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Gravitation and Cosmology with York Time Philipp Roser Clemson University, [email protected]

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Gravitation and Cosmology with York Time Philipp Roser Clemson University, Roser.Philipp@Gmail.Com Clemson University TigerPrints All Dissertations Dissertations 12-2016 Gravitation and Cosmology with York Time Philipp Roser Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_dissertations Recommended Citation Roser, Philipp, "Gravitation and Cosmology with York Time" (2016). All Dissertations. 1821. https://tigerprints.clemson.edu/all_dissertations/1821 This Dissertation is brought to you for free and open access by the Dissertations at TigerPrints. It has been accepted for inclusion in All Dissertations by an authorized administrator of TigerPrints. For more information, please contact [email protected]. GRAVITATION AND COSMOLOGY WITH YORK TIME A Thesis Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Physics by Philipp Roser December 2016 Accepted by: Dr. Antony Valentini, Committee Chair Dr. Murray Daw Dr. Dieter Hartmann Dr. Bradley Meyer ABSTRACT Despite decades of inquiry an adequate theory of ‘quantum gravity’ has remained elu- sive, in part due to the absence of data that would guide the search and in part due to technical difficulties, prominently among them the ‘problem of time’. The problem is a result of the attempt to quantise a classical theory with temporal reparameterisation and refoliation invariance such as general relativity. One way forward is therefore the breaking of this invariance via the identification of a preferred foliation of spacetime into parameterised spatial slices. In this thesis we argue that a foliation into slices of constant extrinsic curvature, parameterised by ‘York time’, is a viable contender. We argue that the role of York time in the initial-value problem of general relativity as well as a number of the parameter’s other properties make it the most promising candidate for a physically preferred notion of time. A Hamiltonian theory describing gravity in the York-time picture may be derived from general relativity by ‘Hamiltonian reduction’, a procedure that eliminates certain degrees of freedom — specifically the local scale and its rate of change — in favour of an explicit time parameter and a functional expression for the associated Hamilto- nian. In full generality this procedure is impossible to carry out since the equation that determines the Hamiltonian cannot be solved using known methods. ii However, it is possible to derive explicit Hamiltonian functions for cosmological scenarios (where matter and geometry is treated as spatially homogeneous). Using a perturbative expansion of the unsolvable equation enables us to derive a quantisable Hamiltonian for cosmological perturbations on such a homogeneous background. We analyse the (classical) theories derived in this manner and look at the York-time de- scription of a number of cosmological processes. We then proceed to apply the canonical quantisation procedure to these systems and analyse the resulting quantum theories. We discuss a number of conceptual and techni- cal points, such as the notion of volume eigenfunctions and the absence of a momentum representation as a result of the non-canonical commutator structure. While not prob- lematic in a technical sense, the conceptual problems with canonical quantisation are particularly apparent when the procedure is applied in cosmological contexts. In the final part of this thesis we develop a new quantisation method based on configuration-space trajectories and a dynamical configuration-space Weyl geometry. There is no wavefunction in this type of quantum theory and so many of the conceptual issues do not arise. We outline the application of this quantisation procedure to gravity and discuss some technical points. The actual technical developments are however left for future work. We conclude by reviewing how the York-time Hamiltonian-reduced theory deals with the problem of time. We place it in the wider context of a search for a theory of quantum gravity and briefly discuss the future of physics if and when such a theory is found. iii ACKNOWLEDGEMENTS First and foremost, I am indebted to my doctoral advisor Antony Valentini at Clemson University for his help in all matters from technical to literary, and his support in en- abling me to pursue the big questions that fascinate me. I am also grateful to Lucien Hardy for many discussions on foundations of quantum theory, and for his hospitality at Perimeter Institute during 2012/2013, where a non-trivial amount of the research in this thesis (especially chapter 11) was carried out. I furthermore wish to thank Lee Smolin, who in early 2012 pointed me toward the literature on the then very recent developments in Shape Dynamics, and Rafael Sorkin, whose critical and novel way to look at ques- tions in both quantum theory and gravity has contributed to my own approach. I owe much of my appreciation of the issues of gravity, quantum theory and quantum theory of gravity to my education in philosophy of physics at Oxford University, in particular to Oliver Pooley, Harvey Brown, Simon Saunders and David Wallace. I am also grateful to the Clemson University Department of Physics and Astronomy and my PhD committee to allow me to pursue my work via somewhat unconventional arrangements: first, through financial support in the form of a minimal assistantship during my time at Perimeter Institute, and second by allowing me to spend the last two years working from my home, some three thousand kilometres from campus. iv Finally, I wish to thank my wife, Jemma, for her patience. v TABLE OF CONTENTS Page 1 Introduction 1 1.1 Quantum theory and gravity . .1 1.2 Outline of this thesis . 15 I Setting the scene 21 2 The problem of time 22 2.1 Time-reparameterisation invariance . 22 2.2 Vanishing Hamiltonians and frozen dynamics . 28 2.3 The problem of time in general relativity . 38 2.4 Other problems of time . 53 3 Physical time 57 3.1 Choosing a physical time: Hamiltonian reduction . 57 3.2 The meaning of the reduced Hamiltonian . 64 3.3 Significance of the choice of time . 66 4 York time 79 vi TABLE OF CONTENTS (CONTINUED) . PAGE 4.1 The initial-value problem of general relativity . 79 4.2 Properties of York time . 90 4.3 The reduced variables and their Poisson structure . 97 4.4 Cosmological history with York time . 102 4.5 York time as a solution to the problem of time . 105 II Classical York-time cosmology 111 5 The homogeneous isotropic universe with York time 112 5.1 The role of cosmology in gravitational theories . 112 5.2 Friedmann-Lemaˆıtre cosmology ‘translated’ into York time . 115 5.3 Reduced-Hamiltonian Friedmann-Lemaˆıtre cosmology . 119 5.4 Inflation . 133 6 A cosmological extension 140 6.1 An end to time? . 140 6.2 The necessity of a cosmological extension . 142 6.3 Dynamics on the other side . 145 6.4 Transition behaviour and classification of potentials . 146 6.5 Discussion . 151 7 Cosmological perturbation theory with York time: preliminaries 155 7.1 The importance of cosmological perturbation theory . 155 vii TABLE OF CONTENTS (CONTINUED) . PAGE 7.2 Classical anisotropic cosmology: Exploration of the Poisson structure . 158 7.3 Mode freezing during inflation . 164 7.4 Re-entry of the Hubble radius in the radiation-dominated universe . 172 8 Cosmological perturbation theory with York time: Hamiltonian reduction 176 8.1 Principles of cosmological perturbation theory via York-time Hamilto- nian reduction . 176 8.2 The perturbative Hamiltonian constraint . 182 8.3 The physical Hamiltonian and the equations of motion . 187 8.4 Analysis of the dynamics . 197 III Canonical quantum cosmology with York time 202 9 Quantum Friedmann-Lemaˆıtre cosmology 203 9.1 Quantum theory of the free scalar field . 203 9.2 Quantum theory with a potential . 212 10 Quantum York-time cosmological perturbation theory 218 10.1 Quantisation, representations and the anisotropic homogeneous quan- tum universe . 218 10.2 Canonical quantisation of York-time cosmological perturbation theory . 233 viii TABLE OF CONTENTS (CONTINUED) . PAGE IV Quantum cosmology without the wavefunction 240 11 Trajectory-Weyl quantisation 241 11.1 Introduction to trajectory-based quantisation . 241 11.2 Relevant aspects of classical mechanics . 251 11.3 Review of Weyl geometry . 258 11.4 Quantum trajectories from an action principle . 262 11.5 Nodes and the phase-like nature of S ................... 270 11.6 Discussion of trajectory-Weyl quantisation . 279 12 Quantum cosmology from trajectory-Weyl quantisation? 284 12.1 Extension of the trajectory-Weyl approach . 284 12.2 Application to gravity and cosmology . 287 293 13 Conclusion 294 ix LIST OF FIGURES Figure Page 2.1 Different ways of foliating spacetime . 55 4.1 Surfaces of constant York time around singularities . 92 4.2 The cosmological timeline with York time . 104 5.1 The scale factor near scalar-field ‘turning points’ . 130 6.1 The extended York timeline with Λ = 0 ................. 146 6.2 The extended York timeline with Λ > 0 ................. 148 x LIST OF TABLES Table Page 6.1 Transition behaviour for potentials bounded during the transition into the extended York-time cosmology . 151 xi CHAPTER 1 INTRODUCTION 1.1 Quantum theory and gravity Cosmology, the study of the history of universe, has a unique role in the investigation of physical reality. Unlike, for example, particle physics or condensed-matter theory it cannot be tested experimentally in the laboratory. All evidence must come from observation of the one and only universe to which we have access. To make matters worse, all our observations are made roughly from a single point in space and time. Yet despite these limitations, cosmology — in particular the study of the very early moments in the history of the universe — is not only fascinating in itself but also ex- ceptionally useful in that it allows us to explore physics at extremely high energies and on very small scales, where both gravitational and quantum effects are expected to play a crucial role.
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