GEOMETRODYNAMICS: SPACETIME OR SPACE? Edward Anderson Submitted for Ph.D. 03 / 2004 Astronomy Unit, School of Mathematical Sciences, Queen Mary, University of London, E1 4NS, U.K. arXiv:gr-qc/0409123v1 30 Sep 2004 1 Acknowledgments To the memory of my Father, who taught me mathematics and much more. To Claire for her caring support, and to my Mother. To all my friends, with thanks. Yves, Becca, Suzy, Lynnette and Mark helped me survive my Cambridge years. Yves, Becca and Suzy have stayed in touch, while I often enjoyed Lynnette and Chris’s company though my PhD years, and Mark was a familiar and welcome figure at QMUL. I also thank the good friends I made toward the end of my Cambridge years: Ed, Matt, Bjoern, Alex, Angela and Alison. And Bryony, for being a good friend during the difficult last year. Thanks also to all my other friends, office mates, and kind and entertaining people I have crossed paths with. To Professors Malcolm MacCallum and Reza Tavakol, with thanks for agreeing to supervise me, and for their encouragement and wisdom. I also thank Dr James Lidsey for supervision, and both Dr Lidsey and Professor Tavakol for active collaborations. I also thank Dr. Julian Barbour and Professor Niall O´ Murchadha for teaching me many things and for collaboration, and the Barbour family for much hospitality. I also thank Brendan Foster and Dr. Bryan Kelleher for many discussions and for collaboration, and Dr. Harvey Brown for many discussions. I thank Professor MacCallum, Dr Barbour and Professor Tavakol for reading, criticizing and commenting on earlier drafts of this thesis. I thank the following people for one or more of: discussion, collaboration, reading final drafts of my papers, providing useful references, providing encouragement. Dr. Martin Bojowald, Professor Bernard Carr, Dr. David Coule, Professor Naresh Dadhich, Dr. Fabio Dahia, Dr. Steven Davis, Dr. Carl Dolby, Dr. Fay Dowker, Dr Henk van Elst, Richard Frewin, Dr. Cristiano Germani, Dr. Domenico Giulini, Dr. Anne Green, Dr Tomohiro Harada, Rachael Hawkins, Dr Gregory Huey, Gian Paolo Imponente, Dr Deborah Konkowski, Dr Janna Levin, Professor Roy Maartens, Dr Joao Magueijo, Professor Shahn Majid, Dr. Marc Mars, Dr. Nikolaos Mavromatos, Dr. Filipe Mena, Simone Mercuri, Professor John Moffat, David Mulryne, Professor James Nester, Professor Don Page, Dr. Alexander Polnarev, Dr. Oliver Pooley, Dr. Istvan Racz, Dr. Carlos Romero, Mike Roper, Professor Misao Sasaki, Dr. Chopin Soo, Professor Raphael Sorkin, Professor Marek Szydlowski, Professor John Stachel, John Taylor, Professor William Unruh, Dr. Raul Vera, Professor James York and Professor Jorge Zanelli. I would like to thank PPARC for funding me, the Astronomy Unit for being generous with my allotted conference money, and the subsidy of the Heraeus Foundation for the Bad Honnef Quantum Gravity Workshop. 1 Abstract The work in this thesis concerns the split of Einstein field equations (EFE’s) with respect to nowhere-null hypersurfaces, the GR Cauchy and Initial Value problems (CP and IVP), the Canonical formulation of GR and its interpretation, and the Foundations of Relativity. I address Wheeler’s question about the why of the form of the GR Hamiltonian constraint “from plausible first principles”. I consider Hojman–Kuchar–Teitelboim’s spacetime-based first principles, and especially the new 3-space approach (TSA) first principles studied by Barbour, Foster, O´ Murchadha and myself. The latter are relational, and assume less struc- ture, but from these Dirac’s procedure picks out GR as one of a few consistent possibilities. The alternative possibilities are Strong gravity theories and some new Conformal theories. The latter have privileged slicings similar to the maximal and constant mean curvature slicings of the Conformal IVP method. The plausibility of the TSA first principles are tested by coupling to fundamental matter. Yang–Mills theory works. I criticize the original form of the TSA since I find that tacit assumptions remain and Dirac fields are not permitted. However, comparison with Kuchaˇr’s hypersurface formalism allows me to argue that all the known fundamental matter fields can be incorporated into the TSA. The spacetime picture appears to possess more kinematics than strictly necessary for building Lagrangians for physically-realized fundamental matter fields. I debate whether space may be regarded as primary rather than spacetime. The emergence (or not) of the Special Relativity Principles and 4-d General Covariance in the various TSA alternatives is investigated, as is the Equivalence Principle, and the Problem of Time in Quantum Gravity. Further results concern Elimination versus Conformal IVP methods, the badness of the timelike split of the EFE’s, and reinterpreting Embeddings and Braneworlds guided by CP and IVP knowledge. 2 Overview Here I explain what is in this thesis and on which of my articles each topic is based. My Introduction serves the following purposes. In I.0-1 I summarize the standard founda- tions of classical physics for later comparison (mostly in Part A); these sections also serve to establish much notation for the thesis. In I.2 I provide the split formulation of GR, to be used both in Part A and Part B. Much of this material was included in early sections of my publications. This section also includes some original material filling up some gaps in the older literature. I.3 briefly provides the quantum physics and quantum gravity needed to understand what is being attempted in Part A, and how Part A and Part B fit into current physical thought. Part A: II considers work by Barbour, Bertotti, O´ Murchadha, Foster and myself (The 3-space Approach), on the foundations of particle dynamics, gauge theory and especially Relativity, starting from relational first principles. I have carefully reworked this material and have added to it both from my papers [5, 6] and as an original contribution to this Thesis. III considers strong and conformal alternative theories of gravity that arise alongside GR in this 3-space approach, and is based on the pure gravity halves of my paper [5] and of my collaboration with Barbour, Foster and O´ Murchadha [11]. IV furthermore includes bosonic matter and is based on the matter halves of the papers [5, 11] and on my paper with Barbour [10]. VI is based on my critical article on the 3-space approach [6]. This comes to grips with what the 3-space approach’s spatial principles really are, and how they may be related to Kuchaˇr’s principles that presuppose spacetime. My modified 3-space 1 principles are capable of accommodating spin- 2 fermions and more. V.1 and VII are original contributions of mine to this Thesis which update the 3-space approach to General Relativity and attempt to relate it to the standard approach’s Relativity Principles and Principle of Equivalence. I hope to publish the new material in II, V.I, and VII as a new article [9]. V.2 provides more material on the conformal alternatives; this or related material in preparation may appear in a solo article as well as in a collaboration with Barbour, Foster, Kelleher and O´ Murchadha [12]. VIII is an incipient account of quantum gravitational issues which underly and motivate motivate much of Part A. I briefly mentioned this in the conference proceedings [7], but it remains work in progress. Part B: This is a separate, largely critical application of the split formulation of GR. I explain in B.1 how the sideways analogue of the usual split, cases of which have recently been suggested in various guises, is causally and mathematically undesirable. This work began when Lidsey and I came across and used a sideways method [13], followed by my criticism of it in my contribution to the article with Dahia, Lidsey and Romero [14], further criticism in my preprint with Tavakol [16], and yet further related work filling in some gaps in the early Initial Value Problem literature, which were most conveniently presented in I.2. [16] also considers a multitude of other treatments of proper and sideways splits for GR-like braneworlds. Some of this material (in B2) was published in my letter with Tavakol [15], some of it will appear in my conference proceedings [8] (a summary of B.3), and some of it will form the basis for further articles that Tavakol and I have in mind. Appendix C on elliptic equations contains further technical material useful in both Parts. 3 Table of Contents Acknowledgments 2 Abstract 3 Overview 4 Table of Contents 5 I Introduction 10 0 Geometry 10 0.1 Geometry and physics 10 0.2 Intrinsic geometry of the 2-sphere 12 0.3 Differential geometry 12 App 0.A Densities and integration 16 1 Classical physics 16 1.1 Newtonian physics 16 1.2 Principles of dynamics 17 1.2.1 Configuration space and the Euler–Lagrange equations 17 1.2.2 Legendre transformations, Jacobi’s principle and Hamiltonian dynamics 18 1.2.3 Dirac’s generalized Hamiltonian dynamics 19 1.2.4 Motivation for use of principles of dynamics 20 1.3 Electromagnetism 21 1.4 Special relativity 22 1.5 General relativity 24 1.6 Many routes to relativity 26 1.7 Other classical matter fields 28 1 1.7.1 Scalars and spin- 2 fermions 28 1.7.2 Electromagnetism, U(1) scalar gauge theory and QED 29 1.7.3 Yang–Mills theory, QCD and Weinberg–Salam theory 30 2 On the split formulation of Einstein’s field equations 33 2.1 Geometry of hypersurfaces 33 2.1.1 Extrinsic curvature 33 2.1.2 Gauss’ outstanding theorem 33 2.1.3 Hypersurface geometry 33 2.2 Split of the EFE’s with respect to nowhere-null hypersurfaces 35 2.3 Analytic approach to the GR “CP and IVP” 37 2.3.1