Trabajo Fin de Grado en Fısica

Warped compactifications andbraneworlds

Jonathan Gilbert Machado RodrıguezUniversidad de Granada

Junio de 2018

Tutor: Bert JanssenDepartamento de Fısica Teorica

Universidad de Granada

Resumen

Las compactificaciones alabeadas fueron propuestas por L. Randall y R. Sundrum, en elmarco de la cosmologıa de branas, como una alternativa al modelo de dimensiones extracompactas y microscopicas de Kaluza-Klein. En los primeros capıtulos discutiremos laposibilidad de dimensiones adicionales, introduciendo dos clases de dimensiones extra,segun los modelos de Kaluza-Klein y de Randall-Sundrum II. Justificaremos la importan-cia del segundo en relacion al confinamiento de la gravedad en la brana y la normalizaciondel graviton. Asimismo, se revisaran conceptos de Relatividad General que apareceranrecurrentemente a lo largo del trabajo. En la segunda parte del proyecto generalizaremosel modelo de Randall-Sundrum, centrandonos primero en la metrica y posteriormenteen la dinamica, y prestando especial atencion al surgimiento de una dinamica efectivacuatrodimensional en la brana.

Abstract

Warped compactifications were proposed by L. Randall and R. Sundrum, in the frameof brane cosmology, as an alternative to the Kaluza-Klein model of compact and micro-scopic extra dimensions. In the first chapters we will discuss the possibilty of additionaldimensions, presenting two sorts of them, in relation with the Kaluza-Klein theory andthe Randall-Sundrum II model. Besides, we will justify the importance of the latter in theconfinement of gravity in the brane and the normalisation of the graviton. Furthermore,we will review certain concepts of General Relativity which will appear throughout thetext. In the second half of the project, we will generalise the metrics and dynamics ofthe Randall-Sundrum model, paying special attention to the emerging four-dimensionaldynamics in the braneworld.

Contents

1 Introduction: A Matter of Scale 4

2 Foundations of General Relativity 6

2.1 Funtamental tensors related to curvature . . . . . . . . . . . . . . . . . . . . 6

2.2 The Einstein’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 The De Sitter Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 The anti-De Sitter Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Randall-Sundrum theory 8

3.1 The metric and the action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Confinement of the graviton . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Considering an arbitrary warp factor f (z) 11

4.1 Λ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2 Λ > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.3 Λ < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Considering a general internal metric gµν 13

6 Considering a general internal metric gµν and an arbitrary warp factor f (z) 14

6.1 Λ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6.2 Λ < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.3 Λ > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.4 Brief summary of the solutions obtained . . . . . . . . . . . . . . . . . . . . . 19

7 A step beyond: Generalizing the Dynamics 20

8 Conclusions 24

A Christoffel symbols and curvature tensors 25

References 26

1 INTRODUCTION: A MATTER OF SCALE 4

1 Introduction: A Matter of Scale

It is widely assumed that we live in a universe with three spatial (probably infinitelylarge) dimensions, which evolve in a temporal “direction”, this is, a (3+1)-dimensionalspacetime. But is it a fair assumption? From our experience, it seems so, as these arethe only dimensions we can perceive, and all the well-established physical theories haveseemingly managed properly in this frame. For instance, Standard Model fields propa-gating (a large distance) in extra dimensions would not be consistent with observations.

Experimentally, it is possible to determine the number (and even the geometry) ofspatial dimensions by studying how the gravitational field decays, as gravity describesthe geometry and dynamics of spacetime. In terms of newtonian mechanics, the grav-itational potential Φ(r) satisfies the Laplace equation, which means that if there are d“conventional” spatial dimensions, the equipotential surfaces are (d− 1)-dimensional hy-perspheres of arbitrary radius r and area

Sd−1(r) =2πd/2rd−1

Γ( d2 )

. (1.1)

In consequence, as the gravity flux through such surfaces is constant,∫Sd−1(r)

~g · d~S = gSd−1(r) = const, (1.2)

the gravitational field, ~g, necessarily decays as ∼ 1rd−1 . Similarly, in the frame of general

relativity, we can study the static spherical solution with mass m in an N-dimensionalspacetime (N = d + 1), this is, the Schwarzschild-Tangherlini solution [2]:

ds2N =

(1− 2GNm

rN−3

)dt2 −

(1− 2GNm

rN−3

)−1

dr2 − r2dΩ2N−2, (1.3)

where GN is the N-dimensional gravitational constant. In this case, the tt component ofthe metric is related to the classical gravitational potential when applying the Newtonianlimit, gtt ∼ 1

rN−3 ∼ Φ(r), which is coherent with what we mentioned before. As the well-known inverse-square law for the gravitational field can be measured (this is, ~g ∼ 1

r2 ),spacetime is necessarily four-dimensional.

All the evidence indicates that we live in (3+1) dimensions, nevertheless, there is a wayout of all these arguments. The fact is that the∼ 1

r decay of the gravitational potential hasbeen experimentally observed for distances no shorter than centimetres. Thus, as gravityis considerably weaker than the other interactions, the evidence for possible additionaldimensions might be “hidden” at shorter scales (submillimetric scales), where there mightbe deviations from the macroscopic behaviour of gravity. It is a matter of scale.

In fact, compact and microscopic additional dimensions of scale R0 might be accept-able provided that R0 is sufficiently small compared to the minimum distance for whichthe inverse-square law has been verified. This kind of extra dimensions are associatedto a factorizable geometry, which essentialy means that the metric of the four commondimensions does not depend on the additional ones. For instance,

ds2 = g(x)dxµdxν − h(z)dz2, (1.4)

5 1 INTRODUCTION: A MATTER OF SCALE

where xµ are the familiar dimensions and z represents a fifth aditional dimension. Thenumber of dimensions observed and their geometry depends on the scale, L, which isdirectly related to the energy E ∼ 1

L studied in an experiment, and the Planck mass,which defines the scale of energies of the graviton. For n extra dimensions with topologyCn:

• L R0 ⇒ 4 dimensions with the topology of a Minkowski spaceM1,3.

• L R0 ⇒ (4+n) dimensions with topologyM1,n.

• L ∼ R0 ⇒ (4+n) dimensions with topologyM1,3 × Cn.

Therefore, the four-dimensional spacetime is a low-energy image, and the n additionaldimensions and their geometry will be noticeable at energies E ∼ 1

R0.

A traditional example of these additional dimensions is the Kaluza-Klein theory (KKtheory) [2, 9]. In 1921, Kaluza proposed an extension of general relativity to five dimen-sions explaining how electromagnetism and gravity can be unified under the conditionthat the five-dimensional metric does not depend on the fifth dimension. Although thisidea was very interesting, there was an objection: there was not a solid justification forsupressing the dependence on the extra coordinate. In 1926, Klein came to the conclusionthat such hypothesis could be dropped as long as the additional dimension was compactand microscopic (this is, with topology S1). Hence, the topology of the spacetime wouldbeM1,3 × S1, rather thanM1,4. The main achievement of the KK theory was explainingelectromagnetism as a consequence of pure gravity in five dimensions so that a four-dimensional observer, this is, which is able to measure at scales L R0, would perceivethe five-dimensional metric as three independent fields: an effective four-dimensionalmetric, a gauge vector potential (which we identify as the electromagnetic potential) anda scalar field (Kaluza-Klein scalar).

Nevertheless, we find a major inconvenience when studying the effective theory infour dimensions associated to the five-dimensional Einstein-Hilbert action: the gravita-tional constant in four dimensions, GN , is not a fundamental quantity, but is related to thefive-dimensional one, GN , and to the compactification radius, R0, [2]

GN =GN

2πR0. (1.5)

This could be a possible way to explain why gravity is ∼ 1032 times weaker than theweak interaction, which is the so-called hierarchy problem [12]. The solution given by theKaluza-Klein theory suggests that this discrepancy between the two interactions mightbe associated to the fact that a four-dimensional observer would measure the effectiveconstant, GN , in stead of the fundamental one, which would be of the order of the weakcoupling constant. But in that case, R0 would have to be macroscopic, and therefore thefifth dimension would be noticeable. Thus, it is evident that the Kaluza-Klein scenario infive dimensions has important phenomenological issues.

As an alternative to compact additional dimensions, in 1999, L. Randall and R. Sun-drum proposed in their article [1] the first model of warped compactification, and provedthat non-compact dimensions are compatible with the experimental observations if theassumption of a factorizable geometry is dropped. They considered that the observable

2 FOUNDATIONS OF GENERAL RELATIVITY 6

universe is restricted to a (3+1)-dimensional hypersurface, a brane, embedded in a five-dimensional space, called the bulk. Unlike the Kaluza Klein theory, here, it is the curva-ture of the bulk which determines the Planck mass and confines gravity to a small regionnear the brane, and not the size of the additional dimension. Plus, in this model, the ef-fective gravitational constant is a fundamental quantity, avoiding the phenomenologicalinconsistencies that we found for the KK theory. In section 3 we will study this modelmore deeply.

2 Foundations of General Relativity

In this section we intend to review briefly some fundamental concepts that will appearthroughout the text, in order to understand the mathematical work behind it.

2.1 Funtamental tensors related to curvature

In General Relativity, it is through the (locally-valid) Equivalence Principle that Einsteinintroduces heuristically the concept of gravitational interaction as a manifestation of thecurvature of spacetime, which is caused by the presence of matter and energy. This is whydifferential geometry, which describes curved spaces, is the proper mathematical frameto develop Einstein’s theory.

One of the main features of a curved space(time) is the fact that if we “parallel trans-port” a vector from an initial point to a final one, the vector resulting from this operationdepends, generally, on the trajectory followed between this two points. When calculat-ing the difference of the differential parallel transport of a vector Vα along two differentdirections xµ and xν, we obtain, [

∇µ,∇ν

]Vα = R α

µνλVλ, (2.1)

where R αµνλ is the Riemann curvature tensor:

R λµνρ = ∂µΓλ

νρ − ∂νΓλµρ + Γλ

µσΓσνρ − Γλ

νσΓσµρ, (2.2)

and Γσµν is the Levi-Civita connection, which can be calculated from the metric gµν:

Γσµν =

12

gσλ(∂µgλν + ∂νgµλ − ∂λgµν

). (2.3)

Hence, the metric gµν describes completely the geometry and curvature of the space.1

We say a space is of constant curvature when it is maximally symmetric, which hap-pens when the Riemann tensor verifies the following condition [2]:

Rµνρλ = K(

gµλgνρ − gµρgνλ

), (2.4)

with K a constant related to the radius of curvature R0. For K < 0, the spacetime turnsout to be of positive curvature, and for K > 0, of negative curvature. If K = 0, then thespacetime is a Minkowski space.

1Thanks to the fact that we are using the Levi-Civita connection, which naturally appears in the frame ofGeneral Relativity.

7 2 FOUNDATIONS OF GENERAL RELATIVITY

Now that we have defined the Riemann tensor, we can introduce the Ricci curvaturetensor Rµν and the Ricci scalar R:

Rµν = R λµλν = ∂µΓλ

λµ − ∂λΓλµν + Γλ

µσΓσλν − Γλ

λσΓσµν, (2.5)

R = gµνRµν. (2.6)

2.2 The Einstein’s equations

In the first part of this text, we focus in solutions of the Einstein’s equations in vacuumwith cosmological constant Λ. The action associated to the system can be expressed as

S =1

2κ

∫dnx√|g|(R−Λ), (2.7)

with n the dimension of the spacetime, and κ = 8πGN , being GN the n-dimensional grav-itational constant. The action is associated, through the variational principle, to the fol-lowing Einstein field equations (or Einstein’s equations):

Rµν −12(R−Λ) gµν = 0, (2.8)

Usually, it is easier to work with the so-called Einstein’s traceless equations:

Rµν =Λ

n− 2gµν. (2.9)

Depending on the sign of the cosmological constant, it can be understood as a repul-sive force if Λ > 0, or an attractive force if Λ < 0. Is is generally interpreted as the energydensity of the vacuum. Since 1998, astronomical observations indicate that the universe isexpanding at an accelerating pace as if there was a positive cosmological constant. There-fore, cosmologists associate Λ > 0 to the dark energy, a kind of energy which has not beenrelated to any matter field so far.

2.3 The De Sitter Space

A n-dimensional De Sitter space2, which can be described by the metric

ds2n = dt2 − e2t/R0 δijdxidxj, (2.10)

is a solution, with constant positive curvature, of the Einstein’s field equations in vacuumwith positive cosmological constant Λ. Hence, a De Sitter space satisfies (2.4),

Rµνρλ = −R−20(

gµλgνρ − gµρgνλ

), (2.11)

with R0 being the De Sitter radius, which is related to the cosmological constant as follows:

R0 =

√(n− 1)(n− 2)

Λ. (2.12)

2The latin indices here refer to the n− 1 spatial coordinates.

3 RANDALL-SUNDRUM THEORY 8

This space represents an empty universe expanding exponentially due to the positivecosmological constant, which acts as a repulsive force. An inhabitant of this universewould observe how the distance between two arbitrary points of the space is increasing.We are particularly interested

2.4 The anti-De Sitter Space

An Anti-De Sitter (AdS) space is a solution, with constant negative curvature, of the Ein-stein’s field equations in vacuum with negative cosmological constant Λ. Thus, an AdSspace verifies (2.4):

Rµνρλ = R−20(

gµλgνρ − gµρgνλ

), (2.13)

where R0 is the so-called anti-De Sitter radius, which is related to the cosmological con-stant

R0 =

√(n− 1)(n− 2)

−Λ. (2.14)

Among all the possible metrics describing an anti-De Sitter space, we find the n-dimensionalmetric:

ds2n = e2z/R0 ηµνdxµdxν − dz2, (2.15)

where µ and ν run from 0 to n− 2, and z is an additional dimension. This metric in fivedimensions is of particular interest here, as it will appear throughout the text.

Although it is not considered as a realistic cosmological model, the anti-De Sitter spacerepresents an empty universe with constant negative curvature. The cosmological con-stant acts here as an attractive force, making particles emitted from a same point recol-lapse. Nevertheless, an anti-De Sitter space is neither expanding nor contracting, it doesnot change over time.

3 Randall-Sundrum theory

As we mentioned in the introductory chapter, L. Randall and R. Sundrum concluded thatnon-compact (large) dimensions are compatible with experimental gravity, and contrar-ily to the microscopic compact ones, the Planck mass is in this case determined by thehigher-dimensional curvature and not by the size of the additional dimension, which inthis model is infinitely large. One of the most important features of this model is thatcurvature allows the existence of a “bound state” of the higher-dimensional graviton. Inthis chapter we intend to introduce the Randrall-Sundrum model, and briefly justify theconfinement of the graviton, although we will not demonstrate the expressions.

The Randall-Sundrum model is a braneworld solution in five dimensions, which meansthat the visible universe is restricted to a (3+1)-dimensional hypersurface, called 3-brane,inside a a higher-dimensional spacetime, known as bulk, which in this case is a five-dimensional anti-De Sitter space. We will refer to the fifth dimension by z from now on.

9 3 RANDALL-SUNDRUM THEORY

3.1 The metric and the action

Let’s consider the Einstein-Hilbert action with a negative cosmological constant3 Λ:

Sgravity =1

2κ

∫d5x√|g|(R− Λ), (3.1)

Among all the possible solutions of the Einstein (traceless) equations,

Rµν =Λ3

gµν, (3.2)

we find the following metrics:

ds2± = e±2z/R0 ηµνdxµdxν − dz2, (3.3)

with R0 =√−12

Λ. The dependence of the exponential prefactor on the additional dimen-

sion causes these metrics to be non-factorisable. As we mentioned in section 2.4, (3.3)describe an anti-De Sitter space, as they satisfy (3.2) and the condition of maximally sy-metric space (2.4), with K = 1

R20. R0 is, then, the anti-De Sitter radius.

The metric in the Randall-Sundrum model [1] is

ds2 = e−2|z|/R0 ηµνdxµdxν − dz2, (3.4)

where e−2|z|/R0 is called the warp factor, as it causes the fifth-dimension to curve, and ηµν

is the internal (Minkowski) metric of the brane (Figure 1). In fact, (3.4) can be constructedby choosing the brane at z = 0 as a singular hypersurface where the coupling of the so-called domain wall metrics (3.3) takes place, imposing what we call the orbifold boundaryconditions.4 The Randall-Sundrum metric (3.4) is a solution of (3.2) for z < 0 and z > 0,but there is a singularity in the brane at z = 0. In order to fix this problem, L. Randall andR. Sundrum added a new term, Sbrane, in the action of the system:

S = Sgravity + Sbrane, (3.5)

whereSbrane =

12κ

∫d5x√|g| (V0δ(z)) , (3.6)

where V0 is called the tension of the brane.

3All 5-dimensional objects and indices running over the five dimensions carry hats. The unhatted indicesrun from 0 to 3.

4We are not interested here in the mathematical and topolgical construction of (3.4) but in the fact theRandall-Sundrum metric allows a bound state of the gravity, as we will discuss in the next section. Moredetails concerning the orbifold boundary conditions can be found in [1].

3 RANDALL-SUNDRUM THEORY 10

Figure 1: The Randall-Sundrum spacetime model.

3.2 Confinement of the graviton

Let’s consider small gravitational fluctuations of the four-dimensional metric,

gµν = e−2|z|/R0 ηµν + hµν(xµ).

If we separate variables in the fluctuation, hµν(xµ) = eipxψ(z), it can be proved that theysatisfy the following equation:[

−m2

2e2|z|/R0 − 1

2∂2

z −2

R0δ(z) +

2R2

0

]ψ(z) = 0, (3.7)

where p2 = m2. If we now make a change of variables,

y ≡ sgn(z)(

e|z|/R0 − 1)

,

hνρ(xµ, y) ≡ hνρ(xµ)e|z|/(2R0),

ψ(y) ≡ ψ(z)e|z|/(2R0), (3.8)

(3.7) becomes an eigenvalue equation:

(−12

∂2y + V(y))ψ(y) = m2ψ(y), (3.9)

where V(y) acts as a potential associated to the curvature which depends on the addi-tional dimension [1]:

V(y) =15/R2

0

8 (|y|+ 1)2 −3

2R0δ(y). (3.10)

The presence of a Dirac delta function in the potential is the reason why there exist asingle normalizable bound state, which we associate to the zero mode or massless gravi-

ton. Its wavefunction, ψ0(y) = R0

(|y|R0

+ 1)−3/2

, confines the graviton in the visible brane(this is, z = 0) by quickly falling off to zero away from it (Figure 2).

11 4 CONSIDERING AN ARBITRARY WARP FACTOR F(Z)

Figure 2: Profile ψ(z) of the massless graviton.

In the following chapters we intend to progressively generalise the metric (3.3) anddynamics (3.1) looking for new braneworld solutions. We will not discuss the normalisa-tion of the graviton in these cases, although this might be an interesting idea for a futureproject.

4 Considering an arbitrary warp factor f (z)

In this chapter, we intend to replace the exponential warp factor, which causes the fifthdimension to curve and, therefore, confine the graviton, by an arbitrary prefactor f (z) inorder to study if there are alternative solutions creating a curved background. Thus, wewill use the following Ansatz:

Minkowski braneworld Ansatz with arbitrary warp factor:

ds2 = f (z)2ηµνdxµdxν − dz2, (4.1)

with f (z) ∈ R, and the vacuum lagrangian L =√|g|(

R− Λ), with arbitrary cosmo-

logical constant Λ.

The non-vanishing components of the Ricci tensor are:

Rzz = 4f (z)f (z)

(4.2)

Rµν = −ηµν

[3 f 2(z) + f (z) f (z)

](4.3)

Plugging this and the Ansatz (4.1) in the equations of motion (3.2) leads to5

4f (z)f (z)

= − Λ3

, (4.4)

5From now on, f (z) ≡ d f (z)dz , for any function depending exclusively on z.

4 CONSIDERING AN ARBITRARY WARP FACTOR F(Z) 12

and

−ηµν

[3 f 2(z) + f (z) f (z)

]=

Λ3

f 2(z)ηµν,

ηµν

[3 f 2(z) + f (z) f (z) +

Λ3

f 2(z)

]= 0. (4.5)

Therefore, we obtain the following set of differential equations:

f (z) +Λ12

f (z) = 0 (4.6)

3 f 2(z) + f (z) f (z) +Λ3

f 2(z) = 0 (4.7)

The solutions fall into two classes depending on the cosmological constant Λ.

4.1 Λ = 0

Equation (4.6) restricts f to the following solutions:

f (z) = Az + B, (4.8)

with A, B ∈ R, and (4.7) is satisfied only if A = 0 :

f (z) = B ∈ R. (4.9)

Hence, the solution is essentially the five-dimensional Minkowski metric:

ds = ηµνdxµdxν − dz2 = ηµνdxµdxν, (4.10)

where the constant factor B2 has been absorbed through a simple change of coordinatesin the braneworld: Bxµ 7−→ xµ. As f does not curve the extra dimension (this is, fdoes not act as a warp factor for Λ = 0), this is not a physically interesting solution, asthere is no confinement of gravity in the brane, and therefore, it is not consistent with themacroscopic measurement of the∼ 1

r decay of the gravitational potential , as we discussedin the introductory chapter.

4.2 Λ > 0

We now define R0 ≡√

12Λ∈ R. Equation (4.6) restricts the warp factor to the following

solutions:f (z) = Aeiz/R0 + Be−iz/R0 , (4.11)

with A and B arbitrary coefficients, and equation (4.7) can only be satisfied if

AB = 0, (4.12)

this is, either A = 0 or B = 0. Hence, we have that

f (z) = Ce±iz/R0 /∈ R, (4.13)

13 5 CONSIDERING A GENERAL INTERNAL METRIC Gµν

which is not an acceptable solution as it is not real.

4.3 Λ < 0

We define R0 ≡√− 12

Λ∈ R. From equation (4.6) we have that

f (z) = Aez/R0 + Be−z/R0 . (4.14)

As in the previous section, equation (4.7) leads to the condition AB = 0, and, therefore,A = 0 or B = 0. We have, then,

f (z) = Ce±z/R0 ∈ R, (4.15)

with C ∈ R, which can be absorbed through a change of coordinates in the brane. Wehave obtained the Randal-Sundrum metrics

ds2± = e±2z/R0 ηµνdxµdxν − dz2, (4.16)

which describe an AdS5 space foliated by Minkowski branes, as we already mentioned.In conclusion, the only physically relevant solution found in this chapter is the Randall-Sundrum solution.

5 Considering a general internal metric gµν

In the Randall-Sundrum model, the brane is flat and does not change over time. If weintroduce a general internal metric (this is, a general four-dimensional metric for the branewhich does not depend on the additional coordinate), gµν(x), we will allow the brane tocurve. This could lead to a different dynamics. As the new five-dimensional metric is nota solution of the Einstein equations, in this chapter we will study under which conditionsthe following Ansatz [3] satisfies (3.2):

Braneworld Ansatz with an arbitrary internal metric and a warp factor:

ds2 = e2z/R0 gµν(x)dxµdxν − dz2, (5.1)

where R0 is a positive constant, and the vacuum lagrangian L =√|g|(

R− Λ), with

negative cosmological constant Λ.

The non-vanishing components of the Ricci tensor are:

Rzz =4

R20

(5.2)

Rµν = Rµν − gµν

[3

R20

e2z/R0 +1

R20

e2z/R0

], (5.3)

where Rµν is the four-dimensional Ricci tensor associated to gµν. Plugging (5.2) and the

6 CONSIDERING A GENERAL INTERNAL METRIC Gµν AND AN ARBITRARYWARP FACTOR F(Z) 14

Ansatz (5.1) in the equations of motion (3.2), we have that

4R2

0= − Λ

3=⇒ R0 =

√−12

Λ, (5.4)

as we expected, from the previous cases. On the other side, the µν components of theEinstein equations (3.2) are

Rµν − gµν

[3

R20

e2z/R0 +1

R20

e2z/R0

]=

Λ3

e2z/R0 gµν,

Rµν − gµν

[3

R20

e2z/R0 +1

R20

e2z/R0 +Λ3

e2z/R0

]= 0, (5.5)

which, if we use (5.4), leads toRµν = 0. (5.6)

As we studied in section 2.1, the Ricci tensor is related to the metric, and therefore, thismight seem a mere geometric condition for the brane. Nevertheless, we have a critical re-sult here: the internal metric, which is initially arbitrary, is restricted by the 5-dimensionaldynamics to the Ricci-flat metrics, this is, to those metrics satisfying the 4-dimensionaltraceless Einsteins equations with cosmological constant Λ = 0. Surprisingly, we haveobtained a dynamical condition for the brane. An effective brane dynamics arises fromthe 5-dimensional dynamics. Eq. (5.6) are the traceless Einstein equations associated toan effective action:

S =1

2κ

∫d4x√|g|R, (5.7)

and offer a rich phenomenology. Among all the possible internal metrics that verify theseeffective equations we find the Schwarzschild solution, which describe the gravitationalfield of spherically symmetric massive objects, e.g., stars or black holes; or small pertur-bartions of the Minkowski metric, gµν = ηµν + εhµν with ε sufficiently small, wich allowthe presence of gravitational waves in the brane.

If we study now which solutions are maximally symmetric, it can be proved that theconstant curvature conditon (2.4) for the bulk is only satisfied, with K = 1

R20, if the brane

is a flate space, this is, if it is a Minkowski space, which is equivalent to require a maxi-mally symmetric brane, as Λ = 0. Hence, the constant curvature solutions for the bulkcoincide with the metrics (3.3) in the Randall-Sundrum model, which describe an AdS5

space foliated by flat branes.

The most important point to take out of this section is that an effective brane dynamicsemerges from the five-dimensional spacetime equations. In the following chapters we willgeneralise the metric (3.3) and dynamics (3.1) keeping this as a leitmotiv.

6 Considering a general internal metric gµν and an arbitrary warpfactor f (z)

We have already justified the generalisation of the warp factor and the internal metricin the previous chapters. For the first generalisation we did not find new interesting

156 CONSIDERING A GENERAL INTERNAL METRIC Gµν AND AN ARBITRARY

WARP FACTOR F(Z)

solutions, whereas the second one lead to a surprising result: there is an effective four-dimensional dynamics in the brane arising from the five-dimensional one. Here we willcombine both generalisations in order to study new possible solutions and their dynam-ics. We will use the following Ansatz [4]:

Braneworld Ansatz with arbitrary internal metric and warp factor:

ds2 = f (z)2gµν(x)dxµdxν − dz2, (6.1)

with f (z) ∈ R, and the vacuum lagrangian L =√|g|(

R− Λ), with arbitrary cosmo-

logical constant Λ.

The non-vanishing components of the Ricci tensor are:

Rzz = 4f (z)f (z)

, (6.2)

Rµν = Rµν − gµν

[3 f 2(z) + f (z) f (z)

]. (6.3)

Replacing (6.2) in the zz component of the Einsten equations, we have that

4f (z)f (z)

= − Λ3

,

f (z) +Λ12

f (z) = 0. (6.4)

Plugging (6.3) in the µν components of the Einstein equations gives the following differ-ential equations:

Rµν − gµν

[3 f 2(z) + f (z) f (z)

]=

Λ3

f 2(z)gµν,

Rµν − gµν

[3 f 2(z) + f (z) f (z) +

Λ3

f 2(z)

]= 0. (6.5)

As the Ricci tensor Rµν and the metric gµν depend exclusively on the coordinates of thebrane, equation (6.5) can only be satisfied if the z-dependence vanishes:

3 f 2(z) + f (z) f (z) +Λ3

f 2(z) ≡ Λ2

, (6.6)

with Λ an arbitrary integration constant which can be interpreted as the four-dimensionalcosmological constant in the braneworld, since equation (6.5) can be written as follows:

Rµν =Λ2

gµν. (6.7)

Here we have one of the most relevant results in this text so far. A five-dimensionalbraneworld Ansatz with arbitrary warp factor and internal metric, leads to the four-dimensional traceless Einstein equations (2.9), with an effective four-dimensional action

S =1

2κ

∫d4x√|g|(R−Λ), (6.8)

6 CONSIDERING A GENERAL INTERNAL METRIC Gµν AND AN ARBITRARYWARP FACTOR F(Z) 16

restricting gµν to the Einstein-type metrics. This is, there is an effective four-dimensionaldynamics, and therefore, an hypothetical physicist living in the brane could eventuallydevelop a theory of gravity in four dimensions based on measurements made inside thebrane. Consequently, this is a solid argument supporting the possibility of additionaldimensions. Let’s think now about the cosmological constant problem or “vacuum catas-trophe” [10], which is essentially the 120 orders of magnitude disagreement between themeasured value of Λ [11], and its theoretical calculation (as the energy density of thevacuum), using quantum field theory. Here, Λ is reduced to a mere integration con-stant resulting from a higher-dimensional dynamics. Therefore, the consequences of thisbraneworld model are doubtlessly important, as it suggests a possible explanation for thisproblem.

The solutions fall into two classes depending on both cosmological constant, Λ and Λ.

6.1 Λ = 0

Equation (6.4) restricts f to the following solutions:

f (z) = Az + B, (6.9)

with A, B ∈ R. If we replace f and its derivatives in equation (6.6) we have that

3A2 =Λ2

=⇒ A = ±√

Λ6

(6.10)

and consequently

f (z) = ±√

Λ6

z + B ∈ R. (6.11)

The condition f (z) ∈ R can only be satisfied if Λ ≥ 0.

If we study the maximally symmetric solutions, we have that the Riemann tensorcomponents are

Rµzρz = gµρ f (z) f (z) = 0, (6.12)

which already verify the constant curvature condition (2.4) with K = 0, and

Rµνρα = f 2(z)Rµνρα +f 2(z)f 2(z)

(gαµ gνρ − gνα gµρ

). (6.13)

Here, the constant curvature condition for the bulk is only satisfied if the brane is maxi-mally symmetric too, this is:

Rµνρα = K(

gµαgνρ − gναgµρ

), (6.14)

withK = −Λ

3.

Therefore, the constant curvature solutions are five-dimensional Minkowski spaces foli-ated by either Minkowski branes (Λ = 0) or de Sitter branes (Λ > 0). The case Λ = 0 wasalready discussed in section 4.1, it is not physically interesting as an observer in the brane,which is flat, would notice the fifth dimension. The case Λ > 0 is a relatively interesting

176 CONSIDERING A GENERAL INTERNAL METRIC Gµν AND AN ARBITRARY

WARP FACTOR F(Z)

case in order to understand the importance of the foliation of the bulk space. Here, aninhabitant of the brane would observe how there is an accelerated exponential expansion,although there would not be an actual expansion in the bulk space, which is flat.

6.2 Λ < 0

We define R0 ≡√− 12

Λ∈ R and β ≡

√−Λ12 . Equation (6.4) restricts the warp factor to

f (z) = Aez/R0 + Be−z/R0 , (6.15)

with A, B ∈ R. Plugging this in equation (6.6), we have that

3R2

0

(Aez/R0 − Be−z/R0

)2+

1R2

0

(Aez/R0 + Be−z/R0

)2+

Λ3

(Aez/R0 + Be−z/R0

)2≡ Λ

2,

3R2

0

(Aez/R0 − Be−z/R0

)2− 3

R20

(Aez/R0 + Be−z/R0

)2≡ Λ

2,

− 12R2

0AB = ΛAB ≡ Λ

2,

AB =Λ

2Λ, (6.16)

where we have used the relationR0 =√− 12

Λin the second and third equalities.

1. Λ = 0From (6.16), we have that either A = 0 or B = 0, then

f (z) = Ce±z/R0 ∈ R, (6.17)

with C ∈ R, which can be absorbed through a change of coordinates in the brane.Thus, we obtain the metrics

ds2± = e±2z/R0 gµνdxµdxν − dz2, (6.18)

with gµν being a solution of the effective dynamical equations Rµν = 0. This is thecase studied in chapter 5.Here, the maximally symmetric solutions coincide with the metrics in the Randal-Sundrum model, (3.3), where the bulk is an AdS5 space foliated by flat branes. Con-sequently, an observer in the brane, would perceive a flat universe, this is, withΛ = 0, whereas the total space is in fact of negative curvature.

2. Λ 6= 0From (6.16), we know that B = Λ

2AΛ, obtaining the solutions

f (z) = Aez/R0 +Λ

2AΛe−z/R0 ∈ R, (6.19)

with A ∈ R.

Let’s consider the maximally symmetric solutions. The non-vanishing components

6 CONSIDERING A GENERAL INTERNAL METRIC Gµν AND AN ARBITRARYWARP FACTOR F(Z) 18

of the Riemann tensor are

Rµzρz =1

R20

gµρ =1

R20

(gµz gρz − gµρ gzz

), (6.20)

which verifies the constant curvature condition (2.4) with K = 1R2

0, and

Rµνρα = f 2(z)Rµνρα +f 2(z)f 2(z)

(gαµ gνρ − gνα gµρ

). (6.21)

It can be proved that the components (6.21) satisfy the same condition only if thebrane is maximally symmetric:

Rµνρα = K(

gµαgνρ − gναgµρ

), (6.22)

withK = −Λ

6. (6.23)

Hence, the maximally symmetric solutions are AdS5 spaces foliated by either AdS4

branes (Λ < 0) or by four-dimensional de Sitter spaces (Λ > 0). In the case Λ < 0an inhabitant of the brane would measure a negative curvature. The case Λ > 0 ismore peculiar, as here the observer would experience an exponential expansion inthe brane, while the five-dimensional spacetime is not actually expanding, but hasconstant negative curvature.

6.3 Λ > 0

1. Λ = 0Again, the conditons in this section coincide with those in section 4, which led to thecomplex warp factor (4.13).

2. Λ 6= 0By replacing B = Λ

2AΛin (6.15) we obtain the following warp factor:

f (z) = Aeiz/R0 +Λ

2AΛe−iz/R0 , (6.24)

with A ∈ C an arbitrary coefficient. If we develop (6.24) in terms of sines andcosines,

f (z) = A[

cos(

zR0

)+ i sin

(z

R0

)]+

Λ2AΛ

[cos

(z

R0

)− i sin

(z

R0

)],

= [Re(A) + iIm(A)]

[cos

(z

R0

)+ i sin

(z

R0

)]+

+Λ

2 |A|2 Λ[Re(A)− iIm(A)]

[cos

(z

R0

)− i sin

(z

R0

)],

=

[1 +

Λ

2 |A|2 Λ

] [Re(A) cos

(z

R0

)− Im(A) sin

(z

R0

)]+

196 CONSIDERING A GENERAL INTERNAL METRIC Gµν AND AN ARBITRARY

WARP FACTOR F(Z)

+i

[1− Λ

2 |A|2 Λ

] [Im(A) cos

(z

R0

)+ Re(A) sin

(z

R0

)],

and impose the condition f (z) ∈ R, we have that

1− Λ

2 |A|2 Λ= 0 =⇒ |A|2 =

Λ2Λ

. (6.25)

It is inmediatly deduced that Λ > 0, as there is no solution otherwise. Considering(6.25), we can write A in terms of a complex phase θ ∈ [0, 2π[,

A =

√Λ

2Λ(cos(θ) + i sin(θ)) . (6.26)

The warp factor can finally be expressed as follows:

f (z) =√

2ΛΛ

cos(

zR0

+ θ

). (6.27)

In this case, the non-vanishing components of the Rieman tensor are:

Rµzρz =−1R2

0gµρ =

−1R2

0

(gµz gzρ − gµρ gzz

), (6.28)

which verify the constant curvature condition (2.4) with K = −1R2

0, and

Rµνρα = f 2(z)Rµνρα +f 2(z)f 2(z)

(gαµ gνρ − gνα gµρ

), (6.29)

which satisfy the same condition only if the internal metric is maximally symmetric:

Rµνρα = K(

gµαgνρ − gναgµρ

), (6.30)

withK = −Λ

3. (6.31)

Hence, the constant curvature solutions are five-dimensional de Sitter spaces foli-ated by four-dimensional de Sitter branes (Λ > 0). An observer in the brane wouldnotice an exponential expansion, while the bulk is expanding as well.

6.4 Brief summary of the solutions obtained

• Both the global and internal metrics satisfy the following Einstein traceless equa-tions:

Rµν =Λ3

gµν, (6.32)

Rµν =Λ2

gµν, (6.33)

where (6.33) represents the effective dynamics in the brane, being the four-dimensionalcosmological constant, Λ, an integration constant which appears as a consequence

7 A STEP BEYOND: GENERALIZING THE DYNAMICS 20

of the five-dimensional dynamics.

• When looking for maximally symmetric solutions in the bulk, the condition requiredis the internal metric to be maximally symmetric. Besides, for this solutions, we seethat there are several combinations of Λ and Λ, and a same bulk space might acceptfoliations of different curvature. For instance, it would possible for an observer inthe brane to observe an expanding universe, while the bulk space is not expandingat all.

Λ Λ f (z) Constant curvature solutions

= 0> 0 ±

√Λ3 z + B, B ∈ R

Minkowski bulk foliatedby De Sitter branes.

< 0 ±i√−Λ

3 z + B /∈ R -

= 0 A, A ∈ RMinkowski bulk foliated

by Minkowski branes.

< 06= 0 Aez/R0 + Λ

2AΛe−z/R0 , A ∈ R

AdS5 bulk foliated eitherby De Sitter branes (Λ > 0) or

AdS4 branes (Λ < 0).

= 0 Ae±z/R0 , A ∈ R

AdS5 bulk foliatedby Minkowski branes.

(Randal-Sundrum)

> 0> 0

√2ΛΛ

cos(

zR0

+ θ)

, θ ∈ R[0, 2π[De Sitter bulk foliated

by De Sitter branes.< 0 No solutions. -= 0 Ae±iz/R0 /∈ R -

Table 1: Different solutions obtained.

7 A step beyond: Generalizing the Dynamics

We have studied so far generalisations of the Randall-Sundrum model (3.3) to non-flatbraneworlds, finding that an effective dynamics arises in the brane, and introducing anintegration constant which acts as a four-dimensional cosmological constant, which inprinciple does not depend on the five-dimensional one. In this chapter we generalise theaction (3.1) by introducing a five-dimensional scalar field φ, called the dilaton. Althoughthis model is more complicated, by adding this field we are considering the possibility of“something else” propagating in the bulk space apart from gravity. We will focus, here,on solutions with non-vanishing effective cosmological constant Λ. We do not intend tostudy all the possibilities here, but notice some remarkable details related to this model.A deeper justification and motivation for dilatonic models is given in [5, 6, 7]. We will usethe following Ansatz [6]:

21 7 A STEP BEYOND: GENERALIZING THE DYNAMICS

Dilatonic braneworld Ansatz with arbitrary internal metric and warp factor:

ds2 = f (z)2gµν(x)dxµdxν − dz2, (7.1)

with f (z) ∈ R, and action

S =1

2κ

∫ √|g|(

R +43(∂φ)2 − e

a3 φΛ

)(7.2)

with φ(x, z) ∈ R being the dilatonic field, arbitrary cosmological constant Λ, and cou-pling constant a.

The equations of motion associated to the system are

Rµν − ea3 φ Λ

3gµν +

43

∂µφ∂νφ = 0, (7.3)

and∇µ∇µφ +

a8

Λea3 φ = 0 (7.4)

which can be obtained from the action (7.2) through the variational principle.6 Equations(7.3) are the traceless Einstein equations associated to gµν, and (7.4) is the dynamical equa-tion associated to the dilaton field φ. Note, as a verification, that imposing φ = 0 in (7.3)leads to Rµν = Λ

3 gµν, which are the equations studied in the previous chapter.

The non-vanishing components of the Ricci tensor are given by the same expressionsthat in chapter 6, since they are obtained from the same five-dimensional metric:

Rzz = 4f (z)f (z)

, (7.5)

Rµν = Rµν − gµν

[3 f 2(z) + f (z) f (z)

]. (7.6)

Introducing (7.5) in the zz component of the Einstein equations (7.3), we have that

4f (z)f (z)

+Λ3

ea3 φ +

43(∂zφ)2 = 0. (7.7)

Similarly, for the µz components of the equations,

43

∂µφ∂zφ = 0, (7.8)

which means that either ∂µφ = 0 or ∂zφ = 0, and for the µν ones:

Rµν − gµν

[3 f 2(z) + f (z) f (z)

]− e

a3 φ Λ

3gµν +

43

∂µφ∂νφ = 0. (7.9)

Due to the complexity of the equations, it is necessary to make some assumption inorder find some solutions. We will use the Ansatz for the scalar field proposed in [6]:

φ(z) = qlog f (z) + φ0, (7.10)

6We will not demonstrate the derivation of the dynamical equations here.

7 A STEP BEYOND: GENERALIZING THE DYNAMICS 22

with q and φ0 arbitrary constants. As we will see in the next paragraphs, the presence ofthe logarithm function considerably simplifies the expressions. (From this point, we willavoid indicating explicitly the z-dependence when writing the warp factor f (z).)

Plugging this Ansatz in (7.4) gives the following differential equation:

∂µ∂µφ + Γµµσ∂σφ +

a8

Λea3 φ = −φ− Γµ

µzφ +a8

Λea3 φ = 0,

−φ− 4ff

φ +a8

Λea3 φ = −q

(ff− f 2

f 2

)− 4q

(ff

)2

+a8

Λea3 φ0 f

a3 q+2 = 0,

q f f + 3q f 2 − Λ8

aea3 φ0 f

a3 q+2 = 0, (7.11)

where we have used the expressions (A.2) for the Christoffel symbols. Analogously, it isstraightforward to see that (7.7) leads to

4ff+

43

q2(

ff

)2

+Λ3

ea3 φ0 f

a3 q = 0. (7.12)

As for the µν components, (7.9), we have that

Rµν − gµν

[3 f 2 + f f + e

a3 φ Λ

3f 2

]= 0,

Rµν − gµν

[3 f 2 + f f +

Λ3

ea3 φ0 f

a3 q+2

]= 0. (7.13)

Since Rµν and gµν depend on the coordinates of the brane, the dependence on z has tovanish, consequently,

3 f 2 + f f +Λ3

ea3 φ0 f

a3 q+2 ≡ Λ

2, (7.14)

and therefore we can write (7.13) as a four-dimensional traceless equation

Rµν =Λ2

gµν, (7.15)

where Λ is an integration constant which acts as a cosmological constant in the brane, aswe know from previous results in this text. Again, an effective dynamics emerges in thebrane as a consequence of the higher-dimensional one.

To sum up, the equations to solve now are the following:

q f f + 3q f 2 − a8

Λea3 φ0 f

a3 q+2 = 0, (7.16)

4 f f +43

q2 f 2 +13

Λea3 φ0 f

a3 q+2 = 0, (7.17)

f f + 3 f 2 +13

Λea3 φ0 f

a3 q+2 =

Λ2

, (7.18)

where we have multiplied (7.12) by f 2. Note that they depend linearly on f f , f 2 andf

a3 q+2, and therefore, proper linear combinations of this three equations are equivalent to

23 7 A STEP BEYOND: GENERALIZING THE DYNAMICS

them. Particularly, it can be proved that (7.16), (7.17) and (7.18) are equivalent to this setof equations: (

43

q2 − 12)(3a + 8q) f 2 = −4 (3a + 2q)

Λ2

, (7.19)

−( a

8+

q3

)Λe

a3 φ0 f

a3 q+2 = −q

Λ2

, (7.20)

f f + 3 f 2 +13

Λea3 φ0 f

a3 q+2 =

Λ2

. (7.21)

From (7.20), we deduce that the exponent, a3 q+ 2, of the warp factor necessarily vanishes7:

a3

q + 2 = 0 =⇒ q = −6a

, (7.22)

and therefore, (7.20) becomes

−(

a8− 2

a

)Λe

a3 φ0 = −q

Λ2

,

which leads to the following relation between both cosmological constants:

Λ =16− a2

24Λe

a3 φ0 . (7.23)

Finally by plugging (7.22) and (7.23) in the equations (7.19) and (7.21), it can be provedthat

f (z) = ± a12

√−Λze

a6 φ0 + C, (7.24)

with C ∈ R an integration constant, and, therefore, we obtain

ds2± =

[± a

12

√−Λze

a6 φ0 + C

]2gµν(x)dxµdxν − dz2,

ds2± =

[± a

12

√−Λz + C

]2e

a3 φ0 gµν(x)dxµdxν − dz2,

where C = Ce−a6 φ0 can be absorbed through a change of coordinates:

ds2± =

[± a

12

√−Λz + 1

]2e

a3 φ0 gµν(x)dxµdxν − dz2, (7.25)

and,

φ(z) = −6a

log[(± a

12

√−Λz + 1

)e

a6 φ0]+ φ0 = −6

alog(± a

12

√−Λz + 1

), (7.26)

where the coordinate z runs from 0 to ∓ 12a√−Λ

. From (7.24) we know that Λ has to benegative, in order to have a real warp factor. (7.23).

Eq. (7.23) expresses the effective cosmological constant in function of the dilaton cou-

7We have considered a non-vanishing coupling constant a in this step. We will omit the cases a = 0, asthe solution obtained in [6] is not relevant here; and a = ±2, since equation (7.19) becomes 0 = 0 for this twovalues of a, and the remaining equations can not be solved analytically.

8 CONCLUSIONS 24

pling a and the cosmological constant of the bulk Λ (in contrast with the previous chap-ter). Consequently, the effective Einstein equations for the brane depends, via the effectivecosmological constant, on the coupling constant. This expresses how effects from the bulkmanifest on the brane. In some way, introducing the dilatonic scalar field in the Ansatzstudied in chapter 6 has allowed a “communication” between the cosmological constants.

8 Conclusions

The Randall-Sundrum model introduces warped compactifications as an attractive andoriginal alternative to the traditional Kaluza-Klein model of compact dimensions, whichis not able to give a satisfactory explanation of the hierarchy problem if the scale of theextra dimension is microscopic. In this frame, large non-compact dimensions are compat-ible with the observations provided that there is a properly curved background “hiding”the additional dimension, this is, confining gravity near the brane and allowing a normal-isation of the massless graviton.

When generalising the Randall-Sundrum model to non-dilatonic curved braneworlds,we have found a surprising result: a four-dimensional effective dynamics arises in thebrane due to the five-dimensional one. The first consequence of this is that an intrin-sically four-dimensional observer could develop a four-dimensional theory for gravitybased on measurements made from the brane. This is consistent with the possibility ofadditional dimensions in our universe. A second important consequence is that the effec-tive four-dimensional cosmological constant (usually interpreted as the energy density ofthe vacuum) is introduced in this context as an integration constant, offering this way apossible explanation to the cosmological constant problem.

Another interesting aspect of this generalisation is that the constant curvature condi-tion for the brane leads to the maximally symmetric solutions for the bulk. Furthermore,we have obtained several combinations of five-dimensional and effective cosmologicalconstants, concluding that, depending on the foliation of the bulk space, an observer inthe brane could actually measure an effective constant curvature Λ completely differentfrom the five-dimensional curvature Λ.

Finally, we have noticed that introducing a dilatonic scalar field in the action of thecurved braneworlds presents potentially interesting properties. For instance, we havefound that the dependence of Λ on the bulk cosmological constant and the coupling dila-ton, expresses how the bulk the influences the brane through the effective Einstein equa-tions.

In conclusion, altough the Randall-Sundrum model was initially proposed as an al-ternative to Kaluza-Klein compactification, its generalisations turn out to be powerfultools to understand some phenomenological problems, as they allow to extract remark-able physical conclusions.

25 A CHRISTOFFEL SYMBOLS AND CURVATURE TENSORS

A Christoffel symbols and curvature tensors

In this appendix we present the non-vanishing Christoffel symbols and components ofthe curvature tensors associated to the general metric:

gµν = f 2(z)gµνdxµdxν − dz2 (A.1)

Although we will not show the calculations here, what we intend with this section is toprovide some algebraic results which might be helpful to the reader when understandinghow the equations are derived from the Einstein traceless equations.

• Christoffel symbols:Γσ

µν = Γσµν

Γzµν = f (z) f (z)gµν

Γσµz = δσ

µ

f (z)f (z)

(A.2)

• Riemann curvature tensor:

Rµνρα = f 2(z)Rµνρα +f 2(z)f 2(z)

(gαµ gνρ − gνα gµρ

)Rµzρz = gµρ f (z) f (z) = gµρ

f (z)f (z)

=f (z)f (z)

(gµz gρz − gµρ gzz

)(A.3)

• Ricci curvature tensor:

Rµν = Rµν − gµν

[3 f 2(z) + f (z) f (z)

]Rzz = 4

f (z)f (z)

(A.4)

• Ricci scalar:

R =R

f 2(z)− 4

[3

f 2(z)f 2(z)

+ 2f (z)f (z)

](A.5)

REFERENCES 26

References

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[2] B. Janssen,Teorıa de la Relatividad General,Universidad de Granada, (2017).

[3] B. Janssen,Curved branes and cosmological (a,b)-models,JHEP 0001 (2000) 044, hep-th/9910077.

[4] N. Alonso-Alberca, P. Meessen, T. Ortın,Supersymmetric Brane-Words,Phys. Lett. B482 (2000) 400, hep-th/0003248.

[5] C. Gomez, B. Janssen, P. Silva,Dilatonic Randall-Sundrum Theory and renormalisation group,JHEP 0004 (2000) 024, hep-th/0002042.

[6] N. Alonso-Alberca, B. Janssen, P. Silva,Curved dilatonic brane-worlds and the cosmological constant problem,Class. Quant. Grav. 17, L163, (2000), hep-th/0005116.

[7] A. Mennim, R. A. Battye,Cosmological expansion on a dilatonic bran-world,Class. Quant. Grav. 18, 2171-2194, (2001), hep-th/0008192.

[8] L. Randall,Extra Dimensions and Warped Geometries,Science, 296(5572), (2002), pp.1422-1427.

[9] M. J. Duff,Kaluza-Klein Theory in Perspective,University of Cambridge, (1994), hep-th/9410046.

[10] S. Weinberg,The Cosmological Constant ProblemsUniversity of Texas, (2000), astro-ph/0005265.

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