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More Space for Unification Kaluza-Klein Theory and Extra Dimensions 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 ´¹º Minkowski metric ½ ¡¹º Christoffel symbol (pseudo-tensor ¤ Cosmological constant §¹º Associated field A¹ EM four-potential F ¹º EM or Faraday tensor g ¹º Spacetime metric G¹º Einstein tensor J ¹ Four-current R Ricci scalar ¸ R¹½º Riemann curvature tensor R¹º Ricci tensor viii T¹º 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 ¹º ¹ J º, (1.1) ¹ Æ 0 @ F 0, (1.2) [¸ ¹º] Æ where F ¹º is the EM field tensor and J º 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.
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