University of Massachusetts Amherst ScholarWorks@UMass Amherst Doctoral Dissertations Dissertations and Theses July 2016 Physical Geometry James P. Binkoski University of Massachusetts Amherst Follow this and additional works at: https://scholarworks.umass.edu/dissertations_2 Part of the Metaphysics Commons, and the Philosophy of Science Commons Recommended Citation Binkoski, James P., "Physical Geometry" (2016). Doctoral Dissertations. 625. https://doi.org/10.7275/8441359.0 https://scholarworks.umass.edu/dissertations_2/625 This Open Access Dissertation is brought to you for free and open access by the Dissertations and Theses at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. PHYSICAL GEOMETRY A Dissertation Presented by JAMES BINKOSKI Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY May 2016 Philosophy c Copyright by James Binkoski 2016 All Rights Reserved PHYSICAL GEOMETRY A Dissertation Presented by JAMES BINKOSKI Approved as to style and content by: Bradford Skow, Chair Phillip Bricker, Member Christopher Meacham, Member Lorenzo Sorbo, Member Joseph Levine, Department Chair Philosophy DEDICATION To Jen. ACKNOWLEDGMENTS Most of what I know about how to do philosophy well, I learned from Phil Bricker, Chris Meacham, and Brad Skow. I thank them for their guidance and support over a period of many years. I consider myself incredibly fortunate to have studied with such top-notch philosophers. I am especially grateful to Brad Skow for agreeing to work with me after his move from UMass to MIT in 2007. From 2008-2011, I was a visiting graduate stu- dent at MIT. I thank the philosophy department at MIT for its generous support and Nina Emery for her great friendship. My first real exposure to the general theory of relativity was in a graduate course taught by Lorenzo Sorbo. I thank Lorenzo for making me feel welcome in the physics department and for agreeing to serve as the outside examiner on this dissertation. Finally, a note of love to my daughter Kate and to my son Ian who probably delayed the completion of this dissertation by a good year or two. v ABSTRACT PHYSICAL GEOMETRY MAY 2016 JAMES BINKOSKI B.A., BOSTON COLLEGE M.A., UNIVERSITY OF CHICAGO Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Bradford Skow All physical theories, from classical Newtonian mechanics to relativistic quan- tum field theory, entail propositions concerning the geometric structure of space- time. To give an example, the general theory of relativity entails that spacetime is curved, smooth, and four-dimensional. In this dissertation, I take the structural commitments of our theories seriously and ask: how is such structure instantiated in the physical world? Mathematically, a property like ‘being curved’ is perfectly well-defined—we know what it means for a mathematical space to be curved. But what could it mean to say that the physical world is curved? Call this the problem of physical geometry. The problem of physical geometry is a plea for foundations—a request for fun- damental truth conditions for physical-geometric propositions. My chief claim is that only a substantival theory of spacetime—a theory according to which space- vi time is an entity in its own right, existing over and above the material content of the world—can supply the necessary truth conditions. vii TABLE OF CONTENTS Page ACKNOWLEDGMENTS .................................................. v ABSTRACT .............................................................. vi LIST OF FIGURES ....................................................... xi CHAPTER INTRODUCTION......................................................... 1 1. CLASSICAL SPACETIMES ............................................ 14 1.1 Leibnizian Spacetime . 14 1.2 Comments . 19 1.2.1 Modal Relationalism . 19 1.2.2 Completeness . 25 1.2.3 Traditional Problems . 29 1.3 Galilean Spacetime. 32 1.4 Comments . 34 2. GALILEAN RELATIONALISM ........................................ 36 2.1 The View . 36 2.2 Inverse Cube Force Laws . 39 2.2.1 Warm-up . 40 2.2.2 Model 1: zero angular momentum . 42 2.2.3 Model 2: non-zero angular momentum . 44 2.3 Supervenience . 47 2.4 Incompleteness . 50 2.5 Conclusion . 54 viii 3. ABSOLUTE ACCELERATION ......................................... 56 3.1 No Well-Posed Initial Value Problem . 57 3.2 Curvature . 59 3.3 Quantitative Galilean Relationalism . 61 3.4 Inertial Structure. 65 4. REGULARITY RELATIONALISM ..................................... 66 4.1 Structure, Ontology, and Mechanics . 67 4.2 The Regularity Account of Relational Spacetime . 72 4.3 Inertial Structure. 74 4.3.1 Frames of Reference . 78 4.4 First Problem: Laws and Inertial Structure . 82 4.4.1 A Newtonian Equivalence Principle . 83 4.4.2 Newton-Cartan Spacetime . 90 4.5 Second Problem: General Covariance . 94 4.6 Third Problem: The Shape of Space. 96 4.6.1 Supervenience . 97 4.6.2 Simplicity . 100 5. DYNAMIC FOUNDATIONALISM ................................... 106 5.1 The Dynamical Approach . 108 5.2 Submetric Structure and the Laws . 113 5.3 Ontological Implications . 118 5.4 Coordinating Dynamical Symmetries . 119 5.5 Conclusion . 126 5.6 Appendix: Explanation and Dependence . 127 5.6.1 Observations . 129 5.6.2 Dependence . 134 5.6.3 Evidence . 138 6. FIELDS .............................................................. 142 6.1 Introduction . 142 6.2 Field Theories . 143 6.3 Interpreting GR . 146 6.4 Physical Geometry . 152 6.5 The Modified Shift Argument . 159 ix 6.6 Leibniz Equivalence . 168 6.7 Conclusion . 169 REFERENCES........................................................... 171 x LIST OF FIGURES Figure Page 2.1 The initial conditions of model 2. 41 2.2 The worldlines of the particles in model 1. 44 2.3 The spatial paths of the particles in model 2. 46 2.4 The worldlines of the particles in models 3-5. 53 3.1 Two curves which differ in curvature at q. As t ! 0, both angles approach 180, with the speed of approach corresponding to the degree of curvature. 64 xi INTRODUCTION All physical theories—from classical Newtonian mechanics to relativistic quan- tum field theory—entail propositions concerning the geometric structure of space- time. This dissertation concerns the ultimate truth conditions for such proposi- tions, and it defends a substantival theory of spacetime according to which space- time is a part of the fundamental ontology of the physical world. To start, consider some of the structural commitments of classical Newtonian mechanics. There are, of course, multiple ways of formulating Newton’s theory. But if your aim is to get clear on what the theory has to say about the space- time structure of the physical world, then you can do no better than the approach taken below. On this approach, the details of which are spelled out in chapter 1, classical Newtonian mechanics is identifiable with its set of spacetime models, ab each of which is a mathematical structure of the form (M; tab; h ; r). Each such model assumes as its basic object a smooth, differentiable manifold M with the lo- cal smoothness structure of R4. Spatiotemporal structure is then added by means of the introduction of smooth tensor fields. In the case of classical Newtonian me- chanics, the added structure induces a division of the manifold into space and time, where space and time are distinct to the point that each demands represen- tation by its own tensor field. Consequently, the theory posits one tensor field, the ab field tab, to represent the structure of time and another, the field h , to represent the structure of space. Beyond this, the theory assumes an affine connection r, the function of which is to ground a distinction between those spacetime curves which are straight and those which are not. 1 As a second example, consider the general theory of relativity (GR), which car- ries its own set of structural commitments. Its models are mathematical structures of the form (M; gab;Tab). As in classical Newtonian mechanics, the theory assumes as its basic object a smooth, differentiable manifold. But whereas classical New- tonian mechanics posits three different fields on M, each representing a different layer of spatiotemporal structure, the general theory of relativity posits a single metric tensor field gab. Moreover, the metric tensor field in GR is a dynamical ele- ment of the theory insofar as it can vary with changes in the distribution of mass and energy in spacetime. That distribution is represented mathematically by the stress-energy tensor field Tab, and its relation to the metric tensor field is captured by the Einstein Field Equation. I take this kind of talk of structure seriously. I take it that the physical world is structured in a determinate, mind-independent way. I take it that the aim of science is to produce truth-conditioned descriptions of the physical world. And I take it that elucidation of the world’s spacetime structure is an important part of the scientific enterprise. A basic assumption behind the work that follows, then, is that the physical world is a geometrically structured object—that it has a determinate spacetime geometry. My question is: How is such structure instantiated in the physical world? Without question, our best guide to the physical world is through physical the- ory. Consequently, the project undertaken in this dissertation is one concerning the interpretation of physical theory. The aim is to engage with the details of a handful of physical theories with the aim of understanding what the world must be like in order for those theories to be true. On this approach, interpretation is the simple task of attempting to spell out what a theory says about the world; in the case of fundamental physics, what it says concerning the world’s most basic objects, properties, and relations. But don’t confuse ‘simple’ with ‘easy’. For example, 2 consider some of the well-known difficulties involved in interpreting quantum mechanics.