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.
- [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; - [Show abstract] [Hide abstract]
ABSTRACT: We study the quasinormal modes of scalar perturbation in the background of five-dimensional charged Kaluza-Klein black holes with squashed horizons immersed in the Goedel universe. Besides the influence due to the compactness of the extra dimension, we disclose the cosmological rotational effect in the wave dynamics. The wave behavior affected by the Goedel parameter provides an interesting insight into the Goedel universe.Physical review D: Particles and fields 04/2009; 79(8):084005-084005.
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
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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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