# Stability and thermodynamics of brane black holes

**ABSTRACT** We consider scalar and axial gravitational perturbations of black hole solutions in brane world scenarios. We show that perturbation dynamics is surprisingly similar to the Schwarzschild case with strong indications that the models are stable. Quasinormal modes and late-time tails are discussed. We also study the thermodynamics of these scenarios verifying the universality of Bekenstein's entropy bound as well as the applicability of 't Hooft's brickwall method.

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**ABSTRACT:**In this paper we investigate the strong gravitational lensing in a charged squashed Kaluza-Klein black hole. We suppose that the supermassive black hole in the galaxy center can be considered by a charged squashed Kaluza-Klein black hole and then we study the strong gravitational lensing theory and estimate the numerical values for parameters and observables of it. We explore the effects of the scale of extra dimension $\rho_0$ and the charge of black hole $\rho_q$ on these parameters and observables.Astrophysics and Space Science 05/2012; 343(2). · 2.06 Impact Factor - SourceAvailable from: Roldao da Rocha[Show abstract] [Hide abstract]

**ABSTRACT:**The perihelion precession, the deflection of light, and the radar echo delay are classical tests of General Relativity here used to probe brane world topologically charged black holes in a f(R) bulk and to constrain the parameter that arises from the Shiromizu-Maeda-Sasaki procedure applied to a f(R) bulk as well. The existing Solar system observational data constrain the possible values of the tidal charge parameter and the effective cosmological constant including f(R) brane world effects. We show that the observational/experimental data for both perihelion precession and radar echo delay make the black hole space of parameters to be more strict than the ones for the Dadhich, Maartens, Papadopoulos and Rezania (DMPR) black hole geometry. Furthermore, the deflection of light constrains the tidal charge parameter similarly as the DMPR black holes due to a peculiarity in the equation of motion.07/2014; - SourceAvailable from: Alan Bendasoli Pavan[Show abstract] [Hide abstract]

**ABSTRACT:**In this work we describe an interesting application of a simple derivative-free optimization method to extract the quasinormal modes (QNM's) of a massive scalar field propagating in a 4-dimensional Schwarzschild anti-de Sitter black hole (Sch-AdS$_4$). In this approach, the problem to find the QNM's is reduced to minimize a real valued function of two variables and does not require any information about derivatives. In fact, our strategy requires only evaluations of the objective function to search global minimizers of the optimization problem. Firstly, numerical experiments were performed to find the QNM's of a massless scalar field propagating in intermediate and large Sch-AdS$_4$ black holes. The performance of this optimization algorithm was compared with other numerical methods used in previous works. Our results showed to be in good agreement with those obtained previously. Finally, the massive scalar field case and its QNM's were also obtained and discussed.03/2014;

Page 1

arXiv:gr-qc/0604033v2 27 Aug 2006

Stability and thermodynamics of brane black holes

E. Abdalla,∗B. Cuadros-Melgar,†and A. B. Pavan‡

Instituto de F´ ısica, Universidade de S˜ ao Paulo

C.P. 66318, 05315-970, S˜ ao Paulo-SP, Brazil

C. Molina§

Escola de Artes, Ciˆ encias e Humanidades, Universidade de S˜ ao Paulo

Av. Arlindo Bettio 1000, CEP 03828-000, S˜ ao Paulo-SP, Brazil

We consider scalar and axial gravitational perturbations of black hole solutions in brane world

scenarios. We show that perturbation dynamics is surprisingly similar to the Schwarzschild case with

strong indications that the models are stable. Quasinormal modes and late-time tails are discussed.

We also study the thermodynamics of these scenarios verifying the universality of Bekenstein’s

entropy bound as well as the applicability of ’t Hooft’s brickwall method.

PACS numbers: 04.70.Dy,98.80.Cq, 97.60.Lf,04.50.+h

I.INTRODUCTION

The extra dimensions idea had its origin in the seminal

works by Kaluza and Klein in the 20’s [1] and gained mo-

mentum in the context of string theory in the last decades

[2]. Recent developments on higher–dimensional grav-

ity resulted in a number of interesting theoretical ideas

such as the brane world concept. The essence of this

string inspired model is that Standard Model fields are

confined to a three dimensional hypersurface, the brane,

while gravity propagates in the full spacetime, the bulk.

The simplest models in this context (abbreviated here as

RSI and RSII), proposed by Randall and Sundrum [3],

describe our world as a domain wall embedded in a Z2-

symmetric five dimensional anti–de Sitter (AdS) space-

time. The RSI model proposes a mechanism to solve the

hierarchy problem by a small extra dimension, while the

RSII model considers an infinite extra dimension with a

warp factor which ensures the localization of gravity on

our brane.

Black holes are important sources of gravitational

waves that are expected to be detected by the cur-

rent and upcoming generation of experiments. This will

open up a new window for testing modifications of gen-

eral relativity. The simplest case of gravitational col-

lapse in the standard four dimensional world is described

by the 4-dimensional Schwarzschild metric.

dimensional scenario it would be natural to ask whether

matter confined on the brane after undergoing gravita-

tional collapse can still be described by a Schwarzschild-

type metric.The most natural generalization in the

RSII model corresponds to a black string infinite in the

fifth dimension, whose induced metric on the brane is

purely Schwarzschild [4]. However, although the cur-

In the 5-

∗Electronic address: eabdalla@fma.if.usp.br

†Electronic address: bertha@fma.if.usp.br

‡Electronic address: alan@fma.if.usp.br

§Electronic address: cmolina@usp.br

vature scalars are everywhere finite, the Kretschmann

scalar diverges at the AdS horizon at infinity, which turns

the above solution into a physically unsuitable object.

It has been argued that there exists a localized black

cigar solution with a finite extension along the extra di-

mension due to a Gregory-Laflamme [5] type of instabil-

ity near the AdS horizon. A class of such a solution has

been found by Casadio et al. [6, 7] using the projected

Einstein equations on the brane derived by Shiromizu et

al. [8]. It has the desired “pancake” horizon structure

ensuring a non-singular behavior in the curvature and

Kretschmann scalars at least until the order of the mul-

tipole expansion considered there. In fact, this solution

belongs to a class of black hole solutions found later by

Bronnikov et al. [9], who also classified several possi-

ble brane black holes obtained from the Shiromizu et al.

projected equations [8] in two families according to the

horizon order. For such spacetimes only horizons of order

1 or 2 are possible, but not higher than that.

In this paper we are interested in the study of black

holes from the point of view of a brane observer, as our-

selves. We analyze some characteristics of Bronnikov et

al. solutions. Firstly, some aspects of the thermody-

namics are studied. Black Hole Thermodynamics was

constructed when Bekenstein proposed the proportional-

ity law between the entropy and the horizon area [10].

The discovery of Hawking radiation validated this pro-

posal and established the proportionality factor 1/4 in

a precise way [11] leading to the well-known Bekenstein-

Hawking formula,

SBH=Area

4G

.(1)

One way to compute the entropy based on a semi-

classical description of a scalar field was proposed by ’t

Hooft [12] and it is known as the brick wall method, which

was frequently used later in several contexts [13]. When

applying this method to a Schwarzschild black hole, ’t

Hooft found that the entropy was proportional to the

area, as expected, but additionally it had a α−2correc-

tion, α being the proper distance from the horizon to

Page 2

2

the wall. This term was later interpreted as a one-loop

correction to the Bekenstein-Hawking formula, since it

can be absorbed as a renormalization of the gravitational

coupling constant G [14].

Another interesting feature of black hole thermody-

namics is the existence of an upper bound on the entropy

of any neutral system of energy E and maximal radius

R in the form S ≤ 2πER, proposed by Bekenstein [15].

This bound becomes necessary in order to enforce the

generalized second law of thermodynamics (GSL).

Besides thermodynamical results, we also consider the

response of a brane black hole perturbation which should

represent some damped oscillating signal. It can be de-

composed with Laplace transformation techniques into a

set of so-called quasinormal modes (QNM). The QNMs

of black holes are important because they dominate in

the intermediate late-time decay of a perturbation and

do not depend upon the way they were excited. They

depend only on the parameters of a black hole and are,

therefore, the “footprints” of this structure. The time-

independent problem for perturbations of a brane black

hole turns out to be quite similar to that for a black hole:

one has to find the solutions of the wave-like equations

satisfying the appropriate boundary conditions, which we

shall further discuss in detail.

The paper is organized as follows, in Sec. II the brane

black holes are presented. Sec. III discusses the ther-

modynamical properties of the solutions thus considered.

Sec. IV treats the question of perturbation and stability

of these objects. Wave-like equations for the perturba-

tions are derived. In Sec. V an analysis of the quasinor-

mal modes and late-time behavior is developed. In Sec.

VI we summarize our results, and some final comments

are made.

II. BRANE BLACK HOLE SOLUTIONS

The vacuum Einstein equations in 5 dimensions, when

projected on a 4-dimensional spacetime and after intro-

ducing gaussian normal coordinates (xµ, with µ = 0...3,

and z), lead to the gravitational equation on the 3-brane

given by [8]

R(4)

µν= Λ4g(4)

µν− Eµν

(2)

where Λ4is the brane cosmological constant, and Eµν is

proportional to the (traceless) projection on the brane of

the 5-dimensional Weyl tensor.

The only combination of the Einstein equations in a

brane world that can be written unambiguously without

specifying Eµνis their trace [6, 7, 9],

R(4)= 4Λ4. (3)

It is clear that this equation, also known as the hamilto-

nian constraint in the ADM decomposition of the met-

ric, is a weaker restriction than the purely 4-dimensional

equation Rµν = 0, which, in fact, is equivalent to Eq.

(2), provided that we know the structure of Eµν.

In order to obtain four dimensional solutions of Eq.

(3), we choose the spherically symmetric form of the 4-

dimensional metric given by

ds2= −A(r)dt2+

dr2

B(r)+ r2(dθ2+ sin2θdφ2).(4)

We relax the condition A(r) = B(r), which is acciden-

tally verified in four dimensions but, in fact, there is no

reason for it to continue to be valid in this scenario. In

this spirit, black hole and wormhole solutions [6, 9, 16, 17]

as well as star solutions on the brane [18] have been ob-

tained in the last years. We should mention that even

without relaxing this condition, previous solutions have

also been found [19, 20].

In this context the hamiltonian constraint can be writ-

ten explicitly in terms of A and B as [6, 9]

B

?

A′′

A

−1

2

?A′

A

?2

+1

2

A′

A

B′

B+2

r

?B′

B+A′

A

??

−2

r2(1 − B) = 4Λ4, (5)

with prime (’) denoting differentiation with respect to r.

We will center our attention in the black hole type

solutions which can be obtained by one of the following

algorithms BH1 and BH2. These are subclasses of the

corresponding algorithms in [9] (where the parameter s

in this reference is set to 1).

First Algorithm (BH1): Specify a function A(r), positive

and analytical in a neighborhood of the event horizon

R[r], in such a way that 4A + rA′> 0 in R[r], and

A ≈ (r − rh) as r → rh. Then B(r) is given by the

general solution of (5) with vanishing brane cosmological

constant,

B(r) =

2Ae3Γ

r(4A + rA′)2

×

??r

rh

(4A + rA′)(2 − r2R)e−3Γdr + C

?

,(6)

where C is an integration constant and

Γ(r) =

?

A′

4A + rA′dr.(7)

For C ≥ 0 we have a black hole metric with a horizon at

r = rh, which is simple if C > 0 and of the order 2+p if

both C = 0, and Q has the behavior,

Q(r) = 2 − r2R ≈ (r − rh)p, near r = rh, p ∈ N. (8)

Second Algorithm (BH2): Specify a function A(r), posi-

tive and analytical in a neighborhood of the event hori-

zon R[r], in such a way that 4A + rA′> 0 in R[r], and

A ≈ (r − rh)2as r → rh. Then B(r) is again given by

(6). The black hole metric appears when C = 0 with

Page 3

3

a horizon at r = rh of the order 2 + p if Q(r) behaves

according to (8).

Both algorithms lead to double horizons in the case

C = 0, if Q(rh) > 0. A case in point of the first algorithm

is the solution with the metric element A(r) having the

usual form of a Schwarzschild black hole found by Casa-

dio, Fabbri, and Mazzacurati (CFM solution) [6, 7] given

by

A(r) = 1 −2M

r

,

B(r) =

(1 −2M

r)(1 −Mγ

1 −3M

2r)

2r

, (9)

where γ is an integration constant. The event horizon is

localized at rh = 2M and the singularity at r = 3M/2

instead of r = 0. Notice that the Schwarzschild solution

is recovered with γ = 3. In this work we are restricted to

the case when γ < 4.

Another interesting example of this algorithm is the

metric with zero Schwarzschild mass [9] given by

A(r) = 1 −h2

r2, h > 0,

B(r) =

?

1 −h2

r2

??

1 +

C − h

√2r2− h2

?

, (10)

whose horizon r = h is simple if C > 0 and double if

C = 0. The singularity occurs at r = h/√2. This exam-

ple shows that in the brane world context a black hole

may exist without matter and without mass, only as a

tidal effect from the bulk gravity. However, there is a spe-

cial situation when h2can be related to a 5-dimensional

mass, namely, C = h. In this case Eq. (10) is the induced

metric of a 5-dimensional Schwarzschild black hole, as de-

scribed in [21], where the chosen background was ADD-

type.

III. BLACK HOLE THERMODYNAMICS

In order to study the thermodynamical properties of

the brane black holes generated by the BH1 and BH2

algorithms, we use the following expressions of the metric

coefficients near the horizon

A(r) = A1(r − rh) + O((r − rh)2)

B(r) = B1C(r − rh) + B2(r − rh)2+ O((r − rh)3),

for BH1 algorithm with A1,B1,B2> 0 and C being an

integration constant that defines the black hole family,

and

(11)

A(r) = A2(r − rh)2+ O(r − rh)3,

B(r) = B3C + B4(r − rh)2+ B5(r − rh)3

+O((r − rh)4),

for BH2 algorithm, being C the family parameter again.

(12)

We will show here the calculation for the BH2 family,

which turns out to be more interesting, since the metric

coefficients expansion (12) is different from the standard

one (11). However, we will display certain quantities for

both families wherever it is relevant.

We first consider the issue of the entropy bound. The

surface gravity at the event horizon is given by

√A1B1C

√A2B3C

κ =

?

1

2

for BH1 family,

for BH2 family.

(13)

Let us consider an object with rest mass m and proper

radius R descending into a BH2 black hole. The con-

stants of motion associated to t and φ are [22]

E = πt,J = −πφ, (14)

where

πt = gtt˙t,

πφ = gφφ˙φ. (15)

In addition,

m2= −πµπµ. (16)

For simplicity we just consider the equatorial motion of

the object, i.e., θ = π/2. The quadratic equation for the

conserved energy E of the body coming from (14)-(16) is

given by

αE2− 2βE + ζ = 0, (17)

with

α = r2,

β = 0,

ζ = A2(r − rh)2(J2+ m2r2).

The gradual approachto the black hole must stop when

the proper distance from the body’s center of mass to the

black hole horizon equals R, the body’s radius,

(18)

?rh+δ(R)

rh

dr

?B(r)

= R.(19)

Integrating this equation we obtain the expression for δ,

δ =

?

1

4CR2B1

R√B3C

for BH1 family,

for BH2 family.

(20)

Solving (17) for the energy and evaluating at the point

of capture r = rh+ δ we have

Ecap≈

?A2(J2+ m2r2

h) δ

rh

(21)

This energy is minimal for a minimal increase in the black

hole surface area, J = 0, such that

Emin=

?

A2mδ . (22)

Page 4

4

From the First Law of Black Hole Thermodynamics we

know that

dM =κ

2dAr, (23)

where Aris the rationalized area (Area/4π), and dM =

Emin is the change in the black hole mass due to the

assimilation of the body. Thus, using (13) we obtain

dAr= 2mR. (24)

Assuming the validity of the GSL, SBH(M + dM) ≥

SBH(M) + S, we derive an upper bound to the entropy

S of an arbitrary system of proper energy E,

S ≤ 2πER. (25)

This result coincides with that obtained for the purely

4-dimensional Schwarzschild solution, and it is also inde-

pendent of the black hole parameters [15]. It shows that

the bulk does not affect the universality of the entropy

bound.

Let us find now the quantum corrections to the classi-

cal BH entropy. We consider a massive scalar field Φ in

the background of a BH2 black hole satisfying the mas-

sive Klein-Gordon equation,

?? − m2?Φ = 0.(26)

In order to quantize this scalar field we adopt the sta-

tistical mechanical approach using the partition function

Z, whose leading contribution comes from the classical

solutions of the euclidean lagrangian that leads to the

Bekenstein-Hawking formula. In order to compute the

quantum corrections due to the scalar field we use the ’t

Hooft’s brick wall method, which introduces an ultravio-

let cutoff near the horizon, such that

Φ(r) = 0 atr = rh+ ε,(27)

and an infrared cutoff very far away from the horizon,

Φ(r) = 0 atr = L ≫ M .(28)

Thus, using the black hole metric (4) and the Ansatz

Φ = e−iEtR(r)Yℓm(θ,φ), Eq.(26) turns out to be

E2

AR +

?

B

A

1

r2∂r

?

r2√

AB∂rR

?

−

?ℓ(ℓ + 1)

r2

+ m2

?

R = 0.(29)

Using a first order WKB approximation with R(r) ≈

eiS(r)in (29) and taking the real part of this equation we

can obtain the radial wave number K ≡ ∂rS as being,

K = B−1/2

?E2

A

−

?ℓ(ℓ + 1)

r2

+ m2

??1/2

. (30)

Now we introduce the semiclassical quantization con-

dition,

π nr=

?L

rh+ε

K(r,ℓ,E)dr.(31)

In order to compute the entropy of the system we first

calculate the Helmholtz free energy F of a thermal bath

of scalar particles with temperature 1/β,

F =1

β

?

dℓ(2ℓ + 1)

?

dnrln(1 − e−βE).(32)

Integrating by parts, using (30) and (31), and performing

the integral in ℓ we have

F = −2

3π

?L

rh+ε

drA−3/2B−1/2r2

?

dE(E2− Am2)3/2

eβE− 1

.

(33)

Following ’t Hooft’s method, the contribution of this

integral near the horizon is given by

F ≈ −2r3

h

3π

? ¯L

1+¯ ε

dy(A2r2

h)−3/2(y − 1)−3

(B3C)1/2

?∞

0

dE

E3

eβE− 1,

(34)

where y = r/rh, ¯ ε = ε/rh, and¯L = L/rh. Notice that

since A goes to 0 near the horizon, the mass term in Eq.

(33) becomes negligible.

Therefore, the leading divergent contribution to F

(with ε → 0) is

Fε= −r2

45β4

hπ3

(A2)−3/2

(B3C)1/2ε2. (35)

The corresponding entropy is then,

Sε= β2∂F

∂β=

4r2

45(B3C)1/2ε2β3.

hπ3(A2)−3/2

(36)

Using the value of the Hawking temperature TH= 1/β =

κ/2π, with κ given in (13) we have

Sε=

?

r2

r2

hB1C/(360ε)

hB3C/(90ε2)

for BH1 family,

for BH2 family.

(37)

We can express our result in terms of the proper thick-

ness α given by

α =

?rh+ε

rh

dr

?B(r)

≈

?

2√ε/√B1C

ε/√B3C

for BH1 family,

for BH2 family.

(38)

Thus,

Sε=

r2

h

90α2,(39)

or in terms of the horizon area Area = 4πr2

h,

Sε=

Area

360πα2,(40)

Page 5

5

which is the same quadratically divergent correction

found by ’t Hooft [12] for the Schwarzschild black hole

and by Nandi et al. [23] for the CFM brane black hole.

Thus, we see that the correction is linearly dependent on

the area.

The calculation of the entropy bound and entropy

quantum correction for the BH1 black hole is similar and

leads to the same results shown in (25) and (40).

IV.

AND GRAVITATIONAL PERTURBATIONS

PERTURBATIVE DYNAMICS: MATTER

For simplicity we model the matter field by a scalar

field Φ confined on the brane obeying the massless

(m = 0) version of the Klein-Gordon equation (26). We

expect that massive fields (m ?= 0) should show rather

different tail behavior, but such cases will not be treated

in the present paper.

Using the decomposition of the scalar field as

Ψ(t,r,θ,φ) = R(t,r)Yℓ,m(θ,φ) in terms of the angular,

radial, and time variables we have the equation

−∂2Rℓ

∂t2+∂2Rℓ

∂r2

⋆

= Vsc(r(r⋆))Rℓ, (41)

with the tortoise coordinate r⋆defined as

dr⋆(r)

dr

=

1

?A(r)B(r).(42)

The effective potential Vscis given by

Vsc= A(r)ℓ(ℓ + 1)

r2

+

1

2r[A(r)B′(r) + A′(r)B(r)] .

(43)

In order to address the problem of black hole stability

under gravitational perturbations, we consider first order

perturbation of Rαβ = −Eαβ, where Rαβ and Eαβ are

the Ricci tensor and the projection of the five dimensional

Weyl tensor on the brane, respectively. In general, the

gravitational perturbations depend on the tidal pertur-

bations, namely, δEαβ. Since the complete bulk solution

is not known, we shall use the simplifying assumption

δEαβ = 0. This assumption can be justified at least in

a regime where the perturbation energy does not exceed

the threshold of the Kaluza-Klein massive modes. Anal-

ysis of gravitational shortcuts [24] also supports this sim-

plification showing that gravitational fields do not travel

deep into the bulk. On the other hand, since we ignore

bulk back-reaction, the developed perturbative analysis

should not describe the late-time behavior of gravita-

tional perturbations. Within such premises we obtain

the gravitational perturbation equation

δRαβ= 0. (44)

We will consider axial perturbations in the brane ge-

ometry, following the treatment in [25]. To make this

section more self contained, we will briefly describe the

treatment used in [25]. The metric Ansatz with sufficient

generality is

ds2= e2νdt2− e2ψ?dφ2− ωdt − q2dθ2− q3dφ2?

−e2µ2dr2− e2µ3dθ2

(45)

where we adopt here a more convenient notation,

e2ν= A(r), e2µ2=

1

B(r), e2µ3= r2, e2ψ= r2sin2θ.

(46)

Axial perturbations in the brane metric (4) are charac-

terized by non-null (but first order) values for ω, q2, and

q3in Eq. (45). We refer to [25] for further details.

In order to decouple the system, it is adopted the

change of variables

Qαβ= qα,β− qβ,α,(47)

and

Qα0= qα,0− ω,α, (48)

with α,β = 2,3. We denote partial differentiation with

respect to t, θ and φ by “,0”, “,1” and “,2”, respectively.

The perturbations are then described by

(e3ψ+ν−µ2−µ3Q23),3e−3ψ+ν−µ3+µ2=

= −(ω,2− q2,0),0, (49)

(e3ψ+ν−µ2−µ3Q23),2e−3ψ+ν+µ3−µ2=

= (ω,3− q3,0),0.(50)

Setting Q(t,r,θ) = exp(3ψ+ν−µ2−µ3)Q23, Eqs.(49)

and (50) can be combined as

r4

?

B(r)

A(r)

∂

∂r

?1

= −sin3θ∂

r2

?

A(r)B(r)∂Q

∂r

?

∂Q

∂θ

− r2

?

B(r)

A(r)

∂2Q

∂t2=

∂θ

?

1

sin3θ

?

.(51)

We further separate variables and write Eq.(51) in the

form of a Schr¨ odinger-type wave equation by introduc-

ing Q(t,r,θ) = rZℓ(t,r)C−3/2

C−3/2

ℓ+2(θ) is the Gegenbauer function. Thus, the axial

gravitational perturbations are given by an equation of

motion with the form given in (41) with the effective po-

tential

ℓ+2(θ), and r = r(r⋆), where

Vgrav(r) = A(r)(ℓ + 2)(ℓ − 1)

r2

+2A(r)B(r)

r2

−1

2r[A(r)B′(r) + A′(r)B(r)] .(52)

Page 6

6

V.SPECIFIC MODELS

A.Overview of the results

The equations of motion for the scalar and axial grav-

itational perturbations give us a tool to analyze the dy-

namics and stability of the black hole solutions in both

the CFM and “zero mass” black hole backgrounds. Of

particular interest are the quasinormal modes. They are

defined as the solutions of Eq. (41) which satisfy both

boundary conditions that require purely out-going waves

at (brane) spatial infinity and purely in-going waves at

the event horizon,

lim

x→∓∞Ψe±iωx= constant, (53)

with Ψ = Rℓ and Zℓ for the scalar and gravitational

perturbations, respectively.

In order to analyze quasinormal mode phase and late-

time behavior of the perturbations, we apply a numerical

characteristic integration scheme based in the light-cone

variables u = t − r⋆ and v = t + r⋆ used, for exam-

ple, in [26, 27, 28]. In addition, to check some results

obtained in “time–dependent” approach we employ the

semi-analytical WKB-type method developed in [29] and

improved in [30]. Both approaches show good agreement

for the fundamental overtone which is the dominating

contribution in the signal for intermediate late-time.

At a qualitative level we have observed the usual pic-

ture in the perturbative dynamics for all fields and ge-

ometries considered here.

regime, it follows the quasinormal mode phase and finally

a power-law tail. In contrast to the 5-dimensional model

in [31], in the present context we do not observe Kaluza-

Klein massive modes in the late-time behavior of the per-

turbations. This is actually expected, since our treat-

ment for gravitational perturbations neglects the back-

reaction from the bulk. Still, as discussed in section IV,

our results should model the quasinormal regime.

A necessary condition for the stability of the geome-

tries we have considered is that these backgrounds must

be stable under the perturbations modelled by the effec-

tive potentials (43) and (52).

If the effective potential (V ) is positive definite, the

differential operator

After the initial transient

D = −∂2

∂r2

⋆

+ V(54)

is a positive self-adjoint operator in the Hilbert space

of square integrable functions of r⋆, and, therefore, all

solutions of the perturbative equations of motion with

compact support initial conditions are bounded.

However, as we will see, the effective potentials may

be non-positive definite for certain choices of the param-

eters in Eqs.(9)-(10). Nevertheless, even when the effec-

tive potential is not positive definite, we do not observe

unbounded solutions.

0

5

10

15

r*

0

3

6

9

12

Vsc

CFM

101

102

103

104

t

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

| Rl=0(t,r*=0) |

γ = -1000

γ = -100

01234

5

-0,3

0

0,3

0,6

~ t-3

Figure 1: (Left) Effective potential for the scalar perturba-

tions in the CFM background for very negative values of γ.

Negative peaks are displayed in detail. (Right) Bounded evo-

lution of the scalar field perturbation with such effective po-

tentials. The dotted line is the late-time power-law tail. The

parameters are ℓ = 0 and M = 1.

-2024

6

8

r*

-4

-3

-2

-1

0

1

2

3

4

Vgrav

CFM

101

102

t

103

10-15

10-12

10-9

10-6

10-3

100

| Zl=0(t,r*=0) |

γ = -500

γ = -100

~ t-7

Figure 2: (Left) Effective potential for the axial gravitational

perturbations in the CFM background for very negative values

of γ. (Right) Bounded evolution of the gravitational field

perturbation with such effective potentials. The dotted line

is the late-time power-law tail. The parameters in the graphs

are ℓ = 2 and M = 1.

Using both high order WKB method and direct nu-

merical integration of the equations of motion a numeri-

cal search for quasinormal modes with positive imaginary

part was performed for scalar and gravitational pertur-

bations. One of the most important results in this work

is that no unstable mode was observed. Furthermore, the

perturbative late-time tails have power-law behavior (in

one case an oscillatory decay with power-law envelope).

Page 7

7

Table I: Fundamental quasinormal frequencies for the scalar perturbation around the CFM black hole for several values of the

parameters γ and ℓ. The black hole mass is set to M = 1.

Direct IntegrationWKB-3thorderWKB-6thorder

ℓγ

Re(ω0)-Im(ω0)Re(ω0)-Im(ω0)Re(ω0)-Im(ω0)

1

1

1

1

1

1

1

-5

-2

0

1

2

3

3.9

0.28580

0.29201

0.29337

0.29400

0.29415

0.29283

0.29076

0.19779

0.16608

0.14138

0.12853

0.11378

0.098045

0.082598

0.208204

0.276143

0.299359

0.298384

0.294679

0.291114

0.287181

0.225080

0.181776

0.154590

0.134468

0.115208

0.0980014

0.0820285

0.305499

0.309350

0.236213

0.252449

0.293168

0.292910

0.289628

0.181027

0.122589

0.165871

0.171879

0.120632

0.0977616

0.0811812

2

2

2

2

2

2

2

-5

-2

0

1

2

3

3.9

0.48053

0.48266

0.48413

0.48449

0.48447

0.48317

0.48178

0.18951

0.16069

0.13815

0.12570

0.11206

0.097097

0.080778

0.488726

0.488518

0.486195

0.485289

0.484420

0.483211

0.481091

0.207518

0.166137

0.139258

0.126043

0.112159

0.09680485

0.08098300

0.433498

0.451478

0.485925

0.485911

0.484691

0.483642

0.481705

0.183580

0.193079

0.142572

0.126758

0.112283

0.0967661

0.0808983

Table II: Fundamental quasinormal frequencies for the axial gravitational perturbation around the CFM black hole for several

values of the parameters γ and ℓ. The black hole mass is set to M = 1.

Direct Integration

Re(ω0)

WKB-3thorder

Re(ω0)

WKB-6thorder

Re(ω0)

ℓγ

-Im(ω0) -Im(ω0) -Im(ω0)

2

2

2

2

2

2

2

-5

-2

0

1

2

3

3.9

0.36409

0.37049

0.37359

0.37457

0.37483

0.37368

0.36961

0.18017

0.16062

0.13539

0.12138

0.10604

0.088957

0.072435

0.401345

0.391442

0.384611

0.381053

0.377306

0.373162

0.368552

0.197274

0.163223

0.137147

0.122624

0.106767

0.0892174

0.0717786

0.418575

0.402747

0.389781

0.383017

0.377126

0.373619

0.371935

0.193276

0.163222

0.139044

0.124656

0.107996

0.0888910

0.0712303

3

3

3

3

3

3

3

-5

-2

0

1

2

3

3.9

0.59476

0.59835

0.59993

0.60033

0.60026

0.59947

0.59700

0.18365

0.15845

0.13527

0.12238

0.10832

0.092690

0.077176

0.608026

0.604567

0.602594

0.601646

0.600614

0.599265

0.597227

0.191340

0.159482

0.135600

0.122520

0.108374

0.0927284

0.0767434

0.613069

0.605032

0.601901

0.601033

0.600375

0.599443

0.597584

0.194241

0.161682

0.136695

0.123070

0.108525

0.0927025

0.0767411

B.CFM brane black holes

We first consider scalar perturbations in the CFM sce-

nario. The tortoise coordinate r⋆after the explicit inte-

gration is

r⋆(r) = T1(r) + T2(r) + T3(r)(55)

with

T1(r) =1

2

?

(2r − γM)(2r − 3M), (56)

T2(r) =M(5 + γ)

4

ln[4r − M(3 + γ) + 2T1(r)], (57)

T3(r) = −

2M

√4 − γ

×ln

?M(5 − γ)r − M2(6 − γ) + M√4 − γ T1(r)

r − 2M

?

.

(58)

Page 8

8

The scalar and axial gravitational effective potentials

for perturbations in the CFM background (respectively,

VCFM

sc

and VCFM

are given by

grav) in terms of the parameters M and γ

VCFM

sc

(r) =

?

1 −2M

r

??ℓ(ℓ + 1)

r2

+2M

r3

+M(γ − 3)(r2− 6Mr + 6M2)

r3(2r − 3M)2

?

(59)

and

VCFM

grav(r) =

?

1 −2M

r

??ℓ(ℓ + 1)

r2

−6M

?

r3

−M(γ − 3)(5r2− 20Mr + 18M2)

r3(2r − 3M)2

Setting γ = 3 we recover the usual expressions for pertur-

bations around the four dimensional Schwarzschild black

hole.

A basic feature of the effective potentials VCFM

VCFM

grav

is that they are not positive definite. Typically,

for negative enough values of the parameter γ (with large

|γ|) a negative peak in the effective potential will show

up. It is no longer obvious that the scalar and gravita-

tional perturbations will be stable.

If the effective potential is not positive definite (and

cannot be approximated by a positive definite one), the

WKB semi-analytical formulae will usually not be appli-

cable, but direct integration techniques will. Using the

later approach an extensive search for unstable solutions

was made. One important result in this work is that even

for very high values of γ, the (scalar and gravitational)

perturbative dynamics is always stable. This is illustrated

in Figs. 1 and 2, where we show non-positive definite ef-

fective potentials and the corresponding (bounded) field

evolution.

The overall picture of the perturbative dynamics for

the effective potentials hereby considered is the usual

one.After a brief transient regime, the quasinormal

mode dominated phase follows, and, finally, at late times

a power-law tail dominates.

For the fundamental multipole mode (ℓ = 0) the effec-

tive potential will not be positive definite for any value

of the parameter γ. Direct integration shows that the

field evolution is always bounded for a great range of

variation of γ. This point is illustrated in Figs. 1 and

2, where we have selected rather large values of γ. In-

deed, it is observed that the decay is dominated from very

early time by the power-law tail. Therefore, it is very

difficult to estimate the quasinormal frequencies directly

from this “time–dependent” approach. The WKB-type

expressions are not applicable if ℓ = 0 for two reasons:

the effective potentials are not positive for r larger than

a certain value, and it is well known that this method

works better with ℓ < n, where n is the overtone num-

ber.

With small but non-zero values of ℓ the quasinormal

frequencies can be accurately estimated. As it is shown in

. (60)

sc

and

Tables I and II, the concordance with the WKB results is

reasonable, except for some values of γ (typically around

γ = 0).

For large values of ℓ an analytical expression for the

quasinormal frequencies can be obtained. Expanding the

effective potential in terms of small values of 1/ℓ and

using the WKB method in the lowest order (which is

exact in this limit) we find

Re(ωn) =

ℓ

3√3M,

(61)

Im(ωn) =

?

2ℓ

3M2

?

n +1

2

?

. (62)

As it can be seen from the data in tables I and II

and Fig. 3, the dependence of the frequencies with the

parameter γ is very weak, although not trivial.

large variation range of γ the absolute value of Im(ω0) is

a monotonically decreasing function, while Re(ω0) typi-

cally has maximum points.

The late-time behavior of the perturbations considered

here can be treated analytically. Far from the black hole

the scalar effective potential in terms of r⋆assumes the

form,

In a

VCFM

sc

(r⋆) ∼

?2M

r3

ℓ(ℓ+1)

r2

⋆

⋆

with ℓ = 0

+4Mℓ(ℓ+1)ln(r⋆/2M)

r3

⋆

with ℓ > 0.

(63)

It is then shown [32, 33] that with the initial data having

compact support a potential with this form has a late-

time tail

RCFM

ℓ

∼ t−(2ℓ+3). (64)

Therefore, at asymptotically late times the perturbation

decays as a power-law tail for any value of the parameter

γ. This is a strong indication that the models are indeed

stable. This point is illustrated in Figs. 1 and 2. It is

reminiscent from a similar behavior of the Gauss Bon-

net term added to Einstein gravity in higher dimensions,

which was recently treated in [34]. Although the result

is formally valid also for the gravitational perturbations

we have considered, it should be noted that in the sim-

plified model developed in this paper the back-reaction

from the bulk, which can modify the tail presented here,

was neglected.

C.“Zero mass” brane black holes

The treatment in section IV is general enough to in-

clude also the case of perturbations around the “zero

mass” brane black hole.

scalar and axial gravitational perturbations are de-

scribed by wave equations similar to Eq.(41) with

Using the metric (10) the

Page 9

9

Table III: Fundamental quasinormal frequencies for the scalar perturbation around the “zero mass” black hole for several values

of the C and ℓ. The parameter h is set to h = 1.

Direct Integration WKB-3thorder WKB-6thorder

ℓC

Re(ω0) -Im(ω0) Re(ω0) -Im(ω0) Re(ω0) -Im(ω0)

1

1

1

1

1

1

1

0-- 0.728358

0.730716

0.732795

0.726005

0.722899

0.719182

0.660519

0.232800

0.247607

0.303801

0.356124

0.368853

0.381471

0.494129

0.746504

0.749727

0.755273

0.751242

0.748461

0.745143

0.732352

0.230697

0.245155

0.301191

0.357060

0.371463

0.385894

0.479133

0.1

0.5

0.9

1.0

1.1

2.0

0.74494

0.75196

0.75437

0.75410

0.75354

0.74548

0.24551

0.30135

0.35023

0.36201

0.37352

0.46711

2

2

2

2

2

2

2

0-- 1.242071

1.243863

1.246290

1.244715

1.243914

1.242968

1.227403

0.230667

0.245695

0.299897

0.347044

0.358055

0.368816

0.456844

1.246220

1.248248

1.251377

1.250452

1.249839

1.249090

1.235779

0.231061

0.246014

0.300033

0.347062

0.358066

0.368835

0.459396

0.1

0.5

0.9

1.0

1.1

2.0

1.24534

1.25534

1.25072

1.24989

1.25377

1.24523

0.21001

0.28301

0.34681

0.35677

0.36031

0.44843

-10 -8 -6 -4 -2 0 2 4

-0.050

0.595

0.600

0.605

0.610

0.615

Re ( ωn )

n = 0

-10 -8 -6 -4 -2 0 2 4

-0.2

0.40

0.44

0.48

0.52

0.56

0.60

n = 1

-10 -8 -6 -4 -2 0 2 4

-0.25

0.1

0.2

0.3

0.4

0.5

0.6

n = 2

-10 -8 -6 -4 -2 0 2 4

γ

-0.250

-0.200

-0.150

-0.100

Im ( ωn )

-10 -8 -6 -4 -2 0 2 4

γ

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-10 -8 -6 -4 -2 0 2 4

γ

-1.50

-1.25

-1.00

-0.75

-0.50

Figure 3: Dependence of gravitational perturbation quasinor-

mal frequencies on γ in the CFM geometry. The results are

qualitatively similar for the scalar perturbation. The param-

eters are ℓ = 3 and M = 1.

effective potentials given by

Vzm

sc(r) =

?

1 −h2

r2

?

??ℓ(ℓ + 1)

1 −5h2

r2

+2h2

r4

−

C − h

(2r2− h2)3/2

r2+2h4

r4

??

,(65)

and

Vzm

grav(r) =

?

1 −h2

r2

?

??ℓ(ℓ + 1)

5 −11h2

r2

r2

−4h2

??

r4

+

C − h

(2r2− h2)3/2

+4h4

r4

.(66)

Again, the effective potentials can be non-positive defi-

nite for specific choices of parameters, as illustrated in

Fig. 4 (left). For example, if ℓ = 0 and C > h, Vzm

not be positive definite. If ℓ > 0, Vzm

non-positive definite for high enough values of C.

Except for C = h, an explicit expression for the tor-

toise coordinate was not found. Nevertheless, the nu-

merical integration is possible. The semi-analytical WKB

approach was also used to compute quasinormal frequen-

cies. The concordance is excellent. The WKB formulas

seem to be more reliable in the present case. With the

choice C = 1 we recover some results considered in [21].

Again, an extensive search for unstable modes was per-

formed. Some calculated frequencies are shown in Tables

III and IV. Our results show that the dynamics of the

scalar and axial gravitational perturbations is always sta-

ble in the “zero mass” background.

point in Figs. 4 and 5.

Analytical expressions for the quasinormal frequencies

for the scalar and gravitational perturbations can be ob-

tained in the limit of large multipole index ℓ. As done in

the CFM geometry, we obtain

sc

will

sc

and Vzm

gravwill be

We illustrate this

Re(ωn) =

ℓ

2h,

(67)

Page 10

10

-10123

r*

4

56

7

0

5

10

15

20

Vsc

zm

101

102

t

103

10-10

10-8

10-6

10-4

10-2

100

| Rl=0(t,r*=0) |

(C - h)/h = -100

(C - h)/h = -5

-1 0 1 2 3 4 5 6 7

-0,1

-0,05

0

0,05

~ t-3

Figure 4: (Left) Effective potential for the scalar perturba-

tions in the “zero mass” black hole background with high

values of C. Negative peaks are displayed in detail. (Right)

Bounded evolution of the scalar field perturbation with such

effective potentials. The dotted line is the late-time power-law

tail. The parameters are ℓ = 0 and h = 1.

-100 10

r*

2030

0,00

0,25

0,50

0,75

1,00

1,25

Vgrav

zm

0 20 40

60

80100

t

10-12

10-10

10-8

10-6

10-4

10-2

100

| Zl=2(t,r*=0) |

C = 0.5

C = 2.0

Figure 5:

tational perturbations in the “zero mass” black hole back-

ground. (Right) Bounded evolution of the gravitational field

perturbation with such effective potentials. The parameters

are ℓ = 2 and h = 1.

(Left) Typical effective potential for the gravi-

Im(ωn) =

?

ℓ

√2h

?

n +1

2

?

.(68)

We expect that the late-time behavior of the gravita-

tional perturbations should be dominated by back scat-

tering from the bulk, not considered here.

tail contribution to the scalar decay can be analytically

treated, at least in the limit where r ≫ h. In this case

if C > h or 0 < C < h, the effective potential Vzm

But the

sc

is

101

102

t

103

10-24

10-18

10-12

10-6

100

| Rl (t,r*=0) |

l = 0

l = 1

l = 2

~ t- 4

~ t- 6

~ t- 8

Figure 6: Bounded evolution of the scalar field perturbation in

the “zero mass” black hole background with C = h for several

values of ℓ. After the quasinormal mode phase a power-law

tail is observed. The tail dependence with ℓ obeys Eq. (73).

In the graphs the parameter h was set to h = 1.

approximated by

Vzm

sc(r⋆) ∼

?C−h

√2r3

ℓ(ℓ+1)

r2

⋆

⋆

with ℓ = 0,

+2(C−h)ℓ(ℓ+1)ln(r⋆/h)

√2r3

⋆

with ℓ > 0.

(69)

Again, we observe that ([32, 33]) with the initial data

having compact support the tail has the form

Rzm

ℓ

∼ t−(2ℓ+3)with 0 < C < h or C > h.

An interesting limit is when C = h. In this case the

explicit expression for the tortoise coordinate according

to the usual definition in Eq. (42) is

(70)

r⋆(r) = r +h

2ln

?r

h− 1

?

with C = h is approximated

−h

2ln

?r

h+ 1

?

. (71)

The effective potential Vzm

by

sc

Vzm

sc(r⋆) ∼

?2h2

r4

ℓ(ℓ+1)

r2

⋆

⋆

with ℓ = 0,

−2ℓ(ℓ+1)h2

r4

⋆

with ℓ > 0.

(72)

In this limit a power-law still dominates the late-time

decay. But its dependence with the multipole index ℓ is

different,

Rzm

ℓ

∼ t−(2ℓ+4)with C = h.(73)

This point is illustrated in Fig. 6.

As observed in the CFM model, for the non-extreme

“zero mass” model the scalar perturbation decays as a

power-law tail suggesting that the model is stable.

The qualitative picture of the field evolution in the

“zero mass” black hole — quasinormal mode followed by

power-law tail — changes drastically when the extreme

case (C = 0) is considered (see Fig. 7 ). If ℓ = 0, we ob-

serve the usual power-law tail dominating the late-time

decay. But when ℓ > 0, the simple power-law tail is re-

placed by an oscillatory decay with a power-law envelope,

Rzm

ℓ

∼ t−3/2sin(ωℓ× t) with C = 0 and ℓ > 0. (74)

Page 11

11

102

102

103

103

10-16

10-16

10-12

10-12

10-8

10-8

10-4

10-4

| Rl (t, r* = 0 ) |

102

102

103

103

10-8

10-8

10-6

10-4

10-4

10-2

100

100

102

102

103

103

t

10-8

10-8

10-6

10-4

10-4

10-2

100

100

| Rl (t, r* = 0 ) |

102

102

103

103

t

10-8

10-8

10-6

10-4

10-4

10-2

100

100

~ t-3

l = 0

~ t-3/2

l = 1

l = 2

l = 3

~ t-3/2

~ t-3/2

Figure 7: Bounded evolution of the scalar field perturbation

in the extreme “zero mass” black hole background (C = 0),

for several values of ℓ. If ℓ = 0, the decay is dominated by

a power-law tail (t−3). If ℓ > 0, the decay is dominated by

an oscillatory tail, whose envelope is t−3/2. In the graphs the

parameter h was set to h = 1.

0

25 5075

100

125 150

i-th zero

0

1000

2000

3000

4000

5000

zero of Rl (t, r* = 0)

l = 1

l = 2

l = 3

01234

56

7

89 10

Multipole Index l

0,0

0,1

0,2

0,3

0,4

Angular Frequency ωl

h = 0.5

h = 1

h = 2

Figure 8:

wave function is zero in the extreme “zero mass” black hole.

Straight lines imply that the period of oscillation is a con-

stant. The parameter h is set to h = 1. (Right) Dependence

of the angular frequency ωℓin Eq.(74) with ℓ for several values

of h.

(Left) Numerical value of t where the scalar

Therefore, for ℓ > 0 the power index (−3/2) is inde-

pendent of the multipole index ℓ. The angular frequency

ωℓfor large times approaches a constant, as we can see

in Fig. 8 (left). The angular frequency is well approxi-

mated by a linear function of ℓ, as indicated in Table V for

some values of h. This result implies that the dominating

contributions in the late-time decay are the modes with

ℓ > 0, i.e., the power-law enveloped oscillatory terms.

We also observe that these tails dominate from very early

times, so that it was not possible to estimate the quasi-

normal frequencies in the extreme case (as indicated in

the first lines of Tables III and IV).

VI.CONCLUSIONS

In this work we studied brane black holes from the

point of view of a brane observer. We considered the

two family solutions found by Bronnikov et al. [9] in

order to derive the Bekenstein entropy bound and the

one-loop correction to the Bekenstein-Hawking formula

using the ’t Hooft brickwall method.

performed scalar and axial gravitational perturbations

in two specific examples of these families. With these

perturbations we were able to analyze the dynamics and

stability of the black hole solutions.

The results of the black hole thermodynamics study

show that the entropy bound continues to be independent

of the black hole parameters. Thus, the presence of the

bulk does not affect the universality of the entropy bound

for a brane observer, as ourselves, reinforcing the Gen-

eralized Second Law. Moreover, applying the ’t Hooft’s

brickwall method to both black hole families we see that

the entropy correction takes the same form as that of

a Schwarzschild black hole when written in terms of its

own black hole parameters. Therefore, as the correction

is linearly dependent on the area, it can be absorbed in

a renormalized gravitational constant.

One of the most important results in this paper came

from the perturbative dynamics. We should stress that

the assumption δEαβ= 0 was necessary in order to solve

the gravitational perturbation equation (44) without any

knowledge of the bulk structure. This vanishing tidal ef-

fect is perfectly justified when the perturbation energy

is lower than the threshold of the Kaluza-Klein massive

modes. Likewise, as we neglect the bulk back-reaction,

our analysis does not describe the perturbation late-time

behavior. Our results show no unstable mode in the

scalar and gravitational analysis. In addition, the late-

time tails display a power-law behavior what enforces

their stability.

In the case of CFM black hole even if the effective

potential is not positive–definite the quasinormal modes

are stable (negative imaginary part). The agreement of

the several methods employed in the calculation is good

for ℓ not too small.

On the other hand, in the case of the “zero mass” black

hole we observe a richer picture. The scalar and gravita-

tional field evolution is always bounded suggesting that

this class of models is stable. But the late time decay

of the matter field strongly depends on the parameters

C and h. If C is non-zero and not equal to h, the late-

time decay is dominated by a power-law tail with the

usual dependence on the multipole parameter ℓ. But if

C = h, this dependence changes. Finally, in the extreme

regime (C = 0) the late-time decay is dominated by os-

cillatory modes with a power-law envelope. This power

index seems to be universal, not depending on ℓ.

Summarizing, thethermodynamics

ofmodels we considered

thedynamics inspecific

in the approachemployed

In addition, we

in theclass

while

stable

work.

is consistent,

backgrounds

inthe

is

present

Page 12

12

Table IV: Fundamental quasinormal frequencies for the axial gravitational perturbation around the “zero mass” black hole for

several values of the C and ℓ. The parameter h is set to h = 1.

Direct IntegrationWKB-3thorderWKB-6thorder

ℓC

Re(ω0)-Im(ω0)Re(ω0)-Im(ω0)Re(ω0)-Im(ω0)

2

2

2

2

2

2

2

0--0.934530

0.935381

0.938066

0.942035

0.943255

0.944551

0.958486

0.191198

0.205749

0.264005

0.318153

0.330875

0.343278

0.442675

0.934386

0.964066

0.958654

0.929971

0.928449

0.928937

0.973998

0.219631

0.215448

0.249590

0.317851

0.334460

0.350243

0.456121

0.1

0.5

0.9

1.0

1.1

2.0

0.94412

0.94783

0.94334

0.94207

0.94069

0.92442

0.21924

0.26533

0.30949

0.31993

0.33016

0.41343

3

3

3

3

3

3

3

0--1.537001

1.537512

1.536138

1.532735

1.531809

1.530883

1.523732

0.214671

0.229197

0.280829

0.325500

0.335949

0.346171

0.430806

1.538619

1.539283

1.538697

1.534597

1.533070

1.531369

1.513596

0.215204

0.229786

0.280900

0.324529

0.334820

0.344973

0.433952

0.1

0.5

0.9

1.0

1.1

2.0

1.50122

1.53958

1.53905

1.53757

1.53594

1.52161

0.25435

0.27762

0.32341

0.33402

0.34411

0.42100

Table V: Oscillatory frequency of the tail in the extreme “zero

mass” (C = 0) black hole for several values of h.

h

Angular Frequency ωℓ (ℓ > 0)

0.5

1.0

2.0

0.02593 + 0.06421 × ℓ

0.01347 + 0.03191 × ℓ

0.006683 + 0.01597 × ℓ

While our results suggest that the brane models

presented are viable, the final check would be the

analysis of the continuation in the bulk of the geometries

presented here.

Acknowledgments

This work was partially supported by Funda¸ c˜ ao de

Amparo ` a Pesquisa do Estado de S˜ ao Paulo (FAPESP)

and Conselho Nacional de Desenvolvimento Cient´ ıfico e

Tecnol´ ogico (CNPq).

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