semi-riemannian geometry and general relativity - s. sternberg

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1、Semi-RiemannGeometryandGeneralRelativityShlomoSternbergSeptember24,200320.1IntroductionThisbookrepresentscoursenotesforaonesemestercourseattheundergraduatelevelgivinganintroductiontoRiemanniangeometryanditsprincipalphysicalapplication,Einstein’stheoryofgeneralrelativity.Thebackgroundassum

2、edisagoodgroundinginlinearalgebraandinadvancedcalculus,preferablyinthelanguageofdifferentialforms.ChapterIintroducesthevariouscurvaturesassociatedtoahypersurfaceembeddedinEuclideanspace,motivatedbytheformulaforthevolumefortheregionobtainedbythickeningthehypersurfaceononeside.Ifwethickenthe

3、hypersurfacebyanamounthinthenormaldirection,thisformulaisapolynomialinhwhosecoefficientsareintegralsoverthehypersurfaceoflocalexpressions.Theselocalexpressionsareelementarysymmetricpolynomialsinwhatareknownastheprincipalcurvatures.Theprecisedefinitionsaregiveninthetext.Thechapterculminateswi

4、thGauss’Theoremaegregiumwhichassertsthatifwethickenatwodimensionalsurfaceevenlyonbothsides,thenthetheseintegrandsdependonlyontheintrinsicgeometryofthesurface,andnotonhowthesurfaceisembedded.Wegivetwoproofsofthisimportanttheorem.(Wegiveseveralmorelaterinthebook.)Thefirstproofmakesuseof“norm

5、alcoor-dinates”whichbecomesoimportantinRiemanniangeometryand,as“inertialframes,”ingeneralrelativity.ItwasthistheoremofGauss,andparticularlytheverynotionof“intrinsicgeometry”,whichinspiredRiemanntodevelophisgeometry.ChapterIIisarapidreviewofthedifferentialandintegralcalculusonman-ifolds,inc

6、ludingdifferentialforms,thedoperator,andStokes’theorem.AlsovectorfieldsandLiederivatives.Attheendofthechapterareaseriesofsec-tionsinexerciseformwhichleadtothenotionofparalleltransportofavectoralongacurveonaembeddedsurfaceasbeingassociatedwiththe“rollingofthesurfaceonaplanealongthecurve”.Cha

7、pterIIIdiscussesthefundamentalnotionsoflinearconnectionsandtheircurvatures,andalsoCartan’smethodofcalculatingcurvatureusingframefieldsanddifferentialforms.WeshowthatthegeodesicsonaLiegroupequippedwithabi-invariantmetricarethetranslatesoftheoneparametersubgroups.Ashort