It’s About Time: Elementary Mathematical Aspects of Relativity

426 Pages·2017·2.96 MB·English

by  Roger Cooke

IITT’’SS AABBOOUUTT TTIIMMEE
EELLEEMMEENNTTAARRYY MMAATTHHEEMMAATTIICCAALL AASSPPEECCTTSS OOFF RREELLAATTIIVVIITTYY
ROGER COOKE
IT’S ABOUT TIME
ELEMENTARY MATHEMATICAL ASPECTS OF RELATIVITY
ROGER COOKE

A M E R I C A N M AT H E M AT I C A L S O C I E T Y
Providence, Rhode Island
2010 Mathematics Subject Classification. Primary89-01, 01-01.
For additional informationand updates on this book, visit
www.ams.org/bookpages/mbk-102
Library of Congress Cataloging-in-Publication Data
Names: Cooke,Roger,1942–
Title: It’sabouttime: elementarymathematicalaspectsofrelativity/RogerCooke.
Description: Providence, Rhode Island : American Mathematical Society, [2017] | Includes
bibliographicalreferencesandindex.
Identifiers: LCCN2016038198|ISBN9781470434830(alk. paper)
Subjects: LCSH: Relativity (Physics)–Mathematics–Textbooks. | Mathematical physics–
Textbooks. | Space and time–Mathematics–Textbooks. | AMS: Relativity and gravitational
theory–Instructionalexposition(textbooks,tutorialpapers,etc.). msc|Historyandbiogra-
phy–Instructionalexposition(textbooks,tutorialpapers,etc.). msc
Classification: LCC QC173.55 .C66 2017 | DDC 530.1101/51–dc23 LC record available at
https://lccn.loc.gov/2016038198
Copying and reprinting. Individual readersofthispublication,andnonprofit librariesacting
for them, are permitted to make fair use of the material, such as to copy select pages for use
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Permissions to reuse
portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink(cid:2)
service. Formoreinformation,pleasevisit: http://www.ams.org/rightslink.
Sendreq[e-mail].
Excludedfromtheseprovisionsismaterialforwhichtheauthorholdscopyright. Insuchcases,
requestsforpermissiontoreuseorreprintmaterialshouldbeaddresseddirectlytotheauthor(s).
Copyrightownershipisindicatedonthecopyrightpage,oronthelowerright-handcornerofthe
firstpageofeacharticlewithinproceedingsvolumes.
(cid:2)c2017bytheauthor. Allrightsreserved.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 222120191817
Contents
Preface vii
Pedagogical Aims ix
Humanistic Aims xii
Special Features of This Book xiii
Other Works on the Subject xvi
Background Necessary to Read This Book xvii
Plan of the Work xviii
Acknowledgments xix
Part 1. The Special Theory 1
Chapter 1. Time, Space, and Space-Time 3
1. Simultaneity and Sequentiality 3
2. Synchronization in Newtonian Mechanics 6
3. An Asymmetry in Newtonian Mechanics: Electromagnetic Forces 19
4. The Lorentz Transformation 20
5. Contraction of Length and Time 26
6. Composition of Parallel Velocities 30
7. The Twin Paradox 32
8. Relativistic Triangles 35
9. Composition of Relativistic Velocities as a Binary Operation* 39
10. Plane Trigonometry* 45
11. The Lorentz Group* 48
12. Closure of Lorentz Transformations under Composition* 52
13. Rotational Motion and a Non-Euclidean Geometry* 57
14. Problems 64
Chapter 2. Relativistic Mechanics 71
1. The Kinematics of a Particle 71
2. From Kinematics to Dynamics: Mass and Momentum 75
3. Relativistic Force 79
4. Work, Energy, and the Famous E =mc2 85
5. Newtonian Potential Energy 87
6. Hamilton’s Principle 92
7. The Newtonian Lagrangian 93
8. The Relativistic Lagrangian 96
9. Angular Momentum and Torque 98
10. Four-Vectors and Tensors* 101
11. Problems 113
iii
iv CONTENTS
Chapter 3. Electromagnetic Theory* 115
1. Charge and Charge Density 116
2. Current and Current Density 118
3. Transformation of Electric and Magnetic Fields 119
4. Derivation of the Curl Equations from the Divergence Equations 122
5. Problems 124
Part 2. The General Theory 125
Introduction to Part 2 127
Chapter 4. Precession and Deflection 129
1. Gravitation as Curvature of Space 131
2. First Analysis: Newtonian Orbits 132
3. Second Analysis: Newton’s Law with Relativistic Force 137
4. Third Analysis: Newtonian Orbits as Geodesics 140
5. Fourth Analysis: General Relativity 155
6. Einstein’s Law of Gravity 161
7. Computation of the Relativistic Orbit 166
8. The Speed of Light 177
9. Deflection of Light Near the Sun 179
10. Problems 183
Chapter 5. Concepts of Curvature, 1700–1850 189
1. Differential Geometry 190
2. Curvature, Phase 1: Euler 197
3. Curvature, Phase 2: Gauss 212
4. Problems 223
Chapter 6. Concepts of Curvature, 1850–1950 225
1. Second-Order Derivations 226
2. Curvature, Phase 3: Riemann 232
3. Parallel Transport 239
4. The Exponential Mapping and Normal Coordinates 248
5. Sectional Curvature 260
6. The Laplace–Beltrami Operator 264
7. Curvature, Phase 4: Ricci 287
8. Problems 297
Chapter 7. The Geometrization of Gravity 303
1. The Einstein Field Equations 304
2. Further Developments 314
3. “Temporonautics” and the G¨odel Rotating Universe 315
4. Black Holes 320
5. Problems 325
Part 3. Historical and Philosophical Context 329
Chapter 8. Experiments, Chronology, Metaphysics 331
1. Experimental Tests of General Relativity 332
CONTENTS v
2. Chronology 336
3. Space and Time 350
4. The Reality of Physical Concepts 361
5. The Harmony Between Mathematics and the Physical World 366
6. Knowledge of Hypothetical Objects: An Example 376
7. Knowledge of the Physical World 380
8. A Few Words from the Discoverers 384
9. Epilogue: The Reception of Relativity 386
Bibliography 389
Subject Index 393
Name Index 401

Preface
And the same year I began to think of gravity extending to ye Orb
of the Moon & (having found out how to estimate the force wch [a]
globe revolving within a sphere presses the surface of the sphere)
from Kepler’s rule of the periodical times of the planets being in
sesquilateral proportion to their distance from the center of their
Orbs, I deduced that the forces wch keep the planets in their Orbs
must [be] reciprocally as the square of their distances from the cen-
ters about which they revolve: & thereby compared the force requi-
site to keep the Moon in her Orb with the force of gravity at the
surface of the Earth, & found them answer [agree] pretty nearly.
All this was in the plague years 1665–1666.
Isaac Newton, 1718. Quoted by Westfall ([84], p. 109).
The present book has grown from what was originally the very small project of
writinganarticlehighlightingthreemutuallyunrelatedareasofspecialandgeneral
relativity, namely the twin paradox (§ 6 of Chapter 1), the relativistic Maxwell
equations(Chapter3),andtheprecessionoftheorbitofMercury(§7ofChapter4).
As the writing proceeded, I realized that, while I was keeping the mathematical
details to a minimum and omitting all the historical context of the physics, this
material would still not be accessible to the audience I had in mind, consisting
of people with mid-level undergraduate preparation in mathematics. One would
still need to know at least the rudiments of calculus of variations and differential
geometry. One thing led to another, and the filling in of those details required
two years of work and expanded this work to its present Brobdingnagian size of
some 400 pages, plus two additional volumes (posted online) of ancillary material.
My vision of the core of the work (Volume 1) remains: it is intended to be a
random set of commentaries on certain aspects of relativity. This book is neither a
technicalintroductiontorelativity,norasystematichistoryofitsdevelopment,nor
yet a professional-quality examination of philosophical issues. The physicists and
historians who vettedit for publication pointedout tome a number of areaswhere
my ignorance and brashness led me to make throwaway comments that describe
research already performed or in progress. As I shall keep reminding the reader, I
am not a specialist in any of these areas; if I make some suggestions that expose
my innocence, that is not the worst fate than can befall an author.
I have had two purposes in mind in writing the present work, one pedagogical,
the other humanistic. The pedagogical purpose is to present some highlights of
the special and general theories of relativity with full mathematical details in a
form accessible to advanced undergraduate mathematics, science, and engineering
majors (and, of course, any interested person who knows a little university-level
vii
viii PREFACE
mathematics). The humanistic purpose is to reflect on what these theories have
meant for the human understanding of the physical world. These two purposes are
intertwined, andthematerialthatfollowswasselectedsothatbothpurposescould
be pursued together. The book constitutes a meandering journey from one funda-
mental equation of mechanics to another, from Newton’s F = mr(cid:3)(cid:3) to Einstein’s
G +Λg =(8πG/c4)T . There are very many digressions along the way. The
μν μν μν
core material in Chapters 5–7is preceded by three chapters discussing the needfor
the Lorentz transformation and its properties (Chapter 1), a standard discussion
of the mechanics of special relativity (Chapter 2), a side excursion to highlight the
fruitful interaction between special relativity and the Maxwell equations (Chap-
ter 3), and a mostly computational chapter (Chapter 4) that shows how general
relativityproduceditsfirsttwomajorsuccesses,allmathematicaljustificationbeing
postponed. These are followed by the three chapters (Chapters 5–7) of exposition
of general relativity, Chapter 5 and Chapter 6 being devoted entirely to the requi-
sitedifferentialgeometry. Chapter7containsaverybriefdiscussion ofnonrotating
black holes, followed by an attempt to make the Einstein field equations appear
natural. Chapter 8, which brings this work to a close, consists of a chronology of
landmarkworksonphysicsingeneralandsomereflectionsonmetaphysicsfromthe
point of view of an interested nonspecialist.
I hope that physicists will be tolerant of my inexperience in their specialty. In
lieu of expertise, what I offer the reader is a selection of topics that I have found
fascinating and wish to share with others. To get all of the apologies out of the
way at the beginning, let me state again that this is not a systematic exposition of
relativity theory, which is an enterprise I leave to the experts. The experts have
notbeenfoundwantinginthisregard. Thebestexampleisthe2013bookEinstein
Gravity by Anthony Zee (see my comments below). Other recent specimens are
the books of Narlikar [62] and Dray ([13], [14]). The purely mathematical heart
ofthe present book—Chapters5and6on curvature—istreatedin more detail and
moresystematicallyintherecentbookofSternberg[78]. Thereaderisencouraged
to look at these books to see how many topics have been omitted here. Generally,
these other sources contain fewer computational proofs and more conceptual ones
than the present book and offer the additional advantage of having been written
by people who do have expertise in this area.
Themanuscripthasbeenvettedbyanonymousreviewers,atleasttwoofwhom
were physicists. These two were both very kind in their criticism but dissatisfied
at many of my omissions, for example the absence of any discussion of the 1923–24
workofElieCartan([6],[7]),whichsubsumesmyamateurattempttodemonstrate
in Chapter 4 that Newtonian mechanics cannot be geometrized in such a way as
to account for the precession of a planetary orbit, and the similar absence of any
mentionofthe1911workofWillemdeSitter(1872–1934)[10],whichsubsumesmy
attempttoadaptspecialrelativitytoaccountforthatprecession. Theyaddedthat
IhavealsoomittedtheworkofAlexanderFriedmann(1888–1925)oncurvedspaces
([30], [31])and thatof Georges Lemaˆıtre(1894–1966) [52] onthe so-called Hubble
expanding universe. I have also not mentioned the original paper1 [45] of Edwin
Hubble (1889–1953) himself. These omissions are, one and all, due to my lack of
familiarity with, and/or complete unawareness of, the topics mentioned. I would
1Hubble’s paperdoes not mentionthe earlierworkofLemaˆıtre; as Hubble was not careless
aboutcitation,onemustassumehehadnotheardofitatthetimehepublishedhisownwork.
PREFACE ix
not venture to write about subjects in which I am so ill-informed. I emphasize
that this book is, as its subtitle indicates, a set of reflections on relativity by a
mathematician who hopes someone else might find them as interesting as I do.
Nevertheless, I think advanced undergraduates can learn something of value about
relativity from reading it.
MytitleisoneIrememberseeingonateleviseddocumentaryonrelativitysome
decades ago. Since completing the work, I have also chanced upon the 1995 book
([9]) by physicist Paul Davies whose title is similar and which discusses relativity
and other issues of modern physics. Indeed, a search through almost any online
catalog for this title will turn up literally thousands of books whose titles contain
the phrase “About Time”. Some of them use it in the sense I intend. For others,
the word “time” is a synonym for a prison term. The reader will recognize that
the title bears a double meaning in all cases. In ordinary speech, “It’s about time”
expresses vexation that a desired result has been delayed and satisfaction that it
has finally arrived. That is not the meaning I intend for it here, since I do not
flatter myself that the world has been waiting impatiently for the present book.
Rather, I have in mind the literal sense that time is precisely what this book is
about and even (with some exaggeration) what the theory of relativity itself is
about. Poincar´e recognized this over a century ago, when he heaped lavish praise
ontheintroductionofwhatHenrikLorentzcalledlocaltime andwhatisnowcalled
(following Hermann Minkowski) proper time. Even in everyday life, if we want to
know the distance to a remote location, we are not usually concerned with the
distance itself, but more often with the amount of time it will take us to go there.
Ourlanguageisfullofexpressionsthatdescribedistanceintermsoftraveltime. In
physics, we can do the same thing by prescribing a fixed velocity v and using the
0
equationd=v ttoconvertadistancedtoatimetandviceversa. Thattechnique
0
leads to some interesting geometrized Newtonian astronomy in Chapter 4 and is
at the heart of the relativistic approach to planetary motion. The central fact on
which relativity is based is that there is an absolute unit of speed that can be used
tocarryoutthisconversion, namelyv =c,thespeedoflightinavacuum,aspeed
0
that is the same for all observers. All of the equations of special relativity depend
on the entanglement of time with a spatial axis of relative motion between two
observers and the resulting distinction between observed time and proper time for
a moving particle. “It” is literally about time.
Somuchforthetitle. Thesubtitleexpressesmypedagogicalpurpose, towhich
I now turn.
Pedagogical Aims
Oneofthehardestquestionsaskedbystudentsstudyinghigh-schoolandcollege-
level algebra is, “What is algebra good for?” They know perfectly well that people
never need to solve even quadratic equations in everyday life, and no one is ever
faced with the problem of computing where two trains will meet setting out in
opposite directions from Chicago and New York at different times. One can tell
them that algebra is the language in which science is written, but the unfortunate
truth is that science seldom needs to solve only algebraic equations; much more
often,theproblemsinvolvedifferential equations. Itistruethatoneneedstoknow
algebraverywellinordertounderstanddifferentialequations,butthatexplanation
is generally lost on people just beginning the study of the subject. It needs to be