Semi-Riemannian Geometry and General Relativity - S. Sternberg.pdf

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Semi-RiemannGeometryandGeneralRelativity
ShlomoSternberg
September24,2003
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0.1Introduction
Thisbookrepresentscoursenotesforaonesemestercourseattheundergraduate
levelgivinganintroductiontoRiemanniangeometryanditsprincipalphysical
application,Einstein’stheoryofgeneralrelativity.Thebackgroundassumedis
agoodgroundinginlinearalgebraandinadvancedcalculus,preferablyinthe
languageofdierentialforms.
ChapterIintroducesthevariouscurvaturesassociatedtoahypersurface
embeddedinEuclideanspace,motivatedbytheformulaforthevolumefor
theregionobtainedbythickeningthehypersurfaceononeside.Ifwethicken
thehypersurfacebyanamounthinthenormaldirection,thisformulaisa
polynomialinhwhosecoecientsareintegralsoverthehypersurfaceoflocal
expressions.Theselocalexpressionsareelementarysymmetricpolynomialsin
whatareknownastheprincipalcurvatures.Theprecisedefinitionsaregivenin
thetext.ThechapterculminateswithGauss’Theoremaegregiumwhichasserts
thatifwethickenatwodimensionalsurfaceevenlyonbothsides,thenthethese
integrandsdependonlyontheintrinsicgeometryofthesurface,andnotonhow
thesurfaceisembedded.Wegivetwoproofsofthisimportanttheorem.(We
giveseveralmorelaterinthebook.)Thefirstproofmakesuseof“normalcoor-
dinates”whichbecomesoimportantinRiemanniangeometryand,as“inertial
frames,”ingeneralrelativity.ItwasthistheoremofGauss,andparticularly
theverynotionof“intrinsicgeometry”,whichinspiredRiemanntodevelophis
geometry.
ChapterIIisarapidreviewofthedierentialandintegralcalculusonman-
ifolds,includingdierentialforms,thedoperator,andStokes’theorem.Also
vectorfieldsandLiederivatives.Attheendofthechapterareaseriesofsec-
tionsinexerciseformwhichleadtothenotionofparalleltransportofavector
alongacurveonaembeddedsurfaceasbeingassociatedwiththe“rollingof
thesurfaceonaplanealongthecurve”.
ChapterIIIdiscussesthefundamentalnotionsoflinearconnectionsandtheir
curvatures,andalsoCartan’smethodofcalculatingcurvatureusingframefields
anddierentialforms.WeshowthatthegeodesicsonaLiegroupequippedwith
abi-invariantmetricarethetranslatesoftheoneparametersubgroups.Ashort
exercisesetattheendofthechapterusestheCartancalculustocomputethe
curvatureoftheSchwartzschildmetric.Asecondexercisesetcomputessome
geodesicsintheSchwartzschildmetricleadingtotwoofthefamouspredictions
ofgeneralrelativity:theadvanceoftheperihelionofMercuryandthebending
oflightbymatter.Ofcoursethetheoreticalbasisofthesecomputations,i.e.
thetheoryofgeneralrelativity,willcomelater,inChapterVII.
ChapterIVbeginsbydiscussingthebundleofframeswhichisthemodern
settingforCartan’scalculusof“movingframes”andalsothejumpingopoint
forthegeneraltheoryofconnectionsonprincipalbundleswhichlieatthebase
ofsuchmodernphysicaltheoriesasYang-Millsfields.Thischapterseemsto
presentthemostdicultyconceptuallyforthestudent.
ChapterVdiscussesthegeneraltheoryofconnectionsonfiberbundlesand
thenspecializetoprincipalandassociatedbundles.
0.1.INTRODUCTION 3
ChapterVIreturnstoRiemanniangeometryanddiscussesGauss’slemma
whichassertsthattheradialgeodesicsemanatingfromapointareorthogo-
nal(intheRiemannmetric)totheimagesundertheexponentialmapofthe
spheresinthetangentspacecenteredattheorigin.Fromthisoneconcludes
thatgeodesics(definedasselfparallelcurves)locallyminimizearclengthina
Riemannmanifold.
ChapterVIIisarapidreviewofspecialrelativity.Itisassumedthatthe
studentswillhaveseenmuchofthismaterialinaphysicscourse.
ChapterVIIIisthehighpointofthecoursefromthetheoreticalpointof
view.WediscussEinstein’sgeneraltheoryofrelativityfromthepointofviewof
theEinstein-Hilbertfunctional.InfactweborrowthetitleofHilbert’spaperfor
theChapterheading.Wealsointroducetheprincipleofgeneralcovariance,first
introducebyEinstein,Infeld,andHomanntoderivethe“geodesicprinciple”
andgiveawholeseriesofotherapplicationsofthisprinciple.
ChapterIXdiscussescomputationalmethodsderivingfromthenotionof
aRiemanniansubmersion,introducedanddevelopedbyRobertHermannand
perfectedbyBarrettO’Neill.ItisthenaturalsettingforthegeneralizedGauss-
Codazzitypeequations.Althoughtechnicallysomewhatdemandingatthebe-
ginning,therangeofapplicationsjustifiestheeortinsettingupthetheory.
Applicationsrangefromcurvaturecomputationsforhomogeneousspacestocos-
mogenyandeschatologyinFriedmantypemodels.
ChapterXdiscussesthePetrovclassification,usingcomplexgeometry,ofthe
varioustypesofsolutionstotheEinsteinequationsinfourdimensions.This
classificationledKerrtohisdiscoveryoftherotatingblackholesolutionwhich
isatopicforacourseinitsown.Theexpositioninthischapterfollowsjoint
workwithKostant.
ChapterXIisintheformofaenlargedexercisesetonthestaroperator.It
isessentiallyindependentoftheentirecourse,butIthoughtitusefultoinclude,
asitwouldbeofinterestinanymoreadvancedtreatmentoftopicsinthecourse.
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Contents
0.1Introduction.............................. 2
1Theprincipalcurvatures. 11
1.1Volumeofathickenedhypersurface................. 11
1.2TheGaussmapandtheWeingartenmap.............. 13
1.3Proofofthevolumeformula. .................... 16
1.4Gauss’stheoremaegregium...................... 19
1.4.1Firstproof,usinginertialcoordinates............ 22
1.4.2Secondproof.TheBrioschiformula............. 25
1.5Problemset-Surfacesofrevolution................. 27
2Rulesofcalculus. 31
2.1Superalgebras. ............................ 31
2.2Dierentialforms. .......................... 31
2.3Thedoperator............................. 32
2.4Derivations............................... 33
2.5Pullback. ............................... 34
2.6Chainrule. .............................. 35
2.7Liederivative.............................. 35
2.8Weil’sformula. ............................ 36
2.9Integration............................... 38
2.10Stokestheorem............................. 38
2.11Liederivativesofvectorfields..................... 39
2.12Jacobi’sidentity. ........................... 40
2.13Leftinvariantforms.......................... 41
2.14TheMaurerCartanequations. ................... 43
2.15Restrictiontoasubgroup...................... 43
2.16Frames. ................................ 44
2.17Euclideanframes............................ 45
2.18Framesadaptedtoasubmanifold. ................. 47
2.19Curvesandsurfaces-theirstructureequations........... 48
2.20Thesphereasanexample....................... 48
2.21Ribbons................................ 50
2.22Developingaribbon.......................... 51
2.23Paralleltransportalongaribbon. ................. 52
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