剑桥大学――广义相对论

GeneralRelativityDr.P.D.D’Eath1Lent19981LATEXedbyPaulMetcalfe–[e-mail]:2.7Date:1998-06-0508:56:46+01Thefollowingpeoplehavemaintainedthesenotes.–datePaulMetcalfeContentsIntroductionv1Outlineofthetheory11.1Curvedspaces..............................11.1.1Geodesics............................21.2Theprincipleofequivalence......................21.2.1Uniquenessoffreefall.....................21.2.2Equivalenceprinciple......................31.2.3Consequencesforlightpropagation..............31.2.4Specialrelativityandgravitation................41.3Outlineofgeneralrelativity.......................41.4StaticspacetimeandNewtoniangravity................51.4.1Staticmetrics..........................51.4.2Newtonianlimit........................62Metricdifferentialgeometry72.1Basictensors..............................72.1.1Examplesoftensors......................82.1.2Operationspreservingtensorproperty.............82.1.3Quotienttheorem........................92.1.4Inversemetrictensor......................92.1.5Raisingandloweringofindices................92.1.6Partialderivativesoftensors..................102.2Lengthsandgeodesics.........................102.3Anglesbetweenvectors.........................102.4Lengthsofcurves............................102.5Geodesics................................112.6CovariantdifferentiationandChristoffelsymbols...........122.6.1Transformationpropertiesof.................132.6.2Actionofonothertypesoftensor.............132.7Differentiationalongacurve:geodesics................142.8Localinertialframes..........................152.9Curvature................................152.10Geodesicdeviation...........................16iiiivCONTENTS3Vacuumgravitationalfields193.1Thevacuumfieldequations.......................193.2TheSchwarzschildmetric........................203.3Gravitationalredshift..........................223.4Particleandphotonpaths........................223.5Perihelionadvance...........................233.6Lightdeflection.............................243.7Blackholesandtheeventhorizon...................254MatterinGeneralRelativity274.1Physicallaws..............................274.2Energy-momentumtensors.......................274.3TheEinsteinfieldequations.......................284.3.1TheBianchiidentities.....................284.3.2Fieldequations.........................28IntroductionThesenotesarebasedonthecourse“GeneralRelativity”givenbyDr.P.D.D’EathinCambridgeintheLentTerm1998.ThesetypesetnotesaretotallyunconnectedwithDr.D’Eath.Therecommendedbooksforthiscoursearediscussedinthebibliography.Othersetsofnotesareavailablefordifferentcourses.Atthetimeoftypingthesecourseswere:ProbabilityDiscreteMathematicsAnalysisFurtherAnalysisMethodsQuantumMechanicsFluidDynamics1QuadraticMathematicsGeometryDynamicsofD.E.’sFoundationsofQMElectrodynamicsMethodsofMath.PhysFluidDynamics2Waves(etc.)StatisticalPhysicsGeneralRelativityDynamicalSystemsPhysiologicalFluidDynamicsBifurcationsinNonlinearConvectionSlowViscousFlowsTurbulenceandSelf-SimilarityAcousticsNon-NewtonianFluidsSeismicWavesTheymaybedownloadedfrom://@lists.cam.ac.uktogetacopyofthesetsyourequire.vCopyright(c)TheArchimedeans,CambridgeUniversity.Allrightsreserved.Redistributionanduseofthesenotesinelectronicorprintedform,withorwithoutmodification,arepermittedprovidedthatthefollowingconditionsaremet:1.Redistributionsoftheelectronicfilesmustretaintheabovecopyrightnotice,thislistofconditionsandthefollowingdisclaimer.2.Redistributionsinprintedformmustreproducetheabovecopyrightnotice,thislistofconditionsandthefollowingdisclaimer.3.Allmaterialsderivedfromthesenotesmustdisplaythefollowingacknowledge-ment:ThisproductincludesnotesdevelopedbyTheArchimedeans,CambridgeUniversityandtheircontributors.4.NeitherthenameofTheArchimedeansnorthenamesoftheircontributorsmaybeusedtoendorseorpromoteproductsderivedfromthesenotes.5.Neitherthesenotesnoranyderivedproductsmaybesoldonafor-profitbasis,althoughafeemayberequiredforthephysicalactofcopying.6.Youmustcauseanyeditedversionstocarryprominentnoticesstatingthatyoueditedthemandthedateofanychange.THESENOTESAREPROVIDEDBYTHEARCHIMEDEANSANDCONTRIB-UTORS“ASIS”ANDANYEXPRESSORIMPLIEDWARRANTIES,INCLUDING,BUTNOTLIMITEDTO,THEIMPLIEDWARRANTIESOFMERCHANTABIL-ITYANDFITNESSFORAPARTICULARPURPOSEAREDISCLAIMED.INNOEVENTSHALLTHEARCHIMEDEANSORCONTRIBUTORSBELIABLEFORANYDIRECT,INDIRECT,INCIDENTAL,SPECIAL,EXEMPLARY,ORCONSE-QUENTIALDAMAGESHOWEVERCAUSEDANDONANYTHEORYOFLI-ABILITY,WHETHERINCONTRACT,STRICTLIABILITY,ORTORT(INCLUD-INGNEGLIGENCEOROTHERWISE)ARISINGINANYWAYOUTOFTHEUSEOFTHESENOTES,EVENIFADVISEDOFTHEPOSSIBILITYOFSUCHDAM-AGE.Chapter1Outlineofthetheory1.1CurvedspacesConsideratwodimensionalcurvedsurfaceinEuclidean,forinstancewiththedefiningequation.Wedistinguishbetweentheextrinsicandintrinsicpropertiesofsuchasurface.Theextrinsicpropertiesdescribetherelationbetweenthesurfaceandthesurround-ing3dime