UNIVERSITY OF CALIFORNIA, IRVINE Similarity and Spacetime: Studies in Intertheoretic Reduction and Physical Significance DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Philosophy by Samuel Craig Fletcher Dissertation Committee: Professor James O. Weatherall, Chair Professor Jeffrey A. Barrett Professor P. Kyle Stanford Professor Christian Wuthrich¨ (UCSD) 2014 Chapter 2 ⃝c 2014 Oxford University Press All other materials ⃝c 2014 Samuel Craig Fletcher TABLE OF CONTENTS Page LIST OF FIGURES iv LIST OF TABLES v ACKNOWLEDGMENTS vi CURRICULUM VITAE viii ABSTRACT OF THE DISSERTATION ix CHAPTER 1: Introduction: Why Spacetime and Similarity? 1 1 CHAPTER 2: Similarity, Topology, and Physical Significance in Relativity Theory 8 2 Introduction . 8 2.1 Similarity, Topology, and Physical Significance . 10 2.2 The Open Topologies . 13 2.3 Continuity in the Geometric Sense and the Compact-Open Topologies . 18 2.4 Methodological Contextualism . 23 2.5 Proofs of Propositions . 28 2.6 CHAPTER 3: On the Reduction of General Relativity to Newtonian Gravitation 35 3 Introduction . 35 3.1 Ehlers’s Frame Theory . 39 3.2 The Newtonian Limit . 45 3.3 Interpretation of the Newtonian Limit . 50 3.4 Topology and Observables . 62 3.5 Conclusions . 67 3.6 Proofs of Propositions . 69 3.7 CHAPTER 4: Global Spacetime Similarity 72 4 Introduction . 72 4.1 Geroch’s Complaint . 75 4.2 The Global Topologies . 84 4.3 Conclusions and Prospects . 98 4.4 CHAPTER 5: Conclusions and Beyond 102 5 Similarity, Topology, and Physical Significance in Relativity Theory . 104 5.1 On the Reduction of General Relativity to Newtonian Gravitation . 110 5.2 Global Spacetime Similarity . 113 5.3 Further Topics . 115 5.4 ii BIBLIOGRAPHY 118 iii LIST OF FIGURES Page FIGURE 3.1 Depicted are four vectors (labeled 0, 1, 2, and 3) at some point q, along with the light cones associated with the λ = 1 member and some λ < 1 member from a family of relativistic spacetimes with κ = λ. The vi(λ) are the speeds that an observer at q with tangent vector proportional to vector 0 would measure of a particle whose worldline at q has (co-directed) tangent vector proportional to vector i. For the figure labeled “counterfactual,” α(λ) = β(λ) = 1, and the relative speed corresponding to each vector does not change, but as λ ! 0 the speed of a light ray increases, and more vectors become timelike. For the figure labeled “factual,” on the other hand, α(λ) = 1 and β(λ) = λ−1=2, and as λ ! 0 the relative speed of each vector decreases, but the speed of a light ray stays the same. More vectors become timelike on for these choices of α and β, too, but as λ becomes sufficiently small their speeds relative to vector 0 also decrease. 56 FIGURE 3.2 Log-plot of the conformal “bump” in eq. 3.22 as a function of t- coordinate. At λ = 1=1000, it is sharply peaked near t = 0 and nearly vanishing elsewhere. 63 iv LIST OF TABLES Page TABLE 3.1 The components and conditions thereon of models of the frame the- ory, and the conditions under which these models specialize to those of Newton-Cartan theory and general relativity. 43 v ACKNOWLEDGMENTS Chapter 2 is a pre-copyedited, author-produced PDF of an article accepted for publication in the British Journal for the Philosophy of Science (BJPS) following peer review. The definitive publisher-authenticated version is available online at http://bjps.oxfordjournals.org/.I am grateful to the editors of the BJPS and to Oxford University Press for their permission to in- clude this material here. While at the University of California, Irvine, I have been supported by a Social Science Merit Fel- lowship from the School of Social Sciences. During the spring quarter of 2010, I was supported by a National Science Foundation Grant (award no. SES-0924135), as part of the project Everett Papers Archive and Commentary, administered by Jeff Barrett. During the 2010–11, 2011–12, and 2013–14 academic years, I have been supported by a National Science Foundation Graduate Research Fellowship, as well as by a National Science Foundation Doctoral Dissertation Improve- ment Grant during the lattermost academic year. I am grateful to all of these funding sources, particularly to Jeff Barrett for his encouragement and generous support, first through his grant and then in acquiring the Graduate Research Fellowship, and likewise to Jim Weatherall regarding the Dissertation Improvement Grant. The work in this dissertation has been shaped by conversations with and advice from many col- leagues, mentors, and friends. Chapter 2 has benefited from the thoughts of three anonymous referees from the British Journal for Philosophy of Science, workshop audiences at Irvine and Pittsburgh, especially Jeff Barrett, Ben Feintzeig, Dennis Lehmkuhl, David Malament, John Man- chak, John Norton, Jim Weatherall and Chris Wuthrich.¨ Questions from David, John, and Jim in particular spurred me to generalize and sharpen previous versions of the chapter’s technical results. The material of chapter 3 has evolved considerably from illuminating discussion with audience members at LMU Munich, London, and Oxford, especially Harvey Brown, Jeremy Butterfield, Neil Dewar, Roman Frigg, Eleanor Knox, Oliver Pooley, Miklos´ Redei,´ David Wallace, and Char- lotte Werndl. Jim Weatherall and Chris Wuthrich¨ deserve special gratitude for their detailed com- ments, and David Malament similarly for providing invaluable copies of the most difficult-to-find of the papers of Jurgen¨ Ehlers, upon whose work I build. My dissertation committee, and the faculty of the Department of Logic and Philosophy of Science more generally, has my warmest thanks for their unflagging enthusiasm and support, and for my transformation from a wayward intellectual outcast, straddling the interstices of the sciences and humanities, into a scholar at home in his community and work. But above all, and in more and deeper ways than any could have expected—academic, professional, and otherwise—my advisor, Jim Weatherall, has been instrumental in this process. His empathy and wisdom, even at my nadir, have been an inextinguishable beacon through the shoals for proposing, crafting, and defending a dissertation. Many others deserve special recognition here. Each has played, in their particular ways, such a pivotal role in my intellectual and professional development that I cannot imagine myself apart from their influence. Anyone familiar with scholarship in the philosophy of physics will recognize vi in this dissertation, and my work more broadly, the influence of David Malament. David has indelibly shaped my approach to scholarship, modeling with humility and grace the gold standard of clarity, force, and mastery to which I aspire. When I was an undergraduate studying physics at Princeton, Hans Halvorson answered my searching inquiry about a senior thesis project, advising me during my last year with patience and compassion. It was he who revealed the path into a kind of philosophy of which I had been unaware, encouraging me to follow it. Many years before that, David Scrofani at Staples High School in Westport, Connecticut, revealed a different and no less important path into physics. In his class, through his enthusiastic instruction, I first caught a glimpse of physics as a systematic, structured, and unified science, revealing the beautiful and intricate subtlety of our world, the search for comprehension of which drives me to this day. Finally, I would like to thank my family, especially my parents, for their unconditional love and support of my intellectual pursuits. They have provided all and more I could have ever needed to thrive, dream, and wonder. vii CURRICULUM VITAE Samuel Craig Fletcher APPOINTMENTS Assistant Professor of Philosophy From August 2015 University of Minnesota, Twin Cities Marie Curie Fellow July 2014 – August 2015 Munich Center for Mathematical Philosophy Ludwig-Maximilians-Universitat,¨ Munchen¨ EDUCATION Doctor of Philosophy in Philosophy 2014 University of California, Irvine Irvine, California Master of Arts in Philosophy 2012 University of California, Irvine Irvine, California Master of Science in Statistics 2012 University of California, Irvine Irvine, California Bachelor of Arts in Physics 2008 Princeton University Princeton, New Jersey SELECTED HONORS AND AWARDS Order of Merit Award for Outstanding Graduate Student Scholarship 2014 School of Social Sciences, University of California, Irvine Graduate Research Fellowship 2010–14 National Science Foundation Robert K. Clifton Memorial Book Prize 2009 University of Western Ontario Maria Goeppert-Mayer Studentship 2008 American Physical Society National Merit Scholarship 2004–08 National Merit Scholarship Corporation viii ABSTRACT OF THE DISSERTATION Similarity and Spacetime: Studies in Intertheoretic Reduction and Physical Significance By Samuel Craig Fletcher Doctor of Philosophy in Philosophy University of California, Irvine, 2014 Professor James O. Weatherall, Chair This dissertation explores in particular a few ways in which similarity bears on general relativity, our best physical theory of space and time. The first way concerns the role of the concept of “phys- icality” in theorizing and using spacetime models, which manifests in at least two different ways. One is whether particular models are (or have properties that are) “physically unreasonable”—they are deemed to be pathological or otherwise artifactual, and so must excluded from the theory. An- other is whether particular properties of those models are “physically significant”—warranted to be inferred about the properties of a physical system they represent. This distinction, which I draw in more detail in chapter 2, separates these two senses of physicality, the former modal-metaphysical and the latter inferential-epistemic. It is this latter that more intimately involves the notion of sim- ilarity, particular notions of which can be encoded through the use of topology. I argue then that there is no canonical topology on the models of general relativity. Rather, the choice of topology must be nontrivially dependent on the context of investigation.