Physics in Higher-Dimensional Manifolds by Sanjeev S. Seahra A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Physics Waterloo, Ontario, Canada, 2003 c Sanjeev S. Seahra, 2003 Author’s declaration for electronic submission of a thesis I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract In this thesis, we study various aspects of physics in higher-dimensional manifolds involving a single extra dimension. After giving some historical perspective on the motivation for studying higher-dimensional theories of physics, we describe classical tests for a non-compact extra dimension uti- lizing test particles and pointlike gyroscopes. We then turn our attention to the problem of embedding any given n-dimensional spacetime within an (n + 1)-dimensional manifold, paying special attention to how any struc- ture from the extra dimension modifies the standard n-dimensional Einstein equations. Using results derived from this investigation and the formalism derived for test particles and gyroscopes, we systematically introduce three specific higher-dimensional models and classify their properties; including the Space-Time-Matter and two types of braneworld models. The remain- der of the thesis concentrates on specific higher-dimensional cosmological models drawn from the above mentioned scenarios; including an analysis of the embedding of Friedmann-Lemaˆıtre-Robertson-Walker submanifolds in 5-dimensional Minkowski and topological Schwarzschild spaces, and an in- vestigation of the dynamics of a d-brane that takes the form of a thin shell encircling a (d+2)-dimensional topological black hole in anti-deSitter space. The latter is derived from a finite-dimensional action principle, which allows us to consider the canonical quantization of the model and the solutions of the resulting Wheeler-DeWitt equation. iii Acknowledgements I would like to take this opportunity to thank my doctoral supervisor Dr. Paul S. Wesson for his help and guidance during the years leading up to this the- sis. I would also like to thank Takao Fukui, Jaime Ponce de Leon, and Hamid Sepangi for collaborations and useful discussions. Along the way, I have benefited from astute commentary from Andrew Billyard, Dan Bruni, Werner Israel, Hongya Liu, Robert Mann, Bahram Mashhoon, Eric Poisson, and William Sajko. I would also like to thank Eric Poisson for his phys 789 course notes, from which much of the style and notation of this document has been shamelessly pilfered; and Tomas Liko for diligently proofreading the entire manuscript. Many calculations in this thesis were assisted by the Maple symbolic computation software, often used in conjunction with the GRTensorII macros (Lake, Musgrave, and Pollney 1995). This work was financially supported by the Natural Sciences and Engineering Research Council of Canada and the Ontario Graduate Scholarships in Science & Technology program. iv For my mother and father, Colleen, and the cat v Contents List of Tables x List of Figures xi Notation and Conventions xiii I Generic Properties of Higher-Dimensional Models 1 1 Why Bother with Extra Dimensions? 2 1.1 The 4th dimensionattheturnofthelastcentury....... 4 1.2Dimensionalityandthequestforunification.......... 7 1.2.1 Oftimeandspace.................... 8 1.2.2 Ofgravityandelectromagnetism............ 9 1.2.3 Of the fundamental forces . .............. 12 1.3 The 5th dimension at the turn of the current century . 15 1.4Anoutlineofwhatistocome.................. 18 BibliographicNotes.......................... 21 2 Test Particles and Pointlike Gyroscopes 22 2.1Observablesinhigher-dimensionaltheories........... 23 2.2Geometricconstruction..................... 25 2.3Covariantsplittingoftheequationofmotion......... 30 2.4Parametertransformations................... 35 2.4.1 Generaltransformations:................. 36 2.4.2 Transformation to the n-dimensional proper time: . 36 2.5Theignorancehypothesis.................... 39 vi 2.5.1 Thefifthforce...................... 40 2.5.2 Thevariationof“restmass”.............. 46 2.5.3 Lengthcontractionandtimedilation.......... 50 2.5.4 Killing vectors and constants of the motion ...... 52 2.6Testparticlesinwarpedproductspaces............ 57 2.7 Confinement of trajectories to Σ hypersurfaces........ 59 2.8Pointlikegyroscopes....................... 64 2.8.1 Aspinningparticleinhigherdimensions........ 66 2.8.2 Decomposition of the Fermi-Walker equation . 68 2.8.3 Observables and the variation of n-dimensional spin . 70 2.9Summary............................. 71 Appendix 2.A Two identities concerning foliation parameters . 73 BibliographicNotes.......................... 74 3 Effective Field Equations on the Σ Hypersurfaces 75 3.1 Decomposition of the higher-dimensional field equations . 76 3.2Fieldequationsinwarpedproductspaces........... 78 3.3 The generalized Campbell-Magaard theorem .......... 82 3.4 Embeddings in higher-dimensional manifolds with matter . 87 3.4.1 Dustinthebulk..................... 87 3.4.2 Ascalarfieldinthebulk................ 90 3.5Summary............................. 92 Appendix 3.A N-dimensionalgravity-mattercoupling....... 93 Appendix 3.B Canonical evolution equation for Eαβ ........ 95 BibliographicNotes.......................... 96 4 Properties of Selected Higher-Dimensional Models 97 4.1Space-Time-Mattertheory.................... 97 4.1.1 Effective4-dimensionalfieldequations......... 98 4.1.2 Testparticles....................... 99 4.1.3 Pointlikegyroscopes................... 101 4.2Thethinbraneworldscenario.................. 101 4.2.1 Effective4-dimensionalfieldequations......... 102 4.2.2 Testparticles....................... 105 vii 4.2.3 Pointlikegyroscopes................... 108 4.3Thethickbraneworldscenario..................110 4.3.1 Effective4-dimensionalfieldequations......... 110 4.3.2 Testparticles....................... 111 4.3.3 Pointlikegyroscopes................... 113 4.4Summary............................. 114 BibliographicNotes.......................... 117 II Our Universe in a Higher-Dimensional Manifold 118 5 Universes Embedded in 5D Minkowski Manifolds 119 5.1PoncedeLeoncosmologies................... 120 5.2 Properties of the Σ hypersurfaces............... 124 5.2.1 Singularpoints...................... 124 5.2.2 Regularpoints...................... 126 5.2.3 Globalstructure..................... 128 5.2.4 Visualization of the Σ hypersurfaces..........129 5.2.5 Onthegeometricnatureofthebigbang........ 135 5.3 Special values of α ........................ 141 5.4 Variation of 4-dimensional spin in a cosmological setting . 142 5.5Summary............................. 146 BibliographicNotes.......................... 146 6 Universes Wrapped Around 5D Topological Black Holes 147 6.1 Two 5-metrics with flrw submanifolds............ 148 6.1.1 TheLiu-Mashhoon-Wessonmetric........... 148 6.1.2 TheFukui-Seahra-Wessonmetric............ 152 6.2 Connection to the 5D topological black hole manifold . 155 6.3Coordinatetransformations................... 156 6.3.1 Schwarzschild to Liu-Mashhoon-Wesson coordinates . 157 6.3.2 Schwarzschild to Fukui-Seahra-Wesson coordinates . 160 6.3.3 Comments.........................162 6.4Penrose-Carterembeddingdiagrams.............. 163 6.5Summary............................. 169 viii Appendix 6.A Thick braneworlds around 5D black holes . 169 BibliographicNotes.......................... 172 7 Classical Brane Cosmology 173 7.1Aneffectiveactionforthebraneworld............. 175 7.2Thedynamicsoftheclassicalcosmology............ 188 7.2.1 TheFriedmanequation................. 188 7.2.2 Exactanalysisofaspecialcase............. 192 7.2.3 Instantontrajectoriesandtachyonicbranes...... 197 7.3Summary............................. 201 Appendix 7.A adm mass of topological S-AdS(d+2) manifolds . 201 Appendix 7.B Velocity potential formalism for perfect fluids . 203 BibliographicNotes.......................... 205 8 Braneworld Quantum Cosmology 206 8.1Hamiltonizationandquantization................208 8.1.1 Theexteriorregion.................... 209 8.1.2 Theinteriorregion.................... 221 8.2ThereducedWheeler-DeWittequation.............223 8.3Perfectfluidmatteronthebrane................ 226 8.4 Tunnelling amplitudes in the wkb approximation....... 232 8.5Summary............................. 235 BibliographicNotes.......................... 235 9 Concluding Remarks 236 9.1Synopsis.............................. 236 9.2Outlook.............................. 240 Bibliography 243 Index 253 ix List of Tables 5.1EquationofstateofinducedmatterinPdLmetric...... 122 5.2 Properties of cosmologies embedded on Σ 4-surfaces in M5 . 129 6.1 Types of cosmologies embedded in the fsw metric...... 154 7.1Qualitativebehaviourofbranecosmologies.......... 197 8.1 The large a limits of U(a) for various model parameters . 225 8.2 Parameter choices characterizing the quantum potential . 229 x List of Figures 2.1Thegeometricinterpretationofthelapseandshift...... 30 2.2 Evolution of the y-coordinates from n-surface to n-surface . 35 2.3Totallygeodesicandnon-geodesicsubmanifolds........ 64 2.4 Behaviour of the spin projection of a confined gyroscope . 65 4.1Gravitationalpotentialneara3-brane............. 107 4.2 The behaviour of a gyroscope spin-basis near a 3-brane . 109 4.3 Interrelationships between higher-dimensional models . 116 5.1aVisualizationofadust-filleduniverse.............