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Foundations of Massive Gravity FOUNDATIONS OF MASSIVE GRAVITY ANDREW MATAS Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Dissertation Advisor: Prof. Claudia de Rham Department of Physics CASE WESTERN RESERVE UNIVERSITY August 2016 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the dissertation of Andrew Matas candidate for the degree of Doctor of Philosophy∗ Committee Chair Claudia de Rham Committee Member Stacy McGaugh Committee Member Glenn Starkman Committee Member Andrew J. Tolley Date of Defense May 19th, 2016 ∗We also certify that written approval has been obtained for any proprietary material contained therein. i Contents List of Figures vii 1 Introduction 1 1.1 Notions of gravity . .1 1.1.1 Gravity is an attractive force . .2 1.1.2 Gravity is curved space-time . .4 1.1.3 Gravity is a fundamental interaction . .5 1.2 A massive graviton and its implications . .9 1.3 Building models of massive gravity . 16 1.3.1 Stability . 17 1.3.2 Strong-coupling and continuity . 19 1.3.3 Extra dimensions and massive gravity . 22 1.4 Overview of dissertation . 24 2 Basics of Massive Gravity 29 2.1 Setting the stage . 31 2.1.1 Consistent m ! 0 limit of a massive spin-1 . 31 ii 2.1.2 Ghost modes . 33 2.1.3 Massless spin-2 . 35 2.2 Toward a theory of massive gravity . 37 2.2.1 Non-interacting massive spin-2 field . 37 2.3 Non-linear formulations of massive gravity . 41 2.3.1 Ghost-free potential . 43 2.3.2 Vielbein formulation . 44 2.3.3 St¨uckelberg for gravity . 45 2.4 Absence of the Boulware-Deser mode . 47 2.4.1 ADM form . 48 2.4.2 Counting degrees of freedom . 49 2.4.3 Mini-superspace . 51 2.5 Minkowski decoupling limit . 52 2.5.1 Interaction scales . 54 2.5.2 Explicit form of decoupling limit . 56 2.5.3 Galileon interactions . 58 2.6 Realization of the Vainshtein mechanism . 59 2.6.1 Vainshtein mechanism . 59 2.6.2 Boulware-Deser ghost . 61 2.6.3 Vainshtein mechanism for stars . 62 2.7 Bi-gravity and multi-gravity . 65 2.8 Summary . 66 iii 3 Dimensional deconstruction 68 3.1 Deconstruction of a scalar . 71 3.2 Deconstruction for gravity . 74 3.3 Deconstruction of the metric . 76 3.4 Consistent deconstruction procedure . 79 3.4.1 Different discretization of derivative . 84 3.4.2 Multi-gravity . 85 3.5 Strong-coupling from deconstruction . 86 3.5.1 Strong-coupling scale at large N ................. 87 3.5.2 The 5D origin of the strong-coupling scale . 91 3.5.3 Gauge fixed continuum theory . 91 3.5.4 Generic gauge for continuum theory . 92 3.6 Outlook . 93 4 Kinetic interactions in massive gravity 95 4.1 Kinetic term for massive spin 1 . 98 4.2 Candidate kinetic interactions . 100 4.2.1 Leading order interactions . 101 4.2.2 Non-linear St¨uckelberg decomposition . 103 4.2.3 Non-linear completions . 104 4.3 Deconstruction-inspired ansatz . 106 4.4 Systematic argument . 109 4.4.1 St¨uckelberg criteria . 110 iv 4.4.2 Decoupling limits . 112 4.4.3 Algorithm . 114 4.4.4 Quadratic order . 116 4.4.5 Cubic order . 116 4.4.6 Quartic order . 119 4.4.7 Extension to higher order . 122 4.5 Relationship to the coupling to matter . 124 4.5.1 Effective vielbein coupling . 124 4.5.2 Effective metric . 126 4.6 Outlook . 128 5 Galileon radiation from binary systems 130 5.1 Galileons with different scales . 133 5.2 Perturbations around a spherically symmetric background . 134 5.2.1 Keplerian source . 134 5.2.2 Background Galileon field . 137 5.2.3 Perturbations for Galileon . 138 5.3 Formula for the power emission . 141 5.4 Power emission for the cubic Galileon . 144 5.5 Power emission for the general Galileon . 147 5.5.1 Mode functions . 148 5.5.2 General form for the power . 149 5.5.3 Power in quadrupole . 151 v 5.5.4 Breakdown of perturbation theory . 152 5.5.5 Hierarchy of masses . 157 5.6 Hierarchy between two strong coupling scales . 159 5.6.1 Power emission . 160 5.6.2 Quadrupole radiation . 162 5.7 Outlook . 162 6 Conclusions 165 6.1 Summary . 165 6.2 Outlook . 167 A Lovelock interactions 169 vi List of Tables 4.1 Coefficients for ghost-free kinetic term at quartic order . 120 vii List of Figures 1.1 Newtonian T-shirt. .3 1.2 Fundamental Forces Comic. .5 1.3 Scattering processes. .7 1.4 Gravitational wave polarizations. 11 1.5 Hulse-Taylor pulsar. 12 1.6 Ghost instabilities. 17 1.7 Vainshtein radius. 20 1.8 Kaluza-Klein compactification. 22 viii Foundations of Massive Gravity ANDREW MATAS General Relativity (GR) is a relativistic theory of gravity which has a large number of theoretical and observational successes. From the perspective of quantum field theory, GR can be thought of as a theory of a massless spin-2 particle called the graviton. It is a fundamental question to ask how the graviton behaves if it has a small but non-zero mass. In this dissertation I shall study the recently constructed theory of ghost-free massive gravity, which avoids the pernicious Boulware-Deser ghost that had thwarted previous attempts to study massive gravity. In Chapter 1 I will give a broad overview of the context, motivation, and model- building issues underlying massive gravity. Then in Chapter 2 I will give a detailed pedagogical introduction to ghost-free massive gravity focusing on concepts that will be used throughout the remainder of the work. In Chapter 3 I will derive the ghost-free structure of massive gravity from an ex- tra dimensional perspective through a process known as Dimensional Deconstruction. Before my work it had been an open question whether Deconstruction could be con- sistently applied to gravity. The key insight relies on using the elegant formulation of General Relativity in terms of the vielbein. Inspired by Deconstruction, in Chapter 4 I will discuss the possibility of non- standard kinetic interactions in massive gravity. I will show that the only consistent derivative interactions for a massive spin-2 particle must be the same as in GR. This is remarkable because there is no known symmetry reason for this to be the case, ix since massive gravity breaks diffeomorphism invariance. Finally, in Chapter 5, in order to connect with observations I shall consider the radiation emitted from binary systems in Galileon theories, which are scalar theories that can mimic the behavior of a massive graviton. This work extends the under- standing of the Vainshtein screening mechanism into a time-dependent situation. x Chapter 1 Introduction In this work I will be concerned with an extension of General Relativity (GR) known as massive gravity. Before diving into the details of this theory, it is useful to recall the basic ideas underlying our modern understanding of gravity. 1.1 Notions of gravity On the one hand, gravity is a familiar phenomenon. Gravity makes apples fall from trees and is responsible for the tides. On the other hand, gravity is mysterious. Gravity connects us to the universe. Newton taught us that the same force that pulls us to the Earth also holds the Earth to the Sun, the Sun to the Galaxy, the Galaxy to the Local Group, the Local Group to the Local Supercluster, and that holds galaxy clusters together in an enormous network of filaments mapped by large scale structure. Gravity collapses stars into black holes, regions so dense that light cannot escape. Under the influence of gravity, the universe 1 expands. The expansion rate has recently been observed to be accelerating, and if this acceleration continues forever then ultimately all matter and energy will be diluted away to almost nothing. In this opening section I will give three different perspectives on the nature of gravity. These different pictures look quite different, but all of these pictures are useful and relevant in different regimes, and whenever two or more are valid they agree in their quantitative predictions. 1.1.1 Gravity is an attractive force The theory of gravity published by Isaac Newton in 1687 states that gravity is an at- tractive force between massive objects. Between any two objects is an attractive force proportional to the masses of the objects, and inversely proportional to the square of the distance between them. The proportionality constant is Newton's gravitational constant G, which sets the strength of the gravitational force. These statements can be summarized in an equation so simple and profound that it can be found on T-shirts (see Figure 1.1). From this one (apparently) simple equation, one can make an enormous array of precise predictions: planets orbit the sun in elliptical orbits and a bowling ball and a basketball dropped simultaneously from the top of the Tower of Pisa will hit the ground at the same time. Despite its success and continued utility, Newton's theory of gravity is unsatisfying in at least one respect. In Newton's theory the gravitational force is instantaneous: 2 Figure 1.1: This T-shirt illustrates the Newtonian view of gravity as a force. Image from http://www.nerdytshirt.com/physics-tshirts.html. the force between two objects at a given time depends on the distance between the objects at that same time. This raises puzzles in extreme situations with fast moving objects. For example, if the sun were to suddenly disappear1, would this information be transmitted to earth instantaneously, or would the information take a finite time to propagate? Intuitively we might say that there should be a finite time.
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