Topics In The Foundations Of General Relativity And .
Topics in the Foundations ofGeneral Relativity and NewtonianGravitation Theory 10 1“530-47773 Ch00 2P.tex” — 1/23/2012 — 17:18 — page i
chicago lectures in physicsRobert M. Wald, series editorHenry J. Frisch, Gene R. Mazenko, and Sidney R. NagelOther Chicago Lectures in Physics titles available from the University of Chicago PressCurrents 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, byRobert M. Wald (1994)Geometrical Vectors, by Gabriel Weinreich (1998)Electrodynamics, by Fulvio Melia (2001)Perspectives in Computation, by Robert Geroch (2009) 10 1“530-47773 Ch00 2P.tex” — 1/23/2012 — 17:18 — page ii
Topics in theFoundations of GeneralRelativity and NewtonianGravitation TheoryDavid B. Malamentthe university of chicago press chicago and london 10 1“530-47773 Ch00 2P.tex” — 1/23/2012 — 17:18 — page iii
David B. Malament is professor in the Department of Logic and Philosophyof Science at the University of California, Irvine. He is the editor of ReadingNatural Philosophy: Essays in the History and Philosophy of Science andMathematics.The University of Chicago Press, Chicago 60637The University of Chicago Press, Ltd., London 2012 by The University of ChicagoAll rights reserved. Published 2012.Printed in the United States of America21 20 19 18 17 16 15 14 13 1212345ISBN-13: 978-0-226-50245-8 (cloth)ISBN-10: 0-226-50245-7 (cloth)Library of Congress Cataloging-in-Publication DataMalament, David B.Topics in the foundations of general relativity and Newtonian gravitationtheory / 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 2012531'.14–dc232011035412 This paper meets the requirements of ANSI/NISO Z39.48-1992(Permanence of Paper). 10 1“530-47773 Ch00 2P.tex” — 1/23/2012 — 17:18 — page iv
To Pen 10 1“530-47773 Ch00 2P.tex” — 1/23/2012 — 17:18 — page v
10 1“530-47773 Ch00 2P.tex” — 1/23/2012 — 17:18 — page vi
ContentsPreface1.ixDifferential Geometry 11.1Manifolds1.2Tangent Vectors1.3Vector Fields, Integral Curves, and Flows171.4Tensors and Tensor Fields on Manifolds241.5The Action of Smooth Maps on Tensor Fields1.6Lie Derivatives1.7Derivative Operators and Geodesics1.8Curvature1.9Metrics174968741.10 Hypersurfaces941.11 Volume Elements2.3542112Classical Relativity Theory 1192.1Relativistic Spacetimes2.2Temporal Orientation and “Causal Connectibility”2.3Proper Time2.4Space/Time Decomposition at a Point and Particle Dynamics2.5The Energy-Momentum Field Tab2.6Electromagnetic Fields2.7Einstein’s Equation2.8Fluid Flow2.9Killing Fields and Conserved Quantities1191281361401431521591672.10 The Initial Value Formulation2.11 Friedmann Spacetimes175181183“530-47773 Ch00 2P.tex” — 1/23/2012 — 17:18 — page vii 10 1
viii / contents3.4.Special Topics 1953.1Gödel Spacetime3.2Two Criteria of Orbital (Non-)Rotation3.3A No-Go Theorem about Orbital (Non-)Rotation195219237Newtonian Gravitation Theory 2484.1Classical Spacetimes4.2Geometrized Newtonian Theory—First Version4.3Interpreting the Curvature Conditions4.4A Solution to an Old Problem about Newtonian Cosmology4.5Geometrized Newtonian Theory—Second Version249266279288296Solutions to Problems 309Bibliography 343Index 347 10 1“530-47773 Ch00 2P.tex” — 1/23/2012 — 17:18 — page viii
PrefaceThis manuscript began life as a set of lecture notes for a two-quarter (twentyweek) course on the foundations of general relativity that I taught at the University of Chicago many years ago. I have repeated the course quite a few timessince then, both there and at the University of California, Irvine, and have overthe years steadily revised the notes and added new material. Maybe now thenotes 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 Schwarzschildsolution and the “classic tests” of general relativity. (And I have always recommended that students who have not already taken a more standard course inthe subject do some additional reading on their own.) My goals instead havebeen to (i) present the basic logical-mathematical structure of the theory withsome 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 havefound 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 thequalitative features that were once thought to be uniquely characteristic of general relativity. On reformulation, Newtonian theory too provides an accountof four-dimensional spacetime structure in which (i) gravity emerges as amanifestation of spacetime curvature, and (ii) spacetime structure itself is“dynamical” in the sense that it participates in the unfolding of physics ratherthan being a fixed backdrop against which it unfolds. It has always seemedto me helpful to consider general relativity and this geometrized reformulation of Newtonian theory side by side. For one thing, one derives a sense ofwhere Einstein’s equation “comes from.” When one reformulates the emptyspace field equation of Newtonian gravitation theory (i.e., Laplace’s equationix“530-47773 Ch00 2P.tex” — 1/23/2012 — 17:18 — page ix 10 1
x / preface 2 φ 0, where φ is the gravitational potential), one arrives at a constraint onthe curvature of spacetime, namely Rab 0. The latter is, of course, just whatwe otherwise know as (the empty-space version of ) Einstein’s equation. And,reciprocally, this comparison of the two theories side by side provides a certaininsight into Newtonian physics. For example, it yields a satisfying solution (ordissolution) to an old problem about Newtonian cosmology. Newtonian theoryin a standard textbook formulation seems to provide no sensible prescriptionfor what the gravitational field should be like in the presence of a uniformmass-distribution filling all of space. (See section 4.4.) But the problem isreally just an artifact of the formulation, and it disappears when one passes tothe geometrized version of the theory.The basic idea of geometrized Newtonian gravitation theory is simpleenough. But there are complications, and I deal with some of them in thepresent expanded form of the lecture notes. In particular, I present two different versions of the theory—what I call the “Trautman version” and the“Künzle-Ehlers version”—and consider their relation to one another. I alsodiscuss in some detail the geometric significance of various conditions onthe Riemann curvature field Ra bcd that enter into the formulation of theseversions.A second special topic that I consider is the concept of “rotation.” It turnsout to be a rather delicate and interesting question, at least in some cases, justwhat it means to say that a body is or is not rotating within the framework ofgeneral relativity. Moreover, the reasons for this—at least the ones I have inmind—do not have much to do with traditional controversy over “absolute vs.relative (or Machian)” conceptions of motion. Rather, they concern particulargeometric complexities that arise when one allows for the possibility of spacetime curvature. The relevant distinction for my purposes is not that betweenattributions of “relative” and “absolute” rotation, but rather that between attributions of rotation that can and cannot be analyzed in terms of motion (in thelimit) at a point. It is the latter—ones that make essential reference to extendedregions of spacetime—that can be problematic.The problem has two parts. First, one can easily think of different criteriafor when an extended body is rotating. (I discuss two examples in section3.2.) These criteria agree if the background spacetime structure is sufficientlysimple—e.g., if one is working in Minkowski spacetime. But they do not agreein general. So, at the very least, attributions of rotation in general relativitycan be ambiguous. A body can be rotating in one perfectly natural sense butnot rotating in another, equally natural, sense. Second, circumstances canarise in which the different criteria—all of them—lead to determinations of“530-47773 Ch00 2P.tex” — 1/23/2012 — 17:18 — page x 10 1
preface / xirotation and non-rotation that seem wildly counterintuitive. (See section 3.3.)The upshot of this discussion is not that we cannot continue to talk aboutrotation in the context of general relativity. Not at all. Rather, we simply haveto 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 candidate for describing our universe, but it is of interest because of what it tellsus about the possibilities allowed by general relativity. It represents a possible universe with remarkable properties. For one thing, the entire materialcontent of the Gödel universe is in a state of uniform, rigid rotation (according to any reasonable criterion of rotation). For another, light rays and freetest particles in it exhibit a kind of boomerang effect. Most striking of all, itadmits closed timelike curves that cannot be “unrolled” by passing to a covering space (because the underlying manifold is simply connected). In section3.1, I review these basic features of Gödel spacetime and, in an appendix tothat section, I discuss how one can go back and forth between an intrinsiccharacterization of the Gödel metric and two different coordinate expressionsfor it.These three special topics are treated in chapters 3 and 4. Much of this material has been added over the years. The original core of the lecture notes—thereview of the basic structure of general relativity—is to be found in chapter 2.Chapter 1 offers a preparatory review of basic differential geometry. It hasnever been my practice to work through all this material in class. I have limitedmyself there to “highlights” and general remarks. But I have always distributedthe notes so that students with sufficient interest can do further reading ontheir own. On occasion, I have also run a separate “problem session” and usedit for additional coaching on differential geometry. (A number of problems,with solutions, are included in the present version of the lecture notes.) Isuggest that readers make use of chapter 1 as seems best to them—as a textto be read from the beginning, as a reference work to be consulted whenparticular topics arise in later chapters, as something in between, or not at all.I would like to use this occasion to thank a number of people who havehelped me over the years to learn and better understand general relativity.I could produce a long list, but the ones who come first, at least, are JohnEarman, David Garfinkle, Robert Geroch, Clark Glymour, Howard Stein, andRobert Wald. I am particularly grateful to Bob1 and Bob2 for allowing thisinterloper from the Philosophy Department to find a second home in theChicago Relativity Group. Anyone familiar with their work, both research andexpository writings, will recognize their influence on this set of lecture notes.“530-47773 Ch00 2P.tex” — 1/23/2012 — 17:18 — page xi 10 1
xii / prefaceErik Curiel, Sam Fletcher, David Garfinkle, John Manchak, and JimWeatherall have my thanks, as well, for the comments and corrections theyhave given me on earlier drafts of the manuscript.Matthias Kretschmann was good enough some years ago to take my handwritten notes on differential geometry and set them in TEX. I took over afterthat, but I might not have started without his push.Finally, Pen Maddy has helped me to believe that this project wasworth completing. I shall always be grateful to her for her support andencouragement. 10 1“530-47773 Ch00 2P.tex” — 1/23/2012 — 17:18 — page xii
1DIFFERENTIAL GEOMETRY1.1. ManifoldsWe assume familiarity with the basic elements of multivariable calculus andpoint set topology. The following notions, in particular, should be familiar.Rn (for n 1) is the set of all n-tuples of real numbers x (x 1 , . . . , x n ).The Euclidean inner product (or “dot product”) on Rn is given by x · y x 1 y 1 . . . x n y n . It determines a norm, x x · x. Given a point x Rn and areal number 0, B (x) is the open ball in Rn centered at x with radius —i.e.,B (x) {y : y x }. Clearly, x belongs to B (x) for every 0. A subsetS of Rn is open if, for all points x in S, there is an 0 such that B (x) S.This determines a topology on Rn . Given m, n 1, and a map f : O Rmfrom an open set O in Rn to Rm , f is smooth (or C ) if all its mixed partialderivatives (to all orders) exist and are continuous at every point in O.A smooth n-dimensional manifold (n 1) can be thought of as a point set towhich has been added the “local smoothness structure” of Rn . Our discussionof differential geometry begins with a more precise characterization.1Let M be a non-empty set. An n-chart on M is a pair (U, ϕ) where U isa subset of M and ϕ : U Rn is an injective (i.e., one-to-one) map from Uinto Rn with the property that ϕ[U] is an open subset of Rn . (Here ϕ[U] isthe image set {ϕ( p) : p U}.) Charts, also called “coordinate patches,” are themechanism with which one induces local smoothness structure on the set M.To obtain a smooth n-dimensional manifold, we must lay down sufficientlymany n-charts on M to cover the set and require that they be, in an appropriatesense, compatible with one another.Let (U1 , ϕ1 ) and (U2 , ϕ2 ) be n-charts on M. We say the two are compatible ifeither the intersection set U U1 U2 is empty or the following conditionshold:1. In this section and several others in chapter 1, we follow the basic lines of the presentationin Geroch [22].1“530-47773 Ch01 2P.tex” — 1/23/2012 — 17:18 — page 1 10 1
2 / differential geometryFigure 1.1.1. Two n-charts (U1 , ϕ1 ) and (U2 , ϕ2 ) on M with overlapping domains.(1) ϕ1 [U] and ϕ2 [U] are both open subsets of Rn .(2) ϕ1 ϕ2 1 : ϕ2 [U] Rn and ϕ2 ϕ1 1 : ϕ1 [U] Rn are both smooth.(Notice that the second makes sense since ϕ1 [U] and ϕ2 [U] are open subsetsof Rn and we know what it means to say that a map from an open subset ofRn to Rn is smooth. See figure 1.1.1.)The relation of compatibility between n-charts on a given set is reflexiveand symmetric. But it need not be transitive and, hence, not an equivalencerelation. For example, consider the following three 1 charts on R:C1 (U1 , ϕ1 ), with U1 ( 1, 1) and ϕ1 (x) xC2 (U2 , ϕ2 ), with U2 (0, 1) and ϕ2 (x) xC3 (U3 , ϕ3 ), with U3 ( 1, 1) and ϕ3 (x) x 3Pairs C1 and C2 are compatible, and so are pairs C2 and C3 . But C1 and C3 arenot compatible, because the map ϕ1 ϕ3 1 : ( 1, 1) R is not smooth (oreven just differentiable) at x 0.We now define a smooth n-dimensional manifold (or, in brief, an n-manifold)(n 1) to be a pair (M, C ) where M is a non-empty set and C is a set of n-chartson M satisfying the following four conditions.(M1) Any two n-charts in C are compatible.(M2) The (domains of the) n-charts in C cover M; i.e., for every p M, thereis an n-chart (U, ϕ) in C such that p U.(M3) (Hausdorff condition) Given distinct points p1 and p2 in M, there exist ncharts (U1 , ϕ1 ) and (U2 , ϕ2 ) in C such that pi Ui for i 1, 2 and U1 U2is empty.(M4) C is maximal in the sense that any n-chart on M that is compatiblewith every n-chart in C belongs to C .“530-47773 Ch01 2P.tex” — 1/23/2012 — 17:18 — page 2 10 1
differential geometry / 3(M1) and (M2) are certainly conditions one would expect. (M3) is included,following standard practice, simply to rule out pathological examples (thoughone does, sometimes, encounter discussions of “non-Hausdorff manifolds”).(M4) builds in the requirement that manifolds do not have “extra structure” in the form of distinguished n-charts. (For example, we can think of thepoint set Rn as carrying a single [global] n-chart. In the transition from the pointset Rn to the n-manifold Rn discussed below, this “extra structure” is washedout.)Because of (M4), it might seem a difficult task to specify an n-dimensionalmanifold. (How is one to get a grip on all the different n-charts that make upa maximal set of such?) But the following proposition shows that the specification need not be difficult. It suffices to come up with a set of n-charts onthe underlying set satisfying (M1), (M2), and (M3), and then simply throw inwholesale all other compatible n-charts.P R O P O S I T I O N 1.1.1. Let M be a non-empty set, let C0 be a set of n-charts on Msatisfying conditions (M1), (M2), and (M3), and let C be the set of all n-charts on Mcompatible with all the n-charts in C0 . Then (M, C ) is an n-manifold; i.e., C satisfiesall four conditions.Proof. Since C0 satisfies (M1), C0 is a subset of C . It follows immediately thatC satisfies (M2), (M3), and (M4). Only (M1) requires some argument. LetC1 (U1 , ϕ1 ) and C2 (U2 , ϕ2 ) be any two n-charts compatible with alln-charts in C0 . We show that they are compatible with one another. Wemay assume that the intersection U1 U2 is non-empty, since otherwisecompatibility is automatic.First we show that ϕ1 [U1 U2 ] is open. (A parallel argument establishesthat ϕ2 [U1 U2 ] is open.) Consider an arbitrary point of ϕ1 [U1 U2 ]. It is ofthe form ϕ1 ( p) for some point p U1 U2 . Since C0 satisfies (M2), there existsan n-chart C (U, ϕ) in C0 whose domain contains p. So p U U1 U2 .Since C is compatible with both C1 and C2 , ϕ[U U1 ] and ϕ[U U2 ] areopen sets in Rn , and the mapsϕ1 ϕ 1 : ϕ[U U1 ] Rn ,ϕ2 ϕ 1 : ϕ[U U2 ] Rn ,ϕ ϕ1 1 : ϕ1 [U U1 ] Rn ,ϕ ϕ2 1 : ϕ2 [U U2 ] Rn ,are all smooth (and therefore continuous). Now ϕ[U U1 U2 ] is open, sinceit is the intersection of open sets ϕ[U U1 ] and ϕ[U U2 ]. (Here we use“530-47773 Ch01 2P.tex” — 1/23/2012 — 17:18 — page 3 10 1
4 / differential geometrythe fact that ϕ is injective.) So ϕ1 [U U1 U2 ] is open, since it is the preimage of ϕ[U U1 U2 ] under the continuous map ϕ ϕ1 1 . But, clearly,ϕ1 ( p) ϕ1 [U U1 U2 ], and ϕ1 [U U1 U2 ] is a subset of ϕ1 [U1 U2 ]. Sowe see that our arbitrary point ϕ1 ( p) in ϕ1 [U1 U2 ] is contained in an opensubset of ϕ1 [U1 U2 ]. Thus ϕ1 [U1 U2 ] is open.Next we show that the map ϕ2 ϕ1 1 : ϕ1 [U1 U2 ] Rn is smooth. (A parallel argument establishes that ϕ1 ϕ2 1 : ϕ2 [U1 U2 ] Rn is smooth.) Forthis it suffices to show that, given our arbitrary point ϕ1 ( p) in ϕ1 [U1 U2 ], therestriction of ϕ2 ϕ1 1 to some open subset of ϕ1 [U1 U2 ] containing ϕ1 ( p) issmooth. But this follows easily. We know that ϕ1 [U U1 U2 ] is an open subset of ϕ1 [U1 U2 ] containing ϕ1 ( p). And the restriction of ϕ2 ϕ1 1 to ϕ1 [U U1 U2 ] is smooth, since it can be realized as the composition of ϕ ϕ1 1(restricted to ϕ1 [U U1 U2 ]) with ϕ2 ϕ 1 (restricted to ϕ[U U1 U2 ]),and both these maps are smooth. Our definition of manifolds is less restrictive than some in that we do notinclude the following condition.(M5) (Countable cover condition) There is a countable subset {(Un , ϕn ) : n N} of C whose domains cover M; i.e., for all p in M, there is an n suchthat p Un .In fact, all the manifolds that one encounters in relativity theory satisfy (M5).But there is some advantage in not taking the condition for granted fromthe start. It is simply not needed for our work until we discuss derivativeoperators—i.e., affine connections—on manifolds in section 1.7. It turns outthat (M5) is actually a necessary and sufficient condition for there to
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. Gravitatio
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