is independent of the initial conditionsof the perturbation and depends only on the black hole parameters and on thefundamental constants of the system (for a recent review see Refs. 15 and 16) andalso through our analytic results we study the stability of the Lifshitz black holevia QNMs.

The paper is organized as follows. In Sec. 2 we calculate the exact QNMs of thescalar perturbations of four-dimensional Lifshitz black hole. Our conclusions are inSec. 3.

2. Quasinormal Modes

The quasinormal modes of scalar perturbations for a minimally coupled scalar fieldto curvature on the background of a Lifshitz black hole in four dimensions aredescribed by the solution of Klein–Gordon equation

�ψ =1√−g∂µ(

√−ggµν∂ν)ψ = m2ψ, (6)

where m is the mass of the scalar field ψ. In order to solve Klein–Gordon equationwe adopt the ansatz ψ = R(r)e−ik·xe−iωt. Thus, Eq. (6) gives

∂2rR(r) − 1

r

1 +

2

1 − r2

r2H

∂rR(r) +

1

1 − r2

r2H

−k2 +

r2ω2

1 − r2

r2H

− m2

r2

R(r) = 0.

(7)

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Performing the change of variables u = rrH

and inserting in Eq. (7) we obtain

∂2uR(u) +

u2 − 3u(1 − u2)

∂uR(u) +r2H

1 − u2

(−k2 +

ω2u2r2H1 − u2

− m2

u2r2H

)R(u) = 0, (8)

now using the changes v = u2 and 1 − v = z, Eq. (8) can be written as

∂2zR(z) +

[1

1 − z+

1z

]∂zR(z)

+r2H

4z(1 − z)

(−k2 +

ω2r2H(1 − z)z

− m2

r2H(1 − z)

)R(z) = 0, (9)

which, under the decomposition R(z) = zα(1 − z)βK(z) with

α± = ± ir2Hω

2, (10)

β± =12(2 ±

√4 +m2), (11)

can be written as a hypergeometric equation for K

z(z − 1)K ′′(z) + [c− (1 + a+ b)z]K ′(z) − abK(z) = 0, (12)

where

c = 1 + 2α, (13)

a =12(−1 ±

√1 − r2Hk

2 − r4Hω2 + 2α+ 2β), (14)

b =12(−1 ∓

√1 − r2Hk

2 − r4Hω2 + 2α+ 2β). (15)

The general solution of Eq. (12) takes the form

K = C1F1(a, b, c; z) + C2z1−cF1(a− c+ 1, b− c+ 1, 2 − c; z), (16)

which has three regular singular point at z = 0, z = 1 and z = ∞. Here, F1(a, b, c; z)is a hypergeometric function and C1, C2 are constants. Then, the analytic solutionfor the radial function R(z) is

R(z) = C1zα(1 − z)βF1(a, b, c; z)

+C2z−α(1 − z)βF1(a− c+ 1, b− c+ 1, 2 − c; z). (17)

So, in the vicinity of the horizon, z = 0 and using the property F (a, b, c, 0) = 1, theradial function R(z) behaves as

R(v) = C1eα ln z + C2e

−α ln z (18)

and the scalar field ψ can be written as

ψ ∼ C1e−iω(t+

r2H2 ln z) + C2e

−iω(t− r2H2 ln z), (19)

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Quasinormal Modes and Stability Analysis

in which, without loss of generality we consider α = α−, the first term in Eq. (19)represents an ingoing wave and the second one an outgoing wave in the black holegeometry. In order to compute QNMs, we must impose boundary conditions on thehorizon, existence of only ingoing waves (this fixes C2 = 0). Then, radial solutionbecomes

R(z) = C1eα ln z(1 − z)βF1(a, b, c; z) = C1e

−iωr2

H2 ln z(1 − z)βF1(a, b, c; z). (20)

Now, we need to implement boundary conditions at infinity (z = 1), before perform-ing this task we shall apply the Kummer’s formula for the hypergeometric functionon Eq. (20) as a connection formula,17

F1(a, b, c; z) =Γ(c)Γ(c− a− b)Γ(c− a)Γ(c− b)

F1(a, b, a+ b− c, 1 − z)

+ (1 − z)c−a−b Γ(c)Γ(a+ b− c)Γ(a)Γ(b)

F1(c− a, c− b, c− a− b + 1, 1 − z),

(21)

with this expression the radial function reads

R(v) = C1e−iω

r2H2 ln z(1 − z)β Γ(c)Γ(c− a− b)

Γ(c− a)Γ(c− b)F1(a, b, a+ b− c, 1 − z)

+C1e−iω

r2H2 ln z(1 − z)c−a−b+β Γ(c)Γ(a+ b− c)

Γ(a)Γ(b)

×F1(c− a, c− b, c− a− b+ 1, 1 − z). (22)

For, β+ > 2 the field at infinity is regular if the gamma function Γ(x) has the polesat x = −n for n = 0, 1, 2, . . . , the wave function satisfies the considered boundarycondition only upon the following additional restriction (a)|β+ = −n or (b)|β+ =−n. In the cases of β− < 0 the field at infinity is regular or null if the gammafunction Γ(x) has the poles at x = −n for n = 0, 1, 2, . . . , the wave function satisfiesthe considered boundary condition only upon the following additional restriction(c − a)|β− = −n or (c − b)|β− = −n. These conditions determine the form of thefrequency of quasinormal modes as

ω = −i(√

4 +m2 + 2n2r2H

+k2r2H +

√4 +m2 + 2n

2r2H(√

4 +m2 + 2n+ 1)

). (23)

The stability of black holes can be analyzed via QNMs and it is known that suchstability is guaranteed if the imaginary part of the QNMs is negative. Note that fromabove equation the stability of the Lifshitz black hole is guaranteed for m2 > −4and for any real value of k and n. It is worth noting, that this condition on the magrees with the analogue to Breitenlohner–Freedman condition, that any effectivemass must be satisfied in order to have a stable propagation.21

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5 10 15 20 25m

–15

–10

–5

n=3

n=2

n=1

n=0

Fig. 1. Im(ω) versus m; rH = 1 and k = 1.

–2 –1 1 2Im(m)

–4

–3

–2

–1

n=3

n=2

n=1

n=0

Fig. 2. Im(ω) versus Im(m); rH = 1 and k = 1.

Now, in Fig. 1, we have plotted the imaginary part of QNMs that we previouslyhave found, for rH = 1 and k = 1, where we show that the imaginary part of QNMsis negative.

Also, we have plotted the behavior of QMNs with respect to imaginary part ofthe scalar field mass in Fig. 2, for rH = 1 and k = 1. In this figure we show thatthe imaginary part of QNMs is negative for −2 < Im(m) < 2.

3. Conclusions

Lifshitz black hole is a very interesting static solution of gravity theories whichasymptotically approach to Lifshitz spacetimes. This spacetime is of interest dueto its invariance under anisotropic scale transformation and its representation asthe gravitational dual of strange metals.18 If z = 1, the spacetime is the usual AdSmetric in Poincare coordinates. Furthermore, all scalar curvature invariants are con-stant and there is a curvature singularity for z �= 1.19–22 In this work we calculated

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Quasinormal Modes and Stability Analysis

the analytic QNMs of scalar perturbations for the four-dimensional Lifshitz blackhole with plane topology by using Dirichlet boundary condition at infinity. First,we have found that the quasinormal modes are purely imaginary i.e. there is onlydamped modes. Second, using our analytic results we studied the stability of thisgeometry under scalar perturbation, we note that the stability is guaranteed if theimaginary part of QNMs is negative. In this sense, our results are summarized inFigs. 1 and 2, where the imaginary part of the modes is plotted with respect to thescalar fields mass. For the cases of positive mass we found, from modes with n ≥ 0the imaginary part of QNMs frequency satisfy the stability condition. Also, we con-sidered a scalar field with imaginary mass and we found that for (−2 < Im(m) < 2),the stability of the Lifshitz black hole is guaranteed. We allow to note, that the con-dition on m that we have used agrees with the analogue to Breitenlohner–Freedmancondition.

Acknowledgments

We thank Bertha Cuadros-Melgar for enlightening discussions and remarks. Y. Vwas supported by Direccion de Investigacion y Desarrollo, Universidad de la Fron-tera, DIUFRO DI11-0071. J. S was supported by COMISION NACIONAL DECIENCIAS Y TECNOLOGIA through FONDECYT Grant 1110076, 1090613 and1110230. This work was also partially supported by PUCV DI-123.713/2011 (JS).P. G acknowledges the hospitality of the Physics Department of Universidad de LaFrontera where part of this work was made.

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