the study of inhomogeneous cosmologies through spacetime … · 2013-10-11 · of a matched...

The Study of Inhomogeneous Cosmologies Through

Spacetime Matchings

by

Dan Giang

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of PhysicsUniversity of Toronto

Copyright c© 2009 by Dan Giang

Abstract

The Study of Inhomogeneous Cosmologies Through Spacetime Matchings

Dan Giang

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2009

Our universe is inherently inhomogeneous yet it is common in the study of cosmol-

ogy to model our universe after the homogeneous and isotropic Friedmann-Lemaıtre-

Roberson-Walker (FLRW) model. In this thesis spacetime matchings are applied to

investigate more general inhomogeneous cosmologies.

The Cheese Slice universe, constructed from matching together FLRW and Kasner

regions satisfying the Darmois matching conditions, is used as a prime example of an

inhomogeneous cosmology. Some observational consequences of this model are presented.

The lookback time verses redshift relation is calculated using a numerical algorithm and

it is shown that the relative thickness of the Kasner regions have the greatest impact on

anisotropies an observer would see. The number of layers and distribution of layers play

a smaller role in this regard. The relative thickness of the Kasner slice should be on the

order of one ten thousandth the thickness of the FLRW regions to have the anisotropies

fall within the observed CMB limit.

The approach to the singularity of a spacetime matching is examined. A criterion

is presented for a matched spacetime to be considered Asymptotically Velocity Term

Dominated (AVTD). Both sides of the matching must be AVTD and each leaf of the

respective foliations mush match as well. It is demonstrated that the open and flat

Cheese Slice universe are both AVTD and the singularity is also of AVTD type.

The Cheese Slice model is then examined as a braneworld construction. The possi-

ii

bility of a Cheese Slice brane satisfying all the energy conditions is shown. However, the

embedding of such a brane into a symmetric bulk is non-trivial. The general embedding

of a matched spacetime into a bulk is investigated using a Taylor series approximation

of the bulk. It is found that the energy-momentum tensor of such a brane cannot have

discrete jumps if the embedding does not have a corner.

A 3+1+1 decomposition of the bulk spacetime is then carried out. With the spacetime

being deconstructed along two preferred timelike hypersurfaces, this becomes a natural

environment to discuss the matching of branes. We find that there are conditions on

the matter content of the branes to be matched if an observer on the brane is to see the

matching surface as a boundary surface with no additional stress energy. Matching more

than two bulks is also examined and shown to allow for more general brane configurations.

iii

Dedication

For grandma.

iv

Acknowledgements

I would like to thank my supervisor, Charles C. Dyer, for his guidance and financial

support toward the completion of this work and his encouragement along the way.

Thanks to my committee members, Michael Luke and Stefan Mochnacki for their

helpful feedback. Special thanks to the external examiner, Charles Hellaby, for going

through the thesis with a fine toothed comb.

To my colleagues in physics Megan McClure, Allen Attard, Johann Bayer, Brian

Wilson, Mitch Thomson, Parandis Khavari, and Alex Venditti, thanks for paving the

way in front of me and showing me how far I’ve come.

I am grateful to my family: my grandma, mom, dad, Amy, Lauren, numerous cousins,

uncles, and aunts, for their support even though they had no idea what I was studying.

Thanks to my friends from Calgary for being a seamless extension of my family: Bernie,

Christine, Duffy, Ellen, Emil, Emily, Hai, Hy, Jeff, Jenny, Joanne, John, Joyce, Justin,

Lan, Maelynn, Monica, Paul, Phuoc, Rishi, Rosita, Sam, Susan, Susan, Tri, Vincent, and

Vivian.

Thanks to my friends in Toronto for being my family away from family: Alex, Athar,

Ben, Ben, Beth, Brenda, Chad, Davin, Ela, Elanna, Geoff, George, Heather, House,

Irena, Janna, Jean-Sebastien, Jenn, Jenny, Julia, Juliet, Karen, Kari, Karine, Kevin,

Kristina, Linda, Lisa, Matt, Moiya, Nisha, Patty, Pascal, Robynne, Sapna, Simon, So,

Sola, Staveley, and Stephanie.

Thanks to the staff and fellows of Massey College for shaping my first few years of

life as a Ph.D. student.

A special thanks goes out to Mario Nawrocki and Margaret Huntley for the special

roles they played.

The financial support was provided by the Natural Sciences and Engineering Research

Council of Canada and the Department of Physics at the University of Toronto.

v

Contents

1 Introduction 1

1.1 Why assume homogeneity? . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The Inhomogeneous Universe . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 About singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 About Spacetime Matchings . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 About Braneworlds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 The Story to Come . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Matching Regions of Spacetimes 12

2.1 Review of Matching Conditions . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Matchings Across a Boundary Surface . . . . . . . . . . . . . . . 15

2.1.2 Matching at a Corner . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.3 Matchings Across Thin Shells . . . . . . . . . . . . . . . . . . . . 22

2.1.4 Null Matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 The Cheese Slice Universe . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Lookback Time and Observational Consequences 30

3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 Null Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.2 Bending Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.3 Calculating the Redshift and Lookback Time . . . . . . . . . . . . 35

vi

3.1.4 Numerical Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 Lookback Time and Redshift Relations . . . . . . . . . . . . . . . 38

3.2.2 Possible CMB Data . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 The Structure of the Singularity 48

4.1 Definition of a Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Classification of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.1 Isotropic Singularities . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.2 Classification Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.3 Strength of a Singularity . . . . . . . . . . . . . . . . . . . . . . . 52

4.3 More General Singularity Structures . . . . . . . . . . . . . . . . . . . . 52

4.3.1 Properties of the Matching . . . . . . . . . . . . . . . . . . . . . . 52

4.3.2 BKL Picture of Cosmological Singularities . . . . . . . . . . . . . 53

4.3.3 Cauchy Horizon Singularities . . . . . . . . . . . . . . . . . . . . 55

4.4 The AVTD Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4.2 AVTD Property of Matched Spacetimes . . . . . . . . . . . . . . 59

4.5 Singularities in the Cheese Slice Universe . . . . . . . . . . . . . . . . . . 60

4.5.1 Case (i) Flat FLRW, k = 0 . . . . . . . . . . . . . . . . . . . . . . 61

4.5.2 Case (ii) Open FLRW, k = −1 . . . . . . . . . . . . . . . . . . . . 62

4.5.3 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . 65

5 Cheese Slice Braneworlds 68

5.1 Braneworld Cosmologies . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.1.1 Randall-Sundrum Braneworlds . . . . . . . . . . . . . . . . . . . . 69

5.1.2 Cosmological Braneworlds . . . . . . . . . . . . . . . . . . . . . . 70

vii

5.1.3 Anisotropic Braneworlds . . . . . . . . . . . . . . . . . . . . . . . 75

5.1.4 Brane Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Constructing an Inhomogeneous Brane . . . . . . . . . . . . . . . . . . . 78

5.2.1 The Cheese Slice Brane . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2.2 Energy Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3 Extending the Matching into the Bulk . . . . . . . . . . . . . . . . . . . 86

5.3.1 The Bulk of the Cheese Slice Brane . . . . . . . . . . . . . . . . . 88

5.4 General Embedding of Matched Branes . . . . . . . . . . . . . . . . . . . 92

5.4.1 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.4.2 An Embedding With no Corners . . . . . . . . . . . . . . . . . . 93

5.4.3 The Bulk Matching Surface . . . . . . . . . . . . . . . . . . . . . 94

5.4.4 Approximation of the Bulk . . . . . . . . . . . . . . . . . . . . . . 96

5.4.5 Matching the Bulk . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.4.6 Consequences of Assuming No Corner . . . . . . . . . . . . . . . . 100

5.5 The 3+1+1 Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.5.1 Defining the Normals, Bases and Metrics . . . . . . . . . . . . . . 102

5.5.2 Fixing the Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 108

5.5.3 Finding the Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.5.4 The Bulk Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.6 The Matching of the Bulk . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.6.1 The Matching Conditions . . . . . . . . . . . . . . . . . . . . . . 115

5.6.2 The Second Fundamental Form and Matter Content . . . . . . . . 118

5.6.3 Matching Four Bulks . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.6.4 Special Cases: Breaking the Angle Condition . . . . . . . . . . . . 124

5.7 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6 Summary and Conclusions 128

viii

A Taylor Expansion of a Tensor Field 132

Bibliography 135

ix

List of Tables

5.1 Positivity of matter density, ρ, as a function of cosmological time, t. . . . 83

x

List of Figures

1.1 The cubic lattice universe. . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 The construction of a matched spacetime. . . . . . . . . . . . . . . . . . 14

2.2 Matching across a corner. . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Finding the corner conditions. . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Illustrations of the Cheese Slice universe with (a) flat FLRW slices and (b)

open FLRW slices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1 Light ray propagating through different regions. . . . . . . . . . . . . . . 35

3.2 Lookback time and redshift relation for an Einstein de Sitter model. . . . 38

3.3 Lookback time and redshift relation for a large Kasner region. . . . . . . 39

3.4 Lookback time and redshift relation for a three slice model . . . . . . . . 40

3.5 Average lookback time and redshift relation for different models. . . . . . 41

3.6 Redshift of the CMB for different models. . . . . . . . . . . . . . . . . . 43

3.7 Changing the position of a thin Kasner slice in a predominantly FLRW

model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.8 Changing the thickness of a thin Kasner slice in a predominantly FLRW

model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.9 Changing the number of Kasner slices while keeping the total ratio of

Kasner to FLRW constant. . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.1 Spacelike foliation of a spacetime. . . . . . . . . . . . . . . . . . . . . . . 57

xi

4.2 Matching two leaves of the foliations across Σ. . . . . . . . . . . . . . . . 60

4.3 The Singularities of the Cheese Slice Model. . . . . . . . . . . . . . . . . 66

5.1 Regions in which ρ is positive. . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2 Region in which the SEC is satisfied. . . . . . . . . . . . . . . . . . . . . 85

5.3 Region in which the DEC is satisfied. . . . . . . . . . . . . . . . . . . . . 87

5.4 Matching of two branes extended into the bulk. . . . . . . . . . . . . . . 98

5.5 Definition of the Z-W plane. . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.6 Illustration of the matching conventions that are being used. . . . . . . . 104

5.7 Matching four different bulks. . . . . . . . . . . . . . . . . . . . . . . . . 123

5.8 Assume one side of the brane is a vacuum. . . . . . . . . . . . . . . . . . 125

xii

Chapter 1

Introduction

1.1 Why assume homogeneity?

It is common practice in the study of cosmology to approximate our universe with a ho-

mogeneous Friedmann-Lemaıtre-Roberson-Walker (FLRW) model. These models were

a result of assuming the Cosmological Principle, which states the the universe is homo-

geneous and isotropic. Homogeneous in this sense refers to spatial homogeneity. For

example the matter density could change in time, but does not depend on spatial coor-

dinates. Isotropy on the other hand, refers to a space that has no preferred direction.

These assumptions taken together translate geometrically to a space that has maximal

symmetry, and thus uniform curvature, that can in general depend on time. The only

metrics that satisfy these conditions are the FLRW metrics. The study of these space-

times led to the conclusion that our universe is not necessarily static and eventually led

to the Big Bang theory. The theory states that our universe is in a state of expansion

from initial conditions characterized by extremely high density and temperature.

The assumption of homogeneity is primarily associated with observations of the Cos-

mic Microwave Background radiation (CMB) and the Copernican principle; the belief

that we are not located at a preferred point in the universe. The CMB is believed to be

1

Chapter 1. Introduction 2

the remnants of the photons from an early period in the development of the universe at

which point the temperature cooled to about 30000K. At this temperature, also called

the time of last scattering, protons and electrons combined to form the first hydrogen

atoms and the universe became transparent to the photons allowing them to propagate to

an observer located at our present position and time. The CMB radiation traces directly

back to the period when these photons last scattered off free electrons and protons. A

comprehensive measurement of the CMB was conducted by COBE [7] and more recently

refined by WMAP [55]. These studies have shown that this radiation is highly isotropic

with variations in temperature on the order of 10−5. Taking these observations of isotropy

coupled with the Copernican belief that we are not occupying any preferred position in

the universe leads to the assumption of homogeneity.

Aside from physical arguments, the main attraction of the FLRW models is their

mathematical simplicity. In General Relativity (GR) the matter content of spacetime is

coupled to its geometry through the Einstein Field Equations (EFE) given by

Gab = κTab, (1.1)

where Tab is the energy-momentum tensor, the cosmological constant, Λ, is assumed to

be zero, Gab is the Einstein tensor given by the Ricci tensor and Ricci Scalar,

Gab ≡ Rab −1

2Rgab (1.2)

and gab is the spacetime metric. The coupling constant κ is related to the Newtonian

gravitational constant by,

κ =8πGN

c4. (1.3)

Since the spatial curvature is uniform the only variables in the metric are the spatial

curvature (which can be either positive, negative or zero) denoted by the variable k and

a scale factor, a(t), that depends only on time. This can be seen through the line element

given by,

ds2 = gabdxadxb = −dt2 + a2(t)

[

dχ2 + S2k(χ)(dθ2 + sin2 θdφ2)

]

(1.4)


where,

Sk(χ) =

sin(χ√k)√

kk > 0

χ k = 0

sinh(χ√

|k|)√|k|

k < 0

. (1.5)

These symmetries allow the EFE to be simplified into two second order differential equa-

tions, known as the Friedmann Equations, that describe the evolution of the scale factor.

This allows for easy comparison of the theory with the observable Hubble constant that

is directly related to the rate of change of the scale factor.

Much of the work in modern cosmology is done on the premise of a homogeneous

background, which is usually accepted as an initial condition. Though the FLRW model

has led to much success it also gives rise to certain problems, such as the horizon problem

[86]. Due to the nature of the FLRW cosmology and the finite speed of light, certain

regions of spacetime are causally unrelated. Therefore it appears to be highly coincidental

that different regions of the universe should have the same temperature or density. This

“problem” should not come as a surprise though because a homogeneous universe is

inherently acausal. To impose homogeneity means to choose a spacelike hypersurface

that is identical at all points. Since the hypersurface is spacelike, each point is causally

unrelated. It is a strong assumption that all these unrelated points are somehow identical.

Some effort has been put into reconciling some of these problems with assuming

a homogeneous universe. Inflation has been suggested as a possible explanation for

homogeneity, that at some period during the evolution of the early universe there was an

period of exponential expansion. Little is understood about the possible mechanisms that

could lead to inflation and the initial conditions before inflation are also not addressed.

Others have appealed to the anthropic principle, which roughly states that the universe

is in its current state because life would not exist otherwise to observe it. However the

author finds such an “explanation” unsatisfying.

This brings us to the question of why should we assume homogeneity. It has been


able to give us some useful results and insights, but it has also introduced some problems

as well. Rather than attempting to explain away these problems or simply accept it as

an initial condition, let us be more prudent and consider the alternative of doing away

with the assumption altogether.

1.2 The Inhomogeneous Universe

Despite the elegance and simplicity of the homogeneous models we know that the universe

is fundamentally not homogeneous: on the scale of galaxies, clusters and super-clusters we

see clumping of matter. It can be argued that on average over a large scale the universe is

homogeneous, but even on the largest observable scales, such as the 2dF Galaxy Redshift

Survey [25], we have seen voids with little matter content. Recent data from the Sloan

Digital Sky Survey (SDSS) continue to assert that our universe is inhomogeneous with

regions of galaxy clusters and voids [107, 52, 99, 57].

There have been attempts at modelling inhomogeneities using perturbations of the

FLRW model. These studies begin with the assumption of a smooth universe with slightly

overdense regions that grow over time to the large scale inhomogeneities observed today.

There is the possibility of perturbing around inhomogeneous models, though admittedly

such an analysis would be more difficult. Thus the Friedmann models are generally used

as the zero point about which to perturb. The problem with this method is explaining

the initial conditions that lead to these perturbations. This is difficult to address and

inflation is often invoked to resolve difficulties with initial conditions. Such perturbative

approaches do not address the possibility that the universe is inherently inhomogeneous

from the onset.

Furthermore, assuming homogeneity introduces problems that might not arise in other

models, such as the horizon problem or the inherent acausal nature of the FLRW models.

Let us assume then that the universe is inherently inhomogeneous and explore the


possibilities. There are perhaps other cosmological models that are anisotropic and in-

homogeneous that can reproduce the observations we see today. For example the cubic

lattice universe, constructed by three sets of mutually perpendicular intersecting planes

of arbitrary thicknesses can be seen as homogeneous and isotropic on large scales. This

construction is depicted in Figure 1.1 However, on the scales that are the same order of

magnitude as the thicknesses of the layers, the model can produce voids, walls and re-

gions of highly dense galaxy distributions. There is no exact solution to the cubic lattice

universe known but exact solutions for many inhomogeneous models do exist.

One of the earliest was discovered by Lemaıtre [73] and later investigated in further

detail by Tolman [101] and Bondi [11]. These solutions represent spherically symmetric

shells that can in general move radially at different velocities. Panek [85] has used these

Lemaıtre-Tolman-Bondi (LTB) solutions to model voids and galaxy clusters. He was able

to show that they do not have an appreciable effect on the anisotropy of the CMB. This

shows that it is possible for an inhomogeneous universe to produce an isotropic CMB.

The Szekeres family of solutions [97] generalizes the LTB models and contain no

symmetries. They can be visualized as non-concentric spheres or non-parallel planes.

Szafron [96] later generalized these solutions further to include pressure. There have

been attempts at using Szekeres models to address inhomogeneities. Bolejko used the

Szekeres solutions to look at CMB data [10] and concluded that inhomogeneities in the

local vicinity of the observer would contribute the largest temperature fluctuations and

fluctuations caused by large scale cosmic structures contribute an insignificant amount

(variations under 10−6). This is further support for the idea that an inhomogeneous

universe can produce an observed isotropic CMB.

The majority of cosmological measurements has so far largely supported the ΛCDM

model of the universe, namely a flat Friedmann universe composed of cold dark matter

and a component of dark energy (or cosmological constant Λ). The dark matter serves

to explain the gravitational interactions of matter that has not yet been detected, such


Figure 1.1: The cubic lattice universe constructed by three mutually perpendicular in-

tersecting planes. The construction can extend indefinitely. On large scales this model

appears homogeneous and isotropic, but locally there are voids, walls and areas of highly

concentrated galaxies where the walls intersect.


as matter required to explain galactic rotation curves. The dark energy drives the accel-

erated rate of expansion of the universe. The focus then turns to explaining what this

dark matter and dark energy is composed of but so far results have been inconclusive.

There remains the possibility that these constructs are not physical, but a result of an

oversimplified cosmology. Rather than adding extra components to fit the observations it

would be prudent to examine the underlying assumptions of the model itself, especially

homogeneity. There is the possibility of fitting the experimental data by locating us at

a void within an LTB model without invoking the need for dark energy [92].

These examples serve to highlight some of the possibilities of using exact inhomo-

geneous models in cosmology but they are only a small selection of the vast amount of

work conducted in this area. The text of Krasinski [65] conducts a comprehensive review

of exact inhomogeneous solutions and argues adamantly for their importance in cosmol-

ogy. We should not allow ourselves to be bound to the FLRW model. If we are to gain

a comprehensive understanding of cosmology we must also investigate inhomogeneous

models.

1.3 About singularities

Singularities are an intriguing aspect of General Relativity. They not only provide an

interesting mathematical problem, but also gives us insight into the heart of General

Relativity itself: in essence the singularity is where the theory itself breaks down. One

can envision the singularity as a point at which the spacetime manifold is undefined.1

Despite this difficulty we are able to describe some of the properties of singularities and

attempt to relate them to physical phenomena. For example the Schwarzschild singularity

is believed to describe the centre of a non-rotating black hole.

Perhaps more important than the singularity itself is understanding how the spacetime

1A more precise definition of singularities will be discussed in Chapter 4.


behaves in the limit as one approaches the singularity. These are the regimes in which

we can make observable predictions that can be tested since the singularity itself is

likely unobservable. It has not been proven that singularities are unobservable, but the

cosmic censorship conjecture, which states that all singularities are hidden within an

event horizon, is widely believed to hold true.

In the context of cosmology the singularities we will be concerned with represent the

initial conditions of the universe. The theorems of Hawking and Ellis [54] have shown,

under very general geometric conditions, that such a singularity is unavoidable. They

argue that our universe satisfies these conditions and conclude that there was a singularity

at some finite time in the past.

Since the universe is observed to be inhomogeneous (at least at some scales) it is

natural to expect that it might be inhomogeneous at all times including at the initial

singularity. It is possible that an inhomogeneous singularity might give rise to the in-

homogeneities observed today. The idea that the initial singularity is homogeneous and

isotropic is likened to the cosmological principle and is merely an assumption. It is a

special case of all the singularities that are possible and there is no physical reason to

believe that our universe had such a high degree of symmetry near the initial singularity.

Belinskii, Khalatnikov and Lifshitz (BKL) [5] led the way in investigating the approach

to the singularity of a generic spacetime. They attempted to find the most general cos-

mological solution to the EFE with an initial singularity. They found that the terms

corresponding to the time evolution dominated over the spatial curvature terms. They

then neglected these terms. What they found was an oscillatory behaviour such that the

spacetime expands and contracts in different directions as the singularity is approached.

Whether or not this behaviour holds true for generic spacetimes is an open question.


1.4 About Spacetime Matchings

There are many cases in the study of GR where one solution of the EFE is not sufficient to

describe the entire spacetime. For instance the Schwarzschild vacuum solution describes

the exterior of a spherical object such as a star, but an interior solution is required to

describe the matter inside the star. Furthermore the star is part of a galaxy within the

universe which requires a cosmological solution to describe it. It is arguable that one

should be able to find a single solution in some coordinate system that can faithfully

describe every region of the spacetime. However, such a prescription would be extremely

difficult and the result will likely be too unwieldy to be of practical value. It is much

easier to find solutions for each region we wish to describe and join them together. The

joining cannot be done arbitrarily as conditions must be met at the matching surface

and much of the difficulty in applying this method is to be able to satisfy these matching

conditions. In some cases the matching might require the use of stress energy along the

surface to make the matching valid. Such surfaces can be thought of as a stress sheet

or thin shell that separates the two regions. More on these matching conditions will be

discussed in Chapter 2.

In the search for inhomogeneous solutions, matching regions of spacetimes is a very

useful method to consider. Voids can be described by a vacuum solution and regions of

dense galaxy distributions might be described by another solution. The case of the cubic

lattice model described earlier in section (1.2) would ideally be constructed through the

matching of different planar solutions. Such a solution though has not been found, but

a more simple layered cosmology has been found through the alternating matchings of

Kasner vacuum solutions with FLRW layers. Such a model has been given the name

Cheese Slice Universe and will be discussed in great detail in the following chapters.


1.5 About Braneworlds

Many authors have studied alternatives to the FLRW universe and a recent trend has been

toward incorporating extra dimensions into cosmology. There have been various theories

proposed involving extra dimensions. One of the earliest was that of Kaluza and Klein

[62, 64] which was originally devised to unify gravity and electromagnetism. In this theory

the extra dimensions are compactified on a microscopic scale in relation to the observed

spatial dimensions thereby recovering the familiar four-dimensional spacetime. More

recently there have been models presented that do not require the extra dimensions to be

compactified. In these models the particles and fields of the standard model are confined

to a four-dimensional spacetime, but gravity can extend into the extra dimensions.

Randall and Sundrum have presented a model in which a four-dimensional spacetime

called the brane, is embedded in a five-dimensional spacetime called the bulk [88, 87].

These models have been termed braneworld models. In the context of spacetime match-

ings, we can view these braneworlds as the matching of two five-dimensional bulks across

a four-dimensional brane. The four-dimensional brane is taken to be the observable

universe. Other authors have since constructed models where the brane reproduces an

FLRW type cosmology. However, like the case of four-dimensional cosmology, there is no

a priori reason why the brane should be homogeneous. Thus even in higher dimensional

theories it is important to consider the implications of an inhomogeneous cosmology.

1.6 The Story to Come

We will examine inhomogeneous cosmology using the concept of spacetime matchings

applied in many different contexts. We discuss the matching conditions in detail in

Chapter 2 and illustrate how the Cheese Slice universe is constructed. We will then use

this “toy model” to discover some of the implications of inhomogeneities in cosmological

models. In Chapter 3 some observational consequences of the Cheese Slice model are


discussed. Chapter 4 goes deeper into the question of the inhomogeneous singularity,

utilizing matching conditions to extend the idea of a “velocity dominated” singularity

inspired by the BKL approach. The higher dimensional braneworld models are examined

in Chapter 5. The possibility of constructing an inhomogeneous brane through matchings

within the brane is tested and the embedding of such a construction into the bulk is

explored.

Throughout this thesis we will be using the signature (− + ++). Greek indices

will refer to three dimensional values; α, β, µ, ν . . . = 0, 1, 2. Latin induces represent

four dimensional values; a, b, c, d . . . = 0, 1, 2, 3 and capital Latin indices refer to five

dimensional values; A,B,C,D . . . = 0, 1, 2, 3, 4. In Chapter 5, lower case latin in-

dicies in the middle of the alphabet will refer to an alternate four dimensional space,

i, j, k, l . . . = 0, 1, 2, 4 The few exceptions to this labelling of indices will be clearly

stated. The partial derivative of u will be represented by u,a and the covariant derivative

by u;a or ∇au. Finally, Lξu, will denote the Lie derivative of u in the direction of the

vector ξ. Geometrized units will be used throughout where G = c = 1.

Chapter 2

Matching Regions of Spacetimes

A central problem in General Relativity is to interpret what occurs at the boundaries

between two regions of spacetime. It is possible that discontinuities in the metric could

appear at the interface of two regions resulting from a number of causes, such as changes

in the energy density, infinitely thin stress sheets or simply different coordinates used on

either side.

Understanding these junctions between spacetimes leads to a useful method of gen-

erating new solutions. One takes two regions of spacetime that are exact solutions of the

EFE and chooses a hypersurface across which to match these two solutions. If certain

conditions are satisfied across this hypersurface, then we have a new solution defined as

the union of the two original regions identified along the hypersurface. The most well

known example of this construction is the Einstein-Strauss “Swiss Cheese” model [36], in

which Schwarzschild voids are matched to dust filled FLRW cosmologies across a time-

like spherically symmetric matching surface. Another situation where matching becomes

necessary is at the surface of a non-rotating star, which can be modelled by matching

the interior solution with the Schwarzschild vacuum in the exterior.

The key to executing this prescription is the matching conditions one must satisfy

along the hypersurface. Several treatments of these junction conditions have been pro-

12

Chapter 2. Matching Regions of Spacetimes 13

posed and we give a brief review in this chapter followed by an example of a smooth

matching in the Cheese Slice universe.

2.1 Review of Matching Conditions

Let us denote two spacetimes by V1 and V2 with corresponding metrics g1ab and g2

ab. In

each spacetime we select a hypersurface, Σ1 and Σ2, that divides each spacetime into two

distinct regions. The regions in V1 will be labelled V +1 and V −

1 . Points on Σ1 are defined

to belong to both V +1 and V −

1 . We will label V2 in an analogous fashion.

Now we define a new spacetime W = V +1 ∪V −

2 with the points on Σ1 and Σ2 identified.

W is then the spacetime that is constructed by the matching of V +1 and V −

2 along the

surface Σ as illustrated in Figure (2.1). This construction cannot be done arbitrarily as

W is not guaranteed to exist. The difficulty lies in identifying Σ1 and Σ2. The conditions

that must exist for this identification to take place are called the matching conditions or

junction conditions.

Note that we can consider V1 = V +1 ∪V −

1 to be a spacetime formed from the matching

of V +1 and V −

1 along the surface Σ1. We know that this can always be done because V1 is

known to exist. Such a matching is referred to as a trivial matching. It can be thought

of as a spacetime “matching with itself”.

Since the choice of V1 and V2 is arbitrary, we will do away with the subscripts and

refer only to V + and V − with the understanding that each is a region of a spacetime

with its corresponding hypersurfaces Σ− and Σ+. In general we will use the superscripts

+ and − to distinguish quantities calculated in either V + or V − respectively.

A hypersurface, Σ, can be defined by specifying a function of the coordinates,

ie. f(xa) = 0. The normal to this surface, na, can by found by taking the first derivative

of this function such that,

na =∂

∂xaf(xa) ≡ f,a. (2.1)


Σ2

Σ = Σ1 = Σ2

V +1

V −2

V −2

W

Σ1

V1 V2

V +2V −

1V +1

Figure 2.1: The construction of W is made from matching together regions of V1 and

V2 and identified along the surface Σ. Matching conditions must be satisfied along Σ to

guarantee the existence of W .


We will call Σ spacelike, timelike or null depending on which of the following conditions

are satisfied.

• If the normal is timelike, nana < 0, then Σ is spacelike.

• If the normal is null, nana = 0, then Σ is also null.

• If the normal is spacelike, nana > 0, then Σ is timelike.

It is possible for a surface to change character between spacelike, timelike or null from

point to point [76]. However the physical interpretation of this is unclear. Throughout

this thesis we will assume all hypersurfaces maintain the same spacelike, timelike or null

character at all points. For example if Σ is spacelike at one point, we will assume it is

spacelike at all points.

We will now take a closer look at the matching conditions that will guarantee the

existence of W .

2.1.1 Matchings Across a Boundary Surface

If W exists and Σ is a timelike or spacelike surface with no additional stress-energy on

the surface, Σ, it is referred to as a boundary layer or boundary surface. Though the

matter content can differ on either side of Σ, the essential feature of a boundary surface

is that there be no additional stress-energy on Σ. One can picture the surface of a star

as a boundary layer. The exterior of the star is a vacuum spacetime and the interior has

matter of some form, yet no additional stress is required on the surface to hold the star

together.

Three sets of junction conditions have been used to determine if such a matching can

exist.

The O’Brien and Synge Conditions [83] require the coordinates to be chosen such that

Σ is defined as x4 = constant, where x4 can be either a timelike or spacelike coordinate.


With this coordinate system V + and V − match if the following conditions are met:

g+ab = g−ab

∂g+αβ

∂x4=∂g−αβ∂x4

and T 4+b = T 4−

b . (2.2)

as calculated on Σ±, were T ab is the energy momentum tensor. Kumar and Singh [66]

have shown that the condition on the energy momentum tensor is not independent of the

other two. Thus the condition can be more succinctly expressed as:

g+ab = g−ab and

∂g+αβ

∂x4=∂g−αβ∂x4

. (2.3)

The Lichnerowicz Conditions [74] states that V + and V − match across Σ if at every

point on Σ there exists an “admissible” set of coordinates such that the metric compo-

nents gab and their first derivatives are continuous across Σ:

g+ab = g−ab and

∂g+ab

∂xc=∂g−ab∂xc

. (2.4)

In both of the above conditions g±ab as functions of the coordinates on Σ are assumed to

be at least twice differentiable along the tangents to Σ.

The Darmois Conditions [28] do not require the same set of coordinates on either

side. Let the coordinates be xa± and the surface defined by,

Σ+ : f+(xa+) = 0 and Σ− : f−(xa−) = 0 (2.5)

of class C2 or higher. Then the unit normals to Σ± can be calculated by,

n+i =

f+,i

√

∣

∣gab+f+,af

+,b

∣

∣

and n−i =

f−,i

√

∣

∣gab−f−,af

−,b

∣

∣

. (2.6)

We also require the parametric representation of Σ±,

xa+ = ha+(u1, u2, u3) and xa− = ha−(u1, u2, u3), (2.7)

where ha± are of class C3 or higher. Then the first and second fundamental forms on Σ

are respectively defined as,

γαβ =∂xa

∂uα∂xb

∂uβgab (2.8)


and

Kαβ =∂xa

∂uα∂xb

∂uβnb;a. (2.9)

The first fundamental form is the inherited metric of the surface and describes its intrinsic

curvature. The second fundamental form is the extrinsic curvature which describes how

the surface bends in the ambient space. With these definitions V + and V − match across

Σ if,

γ+αβ = γ−αβ (2.10)

and

K+αβ = K−

αβ, (2.11)

where γ±αβ and K±αβ are calculated in terms of uα.

Bonner and Vickers [12] have shown that Lichnerowicz conditions are equivalent to

the Darmois conditions. They also show that the O’Brien and Synge conditions are

more restrictive than the other two. It may appear from equations (2.3) and (2.4) that

the Lichnerowicz conditions are more restrictive, but upon closer inspection the O’Brien

and Synge conditions requires one of the coordinates to be constant along the surface

(Σ ≡ x4 = constant). The Lichnerowicz conditions do not require this condition. Any

matching that satisfies the O’Brien and Synge condition necessarily satisfies the Darmois

and Lichnerowicz conditions, but the converse is not true. Thus the O’Brien and Synge

conditions may rule out physically reasonable situations.

For example, in a spherically symmetric comoving coordinate system, the O’Brien and

Synge conditions require that the matter-energy density at a surface of constant radial

coordinate (ie. Σ : r = r0) to be continuous [61]. Thus if we take the Schwarzschild

solution and attempt to match it with an FLRW solution in the these coordinates, it

would require the density of the FLRW side to vanish. This simple model would not be

able to satisfy the O’Brien and Synge conditions.

The Lichnerowicz conditions, while equivalent to the Darmois conditions, are much


more difficult to implement due to the need for admissible coordinates. In general it

could be quite difficult to find such a coordinate system.

This leaves the Darmois conditions as the most convenient and reliable formulation

of junction conditions in general relativity for matching spacetimes across boundary

surfaces. Examples of its use can be found throughout the literature [24, 33, 34, 40,

67, 78, 79]. In section 2.2 we will make use of the Darmois conditions to show that the

Cheese Slice universe is an exact solution of the EFE.

2.1.2 Matching at a Corner

Before we move onto more general matchings we will note a special case of a boundary

surface in which Σ is composed of two distinct components that join non-tangentially

or in other words there is no unique normal where the two components are joined. In

this case the differentiability conditions of equation (2.5) do not need to hold as f(xa)

can be piecewise defined. Such matchings were investigated by Taylor [98] and referred

to as “corners”. In this case Σ can be defined as a three-dimensional manifold with a

two-dimensional submanifold, Λ, such that Σ without Λ has two distinct components.

Let us label these two components Π1 and Π2. To have a well defined corner we assume

that Π1 and Π2 meet at some angle θ 6= π. Then use Σ = Π1 ∪ Λ ∪ Π2 as the surface

over which we attempt to match V + and V −, as illustrated in Figure (2.2). Away from

Λ that matching is straightforward as one can use the Darmois matching conditions,

but on Λ additional conditions must be met for a matching to be valid. The Darmois

conditions are not sufficient on the corner because we do not have a unique normal. Thus

calculating equation (2.11) becomes problematic.

To state the corner conditions we must first make some definitions. Let the coordi-

nates on Λ be ζA. In this section capital indices run over two dimensions (ie. A = 0, 1).

Rather than having a unique normal to Λ there is a two dimensional space of normals.

Let mAa be an orthonormal basis for this space. The first and second fundamental forms


V −

V +

Π1Π2

Λ

θ

Figure 2.2: A corner can be realized if Σ is composed of two distinct components Π1 and

Π2. Since Π1 and Π2 are three dimensional hypersurfaces the point at which they meet

is a two dimensional surface Λ. The angle between the two hypersurfaces as measured

in V ± is denoted θ.

on the corner can then be defined as

γAB =∂xa

∂ζA∂xb

∂ζBgab, (2.12)

KCAB =

∂xa

∂ζA∂xb

∂ζBmCb;a. (2.13)

There is also a torsion vector defined as,

τA =∂xa

∂ζAm1b m2

b;a = −∂xa

∂ζAm2b m1

b;a. (2.14)

The torsion describes the change of one normal, m2b , in the direction of the other normal,

m1b , and can be thought of as a “twisting” of the corner. Finally, let θ± denote the angle

between Π1 and Π2 as measured in V ±.

If we are given a spacetime constructed from V + and V − matched across Σ with a

corner, then it follows that the Darmois conditions are satisfied on Π1 and Π2 and the

following conditions are met on Λ:

γ−AB = γ+AB, (2.15)


Figure 2.3: To find the corner conditions we choose two surfaces, Σ and Π, that satisfy

the Darmois conditions. The corner, Λ, is then defined as the intersection of the two

hypersurfaces.

KC−AB = KC+

AB , (2.16)

τ−A = τ+A (2.17)

and

θ− = θ+. (2.18)

To find where these conditions come from we can envision a spacetime containing

two hypersurfaces, Σ and Π, that satisfy the Darmois conditions. (Figure 2.3). If they

intersect non-tangentially, one can show that the corner conditions, equations (2.15)–

(2.18), are satisfied on the intersection, Λ.

The condition in equation (2.18) is equivalent to stating that no conical singularity

exists around Λ. The condition in equation (2.17) ensures that the torsion of the corner


is identical as seen from both sides. If θ± = π no corner exists and Σ is in a sense

smoothed out. The extremal cases where θ± = 2π or θ± = 0 represents a hypersurface

folded on itself. Σ then appears as a hypersurface that ends on Λ and cannot be used as

a matching surface since it does not separate two regions of spacetime. The conditions

in equations (2.15) and (2.16) are very similar to the Darmois conditions for a two

dimensional surface. In a sense Λ is itself a matching surface across which Π1 and Π2 are

matched. This matching forms Σ.

Though these conditions are interesting, the converse would be much more useful. We

would like to know what are the conditions that are required to ensure that a spacetime

which induces the given structure on the corner does exist. These conditions can then

be used as our matching criteria. The proof has been worked out in detail by Taylor and

we restate the results here.

Two regions of spacetime V + and V − can be matched along a surface with a corner

Σ = Π1 ∪ Λ ∪ Π2, if the following conditions are met,

γ+αβ = γ−αβ as calculated on Π1 and Π2, (2.19)

γ−AB = γ+AB, as calculated on Λ and (2.20)

θ− = θ+. (2.21)

These conditions are quite different from the Darmois conditions, even away from the

corner where Π1 is the only component of Σ. If there were no corner (θ = π), then

equations (2.20) and (2.21) are automatically satisfied. We are left with equation (2.19),

which on its own does not constitute the Darmois conditions, but a matching is still

possible. This will be discussed further in section (2.1.3).

A successful matching of a physically reasonable corner has not yet been found, but

this result is important in that it defines the necessary conditions to match across a

piecewise defined hypersurface. This result will be useful in Chapter 4 when we require

a piecewise defined foliation of a spacetime to determine whether a matched spacetime


is asymptotically velocity term dominated. Mars, Senovilla and Vera have also surmised

that this treatment will be required to describe braneworlds that contain jumps in the

energy-momentum tensor [76]. These types of braneworlds will be the focus of Chapter 5

where we confirm that a corner is indeed required on an inhomogeneous brane.

2.1.3 Matchings Across Thin Shells

In the case where Σ carries stress-energy within it, it is sometimes referred to as a stress-

sheet, thin shell, shock wave or a singular hypersurface. Such a surface allows for greater

flexibility in the possible matchings that can take place. Regions of spacetime that

could not be matched across a boundary surface as defined above, might be realizable if

matched across a stress sheet. This is what happens on the surface of a party balloon;

the rubber acts as a stress-sheet that separate two regions with different air pressure.

One could see the matching conditions across boundary surfaces in section (2.1.1) as a

special case in which the stress-energy of Σ is zero.

Israel [60] formulated a comprehensive treatment of such surfaces1. To begin we note

that the curvature of a spacetime is described by the Riemann curvature tensor, Rabcd,

which disappears only when the spacetime is flat. We define Rabcd as,

Rabcd = Γabd,c − Γabc,d + ΓaecΓ

ebd − ΓaedΓ

ebc. (2.22)

The Christoffel symbols, Γabc are defined by derivatives of the metric tensor,

Γabc =1

2gad (gbd,c + gcd,b − gbc,d) . (2.23)

The Riemann curvature tensor can also be written in terms of quantities on a hyper-

surface within the spacetime using the Gauss-Codacci equations [103],

Rabcdeaαe

bβe

cγedδ = (3)Rαβγδ + ǫ(n) (KαγKβδ −KβγKαδ) (2.24)

1Null surfaces are excluded in this treatment.


and

Rabcdnaebβe

cγedδ = (3)∇δKβγ − (3)∇γKβδ, (2.25)

where (3)Rαβγδ and (3)∇ are the Riemann curvature tensor and covariant derivative in

the three-space of the hypersurface related to the three metric γαβ. Also, ǫ(n) ≡ nanb is

+1 or −1 depending on whether the normal is spacelike or timelike respectively and the

short hand eaα = ∂xa

∂uα is used. These equations relate the curvature in the hypersurface

to the curvature in the spacetime via the extrinsic curvature, Kαβ.

The Gauss-Codacci equations (2.24-2.25) can also be written in terms of the Einstein

tensor defined in equation (1.2),

Gabnanb =

1

2(K2 −KαβK

αβ − ǫ(n)(3)R) (2.26)

and

Gabeaαn

b = (3)∇αK − (3)∇βKβα , (2.27)

where K = Kαα . To define the energy-momentum tensor on Σ we can perform a “pill-box”

integration of the EFE across Σ [75],

Sab = limΣ→0

∫ Σ

−Σ

(

Tab − gabΛ

κ

)

dn =1

κlimΣ→0

∫ Σ

−Σ

Gabdn, (2.28)

where n is the proper distance through Σ in the direction of the normal na. Also Tab is

the energy momentum tensor, Λ the cosmological constant and κ the coupling constant

in the EFE. Sab is the associated 4-tensor of the stress-sheet which vanishes off Σ such

that Sabnb = 0. The corresponding 3-tensor is given by,

Sαβ = eαaeβbS

ab. (2.29)

Then one can show that the jump in extrinsic curvature is directly related to the energy-

momentum of the stress sheet,

limΣ→0

∫ Σ

−Σ

Gabeaαe

bβdn = ǫ(n) ([Kαβ] − γαβ[K]) = κSαβ, (2.30)


where [F ] ≡ F+ −F− denotes the jump in any value across Σ and F± denotes the value

of F in the limit as Σ is approached from either side.

This result is compatible with the Darmois conditions. It is straightforward to see

that if the energy-momentum of Σ is zero then equation (2.11) is satisfied. The Israel

treatment also presupposes equation (2.10) from the outset. Thus we recover a boundary

surface as one would expect.

A distributional method of describing stress-sheets has also been in use and reviewed

by Mansouri and Khorrami [75]. They show that this method is equivalent to the

Darmois-Israel method. However the distributional method relies on the use of a well

defined coordinate system throughout the spacetime in the Lichnerowicz sense of ad-

missible coordinates. Thus the Darmois-Israel formulation is much more convenient to

use.

With stress sheets taken into consideration, one can then ask what the minimum

requirement of matching V + and V − is. Clarke and Dray [22] have addressed this prob-

lem, building on the work of Israel [60], showing that two spacetimes can be matched

if and only if the naturally induced 3-metrics, γ+αβ and γ−αβ, on the hypersurface agree,

essentially echoing equation (2.10) of the Darmois conditions. Furthermore, this result

is true even if Σ is a null surface.

Comparing this result to that of the corner matching conditions in equations (2.19)

to (2.21) discussed in section (2.1.2) we find that they are compatible away from the

corner. We can conclude that the corner matching conditions are a generalization of this

result to surfaces with corners.

2.1.4 Null Matchings

Null hypersurfaces are more difficult to incorporate into the Israel formalism. By defini-

tion, a null hypersurface has a norm that is null. If ka is the normal to a null hypersurface,


we have,

kaka = 0. (2.31)

This means ka is orthogonal to itself and is also tangent to the surface. The induced met-

ric is degenerate and equation (2.9) no longer gives any information about the extrinsic

curvature.

Clarke and Dray [22] have overcome this problem by formulating a definition of vari-

ous fundamental forms for null surfaces. This was later generalized by Barrabes [3] who

proposed a unified formalism to describe singular hypersurfaces whether they be space-

like, timelike or null. Mars and Senovilla [77] developed another method to describe

general hypersurfaces, one that can change from timelike to spacelike to null from point

to point, using what they call a “rigged vector” in place of a normal vector. They then

described the matching conditions one must satisfy to match spacetimes along such a

hypersurface. Their result confirms the results of Clarke and Dray, namely that for a

matching to occur the first fundamental forms must be identical along the hypersurface.

For a matching to occur without any stress along the hypersurface then the addition

quantities, defined in a manner similar to a second fundamental form using the “rigged

vector”, must also match along the hypersurface.

Though the treatment of null and general hypersurfaces is a fascinating area, it is not

immediately applicable to the remainder of our work. Thus we refer to other works, such

as that of Gemelli [44], for a more comprehensive review.

2.2 The Cheese Slice Universe

The Cheese Slice model is a cosmological model constructed by matching together various

layers of FLRW cosmologies and Kasner vacuum solutions. The FLRW line element in

cylindrical coordinates is given by,

ds2F = −dt2 + a2(t)

[

dr2

1 − kr2+ r2dφ2 + (1 − kr2)dz2

]

, (2.32)


with k = −1, 0, 1 for negative, zero and positive spatial curvature. The Kasner line

element is given by,

ds2K = −dT 2 + T 2p1dX2 + T 2p2dY 2 + T 2p3dZ2, (2.33)

with the restrictions

p1 + p2 + p3 = 1 = p21 + p2

2 + p23. (2.34)

Dyer, Landry and Shaver [33] have shown that it is possible to match these two spacetimes

along the surface Σ defined as Σ+ : z = constant and Σ− : Z = constant. This can be

seen by first imposing cylindrical symmetry on the Kasner metric. We choose p1 = p2 = p

and write the metric as,

ds2K = −dT 2 + T 2p

(

dR2 +R2dΦ2)

+ T 2p3dZ2. (2.35)

The only two possible choices of the Kasner exponents are now (p, p3) = (23, −1

3) and

(p, p3) = (0, 1). The latter choice leaves us with a Minkowski spacetime, since

ds2K = −dT 2 + dR2 +R2dΦ2 + T 2dZ2, (2.36)

and

ds2K = −dτ 2 + dR2 +R2dΦ2 + dζ2, (2.37)

are equivalent through the transformations τ = T coshZ and ζ = T sinhZ.

Let the coordinates on Σ be uα = (u, v, w). We parametrize the FLRW coordinates

by t = u, r = v and φ = w. On the Kasner side we choose to parametrize by T = T (u),

R = R(u, v), Φ = w and Z = Z(u). To satisfy the Darmois conditions we first require

γFαβ = γKαβ, (2.38)

where the superscripts F and K will be used to distinguish the FLRW and the Kasner


regions. This implies that,

−1 = γF00 = γK00 = T 2pR2,u − T 2

,u + T 2p3Z2,u (2.39)

0 = γF01 = γK01 = T 2pR,uR,v (2.40)

a2(u)

1 − kv2= γF11 = γK11 = T 2pR2

,v (2.41)

a2(u)v2 = γF22 = γK22 = T 2pR2. (2.42)

From equation (2.41) we see that R,v 6= 0 thus from equation (2.40) we must have R,u = 0.

Combining equations (2.41) and (2.42) we get,

R2,v =

R2

v2(1 − kv2), (2.43)

which can be integrated to give,

R = ± C0v

1 +√

1 − kv2. (2.44)

Equation (2.42) implies that (R/v)2 = a2/T 2p = constant since R is not a function of u.

Thus we must have k = 0. The remaining condition to match the first fundamental form

is,

1 = T 2,u − T 2p3Z2

,u. (2.45)

This can be satisfied if Z,u = 0 and du = ±dT . With this choice we are left with

a2(u) = u2p3 . If (p, p3) = (0, 1) we are left with both sides being Minkowski space.

For a non-trivial solution we choose (p, p3) = (23, −1

3). The FLRW scale factor is now

a(t) = t2/3, which is the pressure-free Einstein de Sitter universe.

To complete the Darmois conditions we find that the normal to Σ is nFa = δ3a on

the FLRW side and nKa = δ3a on the Kasner side. Explicit calculations show that

KFαβ = 0 = KKαβ. This means the matching is possible without any stress-energy along

the matching surface.

An interesting feature of this solution is that there are multiple surfaces of z =

constant and Z = constant along which the matching can take place. Thus this matching


Figure 2.4: (a) The Cheese slice universe constructed by matching together flat FLRW

and Kasner spacetimes along the surface z = constant. (b) The Cheese Slice universe

constructed with open FLRW regions using a different matching surface. Both matchings

can be carried on indefinitely with layers of arbitrary thicknesses.

can be repeated indefinitely with layers of different thicknesses making an inhomogeneous

cosmological model as illustrated in Figure 2.4(a).

Dyer and Oliwa [34] have also found a matching which is possible with an open FLRW

spacetime, k = −1. However, in that case, the matching surface must take the form,

Σ+ : z = ±1

2ln[C(1 + r2)] (2.46)

where C is a positive constant. Refer to Figure 2.4(b) for an illustration of this matching

surface. They also show that this matching is unaffected by the addition of a cosmological

constant.

In the following chapters we will discuss some of the properties of the Cheese Slice

universe including observable properties, the initial singularity and in the braneworld

context. We would like to stress that through these studies we are not attempting to

present the Cheese Slice model as a realistic cosmology. Rather, the choice of using these

models is motivated by the breaking of symmetries of the FLRW universe and being able


to incorporate inhomogeneities in a straightforward manner. Through these studies we

hope to shed some light on the properties of inhomogeneous cosmologies and properties

of spacetime matchings.

Chapter 3

Lookback Time and Observational

Consequences

We investigate the lookback time versus redshift relation for the Cheese Slice model to

find how anisotropic the slices look relative to an observer who resides within one of

the FLRW slices. The relationship between the lookback time and the redshift is highly

dependent on the geometry of the universe. Thus comparing this relation with currently

accepted observations we can get a sense of the limits on the amount of inhomogeneity

allowed in the Cheese Slice model. The goal is to be able to set a limit on the number

and thickness of slices and see under what parameters our model can be considered a

valid cosmology.

The lookback time measures how far back into the history of the universe that we are

observing due to the finite speed of light. If we infer that all galaxies formed at roughly

the same time then the observation of younger galaxies would mean looking back farther

into the history of the universe.

The cosmological redshift (or blueshift) is the change in wavelength that light expe-

riences as the universe expands (or contracts).

We will discuss the details of calculating the lookback time and redshift relations in

30

Chapter 3. Lookback Time and Observational Consequences 31

section (3.1). A numerical algorithm was used to carry out these calculations and results

are presented in section (3.2.1). The CMB is often used as an indication of isotropy in

our universe. Thus in section 3.2.2 we will present our results in terms of possible CMB

data and use this as a comparison with accepted observational values.

3.1 Preliminaries

In the following sections we will use the Einstein de Sitter (ie. spatially flat FLRW)

metric in the form

ds2F = −dt2 + a2t4/3(dr2 + r2dφ2 + dz2) (3.1)

and the Kasner metric in the form

ds2K = −dt2 + a2t4/3(dr2 + r2dφ2) + b2t−2/3dZ2 . (3.2)

The constants a and b are introduced to ensure proper unit bookkeeping. Note that

it is possible to define a consistent time, radial and azimuthal coordinate throughout

many slices, whereas the ‘z’ coordinate is unique in each slice. This inability to define a

consistent ‘z’ coordinate causes some difficulty in defining what is meant by “thickness”.

This will be dealt with in section (3.1.3).

Landry and Dyer have investigated light propagation through the Cheese Slice uni-

verse in detail [68, 69]. Here we follow closely their method.

3.1.1 Null Vectors

To follow the path of the light that reaches the observer we must trace the null geodesics

in our model. A vector tangent to the null geodesic will be denoted ka = dxa/dτ ≡ x

with τ being an affine parameter and the over-dot, ˙ , denotes differentiation with respect

to τ . Let us choose the coordinates such that φ = constant and dφ = 0 and situate the


observer at r = 0. To be null ka is require to satisfy kaka = 0. The geodesic equation,

d2xa

dτ 2+ Γabc

dxb

dτ

dxc

dτ= 0 (3.3)

gives,

t+2

3at1/3(r2 + z2) = 0 (3.4)

tr +4

3tr = 0 (3.5)

tz +4

3tz = 0 (3.6)

in the FLRW regions. Equation (3.5) can be written as,

d

dτ(a2t4/3r) = 0, (3.7)

which can be integrated to give,

r ∝ 1

a2t4/3. (3.8)

Analogously for z we have,

z ∝ 1

a2t4/3. (3.9)

Finally solving for t from equation (3.4) we get,

t ∝ 1

t2/3. (3.10)

Thus in the FLRW regions we have,

kaF =

(

− ξ

t2/3,

ξ√

η2 − 1

ηat2/3, 0, − ξ

ηat2/3

)

, (3.11)

where ξ and η are positive constants yet to be determined. The signs for each component

in equation (3.11) are chosen to describe a time reversed ray (ie. we want ∆t to be

negative). Such a ray originates from the observer and propagates back in time.

Carrying out a similar procedure for the Kasner regions gives,

kaK =

−t−2/3

√

α2 +

(

β

b

)2

t2,α

at4/3, 0, − β

b2t2/3.

, (3.12)


where α and β are constants. Since the geodesics pass through both regions we must

determine how these constants are related. Consider a comoving observer located on the

boundary with normalized 4-velocity ua = (1, 0, 0, 0) as seen from both sides. This is

possible since the coordinates are the same except for z and Z which are both constants.

The 4-velocity must satisfy,

(uaka)F = (uak

a)K and (uAua)F = (uau

a)K = −1. (3.13)

The first is the statement that an observer on the boundary should measure the same

frequency regardless of which side of the coordinates are used. The second ensures that

the observer is timelike. The first condition implies that k0F = k0

K thus,

ξ =

√

α2 +

(

β

b

)2

t2in (3.14)

where tin is the time of entry into the current Kasner slice. Next we can consider a

radially moving observer along the boundary with 4-velocity ua = (√

2, at2/3, 0, 0) again

the same 4-velocity is seen from both sides since the z and Z are constant. This then

implies k1F = k1

K and we can then solve for the constants,

α = ξ

√

η2 − 1

ηand

βtinb

= ξ/η. (3.15)

Equation (3.12) then becomes,

kaK =

− ξ

ηt−2/3

√

(η2 − 1) +

(

t

tin

)2

t2,η√

η2 − 1

ηat4/3, 0, − ξt2/3

ηb2tin.

, (3.16)

3.1.2 Bending Angles

We define ψ to be the angle between the 3-vector tangent to the spatial component of the

null geodesic, kα with α = 1, 2, 3, and a 3-vector normal to the matching surfaces, V α.

We will find the general form of the angle ψ with respects to any surface of z = constant

or Z = constant. This will give us a sense of how the angle evolves as the ray travels


through a region of spacetime. This angle can be calculated from the scalar product in

3-space,

cosψ =V αkα

√

V αVαkβkβ. (3.17)

We have V α = (0, 0, 1) and since ka is null we have kβkβ = gαβkαkβ = k0k0. The angle

can then be simplified to,

cosψ =g33k

3

√

g33(k0k0)=√

|g33|k3

k0. (3.18)

The last equality follows from g33 = |g33| due to our spacetime signature. Since

cos2 ψ + sin2 ψ = 1 we can write,

sinψ = −√

|g11|k1

k0, (3.19)

and also,

tanψ = −( |g11||g33|

)2k1

k3. (3.20)

From equations (3.11) and (3.16) we have

tanψF =√

η2 − 1 = constant and tanψK =tanψF(t/tin)

. (3.21)

The result of ψF being constant is reasonable since the FLRW regions are homogeneous

and isotropic. There should be no preferred direction for the light to bend when it is in

the FLRW region. If we specify the observation angle, ψobs, then ψin can be determined

for each region. This is depicted in Figure (3.1). Using equations (3.21), (3.11) and

(3.16) we arrive at a final form for the null tangent vectors in each region,

kaF =

(

− ξ

t2/3,

ξ sinψinat4/3

, 0, −ξ cosψinat4/3

)

, (3.22)

and

kaK =

−ξ cosψint2/3

√

tan2 ψin +

(

t

tin

)2

,ξ sinψinat4/3

, 0, −ξt2/3 cosψinbtin

, (3.23)


(ψF )out = (ψF )obs

(ψF )obs

(tanψK)out = (tanψF )in

(tout/tin)

(ψK)in = (ψF )out

(ψF )in = (ψK)out

K

F

F

r

z

observer

Figure 3.1: A light ray propagating through different regions is depicted by the thick

line. This ray travels away from the observer. The angles ψ at each point are depicted.

If ψobs is known then all other angles can be calculated.

3.1.3 Calculating the Redshift and Lookback Time

Because each region has its own z-coordinate we must be careful how we choose to

compare relative “thicknesses” of each slice. In each case we can find the distance travelled

by a ray in the z direction as a function of time by integrating dz/dt. In the Einstein de

Sitter regions we have

dz

dt=k3F

k0F

=cosψinat2/3

, (3.24)

which upon integration gives

a(z − zin) = 3t1/3in cosψin

[

(

t

tin

)1/3

− 1

]

. (3.25)

In the Kasner regions we have

dZ

dt=k3K

k0K

=t4/3

btin√

tan2 ψin + (t/tin)2, (3.26)

which gives,

b(Z − Zin) = t4/3in

∫ t/tin

1

(t/tin)4/3d(t/tin)

√

tan2 ψin + (t/tin)2. (3.27)


This integration can be carried out numerically using a Simpson rule integrator. The

relative “thicknesses” will refer to the ratio a(z − zin)/b(Z − Zin). This ratio gives a

sense of distance transversed by the light ray along the z and Z coordinates. This will

not translate to an absolute distance, but will allow us to compare different models using

this ratio as a reference.

For convenience we will work with the redshift factor, x, rather then the redshift

directly,

x ≡ 1 + redshift =(uak

a)observed(uaka)emitted

, (3.28)

where ua is the observer’s 4-velocity. In the Cheese Slice universe, we must treat each

slice separately and therefore replace the observed and emitted times with the time of

entry into the slice, tin and the time of exiting that particular slice, tout. We will assume

a comoving observer with 4-velocity ua = (1, 0, 0, 0). Then from equations (3.23) and

(3.22) we have

xF =

(

tintout

)2/3

. (3.29)

for an Einstein de Sitter region and

xK =cosψin

√

tan2 ψin + (tout/tin)2

(tout/tin)2/3. (3.30)

for the Kasner region. The total redshift factor is the product of the redshift factors from

each slice given by

xtotal =∏

i

xi . (3.31)

From this point forth, we will mean redshift factor when we refer to redshift.

The lookback time, l, is define by

l = temitted − tobserved , (3.32)

but we will work with a normalized lookback time, ℓ, given by

ℓ =temittedtobserved

− 1 . (3.33)


Normalizing the time allows the observer to be at ℓ = 0 and the bang time to be at

ℓ = −1 if the bang time is defined as t = 0.

3.1.4 Numerical Algorithm

The number of slices and relative thicknesses of each slice will always be specified by

the user. Each value entered by the user represents a unique slice and the magnitude of

that value represents the relative thickness. The first slice, the one in which the observer

resides, will always be an FLRW region. The thickness of this slice is defined by the

position of the observer to the first matching surface. Since the ray is travelling in the

positive z-direction, the structure of the model in the negative z-direction (eg. behind

the observer) is irrelevent. The following regions alternate between Kasner and FLRW.

The observation angle is also specified by the user.

Each slice is then divided into one hundred equal points. The lookback time and

total redshift is calculated to each point and plotted. The time spent in each slice is

found from equation (3.25) and by successively integrating equation (3.27) and raising

the upper limit until the required thickness is reached within a preset tolerance. The

integration is carried out using a Simpson rule integrator. Once the time is known the

redshift can be calculated using equations (3.29) and (3.30). Observation angles are

propagated from one slice to the next using equation (3.21).

For Figure 3.5 a slightly different algorithm was used because the goal was to average

over all observation angles. The time between observation and bang-time is divided into

one thousand points. For each point, the distance into the ‘z’ direction is determined

and the number of slices transversed is then known. The redshift is calculated and then

the angle is incremented and calculations repeated for the Simpson rule. The final data

points are then plotted.


Figure 3.2: Lookback time and redshift relation for an Einstein de Sitter model is plotted.

3.2 Results

3.2.1 Lookback Time and Redshift Relations

Recall that the observer must always be in the Einstein de Sitter region because the

Kasner regions are empty. We start with an Einstein de Sitter reference model with no

Kasner regions. The lookback time is plotted in Figure 3.2. Predictably, this relation

does not depend on the observation angle because the Friedmann model is isotropic.

This result could also have been found analytically from equations (3.29) and (3.33).

The normalized lookback time, ℓ, is given by,

ℓ = x−2/3 − 1, (3.34)

since temitted = tout and tobserved = tin.

To see how a Kasner slice would affect this relation, we could build a model with a

thin Friedmann slice and a relatively large Kasner region. We set up a model with the

ratio of thicknesses of FLRW to Kasner to be F : K = (1 : 75), where the ’F’ value refers


Figure 3.3: Lookback time and redshift relation for a thin Einstein de Sitter slice followed

by a large Kasner region with a ratio of F : K = (1 : 75). From left to right, the

observation angle of each curve is ψobs = 0, ψobs = 5, ψobs = 10, followed by increments

of 10 up to ψobs = 80.

to the observer’s local slice and ’K’ the adjacent Kasner region. The thickness of the local

slice is measured from the observer to the first matching surface. The result can be seen

in Figure 3.3. In this case the redshift is highly dependent on the angle of observation.

For an observation angle directly normal to the matching surface we see a blueshift.

This blueshift occurs because the Kasner spacetime contracts in the Z direction while it

expands in the r−φ plane. At about 30 no more blueshift can be observed and at higher

angles the curve begins to resemble that of Figure 3.2. This is expected because as the

observation angle increases the time the ray spends in the FLRW region also increases.

At an angle of ψ = π/2 the ray is entirely in the FLRW region and we recover the result

in Figure 3.2.

Next we try a model with three equal slices. This would give us a sense of how a


Figure 3.4: Lookback time and redshift relation for a three slice model where the Kasner

region is the same thickness as the Einstein de Sitter regions. From left to right the

observation angles are ψobs = 0 followed by increments of 10 up to 80. The curves

overlap in the FLRW regions and are separated as they enter the Kasner region.

distant region of the universe might look if there was a sizable Kasner region between

the source and the observer. The result is plotted in Figure 3.4. We can see that the

Kasner region lowers the redshift for sources at a longer lookback time with the effect

diminishing as the observation angle increases. The result is that for lower angles, it is

possible to have sources with the same redshift but different lookback times.

To get a sense of how the overall redshift would look for the entire sky we can in a

sense “average” out the effect by integrating over the solid angle. We can express this

averaged lookback time and redshift as

L =1

4π

∫ π

0

∫ 2π

0

ℓ sinψdψdθ and X =1

4π

∫ π

0

∫ 2π

0

x sinψdψdθ . (3.35)

If we assume cylindrical and z symmetry about the observer, we can simplify the expres-


Figure 3.5: Lookback time and redshift relation averaged over all observations angles.

The top curve is for an Einstein de Sitter model. The middle curve is for a model with

three equal slices. The bottom curve is for a model with one thin Einstein de Sitter slice

and one large Kasner region.

sions to

L =

∫ π/2

0

ℓ sinψdψ and X =

∫ π/2

0

x sinψdψ . (3.36)

Using a Simpson rule integrator we can numerically plot these values for different models.

The results for the above models (one slice, two slice and three slice models) are plotted

in Figure 3.5. The solid curve in Figure 3.5 represents a model with one slice, which is

equivalent to an Einstein de Sitter universe and serves as an upper bound. As the ratio

of Kasner regions increases the lookback time appears to reach farther into the past for

the same redshift values.


3.2.2 Possible CMB Data

The CMB is often treated as an indication of the isotropy of our universe. If this is true,

then looking at the CMB is a good measure of anisotropy. Could an anisotropic universe

also produce the same degree of isotropy that we see in the CMB? In this section we

attempt to answer that question in relation to the Cheese Slice model. This will allow

us to place a limit on the possible ratio of FLRW regions to Kasner regions. Using our

method to interpret anisotropy in the CMB would mean integrating back to a constant

lookback time, the time of last scattering, and seeing how the redshift varies as one

changes the angle of observation. We will assume that a light ray propagating at 0 will

travel through all the slices in the chosen model. A plot of the results can be seen in

Figure 3.6. If we assume the universe is 13×109 years old [55], and the CMB was formed

when the universe was 4×105 years old, that gives us a normalized time of t = 3.1×10−5

if t = 1 were the present (or ℓ = −0.999969). For an Einstein de Sitter model, we see that

it is isotropic and the redshift is on the order of 1025, which is the same magnitude as

the observed redshift for the CMB [31]. For larger Kasner regions we can see a dramatic

drop in the redshift for lower angles. At higher angles near 90 we see that it always

reaches the Einstein de Sitter limit. This is expected because an observation angle of

90 means the light ray never leaves the observer’s FLRW slice. With this data we see

that a Kasner region of comparable thickness to the Friedman regions would certainly

be noticed in the CMB. However, when the Kasner regions are small in comparison, the

anisotropy is not as noticeable. Thus, we can place a limit on the size of possible Kasner

slices in our universe by imposing a limit of order 10−5 anisotropy that is observed in the

CMB.

To be more rigorous, we can see if the position of a thin Kasner slice would affect

the anisotropy measured in the CMB. We do so by using a three slice model. Using a

relatively thin Kasner slice, we can change the ratio of thicknesses of the two FLRW slices.

In effect we can picture this as “moving” a thin Kasner slice through a predominantly


Figure 3.6: Redshift of the CMB for different models are plotted against observation

angle. A constant lookback time is used to represent the CMB. The top curve represents

the Einstein de Sitter universe, while the dotted curves from top to bottom represent

models with ratios of thicknesses being F : K : F = 0.3 : 0.01 : 2.5, F : K : F : K : F =

0.3 : 0.01 : 0.3 : 0.01 : 2.0 and F : K : F = 1 : 1 : 1. That is, one thin void, two thin

voids and one relatively thick void respectively.


Figure 3.7: Redshift of the CMB are plotted against observation angle for a range of

models with a thin Kasner slice between two thick FLRW slices. The ratios of the

FLRW slices are changed incrementally which is equivalent to changing the position of

a thin Kasner slice in a predominantly FLRW model. From top curve to bottom curve

the ratios of thicknesses are F : K : F = (1 : 0.1 : 5), (2 : 0.1 : 4), (3 : 0.1 : 3), (4 : 0.1 :

2), (5 : 0.1 : 1).

FLRW universe. The result is seen in Figure 3.7. The minimum redshift seen in each

curve, that manifests itself as a dramatic dip, is the angle at which the null ray spends

a maximum amount of time in the Kasner slice. At angles higher than this, the bang

time is reached before the ray has entirely reached the end of the Kasner slice. At angles

near 90 the ray spends its entire time in the local FLRW region and we see the usual

Kasner-free redshift we expect. We can see that there is a slight dependence on position

of the Kasner slice. We can see why this is so from equation (3.30). The redshift is very

much dependent on the time of entry and exit from the Kasner slice. The ratio tout/tin

is smaller for slices that are farther away (time reversed ray tracing). This causes an


Figure 3.8: Redshift of the CMB are plotted against observation angle for models with

three slices. The thickness of the Kasner region is changed incrementally. From top the

curve to bottom the ratio of thicknesses are F : K : F = (3 : 0.1 : 3), (3 : 0.2 : 3), (3 :

0.3 : 3), (3 : 0.4 : 3), (3 : 0.5 : 3).

overall lowering of the redshift.

To see how this effect compares to the effect of changing the thickness of the Kasner

slice, we can keep the Kasner slice at the same “distance” from the observer, but change

its thickness. A plot of the effect can be see in Figure 3.8. In this plot we keep the

thickness of the local FLRW slice constant and change the thickness of the Kasner slice.

We can see that the thickness of the Kasner slice causes a drop in the redshift. Again the

feature of a limit is seen for larger angles. The drop in redshift is greater than the one

found in Figure 3.7. Thus we can conclude that the thickness of the slice plays a more

significant role in determining the redshift than the position of the slice.

Finally we can see if the number of slices affects the redshift when the total ratio of

Kasner to FLRW is kept constant. To do so we choose a three slice model with ratio


Figure 3.9: Redshift of the CMB are plotted against observation angle for models with

different number of slices. The total ratio of thicknesses of Kasner to FLRW regions are

kept constant while the number of slices changes from three (top curve) to five (middle

curve) to seven (lower curve).

F : K : F = (1 : 1 : 1) and compare it to a five slice model with ratio F : K : F : K :

F = (4 : 3 : 4 : 3 : 4). In each case the first ’F’ refers to the observer’s local slice and the

consecutive ’K’s and ’F’s refer to subsequent slices. Taking the scheme one step farther,

we have a seven slice model with ratios F : K : F : K : F : K : F = (3 : 2 : 3 : 2 : 3 :

2 : 3). In all three cases the total thickness of the Kasner region is one half of the total

thickness of the FLRW regions. The result is plotted in Figure 3.9. Again we can see

the feature of a limit for higher angles as in the previous Figures 3.7 and 3.8. What we

see is that there is a lower redshift for the model with more slices, but this could also

be due to the fact that we necessarily need to place these slices father away when there

are more slices. Thus it is difficult to separate the change in redshift due to these two

different effects.


3.3 Summary and Discussion

From the above cases, it is clear that a universe with alternating layers of Kasner and

FLRW regions will have profound observational consequences. Depending on the size of

the Kasner region and the angles of observation, it is possible to observe blueshifts where

a redshift might be expected as was shown in Figure 3.3. We could also observe two

sources at different lookback times with the same redshift as in Figure 3.4.

In terms of the construction of the Cheese Slice model, it appears that the thickness

of the Kasner slices seems to have the most effect on the redshift with the position and

number of slices playing a smaller role. Thus the most important parameter we should

consider is the ratio of the thickness.

The observed temperature anisotropies in the CMB are on the order of 10−5 [41].

To achieve an anisotropy of this magnitude in our model, we would require a very thin

Kasner slice. For instance, in a three slice model, we would require a ratio of F : K :

F = (3 : 0.00001 : 2.5), that is a Kasner slice on the order of ten thousand times

thinner than the FLRW regions. While this may seem like an insignificant ratio it does

fundamentally change the spacetime. Inhomogeneity is introduced and the symmetry

of the FLRW spacetime is broken. This could have profound impacts on some features

of the cosmology. We continue to investigate the Cheese Slice “toy model” to discover

the consequences of breaking these symmetries. In Chapter 4 we look at how the initial

singularity might manifest itself and in Chapter 5 we will consider whether this type of

inhomogeneity could be incorporated in braneworld theory.

Chapter 4

The Structure of the Singularity

Perhaps the most compelling aspect of cosmology is the prediction that our universe

began as a singularity. The theorems of Hawking and Ellis [54] have shown, under very

general conditions, that such a singularity is unavoidable. Thus we are forced to confront

a situation in which the theory of General Relativity (GR) appears to predicts its own

downfall. There are two ways in which this breakdown can occur. The first is through

unbounded physical parameters such as infinite densities or infinite spacetime curvature.

The second is through loss of predictability manifesting itself in the existence of a Cauchy

Horizon. It is widely believed that some theory of quantum gravity could resolve such

singularities, but a satisfactory theory of quantum gravity has yet to be produced. In

addition, any theory of quantum gravity would necessarily have to reproduce the results

of GR in the appropriate limit. Thus it is important that we understand the approach

to these singularities.

Investigating the initial singularity is akin to asking what the initial state of our

universe was. The FLRW models have homogeneous and isotropic singularities. These

symmetries hold true regardless of how close to the singularity we choose to examine.

This can be described as a conformal singularity, one that arises from scaling the entire

space. There is no reason to believe that the initial conditions of our universe contains

48

Chapter 4. The Structure of the Singularity 49

such a high degree of symmetry. The question is then, what happens when this symmetry

is broken? This a difficult question to answer in a general sense, but we can make use

of some toy models to break these symmetries and see what kind of behaviour results.

In this chapter we choose to investigate the singularity of the Cheese Slice universe to

see how this singularity might manifest itself when the spacetime is inhomogeneous and

anisotropic. We begin with some general discussion about singularities and then look at

how a matching can affect the singularity structure. In particular we focus on a property

of singularities that is referred to as asymptotically velocity term dominated (AVTD).

4.1 Definition of a Singularity

Many difficulties arise in trying to define the notion of a singularity [103]. There are

certain phenomena that we would like to include in the definition, such as points in

spacetime where the curvature is unbounded or where physical quantities become infinite.

However using physical properties to define a singularity is not sufficient since there are

examples of spacetimes, such as a cone-like spacetimes [37], where there is a point that

cannot be defined and yet the curvature is zero everywhere. In two dimensions this

singular point would be the vertex of a cone.1 Therefore, there must be a definition of

a singularity that encompasses many different spacetimes that we intuitively consider

singular.

The most satisfactory definition of a singularity, one that encompasses the many

different phenomena, is to define it as a point where timelike and null curves cannot be

extended. One also has to add the condition that the spacetime is inextendible. That

is, the spacetime is not isometric to a proper subset of any other spacetime. This avoids

the complications of artificially creating singularities, by removing a point in Minkowski

space for instance.

1Also known as conical singularities. These spacetimes can be envisioned as Minkowski space with awedge removed and remaining boundary planes identified. The vertex of the wedge becomes undefined.


Mathematically we describe this end of spacetime as geodesic incompleteness. We

can also add the restriction that it must be either timelike or null incomplete. Since

in these cases we would be considering the motion of force-free particles or observers.

An end of the worldline of a particle would mean that we can no longer describe the

motion of a particle after a finite proper time. Spacelike geodesic incompleteness is not

included in this definition since it doesn’t have clear physical implications. The definition

can be made more robust by including all timelike curves. One can define a generalized

affine parameter on such curves and define the spacetime to be b-complete if there is an

endpoint for every curve of finite length as measured by this parameter [54]. A spacetime

is thus singular if it is b-incomplete.

4.2 Classification of Singularities

With the definition of singularities in hand, we can now go ahead and look at the different

types of singularities that can exist.

4.2.1 Isotropic Singularities

The isotropic singularities are well understood and much work has gone into understand-

ing them [38]. They can also be described as conformal singularities, that is a singularity

that arises from scaling the entire space. For instance, the FLRW singularity can be seen

as arising from a space that has scaled down to the point where the density becomes

infinite. It can also be shown that spacetimes with such singularities must be filled with

matter. Near the singularity, this matter will automatically satisfy the strong and weak

energy conditions. The dominant energy conditions can be satisfied, depending on the

equation of state.

It is the high degree of symmetry that allows isotropic singularities to be examined in

such detail. When it comes to more general singularities, the task becomes much more


difficult.

4.2.2 Classification Scheme

Ellis and Schmidt [37] set up a classification scheme to describe different types of singu-

larities.

Quasi-regular singularities are singular points at which there are no curvature ob-

structions to extending the space time. The tensor components Rabcd are well

behaved in a parallel frame along all curves approaching the singularity. The space-

time is well behaved near the singularity. An example of such a singularity is the

vertex of a cone. Such singularities are generally unstable and therefore unlikely to

occur in a physical spacetime.

Curvature singularities are the opposite case where the curvature tensor components

Rabcd are badly behaved. This class of singularities can be further subdivided.

Scalar singularities are singularities where a scalar quantity such as the Kretschmann

scalar, RabcdRabcd, or a physical quantity such as pressure or density, becomes

badly behaved. The Schwarzschild singularity would fit this description as the

Kretschmann scalar diverges.

Non-scalar singularities are singularities in which scalar quantities are well be-

haved yet the curvature is an obstruction to extending the spacetime. One

can perform a conformal transformation on a Taub-NUT spacetime to produce

this type of singularity [94].

These curvature singularities can be further sub divided into divergent singularities

where the relevant components are unbounded, such as the Schwarzschild singular-

ity, or oscillatory singularities where the components are bounded but oscillate


as one approaches the singularity such as in the Bianchi IX, “Mixmaster” type

universes [102].

4.2.3 Strength of a Singularity

One may also classify a singularity according to its strength. An early definition arrived

at by Tipler [100] is generally accepted. Under Tipler’s definition, a strong singularity is

one in which a volume element vanishes at the singularity. A volume element is defined

by taking all the linearly independent spacelike Jacobi fields along a timelike geodesic.

If we choose a geodesic such that it terminates at the singularity, we can then find the

volume element and see if it vanishes.

From this definition many physically relevant singularities such as the Schwarzschild,

Reissner-Nordstrom and dust filled FLRW singularities are all strong. Necessary and

sufficient conditions for the occurrence of strong singularities were given by Clarke and

Krolak [23].

Recently this definition has been refined by Ori [84]. To maintain a distinction from

Tipler’s definition, strong singularities according Ori’s definition are referred to as defor-

mationally strong. A singularity is considered to be deformationally strong if any of the

Jacobi fields become unbounded near the singularity. By this definition a Tipler strong

singularity is a subset of the deformationally strong singularity.

4.3 More General Singularity Structures

4.3.1 Properties of the Matching

Before delving directly into the topic of singularities we first look at properties of a

spacetime that results from a matching. We can ask what kind of properties, if any, do

spacetimes constructed from a matching inherit from the spacetimes used in its construc-

tion? Let us call the matched spacetime W and the constituent spacetimes used in its


construction V + and V − with the appropriate Σ on each side. The Darmois conditions

are imposed across Σ.

It is clear that even if V + and V − are both homogeneous, W in general is not ho-

mogeneous as we have seen with the Cheese Slice universe. However, such a scenario

is possible. McClure and Dyer [79] have shown that a radiation dominated Einstein-de

Sitter universe can be matched to a matter dominated Einstein-de Sitter universe across

a surface of constant time. In this case the matched spacetime, W , is spatially homo-

geneous at any t = t0, with t0 = constant. It appears that whether or not W inherits

homogeneity from V + and V − depends highly on the nature of the matching.

It would be interesting however if there were some intrinsic properties that W must

inherit from V + and V −. For example, any matching requires the intrinsic metric of Σ

to be identical as viewed from either side obeying equation (2.10). This implies that any

symmetries on Σ seen from V − must also be seen from V +. This is evident in the Swiss

Cheese model where the matching surface is spherically symmetric and in the Cheese

slice where the surface is cylindrically symmetric as seen from either side. However, since

matching conditions are strictly local, it is not known if inherited symmetries on Σ have

any consequences on the global structure of the spacetime.

With regard to singularities, we would like to see what sort of singularity might result

when regions with different singularity structure are matched together. If both V − and

V + have initial singularities and we match them along a timelike surface, what can we

surmise about the singularity that is constructed in W?

4.3.2 BKL Picture of Cosmological Singularities

Belinskii, Khalatnikov and Lifshitz (BKL) tried to construct a picture of the approach

to a generic cosmological singularity [5]. By generic, they refer to a solution of the EFE

that has the most number of arbitrary functions. Initially this was an attempt to see

if singularities were a generic feature of spacetimes or if they were a result of overly


simplified FLRW cosmology. The singularity theorems of Penrose and Hawking have

laid this question to rest by showing that singularities can occur given a very general

set of conditions. However these theorems only prove that the singularities exist and do

not indicate the nature of these singularities. Thus the work of BKL remains relevant as

they attempt to describe the general approach to the singularity and shed light on the

behaviour of these singularities.

The BKL approach begins by choosing a frame such that the singularity occurs si-

multaneously for all points in the spacetime. They then found that the terms describing

the time evolution toward the singularity dominated over terms describing the spatial

curvature. By making the assumption that the spatial curvature terms were negligible,

they were able to derive an approach to the singularity such that the spacetime expanded

and contracted in different directions in an oscillatory manner. Furthermore this assump-

tion effectively states that each spatial point evolves independently of any other spatial

point. Any particle’s approach to the singularity is independent of any other particle.

Though they might have influenced each other away from the singularity, these particle

interactions are effectively cut off as one asymptotically approaches the singularity.

In the context of spacetime matchings, this could mean that regions of a matched

spacetime do not effect other as one approaches closer to the singularity.

The BKL approach has not been rigorously proven or refuted [90], but certain space-

times have been demonstrated to satisfy these properties, which came to be known as

“velocity dominated” or “asymptotically velocity term dominated” (AVTD) solutions.

The Gowdy spacetimes [59] and most general Bianchi Types2 [6, 102] were shown to

exhibit these types of singularities. Recent numerical results have also provided some

support for this approach to describing generic singularities [104].

2In particular types VII and IX have oscillatory singularities. The latter is also known as the Mix-master Universe [80].


4.3.3 Cauchy Horizon Singularities

Recently evidence for another generic form of singularities have been seen [17]. These

types of singularities occur inside the inner event horizon of charged or rotating black

holes. Due to the structure of these event horizons in-falling radiation from the entire

history of the universe outside the black hole will accumulate at the horizon. An in-falling

observer will see the entire history of the universe in a finite proper time. This in-falling

radiation is infinitely blue shifted. This causes a divergence in the energy momentum

tensor and in turn, from the EFE, results in a curvature singularity.

These singularities are distinct from the generic BKL variety because they are null

singularities rather than spacelike. Also, current evidence seems to indicate that such

singularities are deformationally weak [16].

4.4 The AVTD Singularity

Attempts have been made to prescribe some precise definition of what it means to sat-

isfy the BKL properties. The assumption that the evolution towards the singularity is

independent of spatial curvature allows the EFE to be simplified. Eardley, Liang and

Sachs (ELS) [35] formulated a definition of this property that they termed “velocity dom-

inated”. They used a Gaussian normal coordinate system with the timelike coordinate,

t, being orthogonal to all spatial coordinates. The spacetime is then foliated by space-

like hypersurfaces of constant time. These hypersurfaces are three-dimensional manifolds

that have an intrinsic metric, (3)gαβ, extrinsic curvature, Kαβ and an intrinsic Einstein

tensor, (3)Gαβ. By neglecting (3)Gαβ they were able to integrate the four-dimensional

EFE and solve for (3)gαβ. Focusing on dust cosmologies, they defined the singularity

to be at the time t = t0 such that the density ρ→ ∞ as t→ t0. The structure of the

singularity is then given by (3)gαβ∣

∣

t0and its evolution depends only on Kαβ. Since Kαβ

describes how the three-surfaces curve in the ambient spacetime (in this case the ∂/∂t


direction), it is often referred to as a “velocity” and thus the term “velocity dominated

singularity” was coined.

This method demonstrates how the structure in the late universe can be directly

attributed to the structure in the singularity itself. However, the ELS approach requires

the use of a time orthogonal coordinate system and is only applicable to dust cosmologies.

One would like to be able to extend this concept of velocity dominated singularity to more

general spacetimes. In that spirit Isenberg and Moncrief has formulated what he refers to

as “Asymptotically Velocity Term Dominated” (AVTD) spacetimes [59]. The definition

of AVTD incorporates more general foliations of spacetimes and matter content. We

begin with a detailed definition of the AVTD property then we define how a spacetime

constructed from a matching can be considered AVTD. Finally, we use the example of

the Cheese Slice universe to demonstrate how this concept can be applied.

Both the FLRW and the Kasner spacetimes have initial singularities that are AVTDS.

We will propose a criterion to determine whether or not a matched spacetime is AVTD

and use this to show that Cheese Slice universe inherits this property and the singularity

is of type AVTDS. We show that this is true in both the matchings with spatially flat

FLRW and open FLRW slices. We conclude by conjecturing that if V − is AVTD and Σ

is a timelike surface, then V + and W are necessarily AVTD as well.

4.4.1 Definitions

Let U be a spacetime with metric gab and coordinates xa. We begin by choosing a

spatial foliation with intrinsic coordinates ξα on each leaf of the foliation as depicted in

Figure (4.1).

Next we identify the intrinsic metric,

γαβ =∂xa

∂ξα∂xb

∂ξβgab, (4.1)


∂∂t

coordinates: xaU, metric: gab

Π0,coordinates: ξαmetric: γαβ

na

t0

t1

t2

t3

Figure 4.1: The spacetime U with coordinates xa is foliated with spacelike surfaces, Πi,

with coordinates ξα. Successive leaves of the foliation are labelled by the time coordinate

t. The timelike foliation vector ∂∂t

is in general different from the normal vector, na.

and extrinsic curvature,

Kαβ =∂xa

∂ξα∂xb

∂ξβnb;a, (4.2)

of the spacelike three-surfaces, where na is the normal to the surface. These values are

defined identical to equation (2.8) and (2.9) however we restate them to emphasize that

this foliation is not identical to the matching surface Σ. The leaves of the foliation are

spacelike surfaces whereas Σ, as we have defined it, is a timelike surface. The mean

curvature is then K = Kαα . The timelike foliation vector, ∂/∂t, where t is a timelike

coordinate that labels successive leaves of the foliation, describes the evolution of the

three surface. If we use the normal to define a frame, we can write this vector in terms

of a component normal to the surface and a component tangential to the surface.

∂

∂t= Nna +Ma. (4.3)

These components are called the lapse, N , and shift, Ma, respectively [106].

The matter density, ρ, momentum, Jα, and spatial stress density, Sαβ, must also be


considered. These quantities must satisfy the EFE written in the form of constraint

equations [59],

(3)R−KαβKαβ +K2 = 2ρ, (4.4)

(3)∇αKαβ − (3)∇βK = −Jβ (4.5)

and evolution equations,

∂

∂tγαβ = −2NKαβ + LMγαβ, (4.6)

∂

∂tKαβ = N

[

(3)Rαβ +KKα

β + Sαβ +1

2γαβ (ρ− Sνν )

]

− (3)∇α(3)∇βN + LMKαβ , (4.7)

where (3)R and (3)Rab are the spatial Ricci scalar and Ricci tensor respectively. (3)∇ is

the three dimensional covariant derivative and LM is the three-dimensional, spatial Lie

derivative in the direction of Ma. Also, geometrized units have been used where 8πG = 1.

Next, the velocity term dominated solutions (VTD) are defined by neglecting all the

spatial derivatives in the field equations. Equations (4.4)-(4.7) then reduce to the VTD

constraint equations,

KαβKαβ + K2 = 2ρ, (4.8)

(3)∇αKαβ − (3)∇βK = −Jβ, (4.9)

and the VTD evolution equations,

∂

∂tγαβ = −2NKαβ, (4.10)

∂

∂tKαβ = N

[

KKαβ + Saβ +

1

2γαβ

(

ρ− Sνν

)

]

. (4.11)

Note that in general the spatial derivatives in ρ, Jα and Sαβ vanish as well. We use

the over-tilde, ˜, to indicate their distinctiveness from the quantities in the Einstein

equations (4.4)–(4.7).

Solutions of the field equations (4.4)–(4.7) are then defined to be AVTD if in the limit

of large t they approach the solutions to the VTD equations (4.8)–(4.11). That is, as


t→ ∞, the values of

γαβ − γαβ

Kαβ − Kαβ

ρ− ρ

Jα − Jα

Sαβ − Sαβ

→ 0. (4.12)

A singularity is said to be an AVTDS if the spacetime is AVTD and the foliation is

chosen such that the singularity is approached as t→ ∞. The AVTDS property is highly

dependent on the choice of foliation. A spacetime that is AVTD in one foliation might

not appear AVTD in another foliation. A spacetime needs only one foliation to satisfy

the AVTD property to be considered AVTD.

4.4.2 AVTD Property of Matched Spacetimes

We will now examine how a spacetime constructed from a matching can be considered

AVTD. The Darmois matching conditions [28], detailed in Chapter 2.1.1 will be used

throughout this section.

Let W be the spacetime constructed from the matching of V − and V + across the

surface Σ. This assumes that the Darmois conditions are satisfied across Σ. Also, let

Π±t± denote leaves of a foliation of V ±, parametrized by t±, such that V ± is AVTD. In

general t− and t+ are different time coordinates. If each leaf of the foliation Π−t− matches

with each leaf of the foliation Π+t+ along the surface Σ, then this constitutes a foliation of

W such that W is AVTD. Note that the corresponding VTD solutions must also match

across the surface Σ in the same manner.

To clarify the matching of the Π−t− with Π+

t+ , let us single out one leaf of the foliation

on each side and call them Π±0 as illustrated in Figure 4.2. Π±

0 are spatial three-surfaces

in V ±. We wish to match Π−0 with Π+

0 across the surface Σ. However, Σ is a timelike

three-surface and the intersection of Π± with Σ is a spatial two-surface. Let us call this


t

V +

Π−0

Σ

θ2

θ1Λ

V −

Figure 4.2: A leaf of the foliation on each side is singled out, Π±0 . The intersection of Π±

0

with Σ is what we refer to as the corner, Λ, which is itself a two-surface. Σ is a timelike

surface while Π is a spacelike surface. The corner can also be seen as the intersection of

Π+0 and Π−

0 . The corner is a subspace of all the depicted hypersurfaces.

two surface the “corner” and denote it by Λ with the coordinates ζA. There is also a

two-dimensional space of normals to Λ. Let mAa be an orthonormal basis for this space.

The matching conditions at a corner have already been thoroughly investigated by

Taylor [98] and reviewed in Chapter 2.1.2. In general Π−0 and Π+

0 could meet at any angle;

thus the corner conditions, equations (2.15)–(2.18) are necessary. If the corner conditions

are satisfied on Λ for all time t± then the union of the foliations, Π ≡ Π−0 ∪Π+

0 , constitutes

a foliation of W such that W is AVTD. We will use this method to show that the Cheese

Slice universe is AVTD and the singularity is an AVTDS.

4.5 Singularities in the Cheese Slice Universe

We will use the FLRW line element in the form,

ds2F = −dt2 + a2(t)

[

dr2

1 − kr2+ r2dφ2 + (1 − kr2)dz2

]

, (4.13)


where k = −1, 0, 1 and the Kasner line element given by,

ds2K = −dT 2 + T 2p1dX2 + T 2p2dY 2 + T 2p3dZ2, (4.14)


p1 + p2 + p3 = 1 = p21 + p2

2 + p23. (4.15)

Both the Kasner and the FLRW spacetimes have an initial singularities (t = 0 = T ). We

look at the cases of the spatially flat FLRW matching and the open FLRW matching in

turn.

4.5.1 Case (i) Flat FLRW, k = 0

The coordinates defined in equation (4.13) and (4.14) single out a natural foliation that

we will use to check the AVTD property for the spatially flat case.

The Kasner spacetime satisfies the VTD equation (4.8)–(4.11) directly; therefore it is

trivially AVTD. With the pressure free FLRW spacetime the spatially flat case satisfies

the VTD equation (4.8)–(4.11) as well with the following quantities,

γαβ = a2 diag (1, r2, 1) , Kαβ = a,ta diag (1, r2, 1) ,

ρ = 3(a,t

a

)2, and Jα = 0 = Sαβ, (4.16)

where ,t = ∂∂t

. Thus both sides are AVTD. Furthermore, we can make the coordinate

transformation τ = − ln t to set the singularity at τ = ∞ and all the conditions of an

AVTDS are satisfied.

To show that the matched spacetime is also AVTD with the chosen foliation we must

check that the corner conditions, equation (2.15)–(2.18), are satisfied.

On the FLRW side the corner is defined as z = z0 and t = t0, with z0 and t0 being

constants. The orthonormal basis of the space normal to the corner can be specified as

m1+a = (1, 0, 0, 0) and m2+

a = (0, 0, 0, 1). (4.17)


Thus we have,

γ+AB = a2 diag (1, r2) , K+1

AB = −a,ta diag (1, r2) ,

K+2AB = 0 and τ+

A = 0. (4.18)

On the Kasner side the corner is defined as Z = Z0 and T = T0, with Z0 and T0 being

constants. The basis for the space normal to the corner can be specified as

m1−a = (1, 0, 0, 0) and m2−

a = (0, 0, 0, 1). (4.19)

Thus we have,

γ−AB = diag (T 2p1 , T 2p2) , K−1AB = T−1 diag (−p1T

2p1 ,−p2T2p2) ,

K−2AB = 0 and τ−A = 0. (4.20)

If we choose the coordinates on the corner as ζA = u, v, parametrize the surface as

r cosφ = u = X and r sinφ = v = Y, (4.21)

we can satisfy equation (2.15)–(2.17). Recall that a = t2/3 and p1 = p2 = 2/3. Further-

more the surfaces defining the corners are orthogonal on both sides and the matching

surface subtends an angle of π as seen from either side and thus equation (2.18) is also

satisfied. Therefore we have a matching at the corner and the flat Cheese Slice universe

is AVTD.

Also, notice that for the matching to take place we have also identified the time

coordinates t = T . With the coordinate transformation τ = − ln t = − lnT we can set

the singularity at τ = ∞ and the conditions for an AVTDS are satisfied.

4.5.2 Case (ii) Open FLRW, k = −1

In general proving the AVTD property is highly dependent on the choice of foliation. A

spacetime that is AVTD in one foliation might not appear to be AVTD in another. Thus


we must be careful in our choice of foliation. To show that the open Cheese Slices can

be AVTD we make the following transformation on the FLRW side,

z = z − 1

2ln(1 + r2). (4.22)

The FLRW metric (4.13) then becomes,

ds2F = −dt2 + a2(t)

[

dr2 + r2dφ2 + 2rdrdz + (1 + r2)dz2]

. (4.23)

On the Kasner side we will make the transformations,

R =√X2 + Y 2, (4.24)

Φ = arctan(Y/X), (4.25)

Z = Z − 9

16b5

[

−3bT1

3

√

1 + b2T2

3 + 2b3T

√

1 + b2T2

3 + 3 ln

(√

1 + b2T2

3 + bT1

3

)]

(4.26)

and

t =3

2b3

[

bT1

3

√

1 + b2T2

3 − ln

(√

1 + b2T2

3 + bT1

3

)]

, (4.27)

where b is a positive constant. With these transformations the Kasner metric (4.14)

becomes,

ds2K = −dt2 + T

4

3

(

dR2 +R2dΦ2)

+ 2bdtdZ + T− 2

3dZ2. (4.28)

The matching now takes place along the surface z = z0 on the FLRW side and Z = Z0 on

the Kasner side, with z0 and Z0 being constants. The coordinates, φ = Φ and t = t, can be

identified along the matching surface. We must also have r = 23bR and a2(t) = 9

4b2T

4

3 (t).

We will use this new foliation to check the AVTD property. Starting with the FLRW

case it is straightforward to check that equation (4.4)–(4.7) are satisfied with the following

quantities,

γ11 = a2,

γ13 = a2r,

γ22 = a2r2,

γ33 = a2(r2 + 1),

K11 = a,ta,

K13 = a,tar,

K22 = a,tar2,

K33 = a,ta(r2 + 1),


ρ = 3

(

a2,t − 1

a2

)

, and Ja = 0 = Sab. (4.29)

The corresponding VTD solution is the spatially flat FLRW solution. We can see that

equation (4.12) is satisfied and thus the open FLRW is AVTD.

Turning to the Kasner case we find that it also satisfies the VTD equation (4.8)–(4.11)

with the lapse and shift being,

N =

√

1 + b2T2

3 and Ma = (0, 0, b) (4.30)

respectively. Therefore it is once again trivially AVTD.

Next we check the corner conditions, equation (2.15)–(2.18). The corners on the

FLRW and Kasner sides are defined as z = z0, t = t0 and Z = Z0, t = t0 respectively

with t0 and t0 being constants. Recall that the coordinates are such that r = 23bR and

Φ = φ. Let us use the superscript, −, to denote the Kasner side and, +, to denote the

FLRW side. The first corner condition, equation (2.15), is satisfied with,

γ−AB = diag(9

4b2T

4

3

0 , RT4

3

0 ) = diag(a20, ra

20) = γ+

AB, (4.31)

where T0 = T (t0) and a0 = a(t0). Let an orthonormal basis of the corner be chosen on

both sides such that,

m−1α = (0, 0, 0, T

− 1

3

0

√

1 + b2T2

3

0 ) m+1α = (0, 0, 0, a0)

m−2α = (1, 0, 0,−b) m+2

α = (1, 0, 0, 0). (4.32)

Then the second corner condition, equation (2.16), is satisfied with,

K−1AB =

3

2bT

2

3

0 diag(1,4

9b2R2) = a0 diag(1, r2) = K+1

AB (4.33)

and

K−2AB =

2

3b2T

1

3

0

√

1 + b2T2

3

0 diag(1,4

9b2R2) = a,t0a0 diag(1, r2) = K+2

AB. (4.34)

The torsion is identically zero on both sides satisfying equation (2.17). On the FLRW

side the foliation is orthogonal to the matching surface and the matching surface itself


subtends an angle of π about the corner. On the Kasner side, the foliation is not orthog-

onal to the matching surface. Fortunately the matching surface also subtends an angle

of π about the corner. This ensures condition equation (2.18) is satisfied on both sides.

Similar to the spatially flat matching, the time coordinate may be transformed as

desired, since it is identical on both sides, to ensure that the singularity is reached as

t→ ∞ and the singularity may be considered an AVTDS.

Let us illustrate how this singularity in the Cheese Slice universe manifests itself. In

the Kasner regions the initial singularity is of a cigar type and at late times the Kasner

regions become pancake-like singularities. In the FLRW slices we have an initial point-like

singularity and no singularities at late times. Thus we can visualize the initial singularity

of the Cheese Slices as an inhomogeneous chain of cigar-like singularities joined by point-

like singularities. At late times, the Cheese Slices become an inhomogeneous matter filled

space with pancake-like singularities throughout, as illustrated in Figure (4.3).

4.5.3 Summary and Discussion

We have proposed a criterion with which we may consider a matched spacetime to be

AVTD. First, both sides of the matched spacetime must be AVTD. Secondly each leaf of

the chosen foliation must also match across the surface at an intersection that we refer

to as the corner. We have also demonstrated this with the example of the Cheese Slice

universe. The flat Cheese Slice satisfies these conditions in a straightforward manner

whereas the open Cheese slice required more effort to find a foliation that satisfies the

AVTD property and the matching conditions. In a general matching it may be difficult

to find a foliation that is consistent with the matching and the AVTD property. However,

as we have shown, it is possible in the case of Cheese Slice universe for the singularity to

inherit the AVTD property from the different spacetimes used in its construction.

Recall that the AVTD property refers to a spacetime in which the evolution towards

the singularity does not depend on the spatial curvature. This becomes very evident in


Figure 4.3: The Singularities of the Cheese Slice Model. The initial singularity is a

chain of cigar singularities, corresponding to the Kasner vacuum regions, and point sin-

gularities, corresponding to FLRW regions. At late time, the vacuum regions become

arbitrarily thin pancake-like singularities.


the Cheese Slice model. The number of layers of FLRW and Kasner and their relative

thicknesses are entirely arbitrary. In each layer we can choose a foliation Πn where n

labels the number of layers. It is straightforward to show that each Πn can satisfy the

corner conditions with each Πn+1. Repeating the process for all n we can build a foliation

Π ≡ Π1 ∪ · · · ∪Πn and show that the entire model is AVTD. The inhomogeneities in the

model exist at all times including at the initial singularities and thus it could be said that

the evolution of the model does not depend on the spatial structure. The inhomogeneities

could be seen as an initial condition arising from the singularity itself.

In addition to modelling inhomogeneities, these models of matched spacetimes are also

very useful in investigating what matching conditions could tell us about the properties

of spacetimes themselves. For example, we conjecture that any spacetime that can be

smoothly matched to an AVTD spacetime, using the Darmois conditions, must necessarily

be AVTD. The resulting matched spacetime would also be AVTD. The general proof of

this remains to be seen and is open to investigation. One possible method of proof

could lie in the use of Lichnerowicz type coordinates. Since Lichnerowicz conditions are

equivalent to the Darmois conditions, as discussed in Chapter 2, we can always find

an admissible coordinate system where the metric is continuous across the matching

surface. One could then use the foliation associated with this coordinate system to test

the AVTD property. In fact the coordinate transformations used in equation (4.24)–(4.26)

were inspired by an attempt to find the associated Lichnerowicz coordinates. A possible

pitfall might occur if the Lichnerowicz coordinates do not correspond to a foliation that

satisfies the AVTD property. It would be interesting to see if a general proof could be

found. If so this could lead the way to using the Darmois conditions to prove AVTD

properties of other spacetimes.

Chapter 5

Cheese Slice Braneworlds

There have been various theories proposed involving extra dimensions. One of the earliest

was that of Kaluza and Klein [62, 64] which was originally devised to unify gravity and

electromagnetism. Extra dimensions were also used in particle physics to incorporate the

fields of the standard model [105]. In these theories the extra dimensions are compactified

on a microscopic scale in relation to the observed spatial dimensions thereby recovering

the familiar four-dimensional spacetime. To compactify an extra dimension, one assumes

that this dimension is closed and has a finite volume such that it is too small to be

observable.

More recently others have suggested that the standard model fields could be confined

to three spatial dimensions by introducing a potential well that is narrow in the extra

dimension [91]. Much interest in these types of theories have been generated from string

theory and M -theory, which predicts eleven dimensions [56]. In these theories six of

the spatial dimensions can be consistently compactified leaving a five-dimensional bulk

spacetime [70, 13]. Our universe is then a domain wall or four-dimensional brane embed-

ded in this bulk. In general these theories are known as braneworld models. Since the

standard model particles are restricted to the brane, the extra spatial dimension need

not be compactified and can even be infinitely large. Randall and Sundrum have shown

68

Chapter 5. Cheese Slice Braneworlds 69

that it is possible to recover Newtonian gravity in this type of scenario at low energies

[87].

In the context of spacetime matchings we can view the braneworld scenarios as a

five-dimensional matching across a stress sheet similar to the situations discussed in

Chapter 2.1.3. In this case, the bulk spacetime is five-dimensional while the matching

surface is a four-dimensional subspace endowed with energy momentum. This energy

momentum is related to the jump in extrinsic curvature of the bulk in the same manner

as equation (2.30).

5.1 Braneworld Cosmologies

The challenge in implementing the braneworld models is to recover observed cosmology

on the brane and predict deviations from standard cosmology that might occur at high

energies such as in the early universe.

5.1.1 Randall-Sundrum Braneworlds

Randall and Sundrum originally proposed a two brane model to solve the hierarchy prob-

lem [88]. In this scenario the observer resides on a brane of negative tension and another

brane of positive tension exists at a finite distance in the fifth dimension. The bulk is a

slice of Anti-de Sitter (AdS) spacetime, that is a spacetime with negative cosmological

constant. Though this solution gave a novel solution to the hierarchy problem, it was

pointed out by Csaki et al. [26] that it could not recover conventional cosmology on the

brane.

It was then suggested that the negative tension brane could be moved to infinity

and an observer on the positive tension brane can recover Newtonian gravity as a low

energy approximation [87]. This one brane model has become the starting point of many


investigations into higher dimensional cosmologies. The five-dimensional metric is,

(5)ds2 = e−2|y|/ℓηabdxadxb + dy2, (5.1)

where the brane sits at y = 0 and ηab is the Minkowski metric. As a reminder, the

index convention we are using is A,B,C . . . = 0, 1, 2, 3, 4), a, b, c . . . = 0, 1, 2, 3),

i, j, k . . . = 0, 1, 2, 4), and α, β, µ . . . = 0, 1, 2). The |y| reflects a symmetry imposed

on the bulk about the brane. The constant ℓ is the curvature scale of the bulk related to

the five-dimensional cosmological constant by,

Λ5 = − 6

κ25ℓ

2. (5.2)

where κ25 is the five-dimensional gravitational coupling constant. This curvature scale,

with the exponential factor, serves to “squeeze” the gravitational field as close to the

brane as desired. A positive tension, σ, is required on the brane to balance the negative

cosmological constant in the form,

σ =6

κ25ℓ, (5.3)

such that an observer on the brane would not observe any cosmological constant. This

is known as the Randall-Sundrum fine tuning condition.

The metric (5.1) is a solution of the five-dimensional EFE,

(5)GAB = κ25

(

TAB − Λ5(5)gAB

)

, (5.4)

with TAB = 0. Thus this solution is a vacuum bulk and has a Minkowski brane, making

it an unrealistic cosmology. It is however, the simplest possible braneworld and serves as

a starting point into investigating braneworld scenarios.

5.1.2 Cosmological Braneworlds

A straightforward way to find a braneworld with realistic cosmology is to impose the

FLRW symmetries onto the brane. It is natural to use a coordinate system focused on


the brane such that the brane is located at y = 0 as in equation (5.1). It is always

possible to write the bulk metric in the form [70],

ds2 = −n(t, y)2dt2 + a(t, y)2γαβdxαdxβ + dy2, (5.5)

where γαβ is the maximally symmetric three-dimensional metric representing the spatial

part of the FLRW brane. Then the FLRW scale factor is a(t, 0) and we can always

rescale the coordinate time such that it corresponds to the cosmic time, n(t, 0) = 1. For

simplicity we can assume the bulk is empty and the energy-momentum only exists on

the brane,

TAB = SABδ(y) = diag (−ρb, P, P, P, 0)δ(y) (5.6)

where the Dirac delta, δ(y), serves to localize the density ρb and pressure P . Rather than

solving the five-dimensional EFE, equation (5.4), directly with this energy-momentum

tensor, it is more convenient to to see the brane as a matching surface across which the

bulk must match. Thus we can recall the Israel conditions from equation (2.30),

[Kab − δabK] = κ2

5Sab . (5.7)

If we assume that the bulk is symmetric about the brane, then the jump in extrinsic

curvature is just twice its value on one side, [Kab] = 2Kab. With the metric (5.5) these

conditions reduce to [71],

(n,yn

)

0+

=κ2

5

6(3P + 2ρb),

(a,ya

)

0+

= −κ25

6ρb. (5.8)

One can then solve the EFE away from the brane and include these conditions as con-

straints. Direct integration of the EFE and considering the junction constraints of equa-

tion (5.8) leads to the energy conservation equation,

ρb,t + 3H(ρb + P ) = 0, H2 ≡a2

0,t

a20

, (5.9)

which is unchanged from conventional four-dimensional cosmology. However, the Fried-

mann equation derived from other components of the EFE appears in a different form,

H2 =κ4

5

36ρ2b +

κ25Λ5

6− k

a2+C

a4, (5.10)


where a0 ≡ a(t, 0) and C is a constant of integration. The most striking difference from

the conventional Friedmann equation is the brane energy density appearing as a quadratic

term. Equation (5.10) is referred to as the modified Friedmann equation.

If we search for the simplest solution, letting C = 0 = H one arrives at,

|κ5|ρb = ±√−6Λ. (5.11)

Thus with Λ = − 6κ25ℓ2

we recover the Randall-Sundrum solution and the fine tuning

condition of equation (5.3) with the only energy content being the tension, −ρb = σ.

To find a more realistic cosmology we could insist that there be matter content in

addition to the constant tension, ρb = σ + ρ. With this energy density the modified

Friedmann equation (5.10) becomes,

H2 =

(

κ45

36σ2 − 1

ℓ2

)

+κ4

5

18σρ+

κ45

36ρ2 − k

a2+C

a4. (5.12)

If now the tension is fine tuned as in equation (5.3) then the first term vanishes. The

tension is then proportional to the Newtonian constant such that,

κ24 ≡ 8πG =

κ45

6σ =

κ25

ℓ. (5.13)

Thus the term linear in ρ is identical to the linear term in the conventional Friedmann

equation and the quadratic term can be seen as a second order correction at high energies.

At very high energies, ρ >> σ, the dynamics is completely dominated by the quadratic

term. The C term behaves like a radiation component and depends on the bulk Weyl

tensor. A detailed investigation of this term can only be carried out with an explicit

solution of the bulk.

In the current context, with the metric of equation (5.5), and assuming the bulk is

static in the fifth dimension, an exact bulk solution has been found [8]. In this case the


scale factor is given by,

a(t, y) =

1

2

(

1 +κ2

5ρ2b

6ρB

)

a20 +

3C

κ25ρBa

20

+

[

1

2

(

1 − κ25ρ

2b

6ρB

)

a20 −

3C

κ25ρBa

20

]

cosh(y/ℓ) (5.14)

− κ5ρb√−6ρBa2

0 sinh(|y|/ℓ)1/2

,

where ρB < 0 is the matter density in the bulk. The functions a0 and C are time

dependent as well as the density on the brane, ρb, and the density in the bulk, ρB. A

similar solution exists for ρB > 0. In the case of ρB = 0 the solution is,

a(t, y) =

a20 −

κ25ρb3a2

0|y| +[

κ45ρ

2b

36a2

0 +C

a20

]

y2

1/2

. (5.15)

In all cases the other metric coefficient is given by,

n(t, y) =a,ta0,t

. (5.16)

An explicit solution for the two brane case has been found to first order and discussed

by Binetruy et al. [9].

An alternate approach to analyzing brane cosmology is to assume a static bulk [58,

30, 29]. Essentially this is a coordinate transformation into a frame where the bulk is

static and the brane moves through the bulk in the extra dimension. The warped nature

of the bulk manifests itself as the cosmological expansion of the brane. While such a

setup is useful for analyzing the nature of the bulk and to make generalizations such

as adding a scalar field or other energy momentum in the bulk, it is not as convenient

when one is interested in the nature of the brane itself. Thus we will continue to use the

coordinate system in which the brane remains at y = 0 in the following sections.

In all cases the resultant cosmological braneworld must be a solution of the EFE.

Since the field equations describe the dynamics of spacetime and how it couples to the

matter content, it is generally assumed that the EFE in the form of equation (5.4)

is valid throughout the bulk. The field equations on the brane are then derived by


projecting the bulk field equations onto the brane via a Gauss-Codacci framework [93].

The generalization of the Gauss-Codacci equations (2.24)–(2.25) to five-dimensions is

given by,

(5)RABCDeAa e

Bb e

Cc e

Dd = (4)Rabcd + ǫ(n) (KacKbd −KbcKad) (5.17)

and

(5)RABCDnAeBb e

Cc e

Dd = (4)∇dKbc − (4)∇cKbd. (5.18)

The four-dimensional Einstein tensor is then given by,

(4)Gab =2κ2

5

3

[

TABeAa e

Bb +

(

TABnAnB − 1

4TAA

)

(4)gab

]

(5.19)

+KKab −KcaKbc −

1

2(4)gab(K

2 −KcdKcd) − Eab,

where

Eab ≡ (5)CABCDnAn

CeBa eDb , (5.20)

and (5)CABCD is the five-dimensional Weyl curvature. Note that the left superscripts have

been used to keep track of the dimensionality of certain tensors. We assume symmetry

about the brane and a metric of the form,

(5)ds2 = dy2 + (4)gabdxadxb (5.21)

where the brane is located at y = 0. Also, let the five-dimensional energy momentum be

of the form,

TAB = −Λ(5)gAB + SABδ(y) with SAB = −σ(5)gAB + τAB, (5.22)

where σ is the brane tension and τAB is the energy momentum on the brane. Then with

the Israel condition, equations (2.30) and (5.19), we arrive at the EFE on the brane,

(4)Gab = −Λ4(4)gab + κ2

4τab + κ25πab − Eab, (5.23)

where

Λ4 =1

2κ2

5

(

Λ +1

6κ2

5σ2

)

, (5.24)


κ24 =

κ45

6σ (5.25)

and

πab = −1

4τacτ

cb +

1

12ττab +

1

8(4)gabτcdτ

cd − 1

24(4)gabτ

2. (5.26)

The values of Eab are taken to be the limiting value as y → 0. This result differs from the

conventional EFE in four-dimensions by the addition of the πab term that is quadratic

in τab. There is also an additional term, Eab, that depends on the Weyl tensor of the

bulk. Thus, it is not possible to fully understand a braneworld solution without explicitly

knowing the bulk solution.

5.1.3 Anisotropic Braneworlds

A Kasner-type braneworld was first discovered by Frolov [43]. This model can be viewed

as the generalization of an isotropic model. Consider the five-dimensional Anti-de Sitter

metric described by,

ds2 = −f(r)dt2 +dr2

f(r)+ r2dσ2

3, (5.27)

where dσ23 represents a three-dimensional spatial metric of uniform curvature. The func-

tion f(r) is given by,

f(r) = k +r2

ℓ2with k = 1, 0,−1. (5.28)

The value of k is determined by the curvature of dσ23, whether it be spherical, planar or

hyperboloid and ℓ is the curvature scale related to the cosmological constant Λ = − 6ℓ2

.

To generalize equation (5.27) into an anisotropic solution we choose k = 0 and allow

3-space to be anisotropic such that equation (5.28) holds and

dσ23 = t2p1dx2 + t2p2dy2 + t2p3dz2. (5.29)

We then arrive at what Frolov calls the Kasner-AdS spacetime. Here the exponents must

satisfy the familiar Kasner restrictions,

p1 + p2 + p3 = 1 = p21 + p2

2 + p23. (5.30)


The coordinate transformation w = −ℓ ln(r/ℓ) can be made to write the metric in the

form,

ds2 = e−2|w|/ℓ (−dt2 + t2p1dx2 + t2p2dy2 + t2p3dz2)

+ dw2, (5.31)

where the brane sits at w = 0 and it is clear that the brane has the same structure as

the Kasner spacetime. The brane must also have a tension and matter-density given by,

σ = ∓ 6

κ25ℓ, and ρ = 0, (5.32)

respectively, which is the same as the Randall-Sundrum tuning condition in equation (5.3).

This means the brane must be empty and has the same fine tuning condition as the

Randall-Sundrum brane. The fact that this brane is a vacuum makes it a poor cosmo-

logical model, but it is important in that it introduces anisotropy into the braneworld

scenario.

Other Authors have investigated anisotropic braneworlds with matter content [1,

4]. Notably, Campos and Sopuerta [20] used dynamical systems techniques to look at

Bianchi-type branes which are homogeneous and anisotropic. 1 However, in these early

studies many assumptions were made about the Weyl term, Eab, due to the lack of an

exact anisotropic bulk solution. This was addressed in [19] for the FLRW and Bianchi I

case and shortly after Campos et al. [18] found a family of exact, anisotropic solutions to

the five-dimensional field equations. Thus they were able to explicitly see the relationship

between the bulk Weyl curvature and the anisotropy on the brane. They found that it is

not possible to have a perfect fluid or scalar field compatible with the anisotropic brane

since the junction conditions require anisotropic stress on the brane. Fabbri et al. [39]

found more exact bulk solutions and agreed that an anisotropic brane cannot support a

perfect fluid if the bulk is static. They found that in some solutions with a non-static

bulk it is possible to have a perfect fluid, but its energy density and pressure is completely

determined by the bulk geometry.

1For a summary of dynamical systems in the context of cosmology, including Bianchi-type cosmolo-gies, refer to [102].


Another interesting feature of anisotropic cosmology on the brane is their apparent

tendency to isotropize at the initial singularity. Harko and Mak [53] investigated Bianchi-

type braneworld behaviour near the singularity and at late times and found that they

tend to isotropize for certain matter content. Dunsby et al. [32] used dynamical systems

techniques and confirmed the idea that braneworlds tend to isotropize as the initial

singularity is approached. Furthermore, they used a perturbative analysis of the FLRW

brane and found similar results [51]. It appears that in the braneworld context there is

a natural mechanism for the initial singularity to be isotropic. This is a drastic change

from conventional cosmology where isotropy is taken to be an initial condition.

5.1.4 Brane Collisions

The first Randall-Sundrum braneworld consisted of two branes at some fixed distance

from each other. In general the distance between two branes does not need to be constant.

Thus we can set up a situation in which two branes moving through the bulk could collide

with each other. In fact any number of colliding branes can produce any number of branes

after the collisions. Such a situation has been investigated by Neronov [82] who looked at

Friedmann type branes colliding in an Anti-de Sitter bulk. He was able to derive a simple

relation between the cosmological constants on either side of the colliding branes that

must be satisfied. This was later followed up by Langlois, Maeda and Wands [72] who

showed that this condition is a form of momentum conservation applied to the colliding

branes.

Novel cosmological models have been constructed from the idea of colliding branes.

Khoury et al. produced a model that they termed the ekpyrotic universe [63, 89]. This

model consisted of two vacuum branes which collide resulting in one brane endowed with

stress energy. The collision point is seen as the big bang from an observer in the resultant

brane. This model was motivated by an attempt to construct a cosmology that does not

require inflation. Another alternative was presented by Gen, Ishibashi and Tanaka [45] in


which a vacuum bubble forms in a false vacuum bulk. This bubble collides with a brane

resulting in a Friedmann type cosmology on the brane. In these models any bulk effects

on the brane were neglected. These models are currently being studied and contested,

but they serve to show the many possible applications of extra dimensions in cosmology.

5.2 Constructing an Inhomogeneous Brane

We now turn to the problem of constructing new exact braneworld solutions using the

existing solutions. In the spirit of Chapter 2, where exact solutions were pieced together

to form the Cheese Slice universe, we wish to do the same in the braneworld context.

With the existence of Kasner type braneworlds and solutions for cosmological FLRW

type branes, it seems natural to attempt to find a braneworld equivalent of the Cheese

Slice universe. Such a solution would provide an exact inhomogeneous solution to model

inhomogeneities in the braneworld context. Also deviations from the conventional Cheese

Slices might appear that could be used to support or refute the braneworld picture.

Finally, insights gained from attempting a Cheese Slice braneworld matching could be

used to search for new matchings that can generate more exact braneworld solutions.

Some attempts at matching on the brane have been attempted. Germani and Maartens

[50] attempted to match a spherical star of uniform density on the brane to an exterior

solution using the Darmois conditions on the brane. They provide two possible non-

Schwarzschild exterior solutions that were able to match to the same interior solution.

The exterior solution is not unique and also not Schwarzschild due to the Weyl tensor

terms, Eab in equation (5.23). Assumptions were made about this term to arrive at the

solutions. In fact they were found completely on the brane and no extension into the

bulk is known. Despite these difficulties they have attempted to investigate gravitational

collapse on the brane [14]. Without knowing anything about the bulk, they were able to

show that the exterior solution to gravitational collapse cannot be static. That is, the


exterior must be non-Schwarzschild due to Weyl tensor terms and a non-zero effective

pressure term on the matching surface.

An attempt at constructing a Swiss Cheese brane has been carried out by Gergely

[49]. In this model, an FLRW brane is punctured by “black string” solutions [21]. The

black strings are characterized by a string-like singularity that extends into the bulk.

The intersection of the black string with the brane forms the familiar Schwarzschild

singularity on the brane. In his treatment Gergely assumes the Weyl tensor term, Eab, is

zero in both the FLRW and the black string regions [48]. Then assuming a perfect fluid

in the FLRW regions and implementing the Darmois matching conditions on a spherical

surface with the Schwarzschild singularity at the centre, he showed that the cosmological

fluid cannot be dust [47, 46]. This is due to the modified Friedmann equations on the

brane. Furthermore, certain values of the cosmological constant required a negative mass

density to ensure a proper matching. Much like the case of a star on the brane, no exact

bulk solution has been found for this construction.

The idea of a black hole colliding into the brane or escaping into the bulk has also

been considered by Flachi and Tanaka [42]. To arrive at a result they neglected any self

gravitating effects of the brane and the brane tension as well.

In all these examples, the bulk was not specified or its effects were neglected alto-

gether. Even though Germani and Maartens took bulk effects into consideration, they

did not consider the requirements to match the respective bulks of the interior and the

exterior of the star. In the following we will attempt to construct an inhomogeneous

brane through a Darmois matching and then look explicitly at how the bulk might affect

the matching.

5.2.1 The Cheese Slice Brane

Despite the difficulties encountered so far in braneworld matchings there is reason to

believe that a Cheese Slice brane will be tractable. Both the Kasner and the FLRW


branes have natural extensions into the bulk with exact solutions known in each case. In

the vacuum regions we will use the Kasner-AdS metric from equation (5.31), with the

Kasner brane located at w = 0. With this form of the bulk we have Eab = 0. Let us

assume that Eab is zero on the FLRW side as well, allowing us to completely describe

the matching from within the brane. Let us also assume a bulk that is symmetric about

the brane. 2 Thus we only require that our brane solutions satisfy the modified EFE in

equation (5.23). We will use the superscripts, −, and, +, to denote the Kasner side and

the FLRW side respectively. The Kasner metric,

ds2− = −dT 2 + T 2p1dX2 + T 2p2dY 2 + T 2p3dZ2, (5.33)


p1 + p2 + p3 = 1 = p21 + p2

2 + p23, (5.34)

satisfies the EFE on the brane in the case of a vacuum. This requires,

τ−ab ≡ 0 and Λ−4 = 0. (5.35)

Turning to the flat FLRW metric we have,

ds2+ = −dt2 + a2(t)(

dx2 + dy2 + dz2)

. (5.36)

Assuming a perfect fluid such that,

τ+ab = (ρ+ P )uaub + Pgab, (5.37)

then directly from equation (5.23) we have the modified Friedmann equations,

(a,ta

)2

=Λ+

4

3+κ2

4ρ

3

(

1 +ρ

2σ

)

(5.38)

and

a,tta

=Λ+

4

3− κ2

4P

2

(

1 +ρ

σ

)

− κ24ρ

6

(

1 +2ρ

σ

)

. (5.39)

2The consequences of these assumptions will be explored in Section 5.3


From Chapter 2.2 we know that a flat FLRW region and a Kasner region can satisfy

the Darmois conditions along a planar surface z = constant. For this matching to occur

we require,

a(t) = t2

3 p1 = p2 =2

3, and p3 = −1

3. (5.40)

Thus it is possible to have a Cheese Slice matching within the brane, though due to the

modified Friedmann equations (5.38) and (5.39), the matter content is non-trivial.

5.2.2 Energy Conditions

We will now investigate the energy conditions of the cosmological fluid in the FLRW

region. In the case of the Swiss Cheese brane [48, 47] it was shown that the matching

required a non-trivial equation of state, we will see a similar result with the Cheese

Slice brane, and emphasize that a non-trivial matter content is in direct contrast with

the four-dimensional Cheese Slice universe in Chapter 2 where the matter content is

necessarily dust. We will show that even though the equation of state is more complex in

the braneworld case, the energy conditions can be satisfied in certain regimes, allowing

for physically reasonable matter content.

The energy conditions are as follows [103],

The Weak Energy Condition (WEC) is the statement that physically reasonable

matter should have non-negative energy density as seen by any timelike observer.

Specifically,

Tabξaξb ≥ 0 (5.41)

for all timelike ξa.

The Strong Energy Condition (SEC) states that,

Tabξaξb ≥ −1

2T, (5.42)

where T = T aa . This is equivalent to the statement that Rabξaξb ≥ 0, through the

four-dimensional EFE.


The Dominant Energy Condition (DEC) states that

−T ab ξb (5.43)

is a future directed timelike or null vector for all future directed timelike ξa. Phys-

ically this condition can be interpreted as stating the speed of energy flow is less

than the speed of light.

The DEC implies the WEC, but otherwise these conditions are mathematically indepen-

dent. These conditions are all assumptions imposed on the spacetime to determine if

they are physically reasonable and are independent from the EFE.

In the case of a perfect fluid these conditions are equivalent to,

WEC

ρ ≥ 0 and ρ+ P ≥ 0, (5.44)

SEC

ρ+ P ≥ 0 and ρ+ 3P ≥ 0, (5.45)

DEC

ρ > |P |. (5.46)

Combining equations (5.38) and (5.39) with equation (5.40) gives the equation of state,

P =

(

Λ+4

κ4

− ρ2

2σ

)

(

1 +ρ

σ

)−1

. (5.47)

We can also solve for the density as a function of cosmological time, t, to get,

ρ

σ= −1 ±

√

1 − 2

κ24σ

(

Λ+4 − 4

3t2

)

. (5.48)

This result is identical to that of Gergely’s black strings cosmology [48]. The positivity

of ρ was investigated in [48] and we quote the result in Table (5.1). The matter density

can always be greater than zero for negative cosmological constant. This can be seen


ρ t < t1 t = t1 t1 < t ≤ t2 t > t2

Λ+4 ≤ 0 + + + +

0 < Λ+4 ≤ κ2σ

2+ 0 - -

κ2σ2< Λ+

4 + 0 - no real solution

Table 5.1: Positivity of matter density, ρ, as a function of cosmological time, t, for

different values of the cosmological constant, Λ+4 . The constants are t1 = 2√

3Λ+

4

and

t2 = 2√

23(2Λ+

4−κ2

4σ)

.

−1 0 1 2 3 0

1

2

3

4

−1 0 1 2 3 0

1

2

3

4Positivity of ρ

Λ+

4

κ24σ

no real solution t2κ24σρ > 0

ρ < 0

Figure 5.1: Regions in which ρ is positive. The lower curve represents ρ = 0 or t2 =

23Λ+

4

. The upper curve represents the boundary over which no real solution exists, t2 =

83(2Λ+

4−κ2

4σ)

. The positivity of ρ is also equivalent to the WEC.


from the second term in equation (5.48). When Λ+4 is negative, the second term is greater

than 1. With the choice of the + sign ρ will be greater than 0. If Λ+4 is positive, ρ will

only be positive for early times. Now if we consider ρ in the regime which it is positive

and considering the equation of state (5.47) we get,

ρ+ P =

(

ρ+ρ2

2σ+

Λ+4

κ4

)

(

1 +ρ

σ

)−1

=4

3t2κ4

(

1 +ρ

σ

)−1

. (5.49)

In the last equality we have used the modified Friedmann equation (5.38) to simplify the

expression. This right hand side of equation (5.49) is positive for all time as long as ρ is

positive. Thus the WEC is satisfied whenever ρ is positive as outlined in Table (5.1) and

illustrated in Figure (5.1).

The SEC does not necessarily require ρ to be positive. Taking a closer look we see

the first condition of equation (5.45) along with equation (5.49) leads to,

4

3t2κ4

(

1 +ρ

σ

)−1

≥ 0, (5.50)

which is equivalent to the condition,

ρ

σ≥ −1. (5.51)

From equation (5.48) we see that this condition is satisfied whenever the term,√

1 − 2κ24σ

(

Λ+4 − 4

3t2

)

, is real and the positive sign chosen. This requires that,

t ≤ 2

√

2

3(2Λ+4 − κ2

4σ)if Λ+

4 >κ2σ

2(5.52)

and is always true if

Λ+4 ≤ κ2σ

2. (5.53)

The second condition of equation (5.45) leads to,

ρ+4

t2κ4

(

1 +ρ

σ

)−1

− 3ρ ≥ 0. (5.54)

Substituting for ρ we find,√

1 − 2

κ24σ

(

Λ+4 − 4

3t2

)

+2

κ24σ

(

Λ+4 − 1

3t2

)

− 1 ≥ 0. (5.55)


−1 0 1 2 3 0

1

2

3

4

−1 0 1 2 3 0

1

2

3

4

Region in which the SEC is satisfied

no real solutionyesno t2κ24σ

Λ+

4

κ24σ

Figure 5.2: The region in which the SEC is satisfied is illustrated. The region includes

both curves on the left and the right, but no points beyond them. The left curve describes

the boundary where equation (5.55) fails and is described byΛ+

4

κ24σ

=−3√y2+y+3y+4

12y, where

y = κ24σt

2. The right curve represents the boundary over which no real solution exists,

t = 2√

23(2Λ+

4−κ2

4σ)

.

As with the WEC we again see a specific range of t and Λ+4 over which the SEC is

satisfied. This region is illustrated in Figure (5.2). Note that equation (5.55) is more

restrictive than either of equations (5.52) and (5.53). The distinctive feature of the SEC

that we can see from Figure (5.2) is that there is no value of Λ+4 for which the SEC is

satisfied for all time. For 0 ≤ Λ+4 ≤ κ2

4σ

2SEC is satisfied for for late times (t → ∞), but

is violated for early times (t→ 0). For Λ+4 >

κ24σ

2the SEC is violated for both early and

late times.

To check the DEC we need to show that ρ ≥ |P |. This necessarily means that ρ is

positive. First we see that the condition,

ρ ≥ −P = ρ− 4

3t2κ4

(

1 +ρ

σ

)−1

(5.56)


is always satisfied for positive ρ. The equality stems from substitution of equation (5.49).

Now we check the condition,

ρ ≥ P =4

3t2κ4

(

1 +ρ

σ

)−1

− ρ, (5.57)

which is equivalent to the condition,

2ρ(

1 +ρ

σ

)

≥ 4

3t2κ4

. (5.58)

Substituting from equation (5.48) and choosing the positive sign, we find that the con-

dition becomes,√

1 − 2

κ24σ

(

Λ+4 − 4

3t2

)

+2

κ24σ

(

1

t2− Λ+

4

)

+ 1 ≥ 0. (5.59)

The range over which the DEC is satisfied is illustrated in Figure (5.3). We can see from

the figure that equation (5.59) is not as restrictive as the condition that ρ be positive.

Therefore, the DEC is satisfied in the same region as the WEC. For 0 ≤ Λ+4 ≤ κ2

4σ

2both

the DEC and the WEC are satisfied for all time while for Λ+4 >

κ24σ

2they are only satisfied

for early times.

5.3 Extending the Matching into the Bulk

It appears so far that there is a consistent matching from within the brane, but to have

a consistent braneworld we must be able to embed the brane into a bulk spacetime.

Finding a bulk solution that can induce the given braneworld structure is in general

quite difficult and in some cases such a bulk might not exist. In this section we will

utilize a first order approximation to extend the FLRW and Kasner branes into the bulk.

This approximation will not provide us with a global solution, but will serve to show

whether or not a bulk matching is possible within the vicinity of the brane. We will

show that the bulk of the FLRW and Kasner brane does not match in a straightforward

manner. We then look at the general case of matching any two bulks and show that the

embedding plays a large role in this matching.


−1−1−1 0 1 2 3 0

1

2

3

4

0 1 2 3 0

1

2

3

4

0 1 2 3 0

1

2

3

4Region in which DEC is satisfied

ρ > 0 ρ < 0 no real solution

Equation (5.59) is valid.

t2κ24σ

Λ+

4

κ24σ

Figure 5.3: The region in which the DEC is satisfied is illustrated. The lower curve

represents ρ = 0 or t2 = 23Λ+

4

. The right curve represents the boundary over which no

real solution exists, t = 2√

23(2Λ+

4−κ2

4σ)

. Equation (5.59) is valid below the central curve

indicated. This curve is given byΛ+

4

κ24σ

=

√3(3y2+8y)+3y+12

12y, where y = κ2

4σt2.


5.3.1 The Bulk of the Cheese Slice Brane

We will use the Darmois matching conditions outlined in Chapter 2 to attempt to match

the bulk. In this case the matching surface will be a four-dimensional surface that

intersects the brane. The procedure must be carried out in two steps. First we have to

show that each bulk can support the respective brane. Then we can check the conditions

to match these two bulks.

We will attempt to keep the bulk as general as possible, assuming only the symmetries

required to produce the FLRW and Kasner branes. On the FLRW side we assume the

same form as in equation (5.5),

ds2+ = −N2(t, y)dt2 + A2(t, y)(

dr2 + r2dφ2 + dz2)

+ dy2 (5.60)

where the fifth coordinate is denoted by y. The brane is located at y = 0 without any

loss in generality. A flat FLRW brane is assumed to ease the calculations.

Let the fifth coordinate on the Kasner side be denoted by w such that the bulk metric

is of the form,

ds2− = −M2(T,w)dT 2 +B2(T,w)(

dR2 +R2dΦ2)

+ E2(T,w)dZ2 + dw2, (5.61)

where the Kasner brane is induced at w = 0. We have assumed a symmetry in the R−Φ

plane as is necessary for the Cheese Slice matching.

For the FLRW bulk we already know the conditions required for a cosmological brane,

namely from equation (5.8) with ρb = ρ+ σ+,

(

N,y

N

)

0+

=κ2

5

6(3P + 2(ρ+ σ+)),

(

A,yA

)

0+

= −κ25

6(ρ+ σ+), (5.62)

where σ+ is the brane tension. We now take the first order expansion, keeping in mind

that the zeroth order terms must be N20 ≡ N2(t, 0) = 1 and A2

0 ≡ A2(t, 0) = t4/3 to

reproduce the flat FLRW cosmology on the brane. The first order expansions are then,

N(t, y) = 1 +N1(t)y and A(t, y) = t2

3 + A1(t)y. (5.63)


The conditions in equation (5.62) are then,

N1 =κ2

5

6(3P + 2(ρ+ σ+)),

A1

t2

3

= −κ25

6(ρ+ σ+), (5.64)

On the Kasner side we must work out explicitly what the conditions on the metric

functions should be. Beginning from the matching condition in equation (2.30) and

assuming symmetry about the brane we have,

K−ab = −κ

25

2

(

S−ab −

1

3S−g−ab

)

, (5.65)

where S−ab is the energy-momentum on the brane, g−ab the four-dimensional Kasner metric

and S = Saa . To reproduce the Cheese Slice we need the Kasner brane to be a vacuum

with the possibility of a brane tension. Thus,

S−ab = −σ−g−ab. (5.66)

Equation (5.65) then implies,

(

M,w

M

)

0+

=

(

B,w

B

)

0+

=

(

E,wE

)

0+

= −κ25σ

−

6(5.67)

Similar to the FLRW case, we now take the first order expansion of the metric functions

with the zeroth order terms being M0 ≡ M(T, 0) = 1, B0 ≡ B(T, 0) = T2

3 and E0 ≡

E(T, 0) = T− 1

3 to produce the Kasner brane. The first order expansions are then,

M(T,w) = 1 +M1(T )w,

B(T,w) = T2

3 +B1(T )w (5.68)

and E(T,w) = T− 1

3 + E1(T )w.

With equation (5.67) we can solve for the first order terms giving,

M1 = −µ B1 = −µT 2

3 and E1 = −µT− 1

3 (5.69)

where µ =κ25σ−

6.


With these conditions on the first order terms in mind, we now turn to the bulk

matching. Let the matching surface be denoted by z − Σ(t, y) = 0. Within the brane

the matching surface must be z = constant, and thus Σ0 ≡ Σ(t, 0) = constant. Let the

parametrization on Σ be such that,

t = u, T = T (u), (5.70)

φ = θ = Φ, (5.71)

r = v = R, (5.72)

y = x, w = w(u, x), (5.73)

z = Σ(u, x) and Z = Z(u, x). (5.74)

Z must be constant in the brane as well, ie. Z(t, 0) = constant. With this parametriza-

tion we can find the first fundamental forms one either side of Σ. Keeping only first order

terms in y and w, we get,

Υ+00 = (u

2

3 + 2A1x)u2

3 Σ2,u − (1 + 2N1x), (5.75)

Υ+03 = (u

2

3 + 2A1x)u2

3 Σ,uΣ,x, (5.76)

Υ+11 = (u

2

3 + 2A1x)u2

3 , (5.77)

Υ+22 = (u

2

3 + 2A1x)u2

3v2, (5.78)

Υ+33 = (u

2

3 + 2A1x)u2

3 Σ2,x + 1 (5.79)

and

Υ−00 = T− 2

3 (1 − 2µw)Z2,u − (1 − 2µw)T 2

,u + w2,u, (5.80)

Υ−03 = T− 2

3 (1 − 2µw)Z,uZ,x + w,uw,x, (5.81)

Υ−11 = T

4

3 (1 − 2µw) , (5.82)

Υ−22 = T

4

3 (1 − 2µw) v2, (5.83)

Υ−33 = T− 2

3 (1 − 2µw)Z2,x + w,x, (5.84)

(5.85)


in the FLRW and Kasner sides respectively.

From the Darmois matching conditions we must have Υ+ab = Υ−

ab. Equating equa-

tions (5.77) and (5.82) we can solve for w to get,

µw =1

2− (u

2

3 + 2A1x)u2

3

2T4

3

. (5.86)

The equality of equations (5.78) and (5.83), Υ+22 = Υ−

22, is now automatically satisfied.

Equating Υ+03 = Υ−

03 and taking the zeroth order term gives the condition,

0 =2A1u(uT,u − T )

3µ2T11

3

. (5.87)

Thus we must have, on integration with respect to u,

T = Cu, (5.88)

where C is a constant of integration. Using equations (5.88) and (5.86) the condition

Υ+00 = Υ−

00 leads to,

N1 = −(

C

u

)2/3

A1. (5.89)

If we recall the matching conditions for the bulk to support an FLRW brane from equa-

tion (5.64) we can now arrive at an equation of state,

0 = 3P + (2 − C2

3 )ρ− (1 − C2

3 )σ+. (5.90)

However, this equation of state contradicts with the one we arrived at in equation (5.47)

by assuming a matching within the brane. Thus it is not possible to match both the bulk

and the brane simultaneously.

The assumptions we made along the way included the symmetry of the bulk as well

as the form of the matching surface z − Σ(t, y) = 0. One could now try to relax these

assumptions to find a bulk that does match, but such a prescription would be mostly trial

and error and there is no guarantee that a solution can be found. Rather than taking

that route we opt to investigate matched branes in general to see what insights can be

gained from the bulk matching.


5.4 General Embedding of Matched Branes

It is clear from section 5.3.1 that a symmetric embedding of the Cheese Slice brane into a

bulk with Eab = 0 is not possible. In this section we investigate what class of matchings

are possible to embed into a bulk and see if there are any restrictions as to what types

of branes are possible.

There are theorems that state an analytic spacetime can be locally embedded into a

higher dimensional Cauchy development given appropriate initial data [27]. Initial data

in this sense consists of a manifold Σ, with an intrinsic metric hab and extrinsic curvature

Kab, all of which are analytic. However, in an inhomogeneous model created from a

matching, there is no guarantee that the spacetime is analytic at the matching point.

Thus these theorems cannot be invoked. Furthermore, we are not only interested in the

existence of the bulk. We would also like to see how the bulk embedding would affect

the brane.

In the following we use the concept of a brane constructed from a matching to inves-

tigate what restrictions the bulk imposes on the brane. By constructing a brane through

a matching we allow for discontinuities in the matter across the matching surface. As

we saw in section 5.2.1 this is entirely consistent within the brane. However we find that

embedding into the bulk severely restricts the brane configurations that are possible. In

particular if the embedding contains no corners, there cannot be discrete jumps in the

matter content on the brane. Conversely this means that if we are to have jumps in the

matter content on the brane, the embedding must allow for a corner to appear at that

point.

5.4.1 Set-up

Let M+4 and M−

4 be two branes with the respective metrics given by g+ab and g−ab. We will

assume that M+4 and M−

4 can be matched along some hypersurface Σ+3 = Σ−

3 = Σ3 using


the Darmois matching conditions. We wish to see the consequences of embedding this

brane into some bulk. More precisely, we embed M+4 and M−

4 into respective bulks M+5

and M−5 and match the bulks along a hypersurface Σ±

4 , which we wish to be an extension

of Σ3. Thus we choose Σ±4 such that, Σ3 = M±

4 ∩ Σ±4 .

Expressions with + or − refer to the respective quantity in either M+5 or M−

5 . In

the following the superscripts will be left off general expressions that apply to both sides

of the matching.

Assuming the Darmois conditions across Σ3 implies

h+AB = h−AB where hAB ≡ gAB −mAmB. (5.91)

and also,

Ω+AB = −Ω−

AB where ΩAB ≡ hDAmB‖D =1

2LmhAB, (5.92)

where mA is the normal to Σ3 in M4.The minus sign arises from our choice that the unit

normals be pointing “inward” in both M+4 and M−

4 . Note that first fundamental form of

Σ3, hAB, and the second fundamental form, ΩAB, are calculated on the three-dimensional

subspace. The brane metric gAB is a four-dimensional quantity. hAB is also the intrinsic

metric and projection operator that projects quantities in M4 onto Σ3. The ‖ denotes

covariant differentiation on M4 associated with gAB and L the Lie derivative.

Let the unit normal to M4 be denoted nA, such that nAnA = 1, when embedded in

M5. Then we can write the bulk metric in the form,

gAB = gAB + nAnB, ds2 = gab(xc, w)dxadxb + dw2. (5.93)

The “5th coordinate”, w is defined normal to the brane such that nAdxA = dw. The

brane is located at w = 0 without loss of generality.

5.4.2 An Embedding With no Corners

We first investigate the consequences of an embedding with no corners. This gives us a

unique normal at all points p ∈ Σ3 ⊂ M4. Let us denote the normal to Σ3 by mA such


that

mAmA = 1 and mA ∈M4. (5.94)

It follows from equation (5.92) that we have

m+A = −m−

A, (5.95)

and also nAmA = 0 by the definition of nA.

We have not yet defined how M+4 and M−

4 meet when they are embedded in M5. For

instance, they could meet at some angle θ measured in M5 forming a corner when viewed

from the bulk, similar to the situation in Figure 2.2 with M±4 in place of Πi and M5 in

place of V . However, in this section we will consider what happens when there is no

corner and insist that

θ = π, (5.96)

which is equivalent to stating,

n+A

∣

∣

p= n−

A

∣

∣

pfor all p ∈ Σ3. (5.97)

5.4.3 The Bulk Matching Surface

The matching surface in the bulk, Σ4, has yet to be determined. However we know that

at w = 0 this surface must coincide with Σ3. This allows us to state some properties of

the normal to Σ4 that will be important when investigating the bulk matching.

Let there be a unique normal to Σ4 at each point, denoted mA, such that mAmA = 1

and

nA∣

∣

p6= mA

∣

∣

pfor all p ∈ Σ3. (5.98)

The normalized projection of mA onto M4 at point p is then equivalent to mA at point p,

gBAmB

‖gBAmB‖

∣

∣

∣

∣

p

= mA

∣

∣

pfor all p ∈ Σ3. (5.99)


To see this we can choose an orthonormal basis, e(B)A , at point p such that e

(α)A lies in Σ3

for α = (0, 1, 2). We can choose e(3)A = mA

∣

∣

pand we are left with e

(4)A = nA

∣

∣

p. Since mA

is perpendicular to Σ3 ⊂ Σ4, the only non-zero components are,

mA

∣

∣

p= m(3)e

(3)A + m(4)e

(4)A . (5.100)

The projection is then,

gBAmB

∣

∣

p= (gBA − nAn

B)mB

∣

∣

p

= (gBA − e(4)A eB(4))(m(3)e

(3)B + m(4)e

(4)B )

= m(3)e(3)A + m(4)e

(4)A − e

(4)A m(4)

= m(3)e(3)A , (5.101)

and upon normalization we have

gBAmB

‖gBAmB‖

∣

∣

∣

∣

p

= e(3)A = mA

∣

∣

pfor all p ∈ Σ3, (5.102)

thus confirming equation (5.99). Let the angle between nA and mA be ψ. Then nAmA = cosψ,

allowing us to simplifying the normalized projection,

gBAmB

‖gBAmB‖=

gBAmB√

gEF mEmF

=mA − nAn

BmB√1 − nEnF mEmF

=mA − nA cosψ√

1 − cos2 ψ

= mA cscψ − nA cotψ (5.103)

Thus at any point p ∈ Σ3 we have,

mA

∣

∣

p= (mA cscψ − nA cotψ)

∣

∣

p(5.104)

or

mA

∣

∣

p= (mA sinψ + nA cosψ)

∣

∣

p. (5.105)


5.4.4 Approximation of the Bulk

The general expression for the Taylor expansion of a tensor is given by,

T (q) = T (p) + (Lξ1T )∣

∣

pλ+ (Lξ2 + L

2ξ1

)T∣

∣

pλ2 +O(λ3), (5.106)

where ξA1 and ξA2 are the generators of the diffeomorphism that are free to be specified

depending on the desired direction of the Taylor expansion. Refer to Appendix A for a

derivation of equation (5.106).

To generate an approximation for the metric tensor of the bulk, gAB, we let nA

generate the first order flow into the bulk and the natural parameterization of this flow

is the coordinate w. With the brane located at w = 0, we have to first order in w,

gAB(w) = gAB∣

∣

p+ (LngAB)

∣

∣

pw +O(w2), (5.107)

where p is a point on Σ3 (ie. w=0). Now we can use the metric form from equation (5.93)

to get,

gAB(w) = (gAB + nAnB)∣

∣

p+ Ln(gAB + nAnB)

∣

∣

pw +O(w2)

= (gAB + nAnB)∣

∣

p+ 2KAB

∣

∣

pw +O(w2), (5.108)

where the last equality follows from the extrinsic curvature of the brane,

KAB ≡ 1

2LngAB and LnnA = 0 (5.109)

Similarly we can also expand the normal to Σ4 in the same way,

mA(w) = mA

∣

∣

p+ (LnmA)

∣

∣

pw +O(w2). (5.110)

Substituting from equation (5.105) we get,

mA(w) = (mA sinψ + nA cosψ)∣

∣

p+ sinψLnmA

∣

∣

pw +O(w2).


5.4.5 Matching the Bulk

We are now prepared to examine the matching conditions in the bulk. The intrinsic

metrics of Σ+4 and Σ−

4 must match, that is,

h+AB = h−AB where hAB ≡ gAB − mAmB. (5.111)

From the first order Taylor expansion of equation (5.108) and equation (5.111) we have,

hAB = (gAB + nAnB) + 2KABw

− [(mA sinψ + nA cosψ) + sinψLnmAw] [(mB sinψ + nB cosψ) + sinψLnmBw]

+O(w2), (5.112)

where it is understood that all the coefficients are evaluated at p. We can rearrange

equation (5.112) to get,

hAB = hAB − (mA cosψ − nA sinψ)(mB cosψ + nB sinψ)

+ 2[

KAB − sinψLnm(A(mB) sinψ + nB) cosψ)]

w +O(w2), (5.113)

where parenthesis, (· · · ), on the indices denote symmetrization. We can now examine

the matching to each order using equation (5.113).

Zeroth Order Matching

The first terms automatically match since

hAB∣

∣

+

p= hAB

∣

∣

−p

(5.114)

from equation (5.91).

For an embedding with no corners we must have,

ψ+ + ψ− = π, (5.115)

as illustrated in Figure (5.4). This follows from equation (5.96) since θ ≡ ψ+ +ψ−. Thus


M+4

M−4

Σ+4

Σ−4

ψ−

nA

m+A

m−A

m−A

m+A

p

ψ+

Figure 5.4: Matching of two branes extended into the bulk. The solid curve represents

the branes M+4 and M−

4 . The bulks are matched across the surface Σ4 represented by

the dashed curve. The angles ψ± are measured between the normals to Σ±4 and M±

4 and

satisfy ψ+ + ψ− = π to avoid a canonical singularity at the point p ∈ Σ3.


we have

cosψ+ = − cosψ− and sinψ+ = sinψ−. (5.116)

Together with equation (5.95) and equation (5.97) we have,

(m+A cosψ+ − n+

A sinψ+)∣

∣

p= (m−

A cosψ− − n−A sinψ−)

∣

∣

p. (5.117)

Thus the second term in equation (5.113) can be matched across Σ4 and the zeroth order

term of the condition in equation (5.111) is satisfied. This zeroth order matching is

essentially the matching of the brane across Σ3.

First Order Matching

For convenience in the first order matching we scale w such that w+ = w−. The first

order condition of equations (5.111) and (5.113) can be written out as,

K+AB−sinψ+

Lnm+(A(m+

B) sinψ++n+B) cosψ+) = K−

AB−sinψ−Lnm

−(A(m−

B) sinψ−+n−B) cosψ−).

(5.118)

Due to equations (5.116) and (5.95), the condition can be simplified into,

K+AB = K−

AB. (5.119)

Since KAB is the second fundamental form of M4, it is related to the matter content on

the brane by,

[Kab] = −κ25

(

Sab +1

3Sgab

)

. (5.120)

The [. . .] denotes the jump in that quantity across the brane and Sab is the energy

momentum tensor on the brane. To describe the jump in KAB across the brane we must

now consider the bulk on the other side of the brane. Let us denote these respective

quantities with a bar, ¯. This gives

[KAB]+ = K+AB −K+

AB (5.121)


and

[KAB]− = K−AB −K−

AB. (5.122)

We expect the barred quantities to obey the same conditions for the matching to be valid

on the other side of the brane as well. This does not imply that the structure of the bulk

is symmetric, it only relies on the matching conditions being the same on either side.

Due to the condition of equation (5.119) we have

[Kab]+p = [Kab]

−p (5.123)

at point p. Combined with equation (5.120) we have

S+ab

∣

∣

p= S−

ab

∣

∣

p, (5.124)

since K+ = K− implies S+ = S−. The stress-energy on the brane must be the same at p

in both branes M+4 and M−

4 . That is, observers in M+4 and M−

4 must agree on the value

of the energy momentum at the matching surface.

5.4.6 Consequences of Assuming No Corner

Since we have assumed that there is no corner, the matching surface Σ3, is not unique.

Any surface that separates the brane into two distinct regions of M+4 and M−

4 can be be

defined as Σ3. We can conclude that the condition of equation (5.124) must hold true

throughout the brane. This imposes a strong constraint on the matter content of the

brane.

Consider a situation in which condition (5.124) would fail. For example the matching

surface of any Cheese Slice or Swiss Cheese model will have vacuum on one side ρ− = 0

and some uniform mass density on the other ρ+ = ρ0 6= 0. If this matching surface is

given by y = 0, then,

ρ+y→0+ 6= ρ−y→0− . (5.125)


This type of discontinuity in the energy-momentum is not permitted in our embedding

of the brane. This confirms the result from section 5.3.1 where we failed to find a bulk

solution for the Cheese Slice brane. In that case we implicitly chose an embedding with

no corner when we assumed symmetry across the brane. When a corner does exist the

bulk is necessarily asymmetric in that the angle of the corner is different when viewed

from either side, θ 6= θ.

Conversely if we wish to construct an inhomogeneous brane with jumps in the energy-

momentum tensor or have any object on the brane, such as star, equation (5.96) must be

broken and θ 6= π. This would need a corner or some form of conical singularity about

Σ3.

5.5 The 3+1+1 Decomposition

The result of section 5.4.6 is a clear indication that we need to allow for a corner at

a point where we wish to have a discrete jump in the matter content. However, using

the method of a Taylor approximation into the bulk, it is not clear how a corner could

be incorporated. This difficulty arises because the corner is required to subtend some

angle, φ 6= π. This adds an additional variable to the condition of equation (5.111).

Furthermore, a matching surface in the bulk might not be needed.

The solution we found to this was to carry out a 3 + 1 + 1 decomposition of the

spacetime. This is inspired by the ADM approach [106], commonly called a 3 + 1 de-

composition, which views a four-space as the timelike evolution of a three-dimensional

hypersurface. This 3 + 1 approach was adapted to the braneworld context by Aliev and

Gumrukcuoglu [2]. They performed a 4 + 1 decomposition, thereby expressing the bulk

spacetime as the spacelike evolution of the brane. We take this one step further and

express the bulk as the evolution of a three-dimensional hypersurface in two spacelike

directions. This three-surface is then taken to be the corner. As we will show, this gives


us a straightforward way to describe the brane with a corner and allows us to draw some

conclusions.

In the following we will focus on the bulk as a continuous region (ie. no bulk matching

surface). We will call the boundaries to the bulk Σ4 and M4. These two boundaries will

later be interpreted as the brane.

5.5.1 Defining the Normals, Bases and Metrics

We begin with a five-dimensional spacetime (M5, gAB) that is covered by the coordinates

xA and define two arbitrary but distinct scalar functionsW (xA) and Z(xA) such thatW =

constant describes a family of non-intersecting timelike four-dimensional hypersurfaces

that cover M5 and likewise for Z = constant. We focus on the W = 0 and Z = 0 surfaces

and insist that they intersect along a three-dimensional timelike surface which we call Σ3

such that Σ3 divides each surface into two distinct regions. We choose one of the W = 0

and one of the Z = 0 regions, calling them (M4, gab) and (Σ4, hij) respectively. The brane

can now be defined as the union of M4 and Σ4 with Σ3 ≡M4 ∩Σ4 being the corner. We

take the brane as the boundary to a region of M5 that we call the bulk. Figure (5.5)

visually depicts this situation.

The initial decomposition

This follows closely the 4+1 decomposition of the bulk described in [2]. We can introduce

two unit spacelike normals to M4 and Σ4,

nA = N∂AZ and mA = M∂AW, (5.126)

which must satisfy,

gABnAnB = 1 = gABm

AmB. (5.127)

The two functions,

N = |gAB∂AZ∂BZ|−1/2 M = |gAB∂AW∂BW |−1/2 (5.128)


Z

M5

Σ3

Σ4, W = 0

M4, Z = 0 W

Figure 5.5: The Z-W plane is defined. The five-dimensional bulk is called M5. Within M5

are two timelike hypersurfaces defined by two functions Z(xA) = 0 and W (xA) = 0. We

call them M4 and Σ4 respectively. The intersection of these two hypersurfaces is a three-

dimensional, timelike hypersurface denoted by Σ3. The highlighted four-dimensional

hypersurfaces are the regions that define the brane.

are defined as the lapses associated with each part of the brane. To realize a corner these

two normals must not be collinear,

gABnAmB = cos θ 6= ±1, (5.129)

where θ is the angle between the two normals. For the later purpose of matching space-

times we insist that both normals be pointing “inward” into the bulk, see Figure 5.6 for

clarity.

The parametric equation of the brane can be defined separately for each side, xA = xA(ya),

xA = xA(ηi) and for the corner, xA = xA(ξα), as well. This allows us to find local frames

for each region given by,

eAa =∂xA

∂ya∈M4, fAi =

∂xA

∂ηi∈ Σ4, and ζAα =

∂xA

∂ξα∈ Σ3. (5.130)

These vectors are thus orthogonal to the normals,

nAeAa = 0 = mAf

Ai , and nAζ

Aα = 0 = mAζ

Aα . (5.131)


(Σ3, γ)

(M−5 , g

−)

φ− φ+

(Σ−4 , h

−) (M+4 , g

+)

(M−4 , g

−) (Σ+4 , h

+)

u+

n+

v+

m+

v−

u−

m−

n−

(M+5 , g

+)

Figure 5.6: Illustration of the matching conventions that are being used. The − brane

uses the same conventions as the + brane to make the notation symmetrical. (M−5 , g

−)

and (M+5 , g

+) are the two bulks to be matched with the respective branes acting as the

matching surface.


They also satisfy the completeness relations,

ebAeAa = δba f jAf

Ai = δji and ζβAζ

Aα = δβα. (5.132)

The induced metrics on the hypersurfaces are then,

gab = gABeAa e

Bb in M4, hij = gABf

Ai f

Bj in Σ4, and γαβ = gABζ

Aα ζ

Bβ in Σ3.

(5.133)

This allows us to write the bulk metric as either,

gAB = gabeaAe

bB + nAnB (5.134)

or

gAB = hijfiAf

jB +mAmB. (5.135)

Four-dimensional indices of tensors on the brane are raised and lowered with gab and

hij depending on which part of the brane the tensor is evaluated in. Three-dimensional

tensors on the corner have the indices raised and lowered by γαβ.

A Recursive Decomposition

Up to this point we have essentially carried out two distinct 4 + 1 decompositions of M5.

To complete the decomposition we make use of Σ3 to perform a recursive decomposition

that could be understood as a (3 + 1) + 1 decomposition. Since Σ3 is defined as the

intersection of M4 and Σ4, it must be a member of a family of surfaces that intrinsically

cover the brane. We introduce normals to Σ3 such that,

ua = U∂aW ∈M4 and vi = V ∂iZ ∈ Σ4. (5.136)

Here U and V are lapse functions within M4 and Σ4 respectively. Analogous to the above

procedure we insist that they be unit normals satisfying,

gabuaub = 1 = hijv

ivj, (5.137)


We can define local frames orthogonal to these normals by,

εaα =∂ya

∂ξαand ǫiα =

∂ηi

∂ξα(5.138)

satisfying,

εaαεβa = δβα = ǫiαǫ

βi and uaε

aα = 0 = viǫ

iα, (5.139)

which allows us to carry out the (3 + 1) decomposition in M4 and Σ4 respectively with

the resulting metrics,

gab = γαβεαaε

βb + uaub (5.140)

and

hij = γαβǫαj ǫβj + vivj. (5.141)

The three-bases εaα, ǫjα and ζAα all span the corner, Σ3, and are related by

εaαeAa = ǫiαf

Ai = ζAα . (5.142)

We choose to use ζAα as the preferred basis for Σ3.

Now since ua is in M4, we can write it in terms of the M4 frame which is orthogonal

to nA such that,

(uaeAa )nA ≡ uAnA = 0. (5.143)

Similarly in Σ4 we have,

(vifAi )mA ≡ vAmA = 0. (5.144)

From our initial assumption that nA and mA are not collinear in equation (5.129), it

follows that uA and vA are also not collinear.

Combining equations (5.134), (5.135) and (5.140) we can express the bulk metric in

its desired form,

gAB = γαβζαAζ

βB + uAuB + nAnB. (5.145)

In Σ4 we can perform the same procedure to get an alternate, but equivalent decompo-

sition,

gAB = γαβζαAζ

βB + vAvA +mAmB. (5.146)


A Note on Sign Conventions

As alluded to after equation (5.129), we must state some sign conventions before we

continue. Our initial construction of the brane leaves some ambiguity in determining the

angle between M4 and Σ4 that we will now clarify.

Let φ be the angle of the corner as measured in the two-space normal to Σ3. This

space is spanned by nA and mA with the angle between them being given by θ,

gABnAmB = cos θ. (5.147)

Being orthogonal to Σ3, uA and vA are also in this two-space and since they are in M4

and Σ4 respectively, the angle between them will give us the angle of the corner,

gABuAvB = cosφ. (5.148)

The ambiguity lies in the relation between θ and φ which is determined by whether the

normals point “outwards” or “inwards”. We will adopt the convention that all normals

point “inward” into their respective spaces as dipicted in Figure 5.6. This results in the

relation,

θ = π − φ with

0 < φ < 2π and φ 6= π

−π < θ < π and θ 6= 0. (5.149)

Since φ is the angle between uA and vA we have,

gABuAvB = cosφ = − cos θ. (5.150)

To further clarify the relation between the various normal vectors we note that mA, vA

forms an orthonormal basis of the two-space. Thus we can express nA as a linear combi-

nation of the basis vectors,

nA = C1vA + C2m

A, (5.151)

where C1 and C2 are constants. Now if we project nA onto Σ4 using gab eAa e

bB as the


projection operator we get,

(gab eAa e

bB)nB = (gAB −mAmB)nB (5.152)

= nA −mA cos θ (5.153)

= C1vA. (5.154)

Only the vA component remains since mA is by definition orthogonal to Σ4. Contracting

equations (5.153) and (5.154) with gABnB gives,

1 − cos2 θ = C1 gABnBvA (5.155)

and contracting equation (5.151) with gABvB gives,

gABnAvB = C1. (5.156)

Therefore C1 = sin θ. Applying the same method with the projection of mA onto M4

allows us to find that 3,

nA −mA cos θ = vA sin θ. (5.157)

Likewise,

mA − nA cos θ = uA sin θ, (5.158)

and it follows that,

gABnAvB = sin θ = gABm

AuB. (5.159)

The relation between uA, νA, mA and nA will become important when we examine

the bulk matching in section 5.6, particularly in Figure 5.6, but first we must fix the

coordinates and find the form of the metrics.

5.5.2 Fixing the Coordinates

Next we define the spacelike vectors WA and ZA such that,

WA∂AW = 1 = ZA∂AZ, (5.160)

3Note the similarity between these definitions and those of section 5.4.3


which can be thought of as the “evolution vectors” off the brane into the bulk. These

vectors are tangent to the congruence of curves intersecting the hypersurfaces that cover

the bulk. In general these vectors are not orthogonal to the brane. However, we can

decompose them using the bases defined in section 5.5.1 giving the components orthogonal

and tangential to the corner,

WA = NnA +NaeAa ZA = MmA +M ifAi

= NnA + (Uua + ναεaα)eAa = MmA + (V vi + µαǫiα)f

Ai

= NnA + UuA + ναζAα = MmA + V vA + µαζAα

(5.161)

Here the 4-vectors Na and M i are known as the shift vectors in the 4+1 decompositions.

M,N,U, and V are the same as those defined in equations (5.126) and (5.136). We make

use of our recursive strategy to further decompose these 4-vectors into “3 + 1-vectors”,

thereby giving the final line of equation (5.161). The 3-vectors, να and µα are the shift

vectors of Σ3 associated with the evolution of WA and ZA respectively. Also, U and V

are the respective lapses of Σ3 within M4 and Σ4.

We now use these vectors to fix the coordinates of M5 such that,

xA ≡ (ξα, Z,W ), (5.162)

ya ≡ (ξα, Z), (5.163)

ηi ≡ (ξα,W ). (5.164)

Then by equations (5.160) and (5.130) we have,

ZA =

(

∂xA

∂Z

)

W=0

= δAZ , (5.165)

WA =

(

∂xA

∂W

)

Z=0

= δAW , (5.166)

ζAα =

(

∂xA

∂ξα

)

Z=W=0

= δAα . (5.167)


5.5.3 Finding the Metrics

We begin with M4, which in the coordinates of xA, can now be defined as the W = 0

hypersurface. In this case we have,

dya =∂ya

∂ξαdξα +

∂ya

∂ZdZ (5.168)

= εaαdξα + eaAZ

AdZ (5.169)

since

∂ya

∂Z=∂ya

∂xA∂xA

∂Z= eaAZ

A. (5.170)

From equation (5.168) we find that,

dyadyb = εaαεbβdξ

αdξβ + εaαdξα(ebBZ

B)dZ + εbβdξβ(eaAZ

A)dZ + eaAebBZ

AZBdZ2. (5.171)

Now we can use equations (5.140) and (5.171)to find the line element.

ds2 = gabdyadyb (5.172)

= (γα′β′εα′

a εβ′

b + uaub)dyadyb (5.173)

= γαβdξαdξβ + 2γαβ′ζβ

′

A ZAdξαdZ

+[

γαβ′ζαAζβ′

B ZAZB + uAuBZ

AZB]

dZ2 (5.174)

= γαβdξαdξβ + 2µαdZdξ

α +[

µαµα + (M sin θ − V cos θ)2

]

dZ2 (5.175)

Equations (5.142) and (5.161) were used, as well as,

uAZA = M sin θ − V cos θ, (5.176)

which results from equation (5.159). We can then read off the brane metric,

gab =

γαβ µα

µβ µαµα + M2

(5.177)

where M ≡ (M sin θ − V cos θ). The inverse is then,

gba =

γαβ +µαµβ

M2− µβ

M2

− µα

M2

1

M2

, (5.178)


Recall that the indices are such that a, b, . . . = 0, 1, 2, 3 and α, β, . . . = 0, 1, 2. γαβ is

therefore three-dimensional. The metric of Σ4 can be found in a similar manner,

hij =

γαβ νβ

νβ νανα + N2

(5.179)

where N ≡ (N sin θ − U cos θ). With the inverse,

hij =

γαβ +νανβ

N2− νβ

N2

− να

N2

1

N2

. (5.180)

Now in these coordinates we have,

nA = (0, 0, 0, 0, N) and mA = (0, 0, 0,M, 0) (5.181)

From equations (5.158) and (5.157) we have,

uA =mA − nA cos θ

sin θ= (0, 0, 0,M csc θ,−N cot θ) (5.182)

and

vA =nA −mA cos θ

sin θ= (0, 0, 0,−M cot θ,N csc θ). (5.183)

Thus,

ua = (0, 0, 0,M csc θ) (5.184)

and

va = (0, 0, 0, N csc θ) (5.185)

Therefore the lapses introduced in equation (5.161) are,

U ≡ M

sin θand V ≡ N

sin θ. (5.186)

Thus we can eliminate U and V .


5.5.4 The Bulk Metric

Now we carry out the same procedure for the bulk spacetime. We have,

dxA = ζAα dηα + ZAdZ +WAdW. (5.187)

The metric can be taken in either form,

gAB = γαβζαAζ

βB + uAuB + nAnB (5.188)

(5.189)

= γαβζαAζ

βB + vAvB +mAmB.

Computing the line element allows us to express the metric in matrix form. With

A,B = 0, 1, 2, 3, 4 and α, β = 0, 1, 2, we arrive at,

gAB =

γαβ µα να

µβ µαµα + M 2 Ψ

νβ Ψ νανα + N 2

, (5.190)

where,

M2 ≡ M2 +N2 sin2 θ (5.191)

N2 ≡ N2 +M2 sin2 θ (5.192)

Ψ ≡ M2 +N2 −MN cot2 θ cos θ . (5.193)

Now we can see that θ, M , N , µα, να and γαβ constitute the 15 arbitrary functions that

we would expect in a general five- dimensional metric. Equivalently one could use Ψ, M ,

N in place of θ, M and N . They are also related to M and N by,

M |W=0 = M and N |Z=0 = N . (5.194)

To find the inverse we also make use of a recursive strategy, we first write the bulk

metric as a 4 + 1 decomposition,

gAB =

gab Pa

Pb PaPa +B2

, (5.195)


where Pa and B we treat as unknown functions. In this form it is easy to read off the

inverse as,

gBA =

gba +P bP a

B2

−P b

B2

−P a

B2

1

B2

, (5.196)

which is identical in block form to equation (5.178). We can in fact explicitly write out gab

from equation (5.178), but we must be careful to use M as the lapse function to ensure

we do not loose the W -dependence of gAB. We must now decompose Pa and find the

contravariant component P a ≡ gabPa. We identify equation (5.196) with equation (5.190)

and find that Pa is given by,

Pa ≡

να

Ψ

. (5.197)

Explicitly performing the calculation for P b gives:

P b =

γαβ +µαµβ

M 2− µβ

M 2

− µα

M 2

1

M 2

να

Ψ

=

νβ −(

Ψ − µαναM 2

)

µβ

Ψ − µαναM 2

(5.198)

=

νβ − Ψµβ

M 2

Ψ

M 2

(5.199)

≡

P β

P 3

,


where we have defined Ψ ≡ Ψ − µανα. Expanding equation (5.196) gives,

gBA =

γαβ +µαµβ

M 2+P βPα

B2− µβ

M 2+P 3P β

B2−P

β

B2

− µα

M 2+P 3Pα

B2

1

M 2+P 3P 3

B2−P

3

B2

−Pα

B2−P

3

B2

1

B2

. (5.200)

To carry on we identify equation (5.190) with equation (5.195) to give,

νανα + N

2 = PaPa +B2 (5.201)

= PαPα + P3P

3 +B2

= νανα − ναµ

αΨ

M 2+

ΨΨ

M 2+B2

= νανα +

Ψ2

M 2+B2 (5.202)

and solve for B2 giving,

B2 =N 2M 2 − Ψ2

M 2. (5.203)

Now all that remains is to substitute equations (5.203) and (5.199) into the expanded

form of gAB (equation (5.200)). After some simplification we arrive at the final form of

the inverse metric,

gBA =

γβα +N 2µβµα + M 2νβνα − 2Ψµ(βνα)

Φ

−N 2µβ + Ψνβ

Φ

−M 2νβ + Ψµβ

Φ

−N 2µα + Ψνα

Φ

N 2

Φ

−Ψ

Φ

−M 2να + Ψµα

Φ

−Ψ

Φ

M 2

Φ

,

(5.204)


where we have defined Φ ≡ M 2N 2 − (Ψ − µγνγ)2 and recall Ψ ≡ (Ψ − µγν

γ). It is

straightforward to verify that equation (5.204) is indeed the inverse by checking that

gAB gBA = I. We could have equally well used hij in the initial 4 + 1 decomposition to

arrive at the same result.

5.6 The Matching of the Bulk

The theorems of Clarke and Dray [22] state that the minimum condition for two space-

times to match is that the intrinsic metric on the matching surface agrees. In addition,

Taylor’s corner conditions [98] insist that the matching at a corner also requires that

observers on either side of the matching agree on the angle of the corner.

From section 5.4.6 we know that jumps in the matter content on the brane are not

possible without a corner in the embedding. The 3 + 1 + 1 decomposition above forms a

natural environment to investigate these corners. When a corner exists, we have an angle

between the branes such that φ 6= π. Thus the coordinate system used in equation (5.162)

are well defined.

Let M4 be part of the brane with some matter content and Σ4 be the other part of

the brane with different matter content, possibly a vacuum. Let the different sides of

the bulk be denoted by, +, and, −, superscripts. The metric of the bulks are then g+AB

and g−AB. This construction is depicted in Figure 5.6. We can now see the advantage of

this method over the method of section 5.4 as it does not require a superfluous matching

surface in the bulk.

5.6.1 The Matching Conditions

We would like to keep the same conventions on either side of the brane when it comes to

defining angles and norms which is discussed in detail in section 5.5.1. To ensure that we

can use the method on both sides we must match Σ−4 to M+

4 and M−4 to Σ+

4 . This ensures


that the sign conventions are the same on either side, making the − bulk completely

analogous to the + bulk, and allowing for an intuitive definition of φ±. Figure 5.6 shows

how the φ+ and φ− are defined.

The matching of the bulk along Σ−4 and M+

4 requires the first fundamental forms to

be identical when calculated on either side. Thus

h−ab = g+ab (5.205)

and the matching along Σ+4 and M−

4 requires that

g−ij = h+ij. (5.206)

From equations (5.177) and (5.179) it follows that,

γ+αβ = γ−αβ µ+

α = ν−α ν+α = µ−

α (M+)2 = (N−)2 and (N+)2 = (M−)2

(5.207)

The final condition required at the corner is that,

φ+ + φ− = 2π, (5.208)

or equivalently,

θ+ = −θ− (5.209)

From the definitions of M , N , equation (5.207) and equation (5.209) we have,

M+ = ςN− (5.210)

M+ sin θ+ − V + cos θ+ = ςN− sin θ− − ςU− cos θ− (5.211)

M+ sin θ+ −N+ cot θ+ = ςN− sin θ− − ςM− cot θ− (5.212)

M+ sin θ+ −N+ cot θ+ = −ςN− sin θ+ + ςM− cot θ+ (5.213)

M+ sin θ+ + ςN− sin θ+ = N+ cot θ+ + ςM− cot θ+ (5.214)

(M+ + ςN−) sin θ+ = (N+ + ςM−) cot θ+ (5.215)

(M+ + ςN−) = (N+ + ςM−) cot θ+ csc θ+, (5.216)


where ς = ±1. Now from the other relation in equation (5.207) involving N+ and M−

we have analogously,

(M− + N+) = (N− + M+) cot θ+ csc θ+ (5.217)

where = ±1.

We can consider two cases:

Case 1; ς = :

In this case we have,

1 = cot2 θ+ csc2 θ+ (5.218)

or equivalently

cos2 θ+ ± cos θ+ − 1 = 0, (5.219)

which, surprisingly, is the equation for the golden ratio. It is interesting that our matching

conditions lead to this specific value. The cosine is then equal to the golden ratio with

the possibility of sign differences,

cos θ+ =1 ±

√5

2or

±1 +√

5

2. (5.220)

In our predefined range given in equation (5.149) we have the possibilities of

θ+ = ±51.8,±128.2 (5.221)

and

φ+ = 51.8, 128.2, 231.8, 308.2 (5.222)

We list the φ values, as they give a more intuitive picture of the matching angle. We can

also solve for M± and N± since cot θ+ csc θ+ = ±1,

(M+ + ςN−) = q(N+ + ςM−) (5.223)

(M− + ςN+) = p(N− + ςM+) (5.224)


where p, q = ±1. If p = q there is no new information. In the case that p 6= q we are

left with the result,

M+ = −ςN− and N+ = −ςM− (5.225)

Case 2; ς = −:

In this case we have,

(M+ + ςN−) = (N+ + ςM−) cot θ+ csc θ+ (5.226)

(M− + N+) = (N− + M+) cot θ+ csc θ+, (5.227)

which is equivalent to

(M+ + ςN−) = (N+ + ςM−) cot θ+ csc θ+ (5.228)

(ςM− −N+) = (ςN− −M+) cot θ+ csc θ+. (5.229)

Unlike case 1, θ can take on any value and we are left with,

(M+)2 − (N−)2 = (N+)2 − (M−)2. (5.230)

The jumps in the lapses squared is equal on either side of the corner.

5.6.2 The Second Fundamental Form and Matter Content

The matching conditions stated in section 5.6 are the minimal conditions to ensure that

a spacetime exists. We have yet to consider what stress energy is on the brane. To do

so we must look at the jump in extrinsic curvature or the second fundamental form.

We define the extrinsic curvature of M4 and Σ4 respectively as,

Kab = nA||BeAa e

Bb ∈M4 (5.231)

and

Ωij = mA||BfAi f

Bj ∈ Σ4, (5.232)


where || denotes covariant differentiation with respect to gAB The extrinsic curvature of Σ3

has two components due to the two-space of normals. We define the second fundamental

form of this space through any two vectors that span this two-space. For example, we

have the two choices,

nωαβ = nA||BζAα ζ

Bβ (5.233)

mωαβ = mA||BζAα ζ

Bβ , (5.234)

where we use the left-superscript to denote which normal is being used to compute the

component of the extrinsic curvature.

In M4 the normal to Σ3 is ua and in Σ4 the normal is vi. Thus the extrinsic curvature

of Σ3 as measured within the brane is,

uωαβ = ua;bεaαε

bβ ∈ Σ3 ⊂M4 (5.235)

vωαβ = vi;jǫiαǫjβ ∈ Σ3 ⊂ Σ4 (5.236)

where ; denotes the covariant derivative with respect to the brane metric (either gab ∈M4

or hij ∈ Σ4). Now we observe that,

ua;b = uA||BeAa e

Bb (5.237)

vi;j = vA||BfAi f

Bj (5.238)

since ua and vi are defined to lie in constant Z and W surfaces respectively. Using

equation (5.142), we can write equations (5.235) and (5.236) as,

uωαβ = uA||BζAα ζ

Bβ (5.239)

vωαβ = vA||BζAα ζ

Bβ (5.240)

Finally, using equations (5.182) and (5.183) we have,

uωαβ =(

mA||B csc θ − nA||B cot θ)

ζAα ζBβ (5.241)

vωαβ =(

nA||B csc θ −mA||B cot θ)

ζAα ζBβ (5.242)


or in terms of Kab and Ωij,

uωαβ = Ωijǫiαǫjβ csc θ −Kabε

aαε

bβ cot θ (5.243)

vωαβ = Kabεaαε

bβ csc θ − Ωijǫ

iαǫjβ cot θ, (5.244)

If the observer on the brane is to see no stress energy on the surface Σ3, then the jump

in extrinsic curvature, as measured intrinsic to the brane, must be zero. This means,

uωαβ = vωαβ. (5.245)

Therefore from equations (5.243) and (5.244)

Kabεaαε

bβ = Ωijǫ

iαǫjβ. (5.246)

Taking the trace of the left-hand side of equation (5.246) gives,

(Kabεaαε

bβ)γ

αβ = Kabεaα(ε

αc g

cb) (5.247)

= Kabgab (5.248)

= K (5.249)

The trace of the right hand side is,

(Ωijǫiαǫiβ)γ

αβ = Ωijǫjα(ǫ

αi h

ij) (5.250)

= Ωijgij (5.251)

= Ω (5.252)

Therefore

K = Ω. (5.253)

From the definitions of Kab and Ωij (equations (5.231) and (5.232)) we also have,

nA||BζAα ζ

Bβ = mA||Bζ

Aα ζ

Bβ . (5.254)


The consequences of this condition on the bulk matching can now be examined. Let

the matter content on the brane be denoted Tab ∈M4 and Sij ∈ Σ4 to distinguish between

either region of the brane. Using the matching situation depicted in Figure (5.6) we have,

K+ab + Ω−

ab = κ25(Tab −

1

3g+abT ) (5.255)

and

Ω+ij +K−

ij = κ25(Sij −

1

3h+ijS). (5.256)

Here we add the extrinsic curvature because they point in different directions. If we now

look only at the components on Σ3, applying equation (5.246), we have,

Tabεaαε

bβ −

1

3γαβT = Sijǫ

iαǫjβ −

1

3γαβS (5.257)

Then the traces of equations (5.255) and (5.256) along with equation (5.253) give,

K+ + Ω− =κ25

3T

Ω+ +K− =κ25

3S

=⇒ T = S, (5.258)

and finally equation (5.257) takes the form,

Tabεaαε

bβ = Sijǫ

iαǫjβ (5.259)

Thus the matter content in each region of the brane, projected onto the matching surface,

Σ3, must be equal. If we have dust on one side, Tab = ρgab, and vacuum on the other,

Sab = 0, we arrive at an inconsistency.

This is in direct contrast to the case of a star on the brane discussed by Germani

and Maartens [50] where they assumed a perfect fluid, constant density star ρ = const

matched to an external vacuum ρ = 0 = P . They then go on to investigate the non-local

effects that the bulk embedding imposes on brane. However, their initial assumption

appears to contradict our result. The same case arises in [47] where a perfect fluid brane

is punctured with Schwarzschild voids.


One difference in our results from those of Germani and Maartens is that they assumed

the non-local bulk effects on the brane were different inside the star and outside the star.

In essence this assumes that the bulk itself is inhomogeneous and has a different structure

in either region. In comparison, our decomposition assumed the bulk (on either side of

the brane) to be one continuous region. This suggests that a combination of the methods

in section 5.4 and section 5.5 is required for a full treatment of inhomogeneous branes.

In contrast, the Swiss Cheese type brane of Gergely assumed that the non-local bulk

effects were zero in either region. This makes a strong assumption about the bulk and

is similar to what we did in section 5.3, where we found this type of assumption to be

too restrictive. This assumption is also stronger than that of Germani and Maartens.

Our results suggest that the initial assumptions of the Swiss Cheese brane need to be

reconsidered.

5.6.3 Matching Four Bulks

The condition of equation (5.257) is rather restrictive. It is possible to have more general

brane constructions using our formalism by considering matching several bulks together.

If we take four different bulk spacetimes, as depicted in Figure (5.7), and apply our

decomposition in each bulk region M i5, where i = 1, 2, 3, 4. We will use integer super-

scripts to distinguish between the four different regions with the variables defined as in

Figure (5.7) The matching conditions can be applied to each region requiring,

h1 = g2, h2 = g3, h3 = g4, h4 = g1, (5.260)

where the tensor indices have been left off for clarity, and the angle condition,

φ1 + φ2 + φ3 + φ4 = 2π. (5.261)

If we choose M14 and M3

4 (equivalently Σ44 and Σ2

4) to define the brane, then the

three-dimensional matching requires,

u1

ωαβ = u3

ωαβ. (5.262)


φ1

n1

m1

u1

γαβ v1

(Σ24, h

2)

n2

m2

u2

φ4

φ3 φ2

v3

u3v2

m3

n3

(Σ44, h

4)

(M34 , g

3)

(Σ34, h

3)

m4

n4v4

u4

M 45

M 35 M 2

5

M 15

(Σ14, h

1)

(M24 , g

2)

(M14 , g

1)

(M44 , g

4)

Figure 5.7: The matching of four different bulks is depicted. The solid lines represent the

brane and the dotted lines represent the matching surface. The 3 + 1 + 1 decomposition

can be applied to each region. The four corners are identified with γαβ being the metric

on the corner.


In terms of the extrinsic curvature of the brane this gives,

Ω1ijǫ

iαǫjβ csc θ1 −K1

abεaαε

bβ cot θ1 = K3

abεaαε

bβ csc θ3 − Ω3

ijǫiαǫjβ cot θ3, (5.263)

This condition is not as restrictive as equation (5.245) and it is evident that breaking up

the bulk allows for more general matchings to occur.

This construction is very similar to the idea of brane collisions which we will discuss

in detail in section 5.7

5.6.4 Special Cases: Breaking the Angle Condition

An interesting case occurs if we are not required to satisfy equation (5.208). This forms

a conical singularity in the bulk; however the brane can still be well defined as we will

show.

Let us examine the trivial case where we assume that one part of the brane is a

vacuum. In this case, there is no jump in extrinsic curvature, Ωij or Kab, across the

brane in that region. Let us take the brane defined by the matching of M−4 and Σ+

4 to

be the vacuum. The normals are co-linear,

nA− = −mA−, (5.264)

and the bulk would appear continuous at that point with no jump in the stress energy.

Thus it no longer matters how we define this part of the brane. We can use the remaining

normals, which are not co-linear, to span the two-space that defines our Z −W plane.

In essence, rather than matching two bulks, we can view this construction as a removal

of a wedge in the bulk and then identifying the resulting boundaries. See Figure 5.8 for

an illustration of this construction. It is then straightforward to apply the 3 + 1 + 1

decomposition by using the normals nA+ ≡ nA and mA− ≡ mA.

The matching conditions, equation (5.205),then require that,

gab = hab. (5.265)


m−

(M+

4 , g+)

φ

φ−

(M−

4 , g−)

u−

n−

m−

v−

(Σ−

4 , h−)

φ+

(Σ+

4 , h+)

u+

n+

v+

m+

(M+

4 , g+)

n+

u+

v−

(Σ−

4 , h−)

Figure 5.8: If we assume that M−4 and Σ+

4 are vacuums then there is no jump in extrinsic

curvature across that region of the brane. We can then apply the 3 + 1 + 1 approach of

section 5.6 to M+4 and Σ−

4 allowing m− and n+ to span the Z−W plane. In essence, this

is identical to removing one wedge out of the bulk and matching the resulting boundaries

to each other. There is a conical singularity at the corner, however Σ3 and the brane

itself is well defined.


It follows that µα = να and N2 = M2. The bulk metric must take the form,

gAB =

γαβ µα µα

µβ µαµα + M 2 Ψ

µβ Ψ µαµα + M 2

. (5.266)

5.7 Summary and Discussion

We have shown that it is possible to match the FLRW and the Kasner regions within

the branes. The solution is similar to the four dimensional Cheese Slice universe with

the exception of the equation of state that had changed through the modified Friedmann

equation. In general the pressure in the FLRW region is non-zero in the braneworld

context, while in the four-dimensional case zero pressure was required for the match-

ing. Through a detailed investigation of the energy conditions, we have shown that the

energy conditions can be satisfied in certain regimes and thus the matter content can

be considered physically reasonable. However we were unable to find a straightforward

bulk for the Cheese Slice brane. It appears that failure to find the bulk arises from the

assumption of bulk symmetry, which necessitates an embedding with no corners.

Through the investigation of general brane matchings we have shown that if there are

no corners in the embedding, the matter content in the brane must be continuous at all

points. This result was arrived at when attempting to match the first fundamental form

of the bulk to first order. It is likely that higher order matchings would impose even

greater constraints on the brane configurations that are possible.

Finally the 3 + 1 + 1 decomposition of the bulk seems to be the most useful and

applicable method. It allowed us to quite easily find the matching conditions that are

required at a corner and draw some conclusions about the matter content of the brane.

It appears that in an inhomogeneous brane, the matter content in each region must agree

when projected onto the matching surface. This is unintuitive as it rules out physically

reasonable spacetimes, like the Swiss Cheese models. We then found that allowing the


bulk to have more structure gave more freedom to the brane configuration. For instance,

the matching of four bulks did not have the same restriction as matching two bulks.

This matching of four bulks relates closely to the idea of brane collisions as each

matching surface can be considered a brane in itself. Our investigation decomposed the

bulk spacetime along two spacelike directions. If one were to repeat our decomposition

along a timelike direction and one spacelike direction then our four step matching can be

interpreted as two branes colliding, producing two other branes. This type of collision

was investigated by Neronov [82] who looked at Friedmann type branes colliding in an

Anti-de Sitter bulk. He was able to derive a conservation law which was later generalized

by Khoury et al. [63] who showed that the conservation law amounted to momentum

conservation. In these cases they all assumed the bulk to be vacuum Schwarzschild-Anti-

de Sitter spacetimes. Though our method was developed to investigate inhomogeneities

on the brane, it could also be used to investigate brane collisions. Our method has the

advantage of not assuming any specific form of the bulk, thereby allowing a more general

way to describe these collisions.

Returning to our matching conditions, our results suggest that inhomogeneities on

the brane require non-trivial embeddings. We emphasize the need to take the bulk into

consideration and carefully consider its implications. As in the case of Germani and

Maartens [50] some sort of structure must be allowed to exist in the bulk to differentiate

the matter filled regions and the vacuum regions of the brane. This structure in the

bulk is likely to induce some sort of non-local effect on the brane as well. Perhaps the

most vital lesson is that we cannot assume an arbitrary matter content on the brane.

Any inhomogeneous brane must be viewed in the context of its bulk embedding and any

non-local effects must be considered as well.

Chapter 6

Summary and Conclusions

Throughout this thesis we have utilized the concept of spacetime matchings in different

contexts to investigate inhomogeneous cosmologies. We first used the Darmois matching

conditions to construct the Cheese Slice universe. This toy model served as the example

of choice in our investigations since it has the advantage of being a mathematically simple

model which can incorporate inhomogeneities.

Through investigating the lookback time verses the redshift relation we were able

to demonstrate that an observer in the Cheese Slice universe will see anisotropies that

depend on the angle of observation relative to the matching plane. The relative thickness

of the Kasner vacuum layers and the matter field FLRW regions plays a dominant role in

determining this anisotropy with the number of layers and distribution of layers playing

a smaller role. When comparing such results with the CMB data we find that the Kasner

regions must be on the order of ten thousand times thinner than the FLRW regions to

fall within limits of the observed CMB anisotropies. Though this might seem like a small

vacuum region, the very existence of such an inhomogeneity breaks the symmetries of

the FLRW cosmology and will affect some properties of the spacetime.

The approach to the singularity is one situation in which even a small inhomogeneity

should be carefully examined. Conventional cosmologies assume an isotropic homoge-

128

Chapter 6. Summary and Conclusions 129

neous singularity and attempt to explain the formation of the large scale structure as

the model evolves. This does not need to be the case as the initial singularity itself

can be inhomogeneous. The concept of an AVTD singularity describes an approach to

the singularity that is independent of the spatial curvature. Thus, the three-space could

have any structure, including an inhomogeneous one, without affecting the evolution to-

wards the singularity. We were able to devise a criterion for a matched spacetime to be

AVTD based on the ability to match the foliations used in the definition of AVTD. This

demonstrates another application of matching conditions, in this case applied to each

hypersurface that constitutes a leaf of the foliation. Using this criterion the Cheese Slice

universe, both with flat and open FLRW regions, were shown to be AVTD. Thus the

structure of the inhomogeneities, the size and distribution of the slices, arises directly

from the singularity itself. This means that structure in the universe could be an initial

condition in addition to any structure that forms during the cosmological evolution.

We conjecture that any spacetime that can be matched to an AVTD spacetime would

also be AVTD. This is based on the fact that the Darmois conditions, which are used

to match the foliations, are equivalent to the Lichnerowicz conditions. The Lichnerowicz

conditions guarantee a coordinate system that is continuous through the matching sur-

faces. In this coordinate system it is likely that a reasonable foliation could be found to

prove the AVTD property. A rigorous proof of this conjecture could be the subject of fu-

ture investigations. These AVTD singularities are of much interest because the approach

to the singularity plays a fundamental role in any cosmology. Our study has offered a

new perspective on these singularities in terms of matching foliations and demonstrates

that an inhomogeneous universe can be AVTD.

In higher dimensional models, matching conditions play a pivotal role. The braneworld

models are essentially the matching of two five-dimensional spacetimes across a brane rep-

resenting the four-dimensional universe. The problem then becomes one of embedding of

the brane within a bulk. Focusing on the Cheese Slice model we were able to show that


it is possible to construct an inhomogeneous brane with matter content that satisfies the

energy conditions in certain regimes. This brane and matter content obeys the modi-

fied Friedmann equations that arise from projecting the five-dimensional EFE onto the

brane. However, we came across difficulties when attempting to embed this brane into a

bulk. We showed that it is not possible to find approximations to a symmetric bulk that

could support such a brane. This serves as a cautionary example that shows one cannot

arbitrarily construct branes, such as a Swiss cheese brane or stars on the brane, without

taking the bulk matching into consideration.

In the more general case, we looked at the conditions for any brane constructed

from a matching to be embedded in a bulk. Using a Taylor expansion of the bulk

in a neighbourhood of the brane we found that if the embedding had no corners, the

brane cannot have discrete jumps in the energy-momentum tensor. This is makes sense

intuitively as the extrinsic curvature is directly related to matter content. Any discrete

jump in energy-momentum should be accompanied by jumps in extrinsic curvature along

the brane. It is clear that if the brane were to be inhomogeneous there must be corners

in the embedding. Thus it would be prudent for those studying braneworld models to

consider the consequences of assuming symmetry about the brane. It is possible to have

corners and also symmetry, but this would require addition structure in the bulk. There

is then no advantage to assuming symmetry in the context of an inhomogeneous brane.

This leads us to the 3 + 1 + 1 decomposition of the bulk inspired by the ADM

decomposition. We were able to find a coordinate system that was adapted to suit the

brane with a corner that can be applied to either side of the bulk. The coordinates were

such that W = 0 represented one region of the brane and Z = 0 defined the other. In fact

this type of coordinates would only be well defined if there is a corner making the W and

Z coordinates distinct. Through this construction we found that the matter content of

each brane had to be equal when projected onto the matching surface. This proved to be

very restrictive and rules out reasonable matter content on the brane, such as the Cheese


Slice brane. It lead us to conclude that an inhomogeneous brane must be embedded into

an inhomogeneous bulk.

It is still possible to apply the 3+1+1 decomposition to an inhomogeneous bulk. This

would require decomposing more regions of the bulk, which lead us to consider a model

consisting of matching four separate bulk regions. The similarity between this type of

model and braneworld collisions became apparent. Our method can be applied to the

collisions by decomposing the spacetime along one timelike and one spacelike direction.

This can allow us to describe brane collisions in more general bulk spacetimes rather then

the Anti-de Sitter bulks that are commonly assumed. Though this wasn’t the initial aim

of our study, it is interesting to see that our method can apply to other braneworld

constructions.

An interesting case occurred when we broke the angle condition when matching

around the corner. We showed that it was possible for a corner to manifest itself as

a conical singularity in the bulk. The brane remains well behaved at the corner.

Ideally one would like to be able to find exact solutions for these inhomogeneous

branes. It is perhaps possible to do so by utilizing the decomposition technique. One

possible method would be to decompose the five-dimensional field equations in this man-

ner. This will provide a natural environment to search for exact solutions with corners.

Exact bulk solutions will allow us to calculate the non-local effects of the bulk on the

perceived matter content of the brane, something that is still elusive in the current un-

derstanding of braneworlds.

Our study has highlighted some of the intricacies of inhomogeneous universes and

the important role of matching conditions in cosmology. We hope that this serves as

another step towards describing our universe using more comprehensive models that do

not require the assumption of homogeneity from the onset.

Appendix A

Taylor Expansion of a Tensor Field

Some authors [15, 95, 81] have formulated a generalized expression for the Taylor expan-

sion of a tensor field on a manifold. This comes in useful as we pursue the matching in

the bulk. Within a neighbourhood of the brane the Taylor approximation of the bulk

metric can give us an impression of which matchings are possible. This will not generate

a global solution, but it will be able to rule out matchings that are not possible since the

first order matching must be satisfied before any higher order matchings can be consid-

ered. We therefore treat the brane as the zeroth order of the Taylor series and expand

the metric into the bulk. We begin with a look at how the Taylor expansion of a tensor

field can be defined. Then we apply it to our brane construction.

Let T (p) be the value of a tensor field at point p ∈ M . The Taylor expansion is an

approximation of the tensor field T (q) at a point q ∈M where q in the neighbourhood of

p. To define it we must introduce a one parameter family of diffeomorphisms φ : D →M

such that, φ(0, p) = p and D = R ×M . We will denote the diffeomorphism by φλ(p) ≡

φ(λ, p) where λ acts as the parameter label.

First we assume that φ is a flow1 generated by a vector field ξ. Then the pullback of

1A flow is defined by φσ+λ = φσ φλ.

132

Appendix A. Taylor Expansion of a Tensor Field 133

a tensor field T can be expanded as a Taylor series,

φ∗λT = T +

d

dλφ∗λT

∣

∣

∣

∣

λ=0

λ+1

2

d2

dλ2φ∗λT

∣

∣

∣

∣

λ=0

λ2 + . . . , (A.1)

where φ∗0T = T was used in the first term. In general φ∗

λT is a tensor-valued function of

the parameter λ. By definition of the derivative we have,

d

dλφ∗λT ≡ lim

λ→0

(

φ∗λT − T

λ

)

= LξT, (A.2)

which is just the Lie derivative in the direction of the flow. Thus the Taylor series can

be more compactly expressed as,

φ∗λT =

n∑

i=0

λi

i!L

iξT, (A.3)

up to order n.

However, not all diffeomorphisms can be expressed as the flow of a vector field. To

incorporate a more general diffeomorphism into this scheme Sonego and Bruni [95] have

formulated the concept of a knight diffeomorphism. They prove that any one parameter

family of diffeomorphisms, denoted by ψλ, can be approximated by a combination of

flows,

ψλ = φ(n)λn/n! · · · φ

(2)

λ2/2 φ(1)λ (A.4)

up to any desired order n. A diffeomorphism defined in this way is termed a knight

diffeomorohism or knight. Each flow, φ(i)

λi/i!is generated by a respective vector field ξi.

This can be understood as a displacement along the integral curves of ξi to a parameter

distance of λ, followed by a displacement along the integral curve of ξ2 to a parameter

distance of λ2/2, and so on.2 Each successive flow, φ(n), can be seen as a correction

to φ(n−1). Thus we can express the Taylor series of any diffeomorphism by combining

equations (A.4) and (A.3),

ψ∗λT =

n−1∑

l=0

λl

l!

∑

Jl

l!

2!j2 · · ·n!jnj1! · · · jn!L

j1ξ1

· · ·L jnξnT + λnR

(n)λ T (A.5)

2Taking the first two orders into consideration, we see a parameter displacement in the direction ofξ1 followed by a smaller parameter displacement in a different direction ξ2. This is reminiscent of themovement of a knight piece in the game of chess, thus inspiring the terminology.

Appendix A. Taylor Expansion of a Tensor Field 134

where R(n)λ T is a remainder term with a finite limit as λ→ 0 and

Jl ≡

(j1, · · · , jn) ∈ Nn∣

∣

n∑

i=1

iji = l

(A.6)

are summation indexes.

The tensor field at point q can be expressed in terms of the knight to the desired

order,

T (q) = T (ψλ(p)) = (ψ∗λT )(p), (A.7)

and thus be expressed as a Taylor expansion. Applying equation (A.5) explicitly to

second order we have,

T (q) = T (p) + (Lξ1T )∣

∣

pλ+ (Lξ2 + L2

ξ1)T∣

∣

pλ2 +O(λ3), (A.8)

where ξA1 and ξA2 are the generators of the diffeomorphism that are free to be specified

depending on the desired direction of the Taylor expansion.

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