quasinormal frequencies of black branes with hyperscaling violation

November 13, 2013 17:31 WSPC/146-MPLA S0217732313501459 1–10

Modern Physics Letters AVol. 28, No. 37 (2013) 1350145 (10 pages)c© World Scientific Publishing Company

DOI: 10.1142/S0217732313501459

QUASINORMAL FREQUENCIES OF BLACK BRANES WITH

HYPERSCALING VIOLATION

NAN BAI∗, YI-HONG GAO†, BU-GUAN QI‡ and XIAO-BAO XU§

State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,

Chinese Academy of Sciences, P. O. Box 2735, Beijing 100190, P. R. China∗[email protected]†[email protected]‡[email protected]§[email protected]

Received 19 May 2013Revised 17 July 2013

Accepted 25 August 2013Published 16 October 2013

We investigated quasinormal frequencies (QNFs) in black brane with hyperscaling viola-tion by using the continued fractions method. We calculate QNF of massless scalar fieldand electromagnetic field both with zero spatial momentum, find that QNFs have neg-ative imaginary frequency suggesting black brane with hyperscaling violation is stableunder those perturbations.

Keywords: Black branes; quasinormal modes; gauge/gravity duality.

PACS Nos.: 04.70.-s, 11.25.Tq

1. Introduction

As is well known, quasinormal modes (QNMs) are the characteristic modes of per-

turbed black hole and black brane. In general, QNMs have complex frequencies,

whose real part represents the perturbation oscillation and whose imaginary part is

associated with the decay timescale of the perturbation (see Ref. 1 and references

therein). When the imaginary part is negative, the amplitude of the perturbation

decays to zero as time grows. Therefore, the quasinormal frequency (QNFs) con-

tains the important information of the stability of the black hole and black brane.

Because QNMs contain information about black holes such as mass and angular

momentum, they are useful in gravitational waves astronomy.2 Black holes pertur-

bation theory and QNMs spectrum of black holes in asymptotically flat spacetimes

have been studied by many authors.3–16

§Corresponding author

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http://dx.doi.org/10.1142/S0217732313501459

mailto:[email protected]





N. Bai et al.

QNMs of black holes in asymptotic AdS spacetimes are interesting from the

view point of AdS/CFT correspondence,17–19 which states that a large black hole

in AdS bulk correspondence to an (approximately) thermal state of the CFT on

the boundary. So, we can study QNMs of black holes in asymptotic AdS space-

times and apply them to strong coupled CFT perturbation theory. Indeed there

are intensive work on it, we lists some of them.20–28 Remarkably, in Ref. 25 they

find that quasinormal spectra of asymptotically AdS5 correspond to poles of the

retarded thermal correlators of dual four-dimensional N = 4 SYM theory according

to AdS/CFT.17–19

In recent years people generalize AdS/CFT to holographic duality whose bulk

geometry is non-AdS geometry, a typical example is that of Lifshitz metric, which

dual to non-relativistic field theory. The corresponding metric is given29

ds2 = − 1

r2zdt2 +

1

r2dr2 +

1

r2dx2 (1)

with a dynamical critical exponent z, where x represents the d spatial directions.

These metric has the following scale symmetry

t→ λzt , r → λr , x → λx . (2)

Among studying Einstein–Maxwell–Dilaton theories (see Refs. 30–38), one finds

larger classes of scaling solutions. Particularly, one may obtain the metric in the

following form:33

ds2 = r2θd

(

− 1

r2zdt2 +

1

r2dr2 +

1

r2dx2

)

, (3)

where the constants z and θ are dynamical and hyperscaling violation exponents,

respectively. The metric is spatially homogeneous and covariant under the following

scale transformation

t→ λzt , r → λr , x → λx , dsd+2 → λθd dsd+2 . (4)

Because of a nonzero θ, the distance is not invariant under the scaling which in the

context of AdS/CFT indicates violations of hyperscaling in the dual field theory.

When in (d + 1)-dimension field theory with hyperscaling violation (dual to back-

ground (3)) the entropy scales as T (d−θ)/z.39,41 In particular, for θ = d − 1 the

holographic entanglement entropy shows a logarithmic violation of the area law im-

plying that the dual theory exhibits an O(N2) Fermi surface.40,41 Due to the null

energy condition, the physically sensible dual field theory imposes the conditions

on the z and θ,42

(d− θ)(dz − θ − d) ≥ 0 , (z − 1)(d+ z − θ) ≥ 0 . (5)

Recently, there has been much interest in studying various properties of the geom-

etry, and attempts to classify the corresponding phases.42–54 Moreover, Hartnoll

and Shaghoulian computed the low energy spectral density of transverse current

operator55 under the geometry with hyperscaling violation byWKB approximation,

which is close to black holes perturbation theory.

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Quasinormal Frequencies of Black Branes

Some authors considered these holographic theories in finite temperature.42,45,49

In this note, we attempt to study the stability of the black brane with hyperscaling

violation under the perturbations in the special case motivated by Ref. 27 which

study QNMs in the Lifshitz black hole.

Here, we are interested in studying z = 3, θ = 1 and the spatial dimensions

d = 2 of black brane with hyperscaling violation for simplicity. The geometry we

used is a solution of the model49 with Q = 0. For the scalar field perturbation, the

equation of motion can be reduced to Heun equation. However, for the parameters

in the equation is not coincident, we cannot analytically solve the QNF like,27

where the analytical QNMs is found elegantly for the special z in the Lifshitz black

hole. Nevertheless, in Ref. 23 taught us how to do with Heun equation numerically,

in fact the authors solved the problem based on the continued fractions method.

This method has been applied to the problem of QNMs in asymptotically flat

spacetimes by Leaver.13 We use the method to calculate QNF of massless scalar

field and electromagnetic field with zero momentum for simplicity. We find the

QNFs have negative imaginary part, indicating the stability of the black brane

with hyperscaling violation which is consistent with its thermodynamic stability.

The paper is organized as follows. In the next section, we briefly Refs. 42

and 49. In Sec. 3, we analyze the QNF of massless scalar perturbation. In Sec. 4,

we discuss the QNF of electromagnetic field with zero momentum. The conclusion

is in Sec. 5.

2. Black Brane with Hyperscaling Violation

The black brane with hyperscaling violation is42

ds2 = r2θd

(

− f(r)

r2zdt2 +

1

r2f(r)dr2 +

1

r2dx2

)

, (6)

f(r) = 1−(

r

rh

)z+d−θ

, (7)

where the boundary is at r = 0. The metric (6) is a special solution with Q = 0 of

the Einstein–Maxwell–Dilaton model.49 We first review the model, then we discuss

the basic thermodynamic quantities briefly.

The action of the theory takes the following form:

S = − 1

16πG

∫

dd+2x√−g

[

R − 1

2(∂φ)2 + V (φ) − 1

4

2∑

i=1

eλiφF 2i

]

. (8)

We consider the following potential

V = V0eγφ . (9)

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N. Bai et al.

Here λ1, λ2, γ and V0 are free parameters of the model. The equations of motion

are

Rµν +V (φ)

dgµν =

1

2∂µφ∂νφ+

1

2

2∑

i=1

eλiφ

(

F ρiµFiρν − gµν

2dF 2i

)

,

∇2φ = −dV (φ)

dφ+

1

4

2∑

i=1

λieλiφF 2

i , ∇µ

(√−geλiφFµνi

)

= 0 .

(10)

Take the following ansatz for the metric, scalar and gauge field

ds2 = r2α(

−r2zf(r)dt2 + dr2

r2f(r)+ r2 dx2

)

, φ = φ(r) , Firt 6= 0 (11)

and assume that the other components of gauge fields are zero. The solution of the

equation of motion is49

ds2 = r−2 θd

(

− r2zf(r)dt2 +dr2

r2f(r)+ r2 dx2

)

,

F1rt =√

2(z − 1)(z + d− θ)eθ(1−d)/d+d√

2(d−θ)(z−1−θ/d)φ0

rd+z−θ−1 ,

F2rt = Q√

2(d− θ)(z − θ + d− 2)e−√

z−1−θ/d2(d−θ)

φ0r−(z+d−θ−1) ,

eφ = eφ0r√

2(d−θ)(z−1−θ/d) ,

(12)

with

f(r) = 1− m

rz+d−θ+

Q2

r2(z+d−θ−1). (13)

This is a charged black brane with charge parameter Q. Note that the boundary of

the geometry (12) is r = ∞. When Q = 0 we get the metric (6). The temperature

and the thermal entropy of black brane (6) are

T =1

4π

|d+ z − θ|rzh

, ST ∼ T(d−θ)

z . (14)

Thus, a positive specific heat imposes the condition

d− θ

z≥ 0 . (15)

3. Quasinormal Frequencies of Massless Scalar Perturbations

In order to get the QNFs, we consider the minimally coupled massless scalar de-

scribed by the Klein–Gordon equation

�Ψ =1√−g ∂µ

(√−ggµν∂νΨ)

= 0 . (16)

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Using Fourier decomposition

Ψ(r, t,x) = ψ(r)e−iωt+ik·x (17)

the equation of motion (16) becomes

ψ′′ +

[

f ′

f+θ + 1− z − d

r

]

ψ′ +

(

r2z−2ω2

f2− k2

f

)

ψ = 0 , (18)

where ′ is the derivative to r, k = |k|. We choose d = 2; z = 3, θ = 1, this choice

satisfies the null energy condition (5) and the thermodynamic stability (15) of the

black brane with hyperscaling violation, then f = 1−(

rrh

)4, Eq. (18) reduces to

ψ′′ +

−4 r3

r4h

1−(

rrh

)4 − 3

r

ψ′ +

(

r4ω2

(

1− ( rrh)4)2 − k2

1−(

rrh

)4

)

ψ = 0 . (19)

Using new coordinate u = 1−(

rrh

)2, Eq. (19) reads

ψ′′ +1 + (1− u)2

u(1− u)(2 − u)ψ′ +

(

(1− u)λ2

4u2(2− u)2− q2

4u(1− u)(2− u)

)

ψ = 0 , (20)

where ′ is the derivative to u, and λ = ωr3h, q = krh. Equation (20) has four regular

singularities at u = 0, 1, 2,∞, the corresponding pairs of characteristic exponents

being respectively {−iλ/4, iλ/4}; {0, 2}; {−λ/4, λ/4}; {0, 0}.QNMs are defined as solutions of Eq. (20) satisfying the “incoming wave” bound-

ary condition at the horizon u = 0 and the vanishing Dirichlet boundary condition

at spatial infinity u = 1. The first condition singles out the exponent ν(1)0 = −iλ/4

at u = 0. Let us make a transformation of ψ

ψ(u) = u−iλ4 (u− 2)−

λ4 φ(u) . (21)

Equation (20) is converted to the standard form of the Heun equation

φ′′ +

[

γ

u+

δ

u− 1+

ǫ

u− 2

]

φ′ +αβu−Q

u(u− 1)(u− 2)φ = 0 , (22)

where the parameters

α = β = −λ(1 + i)

4, γ = 1− iλ

2, δ = −1 , ǫ = 1− λ

2, (23)

Q =q2

4− λ(1 − i)

4+iλ2

8. (24)

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N. Bai et al.

This equation is very similar to Eqs. (2.5)–(2.7) in Ref. 23. We can use the method

employed in Ref. 23 to solve Eq. (22) for the QNMs. We look for solutions obeying

the boundary condition φ(0) = 1, φ(1) = 0. The coefficients of the Frobenius series

at u = 0

φ(u) =

∞∑

n=0

an(λ, q)un (25)

should satisfy the recursion relation

an+2 +An(λ, q)an+1 +Bn(λ)an = 0 , n ≥ 0 , (26)

where

An(λ, q) = − (n+ 1)(2δ + ǫ+ 3(n+ γ)) +Q

2(n+ 2)(n+ 1 + γ), (27)

Bn(λ) =(n+ α)(n+ β)

2(n+ 2)(n+ 1 + γ)(28)

and a0 = 1, a1 = Q2γ . Then, using the recursive relation,

rn ≡ an+1

an= − Bn(λ)

An(λ, q) + rn+1. (29)

Choosing n = 0, we get r0 = Q2γ . To obtain approximate expression for r0 we cut

the recursive relation at some sufficiently large n (n = 200 in our case) by setting

rn = 0,23 this gives a algebraic equation for λ at fixed q.

At q = 0, we find exactly from λ = 2n(±1 − i) of QNFsa by Mathematica

where n = 0, 1, . . . , and the QNMs is given by Heun polynomial which is a poly-

nomial solution of Heun equation and is analytic at three singularities. Because

λ = 2n(1− i) leads to Bn = 0, then from (29) rn+1 = 0, we can see that the series

solution is a polynomial from (26). This result indicates that the black brane with

hyperscaling violation is stable under the massless scalar field perturbation with

zero momentum.

4. Quasinormal Frequencies of Electromagnetic Perturbations

Quasinormal modes for electromagnetic perturbations are physically more interest-

ing than those for scalars because the corresponding fluctuations couple to con-

served symmetry currents in the dual quantum field theory. For example, in N = 4

SYM theory the global currents associated to R-charges map to a vector field in

the bulk AdS5, it is important that quasinormal spectra of asymptotically AdS5

correspond to poles of the retarded thermal correlators of dual four-dimensional

N = 4 SYM theory25 according to AdS/CFT.17–19 So, we would like to calculate

the QNF of electromagnetic perturbations.

aThe symmetry is manifest in Eq. (20).

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The Maxwell equation is

1√−g∂ν[√−ggµρgνσFρσ

]

= 0 , (30)

where Fµν = ∂µAν − ∂νAµ.

We choose the gauge Ar = 0 in the metric (6) and make the Fourier decompo-

sition

Ai(r, t,x) = Ai(r)e−iωt+ik·x , (31)

where i = t, x. We are still interested in d = 2; z = 3, θ = 1, f = 1−(

rrh

)4. We also

choose k = (0, q) along y-direction for simplifying calculation, Eq. (30) becomes

the following set of equations

ωA′t +

qf

r4A′

y = 0 , (32)

A′′t +

2

rA′

t −1

f(qωAy + q2At) = 0 , (33)

A′′y +

f ′

fA′

y −2

rA′

y +r4

f2(ωqAt + ω2Ay) = 0 , (34)

A′′x +

f ′

fA′

x − 2

rA′

x +

(

r4ω2

f2− q2

f

)

Ax = 0 . (35)

Note that ′ is the derivative to r, Eqs. (32) and (33) imply (34). We can reduce

Eqs. (32)–(34) to a single equation by using Eqs. (32) and (33). One obtains

A′′′t +

(

f ′

f+

2

r

)

A′′t +

(

− 2

r2+

2f ′

rf− q2

f+ω2r4

f2

)

A′t = 0 . (36)

So Eqs. (33), (35) and (36) are the relevant equation of the vector field. With the

coordinates transformation u = 1−(

rrh

)2, Eq. (36) turns to

A′′′t +

9(u− 1)2 − 5

2u(u− 1)(u− 2)A′′

t − 1

4u(u− 1)(u − 2)

×(

12(1− u) + l2 + λ2 +λ2

u(u− 2)

)

A′t = 0 , (37)

where ′ is the derivative to u, and λ = ωr3h, l = qrh. Equation (37) is a second-

order differential equation for A′t, we define ψ(u) = A′

t(u) and impose “ingoing

condition” at the horizon. We find Heun equation from (37) by changing ψ(u) to

u−iλ4 (u − 2)

−λ4 φ(u)

φ′′ +

[

γ

u+

δ

u− 1+

ǫ

u− 2

]

φ′ +αβu−Q

u(u− 1)(u− 2)φ = 0 , (38)

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Table 1. The lowest 12 QNFs for l = 0.

n Reλn Imλn

1 ±4.5504968 −6.7878459

2 ±4.9110883 −7.2325897

3 ±5.3058487 −7.6809562

4 ±5.7184483 −8.1502482

5 ±6.1342337 −8.6363504

6 ±6.5489066 −9.1327727

7 ±6.9617735 −9.6355908

8 ±7.3729669 −10.142610

9 ±7.7827530 −10.652549

10 ±8.1913795 −11.164617

11 ±8.5990495 −11.678291

12 ±9.0059236 −12.193213

where the parameters

α =1

4

(

−9−√

33− (4 + 4i)λ+ (1 + i)λ)

,

β =1

4

(

−9 +√

33− (4 + 4i)λ+ (1 + i)λ)

,

γ = 2− iλ

2, δ =

5

2, ǫ = 1− λ

2,

Q =l2

4+ 3− λ

4− 3i

λ

2+ i

λ2

8.

(39)

We require that φ be analytic at the u = 1 which provides a necessary boundary

condition as in Ref. 24. Then we can use the method in the scalar case to calculate

the QNF of electromagnetic perturbations analogously.

For l = 0, the lowest 12 QNFs are listed in Table 1 and shown in Fig. 1.

The imaginary of the QNFs is negative which means that the black brane with

hyperscaling violation is stable against Maxwell field perturbation.

5. Conclusions

In this note, we numerically calculate the QNF of massless scalar field and Maxwell

field for black brane with hyperscaling violation at special parameters by using

the method in Ref. 23. We find that the QNFs of both perturbations have neg-

ative imaginary parts, so imply the stability of the black brane. There are some

problems needed to study. We can consider the spatial momentum k 6= 0 to study

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Re Λn

Im Λn

-5 5

-12

-10

-8

-6

Fig. 1. The lowest 12 QNFs in the complex λ-plane.

the dispersion relation ω(|k|) of the two perturbations. For electromagnetic per-

turbations, when the momentum along y direction q 6= 0 Eqs. (34) and (35) are

different, we must calculate quasinormal spectrum of longitudinal field perturbation

and transverse field perturbation separately. It is important to study the gravita-

tional perturbations of the black brane with hyperscaling violation for the stability.

It would be interesting to investigate the perturbations in the metric with other

special d, z and θ.

Acknowledgments

We would like to thank Emanuele Berti and Vitor Cardoso for their mathematica

codes. We also thank Li Li for helpful discussion.

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quasinormal frequencies of black branes with hyperscaling violation

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