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The Dynamics of Shapes THE DYNAMICS OF SHAPES Henrique de A. Gomes under supervision of Prof. John W. Barrett. Thesis submitted for the title of Doctor of Philosophy at the University of Nottingham August 24, 2011 Abstract This thesis consists of two parts, connected by one central theme: the dynamics of the \shape of space". To give the reader some inkling of what we mean by \shape of space", consider the fact that the shape of a triangle is given solely by its three internal angles; its position and size in ambient space are irrelevant for this ultimately intrinsic description. Analogously, the shape of a 3-dimensional space is given by a metric up to coordinate and conformal changes. Considerations of a relational nature strongly support the de- velopment of such dynamical theories of shape. The first part of the thesis concerns the construction of a theory of gravity dynamically equivalent to general relativity (GR) in 3+1 form (ADM). What is special about this theory is that it does not possess foliation invariance, as does ADM. It replaces that \symmetry" by another: local conformal invari- ance. In so doing it more accurately reflects a theory of the \shape of space", giving us reason to call it shape dynamics (SD). Being a very recent development, the consequences of this radical change of perspective on gravity are still largely unexplored. In the first part we will try to present some of the highlights of results so far, and indicate what we can and cannot do with shape dynamics. Because this is a young, rapidly moving field, we have necessarily left out some interesting new results which are not yet in print and were developed alongside the writing of the thesis. The second part of the thesis will develop a gauge theory for \shape of space"{theories. To be more precise, if one admits that the physically relevant observables are given by shape, our descriptions of Nature carry a lot of redundancy, namely absolute local size and absolute spatial position. This redundancy is related to the action of the infinite-dimensional conformal and diffeomorphism groups on the geometry of space. We will show that the action of these groups can be put into a language of infinite-dimensional gauge theory, taking place in the configuration space of 3+1 gravity. In this context gauge connections acquire new and interesting meanings, and can be used as \relational tools". Acknowledgements I vividly remember being on the beach, at quite a young age, trying to understand my father's explanation of Newton's law of action and reaction. I would like to dedicate this thesis to him, for first getting me interested in physics and thereafter never failing to give me his full support or sharing his enthusiasm. I would like to thank first and foremost my mother, whose words of wisdom always put me back on track whenever I, for one reason or another, lost motivation. I would like to thank Julian Barbour for his interest and immense support for my work, and for reading and extensively commenting a first draft of this thesis. In the great tradition of Einstein, Bohr, Poincar´e,and others, his unwavering focus on foundational principles is an example of what physics has to gain from Natural Philosophy. He is largely responsible for inspiring us to give shape dynamics its due attention. Julian so eloquently argues for shape dynamics that even his son, Boris Barbour { who is not a physicist { was recruited to draw the figures contained in this text, and which I \borrowed". My thanks goes out to Boris as well. My immense gratitude also of course goes out to my good friends Tim and Joy (and Nate and Ruthie) Koslowski for being very gracious hosts to me more than one time. I feel very grateful for having found a collaborator so knowledgeable and yet with whom I can communicate so effortlessly. I would also like to thank Sean Gryb, without whom the \perfect storm" leading to Shape Dynamics would not have been complete. Our heated discussions have enhanced my understanding of shape dynamics a very great deal. My supervisor John Barrett, must also figure prominently in this list. My admiration for his extremely honest approach to physics (and to being a physicist) inspires me to no end. John was somehow able to make sure I was on the right track for my degree while still allowing me to follow my own interests. This is a balance I had never thought possible. Thank you all. In quantum gravity, we are all in the gutter, but some of us are looking at the stars. { Popular saying, pre-cognitively adapted by Oscar Wilde. 1 Contents 1 Introductory remarks6 1.1 A tale of two theories.............................6 1.2 The problem of time.............................7 1.3 A tale of two parts..............................8 1.4 Notation and other warnings.........................9 1.4.1 Other warnings............................9 1.4.2 Notation................................ 10 I Shape Dynamics 12 2 Introducing Shape Dynamics 13 2.1 Technical Background............................ 13 2.1.1 Constrained dynamics........................ 13 2.1.2 Hamiltonian dynamics........................ 14 2.1.3 Poisson brackets and symplectic flows................ 17 2.1.4 Geometric interpretations and the case of a true Hamiltonian... 19 2.1.5 ADM 3+1 split............................ 21 2.1.6 Constraint algebra for ADM..................... 24 2.2 A brief history of 3D conformal transformations in standard general relativity 25 2.2.1 The Weyl connection......................... 26 2.2.2 Lichnerowicz and York's contribution to the initial value problem. 26 2.2.3 Dirac's fixing of the foliation..................... 28 2.3 Barbour et al's work............................. 29 2.3.1 Poincar´e'sprinciple.......................... 29 2.3.2 Best matching............................. 31 2.3.3 Best matching in pure geometrodynamics.............. 32 2.3.4 Introduction of the conformal group and emergence of CMC by best matching............................. 34 2.4 Results and directions for Shape Dynamics................. 36 2.4.1 Brief statement of results....................... 36 3 Linking theory for Shape Dynamics 38 3.1 General trading of symmetries........................ 38 3.1.1 Linking Theories........................... 38 3.1.2 A Construction Principle for Linking Theories........... 40 3.1.3 Diagram of trading.......................... 43 2 4 Trading GR for SD: compact closed Σ case 44 4.1 Construction of the Linking Theory..................... 44 4.2 Recovering General Relativity for compact closed manifolds........ 47 4.3 Recovering Shape Dynamics for compact closed manifolds......... 47 4.3.1 Constraint Surface for Shape dynamics in compact closed manifolds. 49 4.3.2 Constructing the theory on the constraint surface......... 51 4.3.3 Properties of N0 ............................ 52 4.4 Construction of the section in T ∗Riem................... 54 4.4.1 Possibility of bypassing the linking theory construction...... 56 4.5 Comparisons with earlier work........................ 57 4.5.1 Comparison with earlier work: Barbour et al............ 57 4.5.2 Comparison with earlier work: York................. 58 5 Trading GR for SD: asymptotically flat case 59 5.1 Constructing the Linking Theory...................... 59 5.2 Recovering General Relativity in the asymptotically flat.......... 60 5.3 Recovering Shape Dynamics in the asymptotically flat case........ 60 5.3.1 Constraint Surface for Shape Dynamics............... 61 5.3.2 Comparison with Dirac's work.................... 62 5.4 Lagrangian Picture.............................. 63 6 Causal Structure and coupling to different fields 65 6.1 General criteria for coupling.......................... 65 6.1.1 Coupling to the scalar field...................... 65 6.1.2 Coupling to the electromagnetic field................. 68 6.2 Emergence of the causal structure...................... 69 6.2.1 Testing different actions with the reconstruction of the metric... 69 7 Volume expansions and Hamilton-Jacobi approach. 71 7.1 A practical way to calculate the global Hamiltonian............. 72 7.2 Large volume expansion........................... 73 7.3 Hamilton-Jacobi equation for large volume.................. 74 II Gauge Theory in Riem. 76 8 Riem as a principal fiber bundle 77 8.1 Introduction.................................. 77 8.1.1 Principal fiber bundles and gauge theory............... 78 8.2 The 3-diffeomorphism group.......................... 80 8.3 Gauge structures over Riem: Slice theorem................. 81 8.3.1 Constructing the vertical projection operator for the PFB-structure of M0 ................................. 81 8.3.2 Defining Ms, a Riemmanian structure for Ms, and an exp map.. 82 8.3.3 The orbit manifold and splittings.................. 83 8.3.4 The normal bundle to the orbits and construction of the vertical projection operator........................... 86 8.3.5 The Slice Theorem.......................... 88 8.4 The conformal bundle............................. 90 3 8.4.1 York splitting.............................. 90 9 Connection Forms 92 9.1 Introduction.................................. 92 9.2 Connection forms: basic properties..................... 93 9.2.1 An equivariant splitting........................ 94 9.2.2 The connection form obtained from the vertical projection.... 95 9.2.3 Locality of connection forms..................... 97 9.3 Construction of connection forms in M through orthogonality...... 98 9.3.1 Construction through orthogonality and the momentum constraint 98 9.3.2 Using the Fredholm Alternative................... 99 9.3.3 Equivariant metrics.........................
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