Introduction to Differential Geometry & General Re

IntroductiontoDifferentialGeometry&GeneralRelativityTTTThhhhiiiirrrrddddPPPPrrrriiiinnnnttttiiiinnnnggggJJJJaaaannnnuuuuaaaarrrryyyy2222000000002222LLLLeeeeccccttttuuuurrrreeeeNNNNooootttteeeessssbbbbyyyySSSStttteeeeffffaaaannnn:Distance,OpenSets,ParametricSurfacesandSmoothFunctions2.SmoothManifoldsandScalarFields3.TangentVectorsandtheTangentSpace4.ContravariantandCovariantVectorFields5.TensorFields6.RiemannianManifolds7.LocallyMinkowskianManifolds:AnIntroductiontoRelativity8.CovariantDifferentiation9.GeodesicsandLocalInertialFrames10.TheRiemannCurvatureTensor11.ALittleMoreRelativity:ComovingFramesandProperTime12.TheStressTensorandtheRelativisticStress-EnergyTensor13.TwoBasicPremisesofGeneralRelativity14.TheEinsteinFieldEquationsandDerivationofNewton'sLaw15.TheSchwarzschildMetricandEventHorizons16.WhiteDwarfs,NeutronStarsandBlackHoles,byGregoryC.Levine31.PreliminariesDistanceandOpenSetsHere,wedojustenoughtopologysoastobeabletotalkaboutsmoothmanifolds.Webeginwithn-dimensionalEuclideanspaceEn={(y1,y2,...,yn)|yiéR}.Thus,E1isjusttherealline,E2istheEuclideanplane,andE3is3-dimensionalEuclideanspace.Themagnitude,ornorm,||yyyy||ofyyyy=(y1,y2,...,yn)inEnisdefinedtobe||yyyy||=y12+y22+...+yn2,whichwethinkofasitsdistancefromtheorigin.Thus,thedistancebetweentwopointsyyyy=(y1,y2,...,yn)andzzzz=(z1,z2,...,zn)inEnisdefinedasthenormofzzzz-yyyy:DistanceFormulaDistancebetweenyyyyandzzzz=||zzzz----yyyy||=(z1-y1)2+(z2-y2)2+...+(zn-yn)2.Proposition1.1(Propertiesofthenorm)Thenormsatisfiesthefollowing:(a)||yyyy||≥0,and||yyyy||=0iffyyyy=0(positivedefinite)(b)||¬yyyy||=|¬|||yyyy||forevery¬éRandyyyyéEn.(c)||yyyy+zzzz||≤||yyyy||+||zzzz||foreveryyyyy,zzzzéEn(triangleinequality1)(d)||yyyy-zzzz||≤||yyyy----||+||||foreveryyyyy,,,,zzzz,,,,(triangleinequality2)TheproofofProposition1.1isanexercisewhichmayrequirereferencetoalinearalgebratext(see“innerproducts”).Definition1.2ASubsetUofEniscalledopenif,foreveryyyyyinU,allpointsofEnwithinsomepositivedistancerofyyyyarealsoinU.(Thesizeofrmaydependonthepointyyyychosen.Illustrationinclass).Intuitively,anopensetisasolidregionminusitsboundary.Ifweincludetheboundary,wegetaclosedset,whichformallyisdefinedasthecomplementofanopenset.Examples1.3(a)IfaéEn,thentheopenballwithcenteraaaaandradiusrrrristhesubsetB(aaaa,r)={xéEn|||xxxx-aaaa||r}.4Openballsareopensets:IfxxxxéB(aaaa,r),then,withs=r-||xxxx----aaaa||,onehasB(xxxx,s)¯B(aaaa,r).(b)Enisopen.(c)Øisopen.(d)Unionsofopensetsareopen.(e)Opensetsareunionsofopenballs.(Proofinclass)Definition1.4NowletM¯Es.AsubsetV¯MiscalledopeninMMMM(orrelativelyopen)if,foreveryyyyyinV,allpointsofMwithinsomepositivedistancerofyyyyarealsoinV.Examples1.5(a)OpenballsinMMMMIfM¯Es,mmmméM,andr0,defineBM(mmmm,r)={xéM|||xxxx-mmmm||r}.ThenBM(mmmm,r)=B(mmmm,r)ÚM,andsoBM(mmmm,r)isopeninM.(b)MisopeninM.(c)ØisopeninM.(d)UnionsofopensetsinMareopeninM.(e)OpensetsinMareunionsofopenballsinM.ParametricPathsandSurfacesinEEEE3333Fromnowon,thethreecoordinatesof3-spacewillbereferredtoasy1,y2,andy3.Definition1.6AsmoothpathinE3isasetofthreesmooth(infinitelydifferentiable)real-valuedfunctionsofasinglerealvariablet:y1=y1(t),y2=y2(t),y3=y3(t).Thevariabletiscalledtheparameterofthecurve.Thepathisnon-singularifthevector(dy1dt,dy2dt,dy3dt)isnowherezero.Notes(a)Insteadofwritingy1=y1(t),y2=y2(t),y3=y3(t),weshallsimplywriteyi=yi(t).(b)Sincethereisnothingspecialaboutthreedimensions,wedefineasmoothpathinEEEEnnnninexactlythesameway:asacollectionofsmoothfunctionsyi=yi(t),wherethistimeigoesfrom1ton.5Examples1.7(a)StraightlinesinE3(b)CurvesinE3(circles,etc.)Definition1.8AsmoothsurfaceimmersedinEEEE3333isacollectionofthreesmoothreal-valuedfunctionsoftwovariablesx1andx2(noticethatxfinallymakesadebut).y1=y1(x1,x2)y2=y2(x1,x2)y3=y3(x1,x2),orjustyi=yi(x1,x2)(i=1,2,3).Wealsorequirethatthe3¿2matrixwhoseijentryis∂yi∂xjhasranktwo.Wecallx1andx2theparametersorlocalcoordinates.Examples1.9(a)PlanesinE3(b)Theparaboloidy3=y12+y22(c)Thespherey12+y22+y32=1,usingsphericalpolarcoordinates:y1=sinx1cosx2y2=sinx1sinx2y3=cosx1(d)Theellipsoidy12a2+y22b2+y32c2=1,wherea,bandcarepositiveconstants.(e)WecalculatetherankoftheJacobeanmatrixforsphericalpolarcoordinates.(f)Thetoruswithradiiab:y1=(a+bcosx2)cosx1y2=(a+bcosx2)sinx1y3=bsinx2QuestionTheparametricequationsofasurfaceshowushowtoobtainapointonthesurfaceonceweknowthetwolocalcoordinates(parameters).Inotherwords,wehavespecifiedafunctionE2’E3.HowdoweobtainthelocalcoordinatesfromtheCartesiancoordinatesy1,y2,y3?AnswerWeneedtosolveforthelocalcoordinatesxiasfunctionsofyj.Thiswedoinoneortw