Relativity, Gravitation, and Cosmology: A Basic Introduction (Oxford Master Series in Physics)

355 Pages·2005·2.66 MB·English

by  Ta-Pei Cheng

OXFORDMASTERSERIESINPARTICLEPHYSICS,ASTROPHYSICS,ANDCOSMOLOGY
OXFORDMASTERSERIESINPHYSICS
The Oxford Master Series is designed for final year undergraduate and beginning graduate students in physics and
relateddisciplines.Ithasbeendrivenbyaperceivedgapintheliteraturetoday.Whilebasicundergraduatephysicstexts
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specializedtextstendtoberatherdauntingforstudents.Inthisseries,alltopicsandtheirconsequencesaretreatedat
asimplelevel,whilepointerstorecentdevelopmentsareprovidedatvariousstages.Theemphasisisonclearphysical
principleslikesymmetry,quantummechanics,andelectromagnetismwhichunderliethewholeofphysics.Atthesame
time,thesubjectsarerelatedtorealmeasurementsandtotheexperimentaltechniquesanddevicescurrentlyusedby
physicistsinacademeandindustry.Booksinthisseriesarewrittenascoursebooks,andincludeampletutorialmaterial,
examples,illustrations,revisionpoints,andproblemsets.Theycanlikewisebeusedaspreparationforstudentsstarting
adoctorateinphysicsandrelatedfields,orforrecentgraduatesstartingresearchinoneofthesefieldsinindustry.
CONDENSEDMATTERPHYSICS
1. M.T.Dove:Structureanddynamics:anatomicviewofmaterials
2. J.Singleton:Bandtheoryandelectronicpropertiesofsolids
3. A.M.Fox:Opticalpropertiesofsolids
4. S.J.Blundell:Magnetismincondensedmatter
5. J.F.Annett:Superconductivity
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ATOMIC,OPTICAL,ANDLASERPHYSICS
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PARTICLEPHYSICS,ASTROPHYSICS,ANDCOSMOLOGY
10. D.H.Perkins:Particleastrophysics
11. T.P.Cheng:Relativity,gravitation,andcosmology
STATISTICAL,COMPUTATIONAL,ANDTHEORETICALPHYSICS
12. M.Maggiore:Amodernintroductiontoquantumfieldtheory
13. W.Krauth:Statisticalmechanics:algorithmsandcomputations
14. J.P.Sethna:Entropy,orderparameters,andemergentproperties
Relativity, Gravitation,
and Cosmology
A basic introduction
TA-PEICHENG
UniversityofMissouri—St.Louis
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Cheng,Ta-Pei.
Relativity,gravitation,andcosmology:abasicintroduction/
Ta-PeiCheng.
p.cm.—(Oxfordmasterseriesinphysics;no.11)
Includesbibliographicalreferencesandindex.
ISBN0-19-852956-2(alk.paper)—ISBN0-19-852957-0(pbk.
: alk.paper)
1. Generalrelativity(Physics)—Textbooks. 2. Spaceandtime.
3. Gravity. 4. Cosmology. I. Title. II. Series:Oxfordmaster
seriesinphysics;11.
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530.11—dc22
2004019733
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PrintedinGreatBritain
onacid-freepaperbyAntonyRowe,Chippenham
ISBN0 19 852956 2(Hbk)
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Preface
It seems a reasonable expectation that every student receiving a university
degree in physics will have had a course in one of the most important devel-
opmentsinmodernphysics:Einstein’sgeneraltheoryofrelativity.Also,given
the exciting discoveries in astrophysics and cosmology of recent years, it is
highlydesirabletohaveanintroductorycoursewherebysuchsubjectscanbe
presentedintheirproperframework.Again,thisisgeneralrelativity(GR).
Nevertheless,aGRcoursehasnotbeencommonlyavailabletoundergradu-
ates,orevenforthatmatter,tograduatestudentswhodonotspecializeinGRor
fieldtheory.Oneofthereasons,inmyview,istheinsufficientnumberofsuitable
textbooksthatintroducethesubjectwithanemphasisonphysicalexamplesand
simpleapplicationswithoutthefulltensorapparatusfromtheverybeginning.
TherearemanyexcellentgraduateGRbooks;thereareequallymanyexcellent
“popular”booksthatdescribeEinstein’stheoryofgravitationandcosmology
atthequalitativelevel;andtherearenotenoughbooksinbetween.Iamhopeful
that this book will be a useful addition at this intermediate level. The goal is
to provide a textbook that even an instructor who is not a relativist can teach
from. Itisalsointendedthatotherexperiencedphysicsreaderswhohavenot
hadachancetolearnGRcanusethebooktostudythesubjectontheirown.
Asexplainedbelow,thisbookhasfeaturesthatwillmakesuchanindependent
studyparticularlyfeasible.
Students should have had the usual math preparation at the calculus level,
plussomefamiliaritywithmatrices,andthephysicspreparationofcourseson
mechanicsandonelectromagnetismwheredifferentialequationsofMaxwell’s
theory are presented. Some exposure to special relativity as part of an intro-
ductory modern physics course will also be helpful, even though no prior
knowledge of special relativity will be assumed. Part I of this book concen-
trates on the metric description of spacetime: first, the flat geometry as in
specialrelativity,andthencurvedonesforgeneralrelativity.HereIdiscussthe
equation of motion in Einstein’s theory, and many of its applications: the
threeclassicaltests,blackholes,andgravitationallensing,etc.PartIIcontains
threechaptersoncosmology.Besidesthebasicequationsdescribingahomoge-
neousandisotropicuniverse,Ipresentacarefultreatmentofdistanceandtime
in an expanding universe with a space that may be curved. The final chapter
oncosmology,Chapter9providesanelementarydiscussionoftheinflationary
modelofthebigbang,aswellastherecentdiscoverythattheexpansionofour
universeisaccelerating,implyingtheexistenceofa“darkenergy.”Thetensor
formulationofrelativityisintroducedinPartIII.Afterpresentingspecialrela-
tivityinamanifestlycovariantformalism,wediscusscovariantdifferentiation,
paralleltransport,andcurvaturetensorforacurvedspace.Chapter12contains
thefulltensorformulationofGR,includingtheEinstein’sfieldequationandits
vi Preface
solutionsforvarioussimplesituations.Thesubjectofgravitationalwavescanbe
foundintheconcludingchapter.
The emphasis of the book is pedagogical. The necessary mathematics will
be introduced gradually. Tensor calculus is relegated to the last part of the
book. Discussion of curved surfaces, especially the familiar example of a
spherical surface, precedes that of curved higher dimensional spaces. Parts I
andIIpresentthemetricdescriptionofspacetime.Manyapplications(including
cosmology)canalreadybediscussedatthismoreaccessiblelevel;studentscan
reachtheseinterestingresultswithouthavingtostrugglethroughthefulltensor
formulation,whichispresentedinPartIIIofthebook.Afewotherpedagogical
devicesarealsodeployed:
• abulletlistoftopicalheadingsatthebeginningofeachchapterservesas
the“chapterabstracts,”givingthereaderaforetasteofupcomingmaterial;
• matter in marked boxes are calculation details, peripheral topics,
historical tit-bits that can be skipped over depending on the reader’s
interest;
• Review questions at the end of each chapter should help beginning
1Wefindthatthepracticeoffrequentquizzes studentstoformulatequestionsonthekeyelementsofthechapter1;brief
basedonthesereviewquestionsareaneffec- answerstothesequestionsareprovidedatthebackofthebook;
tivemeanstomakesurethateachmemberis • Solutionstoselectedproblemsattheendofthebookalsocontainssome
keepingupwiththeprogressoftheclass.
extra material that can be studied with techniques already presented in
thetext.
Given this order of presentation, with the more interesting applications
coming before the difficult mathematical formalism, it is hoped that the
book can be rather versatile in terms of how it can be used. Here are some
ofthepossibilities:
1. PartsIandIIshouldbesuitableforanundergraduatecourse.Thetensor
formulation in Part III can then be used as extracurricular material for
instructorstoreferto,andforinterestedstudentstoexploreontheirown.
Muchoftheintermediatestepsbeinggivenandmoredifficultproblems
havingtheirsolutionsprovided,thissectioncan,inprinciple,beusedas
self-studymaterialbyaparticularlymotivatedundergraduate.
2. The whole book can be used for a senior-undergraduate/beginning-
graduatecourse.Tofitintoaone-semestercourse,onemayhavetoleave
some applications and illustrative examples to students as self-study
topics.
3. The book is also suitable as a supplemental text: for an astronomy
undergraduatecourseoncosmology,toprovideamoredetaileddiscus-
sionofGR;foraregularadvancedGRandcosmologycourse, toease
the transition for those graduate students not having had a thorough
preparationintherelevantarea.
4. Thebookiswrittenkeepinginmindreadersdoingindependentstudyof
thesubject.Themathematicalaccessibility,andthevarious“pedagogical
devices”(chapterheadings,reviewquestions,andworked-outsolutions,
etc.)shouldmakeitpracticalforaninterestedreadertousethebookto
studyGRandcosmologyonhisorherown.
An updated list of corrections to the book can be found at the website
http://www.umsl.edu/∼tpcheng/grbook.html
Preface vii
Acknowledgments
This book is based on the lecture notes of a course I taught for several years
at the University of Missouri—St. Louis. Critical reaction from the students
hasbeenveryhelpful.DaisukeTakeshita,andalsoMichaelCone,providedme
withdetailedcomments.MycolleagueRicardoFloreshasbeenverygenerous
in answering my questions—be they in cosmology or computer typesetting.
The painstaking task of doing all the line-drawing figures was carried out by
CindyBertram.MyeditorSonkeAdlungatOUPhasgivenmemuchsupport
and useful advice. He arranged to have the manuscript reviewed by scholars
whoprovidedmanysuggestionsforimprovements.ToallofthemIammuch
indebted.Finally,IamgratefultomywifeLeslieforherpatientunderstanding
duringtheratherlengthyperiodthatittookmetocompletethisproject.
Additionalacknowledgment:IwouldliketoexpressmygratitudetoProfessor
Eric Sheldon. He was kind enough to read over the entire book and made
numeroussuggestionsforeditorialimprovements, whichwereadoptedinthe
newprintingsofthisbook.
St.Louis T.P.C.
Thisbookisdedicatedto
ProfessorLing-FongLiofCarnegieMellonUniversity
formorethan30years’friendshipandenlightenment
Contents
PartI RELATIVITY MetricDescriptionofSpacetime
1 Introductionandoverview 3
1.1 Relativityasacoordinatesymmetry 5
1.1.1 FromNewtonianrelativitytoaether 5
1.1.2 Einsteinianrelativity 6
1.1.3 Coordinatesymmetrytransformations 7
1.1.4 Newkinematicsanddynamics 7
1.2 GRasagravitationalfieldtheory 8
1.2.1 Einstein’smotivationsforthegeneraltheory 8
1.2.2 Geometryasgravity 10
1.2.3 Mathematicallanguageofrelativity 11
1.2.4 GRistheframeworkforcosmology 12
Reviewquestions 12
2 Specialrelativityandtheflatspacetime 14
2.1 Coordinatesymmetries 14
2.1.1 Rotationalsymmetry 14
2.1.2 NewtonianphysicsandGalileansymmetry 16
2.1.3 ElectrodynamicsandLorentzsymmetry 17
2.1.4 Velocityadditionruleamended 18
2.2 Thenewkinematicsofspaceandtime 19
2.2.1 Relativityofspatialequilocality 20
2.2.2 Relativityofsimultaneity—thenew
kinematics 20
2.2.3 Theinvariantspace–timeinterval 22
2.3 GeometricformulationofSR 24
2.3.1 Generalcoordinatesandthemetrictensor 24
2.3.2 DerivationofLorentztransformation 28
2.3.3 Thespacetimediagram 30
2.3.4 Time-dilationandlengthcontraction 32
Reviewquestions 35
Problems 35
3 Theprincipleofequivalence 38
3.1 Newtoniangravitationpotential—areview 38
3.2 EPintroduced 39
3.2.1 Inertialmassvs.gravitationalmass 40
3.2.2 EPanditssignificance 41