Scalar field perturbations of a Lifshitz black holein conformal gravity in three dimensions

Marcela Catalán*

Departamento de Ciencias Físicas, Facultad de Ingeniería y Ciencias, Universidad de La Frontera,Avenida Francisco Salazar 01145, Casilla 54-D, Temuco, Chile

Yerko Vásquez†

Departamento de Física, Facultad de Ciencias, Universidad de La Serena,Avenida Cisternas 1200, La Serena, Chile

(Received 31 July 2014; published 3 November 2014)

We study the reflection and transmission coefficients and the absorption cross section for scalar fields inthe background of a Lifshitz black hole in three-dimensional conformal gravity with z ¼ 0, and we showthat the absorption cross section vanishes at the low- and high-frequency limits. Also, we determine thequasinormal modes of scalar perturbations, and then we study the stability of these black holes under scalarfield perturbations.

DOI: 10.1103/PhysRevD.90.104002 PACS numbers: 04.70.-s, 04.70.Bw

I. INTRODUCTION

Three-dimensional models of gravity have attracted aremarkable interest in recent years. Apart from the BTZblack hole [1], which is a solution to the Einstein equationswith a negative cosmological constant, much attention hasbeen paid to topologically massive gravity (TMG), whichgeneralizes three-dimensional general relativity (GR) byadding a Chern-Simons gravitational term to the Einstein-Hilbert action [2]. In this model, the propagating degree offreedom is a massive graviton. TMG also admits the BTZand other black holes as exact solutions. The renewedinterest in TMG results from the possibility of constructinga chiral theory of gravity at a special point of the space ofparameters [3].Another three-dimensional massive gravity theory that

has attracted attention in recent years is known as newmassive gravity (NMG), where the action is the standardEinstein-Hilbert term plus a specific combination of ascalar curvature square term and a Ricci tensor squareterm [4–6], and at the linearized level it is equivalent to theFierz-Pauli action for a massive spin-2 field [4]. NMGadmits interesting solutions; see for instance Refs. [7–12];for further aspects of NMG, see Refs. [13–16]. TMG andNMG share common features; however, there are differentaspects: one of these is the existence in NMG of black holesknown as new-type black holes, which are also solutions ofconformal gravity in three dimensions [17].Lifshitz spacetimes have received considerable atten-

tion from the point of view of condensed matter, i.e., thesearch for gravity duals of Lifshitz fixed points due to theAdS/CFT correspondence [18] for condensed matter

physics and quantum chromodynamics [19,20]. Thereare many invariant scale theories of interest when studyingsuch critical points; such theories exhibit the anisotropicscale invariance t → λzt, x → λx, where z is the relativescale dimension of time and space, and they are ofparticular interest in studies of critical exponent theoryand phase transitions. Lifshitz spacetimes are described bythe metrics

ds2 ¼ −r2z

l2zdt2 þ l2

r2dr2 þ r2

l2d~x2; ð1Þ

where ~x represents a (d − 2)-dimensional spatial vector,with d representing the spacetime dimension while ldenotes the length scale in the geometry. For z ¼ 1, thisspacetime is the usual anti–de Sitter metric in Poincarécoordinates.The metrics of the Lifshitz black holes asymptotically

have the form (1). Several Lifshitz black holes have beenreported [11,12,21–28].In this work, we will consider a matter distribution

outside the horizon of a Lifshitz black hole in three-dimensional conformal gravity with z ¼ 0. It is worthmentioning that for z ¼ 0 the anisotropic scale invariancecorresponds to spacelike scale invariance with no trans-formation of time. The matter is parameterized by a scalarfield, which we will perturb by assuming that there is nobackreaction on the metric. We then obtain the reflectionand transmission coefficients, the absorption crosssection, and the quasinormal frequencies (QNFs) forscalar fields and study their stability under scalarperturbations.Conformal gravity is a four-derivative theory, and it is

perturbatively renormalizable [29,30]. Furthermore, it con-tains ghostlike modes, in the form of massive spin-2

*marceicha@gmail.com†yvasquez@userena.cl

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excitations. However, it has been shown that by using themethod of Dirac constraints [31] to quantize a prototypesecond-plus-fourth-order theory, viz. the Pais-Uhlenbeckfourth-order oscillator model [32], it becomes apparent thatthe limit in which the second-order piece is switched off is ahighly singular one in which the would-be ghost statesmove off the mass shell [33]. In three spacetime dimen-sions, the equations of motion contain third derivatives ofthe metric. AdS and Lifshitz black holes with z ¼ 0 andz ¼ 4 in four-dimensional conformal gravity have beenstudied in Ref. [34], and solutions to conformal gravity inthree dimensions have been studied in Refs. [17,35,36].Several studies have contributed to the scattering and

absorption properties of waves in the spacetime of blackholes. As the geometry of the spacetime surrounding ablack hole is nontrivial, the Hawking radiation emitted atthe event horizon may be modified by this geometry, so thatwhen an observer located very far away from the black holemeasures the spectrum, this will no longer be that of ablackbody [37]. The factors that modify the spectrumemitted by a black hole are known as greybody factorsand can be obtained through classical scattering; their studytherefore allows the semiclassical gravity dictionary to beincreased, and also offers insight into the quantum nature ofblack holes and thus of quantum gravity; for an excellentreview of this topic, see Ref. [38]. Also, see for instanceRefs. [39,40] for the decay of Dirac fields in higher-dimensional black holes.On the other hand, the study of the QNFs gives

information about the stability of black holes under matterfields that evolve perturbatively in their exterior region,without backreacting on the metric [41–46]. Furthermore,according to the AdS/CFT correspondence, the QNFsdetermine how fast a thermal state in the boundarytheory will reach thermal equilibrium [47]. Also, inRefs. [48,49], the authors discuss a connection betweenHawking radiation and quasinormal modes (QNMs),which can naturally be interpreted in terms of quantumlevels. In three dimensions, the QNMs of BTZ black holeshave been studied in Refs. [47,50,51]; scalar and fer-mionic QNMs in the background of new-type black holesin NMG were studied in Refs. [52] and [53], respectively.Studies of QNMs in Lifshitz black holes can be found inRefs. [54–63]. On the other hand, the absorption crosssections for Lifshitz black holes were examined inRefs. [57,64,65], and particle motion in these geometriesin Refs. [66–68]. Fermions on a Lifshitz background havebeen studied in Ref. [69].The paper is organized as follows: In Sec. II, we find a

Lifshitz black hole with z ¼ 0 in three-dimensional con-formal gravity. In Sec. III, we calculate the reflection andtransmission coefficients, the absorption cross section, andthe quasinormal modes of scalar perturbations for the three-dimensional Lifshitz black hole with z ¼ 0. We concludewith final remarks in Sec. IV.

II. LIFSHITZ BLACK HOLE INTHREE-DIMENSIONAL CONFORMAL

GRAVITY WITH z ¼ 0

In this work, we will consider a matter distributiondescribed by a scalar field outside the event horizon of anasymptotically Lifshitz black hole in three-dimensionalconformal gravity with z ¼ 0. In three dimensions, the fieldequations of conformal gravity are given by the vanishingof the Cotton tensor:

Cαβ ¼ ϵρσα∇ρ

�Rσβ −

1

4gσβR

�¼ 0; ð2Þ

where R is the Ricci scalar. The Cotton tensor is a tracelesstensor that vanishes if and only if the metric is locallyconformally flat. Solutions to this theory have been studiedin Refs. [17,35,36].In order to find a Lifshitz black hole solution in three-

dimensional conformal gravity, we consider the followingansatz for the metric:

ds2 ¼ −r2zfðrÞdt2 þ dr2

r2fðrÞ þ r2dθ2; ð3Þ

and from the field equations (2) we obtain

4zðz − 1ÞfðrÞ þ ð−3þ 7zþ 2z2Þrf0ðrÞþ 3ð1þ zÞr2f00ðrÞ þ r3f000ðrÞ ¼ 0: ð4Þ

For fðrÞ ¼ 1 (Lifshitz spacetime), the above equationyields the solutions z ¼ 0 and z ¼ 1. However, as wementioned, z ¼ 1 corresponds to the usual anti–de Sittermetric in Poincaré coordinates.Now, setting z ¼ 0 in the field equation (4), the follow-

ing Lifshitz black hole solution is found:

ds2 ¼ −fðrÞdt2 þ dr2

r2fðrÞ þ r2dθ2;

fðrÞ ¼ 1 −Mr2

: ð5Þ

The requirement rþ ¼ ffiffiffiffiffiM

p> 0 implies thatM > 0. The

Kretschmann scalar is given by

RμνρσRμνρσ ¼ 4þ 20M2

r4; ð6Þ

therefore, there is a curvature singularity at r ¼ 0 forM ≠ 0. Other asymptotically Lifshitz black hole solutionsin four-dimensional and six-dimensional conformal gravityare found in Refs. [34] and [70], respectively.Metric (5) is conformally related to an asymptotically

de Sitter black hole. This can be seen by definingds̄2 ¼ Ω2ds2, where the conformal factor is given by

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Ω ¼ 2r1þ 2Nr2

ð7Þ

and N is a constant. The new radial coordinate r̄ is definedby r̄ ¼ Ωr, and

ds̄2 ¼ −f̄ðr̄Þdt2 þ dr̄2

f̄ðr̄Þ þ r̄2dθ2; ð8Þ

where

f̄ðr̄Þ ¼ −2Nð1þ 2MNÞr̄2 þ 2ð1þ 4MNÞr̄ − 4M; ð9Þ

this metric corresponds to an asymptotically de Sitter blackhole. Metrics of the form (9) were studied in Ref. [17].Notice that the conformal factor is not singular on thehorizon. However, it is singular at the asymptotic regionr → ∞; therefore, the asymptotic regions of the twoconformally related metrics are very different.In the next section, we will determine the reflection

coefficient, the transmission coefficient, and the absorptioncross section of the Lifshitz black hole metric (5) foundin this section. Then, we will compute the QNFs, whichcoincide with the poles of the transmission coefficient, andwe will study the linear stability of these black holes underscalar field perturbations.

III. REFLECTION COEFFICIENT,TRANSMISSION COEFFICIENT, ABSORPTIONCROSS SECTION, AND QUASINORMAL MODES

OF A z ¼ 0 LIFSHITZ BLACK HOLE

The scalar perturbations in the background of anasymptotically Lifshitz black hole in three-dimensionalconformal gravity with dynamical exponent z ¼ 0 aregiven by the Klein-Gordon equation of the scalar fieldsolution with suitable boundary conditions at the eventhorizon and at the asymptotic infinity.In Sec. III A, we will consider a scalar field minimally

coupled to gravity. Then, in Sec. III B, we will study ascalar field nonminimally coupled.

A. Scalar field minimally coupled to gravity

The Klein-Gordon equation in curved spacetime isgiven by

1ffiffiffiffiffiffi−gp ∂μðffiffiffiffiffiffi−g

pgμν∂νÞϕ ¼ m2ϕ; ð10Þ

where m is the mass of the scalar field ϕ, which isminimally coupled to curvature. By means of the ansatz

ϕ ¼ e−iωteiκθRðrÞ; ð11Þthe Klein-Gordon equation reduces to the following differ-ential equation for the radial function RðrÞ:

r2ðr2 −MÞ d2RðrÞdr2

þ 2r3dRðrÞdr

þ�

ω2r4

r2 −M− κ2 −m2r2

�RðrÞ ¼ 0: ð12Þ

Now, by considering RðrÞ ¼ r−1=2GðrÞ and by introducingthe tortoise coordinate x, given by dx ¼ dr

rfðrÞ, the latter

equation can be rewritten as a one-dimensional Schrödingerequation:

½∂2x þ ω2 − VeffðrÞ�GðxÞ ¼ 0; ð13Þ

where the effective potential is given by

VeffðrÞ ¼ fðrÞ − 3

4fðrÞ2 þ κ2

r2fðrÞ þm2fðrÞ: ð14Þ

In Fig. 1, we plot the effective potential for M ¼ 1, m ¼ 1,and different values of the angular momentum κ ¼ 0, 1, 2.Note that when r → ∞, the effective potential goesto 1=4þm2.Performing the change of variable z ¼ 1 − M

r2, the radialequation (12) can be written as

zð1 − zÞ∂2zRðzÞ þ

�1 −

3

2z

�∂zRðzÞ

þ 1

4

�ω2

zð1 − zÞ −κ2

M−

m2

1 − z

�RðzÞ ¼ 0: ð15Þ

Using the decomposition RðzÞ ¼ zαð1 − zÞβFðzÞ, with

α� ¼ � iω2; ð16Þ

β� ¼ 1

4

�1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4ðm2 − ω2Þ

q �; ð17Þ

0 2 4 6 8 10r0.0

0.5

1.0

1.5

2.0

2.5

3.0V r

2

1

0

FIG. 1. The effective potential as a function of r, for M ¼ 1,m ¼ 1 and κ ¼ 0, 1, 2.

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allows us to write (15) as a hypergeometric equationfor FðzÞ:zð1− zÞ∂2

zFðzÞ þ ½c− ð1þ aþ bÞz�∂zFðzÞ− abFðzÞ ¼ 0;

ð18Þ

where the coefficients are given by

a ¼ 1

4þ αþ β � 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

4κ2

M

r; ð19Þ

b ¼ 1

4þ αþ β ∓ 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

4κ2

M

r; ð20Þ

c ¼ 1þ 2α: ð21ÞThe general solution of the hypergeometric equation (18) is

FðzÞ ¼ C12F1ða; b; c; zÞþC2z1−c2F1ða − cþ 1; b− cþ 1; 2− c; zÞ; ð22Þ

and it has three regular singular points at z ¼ 0, z ¼ 1, andz ¼ ∞. 2F1ða; b; c; zÞ is a hypergeometric function, and C1

and C2 are integration constants. Thus, the solution for theradial function RðzÞ is

RðzÞ ¼ C1zαð1− zÞβ2F1ða; b; c; zÞþC2z−αð1− zÞβ2F1ða− cþ 1; b− cþ 1; 2− c; zÞ:

ð23Þ

So, in the vicinity of the horizon, z ¼ 0, and using theproperty Fða; b; c; 0Þ ¼ 1, the function RðzÞ behaves as

RðzÞ ¼ C1eα ln z þ C2e−α ln z; ð24Þand the scalar field ϕ, for α ¼ α−, can be written as follows:

ϕ ∼ C1e−iωðtþln zÞ þ C2e−iωðt−ln zÞ; ð25Þ

where the first term represents an ingoing wave and thesecond one an outgoing wave in the black hole. So, byimposing that only ingoing waves exist at the horizon, thisfixes C2 ¼ 0. The radial solution then becomes

RðzÞ ¼ C1eα ln zð1 − zÞβ2F1ða; b; c; zÞ¼ C1e−iω ln zð1 − zÞβ2F1ða; b; c; zÞ: ð26Þ

The reflection and transmission coefficients depend onthe behavior of the radial function, both at the horizon andat the asymptotic infinity, and they are defined by

R ¼F

outasymp

F inasymp

; T ¼ F in

hor

F inasymp

; ð27Þ

where F is the flux, and is given by

F ¼ 1

2iffiffiffiffiffiffi−g

pgrr

�RðrÞ� dRðrÞ

dr− RðrÞ dRðrÞ

�

dr

�; ð28Þ

whereffiffiffiffiffiffi−gp ¼ 1. The behavior at the horizon is given by

(24), and using (28), we get the flux at the horizon:

F inhor ¼ −ω

ffiffiffiffiffiM

pjC1j2: ð29Þ

Now, in order to obtain the asymptotic behavior of RðzÞ, wetake the limit z → 1 in Eq. (15). Thus, we obtain thefollowing solution:

RðzÞ ¼ B1ð1 − zÞβþ þ B2ð1 − zÞβ− : ð30Þ

Thus, the flux (28) at the asymptotic region is given by

F asymp ¼ −ffiffiffiffiffiM

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2 −m2 −

1

4

rðjB1j2 − jB2j2Þ; ð31Þ

where ω2 ≥ m2 þ 14. On the other hand, by replacing

Kummer’s formula [71] in (26),

2F1ða; b; c; zÞ ¼ΓðcÞΓðc − a − bÞΓðc − aÞΓðc − bÞ 2F1ða; b; aþ b − c; 1 − zÞ

þ ð1 − zÞc−a−b ΓðcÞΓðaþ b − cÞΓðaÞΓðbÞ 2F1ðc − a; c − b; c − a − bþ 1; 1 − zÞ; ð32Þ

and by using Eq. (28), we obtain the flux

F asymp ¼ −ffiffiffiffiffiM

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2 −m2 −

1

4

rðjA1j2 − jA2j2Þ; ð33Þ

where

A1 ¼ C1

ΓðcÞΓðaþ b − cÞΓðaÞΓðbÞ ¼ B1;

A2 ¼ C1

ΓðcÞΓðc − a − bÞΓðc − aÞΓðc − bÞ ¼ B2: ð34Þ

Therefore, the reflection and transmission coefficients aregiven by

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R ¼ jA1j2jA2j2

; ð35Þ

T ¼ ωjC1j2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2 −m2 − 1

4

qjA2j2

; ð36Þ

and the absorption cross section σabs becomes

σabs ¼Tω

¼ jC1j2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2 −m2 − 1

4

qjA2j2

: ð37Þ

Now, we will study the behavior of the reflection andtransmission coefficients and the absorption cross section atthe low- and high-frequency limits.At the low-frequency limit, ω2 ≈m2 þ 1=4, the coef-

ficients A1 and A2 are given approximately by

A1 ≈ C1

Γð1 − iωÞΓ�i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2 −m2 − 1

4

q �

Γ�12− iω

2þ 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 − 4κ2

M

q �Γ�12− iω

2− 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 − 4κ2

M

q � ;

A2 ≈ C1

Γð1 − iωÞΓ�−i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2 −m2 − 1

4

q �

Γ�12− iω

2− 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 − 4κ2

M

q �Γ�12− iω

2þ 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 − 4κ2

M

q � :

ð38Þ

Using the expressions above, the following low-fre-quency limit is obtained for the reflection coefficient

R ≈

Γ�i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2 −m2 − 1

4

q �

Γ�−i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2 −m2 − 1

4

q �2

¼ 1; ð39Þ

where in the last equality we used the fact thatΓðzÞ� ¼ Γðz�Þ. For the transmission coefficient and theabsorption cross section, we find

T ∝1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ω2 −m2 − 14

q Γ�−i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2 −m2 − 1

4

q �2 ¼i

Γ�−i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2 −m2 − 1

4

qþ 1

�Γ�i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2 −m2 − 1

4

q � → 0;

σ ¼ Tω

∝1

Γ�i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2 −m2 − 1

4

q � → 0; ð40Þ

where we have used the properties zΓðzÞ ¼ Γðzþ 1Þ,ΓðzÞ� ¼ Γðz�Þ, and Γ

�i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2 −m2 − 1

4

q �→ ∞.

At the high-frequency limit, ω2 ≫ m2 þ 1=4, the coef-ficients A1 and A2 are given by

A1 ≈ C1

Γð1 − iωÞΓðiωÞΓ�12þ 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 − 4κ2

M

q �Γ�12− 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 − 4κ2

M

q � ;

A2 ≈ C1

Γð1 − iωÞΓð−iωÞΓ�12− iω − 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 − 4κ2

M

q �Γ�12− iωþ 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 − 4κ2

M

q � :

ð41Þ

We will consider the case 4κ2=M < 1. The case4κ2=M > 1 is similar, and the same asymptotic behavioris obtained.Using the asymptotic expansion of the Γ functions for

jyj → ∞ [72],

jΓðxþ iyÞj ¼ffiffiffiffiffiffi2π

pjyjx−1=2e−x−jyjπ=2

�1þO

�1

jyj��

;

ð42Þ

we obtain

jA1j≈ jC1j2πe−1−ωπΓ�1

2þ 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffi1− 4κ2

M

q �Γ�12− 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffi1− 4κ2

M

q �∝

1

eωπ;

jA2j≈ jC1j: ð43Þ

Therefore, at high frequencies,

R ∝1

e2ωπ→ 0;

T → 1;

σ ∝1

ω→ 0: ð44Þ

Now, we will carry out a numerical analysis of thereflection coefficient (35), transmission coefficient (36),and absorption cross section (37) of z ¼ 0 Lifshitz blackholes for scalar fields. So, we plot the reflection andtransmission coefficients and the absorption cross sectionin Fig. 2, with M ¼ 1, m ¼ 1 and κ ¼ 0, 1, 2. We can see,as expected from the above asymptotic analysis, that thereflection coefficient is 1 at the low-frequency limit, and at

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the high-frequency limit this coefficient is null. Thebehavior of the transmission coefficient is the opposite,with Rþ T ¼ 1. Also, the absorption cross section is nullat the low- and high-frequency limits, but there is a range offrequencies for which the absorption cross section is notnull, and it also has a maximum value.

The QNFs are defined as the poles of the transmissioncoefficient, which is equivalent to imposing that onlyoutgoing waves exist at the asymptotic infinity. Thesepoles are given by A2 ¼ 0, and this occurs when c − aþn ¼ 0 or c − bþ n ¼ 0, with n ¼ 0; 1; 2…. Therefore, thequasinormal frequencies are given by

ω ¼ −i−2m2ð1þ 2nÞ þ 1þ 10nþ 24n2 þ 16n3 þ 2 κ2

M þ 4n κ2

M �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − 4 κ2

M

qð1þm2 þ 4nþ 4n2 þ κ2

MÞ3þ 16nþ 16n2 þ 4 κ2

M

: ð45Þ

We observe that for 1 − 4 κ2

M > 0 the QNFs are purelyimaginary, and for 1 − 4 κ2

M < 0 they acquire a real part.Also, we observe that for some values of the scalar fieldmass m, the quasinormal frequencies can have a positiveimaginary part; therefore, we conclude that this black hole

is not stable under scalar field perturbations minimallycoupled to gravity.

B. Scalar field conformally coupled to gravity

In this section, we consider a scalar field nonminimallycoupled to curvature, propagating in the background of theLifshitz black hole (5). In particular, we study the case of ascalar field conformally coupled. The Klein-Gordon equa-tion is given by

1ffiffiffiffiffiffi−gp ∂μðffiffiffiffiffiffi−g

pgμν∂νÞϕ − χRϕ ¼ m2ϕ; ð46Þ

where χ is the nonmininal coupling parameter, andR ¼ −2ð1 − M

r2Þ is the Ricci scalar.In this case, by means of the ansatz ϕ ¼ e−iωteiκθRðrÞ,

we obtain the following radial equation:

r2ðr2 −MÞ d2RðrÞdr2

þ 2r3dRðrÞdr

þ�

ω2r4

r2 −M− ðκ2 þ 2χMÞ − ðm2 − 2χÞr2

�RðrÞ ¼ 0:

ð47Þ

We can see that, due to the simplicity of the Ricci scalar inthis background, it is possible to obtain this equa-tion directly from (12) by means of the followingtransformations;

κ2 → κ2 þ 2χM;

m2 → m2 − 2χ: ð48Þ

Therefore, the reflection and transmission coefficients,the absorption cross section, and the quasinormal modes forthe nonminimal case can be obtained easily from theequations (35), (36), and (37) of the minimal case byusing transformations (48).For a conformally coupled scalar field, m ¼ 0 and

χ ¼ 1=8, one can find that the qualitative behavior ofthe reflection and transmission coefficients and the

1.2 1.4 1.6 1.8 2.0

0.2

0.4

0.6

0.8

1.0

abs

R T

T

R

1.2 1.4 1.6 1.8 2.0

0.2

0.4

0.6

0.8

1.0

abs

R T

T

R

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0.2

0.4

0.6

0.8

1.0

abs

R T

T

R

FIG. 2. The reflection coefficient R (solid curve), the trans-mission coefficient T (dashed curve), Rþ T (thick curve), andthe absorption cross section σabs (dotted curve) as a function of ωð1.12 ≤ ωÞ for M ¼ 1 and m ¼ 1, with κ ¼ 0 (upper figure),κ ¼ 1 (middle figure), and κ ¼ 2 (lower figure).

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absorption cross section is similar to that obtained insection A for a minimally coupled scalar field, and forthe quasinormal frequencies we obtain the followingexpression:

ω ¼ � κ

2ffiffiffiffiffiM

p − i1þ 6nþ 12n2 þ 8n3 þ κ2

M þ 2n κ2

M

2þ 8nþ 8n2 þ 2 κ2

M

;

ð49Þwhich has a negative imaginary part; therefore, the blackhole is stable under conformally coupled scalar fieldperturbations.Also, we note that the effective potential goes to zero at

the asymptotic infinity.

IV. FINAL REMARKS

In this work, we studied scalar field perturbations of anasymptotically Lifshitz black hole in three-dimensionalconformal gravity with dynamical exponent z ¼ 0, wherein this case the anisotropic scale invariance corresponds to aspacelike scale invariance with no transformation of time,and we calculated the reflection and transmission coeffi-cients, the absorption cross section, and the quasinormalmodes. The results obtained show that the absorption crosssection vanishes at the low-frequency limit as well as at thehigh-frequency limit. This means that a wave emitted fromthe horizon, with low or high frequency, does not reachspatial infinity and therefore is totally reflected, because thefraction of particles penetrating the potential barrier van-ishes. We have also shown in Fig. 2 that there is a range offrequencies where the absorption cross section is not null.On the other hand, the reflection coefficient is 1 at the low-frequency limit and null at high frequencies; the behavior ofthe transmission coefficient is the opposite, whereRþ T ¼ 1. Furthermore, we have shown that the absorp-tion cross section decreases for higher values of angularmomentum, and decreases when the mass m of the scalarfield increases; however, for high frequencies the differenceis negligible; see Figs. 3 and 4.Furthermore, we calculated analytically the QNFs of

scalar perturbations, which coincide with the poles of thetransmission coefficient, and we found two sets of quasi-normal frequencies. However, some of these can have apositive imaginary part, depending of the value of m, and

therefore the black hole is not stable under scalar fieldperturbations for a scalar field minimally coupled tocurvature. If the scalar field is conformally coupled tothe curvature, we found that the imaginary part of thequasinormal frequencies is negative, which guaranteesthe linear stability of the black hole in this case.Interesting applications we hope to address in the near

future are the holographic implications of the results ofthis paper.

ACKNOWLEDGMENTS

This work was funded by the Comisión Nacional deInvestigación Científica y Tecnológica throughFONDECYT Grant No. 11121148 (Y. V., M. C.) and alsopartially funded by Dirección de Investigación,Universidad de La Frontera (M. C.).

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m 1

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0 1 2 3 4 50.0

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