THESIS APPROVED BY

g- lq -2-l 2^ø,-* 7. Ðrdl"- Date GintarasK. Duda, Ph.D. Physics Department Chair

ll^*L^?.¿¿rut #nutnu,.P.Iürubel, Ph.D.

Dave L. Sidebottom, Ph.D.

KevinT. FitzGerald, S.J., Ph.D., Ph.D. Interim Dean, Graduate School More Space for Unification Kaluza-Klein Theory and Extra Dimensions

By Joseph Nohea Kistner

Submitted to the faculty of the Graduate School of Creighton University in Partial Fulfillment of the Requirements for the degree of Master of Science in the Department of Physics

Omaha, NE August 13, 2021

Abstract

The purpose of this this thesis is to investigate Kaluza-Klein theory, the history of extra dimensions, and provide a tool to those who want to learn about how they are being used today. In doing so, a re-derivation of the bulk of the theory was carried out because much of the original nomenclature and conventions are not common anymore, creating a barrier to those wanting to learn about it. The re-derivation includes Theodore Kaluza’s

1921 paper where he lays out the impetus for expanding the foundations of General

Relativity to include an extra dimension of space, and one of Oskar Klein’s 1926 papers that answers the problem of the cylinder condition in Kaluza’s theory. Additional sections are provided to introduce tensor mathematics and Riemann Normal Coordinates, the coordinate system used throughout Kaluza’s derivation. The final chapter describes how extra dimensions are used in theories today and how experiments are being used to search for evidence of them.

iii This is dedicated to my loving parents who have always

encouraged me to pursue my happiness.

iv Acknowledgements

First, I would be remiss if I did not thank my adviser, Dr. Gintaras Du¯da. Throughout this process he never failed to be supportive and accommodating. I would not have completed this thesis had it not been for him pushing me forward through all the challenges I faced.

Not the least of which was a global pandemic that kept me from returning to campus to finish my degree as scheduled.

I would also like to thank Dr. Michael Nichols and Dr. Tom Wong for being so kind and allowing me to submit this thesis well after the initially intended date.

Finally, thank you to the Aspen Center for Physics for providing me with a place to work and access to an extensive library throughout the COVID-19 lockdown.

v Table of contents

Nomenclature viii

1 Introducing Extra Dimensions1

1.1 A history of unification ...... 1

1.2 Historical extra dimensions ...... 5

2 Review of Tensors and GR9

2.1 Introduction to tensors ...... 9

2.2 Maxwell’s equations in tensor form ...... 11

2.3 Introduction to GR ...... 14

3 Kaluza-Klein Theory 18

3.1 Kaluza 1921 ...... 19

3.1.1 The spacetime side ...... 19

3.1.2 The energy-momentum side ...... 27

3.2 Oskar Klein and the cylinder condition ...... 29

4 Contemporary Extra Dimensions 32

4.1 Why use extra dimensions ...... 32

vi 4.2 Phenomenological studies ...... 33

4.3 Solutions to other problems ...... 35

4.4 Experimental detection methods ...... 39

5 Conclusions 42

Appendix A Riemann Normal Coordinates 44

A.1 Definitions ...... 45

A.2 Fundamental principles of RNC ...... 46

A.3 General coordinate transformation to RNC ...... 48

References 52

vii Nomenclature

Tensors

ηµν Minkowski metric

ρ Γµν Christoffel symbol (pseudo-tensor

Λ Cosmological constant

Σµν Associated field

Aµ EM four-potential

F µν EM or Faraday tensor

g µν Spacetime metric

Gµν Einstein tensor

J µ Four-current

R Ricci scalar

λ Rµρν Riemann curvature tensor

Rµν Ricci tensor

viii Tµν Energy-momentum tensor

Acronyms / Abbreviations

CC cosmological constant

CMB Cosmic Microwave Background

E6SSM Exceptional(6) Supersymmetric Standard Model

EEP Einstein’s Equivalence Principle

EM Electromagnetic(-ism)

ESC Einstein summation convention

FCC Future Circular Collider

GR General Relativity

GUT Grand Unified Theory

KK Kaluza-Klein

LKP Lightest Kaluza-Klein particle

MOND Modified Newtonian Dynamics

QFT Quantum Field Theory

SM Standard Model

SRN Super-heavy Right-handed Neutrino

ToE Theory of Everything

ix Chapter 1

Introducing Extra Dimensions

1.1 A history of unification

The first time physicists succeeded in unifying seemingly different phenomena was when

James Clerk Maxwell showed that electricity, magnetism, and light all fit into the same mathematical theory. Today we know the resulting equations as Maxwell’s Equations for classical electromagnetism (EM), and they have been since been rewritten in many different notations. One of the simplest, describes Maxwell’s initial four equations simply as,

∂ F µν µ J ν, (1.1) µ = 0 ∂ F 0, (1.2) [λ µν] = where F µν is the EM field tensor and J ν is the four-current density. Though it is possible, it is hard to get any more simple than that, and the other representations exist in notations and formalisms that are not required to read this paper. As a final note on EM unification

1 Introducing Extra Dimensions 1.1 A history of unification

Maxwell did not come up with all the equations that bear his name. Ørsted and Faraday are known to have produced the earliest consequential literature on the connection between electricity and magnetism. Maxwell’s vital contribution was discovering how the equations worked together to produce EM waves and how the multiple forces were actu- ally derived from a single physical object under different circumstances. Work continues to this day under the same ambition of combining seemingly disparate effects under a singular, unified theory.

At the time of Maxwell in the late 1800s, there was no theory which explained why gravity worked the way it does. Newton’s Law of Universal Gravitation described how gravity affected matter, but lacked insight into the mechanism. In the absence of a more fundamental understanding of gravity, it made little sense to attempt unification with electromagnetism. Almost exactly fifty years passed before there was a working theory of gravity. In 1915 Einstein published his General Theory of Relativity (GR) which described gravity geometrically as the curvature of a four dimensional spacetime. Not long after came the first attempts at theories that unified gravity and electromagnetism.

Using GR to inform his direction, Hermann Weyl endeavored to be the first to connect the two known forces in 1918 [1]. He believed that there was an inconsistency in the backbone of GR, which is Reimann mathematics. The fact that two vectors could be simply compared in a dot product even when they did not originate in the same point concerned him. To the casual observer, this may not seem like an issue at all. What difference does it make if my measuring stick and compass are here or there when all we want is a comparison of magnitude and direction? In most situations, in fact, it does not matter because moving a vector in Euclidean space, which Reimannian geometry was developed from, conserves both quantities. However, this is not generally true in

2 Introducing Extra Dimensions 1.1 A history of unification curved geometries. In fact the path over which the vector is moved to another point can change its direction, and in Weyl’s formulation the magnitude could change as well. This is known as non-integrability. Upon correcting the inconsistency, he believed that his equations showed the coupled dependence of both gravity and electromagnetism on the curvature of spacetime. Such claims, that the mathematical foundations of GR were incorrect and that the solution unified the forces, drew a quick response from Einstein [2].

Though he admired the hypothesis, Einstein was quick to point out that changing the length of a vector based on its history would be akin to the frequency of a clock or the length of a rod depending on where it had been. After a few exchanges and comments from other experts [3], which presented justification for both sides, Weyl’s theory was shown to be nonphysical. Eventually, the nuclear forces were discovered and the realm which unification encompassed grew. After that point, efforts to strictly unify gravity with electromagnetism became less popular, and the new focus was a theory for the weak nuclear force.

Unlike the connections between electricity and magnetism which took millennia to realize after their effects were noticed, it took less than two generations from its first proposed observation to accurately characterize the weak nuclear force. In an attempt to describe the process of beta decay, Enrico Fermi postulated the existence of what would eventually develop into the weak force [4]. A fatal flaw in Fermi’s hypothesis was that he made the new force a contact force, meaning the involved particles must be in physical contact with each other at the time of interaction. For many reasons, this did not work for the theory. Most critically, the interaction was far more common than would be predicted for a four particle interaction, and in the high energy limit the scattering cross-section violated unitarity bounds. This was easily remedied mathematically by structuring the

3 Introducing Extra Dimensions 1.1 A history of unification interaction as a particle mediated force, like EM. Then in 1956, an experiment conducted by Chien-Shiung Wu [5] provided evidence of parity symmetry* breaking in the weak in- teraction, a crucial finding as it had never been observed. Further attempts incorporated mediator particles and parity symmetry breaking [6, 7], but even then the hypotheses were incomplete because they only produced massless exchange particles where it was clear that the weak exchange particle had mass. The last major piece of the weak puzzle fell into place when both Abdus Salam and Steven Weinberg discovered structures within the work of Peter Higgs that allowed for massive and massless exchange particles. To their surprise, together, their contributions showed how the weak and electromagnetic forces are one and the same, and are only seen as separate because experiments are conducted in an energy regime below a critical point, characterized by the Higgs particle, where a symmetry is broken [8, 9]. Eventually, Salam, Weinberg, along with Sheldon Glashow won the Nobel prize in 1979 for their electroweak theory, and Peter Higgs likewise received a Nobel prize in 2013 for predicting the mechanism leading to electroweak symmetry breaking and massive particles.

Electroweak theory was the first step of the contemporary unification model, which generally assumes that as interaction energies increase, otherwise separate forces will be shown to be manifestations of a single force. The next step in this progression is combining electroweak and strong forces in a Grand Unified Theory (GUT). As of now, there are a few theories that may work as a GUT, but there is no standout candidate. Two of the mainstream candidates have been composed in part by Salam and Glashow. Jogeshi

Pati and Salam collaborated on a theory containing a fourth flavor of quark and restoring parity symmetry with a high energy weak interaction [10]. The largest limitation to GUT

*Symmetry when referring to a particle interaction implies a quality that typically remains the same and is associated with a conserved quantity. In the case of broken parity symmetry, the weak force weirdly only interacts with what could be called the ‘left’ part of the involved particles (‘right’ for anti-particles).

4 Introducing Extra Dimensions 1.2 Historical extra dimensions formulations is inability to perform experiments at energies that could constrain current hypotheses. Interaction energies on the order of 1013eV [11] at the LHC at CERN are the upper limit for accelerator experiments, currently, and the most energetic cosmic ray ever measured comes in on the order of 1020eV, which is still a far cry from the predicted GUT scale of 1025eV. Glashow and Howard Georgi created a theory that predicts the interesting phenomenon of proton decay that is potentially measurable [12]. However the halflife is longer than the age of the universe making the measurement very difficult to attain.

Strangely, it seems as though theoretical physics has decided to largely to move past GUTs and focus efforts on a unified theory including gravity with the other three fundamental forces.

Such a theory is known as Theory of Everything (ToE) and is thought to have a unifica- tion energy around 1028eV. The one ToE that dominates the science-sphere is Superstring theory (or String theory). There are others, of course, but most of the prominent theories are all different flavors of a hypothesis of 10 or 11 total dimensions of spacetime [13]. Just like the GUTs, String theory suffers from applying to extremely high energy interactions or extremely early epochs in the lifetime of the universe. There are some aspects of Superst- ing theory, particularly the super aspect, that are closer to the reach of current accelerator experiments. The only quality of these theories that can be investigated without requiring super high energies is the structure of spacetime they predict. Though it comes with its own set of challenges, testing for extra dimensions is possible.

1.2 Historical extra dimensions

The first prominent unified field theory to make use of extra dimensions was proposedby the German mathematician and physicist Theodor Kaluza in 1921 (an English translation

5 Introducing Extra Dimensions 1.2 Historical extra dimensions can be found in [14]). Kaluza based his theory off the results of the earlier attempt by Hermann Weyl to unify gravity and electromagnetism [15]. Weyl was the first to explore gauges† in an attempt to explain electromagnetism as a geometric property of spacetime. He hypothesized that working in the correct gauge would reveal the gravity- em connection. What he ended up with was a gravitational tensor and an additional vector that he treated as the EM four-potential. Unfortunately for Weyl, he was unable to reconcile the two forces present, but it primed Kaluza to investigate unification. Kaluza saw promise in what had been done by working within the ansatz of GR, but contrary to Weyl’s method, he chose to pursue a method that would combine gravitation and electromagnetism into a single unified tensor.

By adding a single extra dimension in the formalism of GR, Kaluza was able to re- cover the behavior of the original gravitational metric potential, as well as an object that behaved like the four-vector potential of electromagnetism. In those aspects, he accomplished his goal. However, there were two other aspects of the theory that needed explaining. The first is an explanation for why there is seemingly no measurable inter- action with the fifth dimension that makes the theory possible. Kaluza’s reasoning was fundamental in allowing him to formulate his theory. He called it the "cylinder condi- tion." Imposing this condition freed the components of the metric from a dependence on the fifth dimension providing consistency to the theory. Yet he had no rigorous wayof showing this was a valid imposition, and proving that it was, is why Oskar Klein is now considered equally responsible for Kaluza-Klein (KK) theory. The second point of issue is a product of the mathematics of adding an extra dimension to the metric, and what

†A simple way to understand a gauge is by thinking about how, in most circumstances, the zero point of a potential can be arbitrarily placed without affecting the physics of the system. However, there are certain chosen points that make the physics more clear to the observer and easier to work with mathematically. This is how a gauge works – it is a mathematical tool that can make clear what is happening physically.

6 Introducing Extra Dimensions 1.2 Historical extra dimensions inevitably rendered the theory invalid. It predicted a new scalar potential that did not fit within the bounds of experiment. Both of these points will be discussed in detail further on.

After the failure of KK theory to produce a working model for the universe, extra dimensions were not widely used until the 1960s when an early version of String theory was developed in an attempt to describe the strong interaction of hadrons. It was quickly superseded by Quantum Chromodynamics (QCD) as a model of the strong force, but development continued, and expanded. In 1971 Claud Lovelace showed that (bosonic)

String theory required at exactly 26 dimensions to avoid conformal anomalies [16]. Later, when supersymmetry was included in String theory formulations, it was shown that 10 or 11 dimensions of spacetime were all that were necessary [13]. With that said, the

5-dimensional foundations of KK theory are still actively used in phenomenological studies.

Kaluza’s framework excels as a simple testing ground for new ideas that use extra dimensions to find solutions to fundamental questions. The complexity of String theory makes testing new hypotheses overly complicated and cumbersome when the manip- ulation of a single dimension is all that is necessary for a phenomenological proof of concept. For example, Randall and Sundrum [17] found a possible solution to the hierar- chy problem – the strength disparity between gravity and the other fundamental forces – by warping the fifth dimension and requiring gravity to communicate through it, causing the strength of gravity to dissipate more quickly than other forces. More on the Randall

Sundrum paper and others will be discussed in Chapter 4.

Since KK theory is still used as an important tool, many students studying String theory and extra dimensions will have to engage with it at some point. However, the original

7 Introducing Extra Dimensions 1.2 Historical extra dimensions papers show their age and can be difficult to track down and decipher. In an attempt to make the information more accessible, the majority of this work is a re-derivation of

Kaluza’s paper and one of Klein’s two related papers and it begins with a review of tensors and GR. As part of re-deriving Kaluza’s work it was necessary to include an appendix section on Riemann Normal Coordinates, a particular coordinate system that he uses to simplify his equations without loss of generality. Lastly, it is always good to understand how a theory is being tested. In this case, how extra dimensions are being searched for.

Included at the end of Chapter 4 is a selection of modern methods that are being used to look for extra dimensions and constrain their number and sizes.

8 Chapter 2

Review of Tensors and GR

Kaluza-Klein theory was developed off the back of GR and a tensor formulation of EM. To ensure that the basic understanding of both is met before delving into the re-derivation of

KK theory this section will introduce both topics. First we will review Maxwell’s equations and their tensor form to introduce some notation in a familiar context. Then we will overview the basics of GR.*

2.1 Introduction to tensors

All of tensor calculus takes place on a manifold. In GR, a manifold is the object that contains the points of our spacetime and information on how they relate to each other.

In tensor notation one of the simplest things we can create is a vector as we are used to working with, such as a velocity or acceleration vector. To do so it takes two single-index tensors, one with an upper index, and one with a lower index. The location of the index

*In this chapter SI units are used but the remainder of this thesis uses geomertized units.

9 Review of Tensors and GR 2.1 Introduction to tensors determines how it will transform under a change of basis. For the example here, vµ are the vector components and eµ are the basis of the vector u.

X µ µ 0 1 2 3 u v eµ v eµ v e0 v e1 v e2 v e3. (2.1) = µ = = + + +

The Einstein summation convention (ESC) has been used here to eliminate the need for an explicit summation. The ESC states that any index that is repeated as an upper and lower index will be summed over. This convention greatly simplifies the notation as the tensors gain more indicies and equations become more complex.

The next task is the simple operation of raising or lowering an index. This operation requires a special two-index tensor called a metric tensor. It’s purpose is to contain the topographic information of the manifold, and that enables it to change how a tensor transforms. The notation for raising or lowering an index is:

eµ M µνe , (2.2) = ν v M vν, (2.3) µ = µν

µν where M and Mµν are versions of the same metric, related by the identity

M M σν δσ. (2.4) µν = µ

σ The Kroneker delta, δµ, is the tensor equivalent to the identity matrix. It acts on a tensor to make the two labeled indicies the same. When this operation is carried out on a multi-index tensor in order to sum over two indices it is called a contraction.

10 Review of Tensors and GR 2.2 Maxwell’s equations in tensor form

The final snippet of general tensor notation that needs to be addressed is the partial derivative with respect to the basis components. A partial derivative can be written in a few ways,

∂A µ 1 ∂A ∂A ¶ ∂ρ A A,ρ , . (2.5) ∂xρ = = = −c ∂t ∂xi

Throughout this thesis only the second version is used, because it is compact but the intended operation is still clear. Now, with the basics taken care of, we can get into

Maxwell’s equations.

2.2 Maxwell’s equations in tensor form

The familiar form of Maxwell’s equations that most undergraduate physics students recognize is:

ρ ∂B E , (2.6) E , (2.8) ∇ · = ϵ0 ∇ × = − ∂t µ ∂E¶ B 0, (2.7) B µ0 J ϵ0 . (2.9) ∇ · = ∇ × = + ∂t

These four differential equations explain the coupled behavior of electric fields, E, mag- netic fields, B, charge densities, ρ, and current densities, J. As this is the SI unit formula- tion there is also the permittivity and permeability of free space, ϵ0 and µ0, respectively.

This way of writing Maxwell’s equations is clear and easy to work with if you intend to use one or two equations. There is really no problem with it except that it needs to be in a form that is compatible with GR, which was constructed using tensor calculus.

11 Review of Tensors and GR 2.2 Maxwell’s equations in tensor form

All four of Maxwell’s equations can be represented with the two tensor equations,

∂ F µν µ J ν, (2.10) µ = 0 ∂ F 0, (2.11) [λ µν] = where F µν is the EM tensor and J ν is the four-current. Brackets indicate that it is antisym- metric and is evaluated using the Levi-Civita symbol, ϵγλµν. The details of which can be found in [18]. The EM tensor is defined by,

F µν ∂ν Aµ ∂µ Aν, (2.12) = − where Aµ is the four-potential. The four-current and four-potential are defined by,

J ν ( cρ, J 1, J 2, J 3), (2.13) = − 1 Aµ ( φ, A1, A2, A3), (2.14) = −c where ρ is charge density, J i are current densities, φ is the electric potential, and Ai are the magnetic vector potential components. To evaluate the EM tensor elements the following relationships are necessary,

µ ∂A¶ E φ , (2.15) B A. (2.16) = − ∇ + ∂t = ∇ ×

12 Review of Tensors and GR 2.2 Maxwell’s equations in tensor form

Constructing the EM tensor in matrix form results in:

³ ´  1 ∂t φ 1 ∂t φ ¡ 1 ∂x φ 1 ∂t Ax ¢ ¡ 1 ∂y φ 1 ∂t Ay ¢ ¡ 1 ∂z φ 1 ∂t Az ¢  − c2 + c2 − c + c − c + c − c + c     ¡ 1 t x 1 x ¢ x x x x x y y x x z z x   c ∂ A c ∂ φ (∂ A ∂ A )(∂ A ∂ A )(∂ A ∂ A )  F µν  + − − − , =  ¡ 1 1 ¢   ∂t Ay ∂y φ (∂y Ax ∂x Ay )(∂y Ay ∂y Ay )(∂y Az ∂z Ay )   c + c − − −    ¡ 1 ∂t Az 1 ∂z φ¢ (∂z Ax ∂x Az )(∂z Ay ∂y Az )(∂z Az ∂z Az ) c + c − − −  E  0 Ex y Ez  c c c     Ex   c 0 Bz By  − − . =  Ey   B 0 B  − c − z x    Ez B B 0 − c y − x

This clearly displays how the electric and magnetic field components make up the EM tensor. The next step of finding Maxwell’s equations from (2.10) is shown below using

(2.6) as an example,

∂ F µν µ J ν, µ = 0 ∂ F µ0 ∂ F 00 ∂ F 10 ∂ F 20 ∂ F 30 µ J 0, µ = 0 + 1 + 2 + 3 = 0 Ex Ey Ez ∂t (0) ∂x ∂y ∂z µ0cρ, = − c − c − c = − ∂ E ∂ E ∂ E µ c2ρ, = x x + y y + z z = 0 ρ E . = ∇ · = ϵ0

It took a few steps but it should now be clear how,

µ0 ρ ∂µF E . (2.17) = ∇ · = ϵ0

13 Review of Tensors and GR 2.3 Introduction to GR

Equation (2.9) follows from the same process. The Maxwell’s equations that remain, (2.7) and (2.8), are derived from equation (2.11) which requires a slightly different treatment.

Given that we are only concerned with the inhomogeneous solutions, the process will not be summarized here.

We have now seen how the tensor form of Maxwell’s equations works. From this section we can see that the power of tensors is in how they can simplify a large number of equations into a greatly condensed form. If the objective was to get a numerical answer by hand, however, tensors would not be my method of choice.

2.3 Introduction to GR

General Relativity is the theory of gravity, how it affects mass and energy, and how mass and energy affect it. In GR the effect of gravity is propagated by spacetime curvature.

The more energy there is in a given location, the more drastic the curvature of spacetime there. The more drastic the curvature in a location, the stronger the gravitational pull will be toward that location. Fundamentally, all this information is contained in two tensors with the most important being the spacetime metric, gµν. The metric, as it is generally called in GR, can be thought of as a tensor potential for gravity. It contains all the information about spacetime curvature needed to do calculations in GR. The only other piece needed is the energy-momentum tensor, Tµν. As you may have guessed, it contains the information about energy in all its forms. Together with the derivatives of

gµν, Einstein developed the central equation of GR,

1 8πG Gµν Λgµν Rµν Rgµν Λgµν Tµν. (2.18) + = − 2 + = c4

14 Review of Tensors and GR 2.3 Introduction to GR

The form on the left uses the Einstein tensor,

1 Gµν Rµν Rgµν, (2.19) = − 2

where Rµν is the Ricci tensor and R is the Ricci scalar, and is used mostly as a shorthand.

The cosmological constant term, Λgµν, was added after the initial formulation before being removed again. Today, it is still used sometimes because its effect is analogous to that of dark energy. For the purposes of KK theory, the equation being used is

1 8πG Rµν Rgµν Tµν. (2.20) − 2 = c4

The left hand side of this equation, the spacetime side, is entirely composed of the metric and its derivatives. Hidden within the Ricci tensor and Ricci scalar are the most important objects in GR, when it comes to KK theory. They are Christoffel symbols,

ρ 1 ργ ¡ ¢ Γ g ∂νgγµ ∂µgγν ∂γgµν , (2.21) µν = 2 + − and they contain first derivatives of the metric. Physically, they contain the information about forces, but more importantly they explain how basis vectors will change over a curved manifold. Compared to the EM tensor from the last section, they contain the same type of information. This connection is what Kaluza found worthy of closer inspection and led him to create his unification theory.

15 Review of Tensors and GR 2.3 Introduction to GR

The Ricci tensor generally† contains first and second derivatives of the metric, and is actually derived from the Riemann curvature tensor,

Rλ ∂ Γλ ∂ Γλ Γλ Γσ Γλ Γσ . (2.22) µρν = ρ µν − ν µν + ρσ µν − νσ µρ

The Riemann curvature tensor, with its 265 elements, describes the curvature of space- time. We know this because it is used to determine how a vector’s direction changes as it is moved around in a curved spacetime. To get the Ricci tensor from the Riemann tensor, the Riemann tensor is contracted with the first and third indicies,

Rρ R . (2.23) µρν = µν

The Ricci tensor directly relates to the rate of change of the separation between parallel lines. This may be confusing at first. By definition parallel lines would never get further apart or closer together, right? No. That is only the case in Euclidean geometry. In any curved spacetime it is possible for two particles that start out with the same initial velocity, perpendicular to the line connecting the particles, to converge or diverge. The rate of change of their relative movement is described by the Ricci tensor. Just as the Ricci tensor resulted from a contraction of the Riemann tensor, the Ricci scalar comes from a contraction of the Ricci tensor,

R Rµ g µνR . (2.24) = µ = µν

Note that the metric was used to raise an index of the Ricci tensor to allow it to be contracted. The Ricci scalar reflects the average curvature of the spacetime. Its sign

†In certain coordinates, that will be discussed later, the Riemann tensor and Ricci tensor only contain second derivatives of the metric.

16 Review of Tensors and GR 2.3 Introduction to GR indicates if the spacetime is spherical (positive) or hyperbolic (negative). The last object that needs to be addressed in this review of GR is hidden within the Riemann and Ricci tensors as will be seen in the next chapter.

For a more complete review of tensor calculus for GR see [19] and for GR as a whole see [20].

17 Chapter 3

Kaluza-Klein Theory

In this chapter I will revisit and re-derive the results from the KK theory. Why? On reading the defining works of KK theory I noticed, especially in Kaluza’s paper, that there isa lack of references or explanations for identities and many steps in the derivation are “left to the reader,” so to speak. These identities and gaps may have been inconsequential at the time when GR had just been formulated and before many tools we now use to study it had been developed. Perhaps they were so common that anyone who would be interested in the papers would necessarily be acquainted with the identities already.

After all, these papers are essentially polished versions of letters between the author and another physicist or two. Kaluza, for example, had been corresponding with Einstein for a few years prior to publishing his paper [21]. Another possibility is that they just did not think it was a useful use of text. Whatever the reason, it makes understanding the content rather difficult 100 years later. As this is the case, I feel that expanding the derivations with more commentary will be useful to anyone who wishes to investigate the origins of extra-dimensional unification theories in the future. This also provides a place to introduce the most modern notations to further ease appreciation for KK theory.

18 Kaluza-Klein Theory 3.1 Kaluza 1921

3.1 Kaluza 1921

3.1.1 The spacetime side

In 1921, six short years after Einstein published the field equations for GR, Kaluza had developed a possible method for expanding it to contain electromagnetism, unifying the two known forces. Kaluza was not the first to publish on this matter post GR. He drew inspiration for his theory from Hermann Weyl’s attempt to show that electromagnetism, like gravity, could be expressed as a geometrical component of spacetime [1]. Instead of trying to coax the mathematics into delivering a result through the use of gauges, Kaluza decided to introduce extra dimensions to derive his unification scheme. He began simply with two insights. First, the components of the EM field tensor

F ∂ A ∂ A , (3.1) µν = ν µ − µ ν

where Aµ is the EM four-potential, had the form of truncated Christoffel symbols of the first kind* 1 ¡ ¢ Γ ∂ρ g ∂ gαρ ∂αg , (3.2) αβρ = 2 αβ + β − βγ

gµν being the spacetime metric, the tensor potential of GR. To put this into easy-to-grasp concepts that makes this connection more intriguing, we must look at combinations of the first derivatives of these objects.

As we saw in the previous chapter the EM tensor can be used to express Maxwell’s equations as,†

∂ F µν J ν. (3.3) µ = *Γγ g γαΓ and Γγ Γγ βρ = αβρ βρ = ρβ †F g g F µν αβ = αµ βν

19 Kaluza-Klein Theory 3.1 Kaluza 1921

So we have the relationship between the electromagnetic fields and sources from first derivatives of the EM tensor. With that in mind let us look at Einstein’s equation from

GR. It relates gravitational fields (i.e. spacetime curvature) to its sources. In the common form with no cosmological constant (Λ 0) Einstein’s equation is =

1 Rµν Rgµν 8πTµν. (3.4) − 2 =

The expansion of the Ricci tensor and scalar, which are fundamentally composed of second derivatives of the metric, are done using Reimann Normal Coordinates (RNC).

They are used throughout Kaluza’s derivation and are addressed later in this chapter and in Appendix A. Through a few steps, equation (3.4) can be rewritten it in terms of derivatives of Christoffel symbols,

1 ¡ ρ ρ ¢ 1 © µν ¡ ρ ρ ¢ª Rµν Rgµν ∂ρΓ µν ∂νΓ µρ g ∂ρΓ µν ∂νΓ µρ gµν, − 2 = − − 2 − µ ¶ ¡ ρ ρ ¢ 1 µ ∂ρΓ µν ∂νΓ µρ 1 δ , = − − 2 ν and we see that the gravitational equations of GR are simply first derivatives of Christoffel symbols, µ ¶ ¡ ρ ρ ¢ 1 µ ∂ρΓ µν ∂νΓ µρ 1 δ 8πTµν. (3.5) − − 2 ν =

From equations (3.3) and (3.5) we see that certain combinations of first derivatives of the

EM tensor and the Christoffel symbols result in the same type of equation – relating fields and sources. Taking this one step further by substituting equations (3.1) and (3.2) into equations (3.3) and (3.5), respectively, the correspondence between the two equations

20 Kaluza-Klein Theory 3.1 Kaluza 1921 becomes even clearer,

n o ∂ g αµg βν ¡∂ A ∂ A ¢ J β, (3.6) α ν µ − µ ν = µ ¶ © ρα ¡ ¢ª 1 µ ∂ρ g ∂µgαν ∂αgµν 1 δ 8πTµν. (3.7) − − 2 ν =

The terms match almost one-to-one between the two, when written in terms of the fundamental potentials of the two fields. They are so similar that it made sense to try and incorporate the EM tensor into the metric, and this observation spurred the use of an extra dimension. Because all of the Christoffel symbols in 4D were already being used for gravitational contributions, only by adding an extra dimension could he develop new

Christoffel symbols with EM terms. With an extra dimension came nine new elements in the metric. Only five of these are independent since the metric is symmetric‡, but five was enough to work in the EM four-potential.

The second condition addresses our interaction, or lack thereof, with the additional dimension. Absence of any measured interaction with an extra dimension needed to be maintained and an explanation given. Kaluza’s assumption was simply that there must be very little, if any, dependence on the extra dimension. And he did not offer much of an explanation for why. That portion came later and was the focus of Klein’s contribution. In the limiting case this mathematically manifests as§

∂ g 0. (3.8) 4 r s = ‡A tensor is symmetric about two indicies when they can be reversed without affecting the tensor (e.g. g g ). µν = νµ §Latin indicies run from 0 to 4, with the extra dimension corresonding to 4. Greek indicies are the conventional 0 through 3 indicies of 4D GR.

21 Kaluza-Klein Theory 3.1 Kaluza 1921

In other words, components of the new metric have an explicit independence on the fifth dimension. He termed this the cylinder condition. In spite of the geometric name, at the time, the condition had nothing to do with the topology of the space. It may have been a product of visualization, but there is no mention of how this physically worked until

Klein expanded upon the condition, completing the KK theory.

Making this decision drew a fair bit of criticism [21] because it seemed to contradict the foundations of GR by altering ds2, the spacetime interval, changing it from an observ- able in 4D to an abstract quantity in 5D. Additionally, distinguishing one dimension from the rest through the cylinder condition is not consistent, as GR assumes dimensions are indistinguishable. Nevertheless the cylinder condition was effective. Calculating the 5D

Christoffel symbols using the cylinder condition provided a promising set of equations:

1 ¡ ¢ 1 ¡ ¢ Γγµν ∂νgγµ ∂µgγν ∂γgµν , (3.9) Γ4µν ∂νg4µ ∂µg4ν , (3.11) = 2 + − = 2 + 1 ¡ ¢ 1 Γγ4ν ∂νgγ4 ∂γg4ν , (3.10) Γ44ν Γ4ν4 ∂νg44. (3.12) = 2 − = − = 2

The most important result here is that the gravitational Christoffel symbols (3.9) were unaffected (compare (3.2)). If this had not been true, it probably would have meant an early end to the theory as we know it. The next thing to notice is that equation (3.10) is the form Kaluza was looking for that matched the EM tensor elements (3.1). It was straight forward to then equate (3.1) and (3.10), which produced the relations:

g g 2αA , (3.13) 4γ = γ4 = γ

Γ α¡∂ A ∂ A ¢ αF , (3.14) γ4ν = ν γ − γ ν = γν where α is a coupling constant that was determined later.

22 Kaluza-Klein Theory 3.1 Kaluza 1921

With the g4γ elements set, it seemed appropriate to also equate equation (3.11) to a related field tensor,

Γ α¡∂ A ∂ A ¢ αΣ . (3.15) 4µν = ν µ + µ ν = µν

Dubbed the associated¶ field, it is introduced but not addressed in any detail. Though it seems this could be reason for concern, it appears to simply be a useful shorthand notation as it does not show up in any of the subsequent equations, which is most likely

Kaluza’s reason for neglecting it.

The last undefined term is the terminal diagonal element of the metric, g44. We will define it ||as ,

g 2φ, (3.16) 44 =

Where φ is a scalar field. This makes the Christoffel symbols

Γ Γ ∂ φ. (3.17) 44ν = − 4ν4 = ν

The meaning of the g44 element was undetermined at the time, and a relationship to a physical quantity was never found.

At this point Kaluza had defined all the new terms in his 5D metric. It turned out that the new metric was the gravitational metric potential bordered by the EM four potential,

¶Sometimes referred to as the subsidiary field, depending on the translation of Nebenfeld [21]. ||Kaluza used g 2g. 44 =

23 Kaluza-Klein Theory 3.1 Kaluza 1921 with the unknown φ in the corner. In a short hand, we can write his result as

  g g g g 2αA  00 01 02 03 0      g10 g11 g12 g13 2αA1       gµν 2αAν gr s  g g g g 2αA   . (3.18) =  20 21 22 23 2 =     2αA 2φ   µ  g30 g31 g32 g33 2αA3     2αA0 2αA1 2αA2 2αA3 2φ

With his metric fully defined, Kaluza continued on to calculate the Riemann tensor.

To do this he used RNC, a special, yet general coordinate system that can be applied at any point on a smooth manifold. Essentially, when using RNC, we are justified in saying that locally around that point the space is flat (Euclidean), but the overall curvature is maintained. For a more complete explanation of RNC see [22].

Importantly, using RNC results in the derivatives of the metric equaling zero, which makes the Christoffel symbols equally zero (but the second derivatives remain). There- fore,

∂ g 0, (3.19) a r s = j Γ 0, (3.20) kl = but,

∂ ∂ g 0. (3.21) a b r s ̸=

Conceptually, the Christoffel symbols describe how the basis vectors change in space. In

Euclidean coordinates they remain constant everywhere. As such, it makes sense that

24 Kaluza-Klein Theory 3.1 Kaluza 1921 the Christoffel symbols would be zero near the chosen point. However, since the space is intrinsically curved, the second (and higher order) derivatives will not be zero. This can be expressed as a Taylor series expansion of the metric at the origin,

lˆ mˆ ∂lˆ∂mˆ grˆsˆx x ³ lˆ mˆ nˆ ´ grˆsˆ ηrˆsˆ O x x x . = + 2 +

The hat notation is a common notation for RNC used in many textbooks, but it is unnec- essary here as all expressions in this derivation are conducted with RNC.

Taking the next step in the derivation, Kaluza calculated the Riemann tensor elements which have a simplified form due to the use of RNC, since all Christoffel symbols equal zero:

(arbitrary coordinates) Rλ ∂ Γλ ∂ Γλ Γλ Γσ Γλ Γσ , (3.22) µνρ = ν µρ − ρ µν + νσ µρ − ρσ µν (RNC) Rλ ∂ Γλ ∂ Γλ . (3.23) µνρ = ν µρ − ρ µν

Again we can see in (3.23) the 4D elements remain unaffected by the extra dimension since they do not depend on it in RNC because the summation terms from the arbitrary coordinate representation go away. At this point in the derivation, Kaluza decides to make a clearly noted approximation that simplifies the mathematics immensely, but is abit strange considering he just determined the new elements in his unified metric. He takes all the curvature contributions of his metric and relegates it to a perturbative term such that g detg 1 and Γl Γ . The justification is that the curvature contribution = r s = − r s = − lr s is insignificant and to first order the metric is pseudo-Euclidean [21]. This approximation in mind, the elements with indices of the extra dimension are summarized by,

25 Kaluza-Klein Theory 3.1 Kaluza 1921

Rλ α∂ F λ, (3.24) R4 ∂ ∂ φ, (3.26) µν4 = ν µ µν4 = µ ν

Rλ α∂ F , (3.25) Rλ R4 R4 0. (3.27) 4νρ = − γ νρ µ44 = µ44 = 444 =

These are promising results. Already we see that a contribution from the associated field has gone away and we are left with only the EM tensor elements and φ dependence. The new dimension has added the electromagnetic field equations, naturally, with the only drawback being the prediction of an undetected field, φ. Contracting the Riemann tensor to determine the Ricci tensor, we get:

ρ ρ ρ R R ∂ Γ ∂ Γ . (3.28) µρν = µν = ρ µν − ν µρ

The second term on the on the right hand side vanishes as can bee seen below,

ρ 1 ¡ ¢ ∂νΓ ∂ν ∂ρ gρµ ∂µgρρ ∂ρ gµρ µρ = 2 + − 1 ¡ ¢ ∂ν∂ρ gρµ ∂ν∂µgρρ ∂ν∂ρ gµρ (due to symmetry gρµ = gµρ) = 2 + − 1 ¡ ¢ 1 ¡ ¢ 1 ¡ ¢ ∂ν∂µgρρ ∂ν∂µTr (g) ∂ν∂µ(const) = 2 = 2 = 2 0. =

This leaves,

R ∂ Γρ , (3.29) µν = ρ µν R α∂ F , (3.30) 4ν = − µ µν R φ, (3.31) 44 = −□

26 Kaluza-Klein Theory 3.1 Kaluza 1921 where g µν∂ ∂ is the D’Alembertian. It seemed he had derived a result consistent □ = µ ν with what he had set out to find. The curvature tensor side of Einstein’s equation in5D consisted of the known equations of gravitation (3.29), a version of Maxwell’s equations

(3.30) (compare (3.3)), and an additional element that has the form of Poisson’s equation with a scalar field φ (3.31).

3.1.2 The energy-momentum side

With the curvature side of Einstein’s equation (3.4) determined, Kaluza could now deter- mine the form of the energy-momentum tensor on the right hand side of the equation.

The energy-momentum tensor is defined as,

T r s µ ur us, (3.32) = 0

where µ0 is the rest mass density, and the general five-velocity is

r r dx 2 l m u , ds glmdx dx , = ds = defined using the 5D interval ds. Immediately, an approximation is made in line with the one allowing raising and lowering of the first index of a Christoffel symbol:

T T r s µ ur us, (3.33) r s = = 0

The new elements of the energy-momentum tensor are

R χT , (3.34) 4µ = − 4µ

27 Kaluza-Klein Theory 3.1 Kaluza 1921 where χ is a coupling constant. At this point, Maxwell’s equations and (3.29,3.30,3.31) are used to determine the four-current:

µ µ χ χ 4 µ J ρ0v T4µ µ0u u , (3.35) = = α = α

ρ ρ dx 2 µ ν ρ0 rest charge density, v , dσ gµνdx dx . = = dσ =

Kaluza’s energy-momentum tensor is simply the spacetime energy-momentum tensor with a current density border. His 5D unification scheme was finished except for the con- tribution from Klein, justifying the cylinder condition. After this point, he continues on to make more approximations that he then concedes make his calculations meaningless as they would not hold for electrons. Those calculations also made it clear that the scalar field could not be set to zero as it needed a particular value for the equations toremain consistent.

He left us with a nice looking theory which does not hold for reality though he believed it was possible with some alterations. As we know, it was doomed to fail since the two nuclear forces had yet to be discovered and his theory did not account for them. However its value was not in how well it described fundamental structure but in how it began the exploration into extra dimensions. He proved that extra dimensions expand upon the mathematics of 4D spacetime to produce results that could include more than just the mechanics of gravity. His work still informs papers today and aids in testing new possibilities present in a spacetime with more than four dimension. Without a doubt,

Kaluza has had a lasting impact on fundamental physics.

28 Kaluza-Klein Theory 3.2 Oskar Klein and the cylinder condition

3.2 Oskar Klein and the cylinder condition

While working through Kaluza’s paper in the last section the cylinder condition was glossed over. Because he barely had any idea about how the extra dimension would be made insignificant or uncoupled from the rest of the dimensions, he said little about it other than that he needed it to be that way. Without any experimental evidence for an extra dimension he had to make sure his theory remained consistent. So he took on equation (3.8) as mathematical necessity and left the reasoning to others. Oskar Klein proved to be up to the task.

At the time, Klein was involved in formulating the equations of quantum mechanics that had recently been defined by Shrödinger’s and Bohr’s work. Specifically, he was concerned with the wave equation of electrons, and he found insight in Kaluza’s paper that allowed him to develop an early version of the Klein-Gordon equation [23]. The

Klein-Gordon equation, 1 ∂2 m2c2 2ψ ψ ψ, (3.36) ∇ − c2 ∂t 2 = ℏ2 is a generalized wave equation describing, not electrons, but relativistic scalar particles because it fails to account for spin (a fact that was not made clear for a couple years.) In the paper he reestablishes Kaluza’s five-dimensional spacetime using Hamilton’s method to derive the early, five-dimensional version of the Klein-Gordon equation and a solution which reduced to Shrödinger’s equation under certain assumptions. In doing so he also showed that one could derive a need for a fundamental constant akin to h through a connection to periodic motion in the extra dimension.

With the electron and the quanta of charge still the focus of his efforts, he published a second paper the same year [24]. In it he provided a theory to explain the fundamental

29 Kaluza-Klein Theory 3.2 Oskar Klein and the cylinder condition nature of the electron’s charge as a quantum law. He achieved this by showing that what is typically thought of as a point particle could actually be periodic motion in a small, closed dimension. The tried and true way to conceptualize this is with an ant or another small bug walking on a string. If the ant were to walk down the length of the string a person could see the ant’s movement without much difficulty. If then, the ant decided to stop moving along the length and instead began walking around the circumference of the string, it could be incredibly challenging for a person to determine that the ant had actually not stopped moving. Thus, the ant moving around the string would appear to be standing still just as an apparently stationary point particle could be in motion in a small, closed dimension leading to new and different physical properties.

After making the connection between electronic charge and periodic motion in the fifth dimension, Klein discovered that he could calculate the size of the dimension. Inthe first paper he made the assumption that a proton would have the momentum:

e p0 , (3.37) = cβ where e is the electronic charge, and β is an undetermined constant. Substituting β p2κ = into (3.37), where κ is Einstein’s gravitational constant, he created a generalized classical description of a particle’s momentum in the fifth dimension,

Ne p4 , (3.38) = cp2κ

30 Kaluza-Klein Theory 3.2 Oskar Klein and the cylinder condition where N is an integer quantum number indicating sign and multiple of fundamental charge. Then, considering the quantum mechanical de Broglie momentum relationship,

h p , (3.39) = λ he determined that the momentum could also be described by,

h p4 N , (3.40) = l where N is now a quantum number, positive or negative, indicating the motion in the fifth dimension, and l is the characteristic length and circumference of the extra dimen- sion. Then, by equating the two and solving for l he found the extra dimension to be immeasurably small:

hcp2κ 30 l 0.8 10− cm, (3.41) = e = × providing a reasonable argument for the apparent absence of the fifth dimension [24].

The contributions to Kaluza’s foundation from Oskar Klein in 1926 were very impor- tant to the longevity of the theory, even if just in phenomenological context. His first paper connected Kaluza’s theory to quantum mechanics and explicitly showed that extra dimensions were compatible with it. The second provided a promising mechanism to explain Kaluza’s cylinder condition and gave the field an early bound for the possible size of extra dimensions. In the end these earned him the position as second title author of

KK theory.

31 Chapter 4

Contemporary Extra Dimensions

The lasting impact of KK theory on contemporary theories is mostly conceptual in na- ture. As extra-dimensional theories have developed over the past century the prevailing mathematics has evolved away from that of GR. This is especially true for investigations into how extra dimensions affect particles. Even though the mathematical structure of

KK theory is not common in these applications of extra dimensions, KK particles are often referenced. A KK particle is a general term for a particle that exists in some way in the extra dimension. After all, the largest contribution KK made was by creating the extra-dimensional theory that sparked the innovation that led us to where we are today.

4.1 Why use extra dimensions

At this point it is pertinent to answer why using extra dimensions has gained its popularity.

The answer is simple – it works. Many problems in fundamental physics have more than one possible solution right now. Technologies and experiments are always in develop- ment that strive to one day reach a point where a single solution can be confirmed, but

32 Contemporary Extra Dimensions 4.2 Phenomenological studies they are not there yet. Theories which manipulate the power of extra dimensions are alone when it comes to the scope of problems to which they contribute solutions. Many solutions come in the form of purely phenomenological inquiries that use the ansatz of String theory, and often times they only address one problem at a time. But as they all draw from the same mathematical foundations there is reason to believe that there exists a general solution to all problems that can be constructed within String theory or an other multi-dimensional theory.

4.2 Phenomenological studies

Although we are inevitably operating in a spacetime of 11 dimensions (if Superstring the- ory is correct [25]), it is often valuable to explore phenomenology of lower-dimensional constructs. In this section I will highlight two contemporary theories that provide so- lutions to the Hierarchy problem in Quantum Field Theory (QFT) that employ extra dimensions. The Hierarchy problem is the problem with gravity being extraordinarily weak compared to the other fundamental forces. The first theory, developed by Nima

Arkani-Hamed, Savas Dimopoulos, and Gia Dvali (ADD), incorporates n 2 extra dimen- ≥ sions in a singular 4 n dimensional spacetime [26].The second theory, developed by + Lisa Randall and Raman Sundrum (RS), makes use of a single extra dimension in a Brane

World theory [17].

Arkani-Hamed-Dimopoulos-Dvali (ADD)

Published in 1998 [26], the ADD approach gets rid of the Hierarchy problem by adding n 2 compact extra dimensions with the stipulation that only gravity has a dependence ≥

33 Contemporary Extra Dimensions 4.2 Phenomenological studies on the new dimensions. Increasing the number of dimensions that gravity propagates in, means that the strength of gravity will diminish more quickly than we expect at the scale

1 of the new dimensions. Instead of an r − dependence as expected for a gravitational

(1 n) potential, the potential becomes r − + for distances less than the compactification scale. This changes the conditions by which the gravitational constant is determined. The theory permits it to scale for the number of extra dimensions and sets constraints based

1 on the regime where a r − dependence is measured and Newtonian gravity is found to be valid. The fact that this transition must occur enables the calculation of how large the extra dimensions must be, which depends on the dimensionality. At the time of publishing, they could only rule out the case where n 1 as the extra dimension would be larger than = the solar system. The n 2 case which predicted a characteristic size 1mm, has more = ≈ recently been proven inconsistent by sub-millimeter measurements of gravity [27, 28].

Further experiments will have to measure gravity at nanometer separations to restrict theories with n 3. ≥

Randall Sundrum (RS1)

About a year after ADD published their paper, Lisa Randall and Raman Sundrum proposed a completely different method for resolving the strength disparity between gravity and the

Standard Model (SM) forces [17]. Their model lives in a 5D spacetime using concepts first understood through String theory, particularly the concept of branes. A brane in String theory is an object with dimensionality higher than one to which the one dimensional strings can attach. In the RS1 model, there are two 3-branes - branes of three spacial dimensions. One brane provides the structure for gravity while the other houses the other three fundamental forces. However, they are connected by an extra dimension that only

34 Contemporary Extra Dimensions 4.3 Solutions to other problems gravity can propagate in, which is the only similarity to the ADD model. This sets up a scenario where the distance between the two branes affects the strength of gravity’s interaction with the other forces — thus making it possible for gravity to be much stronger on its local brane, and markedly weaker in the other.

4.3 Solutions to other problems

To further illustrate the scope of extra dimensions this section will cover fundamental problems and give examples of their extra-dimensional solutions. Unlike the previous section not as much detail will be presented on the individual papers. Some of the unresolved conundrums faced in cosmological theory which relate to the composition of space are: dark energy, dark matter, baryon asymmetry, and the horizon problem.

Dark energy

The first of these, dark energy, may be the easiest to explain without resorting toextra dimensions. Dark energy is the term for the energy that drives the apparent accelerating expansion of the universe. Most simply, it could be the result of a constant property of spacetime known as the cosmological constant (CC) Λ, which can trace its origin back to

Einstein’s [29] early formulations of GR

G Λg 8πT . (4.1) µν + µν = µν

Einstein presented it as a method for balancing the inevitable contraction or expansion of his initial formulation,

G 8πT , (4.2) µν = µν

35 Contemporary Extra Dimensions 4.3 Solutions to other problems since he believed the universe to be static. After his initial inclusion of the term, Ein- stein dropped it when Hubble [30] observed extra-galactic nebulae to have a distance dependent radial velocity outward, away from our galaxy. Hubble’s observations indi- cated that the universe was indeed expanding and not static, and consequently Einstein could drop the CC term. However, models with a CC did not cease to exist and recent measurements [31, 32] have shown that there must be a non-zero contribution from a CC term to explain the accelerating expansion of the universe. The fact that this CC property appears to exist is actually a point of contention among String theorists. As a whole,

String theory does not seem to allow for a CC on a scale consistent with observation, but many think that this is nothing to worry about [33–35]. There are phenomenological papers [36–38]that show emergence of CC-like effects using ultralarge extra dimensions.

In a paper produced by Gregory et al. [38], they show how a 5D construction with three

3-branes can have divergences from Newtonian gravity at both microscopic and cosmo- logical scales. The theory is built off of the RS1 model with the same behavior for the CC between branes, but Λ 0 on the other side of an extremal brane. =

Dark matter

Dark matter similarly has many possible explanations that do not require the mecha- nisms provided by extra dimensions. Prior to 2006, when observations of two colliding galaxy clusters all but proved a particle theory of dark matter [39], there were hypothe- ses about modified forms of gravity. Theories such as Modified Newtonian Dynamics

(MOND) [40] predicted that gravity was not comprehensively understood at all scales, and certain modifications would account for the effects of dark matter. The Bullet Cluster weak-lensing observation in 2006 showed a clear disparity between the luminous matter

36 Contemporary Extra Dimensions 4.3 Solutions to other problems distribution and the local curvature of spacetime, confirming the presence of something that has mass yet does not interact with matter [39]. The range of particle dark matter theories is extensive.

This review by Bertone et al. [41] presents a thorough outline of prominent dark matter candidate particles and the theories they accompany, including extra-dimensional candidates. For dark matter particles that use extra dimensions, the common mechanism is to have particle excitation states in the extra dimension. The lowest energy excitation becomes the dark matter candidate as, like the proton, it is the lightest particle of its type and is stable. Due to their historical foundations these are known as the Lightest

Kaluza-Klein particles (LKP).

Baryogenesis

Baryogenesis, if we break it down, ‘baryo’ - referring to baryons, and ‘genesis’ - meaning origin, refers to the origin of baryonic matter. Perhaps it is a bit of a misnomer. Certainly, we care about how baryonic matter was created, but more importantly we want to know how there came to be an imbalance between matter and anti-matter. In this interesting paper by Nevzorov [42], a higher dimensional spacetime provides the conditions for both a dark matter candidate and baryogenesis. Unlike the Bertone et al. example, the contribution of the individual extra dimensions is not as cut and dried as providing an additional dimension for energy to propagate in. The additional degrees of freedom made available by extra dimensions creates a complex mathematics that as a whole allows for the existence of a multitude of symmetries and symmetry breaking conditions. Nevzorov creates a structure denoted as SE6SSM* that predicts superheavy right-handed neutrinos

*‘S’ for his version of a particular Exceptional Supersymmetric Standard Model

37 Contemporary Extra Dimensions 4.3 Solutions to other problems

(SRN) and new families of exotic matter. His analysis reveals how under certain, realistic conditions decaying SRN will lead to baryogenesis indirectly through leptogenesis and sphaleron processes.

Horizon problem

The horizon problem pertains to the homogeneity of the cosmos. On the largest scales the universe has been measured to be uniform in temperature to one part in 100 thou- sand [43] from measurements of the cosmic microwave background (CMB). Such uni- formity is surprising to us because it infers causal contact between regions that are not currently connected in such a way. Popular theory states that causal contact was broken early in the lifetime of our universe before nucleosynthesis occurred [44] during a period called inflation. The inflation hypothesis insists that there was a brief time where the universe expanded at a rate faster than the speed of light before slowing down.

To some, the idea of faster than light expansion doesn’t sit well and they have turned to extra dimensions to explain the horizon problem. One hypothesis by Chung and Freese connects otherwise disconnected regions of space through extra dimensions [45]. In this paper they create a 3-brane, which contains our universe, embedded within a bulk multi-dimensional space. The 3-brane is connected to itself through the extra dimension in addition to how it is classically connected, and because of the curvature of the extra dimensions, the null geodesics on it can be shorter than the null geodesics on the 3-brane.

Since they can be shorter, points on the 3-brane can be causally connected that are not classically in causal contact. Clearly this should cause problems in today’s universe by continuing to break causality. However they address this by explaining that the bulk

38 Contemporary Extra Dimensions 4.4 Experimental detection methods dimensions had a stronger coupling in the early universe. Today the coupling between the 3-brane and bulk dimensions is non existent and therefore causality is maintained.

4.4 Experimental detection methods

The final component to any scientific theory is the ability to test its predictions through experiment and observation. Extra dimensions are no different, and some of the tech- niques are fairly simple conceptually. The barriers are only the precision at which these experiments are conducted.

Torsion pendulums

Conceptually the most simple test of extra-dimensional theories is the torsion pendulum.

Pioneered in 1783 by John Michell, he unfortunately passed away before completing the experiment. His design was handed down to Henry Cavendish who published the results of his first measurements of G, the Newtonian gravitational constant, in 1798 [46].

The method has remained largely unchanged since then, and it is still used to measure the gravitational constant [47–49]. However, there is a slight departure from the torsion pendulums used in those experiments to the ones used to probe for extra dimensions.

Since this method can only hope to find microscopic dimensions, the separation between the interacting massive bodies must be extraordinarily small (i.e. on the scale of the dimension).

The most effective experimental design [27] exchanged masses for a "missing mass" technique that uses the azimuthally symmetric lack of mass to cause a torque on the detector. Both the detector and attractor discs have missing mass with the same symme-

39 Contemporary Extra Dimensions 4.4 Experimental detection methods try except the attractor is composed of two stacked discs that are azimuthally skewed.

Including this offset practically eliminates the applied torque of the rotating attractor disc while Newtonian gravity holds, increasing the sensitivity to non-Newtonian effects.

So far experiments have, with relatively high confidence (95%), ruled out compact extra dimensions with R 44µm [27], which is the finest measurement to date50 [ , 28]. ≥

Optical tweezers

A relatively new technique being used to test gravity on the smallest of scales is optical tweezers. Developed in the early 1980s by Arthur Ashkin, optical tweezers trap particles that are on the scale of a wavelength of visible light [51]. Optical tweezers achieve this trapping effect by focusing a laser beam into a transparent particle, maybe a glass bead or even a bacterium. When the light refracts in the bead it exerts a force on the bead and a stable equilibrium point is created between the refracted light and the light absorbed by the bead. The nature of the refraction force also makes it a restoring force perpendicular to the laser beam since as the particle drifts more light will be directed in the same direction, pushing the particle back towards the equilibrium point [52].

In the search for extra dimensions this apparatus is used similarly to how the torsion pendulum is used by looking for deviations from Newtonian gravity at extremely small scales. One experiment [53] uses a small cantilever that is moved close to a dielectric bead while it is suspended in the laser beam. The cantilever carries an electric potential that attracts the bead and measurements are made of the displacement from center. The displacement is used to determine a force which is then analyzed and compared to theory.

They found no deviation from theory at any of the distances tested down to 20µm [53].

40 Contemporary Extra Dimensions 4.4 Experimental detection methods

High energy colliders

High energy collider experiments such as those at CERN and the Brookhaven National

Lab are the final experimental mode that will be discussed in this paper. While these experiments are searching for new particles and decay modes, they record interaction energies and particle momenta. If some of that energy were to disappear that could be an indication of an extra dimension. The missing energy could have been transferred into an extra dimension and out of the realm that we can detect [54, 55].

A general hypothesis is that at a certain energy an extra dimension may become accessible. This is closely related to how higher energy collisions will produce exotic particles. In this case, a common particle could be excited in an extra dimension or an altogether new particle could be created with energy in the extra dimension. So far, we have not recorded evidence of this. With the highest energy experiments taking place in the LHC at CERN at energy levels near 1.3 1013eV (13 TeV) [11] there is still a vast range × to test if the ToE energy level is around 1023ev (1010 TeV). The next generation particle collider at CERN, the Future Circular Collider (FCC), aims to reach collision energies of

1014eV (100 TeV) [56]. Even with a circumference of 100km, the FCC only brings us one order of magnitude higher in energy. What we will achieve past that is uncertain.

41 Chapter 5

Conclusions

Fundamental physics is exploring the possibilities of extra dimensions today more than ever before. This year marks 100 years since Kaluza published his paper, On the Unifi- cation Problem in Physics [14], and introduced physicists at large to the potential extra dimensions had for solving modern problems in physics. Kaluza’s radical idea that the structure of the Universe may have just one extra dimension has grown into one of the leading theoretical models of the universe requiring 11 dimensions. Even if String theory itself is not the ToE that many think it is, extra dimensions will survive. They are the mechanism for explaining so many disparate phenomena it will be difficult for them to be altogether abandoned.

This aspect of extra dimensions could also be problematic for physics. As future experiments are developed and stricter bounds are put on the size and number of extra dimensions, it will be challenging to show they do not exist. It may take a consistent, novel ToE that does not use extra dimensions to rule them out entirely. On the other hand, because they are ubiquitous in their ability to provide solutions to fundamental problems, proof of their existence could come from a number of sources. Experimentalists searching

42 Conclusions for solutions to all sorts of problems, some that we may not even know we have yet, will have reason to test for extra dimensions.

This review of both theoretical and experimental extra dimensions should provide an introduction to anyone interested in how they are used today and their history. It was developed because I wanted to learn about String theory, but there was quite a knowledge gap between my undergraduate education and even beginning to understand it. Extra dimension are the foundations of String theory and therefore it made sense to get familiar with them in their most simple implementations before proceeding. It is my hope that this work can be utilized by others who want to work on more complex problems to expedite their understanding of extra dimensions and KK theory.

43 Appendix A

Riemann Normal Coordinates

Riemann Normal Coordinates (RNC) are a set of coordinates where the geodesics from a given point to nearby points are unique within a certain limit. This allows us to work with a locally flat metric greatly simplifying many of our equations and allowing usto easily derive some useful identities. Clearly, if generality was lost using this approach it would cease to be useful, but it is valid for any Riemannian manifold. This section will provide an adapted derivation from [57, 58] of RNC’s fundamental properties and an informal proof of their generality by working through a coordinate transformation.

However, before getting to the derivations, a few definitions need to be made.

44 Riemann Normal Coordinates A.1 Definitions

A.1 Definitions

Geodesics in GR

Fundamental to the definition of RNC, a geodesic Gα(s) can be determined at the origin by a tangent vector v vαe and parameterized by s. Thus, it is written as = α

Gα(s) svα. (A.1) =

In GR every geodesic satisfies the following differential equation:

d 2Gα(s) dGβ dGγ 0 Γα . (A.2) = ds2 + βγ ds ds

The geodesic equation is found by examining the equation of motion of a particle in free fall in a gravitational field. When there are no other interactions the particle will havea constant four-velocity, which is the defining characteristic of a geodesic. It is analogous to the Euler-Lagrange equation of Newtonian mechanics.

Einstein’s equivalence principle

Einstein famously came to the conclusion that whether a person was freely falling in a uni- form gravitational potential or at rest in a far off region of space, void of any gravitational or other force, they could not distinguish between the two [59]. The two reference frames are therefore equivalent. Understanding this quality of constant acceleration allowed him to arrive at the equations of special relativity and predict phenomena like the bending of light rays by gravitational wells. In this derivation Einstein’s Equivalence Principle

(EEP) allows for any uniform gravitational acceleration to be treated as if there were no

45 Riemann Normal Coordinates A.2 Fundamental principles of RNC acceleration at all. The generality of this statement extends to all points in spacetime, except for singularities. Even inside a star spacetime can be considered flat at any given point. Mathematically this corresponds to the ability to construct RNC at any point on a spacetime manifold.

A.2 Fundamental principles of RNC

To prove that RNC are applicable to any spacetime, first a display of how they work is in order. The fundamental principles of RNC rely on EEP which tells us that any point in spacetime can be considered free of acceleration or flat. Mathematically this can be written as

(1st Principle of RNC) g (P) η , (A.3) µν = µν which translates to: the metric at point P is the Minkowski metric. The Minkowski metric being the metric that defines a Euclidean spacetime. It only has non-zero elements on the diagonal, which are (-1,1,1,1). Notice that the metric is only specified at point and it is not the metric of a region. Strictly speaking, nothing in this derivation expands RNC to more than a given point. Doing physics at a point may not be the most fruitful endeavor, however the properties of RNC will also apply to a region around the point. In effect, there will always be a region around a point in spacetime where each point lies on a single geodesic (i.e. the geodesics do not overlap), and that quality is adequate for imposing

RNC. For a detailed resource on why that is so please see [57, 58]. When EEP is imposed

46 Riemann Normal Coordinates A.2 Fundamental principles of RNC on the geodesic equation A.2 becomes

dGβ dGγ 0 Γα , (A.4) = βγ ds ds because there are no accelerations. Then, carrying out the differentiation leaves

d(svβ) d(svγ) 0 Γα (A.5) = βγ ds ds Γα dvβdvγ. (A.6) = βγ

For this relation to be true, either the Christoffel symbol or the vector components must be zero. The choice is easy, because constraining the vector components to zero would cause a loss of generality. Therefore, the Christoffel symbol must be zero,

(2nd Principle of RNC) Γα 0. (A.7) βγ =

This makes sense. The Christoffel symbols must be zero, for they encode how basis vectors change over a manifold, and in flat spacetime they do not change. With the second principle in mind, the third may seem like an obvious extension, but it bears showing explicitly. By writing out two Christoffel symbols in terms of derivatives of the metric,

1 ¡ ¢ Γ ∂γg ∂ gαγ ∂αg αβγ = 2 αβ + β − βγ 1 ¡ ¢ Γ ∂αg ∂ gγα ∂γg , γβα = 2 γβ + β − βα

47 Riemann Normal Coordinates A.3 General coordinate transformation to RNC and adding them together, the result is

Γ Γ ∂ g . (A.8) αβγ + γβα = β αγ

As the second principle states, all the Christoffel symbols are zero, so the previous equa- tion simplifies to

(3rd Principle of RNC) ∂ g 0. (A.9) β αγ =

The third principle is that all first derivatives of the metric are zero in RNC. Not so surprising considering the first and second principles, but it is necessary and it comesup in the following generality proof.

A.3 General coordinate transformation to RNC

To prove RNC are generally applicable and do not subvert the properties of KK theory found using them, this section will show a coordinate transformation from arbitrary coor- dinates, x, to RNC, ξ. To begin the proof we assume that there must be a transformation between them,

Ξaˆ(x) X α(ξ), (A.10) ⇐⇒ because both coordinate systems apply to the same point in spacetime. Then, if a gener- ally indeterminable value arises in the transformation, it will prove the transformation is non-general. So, after making the initial assumption we can determine what the transfor- mation equations look like, and the process begins with a Taylor expansion of the general

48 Riemann Normal Coordinates A.3 General coordinate transformation to RNC to RNC transformation equation:

aˆ ¯ 2 aˆ ¯ ˆ ˆ ∂Ξ (x)¯ 1 ∂ Ξ (x)¯ Ξa(x) Ξa(x ) (x x )µ ¯ (x x )β(x x )γ ¯ . (A.11) P P µ ¯ P P β γ ¯ = + − ∂x xP + 2 − − ∂x ∂x xP

Higher order terms unfortunately are non-unique, so they will be neglected in this in- formal proof. Now, the task is to identify each term as a known value or quantity. The zeroth order term is easy. As discussed, RNC are derived at a point, P.This point can be identified in both coordinate systems by

Ξaˆ(x ) ξaˆ. (A.12) P = P

In RNC the point is the origin, but there is no need to make that restriction at this juncture.

Moving to the first order term there is a partial derivative of the transformation equation with respect to the x coordinates. This is another term that can be quickly identified as the Jacobian for this transformation. It is defined by the matrix that relate the metrics in each coordinate system. In equation (A.13),

g g Λa Λb , (A.13) αβ = ab α β the Λ are the general Jacobian matrix. The metric at point P is known in RNC to be the Minkowski metric (A.3), and when substituted into equation (A.13), it defines the

aˆ particular Jacobian which will be denoted as eα:

aˆ bˆ g (x ) η ˆ e e . (A.14) αβ P = aˆb α β

49 Riemann Normal Coordinates A.3 General coordinate transformation to RNC

Updating the transformation expansion with the newly identified Jacobian results in

2 aˆ ¯ ˆ ˆ ˆ 1 ∂ Ξ (x)¯ Ξa(x) Ξa(x ) (x x )µea (x x )β(x x )γ ¯ . (A.15) P P µ P P β γ ¯ = + − + 2 − − ∂x ∂x xP

All that is left is the second order term. To identify this term the fact that it has three indexes leads to the association with Christoffel symbols and the mixing of coordinate systems to the Christoffel symbol transformation equation:

2 aˆ ˆ ∂ Ξ (x) Γα Γaˆ eαeb ecˆ eα. (A.16) βγ = bˆcˆ aˆ β γ + ∂xβ∂xγ aˆ

The indices indicate that this is a transformation from the Christoffel symbols in RNC to those in general coordinates. However, the Christoffel symbols in RNC are zero which simplifies (A.16) to ∂2Ξaˆ(x) Γα eα. (A.17) βγ = ∂xβ∂xγ aˆ

This is almost the equation for the unknown term in equation (A.15). If each side is multiplied by the inverse Jacobian the resulting relation is the final unknown term in the expansion: ∂2Ξaˆ(x) eaˆ Γα . (A.18) ∂xβ∂xγ = α βγ

Rewriting the expansion in its complete form,

aˆ µ aˆ 1 β γ aˆ α Ξ (x) (x xP) e (x xP) (x xP) e Γ , (A.19) = − µ + 2 − − α βγ we have a transformation in terms of knowable quantities of the two bases. This not only shows that is possible to go to RNC from any coordinate system, but that it is also possible to do so without any additional information about the manifold. Most importantly, this

50 Riemann Normal Coordinates A.3 General coordinate transformation to RNC proves that the methods Kaluza used to simplify his mathematics do not violate generality and do not restrict his theory.

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