Scalar field scattering by a Lifshitz black hole under a nonminimal coupling

Samuel Lepe,1,* Javier Lorca,2,† Francisco Pena,2,‡ and Yerko Vasquez2,§

1Instituto de Fısica, Facultad de Ciencias, Pontificia Universidad Catolica de Valparaıso, Casilla 4059, Valparaıso, Chile2Departamento de Ciencias Fısicas, Facultad de Ingenierıa, Ciencias y Administracion, Universidad de La Frontera,

Avda. Francisco Salazar 01145, Casilla 54-D Temuco, Chile(Received 22 June 2012; published 19 September 2012)

We study the behavior of a scalar field under a z ¼ 3 Lifshitz black hole background, in a way which is

nonminimally coupled to the gravitational field. A general analytical solution is obtained along with two

sets of quasinormal modes associated to different boundary conditions which can be imposed on the scalar

field; a nonminimal coupling parameter appears explicitly on these solutions. Stability of quasinormal

modes can be studied and ensured for both cases. Also, the reflection and absorption coefficients are

calculated, as well as the absorption cross section which features an interesting behavior because of being

attenuated by terms strongly dependant on the nonminimal coupling. By a suitable change of variables, a

soliton solution can also be obtained, and the stability of the quasinormal modes are studied and ensured.

DOI: 10.1103/PhysRevD.86.066008 PACS numbers: 04.70.�s, 04.70.Bw

I. INTRODUCTION

It is known that theories such as new massive gravityadmits Lifshitz black holes as solutions, whose particular-ity is that they are invariant under an anisotropic scaletransformation of the form t ! �zt y x ! �x, where z iscalled the relative scale between time and space dimen-sions, specifically z ¼ 3 for the aforementioned theory.These black hole solutions have come to prominence be-cause they might provide a way to extend the AdS/CFTcorrespondence [1] to systems found on nonrelativisticcondensed matter physics which features a very similarbehavior [2–4], where it was proven that the relaxationtime of thermal states of the conformal theory at theboundary is proportional to the inverse of the imaginarypart of the quasinormal modes of the dual gravity back-ground [5].

As a way to consider explicitly an interaction betweengravity and a scalar field, a nonminimal coupling is addedon the equation of motion for the scalar field, motivated inresemblance of those found in theories in which the actionhas this type of coupling [6–9]. As a consequence, theRicci scalar appears directly in the equation of motion;however, in a Lifshitz background the Ricci scalar has arelatively simple form which allows one to obtain ananalytical solution.

In this paper, we focus on the study of the reflection andabsorption coefficients, as well as the absorption crosssection [10–13]. Some efforts have been made for theminimally coupled Lifshitz black hole [14] in this direc-tion. Also, it has been shown that for a spherically sym-metric black hole and a massless minimally coupled scalarfield, the cross section equals the area of the horizon [15];

however, we will show that there is a strong dependence onthe nonminimal coupling parameter in this case. Thisimplies that the absorption cross section is also dependenton it, hence not allowing one to obtain as a result thegeometric area of the black hole in the limit of low ener-gies, unless the nonminimal coupling became null.Gravitational waves predict a non-normal type of oscil-

lation mode where the frequencies become complex or arealso called quasinormal, with the real part representing thefrequency of oscillation and the imaginary part represent-ing the damping [16]. This study has already been done tosome of these black hole solutions [17–20], considering ascalar field moving over a Lifshitz background, where noimaginary parts have been found so far. The presence of thenonminimal coupling does not affect this behavior thanksto the simple form of the Ricci scalar. By relaxing theboundary conditions used, allowing them to be Dirichletand Neumann mixed, one can obtain a new set of quasi-normal modes previously not found, which can be ana-lyzed to study their stability. On the other hand, byperforming two Wick’s rotations between time and spacecoordinates on the metric, it is possible to find a solitonsolution [21], find its quasinormal modes, and study itsstability.This paper is organized as follows. The Sec. II introdu-

ces the formalism, the field equations to be used, andgeneralities on the Lifshitz metric for z ¼ 3. In theSec. III, we find the solution of the Klein-Gordon differ-ential equation for a scalar field on this Lifshitz space-timeand formally treat the suitable boundary conditions to thissolution. Section IV finds the reflection and absorptioncoefficients along with the cross section, which featuresan interesting dependency on the nonminimal couplingparameter. Sections V and VI are committed to the studyof the quasinormal modes, their stability for the blackhole, and its related soliton solution. Finally, we stressthe important results of this paper as final remarks.

*slepe@ucv.cl†j.lorca@ufro.cl‡fcampos@ufro.cl§yvasquez@ufro.cl

PHYSICAL REVIEW D 86, 066008 (2012)

1550-7998=2012=86(6)=066008(9) 066008-1 � 2012 American Physical Society

II. FORMALISM AND FIELD EQUATIONS

Let us consider the typical new massive gravity action

S ðgÞ ¼Z

d3xffiffiffiffiffiffiffi�g

pL; (1)

where

L ¼ 1

2R� 2�0 � 1

�2

�R��R�� � 3

8R2

�: (2)

In this Lagrangian density, we use natural units, i.e.,8�G ¼ c ¼ 1. R denotes the scalar of curvature, andR�� denotes the Ricci tensor. In order to ensure a completecorrespondence to gravitational solutions, the parameters �and �0 must be chosen to be �2 ¼ � 1

2l2(with mass

dimension) and �0 ¼ � 132l2

(the cosmological constant),

respectively.The field equations are obtained by varying the total

action (1) with respect to the metric. We will use the3-dimensional Lifshitz black hole background as a knownsolution from this theory [22],

ds2 ¼ ���

l

�4f2ð�Þdt2 þ 1

f2ð�Þd�2 þ �2d’2; (3)

with f2ð�Þ ¼ ð�l Þ2 �M, where l is the curvature radius of

the Lifshitz space-time and M represents the black holemass. From this final form, the Ricci scalar (for z ¼ 3) canbe calculated to be

R ¼ 8M

�2� 26

l2: (4)

Let us consider a massive scalar field on this background.In a way, it is weakly coupled to the gravitational field, inthe sense that the presence of this field does not perturb thebackground metric, but it is just a field which moves alongthis geometry. A typical equation followed by this scalarfield is

ðh�m2Þ� ¼ d�ð�Þd�

R� dUð�Þd�

; (5)

where � ¼ 12 ð1� ��2Þ and Uð�Þ is the self-interacting

potential density. As was already mentioned, this type ofequation is motivated by the kind of field equation obtainedwhen varying an action with a scalar field with nonminimalcoupling parameter � to the Ricci scalar [9]. For finiteness,consider a null self-interacting potential density (i.e.,Uð�Þ ¼ 0Þ; Eq. (5) takes the form

ðh�m2Þ� ¼ ���R: (6)

III. SOLUTION OF THE DIFFERENTIALEQUATION

Starting from Eq. (6) and using the metric (3), it can beshown that a separation of variables for the field � of theform

�ð�;’; tÞ ¼ Rð�Þ expð�ið!tþ ’ÞÞ (7)

allows one to write the radial equation in the followingform:

@2�Rð�Þ þ ð5 �l2� 3M

�Þð�2

l2�MÞ

@�Rð�Þ þ 1

ð�2

l2�MÞ

�l4!2

�4ð�2

l2�MÞ

� 2

�2�m2 þ �

�8M

�2� 26

l2

��Rð�Þ ¼ 0: (8)

By noting that the horizon of the black hole is located at

�þ ¼ lffiffiffiffiffiM

p, let us define the scaling variable

x ¼ 1���þ�

�2

(9)

so that Eq. (8) can be written in the form

R00ðxÞ þ 1

xð1� xÞR0ðxÞ þ 1

4xð1� xÞ2�l2!2ð1� xÞ3

M3x

� 2ð1� xÞM

� l2m2 þ �ð8ð1� xÞ � 26Þ�RðxÞ ¼ 0:

(10)

Equation (10) is of the Fuchsian type with two regularsingular points located at x ¼ 0 and x ¼ 1 and one irregu-lar singular point located at x ¼ �1, which tell us thatthis equation is somewhat related to the confluent Heunfamily of equations. In fact, the transformation RðxÞ ¼xð1� xÞ�F ðxÞ, justified by Fuch’s theorem around theregular singular points, allows one to identify this assertionbetter, yielding

xðx� 1ÞF 00ðxÞ þ ½ðbþ 1Þðx� 1Þ þ ðcþ 1Þx�F 0ðxÞþ ðdx� �ÞF ðxÞ ¼ 0; (11)

where

� ¼ � i

2M3=2!l; (12)

�� ¼ 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 13

�

2þm2l2

4

s; (13)

b ¼ 2�; (14)

c ¼ 2�� � 2; (15)

d ¼ � l2!2

4M3; (16)

and

� ¼ � þ �� � ð� þ ��Þ2 � 2

4M� l2!2

2M3þ 4

�

2:

(17)

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Using the previous parameters, the solution of Eq. (11) iswritten

F ðxÞ ¼ C1HeunC

�0; b; c; d;� 1

2f1þ cgb� c

2� �; x

�

þ C2x�bHeunC

�0;�b; c; d;� 1

2f1þ cgb

� c

2� �; x

�; (18)

where HeunC are the confluent Heun functions. Finally, thesolution of Eq. (10) is

RðxÞ ¼ C1xð1� xÞ�HeunC

�0; b; c; d;� 1

2f1þ cgb

� c

2� �; x

�þ C2x

�ð1� xÞ�HeunC

��0;�b; c; d;� 1

2f1þ cgb� c

2� �; x

�: (19)

Note that the simplicity of the Ricci scalar for thisbackground allows us to identify directly from Eq. (8)the following transformations from the homogeneousproblem on [20]:

2 ! 2 � 8M� m2 ! m2 þ 26�

l2; (20)

which means that the nonminimal coupling problem isformally equivalent to the homogeneous one. This

transformation is useful when calculating the quasinormalmodes of this black hole.

A. Asymptotic Expressions

In order to obtain the reflection and absorption coeffi-cients, the asymptotic expressions for this solution must beobtained. To incorporate boundary conditions will force usto focus on two distinct points:(i) � ! �þ ) x ! 0. Here, the solution takes the fol-

lowing approximated form:

R ðx ! 0Þ � C1x þ C2x

�

¼ C1 expð lnxÞ þ C2 expð� lnxÞ: (21)

Let us recall that there exists two values for , andso, let us also assume that ¼ � ¼ �i!l

2M3=2 ; then, by

Eq. (21), the condition C2 ¼ 0 arises by consideringjust ingoing flux in the horizon. We recall that choos-ing ¼ þ will derive in the same result because ofconsidering the same flux conditions, which will leadus to set the constant C1 ¼ 0.

(ii) � ! 1 ) x ! 1. This case is a bit more subtle.Here, we have to use the following identities regard-ing the confluent Heun functions [23]:

Heun Cð0; b; c; d; e; 0Þ ¼ 1 (22)

HeunCð0; b; c; d; e; xÞ ¼ D1

�ðbþ 1Þ�ð�cÞ�ð1� cþ kÞ�ðb� kÞ HeunCð0; c; b;�d; eþ d; 1� xÞ þD2ð1� xÞ�c

� �ðbþ 1Þ�ðcÞ�ð1þ cþ sÞ�ðb� sÞ HeunCð0;�c; b;�d; eþ d; 1� xÞ; (23)

where the following set of equations is found to be satisfiedbetween the parameters of the confluent Heun’s functions:

k2 þ ð1� b� cÞk� �� b� cþ d

2¼ 0; (24)

s2 þ ð1� bþ cÞs� �� bðcþ 1Þ þ d

2¼ 0; (25)

� 1

2ð1þ cÞb� c

2� � ¼ e: (26)

Therefore, Eq. (19) takes the following asymptotic form oninfinity:

Rðx ! 1Þ � C1ð1� xÞ�½B1 þ B2ð1� xÞ2�2��þ C2ð1� xÞ�½B3 þ B4ð1� xÞ2�2��;

however, we have already discarded the constantC2 by fluxconditions on the horizon. Hence, the appropriate asymp-totic expression to be used is

R ðx ! 1Þ � C1ð1� xÞ�½B1 þ B2ð1� xÞ2�2��; (27)

where

B1 ¼ D1

�ðbþ 1Þ�ð�cÞ�ð1� cþ kÞ�ðb� kÞ ; (28)

B2 ¼ D2

�ðbþ 1Þ�ðcÞ�ð1þ cþ sÞ�ðb� sÞ : (29)

As a way to check the solution of Eq. (27), the asymp-totic equation on the infinity will be solved. This can bedone by performing in Eq. (10) the following change ofvariable y ¼ 1� x with y ! 0; hence, it follows

y2d2RðyÞdy2

� ydRðyÞdy

þ ½E1 þ E2y�RðyÞ ¼ 0; (30)

where we have only to retain terms up to a second order in

y and E1 ¼ �14 ½ðlmÞ2 þ 26�� and E2 ¼ 1

4 ½8�� 2

M�. Thesolution to this equation can be written

SCALAR FIELD SCATTERING BYA LIFSHITZ BLACK . . . PHYSICAL REVIEW D 86, 066008 (2012)

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Rðy ! 0Þ ¼ yfF1E2�ð1� 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� E1

p ÞJ��ðuÞþ F2E2�ð1þ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� E1

p ÞJ�ðuÞg;where J�ðuÞ are the Bessel’s functions of the first kind,having � ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� E1

pand u ¼ 2

ffiffiffiffiffiffiffiffiE2y

p. Let us use the

following expansion of the Bessel’s functions for u � 1:

J �ðuÞ ¼ u�

2��ð1þ �Þ�1� u2

2ð2�þ 2Þ þOðu4Þ�; (31)

and by restricting the previous expression up to the firstterm, the solution for the asymptotic radial equation can bewritten in the form

Rðy ! 0Þ ¼ F1E1�

ffiffiffiffiffiffiffiffiffi1�E1

p2 y1�

ffiffiffiffiffiffiffiffiffi1�E1

p

þ F2E1þ

ffiffiffiffiffiffiffiffiffi1�E1

p2 y1þ

ffiffiffiffiffiffiffiffiffi1�E1

p: (32)

Note that 1� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� E1

p ¼ ��, so Eqs. (32) and (27) areequivalent; therefore, the corresponding coefficients mustbe equal

F 1 ¼ F1E1�

ffiffiffiffiffiffiffiffiffi1�E1

p2 ¼ C1B1; (33)

F 2 ¼ F2E1þ

ffiffiffiffiffiffiffiffiffi1�E1

p2 ¼ C1B2: (34)

IV. REFLECTION AND ABSORPTIONCOEFFICIENTS

Before going any further, it is convenient to express theequations needed explicitly on the scaling variable x [Eq. (9)].The flux F is known to be defined by Refs. [12,24]:

F ¼ffiffiffiffiffiffiffi�g

pg��

2iðR�ð�Þ@�Rð�Þ � Rð�Þ@�R�ð�ÞÞ;

¼ 2�4þl4

�x

1� x

�ImfR�ðxÞ@xRðxÞg; (35)

and by using Eqs, (32)–(34), the following expression isobtained:

Fasymptotic ¼ 4ð�� 1Þl4

ImfF1ð�þ4F�2Þg: (36)

As it has been discussed by other authors, the problem thatEq. (36) has is that it is impossible to determine wether theflux is ingoing or outgoing; however, it can be written in thefollowing form [10,13,25]:

Fasymptotic ¼ 4hð�� 1Þl4

jFinasymptoticj2

� 4hð�� 1Þl4

jFoutasymptoticj2; (37)

where

F inasymptotic ¼

1

2

�F1 þ i�4þ

hF2

�; (38)

F outasymptotic ¼

1

2

�F1 � i�4þ

hF2

�; (39)

where h is a real parameter with dimension ½L�4. Note thatEq. (37) coincides with the expression (48), this last oneobtained by a different procedure.On the other hand, using Eq. (21), on the horizon, the

following relation results:

Fhorizon ¼ 2�þ4

l4limx!0

�x

1� x

�ImfR�ðxÞ@xRðxÞg;

¼ ��þ4

l4jC1j2 !l

M3=2;

¼ ��þ!jC1j2; (40)

where in the last line, the definition of the black hole masshas been used.By using Eqs. (38), (39), (33), and (34), we can calculate

the reflection and absorption coefficients as

R ���������Foutasymptotic

Finasymptotic

��������¼jB1j2 þ �8

þh2jB2j2 þ 2�4

þh ImðB�

1B2ÞjB1j2 þ �8

þh2jB2j2 � 2�4

þh ImðB�

1B2Þ;

(41)

U���������

Finhorizon

Finasymptotic

��������¼ !l

M3=2jhjj�� 1jðjB1j2þ �8þh2jB2j2� 2�4

þh ImðB�

1B2ÞÞ;

¼ !l4�þjhjj�� 1jðjB1j2þ �8

þh2jB2j2� 2�4

þh ImðB�

1B2ÞÞ; (42)

where as in Eq. (40), the definition of the black hole masshas been used again. As a manner to avoid any divergenceon Eqs. (41) and (42), we will choose negative values for h.Recall that factors B1 and B2 are dependent on Gamma

functions [Eqs. (28) and (29)], which makes it difficult towork with the analytical expressions; however, the absorp-tion cross section is immediate from Eq. (42)

¼ U

!¼ l4�þ

jhjj�� 1jðjB1j2 þ �8þh2jB2j2 � 2�4

þh ImðB�

1B2ÞÞ:

(43)

It is straightforward to verify that in the low energy limitfor an s-wave type of solution ( ¼ 0), an expression forthe absorption cross section can be obtained to be

ð�Þ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 13 �

2

q 1

jfð; �Þj2 2��þ < 2��þ; (44)

where we have chosen jhj ¼ l4

2� and jfð; �Þj2 is the term

appearing on the denominator of Eq. (43) dependent on thecoefficients B1, B2, and h. The presence of the � parameter

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decreases the final value of the absorption cross sectionaway from the geometric area even if we try the low energylimit, i.e., when m ! 0, ! ! 0. Equation (44) is reducedto the geometric area of the black hole ¼ 2��þ onlywhen � ¼ 0 as expected [14].The behavior of the reflection and absorption coeffi-

cients as well as the cross section are shown in Figs. 1–3,respectively.It can be proven that the addition of the reflection and

absorption coefficient adds up to 1, which is consistentwith the election of negative values for h.

V. LIFSHITZ BLACK HOLE QUASINORMALMODES AND THEIR STABILITY

By assuming a null scalar field as a boundary conditionon the infinity, and without loss of generality choosing ¼� ¼ � i

2M3=2 l!, from Eq. (27), it can be seen that the field

is regular in r ! 1 for �� < 0 if 1þ cþ kþ n ¼ 0, for

�þ > 2 if 1� cþ sþ n ¼ 0, regarded � >� l2m2

26 . Using

these conditions, the quasinormal modes are obtainedafter lengthy algebra; however, as it was shown above,this problem is equivalent to the homogeneous one. Thequasinormal modes are easily obtained by using the trans-formations (20), which yields

!1 ¼ 2iM3=2

l

�1þ 2nþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ l2m2 þ 26�

q

��2

2Mþ 7þ 3l2m2

2þ 6nðnþ 1Þ

þ 35�þ 3ð1þ 2nÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ l2m2 þ 26�

q �1=2

�; (45)

and

!2 ¼ 2iM3=2

l

�1þ 2nþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ l2m2 þ 26�

q

þ�2

2Mþ 7þ 3l2m2

2þ 6nðnþ 1Þ

þ 35�þ 3ð1þ 2nÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ l2m2 þ 26�

q �1=2

�; (46)

where n is a zero or a positive integer.Stability of these solutions is restricted specifically by

the termffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ l2m2 þ 26�

p, regarded Imf!g 0. It is

worth noting that this condition is completely equivalentto impose a restriction on the term of squared mass of thescalar field which we have used. To be more precise, if wedefine an effective mass on the form

m2effective ¼ m2 þ 26�

l2;

this redefined effective mass must agree with the analogueof the Breitenlohner-Freedman condition in order to have astable propagation, which was established in Ref. [26].

1 2 3 4 5

0.4

0.6

0.8

1.0Reflection coefficient

FIG. 1. The behavior of the reflection coefficient with respectto ! for the following parameters: M ¼ 1, m2l2 ¼ � 4

5 , l ¼ 1,

¼ 1, h ¼ �1, for three different values � ¼ 0 (fine solid line),� ¼ 0:1 (dashed line), and � ¼ 2 (thick solid line).

1 2 3 4 5

0.2

0.4

0.6

Absorption cross section

FIG. 3. The behavior of the absorption cross section withrespect to ! for the following parameters: M ¼ 1, m2l2 ¼� 4

5 , l ¼ 1, ¼ 1, h ¼ �1, for three different values � ¼ 0

(fine solid line), � ¼ 0:1 (dashed line), and � ¼ 0:2 (thicksolid line).

1 2 3 4 5

0.2

0.4

0.6

Transmission coefficient

FIG. 2. The behavior of the transmission coefficient with re-spect to ! for the following parameters: M ¼ 1, m2l2 ¼ � 4

5 ,

l ¼ 1, ¼ 1, h ¼ �1, for three different values � ¼ 0 (finesolid line), � ¼ 0:1 (dashed line), and � ¼ 0:2 (thick solid line).

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Figure 4 shows that the first set of quasinormal modes isessentially stable, because of having a negative imaginarypart. In fact, Eq. (45) is completely imaginary; therefore,these modes have over damping state of oscillation.Figure 5 shows that the second set of quasinormal modesis instable because of having a positive imaginary part. Inthe same way as before, Eq. (46) turns out to be completelyimaginary, and so these modes are nonconvergent.

By using Eq. (19) with C2 ¼ 0, due to the horizonboundary conditions, and using Eq. (35), we are able toevaluate the flux at infinity in accord with the followingidentity regarding the derivative of the confluent Heun’sfunctions [27]:

d

dxHeunCð0; b; c; d; e; xÞx¼0 ¼ 1

2

�bþ cþ bcþ 2e

bþ 1

�;

(47)

which yields

Fðx ! 1Þ ¼ 2�4þl4

jC1j2Imð2ð�� 1ÞB�1B2

þ ð� v1ÞjB1j2ð1� xÞ2��1

þ ð� v2ÞjB2j2ð1� xÞ3�2�Þ; (48)

where

v1 ¼ 1

2

�cþ bþ bcþ 2eþ 2d

1þ c

�; (49)

v2 ¼ 1

2

��cþ b� bcþ 2eþ 2d

1� c

�; (50)

and B1 and B2 were defined on Eqs. (28) and (29), wherethe flux was evaluated directly from the asymptotic radialequation. Now, imposing the boundary condition of van-ishing flux at infinity, we are able to obtain two sets ofquasinormal modes given by the conditions, 1� cþ kþn ¼ 0 or 1þ cþ sþ n ¼ 0. One set is similar to the onesobtained from the Dirichlet boundary conditions of vanish-ing scalar field; however, we find another set of quasinor-mal modes, which are stable for a range of imaginary scalarmass as it can be seen below. These modes are given by

!3 ¼ 2iM3=2

l

�ð1þ 2n�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ l2m2 þ 26�

qÞ

��2

2Mþ 7þ 3l2m2

2þ 6nðnþ 1Þ

þ 35�� 3ð1þ 2nÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ l2m2 þ 26�

q �1=2

�; (51)

and

!4 ¼ 2iM3=2

l

�ð1þ 2n�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ l2m2 þ 26�

qÞ

þ�2

2Mþ 7þ 3l2m2

2þ 6nðnþ 1Þ

þ 35�� 3ð1þ 2nÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ l2m2 þ 26�

q �1=2

�: (52)

From Eq. (48) and considering �� ¼ 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ m2

effl2

4

q,

wherem2

effl2

4 ¼ m2l2

4 þ 13 �2 , the allowable range for the

parameters are

1<�þ < 2 and 0<�� < 1: (53)

This sets the exponent of the first term on Eq. (48), yielding2�� 1> 0 ! �> 1=2 and 3� 2�> 0 which corre-sponds to � ¼ ��; on the second term, we have �<3=2 ! � ¼ �þ. From here, we haveffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3þ 26�p

< Imfmlg< ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ 26�

p: (54)

Figure 6 illustrates the fact that this last set of modes withimaginary mass are essentially stable.

2 4 6 8 10m

–10

–8

–6

–4

–2

0

2

4

Im

FIG. 4 (color online). The behavior of the black hole’s quasi-normal modes (45) for the following parameters: � ¼ 1

16 ,m ¼ 1,

l ¼ 1, M ¼ 1, ¼ 1, where the thin solid line represents themode n ¼ 0, the dashed line the mode n ¼ 1, the gross solid linethe mode n ¼ 2, and the dashed-dotted line the mode n ¼ 3.

2 4 6 8 10m

–10

0

10

20

30

40

50

60Im

FIG. 5 (color online). Shows the behavior of the black holequasi normal modes (46) for the following parameters: � ¼ 1

16 ,

m ¼ 1, l ¼ 1,M ¼ 1, ¼ 1, where the thin solid line representsthe mode n ¼ 0, the dashed line the mode n ¼ 1, the grosssolid line the mode n ¼ 2, and the dashed-dotted line the moden ¼ 3.

LEPE et al. PHYSICAL REVIEW D 86, 066008 (2012)

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The previous calculation is ensured by the following:consider the Klein-Gordon equation, which is essentiallyEq. (8). Due to the fact that nonminimal coupling onlyredefines the constants and m, this equation can betransformed into a Schrodinger-like equation by makinguse of the tortoise coordinate transformation, defined by

dx ¼ 1

ð�l Þ2ðð�l Þ2 �MÞd�: (55)

Then, the Klein-Gordon equation adopts the form�d2

dx2þ!2 � Vð�Þ

�ð�1=2RðxÞÞ ¼ 0; (56)

where the effective potential can be identified with thefollowing expression:

Vð�Þ ¼ 1

l2

��

l

�2���

l

�2 �M

��2 � 8M�� 3

4M

��m� 26

l2�� 7

4l2

��2

�: (57)

Note that this potential is divergent on � ! 1; therefore,the condition of null flux at this boundary is justified.

VI. SOLITON SOLUTION, QUASINORMALMODES AND ITS STABILITY

A remarkable feature of the Lifshitz black hole is that byperforming the double Wick rotation on the time and spacecoordinates,

t ! � il4

�þ3�’; ’ ! il

�þ�t; (58)

and performing the change of variable,

� ¼ �þ coshðrÞ; (59)

the metric (3) is transformed into the following one:

ds2 ¼ �l2cosh2ðrÞd�t2 þ l2dr2 þ l2cosh4ðrÞsinh2ðrÞd �’2;

(60)

which can be recognized as a soliton metric [21]. Formally,the quasinormal modes of this metric must be obtained inthe same way as was shown in the previous section.However, due to the symmetry of the transformationsinvolved, the quasinormal modes for the soliton can beeasily obtained by performing the following substitution:

! � i�þl

!soliton; ! ! i�þ3

l4soliton: (61)

Inserting Eq. (61) in Eqs. (45) and (46) yields

!soliton ¼ ��4þ 2soliton � 1

22soliton þ l2m2

þ 4nþ 4solitonnþ 4n2 þ 44�

þ 2ð1þ soliton þ 2nÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ l2m2 þ 26�

q �12: (62)

Note that the stability of these solutions only depends on

the termffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ l2m2 þ 26�

p, which has been already

restricted according to the to Breitenlohner-Freedmancondition; the positivity or negativity of rest of the termin Eq. (62) is not transcendent due to the fact that the‘‘minus’’ or ‘‘plus’’ sign can be chosen to ensure stability.The behavior of these modes can be seen in Fig. 7 whichshows explicitly the stability of the solutions presented.In the same way, we find a second set of quasinormal

modes for the soliton. By using the expressions found inEqs. (51) and (52), we have

0.5 1.0 1.5 2.0 2.5 3.0Im m

–4

–3

–2

–1

Im

FIG. 6 (color online). The behavior of the quasinormal modes(51) for the following parameters: � ¼ 0, m ¼ 1, l ¼ 1, ¼ 1,where the thin solid line represents the mode n ¼ 0, the dashedline the mode n ¼ 1, the gross solid line the mode n ¼ 2, and thedashed-dotted line the mode n ¼ 3.

2 4 6 8 10m

–20

–15

–10

–5

0

5

10

FIG. 7 (color online). Shows the behavior of the quasinormalsoliton modes (62) for the following parameters: � ¼ 1

16 , l ¼ 1,

soliton ¼ 1, where the thin solid line represents the mode n ¼ 0,the dashed line the mode n ¼ 1, the gross solid line the moden ¼ 2, and the dashed-dotted line the mode n ¼ 3.

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!soliton¼��4þ2soliton�1

22solitonþ l2m2

þ4nþ4solitonnþ4n2þ44�

�2ð1þsolitonþ2nÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ l2m2þ26�

q �12: (63)

The stability of this case depends on the condition4þ26�

l2> kmk2 which is valid for imaginary masses.

Recall that this is in accordance with the Breitenlohner-Freedman condition. As in the previous case, the stabilityis ensured by the suitable sign election of the term in thesquared root (63). The behavior of these modes can be seenin Fig. 8.

VII. FINAL REMARKS

We can summarize some important features found in thispaper:

(i) The appearance of the coupling parameter explicitlyin the absorption cross section [Eq. (43)] implies thatit is not possible to obtain the geometric area of theblack hole as a result in the limit of low energies andfor an s wave, unless this parameter becomes null.

(ii) As can be seen from this study, nonminimalcoupling does not affect the overall form of thequasinormal modes; moreover, the nonminimalquasinormal modes can be easily obtained by rede-fining the parameters and m. Although this can beconsidered a natural extension from the minimalproblem, this is only possible by the simple formof the Ricci scalar in a Lifshitz background.

(iii) There is a complete new set of stable quasinormalmodes which has been not considered before.These modes are associated with the possibility ofhaving an imaginary mass for the scalar field.These modes are found by imposing the flux to bezero at infinity and requiring the expression to bevalid according to the Breitenlohner-Freedmancondition for stability.

(iv) By performing a doubleWick rotation, it is possibleto change the metric in a way which is isomorphicto a soliton type of metric, and from here, it is easilyseen that the quasinormal modes for this metric canbe obtained from the black hole quasinormal modesby a simple transformation. These modes turn outto be stable.

ACKNOWLEDGMENTS

This research was supported by VRIEA-DI-037.419/2012, Pontificia Universidad Catolica de Valparaıso(S. L.), Fondecyt 1110076 (S. L.), DI12-0006 ofDireccion de Investigacion y Desarrollo, Universidad deLa Frontera (F. P.) and DI11-0071 Direccion deInvestigacion y Desarrollo, Universidad de La Frontera(Y.V.).

[1] J.M. Maldacena, AIP Conf. Proc. 484, 51 (1999).[2] D. Son, Phys. Rev. D 78, 046003 (2008).[3] S. A. Hartnoll, J. Polchinski, and E. Silverstein, J. High

Energy Phys. 04 (2010) 120.[4] S.A.Hartnoll,ClassicalQuantumGravity26, 224002 (2009).[5] G. T. Horowitz and V. E. Hubeny, Phys. Rev. D 62, 024027

(2000).[6] I. L. Buchbinder, S. D. Odintsov, and I. L. Shapiro,

Effective Action in Quantum Gravity Bristol (IOP,

Bristol, 1992).[7] E. Elizalde and S.D. Odintsov, Phys. Lett. B 333, 331

(1994).[8] T. Muta and S.D. Odintsov, Mod. Phys. Lett. A 06, 3641

(1991).

[9] S. Lepe, J. Lorca, F. Pena, and Y. Vasquez, Int. J. Mod.

Phys. D 20, 2543 (2011).[10] D. Birmingham, I. Sachs, and S. Sen, Phys. Lett. B

413, 281 (1997).[11] T. Harmark, J. Natario, and R. Schiappa, Adv. Theor.

Math. Phys. 14, 727 (2010).[12] P. Gonzalez, E. Papantonopoulos, and J. Saavedra, J. High

Energy Phys. 08 (2010) 050.[13] W. T. Kim and J. J. Oh, Phys. Lett. B 461, 189 (1999).[14] T. Moon and Y. S. Myung, arXiv:1205.2317.[15] S. R. Das, G.W. Gibbons, and S. D. Mathur, Phys. Rev.

Lett. 78, 417 (1997).[16] K. D. Kokkotas and B.G. Schmidt, Living Rev. Relativity

2, 2 (1999).

0.5 1.0 1.5 2.0 2.5 3.0 3.5Im m

–8

–6

–4

–2

0

2

FIG. 8 (color online). The behavior of the quasinormal solitonmodes (63) for the following parameters: � ¼ 1

16 , l ¼ 1,

soliton ¼ 1, where the thin solid line represents the mode n ¼0, the dashed line the mode n ¼ 1, the gross solid line the moden ¼ 2, and the dashed-dotted line the mode n ¼ 3.

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[17] B. Cuadros-Melgar, J. de Oliveira, and C. Pellicer, Phys.Rev. D 85, 024014 (2012).

[18] P. Gonzalez, J. Saavedra, and Y. Vasquez, Int. J. Mod.Phys. D 21, 1250054 (2012).

[19] A. Giacomini, G. Giribet, M. Leston, J. Oliva, and S. Ray,Phys. Rev. D 85, 124001 (2012).

[20] Y. S. Myung and T. Moon, Phys. Rev. D 86, 024006 (2012).[21] H. A. Gonzalez, D. Tempo, and R. Troncoso, J. High

Energy Phys. 11 (2011) 066.

[22] E. Ayon-Beato, A. Garbarz, G. Giribet, and M. Hassaine,Phys. Rev. D 80, 104029 (2009).

[23] Y. Kwon, S. Nam, J.-D. Park, and S.-H. Yi, ClassicalQuantum Gravity 28, 145006 (2011).

[24] Y. Satoh, Phys. Rev. D 58, 044004 (1998).[25] J. J. Oh and W. Kim, J. High Energy Phys. 01 (2009) 067.[26] S. Kachru, X. Liu, and M. Mulligan, Phys. Rev. D 78,

106005 (2008).[27] P. P. Fiziev, J. Phys. A 43, 035203 (2010).

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