Table Of ContentRelativity
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Relativity
SPECIAL, GENERAL, AND
COSMOLOGICAL
SECONDEDITION
WolfgangRindler
ProfessorofPhysics
TheUniversityofTexasatDallas
3
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Firsteditionfirstpublished2001
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To my wife Linda
the most generous person
I have ever known
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MakePhysicsassimpleaspossible,butnosimpler.
AlbertEinstein
Theidealistoreachproofsbycomprehensionratherthanbycomputation.
BernhardRiemann
Preface
My earlier book, Essential Relativity, aimed to provide a quick if thoughtful intro-
ductiontothesubjectatthelevelofadvancedundergraduatesandbeginninggraduate
students,while‘containingenoughnewmaterialandsimplificationsofoldarguments
soasnottoboretheexpertteacher.’Butgeneralrelativityhasbynowrobustlyentered
themainstreamofphysics,inparticularastrophysics,newdiscoveriesincosmology
are routinely reported in the press, while ‘wormholes’ and time travel have made
it into popular TV. Students thus want to know more than the bare minimum. The
presentbookofferssuchanextension,inwhichthestyle,thegeneralphilosophy,and
themathematicallevelofsophisticationhaveneverthelessremainedthesame.Any-
onewhoknowsthecalculusuptopartialdifferentiation,ordinaryvectorstothepoint
ofdifferentiatingthem,andthatmostusefulmethodofapproximation,thebinomial
theorem, should be able to read this book. But instead of the earlier nine chapters
therearenoweighteen, andinsteadof167exercises, nowtherearemorethan300;
aboveall,tensorsareintroducedwithoutapologyandthenthoroughlyused.
Einstein’sspecialandgeneralrelativity,thetheoriesofflatandcurvedspacetime
andofthephysicstherein,andrelativisticcosmology,withitsgeometryanddynamics
for the entire universe, not only seem necessary for a scientist’s balanced view of
theworld, butalsooffersomeofthegreatestintellectualthrillsofmodernphysics.
Perhaps the chief motivation in writing this book has been once more the desire to
conveythatthrill,aswellassomeoftheinsightsthatlongpreoccupationwithasubject
inevitablyyields.Itistruethatmanyaspectsofgeneralrelativityhavestillnotbeen
testedexperimentally.Neverthelessenoughhavebeentestedtojustifytheviewthat
allofrelativityisbynowwelloutofthetentativestage.Thatisalsothereasonwhy
theintroductorychaptercontainsanoverviewofallofrelativityandcosmology,so
thatthestudentcanappreciatefromtheverybeginningthelocalcharacterofspecial
relativity and how it fits into the general scheme. The three main parts that follow
dealextensivelywithspecialrelativity, generalrelativity, andcosmology. IneachI
havetriedtoreportonthemostimportantcrucialexperimentsandobservations,both
historicalandmodern,butstressingconceptsratherthanexperimentaldetail.Infact,
theemphasisthroughoutisonunderstandingtheconceptsandmakingtheideascome
alive.Butanequalvalueisputondevelopingthemathematicalformalismrigorously,
andonguidingthestudenttousebothconceptsandmathematicsinconjunctionwith
viii Preface
thetricksofthetradetobecomeanexpertproblemsolver.Avitalpartinthisprocess
shouldbeplayedbytheexercises,whichhavebeenputtogetherrathercarefully,and
whicharemostlyofthe‘thinking’variety.Thoughtheirfullsolutionoftenrequires
someingenuity,theyshouldatleastbelookedat,asasupplementtothetext,forthe
extrainformationtheycontain.
No book ever has enough diagrams. That is one of the luxuries that classroom
teaching has over a book. So readers should constantly draw their own, especially
sincerelativityisaverygeometricsubjectinwhichthefacilitytothinkgeometrically
isagreatasset.Readersshouldalsoconstantlymakeuptheirownproblems,however
trivial:whatwouldhappenif...?Inaninitiallyparadoxicalsubjectlikerelativity,it
isoftenthemostskepticalstudentwhoisthemostsuccessful.
Eachofthethreepartscouldwellbecutshortdrasticallysothatthebookmight
serveasatextforaone-semestercourse.Topresentitfullywilltaketwosemesters,
probablywithmaterialtospare.Butapartfromitsservingasanintroductorytextfor
aformalcourse,Ialsoenvisagethebookashavingsomeuseforthegeneralscientist
whomightwishtobrowseinit,andforthemoreadvancedgraduatestudentinsearch
ofgreenerpastures,asachangefromtherockypinnaclesofmoreseveretexts.
AttheendofthebookthereisanAppendixoncurvaturecomponentsfordiagonal
metrics(inalittlemoregeneralitythantheold‘Dingleformulae’), whichcouldbe
usefuleventoworkersinthefieldwhohavenotreadtherestofthebook.Andfinallya
wordofwarning:inmanysections,asisthecustominrelativity,theunitsarechosen
soastomakethespeedoflightunity,andlatereventomakeNewton’sconstantof
gravitationunity,whichmustbeborneinmindwhencomparingformulae;wherethe
dimensionsseemwrong,c’sorG’saremissing.
I owe much to many modern authors (Sexl and Urbantke, Misner, Thorne and
Wheeler, Ohanian and Ruffini, Woodhouse, etc.), though an exact assignment of
debtwouldbedifficultatthisstage.Ihavealsobenefittedfromthemanysearching
questions of my students over the years, among whom I might perhaps single out
JamesGilsonandJackDenur.ButthegreatestdebtIowe,assooftenbefore,tomy
friend Ju¨rgen Ehlers—discussion partner, scientific conscience, font of knowledge
withoutpeer.
Dallas,Texas W.R.
January2001
Preface to the Second Edition
Ithasbeenmostgratifyingtoseethefavorablereceptionofthefirsteditionofthis
book,totheextentthatasecondeditionisalreadyinorder.
Inthissecondeditionthelastthreechaptersoncosmology,inparticular,havebeen
updatedandrevised.Butthereareadditions,improvements,andsomenewexercises
throughout.
Itismyhopethatreadersofthebookwillenjoyandgiveanequalwelcometothis
amendedversionofit.
Dallas,Texas W.R.
January2006
