Giang Dan 200911 Phd Thesis.Pdf

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The Study of Inhomogeneous Cosmologies Through Spacetime Matchings

Dan Giang

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto

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 cosmology to model our universe after the homogeneous and isotropic Friedmann-Lemaˆıtre-

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 Whyassumehomogeneity?...... 1

1.2 TheInhomogeneousUniverse ...... 4

1.3 Aboutsingularities ...... 7

1.4 AboutSpacetimeMatchings ...... 9

1.5 AboutBraneworlds ...... 10

1.6 TheStorytoCome ...... 10

2 Matching Regions of Spacetimes 12

2.1 ReviewofMatchingConditions ...... 13

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

2.1.2 MatchingataCorner...... 18

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

2.1.4 NullMatchings ...... 24

2.2 TheCheeseSliceUniverse ...... 25

3 Lookback Time and Observational Consequences 30

3.1 Preliminaries ...... 31

3.1.1 NullVectors...... 31

3.1.2 BendingAngles ...... 33

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

vi 3.1.4 NumericalAlgorithm ...... 37

3.2 Results...... 38

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

3.2.2 PossibleCMBData...... 42

3.3 SummaryandDiscussion...... 47

4 The Structure of the Singularity 48

4.1 DeﬁnitionofaSingularity ...... 49

4.2 Classiﬁcation of Singularities ...... 50

4.2.1 IsotropicSingularities...... 50

4.2.2 ClassiﬁcationScheme...... 51

4.2.3 StrengthofaSingularity ...... 52

4.3 MoreGeneralSingularityStructures ...... 52

4.3.1 PropertiesoftheMatching...... 52

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

4.3.3 Cauchy Horizon Singularities ...... 55

4.4 TheAVTDSingularity ...... 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 SummaryandDiscussion...... 65

5 Cheese Slice Braneworlds 68

5.1 BraneworldCosmologies ...... 69

5.1.1 Randall-SundrumBraneworlds...... 69

5.1.2 Cosmological Braneworlds ...... 70

vii 5.1.3 AnisotropicBraneworlds ...... 75

5.1.4 BraneCollisions...... 77

5.2 ConstructinganInhomogeneousBrane ...... 78

5.2.1 TheCheeseSliceBrane...... 79

5.2.2 EnergyConditions ...... 81

5.3 ExtendingtheMatchingintotheBulk ...... 86

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

5.4 GeneralEmbeddingofMatchedBranes ...... 92

5.4.1 Set-up ...... 92

5.4.2 AnEmbeddingWithnoCorners ...... 93

5.4.3 TheBulkMatchingSurface ...... 94

5.4.4 ApproximationoftheBulk...... 96

5.4.5 MatchingtheBulk ...... 97

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

5.5 The3+1+1Decomposition...... 101

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

5.5.2 FixingtheCoordinates ...... 108

5.5.3 FindingtheMetrics...... 110

5.5.4 TheBulkMetric ...... 112

5.6 TheMatchingoftheBulk ...... 115

5.6.1 TheMatchingConditions ...... 115

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

5.6.3 MatchingFourBulks ...... 122

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

5.7 SummaryandDiscussion...... 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 Thecubiclatticeuniverse...... 6

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

2.2 Matchingacrossacorner...... 19

2.3 Findingthecornerconditions...... 20

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

openFLRWslices...... 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 RedshiftoftheCMBfordiﬀerentmodels...... 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

KasnertoFLRWconstant...... 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 ispositive...... 83

5.2 RegioninwhichtheSECissatisﬁed...... 85

5.3 RegioninwhichtheDECissatisﬁed...... 87

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

5.5 DeﬁnitionoftheZ-Wplane...... 103

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

5.7 Matchingfourdiﬀerentbulks...... 123

5.8 Assumeonesideofthebraneisavacuum...... 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 homogeneous Friedmann-Lemaˆıtre-Roberson-Walker (FLRW) model. These models were a result of assuming the Cosmological Principle, which states the the universe is homogeneous 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 coordinates. 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 spacetimes 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

5 with variations in temperature on the order of 10− . 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, 1 G R Rg (1.2) ab ≡ ab − 2 ab and gab is the spacetime metric. The coupling constant κ is related to the Newtonian gravitational constant by, 8πG κ = N . (1.3) c4 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 = g dxadxb = dt2 + a2(t) dχ2 + S2(χ)(dθ2 + sin2 θdφ2) (1.4) ab − k Chapter 1. Introduction 3

where,

sin(χ√k) k > 0 √k  Sk(χ)=  χ k = 0 . (1.5)   sinh(χ√ k ) | | k < 0 √ k  | |  These symmetries allow the EFE to be simpliﬁed into two second order diﬀerential equations, 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 Chapter 1. Introduction 4 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 Chapter 1. Introduction 5

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 Lemaˆıtre [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

Lemaˆıtre-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

6 (variations under 10− ). 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 Chapter 1. Introduction 6

Figure 1.1: The cubic lattice universe constructed by three mutually perpendicular intersecting 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. Chapter 1. Introduction 7

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 inhomogeneous models in cosmology but they are only a small selection of the vast amount of work conducted in this area. The text of Krasi`nski [65] conducts a comprehensive review of exact inhomogeneous solutions and argues adamantly for their importance in cosmology. 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. Chapter 1. Introduction 8 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 inhomogeneities 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 cosmological 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. Chapter 1. Introduction 9

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. Chapter 1. Introduction 10

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 matchings, 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 Chapter 1. Introduction 11 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 or u. Finally, L u, will denote the Lie derivative of u in the direction of the ;a ∇a ξ 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 timelike 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

1 2 Let us denote two spacetimes by V1 and V2 with corresponding metrics gab and gab. 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 V1 and V1−. Points on Σ1 are deﬁned

+ to belong to both V1 and V1−. We will label V2 in an analogous fashion.

+ Now we deﬁne a new spacetime W = V V − with the points on Σ and Σ identiﬁed. 1 ∪ 2 1 2 + W is then the spacetime that is constructed by the matching of V1 and V2− 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 V = V V − to be a spacetime formed from the matching 1 1 ∪ 1 + of V1 and V1− 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,

a ie. f(x ) = 0. The normal to this surface, na, can by found by taking the ﬁrst derivative of this function such that, ∂ n = f(xa) f . (2.1) a ∂xa ≡ ,a Chapter 2. Matching Regions of Spacetimes 14

V1 V2

+ + V1 V1− V2 V2−

Σ1 Σ2 W

+ V1 V2−

Σ = Σ1 = Σ2

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 . Chapter 2. Matching Regions of Spacetimes 15

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

If the normal is timelike, n na < 0, then Σ is spacelike. • a

If the normal is null, n na = 0, then Σ is also null. • a

If the normal is spacelike, n na > 0, then Σ is timelike. • a

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. Chapter 2. Matching Regions of Spacetimes 16

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

+ + ∂gαβ ∂gαβ− 4+ 4 g = g− = and T = T −. (2.2) ab ab ∂x4 ∂x4 b b

a as calculated on Σ±, were Tb 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:

+ + ∂gαβ ∂gαβ− g = g− and = . (2.3) ab ab ∂x4 ∂x4

+ 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:

+ + ∂gab ∂gab− g = g− and = . (2.4) ab ab ∂xc ∂xc

In both of the above conditions gab± 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

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

+ + a+ a Σ : f (x )=0 and Σ− : f −(x −) = 0 (2.5)

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

+ + f,i f,i− ni = and ni− = . (2.6) ab+ + + ab g f,a f,b g −f,a−f,b− q q We also require the parametric representation of Σ±,

a+ a+ 1 2 3 a a 1 2 3 x = h (u , u , u ) and x − = h −(u , u , u ), (2.7)

a 3 where h ± are of class C or higher. Then the ﬁrst and second fundamental forms on Σ are respectively deﬁned as, ∂xa ∂xb γ = g (2.8) αβ ∂uα ∂uβ ab Chapter 2. Matching Regions of Spacetimes 17 and ∂xa ∂xb K = n . (2.9) αβ ∂uα ∂uβ b;a

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,

+ γαβ = γαβ− (2.10)

and

+ Kαβ = Kαβ− , (2.11)

α where γαβ± and Kαβ± are calculated in terms of u . 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 Chapter 2. Matching Regions of Spacetimes 18 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 Π and Π meet at some angle θ = π. Then use Σ = Π Λ Π as the surface 1 2 6 { 1 ∪ ∪ 2} + 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 coordinates 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.

A Let ma be an orthonormal basis for this space. The ﬁrst and second fundamental forms Chapter 2. Matching Regions of Spacetimes 19

V + Λ

Π1 Π2 θ

V −

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

∂xa ∂xb γˆ = g , (2.12) AB ∂ζA ∂ζB ab ∂xa ∂xb Kˆ C = mC . (2.13) AB ∂ζA ∂ζB b;a There is also a torsion vector deﬁned as,

∂xa ∂xa τ = m1b m2 = m2b m1 . (2.14) A ∂ζA b;a −∂ζA b;a

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

1 mb , 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:

+ γˆAB− =γ ˆAB, (2.15) Chapter 2. Matching Regions of Spacetimes 20

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.

ˆ C ˆ C+ KAB− = KAB , (2.16)

+ τA− = τA (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 Chapter 2. Matching Regions of Spacetimes 21

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= Π Λ Π , if the following conditions are met, { 1 ∪ ∪ 2}

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

+ γˆAB− =γ ˆAB, 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 Chapter 2. Matching Regions of Spacetimes 22 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,

Ra =Γa Γa +Γa Γe Γa Γe . (2.22) bcd bd,c − bc,d ec bd − ed bc

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

1 Γa = gad (g + g g ) . (2.23) bc 2 bd,c cd,b − bc,d

The Riemann curvature tensor can also be written in terms of quantities on a hypersurface within the spacetime using the Gauss-Codacci equations [103],

R ea eb ec ed = (3)R + ǫ(n)(K K K K ) (2.24) abcd α β γ δ αβγδ αγ βδ − βγ αδ

1Null surfaces are excluded in this treatment. Chapter 2. Matching Regions of Spacetimes 23 and

R naeb ec ed = (3) K (3) K , (2.25) abcd β γ δ ∇δ βγ − ∇γ βδ where (3)R and (3) are the Riemann curvature tensor and covariant derivative in αβγδ ∇ the three-space of the hypersurface related to the three metric γ . Also, ǫ(n) nan is αβ ≡ b +1 or 1 depending on whether the normal is spacelike or timelike respectively and the − a ∂xa short hand eα = ∂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),

1 G nanb = (K2 K Kαβ ǫ(n)(3)R) (2.26) ab 2 − αβ −

and

G ea nb = (3) K (3) Kβ, (2.27) ab α ∇α − ∇β α

α where K = Kα . To deﬁne the energy-momentum tensor on Σ we can perform a “pill-box” integration of the EFE across Σ [75],

Σ Λ 1 Σ Sab = lim Tab gab dn = lim Gabdn, (2.28) Σ 0 Σ − κ κ Σ 0 Σ → Z− → Z−

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

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

αβ α β ab S = ea eb S . (2.29)

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

momentum of the stress sheet,

Σ a b lim Gabeαeβdn = ǫ(n) ([Kαβ] γαβ[K]) = κSαβ, (2.30) Σ 0 Σ − → Z− Chapter 2. Matching Regions of Spacetimes 24

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

+ if and only if the naturally induced 3-metrics, γαβ and γαβ− , 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, Chapter 2. Matching Regions of Spacetimes 25 we have,

a k ka = 0. (2.31)

This means ka is orthogonal to itself and is also tangent to the surface. The induced metric 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 various fundamental forms for null surfaces. This was later generalized by Barrabes [3] who proposed a unified formalism to describe singular hypersurfaces whether they be spacelike, 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,

dr2 ds2 = dt2 + a2(t) + r2dφ2 + (1 kr2)dz2 , (2.32) F − 1 kr2 − − Chapter 2. Matching Regions of Spacetimes 26 with k = 1, 0, 1 for negative, zero and positive spatial curvature. The Kasner line {− } element is given by,

ds2 = dT 2 + T 2p1 dX2 + T 2p2 dY 2 + T 2p3 dZ2, (2.33) K − with the restrictions

2 2 2 p1 + p2 + p3 =1= p1 + p2 + p3. (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,

ds2 = dT 2 + T 2p dR2 + R2dΦ2 + T 2p3 dZ2. (2.35) K − 2 1 The only two possible choices of the Kasner exponents are now (p,p3)=( 3 , −3 ) and

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

ds2 = dT 2 + dR2 + R2dΦ2 + T 2dZ2, (2.36) K −

and

ds2 = dτ 2 + dR2 + R2dΦ2 + dζ2, (2.37) K −

are equivalent through the transformations τ = T cosh Z and ζ = T sinh Z.

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

F K γαβ = γαβ, (2.38) where the superscripts F and K will be used to distinguish the FLRW and the Kasner Chapter 2. Matching Regions of Spacetimes 27 regions. This implies that,

1 = γ = γ = T 2pR2 T 2 + T 2p3 Z2 (2.39) − F 00 K00 ,u − ,u ,u 2p 0 = γF 01 = γK01 = T R,uR,v (2.40) a2(u) = γ = γ = T 2pR2 (2.41) 1 kv2 F 11 K11 ,v − 2 2 2p 2 a (u)v = γF 22 = γK22 = T R . (2.42)

From equation (2.41) we see that R = 0 thus from equation (2.40) we must have R = 0. ,v 6 ,u Combining equations (2.41) and (2.42) we get,

R2 R2 = , (2.43) ,v v2(1 kv2) − which can be integrated to give,

C v R = 0 . (2.44) ±1+ √1 kv2 − 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 T 2p3 Z2 . (2.45) ,u − ,u This can be satisﬁed if Z = 0 and du = dT . With this choice we are left with ,u ± 2 2p3 a (u) = u . If(p,p3) = (0, 1) we are left with both sides being Minkowski space.

2 1 For a non-trivial solution we choose (p,p3)=( 3 , −3 ). The FLRW scale factor is now a(t)= t2/3, which is the pressure-free Einstein de Sitter universe.

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

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

KF αβ =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 Chapter 2. Matching Regions of Spacetimes 28

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, − 1 Σ+ : z = ln[C(1 + r2)] (2.46) ±2 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 Chapter 2. Matching Regions of Spacetimes 29 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

ds2 = dt2 + a2t4/3(dr2 + r2dφ2 + dz2) (3.1) F −

and the Kasner metric in the form

2 2 2 4/3 2 2 2 2 2/3 2 ds = dt + a t (dr + r dφ )+ b t− dZ . (3.2) K −

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 Chapter 3. Lookback Time and Observational Consequences 32

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

d2xa dxb dxc +Γa = 0 (3.3) dτ 2 bc dτ dτ gives,

2 t¨+ at1/3(r ˙2 +z ˙2) = 0 (3.4) 3 4 tr¨ + t˙r˙ = 0 (3.5) 3 4 tz¨ + t˙z˙ = 0 (3.6) 3

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

d (a2t4/3r˙) = 0, (3.7) dτ which can be integrated to give, 1 r˙ . (3.8) ∝ a2t4/3 Analogously forz ˙ we have, 1 z˙ . (3.9) ∝ a2t4/3 Finally solving for t˙ from equation (3.4) we get,

1 t˙ . (3.10) ∝ t2/3

Thus in the FLRW regions we have,

2 a ξ ξ η 1 ξ kF = 2/3 , 2/−3 , 0, 2/3 , (3.11) −t pηat −ηat ! 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,

2 a 2/3 2 β 2 α β 2/3 k = t− α + t , , 0, t . , (3.12) K − b at4/3 −b2  s   Chapter 3. Lookback Time and Observational Consequences 33 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,

(u ka) =(u ka) and (u ua) =(u ua) = 1. (3.13) a F a K A F a K −

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

0 0 the observer is timelike. The ﬁrst condition implies that kF = kK thus,

β 2 ξ = α2 + t2 (3.14) b in s where tin is the time of entry into the current Kasner slice. Next we can consider a

2/3 radially moving observer along the boundary with 4-velocity ua =(√2,at , 0, 0) again the same 4-velocity is seen from both sides since the z and Z are constant. This then

1 1 implies kF = kK and we can then solve for the constants,

η2 1 βt α = ξ − and in = ξ/η. (3.15) p η b Equation (3.12) then becomes,

ξ t 2 η η2 1 ξt2/3 ka = (η2 1) + t2, − , 0, . , (3.16) K −ηt 2/3 − t ηat4/3 −ηb2t  − s in p in   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 Chapter 3. Lookback Time and Observational Consequences 34 through a region of spacetime. This angle can be calculated from the scalar product in

3-space, V αk cos ψ = α . (3.17) α β V Vαk kβ

α a β α β 0 0 We have V = (0, 0, 1) and since k is nullp we have k kβ = gαβk k = k k . The angle can then be simpliﬁed to,

3 3 g33k k cos ψ = = g33 . (3.18) 0 0 0 g33(k k ) | |k p The last equality follows from g =pg due to our spacetime signature. Since 33 | 33| cos2 ψ + sin2 ψ = 1 we can write,

k1 sin ψ = g , (3.19) − | 11|k0 p and also, g 2 k1 tan ψ = | 11| . (3.20) − g k3 | 33| From equations (3.11) and (3.16) we have

2 tan ψF tan ψF = η 1= constant and tan ψK = . (3.21) − (t/tin) p

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,

ξ ξ sin ψ ξ cos ψ ka = , in , 0, in , (3.22) F −t2/3 at4/3 − at4/3 and

ξ cos ψ t 2 ξ sin ψ ξt2/3 cos ψ ka = in tan2 ψ + , in , 0, in , (3.23) K − t2/3 in t at4/3 − bt  s in in   Chapter 3. Lookback Time and Observational Consequences 35

F (ψF )in =(ψK )out

(ψK )in =(ψF )out K (tan ψ ) = (tan ψF )in K out (tout/tin) F (ψF )obs

z (ψF )out =(ψF )obs

observer r

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 3 dz kF cos ψin = 0 = 2/3 , (3.24) dt kF at which upon integration gives

t 1/3 a(z z ) = 3t1/3 cos ψ 1 . (3.25) − in in in t − " in # In the Kasner regions we have

dZ k3 t4/3 = K = , (3.26) 0 2 2 dt kK btin tan ψin +(t/tin)

which gives, p t/tin 4/3 4/3 (t/tin) d(t/tin) b(Z Zin)= t . (3.27) in 2 2 − 1 tan ψ +(t/t ) Z in in p Chapter 3. Lookback Time and Observational Consequences 36

This integration can be carried out numerically using a Simpson rule integrator. The relative “thicknesses” will refer to the ratio a(z z )/b(Z Z ). This ratio gives a − in − in 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, a (uak )observed x 1 + redshift = a , (3.28) ≡ (uak )emitted 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 t 2/3 x = in . (3.29) F t out for an Einstein de Sitter region and

2 2 cos ψin tan ψin +(tout/tin) xK = 2/3 . (3.30) p(tout/tin) for the Kasner region. The total redshift factor is the product of the redshift factors from

each slice given by

xtotal = xi . (3.31) i Y From this point forth, we will mean redshift factor when we refer to redshift.

The lookback time, l, is deﬁne by

l = t t , (3.32) emitted − observed

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

t ℓ = emitted 1 . (3.33) tobserved − Chapter 3. Lookback Time and Observational Consequences 37

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. Chapter 3. Lookback Time and Observational Consequences 38

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,

2/3 ℓ = x− 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 Chapter 3. Lookback Time and Observational Consequences 39

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 Chapter 3. Lookback Time and Observational Consequences 40

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

1 π 2π 1 π 2π L = ℓ sin ψdψdθ and X = x sin ψdψdθ . (3.35) 4π 4π Z0 Z0 Z0 Z0

If we assume cylindrical and z symmetry about the observer, we can simplify the expres- Chapter 3. Lookback Time and Observational Consequences 41

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

π/2 π/2 L = ℓ sin ψdψ and X = x sin ψdψ . (3.36) Z0 Z0

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. Chapter 3. Lookback Time and Observational Consequences 42

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

5 slices in our universe by imposing a limit of order 10− 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 Chapter 3. Lookback Time and Observational Consequences 43

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. Chapter 3. Lookback Time and Observational Consequences 44

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 Chapter 3. Lookback Time and Observational Consequences 45

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 Chapter 3. Lookback Time and Observational Consequences 46

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. Chapter 3. Lookback Time and Observational Consequences 47

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.

5 The observed temperature anisotropies in the CMB are on the order of 10− [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 a wedge removed and remaining boundary planes identiﬁed. The vertex of the wedge becomes undeﬁned. Chapter 4. The Structure of the Singularity 50

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 understanding 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 Chapter 4. The Structure of the Singularity 51 diﬃcult.

4.2.2 Classiﬁcation Scheme

Ellis and Schmidt [37] set up a classiﬁcation scheme to describe diﬀerent types of singularities.

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

abcd scalar, RabcdR , 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 Chapter 4. The Structure of the Singularity 52

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 deformationally 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 construction? Let us call the matched spacetime W and the constituent spacetimes used in its Chapter 4. The Structure of the Singularity 53

+ 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 homogeneous 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 Chapter 4. The Structure of the Singularity 54 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 simultaneously 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 assumption 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]. Chapter 4. The Structure of the Singularity 55

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 satisfy 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 dominated”. 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

(3) that have an intrinsic metric, gαβ, extrinsic curvature, Kαβ and an intrinsic Einstein

(3) (3) tensor, Gαβ. By neglecting Gαβ they were able to integrate the four-dimensional

(3) EFE and solve for gαβ. Focusing on dust cosmologies, they deﬁned the singularity to be at the time t = t such that the density ρ as t t . The structure of the 0 → ∞ → 0 singularity is then given by (3)g and its evolution depends only on K . Since K αβ t0 αβ αβ

describes how the three-surfaces curve in the ambient spacetime (in this case the ∂/∂t Chapter 4. The Structure of the Singularity 56 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

a Let U be a spacetime with metric gab and coordinates x . 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,

∂xa ∂xb γ = g , (4.1) αβ ∂ξα ∂ξβ ab Chapter 4. The Structure of the Singularity 57

U, metric: gab coordinates: xa

∂ na ∂t

Π0, metric: γαβ coordinates: ξα t0 t1

a Figure 4.1: The spacetime U with coordinates x 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, ∂xa ∂xb K = n , (4.2) αβ ∂ξα ∂ξβ b;a

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

α curvature is then K = Kα . 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.

∂ = Nn + M . (4.3) ∂t a a

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 Chapter 4. The Structure of the Singularity 58 considered. These quantities must satisfy the EFE written in the form of constraint equations [59],

(3)R KαβK + K2 = 2ρ, (4.4) − αβ

(3) Kα (3) K = J (4.5) ∇α β − ∇β − β

and evolution equations, ∂ γ = 2NK + L γ , (4.6) ∂t αβ − αβ M αβ ∂ 1 Kα = N (3)Rα + KKα + Sα + γα (ρ Sν) (3) α(3) N + L Kα, (4.7) ∂t β β β β 2 β − ν − ∇ ∇β M β where (3)R and (3)R are the spatial Ricci scalar and Ricci tensor respectively. (3) is ab ∇

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,

αβ 2 K˜ K˜αβ + K˜ = 2˜ρ, (4.8)

(3) K˜ α (3) K˜ = J˜ , (4.9) ∇α β − ∇β − β

and the VTD evolution equations,

∂ γ˜ = 2NK˜ , (4.10) ∂t αβ − αβ

∂ 1 K˜ α = N K˜ K˜ α + S˜a + γ˜α ρ˜ S˜ν . (4.11) ∂t β β β 2 β − ν 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 Chapter 4. The Structure of the Singularity 59 t , the values of → ∞ γ γ˜ αβ − αβ  ˜ Kαβ Kαβ  −   ρ ρ˜  0. (4.12) −  →  J J˜ α − α  Sαβ S˜αβ  −    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

Πt±± 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 Πt−− matches + with each leaf of the foliation Πt+ 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 Πt−− with Πt+ , let us single out one leaf of the foliation on each side and call them Π0± as illustrated in Figure 4.2. Π0± are spatial three-surfaces

+ in V ±. We wish to match Π0− with Π0 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 Chapter 4. The Structure of the Singularity 60

+ V V −

t θ1 Λ

Π− θ2 0

Figure 4.2: A leaf of the foliation on each side is singled out, Π0±. The intersection of Π0± 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

+ Π0 and Π0−. 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

A two-dimensional space of normals to Λ. Let ma 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 Π0− and Π0 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− Π , constitutes ≡ 0 ∪ 0 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,

dr2 ds2 = dt2 + a2(t) + r2dφ2 + (1 kr2)dz2 , (4.13) F − 1 kr2 − − Chapter 4. The Structure of the Singularity 61 where k = 1, 0, 1 and the Kasner line element given by, {− }

ds2 = dT 2 + T 2p1 dX2 + T 2p2 dY 2 + T 2p3 dZ2, (4.14) K − with the restrictions

2 2 2 p1 + p2 + p3 =1= p1 + p2 + p3. (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,

2 2 2 γ˜αβ = a diag (1, r , 1) , K˜αβ = a,ta diag (1, r , 1) ,

a,t 2 ˜ ˜ ρ˜ = 3 a , 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

1+ 2+ ma = (1, 0, 0, 0) and ma = (0, 0, 0, 1). (4.17) Chapter 4. The Structure of the Singularity 62

Thus we have,

γˆ+ = a2 diag (1, r2) , Kˆ +1 = a a diag (1, r2) , AB AB − ,t ˆ +2 + KAB =0 and τA = 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

1 2 ma− = (1, 0, 0, 0) and ma− = (0, 0, 0, 1). (4.19)

Thus we have,

2p1 2p2 1 1 2p1 2p2 γˆ− = diag (T , T ) , Kˆ − = T − diag ( p T , p T ) , AB AB − 1 − 2 ˆ 2 KAB− =0 and τA− = 0. (4.20)

If we choose the coordinates on the corner as ζA = u, v , parametrize the surface as { }

r cos φ = u = X and r sin φ = v = Y, (4.21)

2/3 we can satisfy equation (2.15)–(2.17). Recall that a = t 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 = ln T 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 Chapter 4. The Structure of the Singularity 63 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,

1 z˜ = z ln(1 + r2). (4.22) − 2 The FLRW metric (4.13) then becomes,

ds2 = dt2 + a2(t) dr2 + r2dφ2 + 2rdrdz˜ +(1+ r2)dz˜2 . (4.23) F − On the Kasner side we will make the transformations,

R = √X2 + Y 2, (4.24)

Φ = arctan(Y/X), (4.25)

9 1 2 3 2 2 1 Z˜ = Z 3bT 3 1+ b2T 3 + 2b T 1+ b2T 3 +3ln 1+ b2T 3 + bT 3 − 16b5 − q q q (4.26) and

3 1 2 2 1 t˜= bT 3 1+ b2T 3 ln 1+ b2T 3 + bT 3 , (4.27) 2b3 − q q where b is a positive constant. With these transformations the Kasner metric (4.14)

becomes,

2 2 4 2 2 2 2 2 ds = dt˜ + T 3 dR + R dΦ + 2bdtd˜ Z˜ + T − 3 dZ˜ . (4.28) K − The matching now takes place along the surfacez ˜ =z ˜0 on the FLRW side and Z˜ = Z˜0 on the Kasner side, withz ˜0 and Z˜0 being constants. The coordinates, φ = Φand t = t˜, can be

4 2 2 9 3 ˜ identiﬁed along the matching surface. We must also have r = 3 bR and a (t)= 4b2 T (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, 2 γ11 = a , K11 = a,ta,

2 γ13 = a r, K13 = a,tar,

2 2 2 γ22 = a r , K22 = a,tar ,

2 2 2 γ33 = a (r + 1), K33 = a,ta(r + 1), Chapter 4. The Structure of the Singularity 64

a2 1 ρ = 3 ,t − , and J =0= S . (4.29) a2 a ab 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,

2 2 N = 1+ b T 3 and Ma = (0, 0, b) (4.30) q 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˜ =z ˜ ,t = t and Z˜ = Z˜ , t˜= t˜ respectively { 0 0} { 0 0} ˜ 2 with t0 and t0 being constants. Recall that the coordinates are such that r = 3 bR 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,

4 4 9 3 3 2 2 + γˆ− = diag( T , RT ) = diag(a ,ra ) =γ ˆ , (4.31) AB 4b2 0 0 0 0 AB where T0 = T (t˜0) and a0 = a(t0). Let an orthonormal basis of the corner be chosen on both sides such that,

1 2 1 3 2 3 +1 m− = (0, 0, 0, T − 1+ b T ) m = (0, 0, 0,a0) α 0 0 α . (4.32) 2 q +2 m− = (1, 0, 0, b) m = (1, 0, 0, 0) α − α Then the second corner condition, equation (2.16), is satisﬁed with,

2 1 3 3 4 2 2 2 +1 Kˆ − = bT diag(1, b R )= a diag(1, r )= Kˆ (4.33) AB 2 0 9 0 AB

and

1 2 2 2 3 2 3 4 2 2 2 +2 Kˆ − = T 1+ b T diag(1, b R )= a a diag(1, r )= Kˆ . (4.34) AB 3b2 0 0 9 ,t0 0 AB q 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 Chapter 4. The Structure of the Singularity 65 subtends an angle of π about the corner. On the Kasner side, the foliation is not orthogonal 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 Chapter 4. The Structure of the Singularity 66

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 singularities, corresponding to FLRW regions. At late time, the vacuum regions become arbitrarily thin pancake-like singularities. Chapter 4. The Structure of the Singularity 67 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 Π Π and show that the entire model is AVTD. The inhomogeneities in the ≡ 1 ∪···∪ n 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 embedded 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 problem [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 Chapter 5. Cheese Slice Braneworlds 70 investigations into higher dimensional cosmologies. The ﬁve-dimensional metric is,

(5) 2 2 y /ℓ a b 2 ds = e− | | ηabdx dx + dy , (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,

6 Λ5 = 2 2 . (5.2) −κ5ℓ

2 where κ5 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, 6 σ = 2 , (5.3) κ5ℓ 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)G = κ2 T Λ (5)g , (5.4) AB 5 AB − 5 AB 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 Chapter 5. Cheese Slice Braneworlds 71 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)2γ dxαdxβ + dy2, (5.5) − αβ

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,

T A = SAδ(y) = diag ( ρ ,P,P,P, 0)δ(y) (5.6) B B − b

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),

[Ka δaK]= κ2Sa. (5.7) b − b 5 b 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],

2 2 n,y κ5 a,y κ5 = (3P + 2ρb), = ρb. (5.8) n 0+ 6 a 0+ − 6 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, 2 2 a0,t ρb,t + 3H(ρb + P ) = 0, H 2 , (5.9) ≡ a0 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, κ4 κ2Λ k C H2 = 5 ρ2 + 5 5 + , (5.10) 36 b 6 − a2 a4 Chapter 5. Cheese Slice Braneworlds 72 where a a(t, 0) and C is a constant of integration. The most striking difference from 0 ≡ 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,

κ ρ = √ 6Λ. (5.11) | 5| b ± −

6 Thus with Λ = 2 2 we recover the Randall-Sundrum solution and the fine tuning − κ5ℓ condition of equation (5.3) with the only energy content being the tension, ρ = σ. − b 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,

κ4 1 κ4 κ4 k C H2 = 5 σ2 + 5 σρ + 5 ρ2 + . (5.12) 36 − ℓ2 18 36 − a2 a4

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,

κ4 κ2 κ2 8πG = 5 σ = 5 . (5.13) 4 ≡ 6 ℓ

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 Chapter 5. Cheese Slice Braneworlds 73 scale factor is given by,

1 κ2ρ2 3C a(t, y) = 1+ 5 b a2 + 2 6ρ 0 κ2ρ a2 ( B 5 B 0 1 κ2ρ2 3C + 1 5 b a2 cosh(y/ℓ) (5.14) 2 − 6ρ 0 − κ2ρ a2 B 5 B 0 1/2 κ ρ 5 b a2 sinh( y /ℓ) , −√ 6ρ 0 | | − B ) 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,

κ2ρ κ4ρ2 C 1/2 a(t, y)= a2 5 b a2 y + 5 b a2 + y2 . (5.15) 0 − 3 0| | 36 0 a2 0 In all cases the other metric coeﬃcient is given by,

a n(t, y)= ,t . (5.16) a0,t

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 Chapter 5. Cheese Slice Braneworlds 74 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)R eAeBeC eD = (4)R + ǫ(n)(K K K K ) (5.17) ABCD a b c d abcd ac bd − bc ad and

(5)R nAeBeC eD = (4) K (4) K . (5.18) ABCD b c d ∇d bc − ∇c bd The four-dimensional Einstein tensor is then given by,

2κ2 1 (4)G = 5 T eAeB + T nAnB T A (4)g (5.19) ab 3 AB a b AB − 4 A ab 1 +KK KcK (4)g (K2 KcdK ) E , ab − a bc − 2 ab − cd − ab

where

E (5)CA n nC eBeD, (5.20) ab ≡ BCD A a b (5) and 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) 2 2 (4) a b ds = dy + gabdx dx (5.21)

where the brane is located at y = 0. Also, let the ﬁve-dimensional energy momentum be

of the form,

T = Λ(5)g + S δ(y) with S = σ(5)g + τ , (5.22) AB − AB AB AB − AB AB

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)G = Λ (4)g + κ2τ + κ2π E , (5.23) ab − 4 ab 4 ab 5 ab − ab

where 1 1 Λ = κ2 Λ+ κ2σ2 , (5.24) 4 2 5 6 5 Chapter 5. Cheese Slice Braneworlds 75

κ4 κ2 = 5 σ (5.25) 4 6 and 1 1 1 1 π = τ τ c + ττ + (4)g τ τ cd (4)g τ 2. (5.26) ab −4 ac b 12 ab 8 ab cd − 24 ab The values of E are taken to be the limiting value as y 0. This result diﬀers from the ab →

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, dr2 ds2 = f(r)dt2 + + r2dσ2, (5.27) − f(r) 3 2 where dσ3 represents a three-dimensional spatial metric of uniform curvature. The function f(r) is given by, r2 f(r)= k + with k = 1, 0, 1. (5.28) ℓ2 −

2 The value of k is determined by the curvature of dσ3, whether it be spherical, planar or

6 hyperboloid and ℓ is the curvature scale related to the cosmological constant Λ = 2 . − ℓ 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

2 2p1 2 2p2 2 2p3 2 dσ3 = t dx + t dy + t dz . (5.29)

We then arrive at what Frolov calls the Kasner-AdS spacetime. Here the exponents must satisfy the familiar Kasner restrictions,

2 2 2 p1 + p2 + p3 =1= p1 + p2 + p3. (5.30) Chapter 5. Cheese Slice Braneworlds 76

The coordinate transformation w = ℓ ln(r/ℓ) can be made to write the metric in the − form,

2 2 w /ℓ 2 2p1 2 2p2 2 2p3 2 2 ds = e− | | dt + t dx + t dy + t dz + dw , (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, 6 σ = 2 , and ρ = 0, (5.32) ∓κ5ℓ 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 cosmological 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 cosmologies, refer to [102]. Chapter 5. Cheese Slice Braneworlds 77

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 Chapter 5. Cheese Slice Braneworlds 78 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 Chapter 5. Cheese Slice Braneworlds 79 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 altogether. 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 Chapter 5. Cheese Slice Braneworlds 80 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,

2 2 2p1 2 2p2 2 2p3 2 ds − = dT + T dX + T dY + T dZ , (5.33) −

with the restrictions

2 2 2 p1 + p2 + p3 =1= p1 + p2 + p3, (5.34) satisﬁes the EFE on the brane in the case of a vacuum. This requires,

τ − 0 and Λ− = 0. (5.35) ab ≡ 4

Turning to the ﬂat FLRW metric we have,

ds2+ = dt2 + a2(t) dx2 + dy2 + dz2 . (5.36) − Assuming a perfect ﬂuid such that,

+ τab =(ρ + P )uaub + Pgab, (5.37)

then directly from equation (5.23) we have the modiﬁed Friedmann equations,

a 2 Λ+ κ2ρ ρ ,t = 4 + 4 1+ (5.38) a 3 3 2σ and a Λ+ κ2P ρ κ2ρ 2ρ ,tt = 4 4 1+ 4 1+ . (5.39) a 3 − 2 σ − 6 σ 2The consequences of these assumptions will be explored in Section 5.3 Chapter 5. Cheese Slice Braneworlds 81

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, 2 2 1 a(t)= t 3 p = p = , and p = . (5.40) 1 2 3 3 −3 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,

T ξaξb 0 (5.41) ab ≥ for all timelike ξa.

The Strong Energy Condition (SEC) states that, 1 T ξaξb T, (5.42) ab ≥ −2 where T = T a. This is equivalent to the statement that R ξaξb 0, through the a ab ≥ four-dimensional EFE. Chapter 5. Cheese Slice Braneworlds 82

The Dominant Energy Condition (DEC) states that

T aξb (5.43) − b

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,

Λ+ ρ2 ρ 1 P = 4 1+ − . (5.47) κ4 − 2σ σ We can also solve for the density as a function of cosmological time, t, to get,

ρ 2 4 = 1 1 Λ+ . (5.48) σ − ± − κ2σ 4 − 3t2 s 4 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 Chapter 5. Cheese Slice Braneworlds 83

ρ tt 1 1 1 ≤ 2 2 Λ+ 0 +++ + 4 ≤ 2 0 < Λ+ κ σ +0 - - 4 ≤ 2 κ2σ + 2 < Λ4 + 0 - no real solution

Table 5.1: Positivity of matter density, ρ, as a function of cosmological time, t, for

+ 2 diﬀerent values of the cosmological constant, Λ4 . The constants are t1 = + and √3Λ4 2 t = 2 + . 2 3(2Λ κ2σ) 4 − 4 q

Positivity of ρ 4

ρ< 0 3

2 2 ρ> 0 no real solution t κ4σ 2

0 −1 0 1 2 3 + Λ4 2 κ4σ

Figure 5.1: Regions in which ρ is positive. The lower curve represents ρ = 0 or t2 =

2 2 + . The upper curve represents the boundary over which no real solution exists, t = 3Λ4 8 + . The positivity of ρ is also equivalent to the WEC. 3(2Λ κ2σ) 4 − 4 Chapter 5. Cheese Slice Braneworlds 84

+ from the second term in equation (5.48). When Λ4 is negative, the second term is greater

+ than 1. With the choice of the + sign ρ will be greater than 0. If Λ4 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,

2 + 1 1 ρ Λ4 ρ − 4 ρ − ρ + P = ρ + + 1+ = 2 1+ . (5.49) 2σ κ4 σ 3t κ4 σ 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,

1 4 ρ − 2 1+ 0, (5.50) 3t κ4 σ ≥ which is equivalent to the condition, ρ 1. (5.51) σ ≥ − From equation (5.48) we see that this condition is satisﬁed whenever the term,

2 + 4 1 2 Λ4 2 , is real and the positive sign chosen. This requires that, − κ4σ − 3t q 2 κ2σ t 2 if Λ+ > (5.52) ≤ 3(2Λ+ κ2σ) 4 2 s 4 − 4 and is always true if κ2σ Λ+ . (5.53) 4 ≤ 2 The second condition of equation (5.45) leads to,

1 4 ρ − ρ + 2 1+ 3ρ 0. (5.54) t κ4 σ − ≥ Substituting for ρ we ﬁnd,

2 4 2 1 1 Λ+ + Λ+ 1 0. (5.55) − κ2σ 4 − 3t2 κ2σ 4 − 3t2 − ≥ s 4 4 Chapter 5. Cheese Slice Braneworlds 85

Region in which the SEC is satisﬁed 4

2 2 no yes no real solution 2 t κ4σ

0 −1 0 1 2 3 + Λ4 2 κ4σ

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

+ 2 Λ4 3√y +y+3y+4 the boundary where equation (5.55) fails and is described by 2 = − , where κ4σ 12y 2 2 y = κ4σt . The right curve represents the boundary over which no real solution exists,

2 t = 2 + . 3(2Λ κ2σ) 4 − 4 q + As with the WEC we again see a speciﬁc range of t and Λ4 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 Λ4 for which the SEC is 2 satisﬁed for all time. For 0 Λ+ κ4σ SEC is satisﬁed for for late times (t ), but ≤ 4 ≤ 2 → ∞ 2 is violated for early times (t 0). For Λ+ > κ4σ the SEC is violated for both early and → 4 2 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,

1 4 ρ − ρ P = ρ 2 1+ (5.56) ≥ − − 3t κ4 σ Chapter 5. Cheese Slice Braneworlds 86 is always satisﬁed for positive ρ. The equality stems from substitution of equation (5.49).

Now we check the condition,

1 4 ρ − ρ P = 2 1+ ρ, (5.57) ≥ 3t κ4 σ − which is equivalent to the condition, ρ 4 2ρ 1+ 2 . (5.58) σ ≥ 3t κ4 Substituting from equation (5.48) and choosing the positive sign, we ﬁnd that the con-

dition becomes,

2 4 2 1 1 Λ+ + Λ+ + 1 0. (5.59) − κ2σ 4 − 3t2 κ2σ t2 − 4 ≥ s 4 4 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.

2 Therefore, the DEC is satisfied in the same region as the WEC. For 0 Λ+ κ4σ both ≤ 4 ≤ 2 2 + κ4σ the DEC and the WEC are satisfied for all time while for Λ4 > 2 they 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. Chapter 5. Cheese Slice Braneworlds 87

Region in which DEC is satisﬁed 4

ρ> 0 ρ< 0 no real solution 3

2 2 2 t κ4σ

Equation (5.59) is valid. 1

0 −1 0 1 2 3 + Λ4 2 κ4σ

Figure 5.3: The region in which the DEC is satisﬁed is illustrated. The lower curve

2 2 represents ρ = 0 or t = + . The right curve represents the boundary over which no 3Λ4 2 real solution exists, t = 2 + . Equation (5.59) is valid below the central curve 3(2Λ κ2σ) 4 − 4 + 2 q Λ4 √3(3y +8y)+3y+12 2 2 indicated. This curve is given by 2 = , where y = κ σt . κ4σ 12y 4 Chapter 5. Cheese Slice Braneworlds 88

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+ = N 2(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,

2 2 2 2 2 2 2 2 2 2 ds − = M (T,w)dT + B (T,w) dR + R dΦ + E (T,w)dZ + dw , (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 κ2 A κ2 ,y = 5 (3P + 2(ρ + σ+)), ,y = 5 (ρ + σ+), (5.62) N + 6 A + − 6 0 0 where σ+ is the brane tension. We now take the first order expansion, keeping in mind that the zeroth order terms must be N 2 N 2(t, 0) = 1 and A2 A2(t, 0) = t4/3 to 0 ≡ 0 ≡ reproduce the flat FLRW cosmology on the brane. The first order expansions are then,

2 N(t, y)=1+ N1(t)y and A(t, y)= t 3 + A1(t)y. (5.63) Chapter 5. Cheese Slice Braneworlds 89

The conditions in equation (5.62) are then,

2 2 κ5 + A1 κ5 + N1 = (3P + 2(ρ + σ )), 2 = (ρ + σ ), (5.64) 6 t 3 − 6

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,

2 κ5 1 K− = S− S−g− , (5.65) ab − 2 ab − 3 ab where Sab− is the energy-momentum on the brane, gab− the four-dimensional Kasner metric

a and S = Sa . To reproduce the Cheese Slice we need the Kasner brane to be a vacuum with the possibility of a brane tension. Thus,

S− = σ−g− . (5.66) ab − ab

Equation (5.65) then implies,

M B E κ2σ ,w = ,w = ,w = 5 − (5.67) M + B + E + − 6 0 0 0 Similar to the FLRW case, we now take the ﬁrst order expansion of the metric functions

2 with the zeroth order terms being M M(T, 0) = 1, B B(T, 0) = T 3 and E 0 ≡ 0 ≡ 0 ≡ 1 E(T, 0) = T − 3 to produce the Kasner brane. The ﬁrst order expansions are then,

M(T,w) = 1+ M1(T )w,

2 B(T,w) = T 3 + B1(T )w (5.68)

1 and E(T,w) = T − 3 + E1(T )w.

With equation (5.67) we can solve for the ﬁrst order terms giving,

2 1 M = µ B = µT 3 and E = µT − 3 (5.69) 1 − 1 − 1 −

2 − κ5σ where µ = 6 . Chapter 5. Cheese Slice Braneworlds 90

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 Σ Σ(t, 0) = constant. Let the 0 ≡ 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 parametrization we can find the first fundamental forms one either side of Σ. Keeping only first order terms in y and w, we get,

2 2 Υ+ = (u 3 + 2A x)u 3 Σ2 (1+2N x), (5.75) 00 1 ,u − 1 2 2 + 3 3 Υ03 = (u + 2A1x)u Σ,uΣ,x, (5.76)

2 2 + 3 3 Υ11 = (u + 2A1x)u , (5.77)

2 2 + 3 3 2 Υ22 = (u + 2A1x)u v , (5.78)

2 2 + 3 3 2 Υ33 = (u + 2A1x)u Σ,x + 1 (5.79) and

2 2 2 2 Υ− = T − 3 (1 2µw) Z (1 2µw) T + w , (5.80) 00 − ,u − − ,u ,u 2 Υ− = T − 3 (1 2µw) Z Z + w w , (5.81) 03 − ,u ,x ,u ,x 4 Υ− = T 3 (1 2µw) , (5.82) 11 − 4 2 Υ− = T 3 (1 2µw) v , (5.83) 22 − 2 2 Υ− = T − 3 (1 2µw) Z + w , (5.84) 33 − ,x ,x (5.85) Chapter 5. Cheese Slice Braneworlds 91 in the FLRW and Kasner sides respectively.

+ From the Darmois matching conditions we must have Υab = Υab− . Equating equations (5.77) and (5.82) we can solve for w to get,

2 2 1 (u 3 + 2A1x)u 3 µw = 4 . (5.86) 2 − 2T 3

+ The equality of equations (5.78) and (5.83), Υ22 =Υ22− , is now automatically satisﬁed.

+ Equating Υ03 =Υ03− and taking the zeroth order term gives the condition,

2A1u(uT,u T ) 0= 11− . (5.87) 3µ2T 3 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

+ Υ00 =Υ00− leads to, C 2/3 N = A . (5.89) 1 − u 1 If we recall the matching conditions for the bulk to support an FLRW brane from equation (5.64) we can now arrive at an equation of state,

2 2 0 = 3P + (2 C 3 )ρ (1 C 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. Chapter 5. Cheese Slice Braneworlds 92

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 investigate 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 M4 and M4− be two branes with the respective metrics given by gab and gab− . We will + + assume that M4 and M4− can be matched along some hypersurface Σ3 = Σ3− = Σ3 using Chapter 5. Cheese Slice Braneworlds 93 the Darmois matching conditions. We wish to see the consequences of embedding this

+ + brane into some bulk. More precisely, we embed M4 and M4− into respective bulks M5 and M5− and match the bulks along a hypersurface Σ4±, which we wish to be an extension of Σ . Thus we choose Σ± such that, Σ = M ± Σ±. 3 4 3 4 ∩ 4 + + Expressions with or − refer to the respective quantity in either M5 or M5−. 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 = h− where h g m m . (5.91) AB AB AB ≡ AB − A B and also, + D 1L ΩAB = ΩAB− where ΩAB hA mB D = mhAB, (5.92) − ≡ k 2 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 M4 and M4−. 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 M onto Σ . The denotes 4 3 k

covariant diﬀerentiation on M4 associated with gAB and L the Lie derivative.

A A Let the unit normal to M4 be denoted n , such that nAn = 1, when embedded in

M5. Then we can write the bulk metric in the form,

2 c a b 2 g˜AB = gAB + nAnB, ds = gab(x ,w)dx dx + dw . (5.93)

th A The “5 coordinate”, w is deﬁned normal to the brane such that nAdx = 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 Σ M . Let us denote the normal to Σ by m such ∈ 3 ⊂ 4 3 A Chapter 5. Cheese Slice Braneworlds 94 that

m mA =1 and m M . (5.94) A A ∈ 4

It follows from equation (5.92) that we have

+ m = m−, (5.95) A − A

A and also nAm = 0 by the deﬁnition of nA.

+ We have not yet deﬁned how M4 and M4− 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 M4± 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 = n− for all p Σ . (5.97) A p A p ∈ 3

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.

A Let there be a unique normal to Σ4 at each point, denotedm ˜ A, such thatm ˜ Am˜ = 1 and

n =m ˜ for all p Σ . (5.98) A p 6 A p ∈ 3

The normalized projection ofm ˜ A onto M4 at point p is then equivalent to mA at point p,

gBm˜ A B = m for all p Σ . (5.99) gBm˜ A p ∈ 3 A B p k k

Chapter 5. Cheese Slice Braneworlds 95

(B) (α) To see this we can choose an orthonormal basis, eA , at point p such that eA lies in Σ3 (3) (4) for α = (0, 1, 2). We can choose eA = mA p and we are left with eA = nA p. Sincem ˜ A is perpendicular to Σ Σ , the only non-zero components are, 3 ⊂ 4

(3) (4) m˜ A p =m ˜ (3)eA +m ˜ (4)eA . (5.100)

The projection is then,

gBm˜ = (˜gB n nB)m ˜ A B p A − A B p B (4) B (3) (4) = (˜gA eA e(4))(m ˜ (3)eB +m ˜ (4)eB ) − =m ˜ e(3) +m ˜ e(4) e(4)m˜ (3) A (4) A − A (4) (3) =m ˜ (3)eA , (5.101) and upon normalization we have

gBm˜ A B = e(3) = m for all p Σ , (5.102) gBm˜ A A p ∈ 3 A B p k k A thus conﬁrming equation (5.99). Let the angle between nA andm ˜ A be ψ. Then nAm˜ = cos ψ, allowing us to simplifying the normalized projection,

B B gA m˜ B gA m˜ B B = g m˜ B gEF m˜ m˜ k A k E F m˜ n nBm˜ = p A − A B E F √1 n n m˜ Em˜ F m˜ − n cos ψ = A − A 1 cos2 ψ − =m ˜ csc ψ n cot ψ (5.103) pA − A

Thus at any point p Σ we have, ∈ 3

m = (m ˜ csc ψ n cot ψ) (5.104) A p A − A p

m˜ A p =(mA sin ψ + nA cos ψ) p. (5.105)

Chapter 5. Cheese Slice Braneworlds 96

5.4.4 Approximation of the Bulk

The general expression for the Taylor expansion of a tensor is given by,

L L L 2 2 3 T (q)= T (p)+( ξ1 T ) pλ +( ξ2 + ξ1 )T pλ + O(λ ), (5.106)

A A where ξ1 and ξ2 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,g ÃB, 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,

L 2 gÃB(w) =g ÃB p +( ngÃB) pw + O(w ), (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,

L 2 g˜AB(w) = (gAB + nAnB) p + n(gAB + nAnB) pw + O(w )

2 = (gAB + nAnB) p + 2KAB pw + O(w ), (5.108)

where the last equality follows from the extrinsic curvature of the brane,

1 K L g and L n = 0 (5.109) AB ≡ 2 n AB n A

Similarly we can also expand the normal to Σ4 in the same way,

L 2 m˜ A(w) =m ˜ A p +( nm˜ A) pw + O(w ). (5.110)

Substituting from equation (5.105) we get,

L 2 m˜ A(w)=(mA sin ψ + nA cos ψ) p + sin ψ nmA pw + O(w ).

Chapter 5. Cheese Slice Braneworlds 97

5.4.5 Matching the Bulk

We are now prepared to examine the matching conditions in the bulk. The intrinsic

+ metrics of Σ4 and Σ4− must match, that is,

+ h˜ = h˜− where h˜ g˜ m˜ m˜ . (5.111) AB AB AB ≡ AB − A B

From the ﬁrst order Taylor expansion of equation (5.108) and equation (5.111) we have,

h˜AB =(gAB + nAnB) + 2KABw

[(m sin ψ + n cos ψ) + sin ψL m w] [(m sin ψ + n cos ψ) + sin ψL m w] − A A n A B B n B + O(w2), (5.112)

where it is understood that all the coeﬃcients are evaluated at p. We can rearrange

equation (5.112) to get,

h˜ = h (m cos ψ n sin ψ)(m cos ψ + n sin ψ) AB AB − A − A B B + 2 K sin ψL m (m sin ψ + n cos ψ) w + O(w2), (5.113) AB − n (A B) B) 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 ≡ Chapter 5. Cheese Slice Braneworlds 98

+ M4 + mA

+ m˜ A

ψ+ p nA

Σ+ ψ− 4

Σ− 4 m˜ A−

mA− M4−

Figure 5.4: Matching of two branes extended into the bulk. The solid curve represents

+ the branes M4 and M4−. The bulks are matched across the surface Σ4 represented by

the dashed curve. The angles ψ± are measured between the normals to Σ4± and M4± and

+ satisfy ψ + ψ− = π to avoid a canonical singularity at the point p Σ . ∈ 3 Chapter 5. Cheese Slice Braneworlds 99 we have

+ + cos ψ = cos ψ− and sin ψ = sin ψ−. (5.116) −

Together with equation (5.95) and equation (5.97) we have,

+ + + + (m cos ψ n sin ψ ) =(m− cos ψ− n− sin ψ−) . (5.117) A − A p A − A p

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 sin ψ L m (m sin ψ +n cos ψ )= K− sin ψ−L m− (m− sin ψ−+n− cos ψ−). AB− n (A B) B) AB− n (A B) B) (5.118)

Due to equations (5.116) and (5.95), the condition can be simpliﬁed into,

+ KAB = KAB− . (5.119)

Since KAB is the second fundamental form of M4, it is related to the matter content on the brane by, 1 [K ]= κ2 S + Sg . (5.120) ab − 5 ab 3 ab

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

[K ]+ = K¯ + K+ (5.121) AB AB − AB Chapter 5. Cheese Slice Braneworlds 100

and

[K ]− = K¯ − K− . (5.122) AB AB − AB

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

+ − Sab p = Sab p, (5.124)

+ + since K = K− implies S = S−. The stress-energy on the brane must be the same at p

+ + in both branes M4 and M4−. That is, observers in M4 and M4− 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 M4 and M4− 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+ = ρ = 0. If this matching surface is 0 6 given by y = 0, then,

+ ρy 0+ = ρy− 0− . (5.125) → 6 → Chapter 5. Cheese Slice Braneworlds 101

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 = θ¯. 6 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 = π. This would need a corner or some form of conical singularity about 6

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 = π. This adds an additional variable to the condition of equation (5.111). 6 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

Gümrük¸cüoglu [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 Chapter 5. Cheese Slice Braneworlds 102

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 functions W (xA) and Z(xA) such that W =

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 M and Σ with Σ M Σ being the corner. We 4 4 3 ≡ 4 ∩ 4

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,

A B A B g˜ABn n =1=˜gABm m . (5.127)

The two functions,

AB 1/2 AB 1/2 N = g˜ ∂ Z∂ Z − M = g˜ ∂ W∂ W − (5.128) | A B | | A B | Chapter 5. Cheese Slice Braneworlds 103

M4, Z = 0 W

Z M5

Σ4, W = 0

Σ3

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,

g˜ nAmB = cos θ = 1, (5.129) AB 6 ±

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,

∂xA ∂xA ∂xA eA = M , f A = Σ , and ζA = Σ . (5.130) a ∂ya ∈ 4 i ∂ηi ∈ 4 α ∂ξα ∈ 3

These vectors are thus orthogonal to the normals,

A A A A nAea =0= mAfi , and nAζα =0= mAζα . (5.131) Chapter 5. Cheese Slice Braneworlds 104

+ + (Σ4−,h−) (M4 ,g )

m− + v− u

n+ + + (M5−, gˆ−) (M5 , gˆ )

+ φ− φ (Σ3,γ)

m+ u− n− v+

+ + (M4−,g−) (Σ4 ,h )

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. (M5−,g−)

+ + and (M5 ,g ) are the two bulks to be matched with the respective branes acting as the matching surface. Chapter 5. Cheese Slice Braneworlds 105

They also satisfy the completeness relations,

b A b j A j β A β eAea = δa fAfi = δi and ζAζα = δα. (5.132)

The induced metrics on the hypersurfaces are then,

A B A B A B gab =g ÃBea eb in M4, hij =g ÃBfi fj in Σ4, and γαβ =g ÃBζα ζβ in Σ3. (5.133)

This allows us to write the bulk metric as either,

a b g˜AB = gabeAeB + nAnB (5.134) or

i j g˜AB = hijfAfB + 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,

u = U∂ W M and v = V∂ Z Σ . (5.136) a a ∈ 4 i i ∈ 4

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,

a b i j gabu u =1= hijv v , (5.137) Chapter 5. Cheese Slice Braneworlds 106

We can deﬁne local frames orthogonal to these normals by, ∂ya ∂ηi εa = and ǫi = (5.138) α ∂ξα α ∂ξα satisfying,

a β β i β a i εαεa = δα = ǫαǫi and uaεα =0= viǫα, (5.139) which allows us to carry out the (3 + 1) decomposition in M4 and Σ4 respectively with the resulting metrics,

α β gab = γαβεa εb + uaub (5.140) and

α β hij = γαβǫj ǫj + vivj. (5.141)

a j A The three-bases εα, ǫα and ζα all span the corner, Σ3, and are related by

a A i A A εαea = ǫαfi = ζα . (5.142)

A We choose to use ζα 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, (uaeA)n uAn = 0. (5.143) a A ≡ A

Similarly in Σ4 we have, (vif A)m vAm = 0. (5.144) i A ≡ A 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,

α β g˜AB = γαβζAζB + uAuB + nAnB. (5.145)

In Σ4 we can perform the same procedure to get an alternate, but equivalent decomposition,

α β g˜AB = γαβζAζB + vAvA + mAmB. (5.146) Chapter 5. Cheese Slice Braneworlds 107

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,

A B g˜ABn m = cos θ. (5.147)

A A Being orthogonal to Σ3, u and v 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,

A B g˜ABu v = cos φ. (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, 0 <φ< 2π and φ = π θ = π φ with 6 . (5.149) −   π<θ<π and θ = 0  − 6 Since φ is the angle between uA and vA we have,

g˜ uAvB = cos φ = cos θ. (5.150) AB −

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 combination of the basis vectors,

A A A n = C1v + C2m , (5.151)

A a A b where C1 and C2 are constants. Now if we project n onto Σ4 using gb ea eB as the Chapter 5. Cheese Slice Braneworlds 108 projection operator we get,

(gaeAeb )nB = (˜gA mAm )nB (5.152) b a B B − B = nA mA cos θ (5.153) − A = C1v . (5.154)

A A Only the v component remains since m is by deﬁnition orthogonal to Σ4. Contracting

B equations (5.153) and (5.154) withg ˜ABn gives,

1 cos2 θ = C g˜ nBvA (5.155) − 1 AB

B and contracting equation (5.151) withg ˜ABv gives,

A B g˜ABn v = C1. (5.156)

A Therefore C1 = sin θ. Applying the same method with the projection of m 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,

A B A B g˜ABn v = sin θ =g ˜ABm u . (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 W A and ZA such that,

A A W ∂AW =1= Z ∂AZ, (5.160)

3Note the similarity between these deﬁnitions and those of section 5.4.3 Chapter 5. Cheese Slice Braneworlds 109

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,

A A a A A A i A W = Nn + N ea Z = Mm + M fi

A a α a A A i α i A = Nn +(Uu + ν εα)ea = Mm +(Vv + µ ǫα)fi (5.161)

A A α A A A α A = Nn + Uu + ν ζα = Mm + Vv + µ ζα

Here the 4-vectors N a 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

A A vectors of Σ3 associated with the evolution of W and Z 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,

∂xA ZA = = δA, (5.165) ∂Z Z W =0 ∂xA W A = = δA , (5.166) ∂W W Z=0 ∂xA ζA = = δA. (5.167) α ∂ξα α Z=W =0 Chapter 5. Cheese Slice Braneworlds 110

5.5.3 Finding the Metrics

A We begin with M4, which in the coordinates of x , can now be deﬁned as the W = 0 hypersurface. In this case we have, ∂ya ∂ya dya = dξα + dZ (5.168) ∂ξα ∂Z a α a A = εαdξ + eAZ dZ (5.169)

since ∂ya ∂ya ∂xA = = ea ZA. (5.170) ∂Z ∂xA ∂Z A From equation (5.168) we ﬁnd that,

a b a b α β a α b B b β a A a b A B 2 dy dy = εαεβdξ dξ + εαdξ (eBZ )dZ + εβdξ (eAZ )dZ + eAeBZ Z dZ . (5.171)

Now we can use equations (5.140) and (5.171)to ﬁnd the line element.

2 a b ds = gabdy dy (5.172)

′ β′ ′ ′ α a b = (γα β εa εb + uaub)dy dy (5.173)

β′ α β ′ A α = γαβdξ dξ + 2γαβ ζA Z dξ dZ

β′ ′ α A B A B 2 + γαβ ζAζB Z Z + uAuBZ Z dZ (5.174) h i = γ dξαdξβ + 2µ dZdξα + µ µα +(M sin θ V cos θ)2 dZ2 (5.175) αβ α α − Equations (5.142) and (5.161) were used, as well as,

u ZA = M sin θ V cos θ, (5.176) A − which results from equation (5.159). We can then read oﬀ the brane metric,

γαβ µα gab = (5.177)  α 2  µβ µαµ + M˜     where M˜ (M sin θ V cos θ). The inverse is then, ≡ − µαµβ µβ γαβ + M˜ 2 −M˜ 2   gba = , (5.178)  α   µ 1     −M˜ 2 M˜ 2    Chapter 5. Cheese Slice Braneworlds 111

Recall that the indices are such that a, b, . . . = 0, 1, 2, 3 and α,β,... = 0, 1, 2 . γ is { } { } αβ therefore three-dimensional. The metric of Σ4 can be found in a similar manner,

γαβ νβ hij = (5.179)  α 2  νβ ναν + N˜     where N˜ (N sin θ U cos θ). With the inverse, ≡ − νανβ νβ γαβ + N˜ 2 −N˜ 2   hij = . (5.180)  α   ν 1     −N˜ 2 N˜ 2    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,

m n cos θ u = A − A = (0, 0, 0, M csc θ, N cot θ) (5.182) A sin θ −

and n m cos θ v = A − A = (0, 0, 0, M cot θ,N csc θ). (5.183) A sin θ −

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,

M N U and V . (5.186) ≡ sin θ ≡ sin θ

Thus we can eliminate U and V . Chapter 5. Cheese Slice Braneworlds 112

5.5.4 The Bulk Metric

Now we carry out the same procedure for the bulk spacetime. We have,

A A α A A dx = ζα dη + Z dZ + W dW. (5.187)

The metric can be taken in either form,

α β g˜AB = γαβζAζB + uAuB + nAnB (5.188)

(5.189)

α β = γαβζAζB + 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, { } { }

γαβ µα να

 α 2  g˜AB = µβ µαµ + M Ψ , (5.190)    α N 2   νβ Ψ ναν +    where,  

M 2 M 2 + N 2 sin2 θ (5.191) ≡ N 2 N 2 + M 2 sin2 θ (5.192) ≡ Ψ M 2 + N 2 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 = M˜ and N = N.˜ (5.194) |W =0 |Z=0 To ﬁnd the inverse we also make use of a recursive strategy, we ﬁrst write the bulk metric as a 4 + 1 decomposition,

gab Pa g˜AB = , (5.195)  a 2  Pb PaP + B     Chapter 5. Cheese Slice Braneworlds 113

where Pa and B we treat as unknown functions. In this form it is easy to read off the inverse as, P bP a P b gba + − B2 B2   g˜BA = , (5.196)  a   P 1   − 2 2   B B    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 ofg ÃB. We must now decompose Pa and find the contravariant component P a gabP . We identify equation (5.196) with equation (5.190) ≡ a and find that Pa is given by,

να Pa   . (5.197) ≡ Ψ     Explicitly performing the calculation for P b gives:

µαµβ µβ γαβ + M 2 −M 2 ν b   α P =    α  Ψ  µ 1   M 2 M 2     −      Ψ µαν νβ − α µβ M 2 = − (5.198)  Ψ µαν  − α  M 2    Ψ˜ µβ νβ M 2  −  = (5.199)    ˜   Ψ   M 2      P β   , ≡ P 3     Chapter 5. Cheese Slice Braneworlds 114

where we have deﬁned Ψ˜ Ψ µαν . Expanding equation (5.196) gives, ≡ − α µαµβ P βP α µβ P 3P β P β γαβ + + + M 2 B2 −M 2 B2 −B2    µα P 3P α 1 P 3P 3 P 3  g˜BA =   . (5.200)  2 + 2 2 + 2 2   −M B M B −B         α 3   P P 1   2 2 2   − B −B B    To carry on we identify equation (5.190) with equation (5.195) to give,

α 2 a 2 ναν + N = PaP + B (5.201)

α 3 2 = PαP + P3P + B ν µαΨ˜ ΨΨ˜ = ν να α + + B2 α − M 2 M 2 Ψ˜ 2 = ν να + + B2 (5.202) α M 2 and solve for B2 giving, N 2M 2 Ψ˜ 2 B2 = − . (5.203) M 2

Now all that remains is to substitute equations (5.203) and (5.199) into the expanded form ofg ÃB (equation (5.200)). After some simplification we arrive at the final form of the inverse metric,

 N 2µβµα + M 2νβνα 2Ψ˜ µ(βνα) N 2µβ + Ψ˜ νβ M 2νβ + Ψ˜ µβ  γβα + − − −  Φ Φ Φ         N 2µα + Ψ˜ να N 2 Ψ˜  g˜BA =  − −  ,  Φ Φ Φ           M 2να + Ψ˜ µα Ψ˜ M 2     − −   Φ Φ Φ         (5.204) Chapter 5. Cheese Slice Braneworlds 115 where we have deﬁned Φ M 2N 2 (Ψ µ νγ)2 and recall Ψ˜ (Ψ µ νγ). It is ≡ − − γ ≡ − γ straightforward to verify that equation (5.204) is indeed the inverse by checking that

BA g˜ABg˜ = 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 spacetimes 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 = π. Thus the coordinate system used in equation (5.162) 6 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 theng ˜+ − AB andg ÃB− . 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 Σ4− to M4 and M4− to Σ4 . This ensures Chapter 5. Cheese Slice Braneworlds 116

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 Σ4− and M4 requires the ﬁrst fundamental forms to be identical when calculated on either side. Thus

+ hab− = gab (5.205)

+ and the matching along Σ4 and M4− requires that

+ gij− = hij. (5.206)

From equations (5.177) and (5.179) it follows that,

+ + + ˜ + 2 ˜ 2 ˜ + 2 ˜ 2 γαβ = γαβ− µα = να− να = µα− (M ) =(N −) and (N ) =(M −) (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) Chapter 5. Cheese Slice Braneworlds 117

+ 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,

1 √5 1+ √5 cos θ+ = ± or ± . (5.220) 2 2

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) Chapter 5. Cheese Slice Braneworlds 118

where p,q = 1. If p = q there is no new information. In the case that p = q we are { } ± 6 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,

+ 2 2 + 2 2 (M ) (N −) =(N ) (M −) . (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,

A B Kab = nA Bea eb M4 (5.231) || ∈ and

A B Ωij = mA Bfi fj Σ4, (5.232) || ∈ Chapter 5. Cheese Slice Braneworlds 119

where denotes covariant diﬀerentiation with respect tog ˜ The extrinsic curvature of Σ || AB 3 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,

n A B ωαβ = nA Bζα ζβ (5.233) || m A B ωαβ = mA Bζα ζβ , (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,

uω = u εa εb Σ M (5.235) αβ a;b α β ∈ 3 ⊂ 4 vω = v ǫi ǫj Σ Σ (5.236) αβ i;j α β ∈ 3 ⊂ 4 where ; denotes the covariant derivative with respect to the brane metric (either g M ab ∈ 4 or h Σ ). Now we observe that, ij ∈ 4

A B ua;b = uA Bea eb (5.237) || A B vi;j = vA Bfi fj (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,

u A B ωαβ = uA Bζα ζβ (5.239) || v A B ωαβ = vA Bζα ζβ (5.240) ||

Finally, using equations (5.182) and (5.183) we have,

u A B ωαβ = mA B csc θ nA B cot θ ζα ζβ (5.241) || − || v A B ωαβ = nA B csc θ mA B cot θ ζα ζβ (5.242) || − || Chapter 5. Cheese Slice Braneworlds 120

or in terms of Kab and Ωij,

uω = Ω ǫi ǫj csc θ K εa εb cot θ (5.243) αβ ij α β − ab α β vω = K εa εb csc θ Ω ǫi ǫj cot θ, (5.244) αβ ab α β − ij α β

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,

u v ωαβ = ωαβ. (5.245)

Therefore from equations (5.243) and (5.244)

a b i j Kabεαεβ = Ωijǫαǫβ. (5.246)

Taking the trace of the left-hand side of equation (5.246) gives,

a b αβ a α cb (Kabεαεβ)γ = Kabεα(εc g ) (5.247)

ab = Kabg (5.248)

= K (5.249)

The trace of the right hand side is,

i i αβ j α ij (Ωijǫαǫβ)γ = Ωijǫα(ǫi h ) (5.250)

ij = Ωijg (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,

A B A B nA Bζα ζβ = mA Bζα ζβ . (5.254) || || Chapter 5. Cheese Slice Braneworlds 121

The consequences of this condition on the bulk matching can now be examined. Let

the matter content on the brane be denoted T M and S Σ to distinguish between ab ∈ 4 ij ∈ 4 either region of the brane. Using the matching situation depicted in Figure (5.6) we have,

+ 2 1 + K + Ω− = κ (T g T ) (5.255) ab ab 5 ab − 3 ab

and

+ 2 1 + Ω + K− = κ (S h S). (5.256) ij ij 5 ij − 3 ij 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,

1 1 T εa εb γ T = S ǫi ǫj γ S (5.257) ab α β − 3 αβ ij α β − 3 αβ

Then the traces of equations (5.255) and (5.256) along with equation (5.253) give,

2 + κ5 K + Ω− = T 3 = T = S, (5.258) 2  + K κ5 S ⇒ Ω + − = 3  and ﬁnally equation (5.257) takes the form, 

a b i j Tabεαεβ = Sijǫαǫβ (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. Chapter 5. Cheese Slice Braneworlds 122

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 i, where i = 1, 2, 3, 4 . We will use integer super- 5 { } 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)

1 3 4 2 If we choose M4 and M4 (equivalently Σ4 and Σ4) to deﬁne the brane, then the three-dimensional matching requires,

u1 u3 ωαβ = ωαβ. (5.262) Chapter 5. Cheese Slice Braneworlds 123

4 4 (M 1,g1) (Σ4,h ) 4

1 4 u 1 1 M5 4 n M n v4 5

1 4 m m 1 φ4 φ (M 4,g4) (Σ1,h1) 4 u4 v1 4 3 γαβ 2 3 3 v u 2 2 (Σ4,h ) (M4 ,g ) φ3 φ2 m3 n2

2 3 n3 2 m 2 M5 v M5 u3

2 2 3 3 (Σ4,h ) (M4 ,g )

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. Chapter 5. Cheese Slice Braneworlds 124

In terms of the extrinsic curvature of the brane this gives,

Ω1 ǫi ǫj csc θ1 K1 εa εb cot θ1 = K3 εa εb csc θ3 Ω3 ǫi ǫj cot θ3, (5.263) ij α β − ab α β ab α β − ij α β

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 M4− and Σ4 to be the vacuum. The normals are co-linear,

A A n − = m −, (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

A+ A A A decomposition by using the normals n n and m − m . ≡ ≡ The matching conditions, equation (5.205),then require that,

gab = hab. (5.265) Chapter 5. Cheese Slice Braneworlds 125

− − + + − − + + (Σ4 ,h ) (M4 , g ) (Σ4 ,h ) (M4 , g )

v− v− m− m− u+ u+

n+ n+

φ− φ+

m+ u− − n v+

− − + + (M4 , g ) (Σ4 ,h )

+ Figure 5.8: If we assume that M4− and Σ4 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 and Σ− allowing m− and n to span the Z W plane. In essence, this 4 4 − 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. Chapter 5. Cheese Slice Braneworlds 126

It follows that µα = να and N 2 = M 2. The bulk metric must take the form,

γαβ µα µα

 α 2  g˜AB = µβ µαµ + M Ψ . (5.266)    α M 2   µβ Ψ µαµ +      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 Chapter 5. Cheese Slice Braneworlds 127 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 towards 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 surfaces. 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 future 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 representing 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 Chapter 6. Summary and Conclusions 130

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 Chapter 6. Summary and Conclusions 131

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 manner. 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 understanding 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 φ (p) × λ ≡ 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

1 A 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, 2 d 1 d 2 φ∗ T = T + φ∗ T λ + φ∗ T λ + ..., (A.1) λ dλ λ 2 dλ2 λ λ=0 λ=0 where φ T = T was used in the first term. In general φ T is a tensor-valued function of 0∗ λ∗ the parameter λ. By definition of the derivative we have,

d φλ∗ T T L φλ∗ T lim − = ξT, (A.2) dλ ≡ λ 0 λ → which is just the Lie derivative in the direction of the ﬂow. Thus the Taylor series can be more compactly expressed as, n i λ i φ∗ T = L T, (A.3) λ i! ξ i=0 X 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,

(n) (2) (1) ψ = φ n φ 2 φ (A.4) λ λ /n! ◦···◦ λ /2 ◦ λ up to any desired order n. A diﬀeomorphism deﬁned in this way is termed a knight

(i) diffeomorohism or knight. Each flow, φλi/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

(n 1) to φ − . Thus we can express the Taylor series of any diﬀeomorphism by combining equations (A.4) and (A.3), n 1 − l λ l! j1 jn n (n) ψ∗T = L L T + λ R T (A.5) λ j2 jn ξ1 ξn λ l! 2! n! j1! jn! ··· J Xl=0 Xl ··· ···

2Taking the ﬁrst two orders into consideration, we see a parameter displacement in the direction of ξ1 followed by a smaller parameter displacement in a diﬀerent direction ξ2. This is reminiscent of the movement 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)T is a remainder term with a ﬁnite limit as λ 0 and λ → n J (j , , j ) Nn ij = l (A.6) l ≡ 1 ··· n ∈ i ( i=1 ) X 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))=(ψλ∗T )(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)+( T ) λ +( + 2 )T λ2 + O(λ3), (A.8) Lξ1 p Lξ2 Lξ1 p

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