UNIVERSITÀ DEGLI STUDI DI TRIESTE - df.units.it Pasquarella.pdf · proiezione del tensore di Weyl su una brana , per la spiegazione dell'origine dei buchi neri pri-mordiali (PBH)

UNIVERSITÀ DEGLI STUDI DI TRIESTE

CORSO DI STUDI IN FISICA

TESI DI LAUREA TRIENNALE

Selected Topics in Gravitational Physics

Laureanda: Relatore:

VERONICA PASQUARELLA Prof. STEFANO ANSOLDI

Anno Accademico 2015/2016

1

Abstract

The present work consists in the analysis of selected topics in the research eld of gravitationalphysics. It consists of three parts: a preliminary revision of the key tools required when dealingwith gravity, hence the derivation of the eld equations from a variational principle, the denitionof the Einstein-Hilbert action, the need for the Gibbons-Hawking boundary term and the Gauss-Codazzi equations for the Riemann tensor decomposition. Then follows the description of higher-dimensional systems, with, both, compactication and warpage: we will concentrate on Kaluza-Klein theory and Randall-Sundrum II model, respectively. We also show the importance of tidalcharge when dealing with the braneworld model and the possibility of explaining the origin ofprimordial black holes (PBH), arising from the contribution of theWeyl tensor's projection on thebrane. The existence of PBH is an essential requirement for the analysis of vacuum metastablityin the very early epochs of the history of the universe. After a brief overview of the work ofthe pioneers in this section, S. Coleman and F. de Luccia, we derive the master equation for thebounce action, hence proving that conical decits may indeed catalyze vacuum decay.

2

Abstract

Il presente lavoro consta nell'analisi di alcuni temi nell'ambilto della sica della gravità. E'strutturato in tre parti: la prima si basa sulla revisione preliminare dei requisiti di base perl'analisi dell'interazione gravitazionale, in particolare, presentiamo la derivazione delle equazionidel campo a partire da un principio variazionale, dunque la denizione dell'azione di Einstein-Hilbert, la necessità di aggiungere il termine di bordo di Gibbons-Hawking e la derivazionedelle equazioni di Gauss-Codazzi per la decomposizione del tensore di Riemann rispetto adun'ipersupercie spacelike. A seguire, la descrizione di sistemi parametrizzati da una dimen-sionalità superiore alle quattro della prima formulazione della relatività generale. In particolare,consideriamo il modello di Kaluza-Klein e Randall-Sundrum II quali esempi di dimensioni com-patticate e warped. Inoltre, mostriamo l'importanza della tidal charge, ottenuta in seguito allaproiezione del tensore di Weyl su una brana, per la spiegazione dell'origine dei buchi neri pri-mordiali (PBH). La loro esistenza è un prerequisito fondamentale per l'analisi della metastabilitàdel vuoto che ha caratterizzato le fasi primordiali dell'evoluzione dell'universo. Dopo un breveriepilogo dei risultati ottenuti dai pionieri del settore, S. Coleman e F. de Luccia, deriviamo laformula generale per la bounce action, dalla quale si evince l'eetto catalitico dei difetti conicinel processo di decadimento del vuoto metastabile.

Contents

1 Introduction 5

2 General relativity: preliminaries 9

2.1 The Einstein-Hilbert action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 The Gibbons-Hawking boundary term . . . . . . . . . . . . . . . . . . . . . . . . 142.3 The Hamiltonian formulation of General Relativity . . . . . . . . . . . . . . . . . 16

2.3.1 Gauss-Codazzi formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.2 The Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.3 The Hamiltonian formulation of General Relativity . . . . . . . . . . . . . 20

3 Higher dimensional systems 29

3.1 Compact and warped extra dimensions . . . . . . . . . . . . . . . . . . . . . . . . 293.1.1 Cartan's Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.2 Kaluza-Klein model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.3 Topological defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.4 Israel's Junction Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 443.1.5 Reissner-Nordstroem solution . . . . . . . . . . . . . . . . . . . . . . . . . 483.1.6 Randall-Sundrum II model . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4 Cosmological phase transitions 73

4.1 Braneworld Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2 False Vacuum Decay with Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3 Adding Conical Decits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.4 Primordial black holes for vacuum decay catalysis . . . . . . . . . . . . . . . . . . 82

5 Conclusion 91

A Background contribution to Euclidean action 93

B Conformal Transformations 97

C Raychaudhuri's Equation 99

3

4 CONTENTS

Chapter 1

Introduction

Mathematics is the language throughout which nature manifests its behavior and indirectlyreveals the beauty of its intrinsic laws. Recursively stated along the centuries, such thoughtis astonishingly contemporary, and in the present work it will appear clear how it can still beextended to the most extreme branches of research. The interplay between dierent branchesof knowledge have lead to great progress towards the understanding of the universe. BeingPhysics part of Philosophy, such progress requires great eorts in conciliating new theories withobservation.

In the present work, we start with some preliminaries concerning the theory of general rela-tivity and also outline the basic tools required when dealing with gravitation. In particular, weprovide the denition of the Einstein-Hilbert action and prove the need of adding the Gibbons-Hawking boundary term after a preliminary digression regarding the geometry of submanifolds.We readily outline the role of the Lagrangian density and the way in which it couples to thevolume element associated to a given manifold, the latter being a nowhere-vanishing k -formassociated to an orientable manifold.

From the denition of the (bulk) Einstein-Hilbert action, we derive the eld equations inboth, vacuo and in presence of a massless scalar eld. From the latter case, it results that theeld acts as a source for the gravitational interaction. The resulting relation will turn out usefulwhen dealing with the topics of the following chapters.

As a further preliminary result, we also derive the Gauss-Codazzi equations, expressing thedecomposition of the Riemann tensor in terms of both intrinsic and extrinsic components w.r.t.a given spacelike hypersurface.

The rest of this work is devoted to the analysis of selected topics that have challenged re-searches in the last few decades, many of which are still a matter of debate. It consists of twochapters: Higher-Dimensional Systems and Cosmological Phase Transitions.

With the ultimate aim of presenting a possible explanation for ination and to describe falsevacuum decay occurring in the very early universe, we outline the importance of adding extradimensions to the original spacetime upon which Einstein's original formulation relies. In doingso, we also analyze some examples of higher -dimensional black holes, in particular, the blackstring, and show how perturbation theory enables to probe the stability of a given metric.

Furthermore, higher dimensional systems enable the study of fundamental interactions: thisis where the geometric nature of gravity emerges with clarity. Branes1 are one such example:they are submanifolds embedded in a higher dimensional setting. Being regular domains, theyare associated to a volume element, which is a (p + 1) form that couples to an antisymmetric

1Branes or p-branes, where p indicates its dimensionality.

5

6 CHAPTER 1. INTRODUCTION

tensor of rank (p+1). They are basically an extension of the coupling of a particle to a potential.According to the Randall-Sundrum model, such setting describes the observable universe as awarped subspace (from which the term braneworld was coined) on which all known interactionsare conned, except for gravity, which can freely propagate through the bulk in the form ofgravitons. The number of polarization states associated to this mediator is what links the twochapters together. When a graviton hits the brane, the exceeding degrees of freedom becomescalar elds. This is what allows to extend the previous results to braneworld cosmology, whichbasically relies upon the notion of conformal transformations for recovering a bulk and braneinationary model in AdS5 starting from a pure gravitational higher -dimensional setting.

The role of conformal transformations is essential in general relativity. In the four -dimensionalformulation, is what allows to dene a (Pseudo)-Riemannian conformally at manifold. Fromthe conformal invariance of theWeyl tensor, which constitutes the trace-free part of the Riemanntensor, we can understand why such term is not mentioned in a four -dimensional description,but, rather turns out to be essential when dealing with a higher dimensionality. Indeed, in thebraneworld model, it is what allows gravity to propagate through the bulk.

In section (3.1.6), when recovering the linearized eld equations on the brane, we have to dealwith the presence of an additional term, arising from the projection (Eab) of the Weyl tensoron the brane, which overcomes the quadratic terms in the brane's energy-momentum tensor atlow energies. The fascinating thing regarding this result is that, even in lack of matter on thebraneworld, the contribution arising from Eab provides a valuable solution to the existence ofprimordial black holes (PBH). The irreducible decomposition of Eab together with the assumptionof a static spherically symmetric solution, lead to a Reissner-Nordstroem type of solution carryingtidal charge. Being at the frontier of current research in cosmology, PBHs are an essentialrequirement when dealing with the topic of cosmological phase transitions2 characterizing thevery early epochs of our universe. In particular, we will show how vacuum decay associated to agiven scalar eld can be catalyzed through the presence of conical decits. After some preliminaryconsiderations concerning the rst semi-classical approach, we describe tunnelling processes whenadding gravity. Although we would expect a more complicated description for this particularsetting, the assumption that gravity preserves the O(4)-symmetry of the problem, allows us touse the Einstein-Hilbert and Gibbons-Hawking boundary terms for recovering the bounce action,where there will also be an additional contribution arising from the wall separating the twovacuum solutions. The location of the wall is determined by the Israel junction conditions, ofwhich we provide a derivation in section (3.1.4).

In what follows, we report the results of the analysis for the Higgs eld case, which is anexample of a double-well potential. After symmetry breaking, hence the birth of the Higgs boson,and the gaining of mass by some of the fundamental particles and quantum mediators, vacuumspacetime experiences a lowering in its energy density. Although we will not provide a detaileddescription of the phenomenon, we remark that it is possible to identify a relation between suchenergy content and the coupling constant associated to the mediator. Hence, probing the rateof change of its magnitude would allow to keep track of the corresponding vacuum transition,meaning that the present universe is still in a metastable state, and could undergo a furthertunnelling process, leading to modications of the present Standard Model.

But is this all? The idea of mediating vacuum decay needs to be combined with the fact thata black hole is characterized by the emission channels of two quantum mediators, i.e. photons,through Hawking radiation, and gravitons. According to the considerations reported above, inthe context of a braneworld scenario, the former are conned on the brane, hence, we shouldbe able to perceive them, although, up to now, because of their weak intensity, researches haveonly been able to study them creating laboratory analogues, relying upon non-linear optics. On

2See chapter 4.

7

the other hand, gravitons can propagate through the bulk, hence leading to an energy loss inthe transverse direction w.r.t. the brane. From the principles of classical mechanics, this meansthat the black hole reacts such that the brane bends, hence leading to a further increase if itssurface tension, which adds to the one provided by the ne tuning condition for an anti-de SitterRandall-Sundrum II model in ve-dimensions (AdS5 RS2); this would ultimately lead to theejection of the black hole from the brane into the bulk. Such process is called recoiling. Hence,when dealing with vacuum tunnelling it is necessary to consider the joined combination of allthree processes. This also shows how progress towards the understanding of dierent phenomena,together with apparently unexplainable observations, are, indeed features of new physics whichis awaiting to be understood.

8 CHAPTER 1. INTRODUCTION

Chapter 2

General relativity: preliminaries

2.1 The Einstein-Hilbert action

In this section we will derive Einstein equations starting from a variational principle. We will rst

consider the case in which only the gravitational eld is present, and obtain Einstein equations in vacuo.

Then, we will consider how matter elds enter into the equations, by using the simpler example of a

minimally coupled massless scalar eld. There is some subtlety in the calculation of the gravitational

action due to the presence of boundary terms: these details are not essential for the derivation of the

eld equations, and will be discussed in detail in the next section.

The Einstein-Hilbert action allows to derive Einstein's eld equations from the least actionprinciple. The construction of a Lagrangian density requires particular care when describinggravitational eects. The structure of a non general relativistic Lagrangian consists of two kindsof terms, precisely, kinetic terms and interaction terms (here, we understand what are also calledpotential terms as self -interaction terms). To emphasize the special case of gravitation let usconcentrate on the kinetic terms: these are usually constructed using the rst order partialderivatives of the elds. However, this is not applicable to the gravitational interaction since therst partial derivatives of the metric are not tensors, thus they are not covariant under coordinatetransformations. Moreover, because of the equivalence principle, it can be seen that no scalarfunction can be constructed from the metric and its rst derivatives only.

In order to consistently dene the gravitational action, it is then also use second order deriva-tives of the metric tensor. Then, the so called Einstein-Hilbert action may be expressed as1

SEH =1

16πG

∫d4x√−g R. (2.1)

The integrand is thus obtained from the curvature of the manifold, which contains second orderderivatives of the metric gµν , as we will soon see. In particular, we call gµν the spacetime metric.It is a type-(0,2) tensor that allows, for instance, to dene the length, L, of a curve xµ(λ) betweenevents P and Q in spacetime as

L =

∫ Q

P

dλ√±gµν xµxν . (2.2)

1We will justify later on the linear dependence from the scalar curvature, R.

9

10 CHAPTER 2. GENERAL RELATIVITY: PRELIMINARIES

It also allows the denition of angles, and it is an essential tool to dene physically measurablequantities. In (2.1) g is the determinant of the metric gµν , and the minus sign under the squareroot is necessary because the metric is Lorentzian, i.e. it is a symmetric matrix with negativedeterminant (coming fromone negative and three positive eigenvalues). We will come back belowto the importance of this term for the action.

Another term in (2.1) that we are now going to discuss is R. To understand the meaning ofthis term we have rst to consider a crucial problem arising from a tensorial description on acurved space (spacetime in our case), that is connected with the concept of derivative. Indeedthe generalization of this concept requires the introduction of an extra term in addition partialderivatives. In the case of a vector eld with components Aα, the covariant derivative ∇βAα(which is often denoted as Aα;β by analogy with the denition ∂βA

α = Aα,β) is dened as

Aα;β = Aα,β + ΓαµβAµ (2.3)

and requires an additional set of functions, the connection Γαµβ , which is a non-tensorial quantity.The result recalled above for vectors can be extended to tensors of arbitrary rank, by adding(subtracting) a term containing Γαµβ for every upper (lower) index of the tensor. For instance,

Aαβ;µ = Aαβ,µ − ΓρµαAρβ − ΓρµβAαρ,

whereasAα

β;µ = Aα

β,µ − ΓρµαAρ

β + ΓβµρAαρ.

Then, under the assumption that the metric is compatible (i.e. that gαβ;γ = 0) and that Γ issymmetric (i.e. Γαβγ = Γαγβ in a basis associated to a givn coordinate system), the Christoelsymbols Γ can be expressed solely in terms of the derivatives of the metric gµν and of the inversemetric gαβ (by convention gασgσβ = δαβ ), as

Γαβγ =1

2gαγ(gµβ,γ + gµγ,β − gβγ,µ). (2.4)

Thanks to the connection, the covariant derivative of a rank (p, q) tensor is again a tensor, of rank(p, q + 1). An important property (or lack thereof) of covariant derivatives, is that, in generalthey do not commute. By considering a vector eld Aα, and using the denitions reported abovewe can readily calculate what happens to the commutator of covariant derivatives. We obtain

Aµ;αβ −Aµ;βα = −RµναβA

µ, (2.5)

whereRρσµν = ∂µΓρµσ − ∂νΓρµσ + ΓρµλΓλνσ − ΓρνλΓλµσ. (2.6)

The Riemann tensor Rµναβ , i.e. the curvature tensor, is what describes this non-commutativity.Since the Riemann tensor has by denition one upper and three lower indexes, it is possible tocontract the upper index with one of the lower ones. As the Riemann tensor is antisymmetric inthe last two indexes, only two dierent kinds of contractions are possible, and the only non-trivialones are those involving one of the two antisymmetric indexes. This allows to dene the Riccitensor as Rµν = Rµαµβ . This tensor can now be further contracted, but only by using the inversemetric gαβ . We obtain in this way the Ricci scalar

R = Rµνgµν . (2.7)

The Ricci scalar, R, will be a good choice for the action, it contains both rst and second orderderivatives of the metric2. Moreover, the latter appear linearly in the Ricci scalar; although in

2This can be seen from the denitions of the connection and of the Riemann tensor reported above.

2.1. THE EINSTEIN-HILBERT ACTION 11

general second order terms can result in equations of motion of fourth order, since second deriva-tives appear linearly in the R density, after an integration by parts all higher order contributionsend up into a boundary term, and do not contribute to the eld equations.

The last term that we need to explain in (2.1) is the volume element. It must be denedin a way consistent with general covariance. Thus, starting from the volume element in specialrelativity, a correction factor, accounting for the more general metric used, must be applied. Thisis related to the Jacobian of the coordinate transformation and leads to d4x

√−g for the volume

element, where we remember that g is the determinant of the metric gµν . The integration in(2.1) is supposed to be extended to all spacetime, which in general relativity is described by apseudo-Riemannian manifold.

Taking R√−g as the Lagrangian density3, we can now proceed to determine the eld equa-

tions. The elds are the components gµν of the metric. As we have already discussed, theaction also contains rst derivatives of the metric (indeed they appear in Γαµν) as well as second

derivatives (because of terms of the form ∂αΓλµν). Although the second derivatives of gµν ap-pear linearly, a fact that can be seen from the denition of the Riemann tensor, from the samedenition we see that non linearities are also present because of the terms containing producs ofΓ's.

We now proceed with the variation of the action. We observe preliminarly that the variationof the Ricci scalar that will appear inside the integral, can be split in two terms,

δR = δRµνgµν +Rµνδg

µν . (2.10)

We then calculate the variation of the action

δS =

∫V

d4xδ(√−gRµνgµν)

=

∫V

d4x[Rµνgµνδ√−g +

√−ggµνδRµν +

√−gRµνδgµν ]

=

∫V

d4x

(Rµν −

1

2Rgµν

)δgµν +

∫V

d4x√−ggµνδRµν . (2.11)

We now need to deal with the last term in (2.11). From the denition of the Riemann tensorgiven in terms of connection its variation reads

δRρσµν = ∂µδΓρνσ − ∂νδΓρµσ + Γρνλ(δΓλνσ) + (δΓρµσ)Γλνσ − (δΓρνλ)Γλµσ − Γρνλ(δΓλµσ)

= ∇µ(δΓρνσ)−∇ν(δΓρµσ), (2.12)

where the denition of covariant derivative has been used: we note, incidentally, that althoughthe connection itself is not a tensor, the variation of the connection is a tensor.

taking the trace of the upper index with the second of the lower indices in (2.12) gives thevariation of the Ricci tensor

δRµν = δRρµρν = ∇ρ(δΓρνµ)−∇ν(δΓρρµ). (2.13)

3It should also be noted that, for a generic matrix M , detM = exp Tr logM , so that

δ detM = (exp Tr logM)M−1δM (2.8)

From this result we obtain

δ√−g = −

1

2√−g

ggαβδgαβ =

√−g2

ggαβδgαβ = −1

2

√−ggαβδgαβ . (2.9)

The last equaliy follows from gαλgλβ = δαβ .


From (2.10) we see that we need a further contraction to get the desired result. Using equationabove we then have

gµνδRµν = gµν∇ρ(δΓρνµ)− gµν∇ν(δΓρρµ)

= ∇ρ(gµνδΓρνµ)−∇ν(gµνδΓρρµ)

= ∇ρ(gµνδΓρνµ)−∇ρ(gµρδΓααµ) = ∇ρ(gµνδΓρνµ − gµρδΓααµ) (2.14)

As we see, the contribution from the last term in (2.11) contributes a total divergence. Afteran integration by parts the term can be transformed into a boundary integral that gives nocontribution to the eld equations, since the boundary metric is assumed to be xed (we willdiscuss this in more detail in the next section, as at the moment this procedure allows us toobtain the correct eld equations). Extremizing the gravitational action in vacuum spacetimeresults then in

δSEHδgµν

=1

16πG

(Rµν −

1

2Rgµν

)= 0, (2.15)

from which we get nothing but Einstein equations in vacuum

Rµν −1

2Rgµν = 0. (2.16)

The left-hand side of these equations is also called, the Einstein tensor, Gµν ,

Gµν = Rµν −1

2Rgµν . (2.17)

Equations (2.16) describe the gravitational eld in the absence of sources. In general, ofcourse, sources are present, so the previous result can be extended to this more general caseby introducing other elds. We will exemplify this process by applying again the Lagrangianapproach that we have followed so far. In particular the Lagrangian densities for these otherelds will have to be added to the Ricci scalar. An example is provided by a massless scalareld. In at spacetime this eld has the Lagrangian

L(M)m =

1

2∂µφ∂

µφ, (2.18)

where we have ∂µφ∂µφ = ηµν∂µφ∂νφ, and ηµν is Minkowski metric. This expression, however is

not covariant. There is a canonical prescription to transform a Lorentz-invariant Lagrangian intoa fully covariant one, called minimal coupling. Minimal coupling consists in replacing ordinarypartial derivatives with covariant derivatives, and Minkowski metric with the general metricgµnu. At the action level, as we discussed already, also the volume element needs to be madecovariant. Following this prescription the Lagrangian density for a massless scalar eld is

Lm =1

2∇µφ∇µφ =

gµν

2∇µφ∇νφ =

gµν

2∂µφ∂νφ (2.19)

with corresponding action

Sφ =

∫V

d4x√−gLm (2.20)

Already using the simplest prescription, i.e. minimal coupling, the presence of other elds inu-ences the spacetime metric, leading to a non-trivial contribution to Einstein equations. We havenow to perform a variation of the action, in the more general form

STOT = SG + Sφ =

∫d4x√−g[

1

16πGR+ Lm

]. (2.21)

2.1. THE EINSTEIN-HILBERT ACTION 13

Two elds now appear in the action, gµν , and φ, so it is necessary to consider both metric andeld variations. We already calculated above the variation of to SG with respect to gαβ , so whatremains is to calculate the variations of Sφ.

Sφ =

∫V

d4x1

2∂µφ∂νφ

√−ggµν . (2.22)

We proceed as follows

δSφ =1

2

∫V

d4x∂µφ(x)∂νδφ(x)√−ggµν +

1

2

∫V

d4x√−g[φ(x),µφ(x),ν −

1

2(∂φ)2gµν

]δgµν ,(2.23)

where the rst term corresponds to the variation of the eld, δφ, while the second comes fromthe variation of the metric, δgµν . Integrating by parts, (2.23) results in

δSφ =

∫∂V

d3x√−g(nµ∂µφ)δφ−

∫V

d4x√−g(

1√−g

∂µ(√−g∂µφ)

)δφ

+1

2

√−g[φ(x),µφ(x),ν −

1

2(∂φ)2gµν

]δgµν . (2.24)

The rst integral does not contribute to the equation of motions, since it is calculated at theboundary. From the remaining terms we can then read the eld equations for φ and the contri-bution of φ to the eld equations for gµν (Einstein equations), respectively,

δSφδφ

= − 1√−g

∂µ√−g∂µφ = −φ , = gµν∇µ∇ν (2.25)

δSφδgµν

=1

2

[φ(x),µφ(x),ν −

1

2(∂φ)2gµν

]. (2.26)

Combining (2.15) with (2.26), we obtain Einstein eld equations in the presence of a masslessscalar eld source

1

16πG

(Rµν −

1

2Rgµν

)=

1

2


1

2(∂φ)2gµν

]. (2.27)

The full system of equations coming from (2.21) is then

φ = 0 (2.28)

Rµν −1

2R gµν = 8πGTµν (2.29)

with Tµν =


1

2(∂φ)2gµν

](2.30)

The last equation denes the energy-momentum tensor (in general, it contains the energy-momentum contribution of all elds permeating spacetime), and acts as the source term inEinstein's eld equations: it shows that it is quite natural for physical elds to couple to thegravitational one, as curvature is aected by the energy and momentum densities and ows.In what follows c will be set to unity, but the gravitational constant will remain explicit. Thereason for this, is that we could be interested to study the problem in dimensionality higher thanfour4, and it may be important to distinguish between the scales set up by gravity in dierentdimensionality.

4Indeed, most of the derivations we did remain substantially unchanged in a higher dimensional setup


2.2 The Gibbons-Hawking boundary term

Above we derived the eld equations from the Einstein-Hilbert variational principle: we haveseen that Einstein equations are obtained from the bulk term arising after a partial integration:this partial integration is especially important in this case, because all higher order terms enterthe boundary term only, and this is the reason that allows for second order eld equations. Theseequations are essential to dene the dynamics of a system, but they are not the only importantresult. As previously emphasized, the gravitational Lagrangian contains second derivatives ofthe metric tensor, while a more traditional Lagrangian density is a function of the elds andtheir rst derivatives only. Technically, because of this fundamental dierence the derivation ofEinstein equations that we discussed above is only correct on a manifold, i.e. spacetime, withoutboundary. When a boundary is present, it is necessary to add further conditions for higher orderLagrangians, and thus also for the gravitational case. This motivates the introduction of anadditional term (called the Gibbons-Hawking boundary term).

To understand this additional contribution it is convenient to rst discuss the geometryof submanifolds, using what is known as the the Gauss-Codazzi formalism. The major featuresneeded for the following discussion will be outlined here. The geometry of a submanifold Σ ⊂M ,where M is equipped with a metric g, can be described by the following quantities (we choose Σspacelike, and the metric signature as before).

1. A unit vector eld nµ normal to Σ, with respect to the metric g.

2. The boundary metric:

habeaαebβ = gαβ + nαnβ ; (2.31)

the tensor hαβ projects all vectors in M into vectors tangent to Σ, a fact that is intuitivelyclear by considering the identity

hαβnβ = gαβn

β + nαnβnβ = nα − nα = 0.

3. the induced metric, which is the intrinsic metric of the hypersurface Σ, and it is denedusing a basis ea

α of vectors tangent to Σ as

hab = gαβeaαeb

β .

4. the extrinsic curvature, which is the covariant derivative of the normal nµ, i.e.

Kαβ = ∇βnα,

which measures the curvature of the embedding of Σ in M. Intuitively, Kµν indicates therate of change of the normal to the hypersurface Σ. Using the basis of vectors tangent toΣ that we already used above, we can also dene

Kab = ∇βnαeαaeβb . (2.32)

From the above denitions, the Riemann tensor for the hypersurface Σ can be decomposedinto two parts, one being the projection of the Riemann tensor of M evaluated on Σ, the othercoming from the extrinsic curvature of Σ into M . We obtain

Rαβγδ = Rµνρσhµαhβ

νhγρhδ

σ − [KγαKβδ −Kδ

αKβγ ], (2.33)

2.2. THE GIBBONS-HAWKING BOUNDARY TERM 15

This identity can be proved directly by remembering the denition of the Riemann tensor. Thenstarting from the denition of hµν , we obtain, for instance

∇µhαβ = nαKβµ + nβKαµ, (2.34)

which shows that extrinsic curvature terms appear in the computation of the covariant derivativeof the boundary metric.

With these denitions we can now go back to the discussion of the boundary term. We willdiscuss what happens when we keep the induced metric xed, and we vary the normal only. Arelated observation concerns the induced metric hab. From its denition we see that xing gµν onthe boundary, indeed implies that hab is held xed during the variation. We can then use theseresults to rewrite the boundary term appearing in the variation of the Einstein-Hilbert action,i.e.

gαβδRαβ = ∇µ(gαβδΓµαβ − gαµδΓβαβ).

Indeed, when we evaluate the bulk integral, we get∫M

d4x√−ggαβδRαβ =

∫M

d4x√−g ∇µ(gαβδΓµαβ − g

αµδΓβαβ)

=

∮Σ=∂M

dΣµ (gαβδΓµαβ − gαµδΓβαβ)

∣∣∣xµ=xµ(ya)

a = 0, 2, 3

=

∮Σ

d3y√h ε (gαβδΓµαβ − g

αµδΓβαβ) nµ, (2.35)

where using Stokes theorem results in a boundary integral over the hypersurface element dΣµ,that can be written in terms of the induced metric in the following way

dΣµ = d3√h ε nµ,

where h stands for the determinant of the induced metric, while ε accounts for the character(spacelike or timelike) of the normal nµ (and it equals −1 in this case, because Σ is spacelike).If we compute the variation of the Christoel symbols on the boundary Σ, we get

δΓµαβ =1

2gµν(δgνα,β + δgνβ,α − δgαβ,ν),

where variation is performed only on the derivatives of the metric, because xing hab on Σ impliesthat δgαβ = 0 = δgαβ on Σ (but normal derivatives of the variation of the metric do not haveto vanish, of course, while tangential derivatives do). Using the above result we can rewrite partof the integrand in (2.35) as

(gαβδΓµαβ − gαµδΓβαβ)nµ = nλgαβ(δgλβ,α − δgαβ,λ).

We can now use the denition of g in terms of h and n. Then

(gαβδΓµαβ − gαµδΓβαβ)nµ = nλ(hαβ − nαnβ)(δgλβ,α − δgαβ,λ)

= nλhαβ(δgλβ,α − δgαβ,λ)− nλnαnβ(δgλβ,α − δgαβ,λ)

= nλhαβ(δgλβ,α − δgαβ,λ), (2.36)

where the second term vanished because the round bracket are antisymmetric in α and λ. Wenow switch from the boundary metric to the induced metric, so that

(gαβδΓµαβ − gαµδΓβαβ)nµ = nλhabea

αeβb (δgλβ,α − δgαβ,λ)

= −nλhabδgαβ,λeaαeβb = −nλhαβδgαβ,λ (2.37)


This is another way to express the contribution from the boundary term.The result in (2.37) shows that the variation of the metric along the normal direction is the

only one which is not required to vanish on the hypersurface. It also suggest the need for aboundary term. Indeed going back to the boundary integral (2.35), we may rearrange the overallEinstein-Hilbert action variation as

δSEH =1

16πG

∫M

d4x√−g Gαβ δαβ −

1

16πG

∮Σ

d3√h ε nµ hαβ δgαβ,µ

At this point, we would like the boundary contribution to vanish, in order to get Einstein's eldequations extremizing the action. We will now prove that the integration of the trace of theextrinsic curvature, K, over the hypersurface Σ, cancels o the second term above. From thedenition of extrinsic curvature we get its trace from contraction with the metric as

K = gαβKαβ = nα;β(hαβ − nαnβ) = nα;βhαβ = hαβ(nα,β − Γναβnν).

When applying the variation to the trace, the result is

δK = −hαβδΓναβnν ,

since the only terms that change are the derivatives of the metric that appear only inside theChristoel symbols. Explicit calculation of the variation of the Christoel symbols leads to thefollowing result

δK = −1

2hαβnµg

µν(δgνα,β + δgνβ,α − δgαβ,ν)

= −1

2hαβ(δgνα,β + δgνβ,α − δgαβ,ν)nν = −1

2hαβδgαβ,νn

ν , (2.38)

which is clearly the result obtained in (2.37), the only dierence being the name of an index.We have thus seen that this last term compensates the one we obtain when performing the

variation of the Einstein-Hilbert action, thereby proving that the trace of the extrinsic curvatureis a good scalar function that we can use to compensate the boundary variation of the overallgravitational action.

We can summarize the results obtained in sections (2.1) and (2.2) throughout the followingexpression accounting for both, the Gibbons-Hawking and Einstein-Hilbert terms

δ

(∫M

d4x√−g R+ 2

∮Σ=∂M

d3x√h K

)=

∫M

d4x√−g Gαβ δg

αβ ,

from which Einstein's eld equations arise.

2.3 The Hamiltonian formulation of General Relativity

The advantage of using the Hamiltonian formalism is well known from analytical mechanics. In the case

of Einstein's eld equations, this approach allows to dene a Cauchy problem that ts the spacetime's

structure, provided a (3+1)-coordinate decoupling may be performed, hence allowing to dene conju-

gate momenta. Setting appropriate initial conditions on spacelike hypersurfaces, means dening both

the induced metric and extrinsic curvature, provided they obey the constraint equations arising from

dierent spacetime projections of the eld equations. First of all, we briey deal with the introduction

of the Gauss-Codazzi formalism, that has already been put to good use when computing the Gibbons-

Hawking boundary term. Then, direct computation of the Hamiltonian action, where we will show it

to be extremized by imposing the Hamiltonian and momentum constraints, which clerly resemble the

constraint equations for the initial value problem.

2.3. THE HAMILTONIAN FORMULATION OF GENERAL RELATIVITY 17

2.3.1 Gauss-Codazzi formalism

To understand the dynamics of spacetime as described by Einstein equations, it is important toidentify the dynamical quantities. Indeed, it is natural to expect that not all the components ofthe metric tensor are dynamical degrees of freedom, as some of them can be xed by appropriatechoices of the reference system.

Some spacetimes can be naturally decomposed into what is called a foliation of spacelikehypersurfaces: the term foliation refers to the fact that it is possible to assign a unique value ofa real parameter to each hypersurface of the family: this parameter can be naturally interpretedas the time of an observer for which each spatial hypersurface is a hypersurface of simultaneity.This decomposition can be useful to understand the dynamical evolution of spacetime as aninitial value problem; under appropriate boundary conditions a unique solution for this problemcan be obtained if we set initial conditions for the dynamical elds and their time derivatives,which, in the case of General Relativity, are the induced metric and the extrinsic curvature. Wealready dened the extrinsic curvature of a given hypersurface int the previous section, where wealso met the idea that four-dimensional quantities can be described in terms of three-dimensionalquantities (intrinsic to a given hypersurface), and of the normal to this hypersurface, which wecall Σ. We will further develop this idea in what follows. In particular, we start by consideringthe covariant derivative of a four-vector. With the same notation that we used before, we canwrite

Aα;βeβb = (εnαnµ + hameαaemµ)Aµ;βe

βb

= ε(nµAµ;βe

βb )nα + ham(Aµ;βe

µme

βb )eαa , (2.39)

where we see that the two terms are normal and tangential to Σ, respectively. Let us now considerthe case in which Aµ is orthogonal to nµ (i.e. Aµ is practically a three-dimensional quantity, i.e.a tangent vector to the hypersurface normal to nµ), the rst term can be rewritten by movingthe covariant derivative on nµ, and we then obtain

Aα;βeβb = −ε(nµ;βA

µeβb )nα + hamAm|beαa

= Aa|beαa − εAa(nµ;βe

µaeβb )nα = Aa|be

αa − εAaKabn

α. (2.40)

The vertical bar denotes the covariant derivative intrinsic to the hypersurface Σ, i.e. the covariantderivative that can be constructed with reference to the metric hab only. It is then clear that,with the above expression, we have decomposed the four dimensional covariant derivative of avector Aµ tangent to an hypersurface normal to nµ in two parts. One is the part that can becalculated by knowing the geometry of Σ only, the other the part that is related to the embeddingof Σ.

From the last expression above applied to the basis vectors tangent to Σ, we can extract whatare called the Gauss-Weingarten equations,

eαa;βeβb = Γcabe

αc − εKabn

α. (2.41)

This expression turns out to be useful when we want to calculate similar decompositions forother tensors, as the curvature. Indeed for our purpose the decomposition of the Riemann tensoris particularly important, and it will allow us to express it in terms of the curvature of Σ andof the normal contributions arising from the extrinsic curvature. This can be achieved startingfrom the denition of the Riemann tensor as the `commutator' of the covariant derivatives and


using (3.45). By calculating the various terms separately, we rst have

(eαa;βeβb );γe

γc = eαa;βγe

βb eγc + eαα;βe

βb;γe

γc

= eαa;βγeβb eγc + eαα;β(Γdbce

βd − εKbcn

β)

= eαa;βγeβb eγc + Γdbc(Γ

eade

αe − εKadn

α)− εKbceαa;βn

β . (2.42)

We now proceed in a similar way for the term in round brackets in the expression above:

(Γcabeαc − εKabn

α);γeγc = Γdab,ce

αd + Γdabe

αd;γe

γc − εKab,cn

α − εKabnα;γe

γc

= Γdab,ceαd + Γdab(Γ

edce

αe − εKdcn

α)− εKabnµ;γe

γc − εKacn

µ;βe

βb .(2.43)

We can now combine the two results above to obtain the nal expression for eαa;βγeβb eγc . We now

need another contribution with the same structure, but with the the last two indexes (i.e. γ and

β) interchanged, and this can be obtain directly from the expression for eαa;βγeβb eγc . We can then

obtain

(eαa;βγ − eαa;γβ)eβb eγc = −Rαµβγeµae

βb eγc

= −Rmabceµm − ε(Kab|c −Kac|b)nµ − εKabn

µ;γe

γc + εKacn

µ;βe

βb , (2.44)

which is an expression for the Riemann tensor in 4-dimensions Rαµβγ is terms of the Riemanntensor of the hypersurface, Rmabc, and of the extrinsic curvature. By projecting this result alongthe hypersurface Σ we nally get

Rαβγδeαaeβb eγc eδd = Rabcd + ε(KadKbc −KacKbd). (2.45)

Projecting (2.44) along the normal direction, we obtain instead

Rµαβγnµeαae

βb eγc = Kab|c −Kac|b. (2.46)

The two results above, (2.45) and (2.46), constitute the Gauss-Codazzi equations: given anhypersurface Σ they allow to express the components of the curvature tensor in terms of thecurvature tensor intrinsic to the Σ, and of the extrinsic curvature of Σ.

To be able to express Einstein equations in a similar fashion, we need rst to extend thisdecomposition to the Ricci scalar. This can be directly computed by applying the denition ofthe Ricci scalar, i.e. by appropriately contracting the Riemann tensor. The Ricci scalar mayrst be rewritten as

R = gαβRαβ = (εnαnβ + hαβ)Rαβ

= εnαnβRαβ + hαβgµνRµανβ

= εnαnβRαβ + hαβ(hµν + εnµnν)Rµανβ

= εnαnβRαβ + hαβhµνRµανβ + εhαβnµnνRµανβ

= εnαnβRαβ + εhαβnµnνRµανβ + habhmneαaeβb eµme

νnRµναβ . (2.47)

The rst two terms in (2.47) may be combined, by observing that

hαβRµανβnµnν = gαβRαµβνn

µnν = Rµνnµnν ;

here we have used the completeness relation and the denition of the Ricci tensor, together withthe fact that Rµανβn

µnαnνnβ = 0, because of the symmetries of the Riemann tensor. If we now


use the denition of the Riemann tensor we can write

2εRαβnαnβ = 2εgµνRµανβn

αnβ

= 2εRµαµβnαnβ

= 2ε[−nµ;µβnβ + nµ;βµn

β ]

= 2ε[−(nµ;µnβ);β + nµ;µn

β;β + (nµ;βn

β);µ − nµ;βnβ;µ]. (2.48)

where the second term is K2 = nµ;µnβ;β and K stands for the trace of the extrinsic curvature.

But we can also express the last term arising in equation (2.48), in terms of the extrinsiccurvature. In order to do so, we need some preliminary considerations, precisely that

nαnα;β =1

2(nαnα);β = 0 ; Kab = nα;βe

αaeβb .

By using these relations, we can rewrite it as

nα;βnβ;α = gβµgανnα;βnµ;ν = Kν;µKµ;ν = Tr(K2) = K

= (εnβnµ + hβµ)(εnαnν + hαν)nα;βnµ;ν

= hβµhανnα;βeαaeβb nµ;νe

µme

νn = hbmhanKabKmn = KabK

ba, (2.49)

thereby also proving that KabKab = KαβK

αβ .Going back to the last term featuring in equation (2.47), we can apply the rst of Gauss-

Codazzi's equations, precisely relation (2.45), therefore obtaining

habhmnRµανβeµme

αaeνneβb = habhmn[Rmanb + ε(KmbKan −KmnKab)], (2.50)

If we subsequently name 3R = habhmnRmanb as the induced Ricci scalar and perform index con-traction between the induced metric and extrinsic curvature, we obtain the ought decompositionof the 4-dimensional Ricci scalar as

R = 3R+ ε(K2 +KabKab) + 2ε(nα;βnβ − nαnβ;β);α. (2.51)

We see that it is determined by 3R,K and Kab and a covariant derivative with respect to thenormal to the hypersurface.

Such result shall turn out useful for to the Hamiltonian formulation of General Relativity.Indeed, it provides the adequate integrand over which we can perform space-time integration.

What we notice, is also that, in doing so, the last term in equation (2.51) can be easily integrated,giving rise to a spatial boundary integral over a hypersurface whose displacement in the manifoldis parametrized by the particular value given to the timelike coordinate. The boundary integralwill turn out essential for determining the asymptotic behavior of a manifold, which in turnenables to consider the background contribution of vacuum spacetime to euclidean action.

2.3.2 The Initial Value Problem

According to Classical Mechanics, solutions to the equations of motion are uniquely determinedby the setting of initial conditions.

As we have already discussed in section (2.1), Einstein's eld equations are second -order.Their corresponding initial conditions require the denition of the metric and its correspondingtime derivative. In order to obtain such parameters, we need to set a spacelike hypersurface Σ,with intrinsic metric hab = gαβe

αaeβb ; as a consequence of this choice, together with the fact that


we are dealing with a four -dimensional manifold, the timelike direction results to be the normalto the chosen hypersurface. Recalling the denition we have given for extrinsic curvature

Kab = nα;βeαaeβb ,

where nα is the normal to the hypersurface, we can actually identify Kab with the variation ofthe metric with respect to the timelike-direction.

The choice of the two symmetric tensor elds hab and Kab is not arbitrary: they are con-strained by the Gauss-Codazzi equations. If we combine the Ricci tensor's and scalar's decom-positions, together with the denition of the Einstein tensor, Gαβ = Rαβ − 1

2Rgαβ , performingindex contraction on (2.45) and (2.47), we can rewrite the Gauss-Codazzi equations in vacuumspacetime as

−2εGαβnαnβ = 3R+ ε(KabKab −K2) = 0, (2.52)

Gαβeαan

β = Kba|b −K,a = 0; (2.53)

together, (2.52 and (2.53) provide the constraint equations that the induced metric and extrinsiccurvature shall have to satisfy in order to constitute valuable initial conditions for a Cauchyproblem in General Relativity.

In the next section, we will show how (2.52) and (2.53) will emerge as two of the fourconditions extremizing the Hamiltonian action.

2.3.3 The Hamiltonian formulation of General Relativity

The Hamiltonian functional allows us to rewrite the eld equations in terms of rst-order partialdierential equations, whose solutions are uniquely determined by the initial values of both, thephysical elds and their conjugate momenta.

The Hamiltonian approach for a Cauchy problem in General Relativity can be obtained byconsidering a foliation of 4-dimensional spacetime in terms of 3-dimensional spacelike hypersur-faces, labeled by a parameter t, where the induced metric and extrinsic curvature satisfy theconstraint equations derived in section (2.3.1). In what follows we will rst discuss this slicingprocedure (foliation).

When we wrote the Einstein-Hilbert action, we integrated over a four -dimensional spacetimevolume V , without considering it explicitly as space and time. In the spirit of a space-time 3 + 1decomposition, however, it is convenient to characterize more precisely the boundary in terms ofthe spacelike/timelike characters of the various components. In particular we dene, [1]

∂V = (−Σt1) ∪ Σt2 ∪ B, (2.54)

where Σt1 and Σt2 are spacelike the initial and nal time part of the boundary (and are charac-terized, respectively, by the parameter values t = t1 and t = t2), while B constitutes the timelikepart of the boundary of the volume, V , and connects the two hypersurfaces. The minus sign in(2.54) makes it explicit that the normals to the Σ's are pointing towards th bulk of the volume.For convenience, we will use Greek letters for the tensors on the original four -dimensional man-ifold, Latin letters for tensors on the Σt and its characteristic parameters, while capital Latinindexes will refer to tensors on St, resulting from the intersection St = Σt ∩ B.

The (3+1)-decomposition can be also considered as a coordinate transformation. In doingso, we dene a coordinate system (t, ya) obtained from the original xα-system, such that xα =xα(t, ya), where ya are the 3-coordinates dening the spacelike hypersurfaces Σt. We can further


dene a vector eld tα, as being the tangent to any geodesics crossing the hypersurfaces; suchvector eld can be decomposed into a timelike and a spacelike term as follows

tα = Nnα +Naeαa ,

where N and Na stand for the lapse function and shift three-vector.This can also be expressed in terms of the partial derivative of the original coordinate system,

at xed values of ya, with respect to the timelike parameter, hence obtaining the following

tα =

(∂xα

∂t

)ya, (2.55)

where nα denotes the unit normal to Σt and eαa are tangent vectors to Σt, so that nαeαa = 0.

Using (2.55) and

dxα = tαdt+ eαadya = (Ndt)nα + (dya +Nadt)eαa , (2.56)

we can rewrite the line element as

ds2 = gαβdxαdxβ = −N2dt2 + hab(dy

a +Nadt)(dyb +N bdt). (2.57)

Using the denition of the cofactor of a matrix element we can write

gtt = cofactor(gtt)/g = h/g, (2.58)

(where g indicates the determinant of the metric gαβ). Combining (2.57) with (2.58), (extractthe time-time component of the metric), resulting in

√−g = N

√h. (2.59)

A similar decomposition can be used to analyze all boundary terms, which will turn out to beuseful for the Hamiltonian description.

The intersection between the two spacelike hypersurfaces, Σt1and Σt2 , and timelike hyper-surface, B, is characterized by an induced metric, σAB = habe

aAe

bB , and a unit normal to the

boundary St, rα, with rαnα = 0, such that the original metric can be expressed as

gαβ = −nαnβ + rαrβ + σABeαAeβB . (2.60)

The extrinsic curvature of the St with respect to the four -dimensional manifold is

kAB = rα;βeαAe

βB . (2.61)

We follow the same procedure for B , which stands for the timelike boundary of V . In thiscase, we use the coordinates zi = (t, θA) for the hypersurface. The induced metric on B, i.e.

γij = gαβeαi eβj , can be written as

γijdzidzj = −N2dt2 + σABdθ

AdθB , (2.62)

where the lapse function is obtained as a partial derivative of xα with respect to the timelikeparameter at xed values of the coordinates θA, describing the two-sphere St, precisely

N = nα

(∂xα

∂t

)θA.


From the metric decomposition, we also get a relation between the metric determinants, precisely

√−γ = N

√σ,

which is similar to (2.59). We may also assign an extrinsic curvature to B with respect to theoriginal four -dimensional manifold. Since St = Σt ∩ B, we may dene

kij = rα;βeαi eβj , (2.63)

and its trace as k = kijγij . With the above notation, the gravitational action can be written as

16πSTOT =

∫V

d4x√−gR− 2

∫Σt2

d3y√hK + 2

∫Σt1

d3y√hK + 2

∫B

d3z√−γk. (2.64)

where the normal to a hypersurface is taken to be positive if pointing towards the inside of thewrapped volume V . By substituting R with expression (2.51), the bulk term in the Hamiltonianaction becomes∫

V

d4x√−gR =

∫ t2

t1

dt

∫Σt

d3y√hN(3R+KabKab −K2)− 2

∮∂V

dΣα(nα;βnβ − nαnβ;β), (2.65)

where, for the integration over the boundary, the conventional - sign has been used, since thenormal is timelike. This term is, basically, the sum of three dierent integrals, two of whichcancel out with the other two boundary terms in equation (2.64), as it can be shown integratingby parts and using the denition for the trace of extrinsic curvature, K, and the fact that theintegration element is dΣα = d3y nα

√h on the hypersurface Σt1 (the same goes for Σt2). Thus,

the only net contribution arises from the B-term, for which, following the previous derivations,the surface element is

dΣα = d3z√−γrα.

Therefore, after an integration by parts, the integral becomes

−2

∫B

d3z√−γnα;βnβrα = 2

∫B

d3z√−γnαnβrα;β .

The overall action turns out to be

16πSTOT =

∫ t2

t1

dt

∫Σt

d3y√h N (3R+KabKab −K2) + 2

∫B

d3z√−γ(k + rα;βn

αnβ).

It is possible to further simplify the second term in this last equation by applying the relationsdescribing the surface B; rewriting the trace of its extrinsic curvature, we get

k = γijkij = γij(rα;βeαi eβj ) = rα;β(γijeαi e

βj ) = rα;β(gαβ − rαrβ). (2.66)

Substituting this in the second term of (??), the integrand can be written as

k + rα;βnαnβ = rα;β(gαβ − rαrβ + nαnβ) = rα;β(σABeαAe

βB) = σABkAB = k.

This means that the boundary integral depends only upon the trace of the extrinsic curvatureof St. By also noticing that

d3z√−γ = d2θdt

√σN


for B, the total action can then be expressed as

STOT =1

16π

∫ t2

t1

dt

[∫Σt

d3y√h N (3R+KabKab −K2)

+1

8π

∮St

d2θ√σN(k − k0)

], (2.67)

where the term k0 is required to avoid divergences when taking the limit St → +∞, and can beinterpreted as background subtraction (it enables the total action to remain nite in the limitof at spacetime5). To rewrite (2.67) explicitly in Hamiltonian form explicit from, we need tointroduce the conjugate momentum to the induced metric on the spacelike hypersurface Σt. Indoing so, we introduce the notion of the Lie derivative, dened as follows

£uAα = Aα,βu

β − uα,βAβ ,

where we have taken the case of a type-(1,0) tensor. We can extend such denition to tensorswith an arbitrary number of indexes; for example, a type-(1,1) tensor follows the rule

£uTαβ = Tαβ,µu

µ − uα,µTµβ + uµ,βT

αµ ,

where we notice that the sign preceding the last two terms on the r.h.s. are dictated by whetherthe index of the tensor that we are considering is either superscript (the former) or subscript (thelatter). We also notice that the advantage of the Lie dierentiation is that it doesn't require theconnection to be dened. If we apply this to the three-metric, we get

hab = £thab

= £t(gαβeαaeβb )

= (£tgαβ)eαaeβb

= (tα;β + tβ;α)eαaeβb = ((Nnα +Nα);β + (Nnβ +Nβ);α))eαae

βb

= (nαN,β +N,αnβ +N(nα;β + nβ;α) +Nα;β +Nβ;α)eαaeβb

= 2NKab +Na|b +Nb|a, (2.68)

where we have used following relations

nαeαa = 0 , Nnα;βe

αaeβb = NKab , Nα;βe

αaeβb = Na|b ,

together with the symmetry of extrinsic curvature and the denition of intrinsic covariant dif-ferentiation, e.g. in Na|b. What emerges from (2.68), is that time evolution of the three-metricis related to the extrinsic curvature. Rearranging this formula, the result is

Kab =1

2N(hab −Na|b −Nb|a), (2.69)

which can be substituted in equation (2.67). This shows that the action depends only on thetime evolution of hab, while both the lapse and shift functions, N , Nα, do not contribute withtime derivatives. Their role is to dene the foliation of the initial volume of integration V intospacelike hypersurfaces Σt.

5In Appendix A, we will briey discuss the method for deriving background contributions.


To go further with the Hamiltonian description, we need to go from the rst time derivativeof hab to the conjugate momentum of the induced metric. This can be obtained as a partialderivative of the Lagrangian density, as usual. Since

16π√−gLG =

√hN [3R+ (hachbd − habhcd)KabKcd], (2.70)

from relation (2.69), we get the conjugate momentum of hab6,

pab =∂STOT

16π∂hab=

√hN

16π

[∂3R

∂hab+∂(KabKab)

∂hab− ∂K2

∂hab

]=

√hN

16π

[1

2NKab +

1

2NKab −

habK

N

]

=

√h

16π[Kab − habK];

its trace results as follows

16πp =√h[K − 3K] ⇒ K = − 16π

2√hp,

from which we get the following relation

√hKab = 16π

(pab − 1

2phab

), (2.71)

where p = pabhab. The Hamiltonian density is obtained performing the Legendre transformationof the Lagrangian

HG = pabhab − L = pab2N1√h

[(pab −

1

2phab

)+Na|b +Nb|a

]+√h N(−3R−KabKab +K2)

= 2N1√h

[(pabpab −

1

2p2

)+ pabNa|b + pabNb|a

]+

1√hN(−3R−KabKab +K2).

If we rewrite Kab and K2 in terms of the conjugate momentum, and then integrate the Hamil-

tonian density, we get

HG =

∫Σt

[d3y

N√h

(pabpab −

1

2p2

)−N√h

3R− 2

√hNa

(1√hpab)|b

]

−2

∮St

d2θ√σ

[N(k − k0)−Na

1√hpabrb

]. (2.72)

To prove that this result is consistent with Einstein's equations, we need to derive the eldequations from equation (2.72), for instance by varying the action with respect to N,Na, haband pab. The variation is restricted by the conditions δN = δNa = δhab = 0. For the lapse andshift functions rst, and noticing that the boundary term does not contribute, the result is

δHG

δN= −3R+

1

hpabpab −

1

2hp2 = 0 ;

δHG

δNa=

(pab√h

)|a

= 0,

6We also notice that in the Lagrangian there is no term accounting for the evolution of the lapse and shift,hence they have no associated momentum: indeed they are not dynamical variables.


which constitute the Hamiltonian and momentum constraints, respectively. We now turn ourattention to the variation with respect to the conjugate variables hab and p

ab. Applying variationwith respect to (pab) 7

δpHG =

∫Σt

d3y

[N√hδp

(pabpab −

1

2p2

)− 2√hNaδp

(1√hpab)|b

]

−2

∮St

d2θ√σ

[N(k − k0)−Na

1√hpabrb

]. (2.73)

Since the second term in the rst integral is a total derivative, we may calculate partial integra-tion, thereby obtaining

δpHG =

∫Σt

d3y2

[N√hδp

(pabpab −

1

2p2

)+N(a|b)

]− 2

∮St

hNaδp

(1√hpab)|b

−2δp

∮St

d2θ√σ

[N(k − k0)−Na

1√hpabrb

]= 2

∫Σt

d3y

[N√hδp

(pabpab −

1

2p2

)+Na|b

], (2.74)

where the last passage justied by the fact that, when we perform variation with respect tomomentum on the surface integral in (2.72), we gain the same lower-dimensional integral arisingfrom partial integration of the rst

∫Σt.

Now, the last thing we have to determine is variation with respect to hab, starting fromequation (2.72), 8

δhHG =

∫Σt

d3y

[−NhNδh

(pabpab −

1

2p2

)δh

1√h

+N√hδh

(pabpab −

1

2p2

)−Nδh(3R

√h)

]−2δh

∮St

d2θ√σNa

1√hpabrb + 2δh

∫Σt

d3y Na|bpab − 2

∮St

d2θ√σN

1

2habδhab,cr

c.(2.75)

Recalling equation (2.9) and applying it to the induced metric

δh√h =

1

2

√hhabδhab,

and noticing that the surface integral vanishes since we have taken the boundary metric to be

7δp(pabpab − 1

2p2)

= 2(pab − 1

2phab

)δpab

8K = nα;α = (εnαnβ + hαβ)nα;β = hαβnα;β = hα(nα,β − Γγαβnγ)

⇒ δK = −hαβ(δgµα,β + gµβ,α − gαβ,µ)nµ = 12hαβδgαβ,µn

µ supposing that the tangential components (to thehypersurface) of the metric don't vary on the boundary.


xed on the hypersurface, the Hamiltonian variation results in9

δhHG =

∫Σt

d3y

[− N√

h

(pcdpcd −

1

2p2

)hab + 2

N√h

(pacp

bc − 1

2ppab

)−Nδh(3R

√h)

]δhab

+2δh

∫Σt

d3y Na|bpab − 2

∮St

d2θ√σN

1

2habδhab,cr

c

= 2

∫Σt

d3y[− N√h

(pcdpcd −

1

2p2

)hab + 2

N√h

(pacp

bc − 1

2ppab

)+N√hGab + 2pcaN|c + 2pcbN|c]δhab +∫

Σt

d3y[−√hN(habδΓcab − hacδΓbab)|c + 2pbcN

dδΓcbd]

−2

∮St

d2θ√σN

1

2habδhab,cr

c. (2.77)

Integrating by parts the rst term of the second integral, we get∫Σt

d3y√hN,c(h

abδΓcab − hacδΓbab)−∮St

d2θ√σN(habδΓcab − hacδΓbab)rc

=

∫Σt

d3y√hN,c(h

abδΓcab − hacδΓbab) +

∮St

d2θ√σNhabδhab,cr

c. (2.78)

The second term cancels o with the boundary integral in (2.77). Now, going through thecomputation of the rst term in (2.78), the result is∫

Σt

d3y√hN,c(h

abδΓcab−hacδΓbab) =

∫Σt

d3y√h

1

2(habN ,d−N ,ahbd)[(δhda)|b+(δhdb)|a− (δhab)|d]

=

∫Σt

d3y√h(habN

|dd −N

|ab)δhab,

where we have once again relied upon partial integration. The last term of (2.77) that remainsto be determined is

2

∫Σt

d3ypbcNdδΓcbd =

∫Σt

d3y pabNd[(δhda)|b + (δhdb)|a − (δhab)|d]

=

∫Σt

d3y√h

1√hpabNd(δhab)|d = −

∫Σt

d3y√h

[1√hpabNd(δhab)

]|d.

By renaming all the relevant terms for the Hamiltonian variation as

1. Hab = 2 N√h

(pabpab − 1

2p2)

+ 2Na|b

2. P ab = − N2√h

(pcdpcd − 1

2p2)hab + 2 N√

h

(pacp

bc − 12pp

ab)

+N√hGab −

√h(N |ab − habN |cc ) +

2pcaN|c + 2pcbN|cN,c −√h(

1√hpabN c

)|c

9Since 3R is the lower-dimensional Ricci tensor

Gab = Rab −1

23Rhab ⇒ δh(3R

√h) = −

√hGabδhab +

√h(habδΓcab − h

acδΓbab)|c. (2.76)


3. C = −3R+ 1hp

abpab − 12hp

2

4. Ca =(pab√h

)|a,

we obtain a more neat expression for the Hamiltonian variation, subtracting the terms in commonwith the term pabhab, we can integrate

δHG =

∫Σt

d3y(P abδhab +Habδpab − CδN − 2CaδN

a),

from which we gain the corresponding action variation as

δSG =

∫dt

[∫Σt

d3y(pabδhab + habδpab − δHG

]=

∫dt

[∫Σt

d3y(hab −Hab)δpab − (pab + P ab)δhab + CδN + 2CaδN

a

]. (2.79)

From (2.79), we are able to determine the action's extremizing conditions

1. hab = Hab

2. pab = −P ab

3. C = 0

4. Ca = 0,

where the rst two are the evolution equations of the induced metric and its conjugate momen-tum, while the last two are the Hamiltonian and momentum constraints, respectively.

For the case of vacuum spacetime, we may dene the gravitational Hamiltonian in terms ofthe surface integral in equation (2.72)

HG = − 1

8π

∮St

d2θ√σ[N(k − k0)− 2Narb(K

ab −Khab)], (2.80)

whose importance becomes relevant for non-compact manifolds; its value takes into account theasymptotic behavior of the domain of integration St and of both lapse and shift. As previouslystated, N and Na are arbitrarily chosen, while the asymptotically at prole of the domain Stindirectly implies that Σt should be spacelike, therefore implying N = 1 and Na = 0, resultingin, [1]

HG = − 1

8πlimSt→∞

∮St

d2θ√σ(k − k0) = M, (2.81)

which may be identied with the gravitational mass of at space-time as a consequence of asymp-totic time behavior. Such result is in accord with classical results, since invariance under timetranslation implies energy conservation. Starting from expression (3.114), it is also possible torecover a formal expression for angular momentum. By setting N = 0 and Na = φa, (3.114)reduces to, [1]

J = − 1

8πlimSt→∞

∮St

d2θ√σrbφa(Kab −Khab). (2.82)

By using (2.81), we are now able to determine the background contribution arising from vacuumspacetime, that we then need to subtract when performing action computation.


Chapter 3

Higher dimensional systems

3.1 Compact and warped extra dimensions

In this chapter, we are going to deal with the topic of extra dimensions and their impact on the description

of physical laws. We will focus on the two long range fundamental interactions and will outline their

description when the new dimensions are either compact or warped. Unication and dark energy provide

valuable examples in which extra dimensions are thought to be playing a key role. At rst, we will

deal with the compactied Kaluza-Klein model, introducing the notion of conformal transformations to

recover the Einstein-Hilbert action. Then, we shall proceed with the warped Randall-Sundrum 2 model,

leading to the introduction of the Weyl tidal charge for brane black holes, and the proposal that they

are good candidates for primordial black holes. In the last section belonging to this chapter, we shall

also provide two ways for deriving the linearized Einstein eld equations, in the setting of the braneworld

model.

3.1.1 Cartan's Formalism

In this section we will provide the basic tools needed for deriving curvature from a given metricassociated to a manifold with arbitrary dimensions. In particular, this provides a way of derivingcurvature from a given metric, ultimately leading to the eld equations. We start computing thecovariant derivative of a vector, [2],

∇v = ∇(vαeα) = vα(∇eα) + (∇vα)eα

= vαΓγβαeγ ⊗ ωβ + vα,βeα ⊗ ωβ = (vα,β + Γαβγv

γ)eα ⊗ ωβ , (3.1)

where eα and ωβ stand for contravariant and covariant basis vectors, respectively; we are con-sidering covariant dierentiation with respect to one of the basis vectors, labeled by the index β.When covariant dierentiation acts on a 0 -form, such as vα, in this case, it reduces to ordinarypartial dierentiation. From the above calculation, we extract the following relation

∇eγ = Γαβγeα ⊗ ωβ ,

which enables to dene the one-form, or spin-connection, as

θαγ = Γαβγωβ . (3.2)

29

30 CHAPTER 3. HIGHER DIMENSIONAL SYSTEMS

Analogously to (3.1), the covariant derivative of the metric, may also be written in terms of(3.2),

∇g = ∇(gαβωα ⊗ ωβ) = ∇(< eα, eβ > ωα ⊗ ωβ) = (< ∇eα, eβ > + < eα,∇eβ >)ωα ⊗ ωβ

− < eα, eβ > Γαδγωδ ⊗ ωβ− < eα, eβ > ωα ⊗ Γβδγω

δ

= (< Γδγαeδ, eβ > ωγ+ < eα,Γδγβeδ > ωγ)ωα ⊗ ωβ

− < eα, eβ > Γαδγωδ ⊗ ωβ− < eα, eβ > ωα ⊗ Γβδγω

δ

= (Γβγαωγ + Γαγβω

γ)ωα ⊗ ωβ− < eα, eβ > Γαδγωδ ⊗ ωβ− < eα, eβ > ωα ⊗ Γβδγω

δ

= (θβα + θαβ)ωα ⊗ ωβ− < eα, eβ > Γαδγωδ ⊗ ωβ− < eα, eβ > ωα ⊗ Γβδγω

δ. (3.3)

In order to simplify the algebra, we choose to work with the Levi-Civita connection, ∇, which is

1. metric compatible, i.e.

∇gαβ = ∇ < eα, eβ >= 0 ⇒ θβα = −θαβ (3.4)

therefore implying that the spin-connection is antisymmetric,

2. torsion-free. Torsion, Ωαβγ , is dened as

Ωαβγ = Γαβγ − Γαγβ − Cαβγ , (3.5)

where the termsCαβγ = ωα([eβ , eγ ])

are called structure constants. It is possible to prove that such formula is consistent. Ageneral k -form ω is a linear alternating map acting on a k -ple of vector elds. Indeed, theone-form ωα acts on a smooth vector eld arising from the Lie brackets, since the latter isa linear map which preserves tensor type. Starting from equation (3.2) and combining itwith (3.5), we obtain1

θαγ ∧ ωγ = Γα[βγ]ω[β ∧ ωγ] =

1

2(Ωαβγ + Cαβγ)ωβ ∧ ωγ

=1

2Ωαβγω

β ∧ ωγ +1

2ωα([eβ , eγ ])ωβ ∧ ωγ (3.6)

where the symbol [. . . ] means anti-symmetrization with respect to the enclosed indexes,which, in turn, follows from (3.4). At this point, we are going to use the relation between theexterior and Lie derivatives. For the purpose of the calculation, we outline the formulationfor a one-form, but it can be extended to more general forms as well. The relation resultsas

dωα(eβ , eγ) = eβωα(eγ)− eγωα(eβ)− ωα([eβ , eγ ]) = −ωα([eβ , eγ ]); (3.7)

2 dωα is a two-form, obtained applying exterior dierentiation to a one-form. Exterior dier-entiation maps a k-form into a (k + 1)-form. Hence ωα can in turn be built applying exteriordierentiation to a 0 -form, f . Hence, when computing the two-form, two wedge products arise.

1Following the rule: A ∧B = A⊗B −B ⊗A, where A and B are covectors.2The notation used here, ωα(eγ) , indicates that the one-form acts as a linear map on a vector. Indeed,

ωα =< eα,− >.

3.1. COMPACT AND WARPED EXTRA DIMENSIONS 31

One of them is null, following from one of the properties of exterior dierentiation, i.e. d(df) = 0.eγ and eβ are two vectors; here they are applied as dierential operators to the scalar eld re-sulting from the action of the covector ωα on the remaining vector, i.e. eβ and eγ , respectively.The equivalence in between the two sides follows from Leibniz's rule for function's composition.The rst two terms on the r.h.s. are null because the argument of the directional derivative lookslike < eα, eγ >, which is a Kronecker -delta, hence its directional derivative, eβ , is null. Torquecan be expressed in terms of the spin-connection from equation (3.6) and the two-form dωα asfollows,

1

2Ωαβγω

β ∧ ωγ = θαγ ∧ ωγ +1

2dωα(eβ , eγ)ωβ ∧ ωγ = θαγ ∧ ωγ + dωα; (3.8)

we notice that the two-form, in the last passage, dωα acts on two basis vectors, leading to a realnumber. But this is compensated by the presence of the wedge product which in turn allows tore-build a two-form. Since here we are dealing with vectors and their dual counterparts, we areleft with the two-form dωα. Setting Ω = 0, from equation (3.5), we obtain the following result

dωα = −θαγ ∧ ωγ . (3.9)

The curvature two-form is dened as, [2],

Rαβ =1

2Rαβγδω

γ ∧ ωδ = dθαβ + θαγ ∧ θγβ , (3.10)

where Rαβγδ is the Riemann tensor.

3.1.2 Kaluza-Klein model

Initially proposed by Theodor Kaluza in 1919 in an attempt to unify General Relativity withMaxwell's eld equations, Kaluza-Klein theory has been later completed by Oskar Klein in 1926,who contributed to the theory providing a geometrical interpretation to the model.

The idea that gravity permeates all spatial dimensions while the other interactions are con-ned to the 4-dimensional spacetime, provides a possible explanation for its relative weaknesswith respect to the others.

In this section we will show how Kaluza-Klein's model extracts four -dimensional gravity froma higher dimensional action, and also how the extra compactied dimension modies the eectivegravitational constant, hence proving what we have stated at the end of section (2.1). This is anexample of an eective eld theory, meaning that it accounts for the degrees of freedom describinga particular length or energy scale, while ignoring substructures that would eventually arise atshorter distances.

Since the metric is a symmetric tensor, for a ve-dimensional description, there are fteenindependent components. Among these, ten are associated to the 4-dimensional spacetime met-ric, and four to the vector potential. There is only one component left, which is associated to anunidentied scalar eld, which assigns a length scale to the fth-dimension.

As a consequence of the connement of interactions, a combined description of, both, grav-itational and electromagnetic interactions requires the addition in the ve-dimensional metric,g5, of the vector potential coupled to the basis of the 4-dimensional metric components.

The rst thing to do is to hypothesize the structure of g5, such that it resembles two distinctsubspaces, characterized by gµν and gφφ respectively. The choice we make for the line elementis the following, [2],

ds25 = gµνdx

µdxν − e2σ(x)[dψ +Aµdxµ]2, (3.11)


where gµν is the metric in 4D, xµ are the four-dimensional spacetime coordinates, ψ indicates thecompactied extra dimension, σ = σ(xµ) and Aµ = Aµ(xµ), while e2σ constitutes the diagonalcomponent gψψ, associated to the massless scalar eld σ. The o -diagonal terms of g5 havebeen rearranged in the second term. From (3.11), we are able to recover the four -metric, just bysuppressing the extra dimension.

Our aim is to build up a 5D action associated with this metric and subsequently compareit with the description we have obtained in (2.1) for the case of a four -dimensional spacetime.In order to do so, we need to evaluate the ve-dimensional Ricci scalar, R5. Such task requiresa certain procedure, that basically relies upon the denitions of connection and curvature. Wemay identify the major steps that have to be followed as being:

1. the identication of an orthonormal basis starting from the given metric (3.11),

2. the computation of the connection one-forms,

3. the curvature two-forms obtained from covariant external dierentiation of the connectionone forms,

4. the identication of the relevant terms arising in the previous point for the Ricci tensor in5D,

5. computation of the Ricci scalar R5 from the above.

At rst, we extract an orthonormal basis from the line element (3.11), for both the 4Dmanifold and the extra compact dimension, which are ωa = eaµdx

µ and ωψ = eσ[dψ + Aµdxµ],

respectively. We now have to follow the steps we have already outlined, in order to get to theve-dimensional Ricci scalar. Using (3.9), we start by expressing the exterior derivatives of theorthonormal basis as, [2],

dωα = −θα0β ∧ ωβ ; dωψ = σ,αωα ∧ ωψ + eσF,

where we have dened eσF = 1/2eσFαβωα∧ωβ for convenience. Their respective spin-connections

follow

θαβ = θα0β +1

2eσFαβ ω

ψ, (3.12)

θψα = σ,αωψ +

1

2eσFαβω

β , (3.13)

where the former takes into account the contribution arising from both the four -dimensionalmetric, gµν , via the term θα0β , and cross-terms coupled to the vector potential, while the latterindicates the spin-connection between the two manifolds. Starting from (3.12), we derive thecurvature two-form, by applying external derivation on the connection one-form. We start fromequation (3.12),

Rαβ = dθα0β +1

2d(eσFαβ ω

ψ) + θαγ ∧ θγβ + θαψ ∧ θ

ψβ ; (3.14)

this example shows how external dierentiation diers from ordinary dierentiation, since itrequires the addition of extra terms featuring as wedge products. There are two such terms in


equation (3.14) because of the additional contribution to the two-form arising from the extra-dimension. Substituting in equation (3.14) the expressions for the connections we have previouslyderived, we get

Rαβ = dθα0β +1

2eσσ,γF

αβ ω

γ ∧ ωψ + +1

2eσFαβ,γω

γ ∧ ωψ + θαγ ∧ θγβ + θαψ ∧ θ

ψβ

= dθα0β +1

2eσσ,γF

αβ ω

γ ∧ ωψ +1

2eσFαβ,γω

γ ∧ ωψ

+1

2eσFαβ (σ,γω

γ ∧ ωψ + eσFγδ) + θα0γ ∧ θγ0β +

1

2eσ(Fαγ ω

ψ ∧ θγoβ + θα0γ ∧ Fγβ ω

ψ)

+1

4e2σFαγ Fβδω

γ ∧ ωδ +1

2eσ(Fαγ ω

γ ∧ σ,βωψ + Fβδσ,αωψ ∧ ωδ). (3.15)

In order to obtain the 5-dimensional Ricci scalar, we need rst to select from equation (3.15) theterms providing a contribution for the evaluation of the four -dimensional Ricci scalar, to whichwe shall have to add the one arising from the interaction with the compactied dimension ψ. Inorder to do so, we have to select the terms leading to the trace components when contracted withthe four -metric, hence ignoring all terms with a ψ-index. The terms with all indexes belongingto the 4-dimensional manifold, are:

1. dθα0β + θα0β ∧ θγ0β

2. 12e

2σFαβ Fγδ + 14e

2σFαγ Fβδωγ ∧ ωδ.

In the former we recognize the background contribution to curvature, inherited by the 4D metricgµν , which, combined together, lead to the background term, Rαoβγδ, while the latter arises fromthe presence of Maxwell's tensor in the connections.

The four terms we have selected are not enough for determining the ve-dimensional Ricciscalar: we also need to identify the cross terms arising in the form Rψαψβ , which, when contracted,lead to the contribution from the extra compactied dimension. In order to do so, we follow thesame procedure starting from equation (3.13) to obtain the curvature two-form labeled by theextra dimension. In this case, the wedge product will feature only once, as a consequence of the ωb

term in the connection. Since the compactication has only one dimension, the wedge producthas to be taken with respect to the connection one-form belonging to the four -dimensionalmanifold with metric gµν . Hence, we obtain

Rψα = dθψα + θψβ ∧ θβα = d

(σ,αω

ψ +1

2eσFβγω

γ

)∧ θβα

= d

(σ,αω

ψ +1

2eσFαβω

β

)+

(σ,βω

ψ +1

2eσFβγω

γ

)∧(θβ0α +

1

2eσF βαω

ψ

)= σ,αβω

β ∧ ωψ + σ,α(σ,γωγ ∧ ωψ + eσF ) +

1

2eσ(σ,γω

γ ∧ ωβFαβ

+Fα,γωγ ∧ ωβ − 1

2Fαθ

β0 ∧ ωγ) + σ,βω

ψ ∧ θβ0α

+1

2eσFβγω

γ ∧ θβ0α +1

4e2σF βαFβγω

γ ∧ ωψ. (3.16)

We now select the mixed terms that have one index, belonging to the four -dimensionalmanifold, and a ψ-component

σ,αβωβ ∧ ωψ ; σ,ασ,γω

γ ∧ ωψ ; σ,βωψ ∧ θβ0α ;

1

4e2σF βαFβγω

γ ∧ ωψ


from which we get the associated Riemann tensor components as

Rψαψβ = −σ,αβ − σ,ασ,β + σ,γΓγβα −1

4e2σF γβ Fαγ ,

where the change in sign is a consequence of index permutation and the Christoel symbolarises from the denition we have provided for spin-connection in (3.2). At this point, wehave to combine the two sets of Riemann components arising from both calculations (3.15) and(3.16), and subsequently perform an index contraction, leading to the following block-diagonalcomponents:

Rψψ = −σ − (∇σ)2 − 1

4e2σF 2 (3.17)

associated to the extra compact dimension, where the d'Alambertian is obtained combining thesecond order partial derivative of the scalar σ together with the Christoel symbol, hence usingthe denition of covariant dierentiation; on the other hand, from Rψαψβ , we get

Rαβ = R0αβ −∇α∇βσ −∇ασ∇βσ +1

2e2σFαγF

γβ , (3.18)

where, in addition to the terms we have selected from (3.15), there are also the ones arising fromRψa that are labeled by the indexes related to the four -dimensional manifold. Contracting theRicci tensor component (3.18) and summing it up with (3.17), we obtain the 5-dimensional Ricciscalar

R5 = R0 +1

4e2σF 2 − 2σ − 2(∇σ)2 = R0 +

1

4e2σF 2 − 2e−σeσ, (3.19)

where the last passage follows from the identication of a total derivative in the last two σ terms.R0 describes the gravitational eld in 4D, 1

4e2σF 2 is related to Maxwell's equations, coupled to

the scalar eld σ, and the last term describes the equation of motion of σ.From previous considerations 3, we may write the 5D action as

S5 = − 1

16πG5

∫V

d5x√g5R5

= −∫dψ

16πG5

∫∂V

d4x√geσ

(R0 +

e2σ

4F 2

)+ C, (3.20)

where∫dψ is the length of the compactied dimension, while C is the total derivative obtained

integrating the last term in equation (3.19). Nevertheless, we may notice that (3.20) diers fromthe denition of the Einstein-Hilbert action for the presence of the prefactor eσ, indicating thecoupling of the scalar to the trace of the energy-momentum tensor.

In the next part of this section we shall analyze the indirect ways in which the extra dimensionsinuences the description of fundamental interactions. We start from extracting the Lagrangiandensity associated to the gravitational eld from (3.20), [2],

L5 = −LeσR0

16πG5

√g,

where L constitutes the length of the compactication (as a result of the integration along theψ-direction), while e−σ(x)G5/L constitutes an eective Newton constant, which depends on the

3See chapter (2.1) for the denition of the Einstein-Hilbert action.


position of the observer in spacetime, through the scalar eld σ. If we want to get back to acanonical Einstein-Hilbert action, we need to apply a conformal transformation on the metricfollowing a non-global rescaling of the metric

gαβ = Ω2(x)gαβ ,

where Ω2(x) constitutes the weighting function, leading to a change in curvature, R4.4 Going

back to our previous computation, we need to choose an adequate weighting function, enablingto cancel out the unwanted prefactor of the Ricci scalar in (3.20) by setting Ω = e−

σ2 . From this,

we can transform the Lagrangian density, hence obtaining, [18],

√−g5R5 = Ω4√−gΩ−2(R0 − 6Ω−1Ω) = e−σ

√−g(R0 + 3σ − 3

2(∇σ)2). (3.21)

As we can see, the e−σ-term cancels o with the eσ-term in the 5D action (3.20). From equation(3.21), we recover the Einstein-Hilbert action accounting for gravitation, the electromagnetic anda massless scalar eld, resulting in the following expression

S =L

16πG5

∫d4x√−g

(−R0 +

1

2(∂φ)2 − e

√3φ

4F 2

), (3.22)

where we have renamed the scalar eld as φ = σ/√

3, [2]. (3.22) makes explicit the degrees offreedom of the system we are dealing with, which are:

1. the graviton, associated to the metric gµν ,

2. the vector potential Aµ,

3. the massless scalar eld, φ, whose quantum counterpart is the dilaton.

Equation (3.22) allows to determine a relation between the gravitational constant and the di-mensionality of the system, hence obtaining G4 = G5/L, as we can see from (3.22).

At the end of this section, we outline some general considerations that will turn out usefulat the end of this chapter. The periodicity of the compactied extra dimension implies that anyfunction of the coordinate ψ may be Fourier -decomposed in the following way

f(xµ, ψ) =

+∞∑n=−∞

fn(xµ)ei2πnψ/L,

where with xµ we denote the usual four dimensional coordinates (with µ = 0, 1, 2, 3), n indicatesthe massive modes, while L stands for the extension of the compactied dimension. In ourcase, we may think of f(xµ, ψ) as resembling a scalar eld, whose dynamical evolution in vedimensions is described by

5f(xµ, ψ) = 0 ⇒ 4fn =

(2πn

L

)2

fn = m2nfn,

where with mn we denote the eective mass of the mode n. This notation will turn out veryuseful in the last section of this chapter, where Fourier modes will be used for estimating thecorrection terms to the linearized gravitational eld equations.

4We refer to the Appendix for a quick overview of how to obtain this result.


3.1.3 Topological defects

According to certain cosmological models, the evolution of the universe at very high red shifts canbe described in terms of rst-order phase transitions where the acquisition of a non-zero vacuumexpectation value involves a change in the energy density of the scalar eld. In the next chapter,we are going to deal with the process of spontaneous symmetry breaking of the Higgs scalar eldand will see how to evaluate its tunnelling rate throughout action computation. Vacuum decayis associated to the lowering of the energy-density content of spacetime.

In order to describe this in terms of the gravitational action, we rely upon a thermodynamiccomparison with specic heat, which is a continuous slowly-varying function on both sides withrespect to the phase transition, but behaves as a δ-function at the transition point, requiring anite amount of energy to produce an innitesimal increase in temperature.

Its cosmological counterpart results in the universe being reheated. A Higgs-like potentialgives a non-zero vacuum expectation value. The acquisition of an energy content is associatedto the formation of topological defects.

The features of the defect are related to, both, the topological defect and the surroundingspace dimensions with respect to the broken symmetry. For a 3-dimensional case, there are fourdierent congurations:

1. Domain Walls, where symmetry breaking implies a discretization and space is divided intoregions separated by barriers with a certain energy per unit area;

2. Strings: these are linear defects;

3. Monopoles: these are point-like defects;

4. Textures: these are delocalized topological defects, which are associated to an extremelyhigh energy content, reason why their study is expected to lead towards a better under-standing of the physics of the early universe.

In the following sections we will deal with the rst three types of defects.

Domain walls

At rst, we will analyze the domain wall case. Indeed, this conguration turns out to be veryuseful when treating the connement of interaction elds on a braneworld.

Perturbative solutions in Quantum Field Theory are obtained by quantizing the eld arounda local minimum. In what follows we will analize the non-perturbative solutions to quantumeld theory. We start by considering the Lagrangian density for a self -interacting scalar eld inMinkowski spacetime as

L =1

2((∂tφ)2 − (∇φ)2)− V (φ), (3.23)

where φ = φ(z), with z being a spacelike direction. We have taken V as resembling a Higgs-likepotential5 with its associated eld equation which are, respectively, [2],

V =λ

2(φ2 − η2)2 ; φ+ 2λφ(φ2 − η2) = 0,

where with λ we indicate the coupling constant characterizing the interaction associated to thescalar eld, while ±η indicate the values of scalar eld providing a null potential. Since the scalar

5With Z2-symmetry.


eld is a function of a single spacelike coordinate, we can rewrite the equation of motion as

−φ′′ + 2λφ(φ2 − η2) = 0 ⇒ −1

2φ′ 2 +

λ

2(φ2 − η2)2 = const., (3.24)

where the last passage is obtained integrating over the equation of motion and multiplying bothterms by φ′. Such potential has two distinct vacua at φ = ±η.

We are interested in nding an interpolating smooth function connecting the two minima,which is also a solution of the eld equations. If we require that φ → ±η as z → ±∞ inMinkowski spacetime, where η is a nite value, a possible solution to the equation of motion isgiven by

φ(z) = η tanh[√λ η(z − z0)], (3.25)

where z0 indicates the location of the wall. This solution, located at z0 = 0, is known as the kinksolution. Noticing that both terms on the l.h.s. of equation (3.24) are proportional to sech4(z),and taking the constant arising from integration as being null, we can rewrite the total derivativeof the solution as

φ′(z) =√λη2 sech2(

√λη z) =

√λ(η2 − φ2) =

√2V .

As suggested by the analytical structure of the solution, the transition between the two vacuarequires an innite energy, because we need to integrate along the whole z-axis. Nevertheless,we may also argue that the tanh(z)-function does not dier too much from unity until z ≈1/(√λη); this allows us to assign a width to the domain wall, which is determined by, both, the

coupling constant and vacuum expectation value, providing the eld's with a natural scale. Thisconstitutes an example of a topological defect , i.e. a non perturbative solution.

When performing quantization around the minima, it is not possible to pass between the twovacua, since the transition would require an innite amount of energy. This confers a topologicalstability to the solution which constitutes a local energy minimum.

The energy-momentum tensor associated to the eld6 is 7

Tµν = φ,µφ,ν − gµνL = ∂µ∂νφ−(

1

2((∂tφ)2 − (∇φ)2)− V

)gµν

= λ η4 sech4(√λ η z) δzµ δ

zν − gµν

(−φ′2

2− V

)= λη4 sech4(

√λ η z)(δzµδ

zν + gµν), (3.26)

where we have used deltas because the eld is parameterized by a single coordinate z, whileindexes of the energy-momentum tensor refer to all spacetime coordinates. From the solutionthat we have obtained, hence the energy-momentum, is strongly localized around z = z0, i.e. thewall's location. We can write explicitly the transverse and parallel components of (3.26) withrespect to the z-axis, hence obtaining, from (3.26), the following components

T 00 = −T xx = −T yy =

1

2φ′2 + V = λ η4 sech4(

√λ η z) = 2V ; T zz =

1

2φ′2 − V = 0, (3.27)

meaning that Tµν almost describes vacuum spacetime. The transverse components of the energy-momentum tensor resemble strongly localized functions around the center of the kink. If weintegrate its components through the kink, we get, [2],∫

dzT 00 =

∫dz λ η4 sech4(

√λ η z) ≈ 4

3

√λη3. (3.28)

6As we have proved in chapter (2.1).7Here we use the following relations: V = φ′(z)/2 and φ′2 = λη2 sech4(

√ληz).


Up to now we have been dealing with at spacetime. In what follows, we will analyze thesame problem when gravity comes into play. Relying upon Einstein's eld equations, we wouldexpect that the apparent divergence8 of the energy value associated to the wall would look like asingularity. Despite appearance, we will show that gravity is non-singular on this hypersurface.

From the homogeneous energy-momentum distribution given by (3.27), the domain wall9 aris-ing from symmetry-breaking has a Lorentz symmetry along the (t, x, y) coordinates, suggestingit has constant curvature.

Under such conditions, the generic line element in aD-dimensional manifold may be expressedin the form, [2],

ds2 = A2(z)ηµνdxµdxν − dz2, (3.29)

where γµν is the (D − 1) dimensional metric with one-timelike and two-spacelike coordinates,i.e. the transverse ones with respect to the z-direction. The line element (3.29) is an example ofwarped compactication; this occurs when the metric, in this case γµν , is preceded by a conformalprefactor which comes as a function of the extra dimension, z. It is important to notice that weare making no particular assumption regarding the structure of the lower-dimensional metric.Our aim is to determine what it actually looks like and see whether the kink could be a valuablesolution to Einstein's eld equations in the higher-dimensional space. We will rely upon Cartan'sformalism, thereby following the same steps of section (3.1.2), showing that it is possible to obtainEinstein's equations from such a geometrical conguration. First of all, the orthonormal basis is

ωz = dz ; ωa = A(z)ωa0 , (3.30)

where ω0 indicates the background basis belonging to the (D−1) dimensional submanifold. Theirexterior derivatives are

dωz = 0 ; dωa =A′

Aωz ∧ ωa − θa0b ∧ ωb,

respectively, where we have used (3.30) in order to obtain an expression in terms of the back-ground one-forms, ω0. As a consequence, we obtain the connection 1-forms

θa b = θa 0b ; θa z =A′

Aωa, (3.31)

from which the curvature two-forms can be obtained as

Ra b = dθa0b + θa 0c ∧ θc 0b + θa z ∧ θz b = Ra0b +

(A′

A

)2

ηbcωa ∧ ωc

=1

2Ra0 bcdω

c0 ∧ ωd0 +

(A′

A

)2

ηbcωa ∧ ωc, (3.32)

(where we have used ηbc to take into account the signs for spacelike/timelike components), and

Ra z =A′′

Aωa ∧ ωz −

A′

Aθa0 z ∧ ωz + θa 0b ∧

A′

Aωb,

where the last two terms cancel out as a consequence of relations (3.31). We may now reconstructthe Riemann tensor, therefore obtaining

Rabcd =1

A2Ra0 bcd +

(A′

A

)2

(δac ηbd − δadηbc) ; Razbz =A′′

Aδab . (3.33)

8The apparent divergence is due to the innite area problem associated to the domain wall. Indeed, in atspacetime, it seems to have an innite extension.

9Orthogonal to the z-direction parameterizing the scalar eld.


For a manifold of arbitrary dimension D resembling the symmetrical properties of (3.29), the zz-term of the trace of the Ricci tensor in the D dimensional manifold is

Rzz = (D − 1)A′′

A.

Furthermore, combining the two terms in (3.33) and contracting the indexes, we obtain thecorresponding Ricci tensor in D-dimensions as

Rµν =1

A2Rµ0ν + (D − 2)

(A′

A

)2

δµν +A′′

Aδµν . (3.34)

The symmetry of the given metric resembles a de Sitter solution.10 In such case, the Ricciscalar is related to the cosmological constant and horizon radius, such that R = 2D/(D− 2)Λ =D(D − 1)/l2 = k/l2, where, according to this last chain of relations, k is dened to be theparameter accounting for the presence of a positive cosmological constant, while l correspondsto the de Sitter radius determining the extension of the isotropically expanding universe. Goingback to the D=4 case [2], we have to evaluate the Einstein tensor components for, both, thez component and the (D − 1) submanifold from equation (3.34). The general formula for theEinstein tensor reads

Gµν = Rµν −1

2Rgµν =

1

A2Rµ0 ν +

(A′

A

)2

δµν +A′′

Aδµν −

1

2

(R0

A2+ 2

(A′

A

)2

+A′′

A

)gµν . (3.35)

According to the coordinate we want to analyze, we get two dierent results by tracing theprevious formula. In particular, we get

G00 =

k

l2A2− A′2

A2− 2

A′′

A= 8πG

(V +

1

2φ′2)∝ sech4(

√λ η z), (3.36)

Gzz = 3

(k

l2A2− A′2

A2

)= 8πG

(V − 1

2φ′2)

= 0, (3.37)

where relations (3.27) have also been used. We also get the wave equation for the scalar eld;noticing that A3 =

√−γ, according to relation (3.19), we get

φ = − 1

A3∂zA

3∂zφ = −2λφ(φ2 − η2). (3.38)

Looking at Einstein's eld equations, we shall now consider what happens near vacuum solutions;in order to do so, we consider the case in which η/MP1 , where η is the vacuum expectationvalue and MP is the Planck mass. The right hand side of equation (3.36) becomes, to lowestorder in 8πGη2,11 we get [2],

8πGV = o(8πGη2λη2). (3.39)

In the above expression, there is λη2 which is indeed proportional to 1/w2, where with w wedenote the width of the wall. This actually sets a scale to the problem we are analyzing. By

10A de Sitter -like solution is the one used by the FLRW description of the evolution of the universe startingfrom an initial singularity.

11Here we have used the following relation φ′2 = 2V ∝ λη4.


setting w = 1, we are left with an energy parameter, i.e. 8πGη2, which, under the assumptionwe have previously made (that the energy scale is well below the Planck scale), we can expandEinstein's equations in terms of this parameter. Hence, equations (3.37) and (3.36) read

A′2 =k

l2;

A′′

A= −8πGη2 V

η2= −8πGη2

2sech4(z),

as obtained expanding w.r.t. 8πGη2. If we integrate the latter once, we obtain

A′ = −8πGη2 tanh z

(1− 1

3tanh2 z

), (3.40)

which is equivalent to equation (3.37) when k = 1 and l = 3/16πGη2, where we recognize thecharacteristic parameters of de Sitter space. When we perform a second integration, we obtain

A = 1− 8πGη2

(2

3ln cosh z − 1

6sech2 z +

2

3ln 2

).

It is now possible to approximate A to leading order, thereby obtaining [2],

A ≈ 1± z

l,

as a consequence of the expansion of the leading term, i.e. sech z .We now go back to the original request of determining whether there are any singularities

in the description of a domain wall. Approaching the edge of de Sitter space, i.e. z → ±l, themetric (3.29) becomes, [2],

ds2 =(

1− z

l

)2(dt2 − l2 cosh2 t

ldΩ2

II

)− dz2, (3.41)

from which we can deduce that a spacetime characterized by constant curvature has a compactextension. dΩII is the solid angle element arising from the (D − 1)-part of the metric (3.29),where, we recall the fact that there are two spacelike coordinates conned; indeed, we hadalready stated that constant curvature implied a spherical symmetry. From the given metric,we can deduce that a spacetime characterized by constant curvature has a compact extension.This also shows how coupling to gravity may solve the divergence problem aforementioned. Ifwe apply a coordinate transformation renaming the variables

ρ = (l − z) cosht

l; τ = (l − z) sinh

t

l(3.42)

they provide a description resembling the one of a at spacetime. Here we are obviously referringto the outer spacetime w.r.t. the wall. By subsequently imposing the conditions

z = const⇒ ρ2 − τ2 = (l − z)2 = const,

andt = const⇒ ρ ∝ τ,

we can deduce that the wall, looks like a hyperboloid in Minkowski spacetime. Inside thehyperboloid lies the lightcone. As z → l ,12 the gap between the hyperboloid and the internal

12Which may appear as a singularity by looking at the metric (3.41).


lightcone tends to zero. With the above parametrization, both regions separated by the domainwall, i.e. z < 0 and z > 0, lie on the inside of the hyperboloid. This means that the wall isactually nite and that there is no physical singularity emerging as a consequence of an inniteenergy (recall the innite area problem). The only singularity we have to consider is a coordinateone, which resembles an event horizon in Rindler coordinates (3.42). This means that gravity isnon-singular. The energy required for jumping across the wall is nite, hence we can interpretthe wall as the boundary capable of patching together two vacuum solutions of the eld equations.We will come back to this topic when dealing with vacuum decay.13

Black Branes and Higher Dimensional Black Holes

We will now apply Cartan's formalism to a particular metric. The result obtained shall allow usto perform conical decit regularization, which is described at the end of this subsection. Thisshall prove to be essential when dealing with vacuum metastability, to which we have devotedthe last chapter of this work. Nevertheless, the particular structure of the metric we are aboutto analyze, actually describes a wide range of other interesting objects, which are p-branes. Wewill come back to this topic in the last two subsections belonging to this chapter, where we alsoprovide the reason of their importance in theoretical physics.

We will consider the case in which the manifold is constituted by two submanifolds of arbi-trary dimensions. The presence of the r-coordinate which features as an extra coordinate andparameterizes the conformal factors, is what allows the two submanifolds to interact with eachother at the level of the curvature two-forms. Such hypothesis allows us to express the lineelement in the following form, [2],

ds2 = A2(r)gµνdxµdxν − dr2 − C2(r)γαβdy

αdyβ , (3.43)

where gµν and γαβ constitute the metrics of the two subspaces, with p and n-dimensions, re-spectively, while A(r) and C(r) are conformal prefactors, such as the one we have dealt withwhen analyzing the kink. Going through the same computations as in (3.1.3), we can extract theconnection one-forms for the two submanifolds, which shall have to consider both, the interiorconnections belonging to each submanifold, and also for their mutual interaction with the radialcoordinate, hence obtaining the following, [2],

θµν = θµ0ν ; θµr =A′

Aωµ,

and

θαβ = θα0 β ; θαr =C ′

Cωα,

where we have conventionally used the index notation µ, ν and α, β to label the tensorial quantitiesproper to the p and n-dimensional manifolds, respectively. At the level of the connection there isno mixing up of the g and γ. In order to describe their reciprocal inuence, we need to computecurvature. For the connection two-forms there are many dierent contributions that we have toconsider; rst, the terms connecting the n-submanifold and the r-coordinate, which are given by,[2],

Rαβ = Rα0β +

(C ′

C

)2

ωα ∧ ωβ , Rα r =C ′′

Cωα ∧ ωr, (3.44)

13See chapter (4).


for the p-submanifold. The same can be done for the n-manifold by changing the indexes α→ µ,β → ν and C → A in equations (3.44). Apart from these, which resemble the result we haveobtained in the previous section, we need to add the cross-terms

Rµ α = θµr ∧ θr α =A′C ′

ACωµ ∧ ωα, (3.45)

arising from the combination between the spin-connections belonging to the two submanifolds.If we now use the denition of the curvature two-form for each of the two submanifolds, [2],

Rab =1

2Rabcd ω

c ∧ ωd,

where Rabcd actually stands for the background in both of them, we can get the Riemann compo-nents from (3.44) as14

Rµνλτ = A−2Rµνp λτ +

(A′

A

)2

(δµλδντ − δν λδµ τ ), (3.46)

Rαβγδ = C−2Rαβn γδ +

(C ′

C

)2

(δαγδβδ − δ

βγδαδ), (3.47)

that constitute the components proper to the two submanifolds, while Rµνp λτ and Rαβn γδ are their

intrinsic contributions. The prefactors A−2 and C−2 arise when we go back to the backgroundorthonormal basis from the ω-ones15; the terms arising from the connection with the radialcomponent are

Rµrνr =

(A′′

A

)δµ ν , Rαrβr =

(C ′′

C

)δαβ ,

while the cross term results as

Rµανβ =A′C ′

ACδµ νδ

αβ .

The Ricci tensor components are obtained from index contraction of the Riemann terms, henceresulting as16

Rµ ν = A−2Rµp ν +

(A′′

A+ (p− 1)

A′2

A2+ n

A′C ′

AC

)δµ ν , (3.48)

Rαβ = C−2Rαn β +

(C ′′

C+ (n− 1)

C ′2

C2+ p

A′C ′

AC

)δαβ , (3.49)

Rr r = pA′′

A+ n

C ′′

C. (3.50)

14We have renamed the background Riemann tensors as Rµνn λτ and Rαβp γδ, where the rst index at the bottom

indicates the number of dimensions belonging to a specic submanifold.15Analogously to what we have done when discussing Kaluza-Klein theory and the domain wall, we can extract

an orthonormal basis from a given metric, from which we can determine the spin and connection two-forms. Forthe metric (3.43), these are ωµ = A(r)ωµ0 , ω

r = dr and ωα = C(r)ωα0 .16The indices that have been used in the following formula may look deceiving. Indeed, with formulae (3.48)-

(3.49), we are expressing the Ricci tensors and not the curvature two-forms.


The terms in equation (3.48)-(3.49) account for the heritage from intrinsic curvature (Rµn ν andRαp β , respectively), the distortion due to their translation along the radial coordinate (i.e. therst two terms inside the brackets) and, nally, their reciprocal interaction.

Deriving such a general description proves to be essential for analyzing black holes in higherdimensions and also for conical decit regularization, which are indeed strongly related. Beforeproceeding with the explicit derivation, we briey describe what a conical decit is. In order todo so, we use cosmic strings, i.e. a straight, innitely long, relativistic string. Their presence isprobed by the deformation they provide on the geometry of the transverse plane with respectto their length. In particular, this results as an angular decit, δ, such that its circumference,C, for the case of a two-dimensional transverse plane, would result as C = (2π − δ)r, where ris the radius of the string's section, while δ is dictated by, both, the string's tension, T , andthe gravitational constant, according to the following relation, δ = 8πGT/c4. The term conicalarises from the fact that the transverse space is indeed a cone obtained by cutting o from a atcircle a section of angle δ and subsequently patching up the two edges. The cosmic string henceruns trough the vertex of such cone. Regularizing such defects is necessary in case of actioncomputation, such as the one requested for vacuum tunneling mediated by black holes17. Westart from a line element of the form, [12],

ds2 = dρ2 +A2(ρ)dχ2 + C2(ρ)dΩ2H . (3.51)

where H denotes the surface of the hypersphere, constituting a transverse space with respect tothe radial coordinate ρ. We now suppose the conical decit lies on a 2-plane parameterized bylocal coordinates (ρ, χ). and that the transverse metric does not depend on these parameters inthe limit ρ→ 0. The area of the conical defect is

ACD = C(0)nAΩH , (3.52)

where n is the dimensionality of the hypersurface H, transverse to the radial direction ρ, whileρ = 0 stands for the position of the conical decit in the manifold.

In order to regularize the defect, we need to take A as a smooth function. We also set, [12],

A′(0) = 1 ; A′(ε) = 1− δ,

where derivation is performed with respect to the variable ρ where 2πδ constitutes the conicaldecit. On the other hand, the function C(ρ) is assumed to be regular at the decit's locationand with a null rst derivative with respect to the radial coordinate, therefore we are allowed tochoose it as being of the form, [12],

C = C0 + ρ2C2 ⇒ C ′′ = 2C2,

where C0 and C2 are set as constant values. As a consequence, the overall expression for theradial component of the Ricci scalar in the vicinity of the defect, i.e. at ρ = 0, is

R =A′′

A+

2nC2

C0 + ρ2C2. (3.53)

By using the inverse relation between C and C0 , and applying Bernoulli's approximation, thesecond term in equation (3.53) becomes

2nC2

C0 + ρ2C2=

2nC2

C0− 2nC2

2ρ2

C20

=2nC2

C0− 2nC2(C − C0)

C20

=4nC2

C0− 2nC2C

C20

.

17See section (4.4).


Since C(ρ) behaves as a smooth function while approaching the singularity, if we set the initialradial value as ρ = ε, by letting ε→ 0, as a consequence of the conditions we have set, the onlyterm in equation (3.53) that gives a signicant contribution to the Ricci scalar is the rst one.

After such considerations, we are now able to evaluate, both, the Einstein-Hilbert andGibbons-Hawking boundary terms, thereby obtaining the overall action contribution relatedto the conical decit. The bulk contribution is

SEH =1

16πG

∫d4x√−gR ≈ − 1

16πG

∫dρ dχA dΩHC

A′(0)−A′(ε)A ε

where we have expressed the second derivative of the function A with respect to the radialcoordinate; when we integrate over the solid angle, we get the decit's area ACD and we can alsosimplify many factors, to obtain the following expression, [12],

SEH ≈ −1

16πG(ACD[A′(0)−A′(ε)]) = − 1

4GδACD;

in the last passage we have used the denitions provided when we dened the properties of thefunctions appearing in the metric. On the other hand, for the boundary term, we notice thatthe only relevant normal18 direction for the computation of the extrinsic curvature, is n = −dρ(this follows from the symmetry properties of the metric). Its trace is

K = ∇ana = −A′

A− 2n

C ′

C.

As a consequence, the Gibbons-Hawking boundary term reads, [12],

SGH =1

8πG

∫ρ=ε

d3x√hK =

1

8πG

∫ρ=ε

dχdΩHACnK

≈ − 2π

8πGACDA

′(ε) = − 1

4GACD(1− δ),

where we have used an analogous procedure as for the previous integral. Finally, we obtain thetotal action as, [12],

STOT = SEH + SGH = −ACD4G

,

from which we notice that the contribution to the total action is completely determined by thearea of the conical decit and by the gravitational constant which is linked to the dimensionalityof the manifold.

3.1.4 Israel's Junction Conditions

In the previous sections, we have shown how it is possible to use the Cartan formalism, fordetermining the curvature two-forms for a given metric.

We will now recover the conditions that are are necessary to join two parts of two manifoldsacross, for dening a generic hypersurface ; we will do so by analyzing the jump of the geometricalproperties characterizing the manifold, and derive Israel's junction conditions.

In chapter (2.2), we have already outlined the basic ideas at the heart of intrinsic and extrinsicgeometry, when dealing with the geometry of submanifolds.

In an analogous way to what we have discussed before, the location of an hypersurfacemay be parametrized by a coordinate `, that, could stand for either a spacelike or a timelike

18The - sign comes from the fact that the normal has been chosen as inward pointing.


coordinate, depending on the spacelike or timelike character of the hypersurface (we excludehere the lightlike case). If we set the parameter value ` = 0 as the position of along geodesicsnormal to the hypersurface, andA hypersurface embedded in a higher -dimensional manifold maybe parameterized by a certain coordinate `. and conventionally assign ` > 0 for V+ and ` < 0for V−, where V± indicate the two manifolds, we are able to discuss the compound manifoldV−∪Σ∪V+ of the two representations in terms of distributions. For the purpose of our analysis,the Heavyside function appears to be more adequate, because of its symmetry properties. Wewill see that the advantage of using this particular distribution shall introduce singularities thathave to be compensated, therefore leading to the junction conditions.

Since we are interested in determining the junction of two submanifold that are solutions tothe eld equations, we introduce here a convenient notation to indicate the jump of a tensorialquantity A when crossing the junction hypersurface, according to the following denition

[A] = A(V+)|Σ −A(V−)|Σ. (3.54)

In the present case the solution of Einstein equations will be written as

gαβ = Θ(`)g+αβ + Θ(−`)g−αβ . (3.55)

Of course the expression above is not well behaved at the hypersurface. Because of these, both,the connection and Riemann tensor behave as distributions at ` = 0, i.e. on Σ.

Our aim is now to recover the eld equations of which (3.55) provide a valuable solution.Let us rst dierentiate equation (3.55), obtaining, [1],

gαβ,γ = Θ(`)g+αβ,γ + Θ(−`)g−αβ,γ + εδ(`)[gαβ ]nγ , (3.56)

where the parameter ε = ±1 according to whether the normal to the hypersurface is eitherspacelike or timelike. We need to eliminate the singularity appearing in the last term. Thisimplies that

[gαβ ] = 0 ⇒ [hαβeαaeβb ] = 0 ⇒ [hab] = 0, (3.57)

which constitutes the 1st Junction Condition, where with hab we denote the three-metric of thehypersurface Σ. For the three-metric, we use the Latin indexes, to distinguish them from theones related to the four -metric.

In order for the connection symbols not to diverge, we need to eliminate the singularityappearing in the last term. This implies that

Γαβγ = Θ(`)Γα+βγ + Θ(−`)Γα−βγ . (3.58)

If we now dierentiate (3.58), we get

Γαβγ,δ = Θ(`)Γα+βγ,δ + Θ(−`)Γα−βγ,δ + εδ(`)[Γαβγ ]nδ. (3.59)

For the Riemann tensor, we now get

Rαβγδ = Θ(`)Rα+βγδ + Θ(`)Rα−βγδ + δ(`)Aαβγδ, (3.60)

where we have dened the tensor quantity

Aαβγδ = ε([Γαβδ]nγ − [Γαβγ ]nδ),

that respects the antisymmetry of the Riemann tensor in the last two indexes. In a similar manneras when discussing variation of the metric when computing the Gibbons-Hawking boundary term,


from the continuity of the tangential derivatives of the metric, we are left with the followingcondition

[gαβ,γ ] = καβnγ , (3.61)

meaning that if there are any discontinuities in the metric, they can only be directed along thenormal to the hypersurface. In (3.61), nγ stands for the normal vector to Σ, while καβ indicatesa tensorial eld which can be computed inverting equation (3.61)

καβ = ε[gαβ,γ ]nγ (3.62)

where ε is the prefactor accounting for whether the normal to the hypersurface is timelike orspacelike. Recalling the relation between connection and the metric, we get

Γαβγ =1

2gαµ(gµβ,γ + gµγ,β − gβγ,µ);

if we now substitute the relation for normal discontinuity (3.62), the δ-term in equation (3.59)turns into

[Γαβγ ] =1

2(καβnγ + καγnβ − κβγnα),

from which we get

Aαβγδ =ε

2(καδ nβnγ − καγnβnδ − κβδnαnγ + κβγn

αnδ),

by subtracting two such terms. Since this last equation accounts for the δ-term in the Rie-mann tensor, by applying index contraction, we get the contribution arising from the singularityterm in both the Ricci tensor and Ricci scalar, ultimately leading to the determination of theircorresponding corrective terms in Einstein's eld equations.

In practice, a contraction of the rst and third indexes with respect to the four -metric leadsto

Aαβ = Aµαµβ =ε

2(κµβnαnµ − κ

µµnαnβ − καβ − καµnµnβ). (3.63)

When performing a further contraction, we reduce expression (3.63) in the following way

A = Aαα = ε(κµνnµnν − κ). (3.64)

Analogously to equations (3.55), (3.58), (3.60), the energy-momentum tensor can be expressedas

Tαβ = Θ(`)T+αβ + Θ(`)T−αβ + δ(`)Sαβ (3.65)

where the last term is, explicitly

8πSαβ = Aαβ −1

2Agαβ , (3.66)

which, precisely, represents the stress-energy tensor associated to the hypersurface Σ.If we now substitute in equation (3.66) relations (3.63) and (3.64), we obtain

16πSαβ =ε

2(κµβnαnµ − κ

µµnαnβ − καβ − καµnµnβ)− ε(κµνnµnν − κ). (3.67)


Being the stress-energy tensor conned to the hypersurface, we have

Sαβnβ = 0 ⇒ Sab = Sαβe

αaeβb .

We can now rewrite equation (3.67) with fewer terms as a consequence of the projection we haveapplied, hence obtaining

16πSab = −καβeαaeβb − ε(κµνn

µnν − κ)hab

= −καβeαaeβb − κµν(gµν − hmneµmeνnhab) + κhab

= −καβeαaeβb + hmnκµνe

µme

νnhab. (3.68)

It is now possible to rewrite this last result in terms of the extrinsic curvature by simply re-membering its denition, given in chapter (2.1), and then calculating explicitly the jump in theextrinsic curvature. It is now possible to prove that we can use extrinsic curvature to rewritesome of the terms emerging in equation (3.68). From its denition, given in section (2.2), thejump in extrinsic curvature

[Kab] = [nα;β ]eαaeβb ,

where

[nα;β ] = −[Γγαβ ]nγ

= −1

2(καβnγ + καγnβ − κβγnα)nγ . (3.69)

If we now rearrange these three terms in the expression for the jump in the extrinsic curvature,and project onto the hypersurface, we obtain

[Kab] =ε

2καβe

αaeβb

where only the rst term from equation (3.69) has a non-vanishing projection onto the hyper-surface. This allows to express equation (3.68) as

Sab = − ε

8π([Kab]− [K]hab). (3.70)

If we now contract equation (3.70) with the three-metric hab, we get

Sabhab = S = − ε

8π([K]− (D − 1)[K]) ⇒ [K] = 8π

1

D − 2S,

where D denotes the number of dimensions of the manifold. From such result, we get the 2nd

junction condition

[Kab] = 8π

(Sab −

1

D − 2Shab

). (3.71)

Equation (3.71) is also known as Israel junction condition; it describes how the hypersurfacelocalized stress-energy tensor results in a jump of the extrinsic curvature. This result will be putto good use when dealing with the braneworld model, at the end of this chapter.


3.1.5 Reissner-Nordstroem solution

In this section we will consider in detail a solution to Einstein's equations in presence of elec-tromagnetic eld: this solution will also describe charged black holes. In particular, it describeselectrically-charged non-rotating black holes. We have already dealt with the interplay betweengravity and electromagnetism when discussing the Kaluza-Klein model. Here we will go back tothe four -dimensional spacetime.

Although charged black holes are not astrophysically relevant, they are interesting theoreticallaboratories to discuss various aspects of general relativity, as they provide exact examples ofsolutions in presence of other elds, and have a peculiar causal structure.

The no-hair theorem states that a black hole solving, both, Einstein's and Maxwell's equa-tions, has only three externally observable classical parameters, which are: its mass, electriccharge and angular momentum.

The Reissner-Nordstroem solution to Einstein's equations is able to describe a static spheri-cally symmetric, hence non-rotating, and electrically charged black hole. In this subsection, weprovide the derivation of this solution as a preliminary result for the following section, where weshall deal with braneworld black holes carrying tidal charge.

According to Birkho's theorem, a spherically-symmetric solution of the eld equations invacuum spacetime, has to be static and asymptotically at ; for such a symmetry, we have thefollowing structure

ds2 = −e2α(r,t)dt2 + e2β(r,t)dr2 + r2dΩ2, (3.72)

where α(r, t) and β(r, t) constitute the functions that we have to determine in order for the metricto satisfy Einstein's equations, while Ω denotes the solid angle on a two-sphere. In particular, suchfunctions will explicitly carry the relevant charges characterizing a specic solution ; accordingto the particular problem we are dealing with they may come in the form of any combination ofeither mass, cosmological constant, electric or magnetic charge.

We start with the denition of the energy-momentum tensor associated to the electromagneticpotential, [3],

Tµν = FµρFρν −

1

4gµνFρσF

ρσ, (3.73)

whose trace is null, as it can be veried contracting with the metric,

T = gµνTµν = gµνFµρFρν − 1

4gµνgµνFρσF

ρσ = 0,

where, in the second passage we have used the property gµνgµν = 4.As a consequence of (3.73), the Ricci scalar also vanishes, therefore leading to an overall

simplication of Einstein's equations; indeed, they can be written in the form

Rµν = 8πTµν , (3.74)

that will enable us to simplify the following calculations.In our setup Einstein equations are coupled with Maxwell equations

gµν∇µFνσ = 0 ; ∇[µFνρ] = 0, (3.75)

where the former account for Gauss-Ampere's laws (divergence of ~E and rotor of ~B) while the

latter for Gauss-Faraday's laws (rotor of ~E and divergence of ~B). Under the assumption that


the system is spherically symmetric, the only spacelike non-vanishing components of the electro-magnetic eld are the radial ones. For the electric eld, these should be of the form

Er = Ftr = −Frt = f(r, t), (3.76)

where f(r, t) is an unknown function.On the other hand, since the magnetic eld is the rotor of the vector potential, it is necessary

to introduce the Levi-Civita tensor, εtrµν , to keep track of the changes in sign resulting fromindex permutations19 , therefore leading to its radial component as

Br = grrεtrµνFµν =

grr√|g|εtrµνFµν

=grr√|g|

(εtrθφFθφ + εtrφθFφθ) = 2grr√|g|Fφθ. (3.77)

The indexes that have been used in this last passage are justied by the fact that B is radial.Thus, the only term on the r.h.s. that can compensate the angular content of the determinantof the metric, are the angular components of F , which is precisely of the form

Fθφ = −Fφθ = g(t, r)r2 sin θ, (3.78)

where g(t, r) plays the same role as the f(r, t) function that we previously dened for the electriceld.

Having determined all of its components, we may now write down the electromagnetic anti-symmetric tensor as

Fµν =

0 f(t, r) 0 0

−f(t, r) 0 0 00 0 0 g(t, r)r2 sin θ0 0 −g(t, r)r2 sin θ 0

. (3.79)

The next step consists in determining the functions f(r, t) and g(r, t), that will subsequentlyallow to identify the metric functions in (3.72). In order to do so, we basically have to solveMaxwell's equations.

We rst deal with the electric eld. Combining equation (3.76) with the radial componentof the rst set of Maxwell's equations, the covariant derivative with respect to the timelikecoordinate is

∂tFtr − ΓαttFαr − ΓαtrFtα = ∂tFtr − Ftr(Γttt + Γrtr) = 0, (3.80)

where the index change α → r indicates we have selected a particular coordinate belonging tothe metric.

Before solving the electromagnetic eld equations, we need to go through the determination ofthe connection and curvature non-vanishing components; then, provided the trace-free condition(3.73), we shall obtain the energy-momentum components that are required for determiningfunctions f and g.

From Birkho's formula for a spherically symmetric solution (3.72), the Christoel symbolsare obtained performing partial dierentiation with respect to the coordinates of the metric.Since the computation is quite long we have not reported it here; in what follows we indicate thenon-vanishing terms, [3],

Γttt = ∂tα ; Γttr = ∂rα ; Γtrr = e2(β−α)∂tβ

19The Levi-Civita tensor is obtained as ε =√|g|ε, where ε is the density tensor, while |g| = det gµν = r4sin2θ.


Γrtt = e2(α−β)∂rα ; Γrtr = ∂tβ ; Γrrr = ∂rβ

Γθrθ =1

r; Γrθθ = −re−2β ; Γφrφ =

1

r

Γrφφ = −re−2β sin2 θ ; Γθφφ = − sin θ cos θ ; Γφθφ =cos θ

sin θ.

We can subsequently apply the denition of the Riemann tensor, therefore obtaining the followingcomponents, [3],

Rtrtr = e2(β−α)[∂2t β + (∂t)

2 − ∂tα∂tβ] + [∂rα∂rβ − ∂2r − (∂rα)2]

Rtθtθ = −re−β∂rα ; Rtφtφ = −re−β sin2 θ∂rα ; Rtθrθ = −re−α∂tβ

Rtφrφ = −re−β sin2 θ∂tβ ; Rrθrθ = re−β∂rβ ; Rrφrφ = re−β sin2 θ∂rβ

Rθφrφ = (1− e−β) sin2 θ.

Performing index contraction, we subsequently get the Ricci tensor components, [3],

Rtt = [∂2t β + (∂tβ)2 − ∂tα∂tβ] + e2(α−β)

[∂2rα+ (∂rα)2 − ∂rα∂rβ +

2

r∂rα

]

Rrr = −[∂2rα+ (∂rα)2 − ∂rα∂rβ −

2

r∂rβ

]+ e2(β−α)[∂2

t β + (∂tβ)2 − ∂tα∂tβ]

Rtr =2

r∂tβ ; Rθθ = e−2β [r(∂rβ − ∂rα)− 1] + 1 ; Rφφ = Rθθ sin2 θ.

Remembering that our aim is the one of solving Einstein's equations with respect to the functionsf and g, we now need the e.m. energy tensor components, that shall have to be proportional tothe Ricci tensor components we have reported above, as a consequence of Tµν being trace-free.

Combining the denition (3.73) with the matrix representation (3.79), we get the followingcomponents, [3],

Ttt =f(r, t)2

2e−2β(r,t) +

g(r, t)

2e2α(r,t) ; Trr = −f(r, t)2

2e−2β(r,t) − g(r, t)

2e2α(r,t)

Ttr = 0 ; Tθθ =r2g(r, t)2

2+r2f(r, t)2

2e−2(α(r,t)+β(r,t)) ; Tφφ = Tθθ sin2 θ.

From the t-r -component, we get

Rtr =2

r∂tβ = 0 ⇒ β = β(r);

hence, β is time-independent. As a further step toward determining the unknown functionsα(r, t) and β(r), we select two of the above components of the Ricci scalar; in particular

e2α(r,t)Rrr + e2β(r)Rtt = e2α(r,t)[−2∂2rα(r, t) + (∂rα)2 − ∂r∂rβ +

2

r∂rβ]

e2α(r,t)[∂2rα(r, t) + (∂rα)2 − ∂r∂rβ +

2

r∂rα] = 0

⇒ α(r, t) = −β(r). (3.81)

Since we are dealing with a static spherically-symmetric solution, we have

∂tRθθ = ∂t(e−2β(r)[r(∂rβ(r)− ∂rα(r, t))− 1] + 1)0⇒ ∂t∂rα(r, t) = 0. (3.82)


It is thus possible to deduce that α depends on the radial and timelike coordinates in a formresembling a linear combination of the two, such as α(r, t) = l(r) + n(t), such that (3.82) issatised.

We may write the rst term in the metric (3.72) as −e2l(r)e2n(t)dt2. It is then possible toredene the time coordinate via the following substitution dt→ e−n(t)dt, which allows to denea certain parameter value, t, such that g(0) = 0, therefore leading to α(r) = l(r).

What we have proved with this digression is that, calculating the Ricci tensor and energy-momentum components, the exponents α and β are independent from the time coordinate. Thismeans that any system sharing the symmetry of this problem has a timelike Killing vector eld20,and is thus called stationary. From this follows that the connection terms in equation (3.80) alsovanish, leaving the partial derivative

∂tFtr = 0⇒ Ftr = f(r), (3.83)

implying that f is also time-independent. At this point, we would like to determine the explicitform of the radial function. In order to do so, we need to solve Maxwell's eld equations (3.75).We shall rst deal with the former.

The covariant derivative of Maxwell 's tensor may be written in the form 21

∇µFµν =1

r2 sin θ∂µ(r2 sin θFµν),

⇒ ∂r(r2F rt) = ∂r(r

2grrgttFrt) = ∂r(r2f) = 0⇒ f(r) ∝ 1

r2, (3.84)

where we proceeded in the same manner as for the electric eld, (3.77), such that the deter-minant of the metric, involving an angular contribution would be compensated by an adequatecomponent of the tensor, in order to obtain an overall radial expression.

Setting the constant of proportionality as Q/√

4π, where Q stands for the black hole's electriccharge, thus enabling us to write the tensor components associated to the electric eld as

F tr =Q√4πr2

.

It is possible to obtain a surface integral representation for the electric charge, as for the deni-tions of mass and angular momentum provided in section (2.3.3),

Q =

∫V

dVαFαβ;β =

∮S

dSαβFαβ

which remains valid also for a singular charge distribution.Now that we have found the expression for f , we need to nd g. In order to do so, we

need to consider the magnetic eld. Going back to the second set of equations in (3.75), andremembering that the connection is taken to be torsion-free, we get, explicating the dierentcontributions, we get

∇rFθφ +∇θFφr +∇φFrθ = ∂rFθφ + ∂tFφr + ∂tFrθ = 0. (3.85)

20This means that the Lie derivative evaluated along this vector eld is null.21Γµµν = 1√

|g|∂ν√|g| ⇒ ∇µTµν = ∂µTµν + ΓµµαT

αµ + ΓνµαTµα = ∂µTµν + 1√

|g|∂α(√|g|)Tαν + ΓνµαT

µα =

1√|g|∂µ(√|g|Tµν) where the last passage follows from Tµν being antisymmetric and the connection coecients

torsion-free.


By looking at the matrix components in (3.79), only the rst term in equation (3.85) is non-zero;as a consequence, g is also a function of the radial coordinate only, as can be checked throughthe following steps

∂rFθφ = ∂r(r2g(r)) = 0⇒ g(t, r) = g(r)

g(r) ∝ 1

r2⇒ g(r) =

P√4πr2

where P is an integration constant that can be identied with the magnetic charge. Maxwell'stensor may thus be expressed in terms of the radial functions we have determined

Fµν =1√4π

0 Q/r2 0 0

−Q/r2 0 0 00 0 0 P sin θ0 0 −P sin θ 0

. (3.86)

There is only one term left to determine, which is α(r). From the calculation of the terms ofthe trace of the Ricci energy tensors, we proved that α(r) = −β(r). In order to determine theirvalues, one of Einstein's equations has to be solved in terms of one of the two parameters, forexample α. We have chosen to solve the following

Rθθ = 8πGTθθ

⇒ e−2β(r)[r(∂rβ(r)− ∂rα(r))− 1] + 1 = 8πG

[r2g(r)2

2+r2f(r)2

2e−2(α(r)+β(r))

].

If we substitute the expressions for g(r) and f(r) obtained in the previous passages, this lastequation becomes

∂r(re2α) = 1− G

r2(Q2 + P 2) ⇒ e2α = 1− 2M

r+G

r2(Q2 + P 2),

where M denotes the black hole mass, arising when integrating the former of the last two equa-tions. Having determined the conformal prefactors, the line element (3.72) acquires the form

ds2 = −[1− 2M

r+G

r2(Q2 + P 2)

]dt2 +

dr2

1− 2Mr + G

r2 (Q2 + P 2)+ r2dΩ2.

For the moment, we will focus on the electric charge. Magnetic monopoles will turn out useful inlater sections, precisely when dealing with the topic of tunnelling processes mediated by chargedblack holes.

Setting P = 0, the metric solution leads to the Reissner-Nordstroem metric

ds2 = −(

1− 2M

r+Q2

r2

)dt2 +

(1− 2M

r+Q2

r2

)−1

dr2 + r2dΩ2, (3.87)

while for P = Q = 0, we get the Schwarzschild metric22

ds2 = −(

1− 2M

r

)dt2 +

(1− 2M

r

)−1

dr2 + r2dΩ2. (3.88)

22A brief discussion on the Schwarzschild solution is provided in Appendix A. In this chapter we are focusingon the electrically charged solution. The neutral solution will be considered when dealing with cosmological phasetransitions.


Equation (3.87) is solution to the eld equations coming from the 4D action

S =1

16πG

∫d4x√−g(R− FµνFµν) (3.89)

which accounts for two long range interactions simultaneously.The fascinating thing about the Reissner-Nordstroem solution is that, according to the par-

ticular value of the ratio Q/M , the internal structure varies. Coordinate singularities allow theidentication of the event horizon and a Cauchy horizon, i.e. a light-like boundary dening thedomain of validity of a certain Cauchy problem.

Using Natural Planck Units, we can compute th roots of the metric function in equation(3.87), in order to determine the horizon's location, precisely . In particular

1− 2M

r+Q2

r2=

(r − r+)(r − r−)

r2, (3.90)

from which the location of the two horizons can be determined, as r± = M ±√M2 −Q2 are

the locations of the event and inner (or Cauchy) horizons, respectively.23 For r < r− equation(3.87) describes a space-like singularity which can therefore be avoided by an in falling observer.The Cauchy horizon r = r− represents the limit of unique evolution of a spacelike hypersurface.Such inner horizon is susceptible of energy perturbations and, as it stores all in falling matter, itscurvature tends to increase. Furthermore, the inner structure must avoid violation of the cosmiccensorship hypothesis, formulated by R. Penrose. It is supposed to hold when singularitiesappear in solutions to Einstein's eld equations. There are two version of the cosmic censorshiphypothesis:

1. the weak cosmic censorship hypothesis states that the maximal Cauchy development pos-sesses a complete future null innity, meaning that it is not possible for an external observerto perceive a naked singularity;

2. the strong cosmic censorship hypothesis asserts that the maximal Cauchy developmentof generic compact or asymptotically at initial data is locally inextensible as a regularLorentzian manifold, i.e. that general relativity is deterministic.

In terms of the Reissner-Nordstroem solution, this would mean that Q2 ≥ M2. In orderto avoid this problem, mass ination occurs, turning the Cauchy horizon into a null curvaturesingularity, therefore acting as an impenetrable barrier24. Its back reaction also leads to modi-cations of the black hole's causal structure. For a globally neutral solution, opposite charges maybe considered to lie on the two coordinate singularities identied via 3.90). If Q = M , equation(3.90) has two coinciding roots r± = Q , meaning that the two surfaces have joined together,leading to the formation of an extremal black hole. In such case, the metric may be expressed as

ds2 = −(

1− Q

r

)2

dt2 +

(1− Q

r

)−2

dr2 + rdΩ2.

If we subsequently apply a coordinate transformation ρ = r − Q and h = 1 + Q/ρ , this lastequation turns into

ds2 = −h−2(ρ)dt2 + h2ρ(dρ2 + ρ2dΩ2). (3.91)

23It is important to bare in mind that both horizons lie within the Schwarzschild radius. This is necessary forthe General Theory of Relativity, but there are some interesting exceptions (outlined in following section), thatprove to be essential in other circumstances.

24This hints at the second part of this work, which is devoted to vacuum tunnelling processes described interms of junction conditions, once gravity is taken into account.


By rescaling the variables ρ = ε ρ and t = t Q2/ε , and taking the limit ε→ 0, the line elementbecomes

ds2 = Q2

(−ρ2dt2 +

dρ2

ρ2+ dΩ2

), (3.92)

which resembles a 2-sphere in an AdS2 spacetime associated to the Schwarzschild solution: in-deed, such coordinate transformation enables the horizon to acquire a higher symmetry, resem-bling the neutral solution.

Although we shall deal with Hawking radiation in the next chapter, when discussing BlackHole Thermodynamics, we expect there to be some kind of competition between mass evaporationand discharge. According to the value of the ratio, Q/M , a black hole may either become extremalor neutrally charged. The main feature of the former limit is the ceasing of thermal emission, asthe inner and outer horizons join together. In the second case, instead, the inner horizon shrinks,hence becoming a space-like singularity, described by Schwarzschild's solution.

The last case, Q > M , violates the cosmic censorship hypothesis, since it indicates that thesingularity becomes timelike, meaning it should be perceived by a distant observer; consequently,it does not describe a real solution. In conclusion the evolution of an isolated black hole isdetermined by two quantum eects: Hawking radiation and discharge. At a rst stage, Hawkingradiation constitutes the dominant process, leading to a higher Q/M ratio.

As M → Q , the black hole becomes near -extremal, implying Q and M decay followingthe same rate. Once Q becomes suciently small, discharge overcomes mass loss, leading toa spherically neutral black hole remnant. As already stated, in the limit Q → 0, the Cauchyhorizon collapses, therefore becoming a singularity. While approaching this nal state, we canconsider the horizon's section as resembling the throat of a wormhole.

Although we are not going to discuss explicitly about wormholes, it is interesting to noticethat they are thought to have been generated by the intersection of vacuum bubbles in the earlyepochs of our universe. Indeed, they have been conjectured to bridge between dierent truevacuum universes arising from cosmological phase transitions, of which we shall discuss in thelast chapter.

3.1.6 Randall-Sundrum II model

Warped Compactication, Tidal Charge and Primordial Black Holes

Another way of adding extra dimensions, dierent from Kaluza-Klein theory, is provided bybraneworld models, where matter elds are conned on a brane while gravity is allowed topropagate in the bulk in the form of gravitons. The fascinating thing about branes (also referredto as p-branes, where p indicates their dimensionality), is that they provide a valuable setting forthe study of fundamental interactions; they rely upon the interplay between exterior algebra andphysical understanding. Indeed, since branes are orientable submanifolds, they have a nowherevanishing (p+1)-form which provides a volume element which in turn couples to antisymmetrictensors whose rank is compatible with the dimensionality of such volume element. So, they canbe thought of as being a generalization of a particle interacting with an antisymmetric tensorof the adequate rank. The complementary part to the brane is the bulk, which is of coursecharacterized by a higher dimensionality. An example of a p-brane is provided by the rst partof the metric (3.43).

Einstein's theory may loose its validity at high energy regimes; as a possible explanationof this, various candidate quantum gravity theories have been developed. Among them, stringtheory requires a higher dimensional spacetime: in these theories it is important to recover thefour-dimensional description at low energy. It is then interesting to study the eective low energy


theory, and here we will see this realized in a specic brane model. This model is related to whatis usually called a warped geometry.

One possible way of describing a warped metric is realized by Randall-Sundrum's model,consisting in a domain wall in anti− deSitter5 (AdS5)

25, with Z2-symmetry. In coordinates themetric can be written as

ds2 = e−2|z|/lηµνdxµdxν + dz2, (3.93)

where ηµν stands for the braneworld metric, transverse to the extra dimension, z, while the warpfactor e−2|z|/l is responsible for squashing the 4D hypersurface when transporting it along the zdirection. l is the anti-de Sitter (AdS)26 radius. The main feature of the given metric is that itis dominated by the transverse direction.

The model that we are considering is one of the two Randall-Sundrum models that arediscussed in the literature. It is called Randall-Sundrum 2 (RS2) and it can be seen as a particularcase of a more general model, the Randall-Sundrum 1 (RS 1) model.

The RS 1 model is based on a pair of branes with opposite surface tension, σ, separated bya 5D bulk with negative energy density parameterized by a cosmological constant, Λ5. For thesystem to be in equilibrium, the magnitude of σ on either brane is required to match the bulk'scosmological constant's value; such compatibility is called ne tuning. From such condition, wegain the following relation, [7],

Λ5 = −4πσ2

3M3p

.

In the previous expression, we have used MP for denoting the Planck mass associated to theve-dimensional description. The Planck scale is a set of critical values for mass, length andtime which can be determined when the Schwarzschild radius equals the Compton radius, atwhich quantum gravity eects become relevant. The formula reads

GMp

c2= ~Mpc ⇒ Mp =

√~cG

;

(3.94)

recalling what we have discussed when dealing with Kaluza-Klein theory, the presence of thegravitational constant implies that the Planck mass diers according to the number of dimensionswe are dealing with and also to the extension of the extra dimensions. Being the Planck masspositively dened, the bulk cosmological constant, Λ5, is negative, hence the bulk is requiredto be an AdS5. The brane with positive tension resembles our four -dimensional world, wherematter is conned. On the other hand, the one characterized by a negative tension accounts forall the unknown elds that we are unable to perceive. If we move the latter brane to innity,we are left with a single brane with positive tension, i.e. our world, still embedded in an AdS5

bulk, hence obtaining the RS 2 model, which will be the setting of our discussion in the followingpages.

In the RS model, the brane becomes a self-gravitating object, which is allowed to interactdynamically with the bulk. In this model, linearized gravity in 4D is recovered thanks to thebrane's tension, corresponding to nite curvature.

The warp factor in the given metric encodes the mutual interaction between the brane andthe bulk. By using Israel 's equations, we can compute the stress-energy tensor and see that itis directly proportional to the warp factor k.

25The index 5 indicates the dimensionality of the overall manifold.26An AdS spacetime is characterized by a negative bulk cosmological constant.


As derived in [2], the jump in extrinsic curvature reads

[Kab]− [K]ηab = 8πGSab,

where the Latin indexes indicate we are referring to properties belonging to the brane, whilewe shall use the Greek letters for denoting the properties of the bulk. With ηab we indicate theintrinsic metric w.r.t. the brane.

Since we are dealing with a Z2-symmetric conguration, it is enough to evaluate the extrinsiccurvature on one side of the hypersurface and then double the result to express the left handside of the equation above

K+ab = nα;βeαaeβb = −Γγαβnγe

α,ae

β,b = − k

lηab,

where the + sign has been added in the above notation in order to indicate that we are computingthe outer extrinsic curvature

⇒ [Kab] = −2k

lηab ⇒ [K] = −8

k

l. (3.95)

If we now apply the denition of the surface stress-energy given by Israel's junction conditions,the result is

Sab =3

4πG

k

lηab. (3.96)

From this last equation, we notice that the tension is not arbitrary, but is rather linked to thecosmological constant of the 5-dimensional manifold from equation (3.96)

Λ5 = −6k2

l2.

This means that the presence of a cosmological constant in the bulk gravitationally aects thebraneworld metric.

When dealing with extra dimensions, we have to check whether linearized gravity can berecovered on the braneworld in the form of an eective eld theory. There are two possible waysof doing so:

1. either perturbatively by the addition of a corrective term to the metric, and then analyzehow it aect the exact solution 27;

2. or via the Gauss-Codazzi formalism.

We shall now deal with the latter, applying it to the RS 2 model. The system is Z2-symmetric,hence we can identify a normal to the hypersurface that allows to perform a (4+1)-decompositionof Einstein's equations. In the bulk the eld equations are [6]

5Gαβ =5 Rαβ −1

25R gαβ = κ2

5Tαβ , (3.97)

where gαβ is the ve-metric, while κ25 contains the bulk gravitational constant. The gravitational

constant is set by the Planck mass.

27We shall describe this approach in the following section.


Applying Gauss-Codazzi's equations to this 5-dimensional description, we get

4Rmanb = 5Rµανβemµ e

αaeνneβb +Km

n Kab −Kmb Kan ; Kab|b −K|b = 5Rαβ n

α eβb (3.98)

from which we obtain28 the brane's Ricci tensor and scalar

4Rab = 5Rαβeαaeβb −

5 Rµανβnµeαan

νeβb +KKab −KcbKac,

4R = 5R− 2 5Rαβnαnβ + 5Rµανβnµn

αnνnβ +K2 −KabKab.

It is important to notice that we are dealing with D ≥ 4; for D < 4, the Riemann tensor isuniquely determined by the Ricci tensor. For the case of the braneworld model, computationrequires the introduction of the Weyl tensor, 5Cµανβ

29

It is a conformally-invariant tensor, reason why, in order to have a conformally at30 Riemannmanifold, the Weyl tensor vanishes. This also explains why we have not dealt with it whenoutlining the fundamentals of general relativity in chapter (2.2). The Weyl tensor emerges as alinear term in the generalized denition of the Riemann tensor, from which also follows the factthat they share the same symmetry properties. The denition reads

5Rµανβ =2

3(gµ[ν

5Rβ]α − g 5α[ν Rβ]µ)− 1

6gµ[ν g

5β]α R+5 Cµανβ , (3.99)

The Gauss-Codazzi equations can be can be used to decompose the Einstein tensor. We justhave to add the terms containing the Ricci scalars 4R and 5R. This is responsible for the additionof other extrinsic curvature terms, preceded by a factor of 1/2. Hence we get, [14],

4Gab = 5Gαβeαaeβb +KKab −Kc

aKbc −1

2ηab(K

2 −KabKab)−5 Rµανβnµeαan

νeβb

=

[5Rαβ −

1

2R gαβ

]eαae

βb +5 Rαβn

αnβηab +KKab −KcaKbc

−1

2ηab(K

2 −KabKab)−5 Rµανβnµeαan

νeβb

=2

3κ2

5

[Tαβe

αaeβb +

(Tαβn

αnβ − 1

4T

)]ηab +KKab −Kc

aKbc

+1

2(KadKcd −K2)ηab − Eab, (3.100)

where Eab is the traceless Weyl tensor's projection on the brane. Combining the second ofGauss-Codazzi's equations with (3.97), we obtain the following result

Kab|a −K|b = 5Rαβ nα eβb = κ2

5Tαβnαeβb .

Since we are dealing with a Z2-symmetrical braneworld, we are able to parameterize its locationthrough a certain value of the coordinate z. As a consequence, we take the 5D energy-momentum

28The trace of the 5D Einstein equation leads to 5R = −2/3κ255T . Then, for its projection along the normal to

the hypersurface, the result is R = −κ25T , hence we can substitute this value in all the Ricci scalar terms arisingfrom the general formula of the Riemann tensor.

29We can express the Weyl tensor's projection as Eab = Cµaνβnµeαan

νeβb .30Conformally at means that, given a (pseudo)-Riemannian manifold with a metric, for every point lying on

the manifold there exists an open neighborhood and a smooth map that denes a new dierentiable manifold overwhich curvature vanishes.


tensor as constituted by two parts: a term coming for the bulk vacuum energy and the otherfrom the brane

Tαβ = −Λgαβ + Sαβδ(z) = −Λgαβ + (−ληαβ + ταβ)δ(z), (3.101)

where Λ stands for the bulk cosmological constant, while Sαβ indicates the contribution fromthe stress-energy tensor characterizing the brane. The latter is in turn composed of a vacuumenergy, λ, and a 4D energy-tensor ταβ .

According to Israel junction conditions, the induced metric and extrinsic curvature are re-quired to satisfy the following relations

[ηab] = 0 ; Sab = − 2

3κ25

([Kab]−

1

3[K]ηab

)⇒ K+ab = −4πG5

(Sab −

1

3Sηab

). (3.102)

Relations (3.101) and (3.102), enable us to rewrite equation (3.100) in terms of the brane'sproper energy-momentum tensor, τab, and vacuum energy, λ. Hence, Einstein's eld equationson the brane become 31 [6]

4Gab = −1

2κ2

5

(Λ5 +

1

6κ2

5λ2

)ηab +

κ45λ

48πτab

+κ45

[−1

4τacτ

cb +

1

12ττab +

1

8ηabτcdτ

cd − 1

24ηabτ

2

]− Eab

= −Λ4ηab + 8πG4τab + κ45πab − Eab. (3.103)

This last expression shows that linearized eld equations on the brane require the addition ofthe Weyl tensor's projection, Eab, which is responsible for transmitting tidal gravitomagnetic32

and gravitational wave eects to the brane from the 5-dimensional non-local gravitational eld.For linearization to be explicitly recovered, we need to set the bulk's cosmological constant tozero, while keeping the four -dimensional gravitational constant nite.

It is essential to bare in mind that the decomposition of the brane's stress-energy tensor, givenin (3.101), is not unique. Indeed, there are circumstances, such as the ones describing the veryearly epochs of our universe, in which the energy content is so high that prevents to distinguishλ from τµν . In such circumstances, the terms of second order in τ would be the most relevant forthe eld equations (3.103). On the other hand, for the low-energy case, they can be neglected,hence linearized gravity is recovered, but there is the extra term arising from the Weyl tensor'sprojection on the brane. It is important to understand the way in which it aects the energycontent of the braneworld.

At rst, we need to nd a relation between the brane's energy-momentum and Eµν . In orderto do so, we use Codazzi 's equation. From the second junction condition (3.102), we are nowable to express (3.98) in terms of the brane's energy-momentum tensor, leading to

Kab|b −K|b ∝ τab |a = 0,

where the last equivalence follows from (four -dimensional) conservation of energy-momentum;we recall that the covariant derivative we have used in this last passage is indeed the intrinsicone, with respect to the brane.

31Where Λ4 = 12κ25(Λ5 + 1

6κ25λ

2)stands for the cosmological constant on the braneworld, G4 =

κ45λ

48π,

πab = − 14τacτcb + 1

12ττab + 1

8eabτcdτ

cd − 124eabτ

2.32We will discuss about tidal charge at the end of this section.


From the Bianchi identities,4Gab|a = 0,

only three of the terms featuring in (3.103) will be left, leading to the following relation

Eab|a = Kcd(Kcd|b −Kbc|d) =1

4κ2

5

[τ cd(τcd|b − τbc|d) +

1

3(τab − τgab)τ|a

].

The meaning of this result is that: the Weyl tensor is linked to the matter content characterizingthe brane. We should have expected such a thing to happen, as a consequence of some kindof back-reaction of the bulk in presence of the brane itself. We shall analyze bulk uctuationsin the following subsection, and show how the braneworld is aected by them even for vacuumbraneworld settings.

The decomposition of the Weyl tensor's projection, can, in turn, be decomposed into atransverse-traceless part and a longitudinal component, where the former is related to gravi-tational waves, while the latter are determined by the matter content.

Because of the presence of the Weyl tensor, the eld equations (3.103) are not a closed setof equations: we need to determine the evolution equations for the transverse traceless part ofEµν , which we can think of as being a 5-dimensional propagating graviton.

We can identify two orthogonal components, precisely an electric and a magnetic one, whichare, [6],

Eab = Cµaνbnµnν ; Babγ = eαae

βbCαβγδn

δ,

respectively. They both share the property of being trace-free, hence Eαα = 0, Bµµα = 0; whatmakes them dierent, is that the former is symmetric, while the latter is antisymmetric in therst two indexes.

Given the general decomposition of the Riemann tensor of equation (3.99), the two com-ponents can be expressed in terms of the Riemann and Ricci bulk tensors, therefore obtaining,[6],

Eab = 5Rµaνbnµnν − 1

3ηab

5Rµνnµnν − 1

3eαae

βb

5Rαβ +1

12ηab

5R,

where we have used its symmetry properties, and, [6],

Babγ = eαaeβb

5Rαβγδnδ +

2

3[(∇eKe

a −∇aK)eγb − (∇eKeb −∇bK)eγa], (3.104)

where only the Riemann-tensor terms are non-vanishing: indeed, if there would be any Ricci-scalar term, the magnetic component would contribute to the brane's energy-momentum content,implying that equation (3.103) is incomplete.

If we now apply Israel junction conditions to equation (3.104), the jump in the extrinsiccurvature is33

[eαaeβb

5Rαβγδnδ] = 2∇[a[Kb]γ ] = −2κ2

5∇[a

(τb]γ −

1

3ηb]γτ

). (3.105)

The terms after the last equal sign in (3.105) come because of the same contribution from bothsides of the ve-dimensional manifold, although with opposite sign. Substituting this in equation(3.104), the magnetic component on one side of the brane, B+

abγ , becomes

B+abγ = 2∇[a[K+

b]γ ] +2

3[(∇dK+d

a −∇aK+)ηbγ − (∇dK+db −∇bK

+)ηaγ ]. (3.106)

33We have used the same notation as the one adopted for deriving the junction conditions. Precisely, thesquared brackets are used when evaluating changes in a physical quantity, induced by the presence of the brane.


Substituting (3.105) in (3.106), we get

B+abγ = (eαae

βb

5Rαβγδnδ)+, (3.107)

which provides the boundary conditions on the brane that are required to solve the evolutionequations for Eµν .

In order to get the evolution equations, we start from the generalization of the Bianchiidentities in our ve-dimensional description, which are given by the following expression

∇[a5Rbγ]δσ = 0; (3.108)

since we have used the Riemann tensor in the denitions of the electric and magnetic compo-nents of the Weyl tensor, we can use (3.108) to derive the way in which the two componentsmutually interact during their evolution. If we split the covariant derivative into its intrinsicbrane component34, and the Lie derivative with respect to the vector eld normal to the brane,equation (3.108) returns four sets of coupled equations, which are, precisely, [6]

D[aBδbγ] +Kσ

[a4Rbγ]σ

δ = 0,

£nBabγ + 2 D[aEb]γ] −KσγBabσ + 2Bγσ[aK

σb] = 0, (3.109)

£n4Rabαδ + 2 4Rabσ[γK

σδ] + 2 DaBγδb − 2 DbBcda = 0, (3.110)

D[a4Rbγ]δσ = 0,

where we have used the identity (3.107) and have renamed some of the terms so that the equationsare expressed in terms of, both, electric and magnetic components. The evolution equations areobtained by inverting equations (3.109) and (3.110), resulting in

£nBabγ = −2 D[aEb]γ +KσγBabσ − 2Bγσ[aK

σb] (3.111)

and

£nEγδ = DaBa(γδ) +1

6κ2

5Λ5(Kγδ − ηγδK) +Kab 4Raγbδ

+3Ka(γEδ)a −KEγδ + (KγaKδb −KγδKab)K

ab. (3.112)

For the latter, we have used the linearized expression for the eld equations in four -dimensions(3.103). Equations (3.111), (3.112) together with (3.103) constitute a closed system of equations.

When dealing with Kaluza-Klein's model, we have discussed how the gravitational constantis related to the Planck scale through the relation MP =

√8π/GN =

√8π/κN , where GN

and κN are in turn related to the number of dimensions describing a particular manifold. Insection (3.1.2) we saw, for example, that G4 = G5/L, where L was the size of the compactieddimension. It is now possible to determine a relation between gravitational and cosmologicalconstants associated to spacetimes with dierent dimensionalities. For the case we are interestedin, the relation between their Planck scales and cosmological constants are [8]

MP =

(3

4πλ

)1/2

M3P ; Λ4 =

4π

M3P

[Λ5 +

4π

3M3P

λ2

](3.113)

34We use the symbol Da to indicate the intrinsic covariant derivative with respect to the brane.


where MP is associated to the braneworld while MP to the bulk. These relations are a conse-quence of the substitution of κ5 and previous calculation (see footnote on page 45 for the explicitform).

For the purpose of this work, we are interested in the description of a vacuum brane, i.e.Λ = 0, and in determining its corresponding eld equations starting from (3.103). Dealing witha 4D vacuum, we obtain that

Λ5 = −4πλ2

3M3P

⇒ Raa = Eaa = 0 ⇒ Rab = −Eab, (3.114)

where the traceless property of the latter has been used. As already stated when rst introduced,the Weyl tensor encodes gravitational degrees of freedom, including the tidal contributions of thegravitational eld. Equations (3.114) together with the divergence constraint ∇aEab = 0 givenby boundary conditions, determine a close set of equations on the brane. It is thus possible togenerate a vacuum braneworld solution in 4D starting from the description of gravity in 5D; inorder to do so, we need to identify the stress-energy tensor Tab with the Weyl term Eab, therebymaking the constraint equation coincide with the conservation law ∇aTab = 0 for the brane'senergy-momentum. A stationary general relativity solution with a trace-free energy-momentumtensor gives rise to a vacuum braneworld solution in ve-dimensional gravity. This is why wecan build up from the braneworld model an analogue to Reissner-Nodstroem's solution. In whatfollows we will use (in the brane), the spherical symmetry proper to the exact. Indeed, suchproperty belongs to a lower -dimensional description, reason why we can apply it even whendealing with a Z2-symmetry.

By selecting an arbitrary 4-velocity vector eld uα, such that hab is the three-metric given bythe following hab = ηab + uaub, we can decompose the brane projection of the Weyl tensor intothree irreducible parts, hence obtaining the following expression [8]

Eab = −(κ5

κ4

)4 [U

(uaub +

1

3hab

)+ Pab + 2Qaub + 2Qbua

],

where U = −Eabuaub stands for the eective energy density on the brane, resembling the de-nition we have given for Sab, Pab is the non-local anisotropic stress part, and Qa is the eectiveenergy ux, and hab is the three-metric given by the following hab = gab + uaub.

Pab = −(κ4

κ5

)4 [hc(ah

db) −

1

3habh

cd

]Ecd ; Qa =

(κ4

κ5

)4

hcaEcbub.

For a static vacuum solution, uα can be chosen as a timelike Killing vector eld, hence thedivergence constraint equation becomes a spatial equation

∇aEab =1

3U|b +

4

3Uua∇bua + Pab|b + ua∇aubPab = 0, (3.115)

since Qa = 0. The second term on the r.h.s. is a 4-acceleration. In a static spherical symmetry,we can express, both, Aα and Pαβ , in terms of a radial distance, r.35 Hence we get

Aa = ub∇bua = A(r)ra,

while rα stands for a unit radial vector. As reported above, here we are dealing with a lowerdimensional description from that of the braneworld, which makes it legitimate to describe a

35As a consequence of the symmetry of the solution on the brane.


static spherically symmetric solution. We may also rewrite the anisotropic term in terms of theradial distance, hence leading to the expression

Pab = P (r)

[rarb −

1

3hab

].

The overall constraint equation, thus turns out to be

1

3DaU +

4

3UAa +DbPab +AbPab = 0, (3.116)

where Da is the projection, orthogonal to the vector eld uα, of the brane covariant derivative,∇α. We can now solve equation (3.116) by splitting it into two distinct rst-order dierentialequations, U can be expressed in terms of P

U = −1

2P =

(κ4

κ5

)4N

r4,

where the last passage has been made to resemble the Reissner-Nordstroem solution (with theintroduction of an electric charge N . Such solution can be proved to be adequate for the assignedeld equations, just by following the same procedure we have already outlined for deriving theexact R.-N. solution. In doing so, we should consider the braneworld with metric, [8],

ds2 = −A(r)dt2 +B(r)dr2 + r2(dθ2 + sin2θdφ2),

and verify that

A = B−1 = 1 +α

r+β

r2,

where the energy-momentum tensor non-vanishing components are

Ett = Err = −Eθθ = −Eφφ =β

r4.

The metric functions are characterized by the same power-law in r with respect to the Reissner-Nordstroem solution. The timelike component of the metric solution results of the form, [8],

−gtt =1

grr= 1−

(2M

M2P

)1

r+N

r2= 1−

(2M

M2P

)1

r+

q

M2P r

2(3.117)

where q = NM2P indicates the dimensionless tidal charge conferred by the Weyl tensor to the

brane. Indeed, we may consider it as a ve-dimensional mass-parameter.We are now interested in comparing this result with the electrically charged solution obtained

in 4-dimensional GR. The roots of (3.117) are related to the tidal charge via the followingexpression

r± =M

MP

[1±

√1− q

M4P

M2M2P

], (3.118)

which for the case q ≥ 0 resembles the Reissner-Nordstrom solution; indeed, both horizons lieinside the Schwarzschild radius rSch = 2M/M2

p . This implies that at xed M the tidal chargemust have an upper limit, in order not to make the root argument drop to negative values;


the parameter range in which this solution corresponds exactly to the four-dimensional solution,therefore, is

0 ≤ q ≤ M2PM

2

M4P

. (3.119)

From (3.118), we can see that brane black holes may have a negative tidal charge, q < 0,which is prohibited for the exact 4D solution. A negative tidal charge implies that the horizonlies outside the Schwarzschild radius, meaning that bulk eects increase entropy and tend tostrengthen gravity, as follows from the Second Law of black hole mechanics.

We can summarize the result that we have obtained in this section by saying that the freepart of the bulk stress-energy tensor is responsible for connement on the brane. From the eldequations in 5D, (3.97), tidal acceleration on the brane in the o-shell direction may be expressedin terms of the Riemann tensor's projection as

−5Rµaνbnµuanνub = −Eabuaub +

1

6κ2

5Λ5 =

(κ5

κ4

)4

U +1

6κ2

5Λ5.

Since the cosmological constant is negative in the Randall-Sundrum model, the only way toenhance connement on the brane is via U < 0. Setting M = 0, the prefactor in the line element(3.117) turns into

−gtt =1

grr= 1 +

q

M2P r

2, (3.120)

with an associated horizon located at rh =√−q/MP , therefore showing that the existence of

(negative) charge is allowed even in the absence of matter on the braneworld.Tidal corrections are relevant in strong gravity regimes and are also thought to be responsible

for the birth of primordial black holes: indeed, the absence of collapsing matter in the very earlyepochs of the universe could have been compensated by tidal-bulk eects.

Gravitational Perturbation Theory, Black String's Instability and Spontaneous Sym-

metry Breaking in GR

Since gravity is a non-linear theory, perturbation theory constitutes an essential tool that allowsto test the stability and evolution of a given metric solution. Since we are interested only inlinear perturbations, in the following steps we shall keep only the terms that are relevant in thisapproximation.

The addition of a perturbation to the metric , resulting in the form η′ab = ηab + hab, impliesa variation of the Christoel symbols

δΓabc =1

2η′an((ηnb + hnb),a + (ηna + hna),b − (ηab + hab),a) =

1

2(∇chab +∇bhac −∇ahbc),(3.121)

where we have used ηµν for raising and lowering the indexes of the tensor perturbation, followingthe relations hab = ηachcb and h

ab = hacηbdhab. From the denition of the Ricci

Rab = Γcab,c − Γcac,b − ΓdacΓcdb + ΓdabΓ

cdc (3.122)

we can neglect the ΓΓ-terms for a rst-order approximation; hence, we get

δRab = ∇cδΓcab −∇bδΓcba =1

2∇c∇ahcb +

1

2∇c∇bhca −

1

2∇c∇chab −

1

2∇b∇ah.


If we now use the Riemann Identity to swap the indexes of the covariant derivatives featuring inthe rst two terms, we get a Riemann tensor for each exchange of indexes, hence leading to

δRab =1

2∇a∇chcb +

1

2Rcdcah

db +Rbdcah

cd +1

2∇b∇chca +

1

2Rcdcbh

da −

1

2hab −

1

2∇b∇ah

= −1

2(hab + 2Rabcdh

cd − 2Rd(ahb)d −∇(a∇dhb)d) = −1

2∆Lhab, (3.123)

where we have used a new variable, dened as

hab = hab −1

2hηab, (3.124)

which is still trace-free. The symbol ∆L in (3.123) is the Lichnerowicz operator, which constitutesa curved spacetime wave operator. In at spacetime, the Ricci tensor vanishes. Hence, theequation describing the perturbation reduces to, [2]-[4],

∆Lhab = hab + 2Rabcdhcd = 0, (3.125)

where the divergence-free gauge has been applied, i.e.

∇ahab = 0. (3.126)

Coordinate transformations in Group Theory are described by operators. For an innitesimalcoordinate transformation, we can expand the operator around unity. Locally, at leading order,we get a linear contribution. So, we can think of coordinate transformations as being capableof acting on a given metric, adding a perturbation term. For such reason, when dealing withgravity, we need to consider only coordinate transformations occurring along a Killing vectoreld, along which the Lie derivative of the metric is null. Therefore, we can rewrite the metricby adding a shift proportional to the aforementioned one, obtaining, [2],

ηab → ηab + £ξηab ⇒ ηab + 2∇aξb = ηab + hab, (3.127)

where ξ is a Killing vector eld. From this, (3.124) results

hab = 2∇aξb − (∇ξ)gab. (3.128)

By computing its covariant derivative, the result is

∇ahab = ξb +∇a∇bξa −∇b∇aξa = ξb +Rcbξc; (3.129)

using, once again, the Riemann Identity. Now equation (3.129) is a hyperbolic equation. Wecan further simplify it, hence obtaining the evolution equation for ξb, by choosing the followinggauge

∇ahab = 0 ⇒ ξb = −Rcbξc. (3.130)

Gravitational perturbations are characterized by a certain number of degrees of freedom, whichcan be understood as the polarizations of gravitational waves. By denition, hab is a symmetricrank-2 tensor, therefore it has D(D + 1)/2 degrees of freedom. At the same time, the gaugecondition (3.130) is equivalent to a system of D-(rst order) dierential equations, to which wehave to add the degrees of freedom allowing the choice of an arbitrary coordinate transformation,


i.e. ξ, which are still in number of D in order to be compatible with hab in the previous relations.Thus, the remaining degrees of freedom are, [2],

n =D(D+1)

2− 2D =

D(D − 3)

2

If D = 4 ⇒ n = 2, meaning that the perturbation is a massless spin-2 eld with 2 degrees offreedom, i.e. a quadrupole radiation, alternating contractions to stretching. If D = 5 ⇒ n = 5;from these examples, we can deduce that the graviton's degrees of freedom increase as thespacetime's number of dimensions rises.

The surface topology of a four -dimensional black hole is bound to be spherical. But, when westart dealing with extra dimensions, we loose such a constraint, and more exotic shapes appearto be equally valid, since they share the same charges.

From the Laws of Black Hole Mechanics, we know that the surface area is associated to theentropy. As a consequence, the solution with greatest entropy should be the most stable. Westart considering the ve-dimensional case, where the solution might be either a hypersphericalone [4]

ds2 = −V5(r)dt2 + V5(r)−1dr2 + r2dΩ23,

resembling Schwarzschild's solution, with associated potential, [4],

V5(r) = 1− r25

r2,

or a black string, obtained by extending the 4-dimensional sphere along the extra straight di-mension, z, which constitutes a Killing symmetry of the full solution,

ds2 = −V (r)dt2 + V (r)−1dr2 + r2dΩ22 + dz2.

The mass associated to the hyperspherical black hole is MBS = rSch/2G4 = rSchL/2G5.Entropy is recovered from the rst law of black hole thermodynamics as S = A/4G. Computingit for the two possible solutions, we get

MBS =r+

2⇒M5 =

3πr25

8L= MBH

since the relation between the Planck masses in dierent dimensions isM2P = VD−4M

D−2D . Their

associated entropies result to be

SBS = πr2+ ; SBH =

π2r35

2L⇒ SBS = SBH

√27πr+

16L.

This proves that, at xed mass, SBH drops as the compactication length increases. Thus, fora given thickness/4D-section there is a critical length above which the black hole is, indeed, apreferred conguration; the 4D section would be too thin with respect to the length of the 5th

dimension. Indeed, perturbations along the extra dimension would eventually generate blackholes, as a consequence of the intersections along the boundary of length L.

The procedure we have described would be enough for proving that black holes are morestable than black strings in a higher -dimensional description, if it were possible to dene theinternal structure of a black hole, which is obviously incorrect for the presence of the eventhorizon. From this follows the need to check for the stability of the black string with the aid ofperturbation theory.


When rst analyzed by R.A.W. Gregory and R. Laamme, they conjectured that the nalstate should cease to be translationally invariant along the extra dimension, ultimately leadingto the black string breaking up. Such hypothesis was based on the possibility to express per-turbations in terms of the background 's symmetries36. For the case of a 5D black string theyare: the two Killing vectors ∂/∂t , ∂/∂z and the SO(3) Killing group. Hence, the instabilitywas expected to show as an S-wave, i.e. a constant deformation of the 2-sphere. Thanks to thesymmetries of the background, we can perform separation of variables in order to solve equation(3.125). In this way, three separate modes arise, denoted as

1. hzz , a scalar

2. hza, a vector

3. hab(r), a tensor,

therefore allowing to factorize the solution as, [4],

hµν = hab(r)eΩteiµz, (3.131)

where hab(r) constitutes the radial prole eigenfunction, while the two exponents are Fouriermodes resulting from the t and z symmetries, accounting for the oscillatory behavior, eiµz, andthe growing mode eΩt (assuming the perturbation to be unstable); m and Ω are two parametersthat adequately compensate the dimensionality of z and t, respectively.

Now, it is just a matter of solving equation (3.125) for the three modes. Starting from thehzz-scalar mode, its corresponding equation is of the form, [2]-[4],

∆Lhzz = (4 + µ2)hzz, (3.132)

which resembles the wave equation of a massive scalar eld, where µ is inversely proportionalto the black string's length. When discussing about Kaluza-Klein's model, z was taken to be acompactied extra dimension, thus there was no dependence from h in that direction. Instead, inthis case, where its inuence is assumed to be relevant, the application of the operator ∂2/∂z2,adds to the overall solution a µ2-term which looks like a mass-term, associated to a certain mode,from a 4D prospective, being inversely proportional to the length of the compactied dimension,[µ] = L−1 = M . Computing (3.132), leads to a second order dierential equation of the form,[2]-[4],

∆Lhzz =− V h′′zz − 2r −GM

r2h′zz +

(µ2 +

Ω2

V

)h55, (3.133)

where V = 1 − 2GM/r. Since there are no analytic solution, the next step will be to considerthe equation's behavior near the black string's horizons, hence leading to the following result [4],

r → +∞⇒ hzz ∝ e±√

Ω2+µ2r → 0 (3.134)

r → 2GM ⇒ hzz ∝ (r − 2GM)2GMΩ → 0. (3.135)

Finding a perturbative solution for the scalar mode, reduces to the problem of identifying aturning point satisfying the conditions

h′ = 0 ; h′′zz < 0,

36Corresponding to the irreducible representation of the topology S2 × S1.


from which equation (3.133) would reduce to,

−V h′′zz +

(µ2 +

Ω2

V

)hzz = 0. (3.136)

Within the range 2GM < r < +∞, there is no unstable hzz, because V > 0. The same conclusionis obtained for the vector mode hza, meaning that the only possible way in which the instabilitymay arise is throughout the tensor -mode37 hab , which is a 4x4 matrix of rank 3 with a blockdiagonal structure describing the mutual interaction between the time and radial components ofthe mode (via the divergence condition), and one angular component (diagonal), indicating thatthe 3-sphere either grows or shrinks, preserving its spherical symmetry. Moreover,

hµν = eΩteiµz

htt htr 0 0hrt hrr 0 00 0 ka(t, r) 00 0 0 0

(3.137)

where k is some constant of proportionality and a(t, r) is a generic function accounting for thedependence of the angular piece from the timelike and radial coordinates, [4]. At this point,hab(r) obeys a second order dierential equation in terms of the parameter variables (µ,Ω).Thanks to numerical simulations, it was possible to prove that, for a certain range of parametervalues, a solution does exist, therefore proving the growth of a perturbation. Because of thepresence of the Fourier mode in z, the black string's horizon starts to ripple at linear order andsuch conditions were at rst believed to cause the birth of hyperspherical black holes connectedby very thin black strings that would eventually pinch o as a result of quantum eects producedby an increasing horizon curvature.

Recent developments on this topic have put forward an alternative possibility. Instead ofbreaking up, new perturbation could originate at its center resembling a spherically symmetricblack hole. G.T. Horowitz and K. Maeda analyzed the process in one of their recent works, anddescribed such phenomenon as a case of spontaneous symmetry breaking in general relativity 38.

In order to determine how the new stable solution may look like, some preliminary consider-ations must be outlined. The reason that lead to such proposal is that pinching would violatethe Cosmic Censorship Hypotheses, of which we have previously reported the two assertionsit consists of when dealing with the Reissner-Nordstroem solution. The possibility for higherdimensional black holes to rise from a black string tearing apart, would imply the violation ofsuch conjectures.

Such a thing could be explained using Raychaudhuri 's equations, whose derivation is providedin the Appendix. Horowitz and Maeda generalized this relying upon a theorem, stating that eventhorizons cannot have any collapsing S1-sphere [5]. To prove this, we take a horizon metric of theform, [5],

ds2 = e2φ(dy +Aµdxµ)2 + ηµνdx

µdxν ,

which resembles the one we used for Kaluza-Klein model; λ is an ane parameter along the nullgeodesic generators, such that `µ = (∂/∂λ)µ. Then, we can write Raychaudhuri's equation as

θ = −BµνBµν −Rµν`µ`ν ,

37As previously conjectured calculating entropy.38It is important to notice that, in particle physics the broken symmetry is an internal one, while here it is

related to spatial translations.


where θ is the expansion scalar39which is associated to the trace of a given tensor. In this case,such tensor is Bµν : it indicates the evolution of the deviation vector between geodesics. Fromthe given metric, we get

BµνBµν = hαµh

βν∇α`β = φ2 +

1

2e2φ||Aµ||2 +

1

4||γµν ||2.

Provided the weak energy condition holds, stating that for every timelike vector eld `µ, thematter density observed by the corresponding observers is always non-negative, we get, [5],

ρ = Tµν`µ`ν ≥ 0 ⇒ Rµν`

µ`ν ≥ 0 ⇒ θ ≤ −φ2,

meaning that the parameter φ doesn't reach an innite negative value for a nite ane valueλ. This proves that the black string is not allowed to split-up, but rather generates othersmaller black holes located at the centre of these very thin necks. This implies that the moststable solution is obtained violating translational invariance, thereby constituting an exampleof spontaneous symmetry breaking in General Relativity. Further studies, conducted in the lastdecade, have analyzed the possibility that a black string could become stable if one of its endslies on a brane.

In the previous subsection, we have dealt with the Randall-Sundrum 2 model for determiningthe inuence of the bulk on the brane and have shown how it is possible to recover Einstein'slinearized eld equations in the Z2-symmetry. At the end of this chapter, we provide an alter-native derivation. In particular, we shall use the notion of Kaluza-Klein modes for constructingGreen's functions for the evolution equations for a given perturbation. We start by adding aperturbation to the brane's metric, resulting in, [7],

ds2 = (e−2|z|/lηab + hab)dxadxb + dz2. (3.138)

We notice that the line element resembles the kink analyzed in chapter (3.1.3), thus we refer tothe results that we have obtained then for the computation of the tensor components. Einstein'sequations on the brane, at linear order in the perturbation, are given by [7]

Gab = Rab −1

2Rη′ab = Rab −

1

2R(e−2|z|/lηab + hab)

=1

2e2|z|/l(2∂c∂(bh

ca) −hab − ∂a∂bh)− 1

2∂2zhab +

2

l2hab + ηab

(h

l2+∂zh

2l

)= 0, (3.139)

where we notice that the rst three terms in brackets 40 are indeed the terms we obtained whencomputing the variation of the Ricci tensor at the beginning of this section. The perturbationlies on the brane, hence, we are free to choose it as being transverse, i.e. hµz = 0, and divergence-free, meaning ∂νh

νµ = 0. Such gauge choice enables us to simplify equation (3.139), leading to

the following expression, [7], (e2|z|/l + ∂2

z −4

l2

)hab = 0. (3.140)

In order to solve this second -order partial dierential equation, we can use a rst-order relation,which is provided by the jump in extrinsic curvature, i.e. the junction conditions. Under a

39See Appendix C for the derivation of Raychaudhuri equation, where a denition of the parametrs needed hereis provided.

40Indeed they are four, but two of them are symmetric in the µ and ν indexes, which is why we have used thenotation with brackets.


coordinate transformation, hab → ξab, we can express the perturbation in terms of Gaussiannormal41 coordinates

z = z + bz(xa) ; xa = xa + ba(xa), (3.141)

where a = 0, 1, 2, 3. Since the two corrections in this particular coordinate transformation aretaken to be independent on the extra dimension z, we can relate them in the following way, [7],

ξaz = 0 ⇒ −2

le−2|z|/l(ba +Ma(xb)) = ηac∂cb

z(xb),

resulting from the change in the warp factor of the metric (3.138). Here we denote with Ma(xa)an arbitrary function of the brane's coordinates xa: it basically confers a gauge-freedom tothis description. Being hab a symmetric tensor, it has ten independent components. By xingthe location of the brane z = 0, this number drops to ve. Since we are dealing with a ve-dimensional description, gravity can only have 5 independent components. This is why we haveto consider an additional function, Ma(xa), accounting for all possible metric transformationson the brane. Combining these relations together with the RS gauge, we can write an expressionlinking the perturbation metrics as follows, [7],

ξab = hab + l∂a∂bbz +

2

le−2|z|/lηabb

z +Ma(xb). (3.142)

In this way, the extrinsic curvature terms turn out to be equal to the Christoel symbols but withopposite sign, hence obtaining, by partial dierentiation of the perturbed brane metric (3.138)

−2

lηab + ∂zξ = −8πG5

(Sab −

1

3(ηab + ξab)S

), (3.143)

where, ξ is the perturbation, while Sab is the usual stress-energy tensor associated to the brane.Recalling equation (3.101), we can rewrite equation (3.143) in terms of the brane's energy-momentum tensor, as follows

Sab = −λ(ηab + ξab) + Tab,

where λ indicates the stress exerted on the brane by the bulk's negative cosmological constant.From this, we get

−2

lηab + ∂zξ = −8πG5

(− 3

4πlG5ηab −

3

4πlG5ξab + Tab −

1

3(ηab + ξab)

(− 3

πlG5+ T

))

=6

l(ηab + ξab)− 8πG5

(1

πlG5ηab −

1

3ηabT +

1

πlG5ξab

).

In computing the trace for Sab, we have used the RS gauge requiring the perturbation to be trace-free; such gauge is preserved under the coordinate transformation we have performed, being thenormal to the brane a Killing vector.

We can now simplify the expression for the junction condition, hence obtaining(2

l+ ∂z

)ξab = −8πG5

(Tab −

1

3ηabT

).

41The reason for using Gaussian normal coordinates is that they resemble a at metric of the form ds2 =−dy2 + hij(x, y)dxidxj . Hence, they allow to recover a at hypersurface, modeling the metric such that alldeformations are removed.


If we now use equation (4.9), we obtain the junction conditions back in the original coordinates,(2

l+ ∂z

)hab = −8πG5

(Tab −

1

3ηabT

)− 2∂a∂bb

z, (3.144)

where only the rst term remains, since the expression is evaluated on the brane; therefore,the terms containing the parameter z give no contribution. Performing an index contraction on(3.144), leads to the equation of evolution for the z-component of the perturbation's transforma-tion,

haa = 0 ⇒ 8πG5

3T = 2∂a∂

abz ⇒ 4bz =4πG5

3T. (3.145)

From this, we have proved that matter conned on the brane, denoted by T , is responsible forits bending.

We can also rewrite equation (3.140) adding a delta-function centered at the brane's location,hence obtaining a more general expression[

e2|z|/l + ∂2z −

4

l2+

4

lδ(z)

]hab = −2Σabδ(z). (3.146)

In order to recover the linearized metric perturbation on the brane, we need to use Green'sfunctions. For the symmetry of the system we are analyzing, they are dened as solutions to thefollowing equation[

e2|z|/l + ∂2z −

4

l2+

4

lδ(z)

]G(x, z;x′, z′) = δ4(x− x′)δ(z − z′).

Green's functions are obtained starting from a complete set of eigenstates, which in our case arerelated to the massive modes; thus, we have to distinguish between the zero-mode associated tothe massless graviton lying on the brane and all the Kaluza-Klein modes according to how faro from the brane we are actually evaluating the perturbation. Its structure is

G(x, z;x′, z′) = −∫

d4k

(2π)4eikµ(xµ−x′µ)

[e(−2|z|+|z′|)/l

l(k2 − (ω + iε)2)+

∫ ∞0

dmum(z)um(z′)

m2 + k2 − (ω + iε)2

],

where kµ = (ω,k), while um contain the Bessel functions. By substituting G into (3.146), weget the perturbation on the brane

hab = −2

∫d4x′dz′G(x, z;x′, z′)Σabδ(z

′).

If we expand Bessel 's functions in the modes, we get a lower -order term that is linked to themassless graviton, which is bound to the brane, plus additional corrective terms arising fromKaluza-Klein's massive modes, that become relevant only when dealing with high energies.

We can now appreciate the importance of the brane's bending. If we go back to Gaussiannormal coordinates, where the perturbation in the metric is described by

ξab = hmab + hbab +2

le−2|z|/lηabb

4,

where the terms hmab and hbab stand for the matter content and brane bending contributionsrespectively. if we then substitute Green's functions, we get

ξab = −16πG5

∫d4x′G(x, z;x′, 0)

(Tab −

1

3ηabT

)− 4

∫d4x′G(x, z;x′, 0)∂a∂bb

4 +2

lηabb

4,


where the second integral simplies for z = 0. For the zero-mode truncation, the result is

ξab = −16πG5

l

∫d4x′

14

(Tab −

1

3ηabT

)+

4πG5

3

∫d4x′

14

T

= −16πG5

l

∫d4x′

14

(Tab −

1

2ηabT

); (3.147)

the change of the prefactor from 1/2 to 1/3, shows that linearized gravity on the brane is recoveredat low energies thanks to the graviton's propagator in 4D arising from the bending of the brane.What actually happens, is that the fth polarization of the graviton turns into a 4D scalar eld.This observation hints at the topic we shall discuss about in the next chapter, [9].


Chapter 4

Cosmological phase transitions

In the previous chapter we have dealt with topological defects with dierent dimensionality. In particular,we have focused on domain walls, and discussed the braneworld model.

In this last chapter, we are going to describe how cosmological phase transitions can be mediatedby primordial vacuum uctuations, by discussing a relation between the coupling constants of the Higgseld with matter and the cosmological constant.

Starting fom the semi-classical approach described by instanton solutions, we will show how the

tunnelling processes of vacuum spacetime can be catalyzed thanks to primordial black holes, as described

in the form of projections of bulk vacuum uctuations in a braneworld model.

4.1 Braneworld Cosmology

Ination may be used to probe for the existence of extra dimensions. Such a process can beeither conned to the braneworld, or resulting from both, bulk and brane, in a higher dimensionaldescription.

Both perspectives have been analyzed in the last two decades, the latter being more recent,although they still suer from problems arising when considering high energy regimes. Oneimportant reason to understand the very early universe is in connection with the origin of thelarge scale structure. After cosmic ination, density perturbations are thought to have arisen,ultimately leading to the formation of topological defects.

In the previous chapter, we have dealt with the construction of the 5D black string startingfrom the 4D exact Schwarzschild solution. In this case, we will follow a very similar procedure;the major dierence resides in that we shall start from a (D+ 1)-dimensional empty bulk space,where D + 1 = d + 2 + m; d and m indicate the uncompactied and compactied dimensions,respectively, w.r.t. the brane in a braneworld setting. Provided the metric to be of a particularkind, performing a dimensional reduction allows us to describe the lower -dimensional RS IImodel (of (d + 2)-dimensions) embedded in the higher-dimensional manifold, with the presenceof scalar elds coming from the extra m-dimensions.

In this section, we provide, at rst, a brief overview concerning the generalization of theprevious statement for an arbitrary number of extra-dimensions; then we shall concentrate onthe case of a unique scalar eld, which has been proposed for describing cosmological ination.

73

74 CHAPTER 4. COSMOLOGICAL PHASE TRANSITIONS

We start dening the line element as being [9]

ds2D+1 = gαβdX

αdXβ = gabdxadxb +

∑i

e2φidσ2i , (4.1)

where Latin indexes a and b range from 0 to d + 1, with d as previously dened. gαβ standsfor the metric of the empty bulk, gαβ for the metric of the joined bulk and brane model weare aiming to describe, φi are the scalar elds which are functions of the coordinate xα, whileσ2i are line elements of a constant curvature compactied space, with an associated volume Vi;

this expression recalls what we have previously seen when analyzing the compactied dimensionin Kaluza-Klein's model. Indeed, the procedure we will now follow is similar. To describe theembedded braneworld, we analyze the action associated to the eld equations of which (4.1) isa solution. In doing so, we basically have to integrate over all the spacelike dimensions that arecompactied on the brane. When computing the action, there are two dierent contributionsthat have to be taken into account, precisely a bulk and a boundary term. The former is, in ageneral formulation

Sd+2 =1

2κ2d+2

∫dd+2x

√−g(d+2R− 2Λd+2)

=1

2κ2d+2

∫dd+2x

√−g e

∑i jiφi [d+2R−

∑i

jigab∂aφi∂bφi + gab(∂a

∑i

jiφi)(∂b∑i

jiφi)

−2Λd+2 +∑i

Ξiji(ji − 1)e−2φi ], (4.2)

where the last passage follows from dimensional reduction; for AdS5, d + 2 = 5. Ki are thesignatures of the curvature of the metric, dσ2

i describing either a closed, at or open universe,while κi is related to the gravitational constant, in turn dictated by the overall number of dimen-sions; d+2R, Λd+2 and κd+2 are the Ricci scalar, the cosmological and gravitational constants ofthe (d+2)-bulk, respectively; indeed, this formula resembles the one obtained in chapter (3.1.2),where σ =

∑i jiφi, and σ =

∑i Ξiji(ji − 1)e−2φi . Ξi indicates the sign of the curvature of the

compactied dimensions.Once again, after a conformal transformation, i.e. γab = e2

∑i /dgab, where γab is still the

(d+2)-dimensional braneworld metric, we recover the Einstein-Hilbert action

Sd+2 =1

κ2d+2

∫dd+2x

√−γ[d+2R−

∑i

jiγab∂aφi∂bφi −

1

dγab(∂a

∑i

jiφi)(∂b∑i

jiφi)

−2ΛD+1e−2

∑i /d +

∑i

Ξiji(ji − 1)e−2φi−∑i jiφi/d]. (4.3)

The brane's contribution undergoes the same conformal transformation; in the RS II model, thecontribution to the action arising from the brane is related to its surface tension through thefollowing relation

Sbrane = −∫dn+1x

√−ησ (4.4)

where η is the determinant of the induced metric on the brane, while σ is its surface tension,set by the ne-tuning condition. We can extend this to the more general case we are dealingwith now, i.e. a braneworld model in a higher-dimensional setting, hence obtaining the followingexpression

Sbrane = −∫dd+1x

√−gσ ⇒ Sbrane = −

∫dd+1x

√−gλe−

∑i jiφi/d, (4.5)

4.2. FALSE VACUUM DECAY WITH GRAVITY 75

where λ = σ∏1,mi Vi stand for the brane surface tension before and after having performed a

conformal transformation. We also recall that the Vi constitute the volume of the compactieddimensions, and that, the procedure we need to follow for recovering such rescaling is completelyanalogous to the one performed when dealing with Kaluza-Klein theory. The overall action is,obviously, the sum of the brane and bulk terms.

Now, we can restrict this formulation for the case of a single bulk scalar eld. In doing so,we dene it in the following manner

Φ =1

κd+2

√m(m+ d)

dφ,

where we recall that m and d indicate the number of compactied and uncompactied dimensionson the brane, respectively.

Stot = Sbrane + Sbulk

=

∫bulk

dd+2x√−γ[

1

2κ2d+2

d+2R− 1

2γab∂Φa∂Φb − V (φ)

]−∫brane

dd+1x√−g U(Φ),(4.6)

where γ is the determinant of the metric γab previously dened. In this last expression, the twopotentials are [9]

V (Φ) = − (m+ n)(m+ n+ 1)

2κ2d+2`

2e−2√

2(d+1)/[d(d−1)]κd+2Φ − Ξm(m− 1)

2κ2d+2

e−κd+2φ/d√m/[d(m+d)],

U(φ) = λe−√m/[d(m+d)]κd+2Φ,

in the bulk and on the brane, respectively. This proves that it is possible to resemble a joinedbrane and bulk ination process by applying a dimensional reduction to a pure gravitationalsetting.

4.2 False Vacuum Decay with Gravity

In the present and following sections, we will describe the application of quantum tunnelling tofalse vacuum, and discuss cosmological phase transitions.

Instantons are Euclidean classical solutions in quantum eld theory, interpolating betweentwo dierent vacua. They are geometrically described in terms of bubble nucleation at theemergence point after having crossed a potential barrier with negative kinetic energy. In orderto understand such assertion, we rely upon the quantum mechanical process of tunnelling througha potential barrier.

When a potential is added to the motion of a free particle, we need to subtract its contributionto the energy associated to the free conguration. For the case of a potential barrier exceedingthe particle's kinetic energy, we get an overall negative energy. From the relation between kineticenergy and linear momentum, we deduce that an imaginary time coordinate, τ , is required.

In the case of cosmological phase transitions, a full understanding requires a theory of coupledquantum and gravitational eects, which is not yet available. For this reason we will considerthe rst semiclassical approach, formalized by Coleman and de Luccia. After explaining the mainparameters describing tunnelling probabilities in at spacetime, we will consider the changes thatare required to include gravitational eects. In particular, we will show one way in which thebounce1 action can be dened consistently with, both, the quantum and gravitational principles.

1We will provide its denition in the following pages.


Vacuum phase transitions can arise from quantum uctuations. They can be described in termsof the decay probability per unit time per unit volume, represented by a two-parameter function[10]

Γ/V = Ae−B/~(1 +O(~)), (4.7)

where both parameters, A and B, depend on the characteristics of the potential constituting thebarrier. In particular, B stands for the action associated to a certain tunnelling process andalso features in the prefactor, which in turn depends on the dimensionality of the space we areconsidering. For the case of a four -dimensional spacetime, A is of the form [10]

A =B2

4π2~2

∣∣∣∣ det′(−∂2 + U ′′(φ))

det(−∂2 + U ′′(φ+))

∣∣∣∣−1/2

(1 +O(~)). (4.8)

Although we do not report here the full derivation of (4.8), we can still provide some explanationregarding its structure, which will turn out useful when dealing with the following discussion. Inthe path integral approach, the bounce action is obtained by integrating over all possible pathstunnelling through a classically-prohibited region. Under a pure quantum mechanical approach,we can describe the transition between two quantum states in terms of a functional integral, asfollows

< b|e−Ht/~|a >= N

∫Dx e−S/~, (4.9)

where with |a > and |b > we indicate the initial and nal quantum states, respectively, while∫Dx indicates the functional integral, i.e. the sum over all possible paths connecting the two

states, while N is a normalization parameter. Each path has an associated weight, e−S/~, whereS is the action associated to the transition. From the above expression, we can deduce that thepath associated to the lowest action is more likely to be taken. In the semi-classical approach,i.e. taking ~ 1, equation (4.9) is dominated by stationary points. From the steepest-descentapproximation, the transition probability becomes a functional integral over Gaussians, precededby the normalization factor, N , and also by an additional exponential coecient arising fromthe zeroth-order term of the above expansion. Of course the rst-order term is null from thedenition of stationary point. In computing the integral, the result is proportional to a powerlaw of the product of all the second derivatives of the action, evaluated at the stationary point.The number of such factors is dictated by the spectrum of the Hamiltonian dening the evolutionoperator in (4.9). This explains the presence of the determinant in denition (4.8). To explainthe presence of det′, we have to notice that, the above considerations are legitimate under theassumption that all energy eigenvalues are positive. Indeed, violating such condition, the integralwould diverge, and the same would be for the transition probability. Now, the bounce action,B, is dened as being associated to a null energy eigenvalue and also as being an approximatestationary point for the functional integrand, reason why, when evaluating the above determinantthis term has to be excluded from the spectrum, and replaced by a factor (B/2π~)1/2 for eachdimension characterizing the system we are analyzing, [11]. This also explains the prefactor in(4.8). Going back to the formula for the tunnelling rate, (4.7), the action is, [10]

B = 2

∫ ~σ

~q0

ds√

2V ,

where V is a given potential. The bounce action is a saddle point. It is associated to a nullenergy eigenvalue. The extremes of integration ~q0 and ~σ stand for the location of the classical


minimum and of the emergence site, respectively; for a four -dimensional description, they are nolonger points, but rather a set of coordinates on a hypersurface. Integration is performed overthe path for which the action is extremized.

The classical equilibrium point ~q0 can only be reached by the particle for τ → −∞, whichconstitutes a boundary condition.

The system is invariant under time translation, hence, we can set the instant τ = 0 as themoment in which the system reaches the emerging point ~σ.

For τ > 0, the motion is the time-reversal of the one previously described. As a consequence,it is possible to deduce that the particle bounces o the wall where ~σ lies and gets back to itsinitial position in the limit τ → +∞,

B =

∫ +∞

−∞dτLE = SE . (4.10)

In a multi -diensional description, the euclidean action can be extremized by more than onetrajectory. This is why an accurate determination of the transition rate Γ requires integrationover the symmetry group of the system. Symmetry changes when passing from Lorentzian toEuclidean description, i.e. from real to imaginary time. In particular, this transition raises thesymmetry of the problem from O(3, 1) to O(4). In geometrical terms, this corresponds to thetransmutation of a hyperboloid into a hypersphere.

We start our analysis with a massless scalar eld, with Lagrangian density including, both,a kinetic and a self-interacting potential,

L =1

2∂µφ∂

µφ− V (φ),

where the double-well potential is given in the form

V (φ) =λ

2(φ2 − η2)2 − ε

2η(φ− η); (4.11)

we have already encountered some of the parameters featuring in equation (4.11), when dealingwith the kink. We briey recall their denition here: λ is the coupling constant, associated to thewidth of the wall, φ is the potential, η is the vacuum expectation value of φ. The presence of asmall perturbation, parametrized by ε, indicates we are dealing with a transition from ametastable(false) to an eective (true) vacuum, which constitute the two minima of the potential associatedto the scalar eld φ. Under the assumption that ε 1, from expression (4.11), the false vacuumcorresponds to φ = −η and V ≈ ε, while the true vacuum is associated to φ = η and V ≈ 0.From (4.11), we can see there is a unique absolute minimum, while the other is just a local one.The form of the potential resembles a Higgs-like potential. In what follows we shall refer to theresults obtained when dealing with the kink in section (3.1.3). In a four -dimensional description,the action is

S =

∫d4x

[1

2∂µφ∂

µφ− V (φ)

]. (4.12)

The sign preceding the timelike part of the d'Alambertian changes according to whether we aredealing with Euclidean or real time.

The scalar eld, φ, is required to t the following properties, [19]:

1. it approaches the false vacuum at Euclidean innity

2. φ is not a constant


3. φ has an action which is less than or equal to that of any solution provided by the rst twocases.

We need to calculate the bounce action in order to determine the tunnelling probability forvacuum decay, since it coincides with the most likely process to occur. The bounce as a solutionin Euclidean time, τ , is characterized by a hyperspherical symmetry, O(4), with four -radius

ρ =√τ2 + ~x 2.

Using imaginary time implies a change in sign in the line element, w.r.t. the Lorentzian metric.Indeed, this allows us to consider the timelike coordinate as being indistinguishable from thespacelike ones, hence determining a hyperspherical symmetry. The Euclidean action may bere-expressed in terms of the new parameter ρ, therefore obtaining

SE = 2π2

∫ ∞0

dρρ3

[1

2(φ′)2 + V

], (4.13)

where the total derivative of the scalar eld is performed w.r.t. the scalar parameter ρ. Theequation of motion associated to (4.13) reads

φ′′ +3

ρφ′ =

dV

dφ. (4.14)

It is possible to have an idea of the analytical structure of the solution going through someapproximations. First of all, if the potential dierence between the two minima is very small,equation (4.14) can be combined with (4.11), hence obtaining

φ′′ +3

ρφ′ ' 2λφ (φ2 − η2). (4.15)

If we subsequently impose the condition ρ η/√λ, equation (4.15) becomes

φ′′ = 2λφ (φ2 − η2),

with solutionφ(ρ) = η tanh

√λ η(ρ− ρ0), (4.16)

where ρ0 indicates the radius of the bubble of false vacuum. This solution clearly resembles thekink, one of the topological defects analyzed in section (3.1.3).

We can thus obtain the bounce action by subtracting from the Euclidean action, associatedto the bubble wall, the background contribution, coming from the false vacuum, hence resultingin [2]

B = IW − IFV =

∫d4x

(1

2φ′2 + V

)− π2

2ερ4

0 = 2π2

∫dρ32V − π2

2ερ4

0 = 2π2ρ30σ −

π2

2ερ4

0,(4.17)

where σ denotes the tension of the wall. The reason for the rst passage in equation (4.17) isthat, the energy loss due to the crossing of the barrier is, in part, compensated by the acquisitionof the energy associated to the false vacuum characterizing the initial conguration. Extremizingthe bounce action with respect to the bubble radius, we get [10]

∂B

∂ρ0= 6π2ρ2

0σ − 2π2ρ30ε = 0 ⇒ ρ0 = 3

σ

ε⇒ B =

27π2

2

σ4

ε3.


From this last expression, we can deduce that the bounce action outlines the competition betweenthe wall's tension and the energy dierence in between the two vacua.

From the particular choice of the double-well potential provided in equation (4.11), the energy-momentum tensor associated to the false vacuum conguration reads

Tµν = φ,µφ,ν − Lgµν = εgµν ,

where the indexes µ and ν refer to, both, space and time coordinates. The partial derivativesof the scalar eld are null, as follows from the fact that we are analyzing the false vacuum only,hence there is no time nor space-dependence for the scalar eld. The only contribution from theLagrangian density is provided by the potential evaluated at φ = φFV , i.e. the eld congurationin false vacuum. At rst approximation, its value is V ≈ ε. As a consequence, we are left withan expression resembling de Sitter spacetime2, which is, by denition characterized by a positivecosmological constant, Λ = 3/l28πGε, where l stands for the horizon radius. This proves that thefalse vacuum is not an absolute minimum of the potential, but rather a local one, otherwise theenergy-momentum tensor would have been null. On the other hand, at spacetime is associatedto the true vacuum.

When gravity comes into play, things may look more complicated, because, in addition tothe scalar eld, we also need to consider the ten independent components of the metric tensor.Nevertheless, it is possible to assume that adding gravity does not break the symmetry of theproblem, [19], implying that the bounce is invariant under four -dimensional rotations. From thisassumption, we can deduce that the Euclidean metric has to be rotationally invariant. Such anexample is provided in the following discussion.

Passing from Field Theory to Gravity, requires the addition of a cosmological constant, whichis related to the false vacuum potential via the relation Λ ≈ V (φFV )/8πG. The backgroundspacetime, with respect to the nucleation process can either be de Sitter, Minkowski or Anti desitter, and the same goes for the true vacuum: we need a cosmological constant gradient in orderto make the tunnelling process possible.

Coleman and De Luccia, have been the rst to analyze the bounce action in presence of gravityfor the transition from de Sitter to Minkowski spacetime. Associating the false vacuum energy, ε,

to the de Sitter cosmological constant, Λ+ = 8πGε, with cosmological radius l =√

38πGε , and the

true vacuum energy (Λ− = 0) to Minkowski spacetime, we can use the Einstein-Hilbert actionto calculate the transition probability between the two states.

The true vacuum conguration can be described by a metric of the form

ds2 = dρ2 + ρ2d2ΩIII ,

which basically resembles a hypersphere, where ρ parameterizes its radius, while ΩIII is the solidangle in three dimensions, with outward -pointing normal ∂/∂ρ, which allows us to calculate theextrinsic curvature as follows [2]

K−αβ = −Γραβ =1

ρ0gαβ , (4.18)

where ρ0 is the radius of the bubble of false vacuum, at the time of nucleation, while the de−Sittervacuum, is of the form

ds2 = l2(dχ2 + sin2 χdΩ2III), (4.19)

2In de Sitter space, the energy-momentum tensor is directly proportional to the metric.


describing an isotropically expanding universe. l indicates the horizon radius, hence, it is avariable, χ is a dimensionless parameter, while ΩIII is dened analogously to the former lineelement. We indicate the wall's position with the parameter χ0; the normal to the wall is takento be ∂

∂(lχ0) , therefore, its associated extrinsic curvature becomes [2]

K+αβ = −1

lΓχαβ =

cotχ0

lgαβ . (4.20)

From Israel's junction conditions, we know that the induced metrics on both sides of the wallhave to match. We also need the radial coordinates to be related via the following expression

ρ0 = l sinχ0;

the reason for such requirement is that it allows for the two vacuum bubbles to match at t = τ = 0,i.e. at the time of nucleation, where t indicates time in the Lorentzian metric. The jump inextrinsic curvature is proportional to the surface stress-energy tensor, hence, combining (4.18)and (4.20), we get

K+αβ −K−αβ = −2 cotχ0

l+

2

ρ0= 8πGSab ⇒ 1− cosχ0 = 4πGσρ0,

where the last result constitutes a constraint equation for the size of the bubble, ρ0, indicatingthat it is set by the surface tension, σ, of the wall.

Computing the Einstein-Hilbert action and performing background subtraction just as pre-viously done for the pure eld theory case, the bounce becomes

B = −∫χ<χ0

R+ 2Λ

16πG+

∫W

(∆K

8πG+ σ

)where the domain of the bulk integration is just the interior of the bubble: the outer part cancelso with the term accounting for background contribution. In a 4D de Sitter space, the scalarcurvature is R = 2nΛ/(n− 2) = 4Λ, therefore the bounce becomes [2]

BCDL =2Λl4

16πG

∫ χ0

0

dχ 2π2 sin3 χ0 +

(−3σ

2+ σ

)2π2ρ3

0

= 2πεl4[

1

3cos3 χ0 − cosχ0 +

2

3

]− π2ρ3

0σ

=2l2

4G[cos3 χ0 − 3 cosχ0 + 2− sin2 χ0(1− cosχ0)] =

πl2

4G(1− cosχ0)2

=πl2

G

(4πGσl)4

[1 + (4πGσl)2]2⇒ lim

G→0BCDL ∝

σ4

ε3(4.21)

where taking the limit for G→ 0 allows to recover the eld theory result from Coleman-de Luc-cia's model, therefore proving it to be an adequate method for describing spacetime nucleation.

4.3 Adding Conical Decits

We have already proved in section 3.1 that conical decit regularization leads to a contributionto action computation which is proportional to the area of the decit. We can now use this to

4.3. ADDING CONICAL DEFICITS 81

express the Euclidean action associated to a bubble wall. Recalling the Ricci scalar decomposition,given by

R = 3R+ ε(K2 +KabKab) + 2ε(nα;βnβ − nαnβ;β);α, (4.22)

the euclidean action becomes [12]

I± = − 1

16πG

∫ β

0

dτ

∫Στ

(3R−K2 +KabKab)−1

8πG

∫W

K±

− 1

8πG

∫W

K± +1

8πG

∫W

(nα;βnβ − nαnβ;β);α (4.23)

where with Στ we denote the spacelike hypersurface at constant time τ , W is the wall, whilenα = ±(τ dr− rdτ) are the normals to the wall. The total derivatives featuring in the denitionof the normals is performed w.r.t. proper time of an observer co-moving with the wall.

Israel's junction conditions imply that

[K] = K+−K− = −4πGσ, (4.24)

where σ is the wall's surface tension, where, according to the thin wall approximation, the tracesof extrinsic curvature must be equal on both sides of the wall, therefore implying that K+ = K−.Assuming that the timelike vector led normal to the wall is a Killing vector, the rst integrandvanishes3. The last term, instead, may be rewritten in terms of the surface gravity, κ. In orderto do so, we briey explain what surface gravity consists of. We start considering a body withunit mas; then, we consider the line element for a static spherically symmetric spacetime, whichreads

ds2 = −fdt2 + f−1dr2 + r2dΩ2. (4.25)

The body is located at a certain radial distance r from a black hole described by the given metric,and is kept stationary by an observer located at innity. Its four -velocity and four -accelerationresult from (4.25)

uα = f(r)−1/2tα ⇒ aα(r) = uα;βuβ =

1

2f(r)−1/2f ′(r),

where, for the symmetry of the problem, only the radial components of the four -vector eldsare relevant. The work done to keep the body in place is converted into radiation and reachesthe observer at innity after being redshifted by a scale factor f1/2. For energy conservation tohold, the two observables (i.e. the work done and the detected radiation) have to match, henceobtaining

a∞(r) = f1/2a(r) =1

2f ′(r)⇒ a∞(r+) =

1

2f ′(r+), (4.26)

where with r+ we denote the location of the conical decit, in this case the black hole eventhorizon, while a∞ is the force exerted at innity to held the body on the event horizon. Recallingthe result obtained in the previous chapter, the roots of the metric functions, i.e. f , allow usto determine the location of the event horizon. According to the particular cosmological model

3We can see this by using the Hamiltonian description given in terms of the wall's induced metric and conjugatemomenta, together with the Hamiltonian and momentum constraints.


that is being used, they can either be one or two. They contribute with an additional term whenperforming action computation. For the de Sitter case, the nal result for (4.23) is

I = − 1

4G(Ah +Ac)−

1

4

∫W

σ − 1

16πG

∫W

(f ′+τ+ − f ′−τ−), (4.27)

where Ac and Ah constitute the surface areas of the conical decits at the event and cosmologicalhorizons, respectively.

4.4 Primordial black holes for vacuum decay catalysis

The CDL instanton describes the tunnelling between two spacetimes, characterized by dieringcosmological constants, where no inhomogeneity is taken under consideration.

We are now going to deal with the transition between two Schwarzschild-de Sitter spacetimes.There are two conical singularities, at the event and cosmological horizons. We will show thatthe addition of their contribution has the eect of lowering the bounce action, hence leading toa higher tunnelling probability for the false vacuum, proving that impurities act as nucleationsources.

The aim of this section is to provide an insight regarding the analysis of vacuum bubbleevolution. With the aim of adding the contribution arising from topological defects, we will relyupon the complementary Euclidean and Lorentzian description of the same phenomenon. Asalready mentioned, the passage in between the two occurs with Wick rotation of Lorentziantime. In what follows, we will also rely upon the dynamics of thin shells, which in turn nds itsbasis in the Israel junction conditions. This approach reduces the problem to the study of a one-dimensional parameter, R, i.e. the vacuum bubble's radius, which is a function of proper time,λ, i.e. the time coordinate associated to an observer co-moving with the wall. The evolutionequation governing this parameter, resembles that of a particle in a one dimensional potentialwell, where the potential, V is in turn a function of the radial coordinate R. Starting from theinduced metric on the bubble surface, we calculate the components of the trace of its extrinsiccurvature; from spherical symmetry, we obtain two expressions associated to the surface energydensity, σ, and surface pressure, P , i.e. the trace components of the stress-energy tensor Sab.Form the former, we can extract the evolution equation for the radial parameter R in terms ofλ: we gather all the other terms and dene the potential V (R). The resulting equation is in theLorentzian description. In order to get its Euclidean counterpart, we need to Wick -rotate propertime; indeed, from special relativity, we know that λ ad t are directly proportional and also thatthey share the same sign. From this, the evolution equation ips the sign preceding the termR2. To get a better idea of what is actually happening, we could equivalently assert that it looksas if the potential has changed sign. According to the example provided in section (4.2), we passfrom a potential well to a potential barrier. Since both, R and V can be thought of as beingtwo-parameter functions, there is a certain gauge in the choice of the set of parameters. But,we can also notice that, the variation of either of them can lead to a lowering of the potential,which means that the range of allowed R values, i.e. the radius of the nucleating vacuumbubble, widens. As a consequence of the intuitive ip in sign of the potential, for each Euclideansolution, there are two Lorentzian solutions. The minimum of the potential has now turned intoits maximum. This means, that, the bubble radius associated to this particular potential value,constitutes a critical point, separating the transmitting from the bouncing-o solutions, whichare, thus associated to a Euclidean (i.e. barrier-penetrating) solution.

In conclusion to this brief overview of what follows, we can appreciate the following require-ment: the metric functions featuring in the line elements of either bulks must have at least one

4.4. PRIMORDIAL BLACK HOLES FOR VACUUM DECAY CATALYSIS 83

positive root. This case, actually coincides with the aforementioned critical conguration, whilethe case of two positive roots means we can identify an event and cosmological horizon. This alsoexplains why there are two Lorentz solutions for each Euclidean one: there is a bounce for eachhorizon. The two parameters we have set for dening the potential are such that they resemblethe remnant and mass-dierence between two bulk Scwarzschild black holes. By determiningthe critical radius, R∗, we get the corresponding relic black hole mass, which provides an upperboundary for such parameter. The reason for this is provided by the denition of surface tension.Indeed, when the radius reaches its maximum, the remnant mass is such that the overall tensionresults negative, hence ceasing the bubble's expansion and ultimately leading to its re-collapse.On both sides of the wall, the Lorentzian metric reads [13]

ds2 = −f±(r)dt2± +dr2±

f±(r)+ r2dΩ2

II , (4.28)

4 where the metric functions are taken as [12]

f±(r) = 1− 2GM

r− Λ±r

2

3= 1− 2GM

r− r2

l2; l2 =

3

8πGε. (4.29)

In equation (4.28) ε represents the vacuum energy density, while Λ± stand for the inner (-)and outer (+) cosmological constants, respectively. In order for a tunnelling process to occur,the transition should be between a higher to a lower cosmological constant, hence leading to alower vacuum energy content. In the line element reported above, we have added the index ±in order to explicit the coordinates that actually vary when crossing the shell. We can thus seethat the two-sphere is equivalently described on both sides. The radial coordinate indicates thedisplacement of the bubble's surface w.r.t. the roots of the metric functions. For a consistentdescription of the wall's position, relying upon the spherical symmetry of the problem, we candene the following parameter R(λ) = r+ = r−, where λ stands for proper time w.r.t. thebubble's surface.

For the case we are interested in, vacuum decay can be considered as the tunnelling betweentwo Schwarzschild bulks diering in terms of, both, their cosmological constants, and associatedmass, separated by a thin wall of constant tension and null surface pressure.

Under these hypotheses, it is enough to examine the evolution of the system in terms of twocoordinates, t and r, being the angular coordinates analogue for both parts.

First of all, we need to outline the main geometrical quantities accounting for a completedescription of such a parametrization.

In the thin wall approximation, the trajectory of the wall is described by local coordinates

xa± = (t±(λ), r±(λ), θ, φ)

where λ is the wall's proper time. The intrinsic metric results as

ds2 = −dλ2 +R2(λ)[dθ2 + sin2 θdφ2].

We also dene the normal on both sides of the wall, as [12]

n± = −r±dτ± + τ±dr±

both pointing towards increasing values of r, where the derivation is done w.r.t. λ from whichwe can compute their associated extrinsic curvatures

K±ab = eα±,aeβ±,b∇αn±β ,

4τ± is the Euclidean time obtained by Wick -rotating Lorentzian-time, t = −iτ .


where indexes a and b range in the parameter values λ, θ, φ. From Israel junction conditions weget [12]

f+t+ − f+t+ = −4πGσ. (4.30)

In order to make some qualitative analysis regarding vacuum tunnelling, we rely upon the for-malism of the dynamics of thin shells; in doing so, it turns out useful to work in terms ofthe discontinuity of the extrinsic curvature, that we introduced for the deriving the junctionconditions. Recalling the following notation

[Kab] = K+ab −K−ab, (4.31)

where the ± signs have been added to indicate the properties of the two separate vacua, withthe same convention as before. In terms of (4.31) we can rewrite the expression for the surfacestress-energy tensor derived in section (3.1.4) as

Sab = − 1

8π([Kab]− [K]hab),

which are known as the Lanczos equations, where hab is the shell's intrinsic metric. Since we aredealing with a spherically symmetric system, the discontinuity parameters (4.31) are all diagonalterms, hence enabling to write the associated tensor as [Kab] = diag([Kλλ], [Kθθ], [Kθθ]), wherethe last two components are equivalent on the two-sphere. For the metric (4.28), the diagonalterms are [13]

[Kλλ] =d

dλ

√f+ + R2

R−

√f− + R2

R

[Kθθ] =

√f+ + R2

R−

√f− + R2

R.

The stress-energy tensor is, in turn, diagonal

Sab = diag(−σ, P, P ),

where σ is the wall's tension, as previously remarked, while P is the surface pressure. They aredened by the following expressions [13]

σ = − 1

4πG[Kθθ] P =

1

8πG[Kθθ] +

1

8πG[Kλλ].

From the surface tension, we get√f+ + R2 =

√f− + R2 − 4πGσR, (4.32)

from which we can obtain the shell's equation of motion

R2 + V (R) = 0, (4.33)

where V (R) contains all terms other than R obtained by evaluating the square of equation (4.32).Hence, (4.33) resembles the motion of a particle in a one-dimensional potential well.

From the above, together with the denition we have provided for the metric functions, wecan extract the explicit evolution equation of R as follows, [12]

R2 = 1−(

4π2G2σ2 +Λ+ + Λ2

−16π2G2σ2

)R2

−2G

R

(M+ +M−

2+

(M+ −M−)(Λ+ − Λ−)

96π2G2σ2

)− (G(M+ −M−))2

16π2G2σ2R4, (4.34)


where, we recall that derivation is performed w.r.t. λ. Now, we introduce a set of parameters,that enable to write more simply equation (4.34), [12]

`2 =3

∆Λ; γ =

16π2G2σ2`2

1 + 16π2G2σ2`2; α2 = 1 +

Λ−γ2

3

κ1 =2αGM−

γ+

(1− α)αG∆M

2πGσγ2; κ2 =

α2G∆M

4πGσγ2. (4.35)

We can rewrite (4.34) as

R′2 = 1−(R+

κ2

R2

)− κ1

R= −V (R), (4.36)

where we have rescaled the radial coordinate as R = αR/γ, τ = ατ/γ, λ = αλ/γ. We are nowable to study the evolution of the bubble wall in terms of two parameters (κ1, κ2). Their valuesare determined by three conditions, two of which are linked to the evolution equations of thevariables R and τ± and one to the positivity of M±. Later, we will refer to them as the seed(+), and remnant (-) mass, respectively. In particular, the solution to the evolution equationof the radial parameter, R, depends only on κ1 and κ2 and that the shell's tension is linked tothe positivity of time through the junction conditions. As proved after performing numericalsimulation, [14], at the level of the potential V , we could increase either parameter and thepotential would still be lowered, meaning that the range of possible R values would be wider.The critical value for R is obtained by solving the system provided by the following equations5

V = V ′ = 0, (4.37)

which means evaluating the bubble's radius associated to the maximum (minimum) in the Eu-clidean (Lorentzian) description. By analyzing the trace components of the stress-energy tensor,σ and P , we can deduce that, if M− < M+, then σ > 0, hence σ+P > 0, which in turn satisesthe null energy condition. It also satises the strong and weak energy conditions. On the otherhand, if M− > M+, then σ < 0, and the surface pressure would be negative, hence violating theenergy conditions. Since σ is determined by the evolution equation of the bubble's radius, weneed to determine its value in the static case, i.e. R∗. We will not go into a detailed calculation,but only provide a qualitative analysis of some basic features. We have already stated thatequation (4.33) resembles a particle subject to a one-dimensional potential. Performing a Taylorexpansion around the static point R∗, at rst approximation, we would be left with the termcontaining the second-order total derivative of V . Now, according to its sign, we could eitherhave an unstable or a stable conguration.

In order for the Euclidean bubble to be a regular instanton, there are constraints we have tomake on κ1 and κ2, which in turn dictate the evolution equation of the bubble. They shouldallow for a range of radial parameters R such that V > 0 and also provide the positivity of thearrow of time, hence satisfying the aforementioned energy conditions, which, in turn, providesa lower boundary for the parameter value κ1, since it is directly linked to the remnant mass.Solving the system (4.37), leads us to two polinomial constraints

R6∗ −

(κ2 +

κ1

2

)R3∗ − 2κ2

2 = 0 ; R3∗ −

2

3R∗ + κ2 +

κ1

2= 0.

5The rst derivative of the potential is performed w.r.t. the radial parameter R.


From the latter we can extract one of the parameters,

κ1 = κ∗1 = −2κ2 +4

3R∗ − 2R3

∗, (4.38)

where κ∗1 denotes the parameter associated to the critical solution R∗, while the former, i.e. thepolinome of sixth degree in R∗, can be solved as a polinome of second degree in R3, of which wecan extract the cubic root, therefore obtaining

R∗ =

[−κ2

2− κ1

4+

√1

4

(κ2 +

κ1

2

)2

− 8

272

]1/3

. (4.39)

Substituting this result in equation (4.38), and solving it with respect to κ1, the maximum6

parameter value corresponding to the peak of the potential is [12]

κ∗1 = −2κ2 +2

9[1 + 81κ2

2 −(−1− 5(27κ2)2 +

(27κ2)4

2+

27κ2

2(4 + (27κ2)2)3/2

)1/3

+

(1 + 5(7κ2)2 − (27κ2)4

2+

27κ2

2(4 + (27κ2)2)3/2

)1/3

]1/2 (4.40)

On the other hand, to get the minimum value for κ1, the other two conditions must be takeninto account. Starting from the black holes' masses M±, they may be written in terms of thepreviously dened parameters (4.35), as

GM− =γ

2α

(κ1 − 2

(1− α)

ακ2

)GM+ =

γκ1

2α+κ2γ

α2(α− 1 + 4πGσγ) (4.41)

which, in order to be positively dened, require that

GM− ≥ 0 ⇒ κ1 ≥ 2(1− α)κ2

αGM+ ≥ 0 ⇒ κ1 ≥ 2(α− 1 + 4πGσγ)

κ2

α(4.42)

The evolution equations of the timelike parameter, on both sides of the wall are, are obtainedimposing the condition of positivity of the arrow of time, i.e. τ± ≥ 07

f+˙τ+ =

κ2

R2+R

α(1− 4πGσγ) ≥ 0 ⇒ R+ =

(ακ2

4πGσγ − 1

)1/3

. (4.43)

A lower bound for κ2 is obtained from the condition M− = 0, i.e. no remnant, which is obtainedwhen ακ∗1(κ2) = 2(1 − α)κ2, while the lower bound occurs when the range of κ1 closes o forκ2 < 0, i.e. at a static point of the potential V (R+) = V ′(R+) = 0, [14].

In the second part of this section, we calculate the bounce action by subtracting the backgroundcontribution to the Euclidean action, given by, [12]

IE = − 1

16πG

∫M+

√g(R+ − 2Λ+)− 1

16πG

∫M−

√g(R− − 2Λ−)

+1

8πG

∫∂M+

√hK+ −

1

8πG

∫∂M−

√hK− +

∫W

√hσ, (4.44)

6This is because, when the bubble radius reaches its critical value, σ decreases, hence the bubble contracts,hence lowering the value of the remnant mass.

7This allows for the surface tension of the wall to be positive, as a consequence of the Israel junction conditions,relating the evolution of the time coordinate on either side with , which leads to the following


where M± denote the two vacuum bulks, while ∂M± and W indicate the boundaries of the bulksand the bubble wall, respectively. With g and h we denote the determinants of the bulk andboundary metrics, respectively, while R± and K± are the Ricci scalars and extrinsic curvaturetraces, respectively. For both sides, we have R± = 4Λ±, since we are dealing with two de Sittervacuum solutions. 8 Considering one at a time the bulk terms and the wall, we have

1. The inner bulk term, M−, associated to the true vacuum, resulting in

IM− = −A−4G

+1

4G

∫dλR2f ′−τ−, (4.45)

where R denotes the position of the wall9, while A− indicates the area of the event horizonof the inner black hole. The second term on the right is an integration w.r.t. proper time,which emerges after performing the integration with respect to the intrinsic angular coor-dinates, i.e. two of the three bubble's intrinsic coordinates. Being the system sphericallysymmetric (de-Sitter), then the solid angle is 4πR2. We also recall that the derivative ofthe metric function is taken w.r.t. the radial parameter r−, while the derivative of theEuclidean time is done w.r.t. λ.

2. The outer bulk term M+ requires extra care. According to whether the false vacuum isof dS, Schwarzschild or AdS -type, we need to subtract dierent background terms. In theformer case, one of the roots of equation (4.29) is the cosmological horizon with a naturalhorizon periodicity10, leading to the Euclidean contribution

IM+ = −Ac4G− 1

4G

∫dλR2f ′+τ+, (4.46)

where Ac indicates the area of the cosmological horizon. In this case, the backgroundcontribution is just

IM+0= −Ac

4G− A+

4G, (4.47)

where A+ denotes the event horizon of the outer black hole. For the other two cases, wehave to deal with a radial coordinate extending to innity. To prevent the action fromdiverging, we use the result obtained in the appendix. We basically x an arbitrary radiusr = r0 and calculate the Gibbons-Hawking boundary term associated to it. The subtractionterm is

IM+0 =1

8πG

∫r=r0

√h(K −K0) =

βM+

2= βM+ −

1

4G

∫dλf ′+R

2τ+ (4.48)

where β = 8πGM+ is the Euclidean periodic time computed in the Appendix, while K0 isthe background curvature).

3. The wall termW has two contributions, arising from the curvatures and the surface tension.Using Israel's junction conditions, we can rewrite the integrands in terms of the wall's

8We also notice the presence of the cosmological constants in the Einstein-Hilbert actions; in chapter (2.1),we provided the denition of the Einstein-Hilbert action in vacuo. When adding Λ±, the eld equations resultRµν − 1/2Rgµν + Λ±gµν = 8πGTµν . They can be derived by applying the variational principle on the moregeneral action S = (1/16πG)

∫M±

√g(R± − 2Λ±)

9Not to be confused with the Ricci scalars, reason why we have added the ± signs.10This follows from the denition of conical decits, provided in the previous chapter.


tension as follows,

IW = ± 1

8πG

∫∂M±

√hK +

∫W

√hσ = −

∫W

√hσ

2=

1

2G

∫dλR(f+τ+ − f+τ+), (4.49)

where σ is the tension of the wall.

Subtracting the background contributions, M+0, to the euclidean action, we get the tunnellingaction as, [12]

B =A+

4G− A−

4G+

1

4G

∮dλ[(2Rf+ −R2f ′+)τ+ − (2Rf− −R2f ′−)τ−], (4.50)

where Ac has vanished after performing background subtraction between equations (4.46) and(4.47). We also notice that the bounce action is independent from either cosmological constantsand time periodicity. This follows from the junction conditions for the metric functions; sincespherical symmetry is preserved, the same goes for the bubble's surface tension. On the otherhand, for their derivative, we can argue that, since the motion is parametrized by proper timew.r.t. an observer co-moving with the wall, we can dene a timelike Killing vector eld, tµ,which is also normal to the bubble's surface. By denition, there exists a scalar eld, κ, suchthat (−tµtµ);α = 2κtα. Such scalar eld is exactly surface gravity, which, for the case of aSchwarzschild black hole, is κ± = 1/4M±, [1], hence is independent from the cosmologicalconstant. The important thing is that the bounce action depends on the area of the conicaldecits. This means that the decay probability of the false vacuum is aected by the presenceof conical singularities.

In what follows, we will compare the bounce solution to the CDL-action, obtained in theabsence of black holes. The CDL bubble wall satises the radial and timelike evolution equationswith the solution we have previously found, and can be recovered from the general formulationby setting the parameter values κ1 = κ2 = 0.

By solving numerically the wall's trajectories, it was found [14] that the bounce action in-creases as κ2 drops, hence meaning that, the preferred instanton is the one with a bigger dierencein between the seed and remnant mass.

Furthermore, the ratio B/BCDL does not vary with respect to a change in the cosmologicalconstant, Λ+. The only solutions that can be analytically computed are the static and critical11

bubble solution, which are, respectively, are, [14]

Bs =A+

4G− A−

4G; Bc = 4πGM2

c ∝π`2

G(σ`)6 ∝ (σ`)2BCDL;

they can be obtained from the general expression (4.50) without considering the evolution termfor the rst case, and setting the seed mass equal to the critical one for the latter. The criticalmass is dened as the value of the seed mass, M+, such that there is no remnant, i.e. M− = 0,hence the black hole leaves behind an empty vacuum. This parameterizes the point at which thebounce action reaches its minimum, hence corresponds to the maximum tunnelling probability.IfM+ > Mc, then the dominant process will be a static bubble with a remnant black hole, while,for M+ < Mc, there will be no remnant.

It is not possible to determine analytically which is the dominating bubble solution. Numeri-cal simulations, [12], with dierent samples ofM− have proved that the static bubble contributesthe most, while the critical one dominates for M− = 0.

11By varying the value of the surface tension, it has been possible to determine a critical mass, preciselyGMc = 2

27M2pσγ2.


Application to the Higgs potential

One of the most well known double-well potentials is the one associated to the Higgs eld. It isan interesting example, that allows to see how the presence of primordial black holes could leadto vacuum decay in the context of the Standard Model, hence indicating that the universe couldbe in a metastable state.

The main features of such conguration are

1. the barrier height h = φTVMp

2. the separation between the minima ε = −V (φTV ) where φTV is the eld congurationassociated to the true vacuum

3. the energy of the true vacuum, ζ.

We recall that the analysis of vacuum decay was based on the thin wall approximation, meaningthat the energy dierence is very low. When dealing with phase transitions in the bulk ina-tion model, we need to add to the scalar potential some corrective terms, arising from scalarperturbations.

At rst, we need the back-reaction parameter, dened as σ`, which is obtained from theparameters characterizing the eld as follows, [14]

1

4M2p

σ` =

√3

4κ h

(ζ

ε

)1/2

,

from which the CDL action turns out to be

BCDL =27π2σ4

2ε3(

1− 14M4

pσ2`2

) =27κ4π2

2

(h4M4

p

ε

)(ζ

ε

)2(1− 1

4M4p

σ2`2)−2

.

Studying the value of the CDL action after varying the parameter values of the eld, has shownthat the timescales for vacuum decay are much longer than the age of the universe. Once weintroduce a seed black hole, with critical mass, obtained by knowing the back-reaction parameter,[14]

Mc ≈128

27

`

G

(1− 1

4M2p

σ`

)3

=16

3πκ3

(h4M4

p

ε

)1/2 (ζε3/2Mp

),

its corresponding nucleation rate becomes Bc = 1/2(MC/Mp)2. According to the relation be-

tween the critical and the Planck mass, the tunnelling probability either drops or enhances. Inparticular, for Mc < Mp, the tunnelling process is accelerated, hence proving that nucleation oftrue vacuum requires the existence of light or primordial black holes. At his point we have to besure that the evaporation, in the form of Hawking radiation, does not overcome the tunnellingprocess. Hence, we have to compare the two decay rates. As it was studied by R. Page, theHawking evaporation process is related the black hole's mass. Hence as the black hole radiates,the mass will reach a critical value, that will enable the black hole to catalyze vacuum decay.

Hawking Radiation and Gravitons

In a braneworld scenario, we have shown that bulk uctuations lead to the birth of tidal -chargedblack holes on the brane and how they constitute a valuable proposal for primordial black holes.


At this point is seems reasonable to question how they interact with the brane after their forma-tion, [15]-[17].

We have also presented that the brane/domain wall conguration is a topological defect.Hence, we can think of braneworld black holes as a valuable setting for testing the mutualinteraction between dierent topological defects; an example is indeed provided by the falsevacuum decay we have been discussing.

As opposed to tunnelling processes, describing the dynamics of their intersections, black holesare also characterized by two emission channels in the form of radiation on the brane or gravitons,[16], that, unlike photons, are able to travel through the bulk, resulting in an apparent violationof energy conservation. Obviously, this is not so because of the contribution of the Weyl tensorwe have discussed about in the previous chapter.

Being the majority of such emissions in the form of gravitons, we would expect the blackhole to acquire a momentum component along the normal direction w.r.t. the brane, eventuallyescaping in the bulk. Such process is called recoiling, [16]-[17].

Chapter 5

Conclusion

Through the wide range of topics we have been discussing in this work, we have tried to outlinesome of the most recent developments and applications of gravity in, both, theoretical physicsand cosmology. The introduction of the braneworld model for the understanding of processessuch as ination and the existence of primordial black holes, seeds the request for a betterunderstanding of the dynamical evolution of the universe. The idea that gravity coincides withgeometry permeates the topics we have been dealing with. Spontaneous symmetry breakingin translational invariance and conical decits are two examples of the concreteness of suchfundamental interaction, that beautifully reconciles with what appears to be at the boundary ofthe most abstract and unconceivable formulations.

91

92 CHAPTER 5. CONCLUSION

Appendix A

Background contribution to

Euclidean action

In this appendix, we provide the computation of the Euclidean action associated to a Schwarzschildsolution. We shall consider the background contribution in order to prevent a divergence to arisewhen considering the domain of integration for the Gibbons-Hawking boundary term. This resultis essential for computing the bounce action in the last chapter. For a Euclidean Schwarzscildsolution, i.e. the one obtained after Wick rotation, the line element is of the form

ds2 =

(1− 2GM

r

)dτ2 +

(1− 2GM

r

)−1

dr2 + r2dΩ2II . (A.1)

In the Lorentzian case, the tt and rr components of the metric have opposite singes, allowingfor t and r to swap their roles outside and inside the event horizon; instead, for the EuclideanSchwarzschild case, crossing the event horizon does not preserve the signature of the metric.Hence, the manifold is required to have a boundary at r = 2GM. Considering the rst two termsof (A.1) (

1− 2GM

r

)dτ2 +

(1− 2GM

r

)−1

dr2, (A.2)

and then introducing a new variable ρ, dened as ρ2 = λ(r − 2GM), where λ is a constant ofproportionality dierentiating, leads to 2ρdρ = λdr. Hence, the radial part of the metric becomes

dρ2 =λ2dr2

4ρ2=

λdr2

4(r − 2GM);

by setting λ = 8GM , we can see how dρ2 can adequately replace the second term in (A.2), whilethe timelike part of the metric results

gtt =

(1− 2GM

r

)→ ρ2

8GM

1

2GM.

Now, (A.2) may be written in terms of the new variable

ρ2d( τ

4GM

)2

+ dρ2,

93

94 APPENDIX A. BACKGROUND CONTRIBUTION TO EUCLIDEAN ACTION

that we may consider as a polar coordinate system, where τ4GM constitutes an angular coordinate.

Dening a new variable as

χ =τ

4GM

and, requiring periodicity

χ+ 2π = χ,

we get τ + 8πGM → τ. In Field Theory, transition amplitudes describing the evolution of asystem from an initial to a nal state at a given temperature, T , are mathematically representedby Green's functions. By making the time variable complex, the function acquires periodicityβ = 1

T , therefore the thermal parameter may be expressed in terms of Euclidean time as

T =1

β=

1

8πGM

for the Schwarzschild solution. According to the particular solution to Einstein's eld equations,time periodicity varies. Nevertheless, we may dene a general formula

β =4π

|V ′+|,

where |V ′+| stands for the rst derivative of the metric function of the tt component of the metricwith respect to the radial parameter featuring in the potential, obtained throughout Taylorexpansion near the horizon.

Combining the notion of Euclidean time with the Hamiltonian denition for mass, it is possibleto determine the Euclidean action I in vacuum spacetime starting from a given metric. The eventhorizon of a black hole is a 2-sphere and we can consider it as a boundary of the manifold M ,∂M . The extrinsic curvature of ∂M can, thus, be computed from its metric

ds2 =

(1− 2GM

R

)dτ2 +R2dΩ2

II , (A.3)

where r = R indicates the distance of the manifold's boundary from the origin of the coordinatesystem. Its normal n points inwards , therefore

n = −√

1− 2GM

R

∂

∂r,

from which we obtain extrinsic curvature as

K = ∇ana =1

r2∂rr

2nr =2

rnr +

GMnr

r(r − 2GM). (A.4)

The volume element in the Gibbons-Hawking boundary term is

√h =

√1− 2GM

rR2 sin θ, (A.5)

where R2 sin θ constitutes the contribution from the angular part of the metric. For vacuumsolutions, the Ricci scalar vanishes and, as a consequence, the Einstein-Hilbert action too. Hence,

95

the only contribution we are left to deal with for computing the Euclidean action arises from theboundary term. By using (A.4) and (A.5)

I =

∫∂V

d3xK√h = −4πβR2

(√1− 2GM

R

)2 [2

R+

GM

R(R− 2GM)

]= −4πβ(2R− 3GM)→ +∞ as R → +∞, (A.6)

where β stands for time periodicity in the euclidean metric (A.3).1 The presence of a divergencemight be accepted from a mathematical point of view, but it makes no sense for physics. Inorder to obtain a nite action, we need to perform background subtraction. In the Euclideanpicture, we can think of at vacuum space as a cylinder over which the length of the time circle,T , is required to be constant, since it has constant temperature. The resulting metric associatedto background spacetime thus turns out to be

ds2 =

(1− 2GM

R

)dT 2 +R2dΩ2

II . (A.7)

It is important to notice that, for the background,(1− 2GM

R

)is a constant.

For the two metrics (A.3) and (A.7) to match, we need the boundary metric to be the same,hence requiring for the lengths of the time circles (i.e. their temperatures) to converge on theboundary.

For the background, extrinsic curvature and the normal are K0 = − 2R , and n = − ∂

∂r ,respectively. As a consequence, we obtain

I0 =

∫∂V

d3xK0

√h = −4πβR2

√1− 2GM

R

2

R= −4πβ2R

(1− 2GM

R

)⇒ I0 ≈ −4πβ[2R− 2GM ],

where we have used binomial expansion in the last passage. As it was expected to be, theeuclidean background action also diverges as R→ +∞. Subtracting the two contributions,

I − I0 =1

8πG[−4πβ(−GM)] =

βM

2, (A.8)

we obtain an eective nite action.

1We have renamed the boundary by ∂M = ∂V , in order to be consistent with the notation in chapter (??).

96 APPENDIX A. BACKGROUND CONTRIBUTION TO EUCLIDEAN ACTION

Appendix B

Conformal Transformations

A conformal transformation is a transformation realized by a non-zero dierentiable function Ω,such that gαβ = Ω2gαβ . In this appendix we outline the key features of these transformations,that we have used in chapter (3.1) when obtaining the Einstein-Hilbert action from its higher-dimensional counterpart. The results obtained may also be used when dealing with AdS/CFTcorrespondence correspondence, and in many other recent formulations of gravity theories.

Our goal is to nd the relation between the Ricci scalars of the two metrics and the conformalfactor. In order to do so we just have to go through the computation of the Christoel symbolsat rst, and then evaluate the Riemann tensor. Hence, we get

Γabc =1

2gae(geb,c + gec,b − gbc,e) =

1

2Ω−2gae(Ω2geb),c + (Ω2gec),b − (Ω2gbc),e)

=1

2Ω−2gae

(∂Ω2

∂xcgeb + Ω2geb,c +

∂Ω2

∂xbgec + Ω2geb,c −

∂Ω2

∂xegbc − Ω2gbc,e

).

We can notice that three of the terms that appear in the last line can be combined in theChristoel symbols of the g metric ; rearranging the terms we get

Γabc = Γabc + Ω−1

(δab∂Ω

∂xc+ δac

∂Ω

∂xb− gaegbc

∂Ω

∂xe

).

The next step is to calculate the derivatives of the connection symbols and then combine themwith the above in the expression for the Riemann tensor

Rabcd = ∂cΓabd − ∂dΓabc + ΓaecΓ

edb − ΓaedΓ

ecb;

as a result we get

Rabcd = Ω−2Rabcd + δ[a[cΩ

b]d],

where, [18],Ωab = 4Ω−1(Ω−1);beg

ae − 2(Ω−1);c(Ω−1);dg

cdδab .

Then, contracting on the rst and third index, the Ricci tensor becomes

Rbd = Ω−2Rbd + (n− 2)Ω−1(Ω−1);dcgbc − 1

n− 2Ω−n(Ωn−2);acg

acδbd;

n indicates the number of dimensions characterizing a given manifold. The Ricci scalar, can thusbe expressed as

R = Ω−2R− 2(n− 1)Ω−3Ω;cdgcd − (n− 1)(n− 4)Ω−4Ω;cΩ;dg

cd,

97

98 APPENDIX B. CONFORMAL TRANSFORMATIONS

where the last term is null for n = 4, such as the situation that we considered when we discussedthe Kaluza-Klein model.

Although we do not enter into the details, an important consequence of this result combinedwith the Bianchi identities is that the Weyl tensor is conformally invariant.

Appendix C

Raychaudhuri's Equation

In the study of the physical signicance of curvature we are interested in the eects that curvaturecan have on test particles. One way to study this is to consider geodesics. Another is to studythe eect of curvature on congruences of curves. This is a kinematic description, that can beachieved by introducing a 2-tensor, dened as the covariant derivative of the tangent vector to acongruence of (timelike) curves, u, [1],as

Bαβ = uα;β .

A congruence is a family of curves such that there is a unique one that passes through each pointin spacetime. B actually describes the evolution of the deviation between one given particlemoving along one of the lines of the congruence, and a nearby one, since

ξα;βuβ = uα;βξ

β ⇒ ξα;βuβ = Bαβ ξ

β .

Thus, B measures the failure of ξα to be parallel transported along the congruence. The identi-cation of the vector eld uα allows to dene a transverse metric, hαβ , such that hαβ = gαβ+uαuβ ,where gαβ is the overall spacetime metric. An example of application of Raychaudhuri equationis the dynamical evolution of horizons and singularity theorems. Let us rst dene some quan-tities that will turn out to be useful to provide a physically meaningful decomposition of Bαβ ,[1]. These are:

1. the expansion scalar, θ = Bαα = uα;α

2. symmetric trace-free part, or shear, σµν = B(αβ) − 13θhαβ

3. the antisymmetric part, or rotation, ω = B[αβ].

Hence, we get

Bαβ =1

3θhαβ + σαβ + ωαβ , (C.1)

deriving (C.1), leads to the evolution equation for the expansion scalar θ

Bαβ;µuµ = uα;βνu

µ.

By subsequently applying the Riemann Identity (in the case of an orthogonal metric), the evo-lution equation becomes

Bαβ;µuµ = (uα;βµ −Rανβµuν)uµ = (uα;µu

µ);β − uα;µuµ;β −Rανβµu

µuν

= −BαµBµβ −Rανβµuµuν (C.2)

99

100 APPENDIX C. RAYCHAUDHURI'S EQUATION

from which we derive Raychaudhuri's equation for a congruence of timelike geodesics

dθ

dτ= −1

3θ2 − σαβσαβ + ωαβωαβ −Rαβuαuβ ,

where τ is proper time of an observer moving along a given geodesic belonging to the congruence.

Bibliography

[1] E. Poisson. A Relativist's Toolkit (The Mathematics of Black Hole Mechanics). 1st ed., Cam-bridge University Press, 2004;

[2] R. Gregory. Gravitational Physics Course. Perimeter Institute (PIRSA) Academic Year(201472015);

[3] G. Mammadov. ′′Reissner-Nordstroem Metric′′. 4th May 2009;

[4] G.T. Horowitz. Black Holes in Higher Dimensions. 1st ed. Cambridge university Press, 2012;

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[6] T. Shiromitzu, K. Maeda, M. Sasaki. ′′The Einstein Equations on the 3-Brane World.Phys.Rev. D62: 024012, 2000 (arXiv:gr-qc/991007v3. 17 January 2000)

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[10] S. Coleman. ′′Fate of the False Vacuum: Semiclassical Theory′′. Physical Review D, volume15, number, number 10. 15 May 1977

[11] C. Callan, S. Coleman. ′′The fate of the False Vacuum II: First Quantum Corrections′′.HUTP-77/A032;

[12] R. Gregory, I.G. Moss, B. Whithers. ′′Black Holes as Bubble Nucleation Sites′′. JHEP 03(2014) 01

[13] F.S.N. Lobo, P. Crawford. ′′Stability Analysis of Dynamic Thin Shells′′. Class. Quant. Grav.22(2005)4869-4886 (arXiv:gr-qc/0507063)

[14] P. Burda, R. Gregory, I.G. Moss. ′′Vacuum Metastability with Black Holes′′. 10.1007/JHEP08(2015) 114 (arXiv: 1503.07311v1 [hep-th] 25 March 2015)

[15] T. Tanaka. ′′Classical Black Hole Evaporation in Randall-Sundrum Innite Braneworld′′.Progress of Theoretical Physics Supplement No. 148: 307-316, 2003

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[16] V. Frolov, D. Stojkovic. ′′Black Hole Radiation in the Brane World and Recoil Eect′′.Phys.Rev. D 66, 084002 (arXiv: hcp-th/020646v3. 3 August 2002)

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