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Topics in the Foundations of General Relativity and Newtonian Gravitation Theory

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Topics in the Foundations of General Relativity and Newtonian Gravitation Theory Topics in the Foundations of General Relativity and Newtonian Gravitation Theory −1 0 +1 “530-47773_Ch00_2P.tex” — 1/23/2012 — 17:18 — page i chicago lectures in physics Robert M. Wald, series editor Henry J. Frisch, Gene R. Mazenko, and Sidney R. Nagel Other Chicago Lectures in Physics titles available from the University of Chicago Press Currents and Mesons, by J. J. Sakurai (1969) Mathematical Physics, by Robert Geroch (1984) Useful Optics, by Walter T. Welford (1991) Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics,by Robert M. Wald (1994) Geometrical Vectors, by Gabriel Weinreich (1998) Electrodynamics, by Fulvio Melia (2001) Perspectives in Computation, by Robert Geroch (2009) −1 0 +1 “530-47773_Ch00_2P.tex” — 1/23/2012 — 17:18 — page ii Topics in the Foundations of General Relativity and Newtonian Gravitation Theory David B. Malament the university of chicago press • chicago and london −1 0 +1 “530-47773_Ch00_2P.tex” — 1/23/2012 — 17:18 — page iii David B. Malament is professor in the Department of Logic and Philosophy of Science at the University of California, Irvine. He is the editor of Reading Natural Philosophy: Essays in the History and Philosophy of Science and Mathematics. The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London © 2012 by The University of Chicago All rights reserved. Published 2012. Printed in the United States of America 21201918171615141312 12345 ISBN-13: 978-0-226-50245-8 (cloth) ISBN-10: 0-226-50245-7 (cloth) Library of Congress Cataloging-in-Publication Data Malament, David B. Topics in the foundations of general relativity and Newtonian gravitation theory / David Malament. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-226-50245-8 (hardcover : alkaline paper) ISBN-10: 0-226-50245-7 (hardcover : alkaline paper) 1. Relativity (Physics) 2. Gravitation. I. Title. QC173.55.M353 2012 531'.14–dc23 2011035412 ∞ This paper meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). −1 0 +1 “530-47773_Ch00_2P.tex” — 1/23/2012 — 17:18 — page iv To Pen −1 0 +1 “530-47773_Ch00_2P.tex” — 1/23/2012 — 17:18 — page v −1 0 +1 “530-47773_Ch00_2P.tex” — 1/23/2012 — 17:18 — page vi Contents Preface ix 1. Differential Geometry 1 1.1 Manifolds 1 1.2 Tangent Vectors 7 1.3 Vector Fields, Integral Curves, and Flows 17 1.4 Tensors and Tensor Fields on Manifolds 24 1.5 The Action of Smooth Maps on Tensor Fields 35 1.6 Lie Derivatives 42 1.7 Derivative Operators and Geodesics 49 1.8 Curvature 68 1.9 Metrics 74 1.10 Hypersurfaces 94 1.11 Volume Elements 112 2. Classical Relativity Theory 119 2.1 Relativistic Spacetimes 119 2.2 Temporal Orientation and “Causal Connectibility” 128 2.3 Proper Time 136 2.4 Space/Time Decomposition at a Point and Particle Dynamics 140 2.5 The Energy-Momentum Field Tab 143 2.6 Electromagnetic Fields 152 2.7 Einstein’s Equation 159 2.8 Fluid Flow 167 2.9 Killing Fields and Conserved Quantities 175 2.10 The Initial Value Formulation 181 2.11 Friedmann Spacetimes 183 −1 0 +1 “530-47773_Ch00_2P.tex” — 1/23/2012 — 17:18 — page vii viii / contents 3. Special Topics 195 3.1 Gödel Spacetime 195 3.2 Two Criteria of Orbital (Non-)Rotation 219 3.3 A No-Go Theorem about Orbital (Non-)Rotation 237 4. Newtonian Gravitation Theory 248 4.1 Classical Spacetimes 249 4.2 Geometrized Newtonian Theory—First Version 266 4.3 Interpreting the Curvature Conditions 279 4.4 A Solution to an Old Problem about Newtonian Cosmology 288 4.5 Geometrized Newtonian Theory—Second Version 296 Solutions to Problems 309 Bibliography 343 Index 347 −1 0 +1 “530-47773_Ch00_2P.tex” — 1/23/2012 — 17:18 — page viii Preface This manuscript began life as a set of lecture notes for a two-quarter (twenty- week) course on the foundations of general relativity that I taught at the Uni- versity of Chicago many years ago. I have repeated the course quite a few times since then, both there and at the University of California, Irvine, and have over the years steadily revised the notes and added new material. Maybe now the notes can stand on their own. The course was never intended to be a systematic survey of general relativity. There are many standard topics that I do not discuss—e.g., the Schwarzschild solution and the “classic tests” of general relativity. (And I have always recom- mended that students who have not already taken a more standard course in the subject do some additional reading on their own.) My goals instead have been to (i) present the basic logical-mathematical structure of the theory with some care, and (ii) consider additional special topics that seem to me, at least, of particular interest. The topics have varied from year to year, and not all have found their way into these notes. I will mention in advance three that did. The first is “geometrized Newtonian gravitation theory,” also known as “Newton-Cartan theory.” It is now well known that one can, after the fact, reformulate Newtonian gravitation theory so that it exhibits many of the qualitative features that were once thought to be uniquely characteristic of gen- eral relativity. On reformulation, Newtonian theory too provides an account of four-dimensional spacetime structure in which (i) gravity emerges as a manifestation of spacetime curvature, and (ii) spacetime structure itself is “dynamical” in the sense that it participates in the unfolding of physics rather than being a fixed backdrop against which it unfolds. It has always seemed to me helpful to consider general relativity and this geometrized reformula- tion of Newtonian theory side by side. For one thing, one derives a sense of where Einstein’s equation “comes from.” When one reformulates the empty- −1 space field equation of Newtonian gravitation theory (i.e., Laplace’s equation 0 +1 ix “530-47773_Ch00_2P.tex” — 1/23/2012 — 17:18 — page ix x / preface ∇2φ = 0, where φ is the gravitational potential), one arrives at a constraint on = the curvature of spacetime, namely Rab 0. The latter is, of course, just what we otherwise know as (the empty-space version of ) Einstein’s equation. And, reciprocally, this comparison of the two theories side by side provides a certain insight into Newtonian physics. For example, it yields a satisfying solution (or dissolution) to an old problem about Newtonian cosmology. Newtonian theory in a standard textbook formulation seems to provide no sensible prescription for what the gravitational field should be like in the presence of a uniform mass-distribution filling all of space. (See section 4.4.) But the problem is really just an artifact of the formulation, and it disappears when one passes to the geometrized version of the theory. The basic idea of geometrized Newtonian gravitation theory is simple enough. But there are complications, and I deal with some of them in the present expanded form of the lecture notes. In particular, I present two dif- ferent versions of the theory—what I call the “Trautman version” and the “Künzle-Ehlers version”—and consider their relation to one another. I also discuss in some detail the geometric significance of various conditions on a the Riemann curvature field R bcd that enter into the formulation of these versions. A second special topic that I consider is the concept of “rotation.” It turns out to be a rather delicate and interesting question, at least in some cases, just what it means to say that a body is or is not rotating within the framework of general relativity. Moreover, the reasons for this—at least the ones I have in mind—do not have much to do with traditional controversy over “absolute vs. relative (or Machian)” conceptions of motion. Rather, they concern particular geometric complexities that arise when one allows for the possibility of space- time curvature. The relevant distinction for my purposes is not that between attributions of “relative” and “absolute” rotation, but rather that between attri- butions of rotation that can and cannot be analyzed in terms of motion (in the limit) at a point. It is the latter—ones that make essential reference to extended regions of spacetime—that can be problematic. The problem has two parts. First, one can easily think of different criteria for when an extended body is rotating. (I discuss two examples in section 3.2.) These criteria agree if the background spacetime structure is sufficiently simple—e.g., if one is working in Minkowski spacetime. But they do not agree in general. So, at the very least, attributions of rotation in general relativity can be ambiguous. A body can be rotating in one perfectly natural sense but not rotating in another, equally natural, sense. Second, circumstances can −1 arise in which the different criteria—all of them—lead to determinations of 0 +1 “530-47773_Ch00_2P.tex” — 1/23/2012 — 17:18 — page x preface / xi rotation and non-rotation that seem wildly counterintuitive. (See section 3.3.) The upshot of this discussion is not that we cannot continue to talk about rotation in the context of general relativity. Not at all. Rather, we simply have to appreciate that it is a subtle and ambiguous notion that does not, in all cases, fully answer to our classical intuitions. A third special topic that I consider is Gödel spacetime. It is not a live can- didate for describing our universe, but it is of interest because of what it tells us about the possibilities allowed by general relativity. It represents a possi- ble universe with remarkable properties.
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