General relativity : a concise introduction Carlip Visit to download the full and correct content document: https://ebookmass.com/product/general-relativity-a-concise-introduction-carlip/
More products digital (pdf, epub, mobi) instant download maybe you interests ...
Window on Humanity: A Concise Introduction to General Anthropology 8th Edition, (Ebook PDF)
https://ebookmass.com/product/window-on-humanity-a-conciseintroduction-to-general-anthropology-8th-edition-ebook-pdf/
Window on Humanity: A Concise Introduction to General Anthropology 8th Edition Conrad Kottak
https://ebookmass.com/product/window-on-humanity-a-conciseintroduction-to-general-anthropology-8th-edition-conrad-kottak/
Window on Humanity: A Concise Introduction to General Anthropology 7th Edition – Ebook PDF Version
https://ebookmass.com/product/window-on-humanity-a-conciseintroduction-to-general-anthropology-7th-edition-ebook-pdfversion/
Introducing General Relativity Mark Hindmarsh
https://ebookmass.com/product/introducing-general-relativitymark-hindmarsh/
Special relativity, electrodynamics, and general relativity : from Newton to Einstein Second Edition Kogut
https://ebookmass.com/product/special-relativity-electrodynamicsand-general-relativity-from-newton-to-einstein-second-editionkogut/
Relativity Made Relatively Easy. Volume 2: General Relativity and Cosmology Andrew M. Steane
https://ebookmass.com/product/relativity-made-relatively-easyvolume-2-general-relativity-and-cosmology-andrew-m-steane/
(eBook PDF) Weather: A Concise Introduction
https://ebookmass.com/product/ebook-pdf-weather-a-conciseintroduction/
A Concise Introduction to Logic – Ebook PDF Version
https://ebookmass.com/product/a-concise-introduction-to-logicebook-pdf-version/
A Concise Introduction to Logic 13th Edition, (Ebook PDF)
https://ebookmass.com/product/a-concise-introduction-tologic-13th-edition-ebook-pdf/
GeneralRelativity GeneralRelativity AConciseIntroduction StevenCarlip
TheUniversityofCaliforniaatDavis
GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom
OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries
c StevenCarlip2019
Themoralrightsoftheauthorhavebeenasserted
FirstEditionpublishedin2019
Impression:1
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove
Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer
PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData
Dataavailable
LibraryofCongressControlNumber:2018957383
ISBN978–0–19–882215–8(hbk.) ISBN978–0–19–882216–5(pbk.)
DOI:10.1093/oso/9780198822158.001.0001
Printedandboundby
CPIGroup(UK)Ltd,Croydon,CR04YY
LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork.
InmemoryofBryceDeWittandCecileDeWitt-Morette
Preface Generalrelativitywasbornin1915,theculminationofeightyearsofwork,byEinstein andothers,aimedatreconcilingspecialrelativityandNewton’saction-at-a-distance gravity.Inmuchofthecenturythatfollowed,mostphysicistsviewedtheresultwith ambivalence.Generalrelativitywasseenasabeautiful,eleganttheory,amodelof whatphysicsshouldbe;butatthesametime,itwasatheorythatseemedalmost completelydivorcedfromtherestofphysics.
Thebeautywasobvious.Einsteinhadidentifiedanassumptionthathadbeen takenforgranted,thatspacetimewasflatandnondynamical;hehadchangeditinthe simplestwaypossible;andoutofthatsinglestephadsprungallofNewtoniangravity, smallbutmeasurablecorrections,andacompletelynewviewofcosmology.Butthe irrelevancealsoseemedobvious.Generalrelativityremainedstubbornlyoutsidethe quantumrevolutionthatwassweepingthroughphysics,and,asapracticalmatter,the bestavailabletechnologyhadtobestretchedtoitslimitstodetectthetinydeviations fromNewtoniangravity.
Thingshavechanged.Generalrelativityisstillwidelyviewedasthemodelofeleganceinphysics,thoughsomearguethatitssimplicitymaybeanaccidentallow energymanifestationofamorecomplicatedhighenergytheory.Buttheseparation fromtherestofphysicshasended.Cosmologistscannolongerrelyonafewsimple solutionsoftheEinsteinfieldequations;theymustunderstandperturbations,gravitationallensing,andalternativetheoriesofgravity.Gravitationalwavesareopening upanentirelynewwindowintotheUniverse,allowingustoobservephenomenasuch asblackholemergersthatwouldotherwisebecompletelyinvisible.Highenergytheoristsfinditincreasinglydifficulttoescapethequestionofan“ultravioletcompletion,” ahighenergylimitthatwillalmostcertainlyhavetoincorporatequantumgravity. Evensomecondensedmatterandheavyionphysicistsarelookingatthepeculiarlinks betweentheirfieldsandgravitysuggestedbytheAdS/CFTcorrespondence,andnew researchonevaporatingblackholesispointingtowardsurprisingconnectionsbetween generalrelativityandquantuminformationtheory.
Thismakesadifferenceinhowweteach,andlearn,generalrelativity.Acourse shouldbeajumping-offpointforpeoplegoinginmanydifferentdirections.Itshouldn’t bemathematicallysloppy—studentswillstillneedtoread,andperhapswrite,mathematicallysophisticatedpapers—butitshouldmoveasquicklyaspossibletophysics.It shouldtantalize,offeringstudentsglimpsesofthevastlandscapeofscienceconnected togeneralrelativitywithouttryingtoexplaineverythingatonce.
ThisbookhasgrownoutofanintroductorygraduatecourseI’vetaughtatthe UniversityofCaliforniaatDavissince1991.Davisoperatesonaquartersystem—in practice,about25hoursofinstructionpercourse—andI’vetriedtowriteatextbook thatcouldbeusedforsuchacourse,althoughinstructorswithmoretimeshouldfind
Preface
iteasytoaddmaterial.Thebookisdirectedprimarilytowardgraduatestudents,but myclasseshaveoftenincludedafewundergraduates,whohavekeptupwithouttoo muchextrawork.Iassumebasicknowledgeofspecialrelativity(includingfour-vectors andLorentzinvariance),somefamiliaritywithLagrangiansandvariationalprinciples, areasonablelevelofcomfortwithpartialdifferentialequationsandlinearalgebra,and anacquaintancewithasmallbitofsettheory(opensets,intersections,andthelike), butnopriorknowledgeofdifferentialgeometryortensoranalysis.
Myassumptionisthatmoststudentsusingthisbookwillbephysicsstudents.I takethe“physicsfirst”approach,popularizedbyJimHartle,inwhichaclassmoves veryquicklytocalculationsofgravityintheSolarSystemandonlylaterreturnstoa moresystematicdevelopmentofthenecessarymathematics.Ihaveincludedashort introductiontotheHamiltonianformulationofgeneralrelativity,atopicoftenleftout ofintroductorycourses,andIclosewitha“bonus”chapterthatbrieflydescribessome ofthemanydirectionsonecouldgofromhere.
IlearnedgeneralrelativityfromBryceDeWittandCecileDeWitt-Morette,to whomIoweadeepdebtofgratitude.Mystudentsoverthepast25yearstaughtme more.Oneofthem,JosephMitchell,undertookaverycarefulreadingofadraft.I thankmysons,PeterandDavidCarlip,fortheirhelpineditingandproofreadingthis book,andmybrother,WalterCarlip,forextensiveproofreadingandinvaluablehelp withtypography.ThisworkwassupportedinpartbytheU.S.DepartmentofEnergy undergrantDE-FG02-91ER40674.
Gravityasgeometry Generalrelativityisanelegantandpowerfultheory,butitisalsoastrangeone. AccordingtoEinstein,thephenomenonweusuallythinkofastheforceofgravityis reallynotaforceatall,butratherabyproductofthecurvatureofspacetime.Although wehavebecomeaccustomedtothisideaovertime,itisstillapeculiarnotion,and it’sworthtryingtounderstandwhatitmeansbeforeplungingintothedetails.
Startwiththefoundationofmechanics,Newton’ssecondlaw,
Studentswhofirstseethisequationsometimesworrythatitmightbeatautology. True,forcescauseaccelerations;butthewaywerecognizeandmeasureaforceisby theaccelerationitcauses.Isthisnotcircular?
Lookingdeeper,though,wecanseethatthesecondlawisrichwithinsight.Newton hassplittheUniverseintotwopieces:theright-handside,theobjectwhosemotionwe wanttounderstand,andtheleft-handside,therestoftheUniverse,everythingthat mightinfluencethatmotion.HetellsusthattherestoftheUniversecausesaccelerations,secondderivativesofposition—notvelocities(firstderivatives),not“jerks” (thirdderivatives),notanythingelse.Thisbehaviorisreflectedthroughoutphysics, wheresecondorderdifferentialequationsarefoundeverywhere;itisonlyrecentlythat wehavebeguntounderstandtheunderlyingreasonforthispattern(seeBox1.1).
Beyondthis,Newton’ssecondlawtellsusthattheresponseofabodytothe Universehastwoseparateelements:itsacceleration,butalsoitsinertia,orresistance toacceleration,asexpressedbyasingle“inertialmass” m.Thisallowsustoextract informationabouttheforcebymeasuringtheresponseofotherwiseidenticalbodies withdifferentmasses.Sophysicsreallyisaboutforces.
Butthereisoneexception.Considerthemotionofanobjectinagravitational field.GoingbackagaintoNewton—nowtohislawofgravity—wehave
Themasses m ontheleft-handandright-handsidecancel,andweareleftonlywith acceleration.
Thisdidnothavetobe.Themass m ontheleft-handsideof(1.2)isinertial mass,ameasureofresistancetoacceleration,whilethemass m ontheright-handside isgravitationalmass,akindof“gravitationalcharge”analogoustoelectriccharge. FromthepointofviewofNewtoniangravity,thereisnoreasonforthesetobethe
GeneralRelativity:AConciseIntroduction.StevenCarlip c StevenCarlip2019. Publishedin2019byOxfordUniversityPress.DOI:10.1093/oso/9780198822158.001.0001
Box1.1Ostrogradsky’stheorem TheimportanceofaccelerationinNewtonianmechanicsisechoedthroughout physics:almostallfundamentalinteractionsaredescribedbyequationswithat mosttwoderivatives.Wenowunderstandthatagoodpartoftheexplanation liesinan1850theoremofOstrogradsky[1].
StartingwiththeLagrangianformalism,Ostrogradskyexploredtheproblemof constructingaHamiltonianforatheorywithmorethantwotimederivatives. Inanordinarytwo-derivativetheory,eachgeneralizedposition q isaccompanied byageneralizedmomentum p,whichisrelatedtothetimederivative˙ q.For higher-derivativetheories,newgeneralizedmomentaappear,relatedtohigher derivativesofthegeneralizedpositions.Ostrogradskyshowedthat,withaclearly demarcatedsetofexceptions,theresultingHamiltoniansarelinearinatleastone ofthesenewmomenta.Thismeanstheenergyisnotboundedbelow:sincethe momentumcanhaveanysign,thereareconfigurationswitharbitrarilynegative energies.Suchtheorieshavenostableclassicalconfigurations,andnoquantum groundstates;theycannotdescribearealworld.
same.Buttheequalityofthesetwomasses—aformofwhatiscalledtheequivalence principle—hasbeenverifiedexperimentallytobetterthanapartin1014
Forgravity,then,physicsisreallynotaboutforces,butaboutaccelerations,or trajectories.Anysufficientlysmall“testbody”willmovealongthesamepathina gravitationalfield,nomatterwhatitsmass,shape,orinternalcomposition.Agravitationalfieldpicksoutasetofpreferredpaths.
Butthisiswhatwe mean byageometry.Euclideangeometryisultimatelya theoryofstraightlines;itconsistsofthestatement,“Thesearethestraightlines,” andadescriptionoftheirproperties.Sphericalgeometryconsistsofthestatement, “Thesearethegreatcircles,”andadescriptionoftheirproperties.Ifagravitational fieldpicksoutasetofpreferred“lines”—thepathsoftestbodies—andtellsustheir properties,itisdeterminingageometry.
TheequivalenceprinciplewasalreadyknowntoGalileo.Thefamousexperiment inwhichheissaidtohavedroppedtwoballsofdifferentmassesofftheLeaningTower ofPisamaynothavetakenplace—itwasfirstdescribedyearsafterhisdeath—buthe knewfrommanyotherexperimentsthatbodieswithdifferentmassesandcompositions respondedidenticallytogravity.Couldhehaveformulatedgravityasgeometry?
Probablynot:thereisonemoresubtletytotakeintoaccount.InEuclideangeometry,twopointsdetermineauniqueline.Foranobjectmovinginagravitationalfield, ontheotherhand,themotiondependsnotonlyontheinitialandfinalpositions,but alsoontheinitialvelocity.Icandropacointothefloor,orIcanthrowitstraightup andallowittofall.Ineithercase,itwillstartandendatthesameposition.
Butthecoinwillreachtheflooratdifferent times.IfIspecifytheinitialandfinal positions and times,thetrajectoryisunique.Gravitydoes,indeed,specifypreferred
Box1.2Equivalenceprinciples Theprincipleofequivalencehastwocommonformulations.Thefirstis“universalityoffreefall,”thestatementthatthetrajectoryofasmallobjectin agravitationalfieldisindependentofitsmassandinternalpropertiessuchas structureandcomposition.Thesecondis“weightlessness,”thestatementthat inasmallenoughfreelyfallinglaboratory,nolocalmeasurementcandetectthe presenceoftheexternalgravitationalfield.Thetwoformulationsareessentially equivalent:wedetectedgravitybymeasuringaccelerations,butuniversalityof freefallimpliestheabsenceofrelativeaccelerationsinafreelyfallinglaboratory. Therearenuances,though,dependingonexactlywhatobjectsareincluded. The“weakequivalenceprinciple”appliestoobjectswhoseinternalgravitational fieldscanbeneglected.The“Einsteinequivalenceprinciple”addslocalposition invarianceandlocalLorentzinvariance,thestatementthattheoutcomeofany experimentinafreelyfallingreferenceframeisindependentofthelocationand velocityoftheframe.The“strongequivalenceprinciple”extendstheseclaims toobjectswhoseself-gravitationcannotbeignored.
Precisiontestsoftheweakequivalenceprincipledatebacktoexperimentsby E¨otv¨osintheearly20thcentury.Today,theprinciplehasbeentestedtoexquisite accuracy:relativeaccelerationsofbodieswithdifferentcompositionshavebeen showntobeequaltoaboutapartin1014.Progresshasbeenmadeontesting thestrongequivalenceprincipleaswell(seeBox8.1).
“lines,”butthelinesarenotinspace,theyareinspacetime.GalileoandNewtondidn’t havespecialrelativity—theyhadnounifiedtreatmentofspaceandtime.Einstein did.Oncethatframeworkwasavailable,thepossibilityofunderstandinggravityas geometrybecameavailableaswell.
Furtherreading Agoodintroductiontothevariousformsof theprincipleofequivalenceandtheirtests canbefoundinWill’sbook, TheoryandExperimentinGravitationalPhysics [2].The first“modern”testsoftheequivalenceprinciplewereperformedbyE¨otv¨osintheearly 1900s[3].Moreup-to-dateinformationon
experimentaltestscanbefoundin,forinstance,[4].AbeautifulreviewofOstrogradsky’stheoremisgivenin[5].Foranonmathematicalbutdeepandthought-provoking introductiontotheideaofgravityasspacetimegeometry,seeGeroch’sbook, General RelativityfromAtoB [6].
Geodesics Letusnowlookforamathematicalexpressionofthisideaofgravityasgeometry. Todoso,weneedtogeneralizetheideaofa“straightline”toarbitrarygeometries. InordinaryEuclideangeometry,astraightlinecanbecharacterizedineitheroftwo ways:
• Astraightlineis“straight,”orautoparallel:itsdirectionatanypointisthesame asitsdirectionatanyother.Thisturnsouttobecomplicatedtogeneralize—it requiresustoknowwhethertwovectorsatdifferentlocationsareparallel—and wewillhavetowaituntilChapter5foramathematicaltreatment.
• Astraightlineistheshortestdistancebetweentwopoints.Thisismucheasier togeneralize;it’sexactlythekindofquestionthatthecalculusofvariationswas designedfor.
Thischapterwilldevelopthegeneralequationfortheshortest,ormoregenerallythe extremal,linebetweentwopoints,a“geodesic.”
2.1Straightlinesandgreatcircles Westartwiththesimplestexample,astraightlineinatwo-dimensionEuclidean spacewithCartesiancoordinates x and y.Chooseinitialandfinalpoints(x0,y0)and (x1,y1).Amongallcurvesbetweenthesetwopoints,wewanttofindtheshortest. Tospecifyacurve,we could give y asafunctionof x,or x asafunctionof y.Butthis iscontrarytothespiritofgeneralrelativity,inwhichallcoordinatesshouldbetreated equally.Foramoredemocraticdescription,wechooseaparameter σ ∈ [0,σmax],and describethecurveasapairoffunctions(x(σ),y(σ))with (x(0),y(0))=(x0,y0), (x(σmax),y(σmax))=(x1,y1) .
Mathematically,wearedescribingacurveasamapfromtheinterval[0,σmax]to R2 Physically,wecanthinkof σ asasortof“time”alongthecurve,thoughnottheusual timecoordinate,whichweshouldtreatasacoordinatelikeanyother.
Wenextneedthelengthofsuchacurve.Ifthecurveissmoothenough,ashort enoughsegmentwillbeapproximatelystraight,andwecanusePythagoras’theorem towrite
foran“infinitesimaldistance” ds,oftenreferredtoasa“lineelement.”Mathematically inclinedreadersmaybeuncomfortablewiththisequation;wewillseeinChapter4 howtomakeitintoamathematicallywell-definedstatementaboutcovarianttensors.
GeneralRelativity:AConciseIntroduction.StevenCarlip c StevenCarlip2019. Publishedin2019byOxfordUniversityPress.DOI:10.1093/oso/9780198822158.001.0001
Fornow,wewilltakethephysicists’approachandthinkof“ds”asrepresentinga smallbutfinitedistance,takinglimitswhentheymakesense.
Letusdefine
Thelengthofourcurveisthen
Weextremizebysettingthevariationtozerowhileholdingtheendpointsfixed:
It’shelpfultorememberthatthevariation δ isreallyjustaderivative,albeitaderivativeinaninfinite-dimensionalspaceoffunctions.Inparticular,theusualproductrule andchainruleofcalculuscontinuetohold.I’vealsousedthefactthat
whenavariationchanges x(σ)to x(σ)+ δx(σ),thederivativechangesaccordingly.
Tofinish,weintegrate(2.4)bypartstoisolate δx and δy.Ingeneral,integration bypartsleadstoboundaryterms,butherethesetermsareproportionalto δx and δy,whicharezeroattheendpoints,sinceweareholding(x0,y0)and(x1,y1)fixed. Hence
Sincethevariations δx and δy arearbitrary,theircoefficientsmustbezero,givingus theequationsweseek.
Onefinaltricksimplifiesthisresult.Sofar, σ hasbeenanarbitrarylabelforpoints onthecurve.Nowthatwehaveaparticularcurve,though,wecanmakethespecific choice σ = s,labelingpointsbytheirdistancefromtheinitialpoint.(Thereadermay checkthatmakingthisidentification before thevariationleadsnowhere.)Withthis parametrization, E =1,andtheequationsfortheextremalcurvearesimply
(2.6) theexpectedequationsforastraightline.
Thismayseemtobearathercomplicatedwaytogetasimpleresult.Theadvantageisthatitgeneralizeseasily.Consider,forexample,thesameprobleminpolar coordinates, x = r cos θ,y = r sin θ. (2.7)
Substituting dx =cos θdr r sin θdθ and dy =sin θdr +r cos θdθ intothelineelement (2.1),wefind
Now E =
,andrepeatingthestepsthatledto(2.5),wehave
Notethat E nowdependsexplicitlyon r,andnotjustitsderivative;thisfactorof r mustalsobevaried.Againchoosing σ = s,weobtain
Theseequationsareeasyenoughtointegrate,butwecanalsointroduceanew trick.From(2.8),wehavetheidentity
Thisisnotanindependentequation—wewillseebelowthatitisa“firstintegral”of (2.10)—butitsavesusastep.
Solvingthesystem(2.10)–(2.11)isnowstraightforward.From(2.10),
where b isanintegrationconstant.Substitutinginto(2.11)andintegrating,weobtain
andthus r cos(θ θ0)= b.Thisisonceagainastraightline,asofcourseitmustbe. Again,thismayseemaroundaboutwaytofindanobviousresult.Foralesstrivial example,considera“straightline”onasphereofradius r0, x2 + y2 = r2 0 .Toobtain thelineelement,westartwithflatthree-dimensionalspaceinsphericalcoordinates
forwhichitiseasytocheckthat
Wenowrestricttothesurface r = r0,toobtainthelineelementforasphere,
Box2.1Coordinates:partI
Thelineelements(2.1)and(2.8)lookverydifferent,buttheydescribethe sameflatspace.Similarly,thegeodesicequations(2.6)and(2.10)givedifferent descriptionsofthesamestraightlines.Thisisafirstsignofarecurringtheme ingeneralrelativity,theneedtoseparateouttherealphysicsfrom“coordinate effects.”
Forflatspace,thereisapreferredcoordinatesystem,Cartesiancoordinates. Thiscangiveusafalsesenseofsecurity:wemaythinkweunderstandwhat coordinatesmean,especiallyifthey’recalled x and y,or r and θ.Butcurved spacetimestypicallyhavenopreferredcoordinates,andagoodpartoftheworkis tofigureoutwhatsomechosencoordinatesreallymean.Muchofthemathematics developedinChapters4and5isdesignedtoensurethatthefinalresultsdon’t dependoncoordinates,buteventhenitcanbetrickytointerpretoutcomesin termsofhonestphysicalobservables.
Proceedingasbefore,with
Setting σ = s,weobtaintheequations
andfrom(2.16),afirstintegral
Thefirstequationin(2.18)gives
whichallowsustointegrate(2.19):
,wehave
Tointerpretthissolution,wecangobacktotheCartesiancoordinates(2.14).Itis straightforwardtocheckthat
whichmayberecognizedastheequationforaplanethroughtheorigin.Thegeodesics arethusintersectionsofthespherewithaplanethroughtheorigin.Bydefinition, thesearepreciselythegreatcircles.
Thereisashortcuttothissolution,whichwewilluseinthenextchapter.Note firstthatifwechoosetheintegrationconstants b =1/r0, ϕ0 = π/2,then(2.21)isthe greatcirclearoundtheequator, θ = π/2,ϕ =(s s0)/r0.Thisisaspecialcase,but sincethelineelementissphericallysymmetric,wecanalwaysrotateourcoordinates sothat θ = π/2and dθ/ds =0attheinitialpoint s =0.Thenfrom(2.18)–(2.19),
Since(2.18)isasecondordersystemofdifferentialequations,weareguaranteedthat thehigherderivativesaddnoinformation,so
isasolutionforall s.Bynowundoingtherotationofcoordinates,wecanobtainany othergreatcircle,reproducingthegeneralsolution.
2.2Spacetimeandcausalstructure Intheprecedingsection,wesawsomesimpleexamplesofgeodesicsinflatandcurved two-dimensionalspaces.Theseexampleswereall spatial geodesics,though,geodesics inspaceswhoselineelementswerepositivedefinite.Weareinterestedinspacetime, notjustspace,andthiswillrequireafurtherstep.
Tostart,weneedageneralizationofthelineelementtospacetime.Fortheflat spacetimeofspecialrelativity,weknowtheanswer.Inspecialrelativity,neitherintervalsintimealonenorinspacealonehaveanyinvariantmeaning.ButasEinstein andMinkowskitaughtus,thereisaninvariantcombination,thespacetimeinterval (orpropertime)
alsoknownasthe“Minkowskimetric.”Wewillnormallyuseunitsinwhichthespeed oflightis c =1:ifwemeasuretimeinseconds,wemeasuredistanceinlight-seconds. Wecanthuswrite
Fig.2.1 Thelightconeofthepoint p.Theexpandingcircularcross-sectionsofthefuturelightcone,picturedinthreedimensions,representanexpandingsphereoflightinfour dimensions.
Thecalculationofgeodesicsforthelineelement(2.26)ismathematicallyidentical tothecalculationinflatspace,andthegeodesicsareagainstraightlines.Thephysics, though,isratherdifferent.Ageodesiccannowhavea“length” s thatispositivereal, zero,orimaginary.If ds2 ispositive,ageodesic,ormoregenerallyasegmentofa curve,is“timelike.”Asimpleexampleisthetrajectory x = y = z =0, t = s,for whichanobjectonly“moves”intime.If ds2 isnegative,thesegmentis“spacelike.” Atypicalexampleisacurveinspaceatafixedtime t =0.If ds2 =0,thesegment is“lightlike”or“null.”Atypicalexampleisthetrajectory x = ct.Moregenerally,a nullcurveisapossiblepathofapulseoflight,having“distanceinspace”equalto c timesa“distanceintime.”
Apathiscausalifitistimelikeornulleverywhere.Suchapathcanbetraversed byanobjectmovingnofasterthanlight.Apoint q issaidtobeinthefutureofthe point p ifthereisfuture-directedcausalpathfrom p to q;itisinthepastof p ifthere isafuture-directedcausalpathfrom q to p.Thelightconeof p—thesetofnullcurves passingthrough p—dividesspacetimeintothreeregions,thefuture,past,andpresent of p (seeFig.2.1).Thepathofamassiveobject,calledits“worldline,”alwayslies insideitslightcone.Thisstructureisfamiliarinspecialrelativity,butitexistslocally incurvedspacetimesaswell,althoughtheglobalstructurecanbemorecomplicated.
Thetermsspacelike,timelike,andnullcanalsobeappliedtocertainsubspaces. Definea“hypersurface”tobeasubspaceofspacetime—technically,asubmanifold— thathasadimensiononelessthanthedimensionofspacetime(d =3forourfourdimensionalspacetime).Ahypersurfaceissaidtobespacelikeifitsnormalvectoris timelike,timelikeifitsnormalisspacelike,andnullifitsnormalisnull.
2.3Themetric Wearenowreadytotackletheproblemoffindinggeodesicsinanarbitrarycurved spacetime.Thestartingpointisagainthelineelement.Thelessonwetakefrom
Box2.2Conventions:partI Aspacetimelineelementrequiresachoiceofsign.Thepropertime(2.26)is positiveforpointswithtimelikeseparations(seeSection2.2)andnegativefor pointswithspacelikeseparations.Butwecouldjustaswellhavestartedwith properdistance,
whichdiffersinsign.Option(2.26)iscalledthe“mostlyminuses”or“West Coast”convention;theoppositeis“mostlypluses”or“EastCoast.”Todescribe motioninagravitationalfield,theWestCoastconventionisconvenient,since theparameter s isthenpropertimealongatrajectory.Particlephysicistsalso tendtopreferthatconvention,sinceitmakesthesquareofthefour-momentum positiveonshell.Incosmology,ontheotherhand,theEastCoastconventionis easier,sincethemetricofspaceatconstanttimeisthenpositivedefinite.With onepossibleobscureexception[7],eitherchoicecanbemade.Whatmatters,here andwithotherconventions,isconsistency.
ourexamples—thehallmarkofwhatmathematicianscallRiemannianorpseudoRiemanniangeometry—isaversionofPythagoras’theorem,theprinciplethatthe squareofthedistancebetweentwopointsisquadraticinthecoordinatedifferences. Ina d-dimensionalspacetime,thismeans
forsomesetofcoordinates {xµ} andsomefunctions gµν (x).Notethattheindicesused tolabelcoordinatesinspacetimeconventionallyrangefrom0to d 1,where x0 is normallythe“time”coordinate.Thepositionoftheindices—upforcoordinateslike xµ,downfor gµν —encodesamathematicaldistinctionbetweentangentandcotangent vectors.WewillexplorethisinChapter4;fornow,it’sjustanotationalconvention. Wecanalwaysassumethat gµν issymmetric,thatis, gµν = gνµ,sinceit’seasyto checkthatanyadditionalantisymmetricpiecewouldcanceloutinthesum(2.27).
Thefunctions gµν arecalledthecomponentsofthemetrictensorinthecoordinate system {xµ}.Thisratherlongappellationissometimesabbreviatedto“themetric.” Theshortenedterminologycanbeabitmisleading,though.Wehaveseenthattheline elementinflattwo-dimensionalspacemaybewritteneitheras dx2 + dy2 inCartesian coordinatesoras dr2 + r2dθ2 inpolarcoordinates.Thecomponentsofthemetricare clearlydifferent,butthegeometryisthesame.WeshallseeinChapter4thatthe metrictensor,properlydefined,isindependentofthechoiceofcoordinates;wehave merelyexpressedasingleobjectintwodifferentways.
Equation(2.27),andmanyothers,becomesimplerifweadopttheEinsteinsummationconvention.Thisconventionhastwoparts:
Box2.3Indexgymnastics Apractitionerofgeneralrelativitymustbecomeadeptatjugglingindices.Here areafewtricks(makesuretoworkoutwheretheycomefrom)!
• Anindexthatissummedoveriscalleda“dummyindex.”Anylettermaybe usedforadummyindex,provideditisn’tusedelsewhereinthesameterm:
sincethesumsconsistofexactlythesameterms.
• Suppose Aµν isantisymmetricinitsindices, Sµν issymmetric,and T µν has noparticularsymmetries.Then Aµν Sµν =0and
• If δµ ν istheKroneckerdelta,then
= A
and
= d,where d isthe dimensionofthespaceorspacetime.
1.Agivenindexcanappearatmosttwiceinanytermofanequation.Ifitappears twice,thismustbeonceasalowerindexandonceasanupperindex.
2.Ifanindexappearstwiceinaterm,summationoverthatindexisimplied,and noexplicitsummationsignisneeded.
Thus(2.27)becomes
Thecomponents gµν canbeviewedasasymmetric d × d matrix,withthefirst indexlabelingtherowandthesecondthecolumn.Suchamatrixhas d eigenvalues. Weassumethat gµν isnondegenerate,thatis,thattheeigenvaluesareallnonzero. Thenumbers p ofpositiveeigenvaluesand n ofnegativeeigenvaluesdeterminethe “signature”ofthemetric,whichcanbedenoted(p,n)or p + n,or,mostcommonly inphysics,(+... ...).Thelineelement(2.26),forinstance,hassignature(1, 3) or1+3or(+ ).Whileitisnotobvious,thesignatureisindependentofthe choiceofcoordinates.Alineelementwithsignature(d, 0),like(2.8)or(2.16),iscalled Riemannian;onewithsignature(1,d 1)or(d 1, 1),like(2.26),iscalledLorentzian.
Thematrixinverseof gµν ,looselyreferredtoastheinversemetric,iswrittenwith upperindices,as gρσ.Itsexistenceisguaranteedbythenondegeneracyofthematrix ofcomponents.Bydefinition,theinversemetricsatisfies
where δµ ν istheusualKroneckerdelta.(Readersshouldcheckthatthisis,infact,usual matrixmultiplication.)
2.4Thegeodesicequation Startingwiththegenerallineelement(2.28),wecannowderivethegeodesicequation foranarbitraryspacetime.TheprocedureisexactlythesameasinSection2.1:
whereIhaveusedthenotation
Notethatthevariationhereis not avariationofthemetriccomponents gµν ,but ratherofthepath xµ(σ).But“gµν ”in(2.30)meansthevalue gµν (x(σ))onthepath, sobythechainrule,whenthepathchanges, δgµν = δxρ∂ρ
Thefirsttwotermsinthesecondlineof(2.31)areactuallyequivalent:theydifferonlybyarenamingofthedummyindices µ to ν and ν to µ.Combiningthem, integratingbyparts,andrenamingsomedummyindices,weseethat
Asbefore,thevariation δxρ isarbitrary,soitscoefficientmustvanish:
Equation(2.34)isthegeneralformofthegeodesicequation.Forspacelikeand timelikegeodesics,wecanperformthesametrickweusedinSection2.1,andchoose σ = s.Then
Fornullgeodesics,though, s =0and E =0.Herewehavetwoalternatives.Wecan returntotheideaofageodesicasanautoparallelline—wewilldosoinChapter5—or wecantakealimitof(say)atimelikegeodesic,definingaparameter
andletting E gotozero.Ineithercase,theresultingequationis
Aparameter λ forwhichthegeodesicequationtakesthisformiscalledanaffine parameter.
Thereisanothercommonformforthegeodesicequation,obtainedbydifferentiatingthefirsttermin(2.35)or(2.36)andusingtheproductrule.Themetriccomponents gµν dependon s throughtheirdependenceon x(s),so
Usingtheinversemetric(2.29),itiseasytocheckthat
with
TheΓρ µν arecalledthecomponentsoftheChristoffelortheLevi-Civitaconnection, orsometimesthe“Christoffelsymbols”;inolderpaperstheymaybewrittenas {ρ µν }. Finally,letusconsiderthegeneralizationofthefirstorderequations(2.11)and (2.19).Itisevidentfromthedefinitionofthelineelementthat
Thesearenotindependentequations,though.Theirfirstderivativesgivebackalinear combinationofthegeodesicequations:forinstance,
Inthatsense,(2.41)are“firstintegrals,”obtainedbyintegratinganappropriatecombinationofthegeodesicequations,butwithfixedintegrationconstantsdetermined fromthedefinitionsoftheparameters s and λ
2.5TheNewtonianlimit
Generalrelativityisatheoryofphysics,notmathematics,andweoughttocheckthat thisdescriptionofgeodesicsmakesphysicalsense.Aminimaltestisthat,atleastto averygoodapproximation,wemustbeabletoreproducetheremarkablesuccessesof
Newtoniangravity.Thatis,wemustbeabletofindametricwhosegeodesicsduplicate thetrajectoriesthatobjectsfollowinaNewtoniangravitationalfield.
Forthis,weneedthespacetimemetricintheNewtonianapproximation.Such ametriccomesasasolutionoftheEinsteinfieldequations,whichwewillnotsee untilChapter6.Fornow,letusanticipatetheresultofSection8.2.Asonemight expect,theapproximatemetricdependsontheNewtoniangravitationalpotentialΦ. Itis,further,a“slowmotion”solution—allspeedsaremuchlessthan c—anda“weak field”solution—Φ/c2 isoforder v2/c2,asistypicalforsystemsofgravitationally boundbodies.Thelineelementthentakestheapproximateform
Thegeodesicequationsforthislineelementaresimple.Equation(2.41)becomes
while(2.35)becomes
Ifwenowneglecttermsoforder v2/c2—theorderatwhichordinaryspecialrelativistic correctionsbecomeimportant—theseequationsreduceto
whicharejusttheNewtonianequationsofmotionforanobjectinagravitationalfield. TheappropriatespacetimegeometrycanthusreproduceNewtonianmotion.
Furtherreading ThegeodesicequationcanalsobederivedbymeansoftheusualEuler-Lagrange equations:see,forexample,Section7.6of d’Inverno’s IntroducingEinstein’sRelativity [8].Anintroductiontothelineelement(2.26)inspecialrelativity,focusing onitsgeometricsignificance,maybefound inTaylorandWheeler’sclassic, Spacetime
Physics [9],orinChapter1ofthetextbookbySchutz, AFirstCourseinGeneralRelativity [10].TheinsidecoverofMisner,Thorne,andWheeler’sfamoustextbook Gravitation [11]hasatableofconventionsusedbyvariousauthors;whilesomewhatoutdated,itgivesanoverviewofthe varietyofchoices.
GeodesicsintheSolarSystem Attheendoftheprecedingchapter,wesawthatgeneralrelativitycansuccessfully reproducetheequationsofmotionofNewtoniangravity.Itdoesmorethanthat, ofcourse:evenintheSolarSystem,itpredictssmallbutobservablecorrectionsto Newtonianphysics.Thefour“classicaltests”ofgeneralrelativity—theprecessionof planetaryorbits,thedeflectionoflightbyagravitationalfield,thegravitationaltime delayoflight,andgravitationalredshiftandtimedilation—canalsobeobtainedfrom thegeodesicequation.
3.1TheSchwarzschildmetric Toinvestigatethesepredictions,weneedalineelementthatismoreaccuratethan theweakfieldapproximation(2.43).Thespacetimegeometryoutsideanonrotating sphericalmasswasfirstworkedoutbyKarlSchwarzschildin1916,verysoonafter Einsteinpublishedthefieldequationsofgeneralrelativity.Wewillderivetheresulting SchwarzschildmetricinChapter10,wherewewillobtainalineelement
Notethatwhen m =0,thisreducestotheflatMinkowskimetric(2.26)inthespherical coordinatesofeqn(2.15).Similarly,as r approachesinfinity,themetricapproaches theMinkowskimetric;theSchwarzschildmetricis“asymptoticallyflat.”
Theparameter m isproportionaltothemassofthegravitatingobject,
where M istheordinarymassand G isNewton’sgravitationalconstant.Thequantity m hasunitsoflength,withavalueofabout4.5mmfortheEarthand1.5kmforthe Sun.Theratio m/r in(3.1)isthereforetypicallyverysmall;evenatthesurfaceofthe Sun,itisonlyabout2 × 10 6.TheSchwarzschildlineelementfortheSolarSystem thusdeviatesonlyveryslightlyfromthelineelementofflatspacetime.Nevertheless, thissmalldifferenceisresponsibleforthemotionoftheplanets.
3.2GeodesicsintheSchwarzschildmetric Letusnowcalculatethegeodesicsforthelineelement(3.1).Wewillstartwiththe timelikegeodesicsthatdescribethemotionofplanets,andthenturntothenull
GeneralRelativity:AConciseIntroduction.StevenCarlip c StevenCarlip2019. Publishedin2019byOxfordUniversityPress.DOI:10.1093/oso/9780198822158.001.0001
Box3.1Coordinates:partII
Itistemptingtothinkofthecoordinate r asthefamiliarradialcoordinateof flatspace.Forlarge r,thisisagoodapproximation,sincethelineelementis nearlyflat.Forsmall r,though,thispicturebreaksdown.Asweshallseein Chapter10,theflatspacecoordinate r servesmanydifferentfunctions—itis, forexample,botharadialdistanceandasquarerootoftheareaofasphere.In acurvedspacetime,thesearenolongerequivalent,anddifferentchoicesleadto differentformsofthelineelement.
Forinstance,in(3.1),coordinatedistancesintheradialdirectiondifferfrom thoseinangulardirections.Butifwetransformto“isotropiccoordinates,”
thelineelementbecomes
with“Cartesian”coordinates x, y,and z thatcontributesymmetricallyto isotropicdistance¯ r =(x2 + y2 + z2)1/2.Atlargedistances,thedifferencebetween r and¯ r isinsignificant,andtofirstorderin m/r thepredictionsforphenomena suchasthedeflectionoflightareidentical.Atthenextorder,though,theequationsdependonwhich“r”weuse.Thisisnotatruephysicaldifference,butto seethistakessomecarefulwork[12]:wehavetoreexpresstheresultsinterms ofunambiguouslymeasurablecoordinate-independentquantities.
geodesicsthatdescribethepathsoflightrays.From(3.1),thenonzerocomponentsof themetricare
wheretheindexlabels t,r,θ,ϕ meanthesamethingas0, 1, 2, 3.Inwritingoutthe geodesicequation,it’shelpfultokeeptwoshortcutsinmind:
• Themetricisdiagonal,soinsumslike gρµ dxµ ds onlytermswith µ = ρ appear.
• Thecomponents(3.3)dependonlyon r and θ,so ∂µgστ iszerounless µ is r or θ. Wenowlookatthefourequations(2.35)andthefirstintegral(2.41):
µ = r :complicatedexpression;we’llusethefirstintegralinstead ,
Letusstartwith(3.7).AsinSection2.1,wecanrotateourcoordinatessothatatan initialtime, θ = π/2and dθ/ds =0.Then(3.7)impliesthat
foralltimes.JustasinNewtoniangravity,trajectoriesremaininasingleplane. Next,wecanintegrate(3.6)and3.8):
where E and L areintegrationconstants.(Thetildesareaconventiontodistinguish timelikegeodesics,thecasehere,fromnullgeodesics,whichwillcomelater.)Weare leftwith(3.9),whichnowsimplifiesto
Equation(3.13)isintegrable,andinprinciple(3.10)–(3.13)giveacompletedescriptionofthegeodesicsoftheSchwarzschildgeometry.Buttheresultscanonlybe expressedintermsofcertainspecialfunctions,ellipticintegrals,thatareunfamiliar tomostphysicists.Soeventhoughexactsolutionsareavailable,itisusefultofind approximationsthatinvolvemoreeasilyunderstoodfunctions.
