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Semi-Riemann Geometry and General Relativity
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Title: Semi-Riemann Geometry and General Relativity
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Creator: Sternberg, Shlomo
Publication Date: 9/4/2003
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Subjects / Keywords: Bundles, Calculus, Die Grundlagen, Der Physik, Frames, Gauss's Lemma, Levi-Civita, Petrov, Principal Curvatures, Relativity, Star, Submersions, ogt+ isbn: 9781616100674
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Abstract: This free textbook by Harvard Professor Shlomo Sternberg provides a one semester course at the undergraduate level giving an introduction to Riemannian geometry and its principal physical application, Einstein’s theory of general relativity. The background assumed is a good grounding in linear algebra and in advanced calculus, preferably in the language of differential forms. Contents: 1) The Principal Curvatures. 2) Rules of Calculus. 3) Levi-Civita Connections. 4) The Bundle of Frames. 5) Connections on Principal Bundles. 6) Gauss's lemma. 7) Special Relativity. 8) Die Grundlagen der Physik. 9) Submersions. 10) Petrov types. 11) Star.
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Semi-RiemannGeometryandGeneralRelativityShlomoSternbergSeptember24,2003

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20.1IntroductionThisbookrepresentscoursenotesforaonesemestercourseattheundergraduatelevelgivinganintroductiontoRiemanniangeometryanditsprincipalphysicalapplication,Einstein'stheoryofgeneralrelativity.Thebackgroundassumedisagoodgroundinginlinearalgebraandinadvancedcalculus,preferablyinthelanguageofdierentialforms.ChapterIintroducesthevariouscurvaturesassociatedtoahypersurfaceembeddedinEuclideanspace,motivatedbytheformulaforthevolumefortheregionobtainedbythickeningthehypersurfaceononeside.Ifwethickenthehypersurfacebyanamounthinthenormaldirection,thisformulaisapolynomialinhwhosecoecientsareintegralsoverthehypersurfaceoflocalexpressions.Theselocalexpressionsareelementarysymmetricpolynomialsinwhatareknownastheprincipalcurvatures.Theprecisedenitionsaregiveninthetext.ThechapterculminateswithGauss'Theoremaegregiumwhichassertsthatifwethickenatwodimensionalsurfaceevenlyonbothsides,thenthetheseintegrandsdependonlyontheintrinsicgeometryofthesurface,andnotonhowthesurfaceisembedded.Wegivetwoproofsofthisimportanttheorem.Wegiveseveralmorelaterinthebook.Therstproofmakesuseofnormalcoor-dinates"whichbecomesoimportantinRiemanniangeometryand,asinertialframes,"ingeneralrelativity.ItwasthistheoremofGauss,andparticularlytheverynotionofintrinsicgeometry",whichinspiredRiemanntodevelophisgeometry.ChapterIIisarapidreviewofthedierentialandintegralcalculusonman-ifolds,includingdierentialforms,thedoperator,andStokes'theorem.AlsovectoreldsandLiederivatives.Attheendofthechapterareaseriesofsec-tionsinexerciseformwhichleadtothenotionofparalleltransportofavectoralongacurveonaembeddedsurfaceasbeingassociatedwiththerollingofthesurfaceonaplanealongthecurve".ChapterIIIdiscussesthefundamentalnotionsoflinearconnectionsandtheircurvatures,andalsoCartan'smethodofcalculatingcurvatureusingframeeldsanddierentialforms.WeshowthatthegeodesicsonaLiegroupequippedwithabi-invariantmetricarethetranslatesoftheoneparametersubgroups.AshortexercisesetattheendofthechapterusestheCartancalculustocomputethecurvatureoftheSchwartzschildmetric.AsecondexercisesetcomputessomegeodesicsintheSchwartzschildmetricleadingtotwoofthefamouspredictionsofgeneralrelativity:theadvanceoftheperihelionofMercuryandthebendingoflightbymatter.Ofcoursethetheoreticalbasisofthesecomputations,i.e.thetheoryofgeneralrelativity,willcomelater,inChapterVII.ChapterIVbeginsbydiscussingthebundleofframeswhichisthemodernsettingforCartan'scalculusofmovingframes"andalsothejumpingopointforthegeneraltheoryofconnectionsonprincipalbundleswhichlieatthebaseofsuchmodernphysicaltheoriesasYang-Millselds.Thischapterseemstopresentthemostdicultyconceptuallyforthestudent.ChapterVdiscussesthegeneraltheoryofconnectionsonberbundlesandthenspecializetoprincipalandassociatedbundles.

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0.1.INTRODUCTION3ChapterVIreturnstoRiemanniangeometryanddiscussesGauss'slemmawhichassertsthattheradialgeodesicsemanatingfromapointareorthogo-nalintheRiemannmetrictotheimagesundertheexponentialmapofthespheresinthetangentspacecenteredattheorigin.FromthisoneconcludesthatgeodesicsdenedasselfparallelcurveslocallyminimizearclengthinaRiemannmanifold.ChapterVIIisarapidreviewofspecialrelativity.Itisassumedthatthestudentswillhaveseenmuchofthismaterialinaphysicscourse.ChapterVIIIisthehighpointofthecoursefromthetheoreticalpointofview.WediscussEinstein'sgeneraltheoryofrelativityfromthepointofviewoftheEinstein-Hilbertfunctional.InfactweborrowthetitleofHilbert'spaperfortheChapterheading.Wealsointroducetheprincipleofgeneralcovariance,rstintroducebyEinstein,Infeld,andHomanntoderivethegeodesicprinciple"andgiveawholeseriesofotherapplicationsofthisprinciple.ChapterIXdiscussescomputationalmethodsderivingfromthenotionofaRiemanniansubmersion,introducedanddevelopedbyRobertHermannandperfectedbyBarrettO'Neill.ItisthenaturalsettingforthegeneralizedGauss-Codazzitypeequations.Althoughtechnicallysomewhatdemandingatthebe-ginning,therangeofapplicationsjustiestheeortinsettingupthetheory.Applicationsrangefromcurvaturecomputationsforhomogeneousspacestocos-mogenyandeschatologyinFriedmantypemodels.ChapterXdiscussesthePetrovclassication,usingcomplexgeometry,ofthevarioustypesofsolutionstotheEinsteinequationsinfourdimensions.ThisclassicationledKerrtohisdiscoveryoftherotatingblackholesolutionwhichisatopicforacourseinitsown.TheexpositioninthischapterfollowsjointworkwithKostant.ChapterXIisintheformofaenlargedexercisesetonthestaroperator.Itisessentiallyindependentoftheentirecourse,butIthoughtitusefultoinclude,asitwouldbeofinterestinanymoreadvancedtreatmentoftopicsinthecourse.

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Contents0.1Introduction..............................21Theprincipalcurvatures.111.1Volumeofathickenedhypersurface.................111.2TheGaussmapandtheWeingartenmap..............131.3Proofofthevolumeformula.....................161.4Gauss'stheoremaegregium......................191.4.1Firstproof,usinginertialcoordinates............221.4.2Secondproof.TheBrioschiformula.............251.5Problemset-Surfacesofrevolution.................272Rulesofcalculus.312.1Superalgebras.............................312.2Dierentialforms...........................312.3Thedoperator.............................322.4Derivations...............................332.5Pullback................................342.6Chainrule...............................352.7Liederivative..............................352.8Weil'sformula.............................362.9Integration...............................382.10Stokestheorem.............................382.11Liederivativesofvectorelds.....................392.12Jacobi'sidentity............................402.13Leftinvariantforms..........................412.14TheMaurerCartanequations....................432.15Restrictiontoasubgroup......................432.16Frames.................................442.17Euclideanframes............................452.18Framesadaptedtoasubmanifold..................472.19Curvesandsurfaces-theirstructureequations...........482.20Thesphereasanexample.......................482.21Ribbons................................502.22Developingaribbon..........................512.23Paralleltransportalongaribbon..................525

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6CONTENTS2.24SurfacesinR3.............................533Levi-CivitaConnections.573.1Denitionofalinearconnectiononthetangentbundle......573.2Christoelsymbols...........................583.3Paralleltransport...........................583.4Geodesics................................603.5Covariantdierential.........................613.6Torsion.................................633.7Curvature................................633.8Isometricconnections.........................653.9Levi-Civita'stheorem.........................653.10Geodesicsinorthogonalcoordinates.................673.11Curvatureidentities..........................683.12Sectionalcurvature..........................693.13Riccicurvature.............................693.14Bi-invariantmetricsonaLiegroup.................703.14.1TheLiealgebraofaLiegroup................703.14.2ThegeneralMaurer-Cartanform...............723.14.3Leftinvariantandbi-invariantmetrics............733.14.4Geodesicsarecosetsofoneparametersubgroups......743.14.5TheRiemanncurvatureofabi-invariantmetric......753.14.6Sectionalcurvatures......................753.14.7TheRiccicurvatureandtheKillingform..........753.14.8Bi-invariantformsfromrepresentations...........763.14.9TheWeinbergangle......................783.15Frameelds..............................783.16Curvaturetensorsinaframeeld..................793.17Frameeldsandcurvatureforms...................793.18Cartan'slemma............................823.19Orthogonalcoordinatesonasurface.................833.20ThecurvatureoftheSchwartzschildmetric............843.21GeodesicsoftheSchwartzschildmetric...............853.21.1Massiveparticles........................883.21.2Masslessparticles.......................934Thebundleofframes.954.1Connectionandcurvatureformsinaframeeld..........954.2Changeofframeeld.........................964.3Thebundleofframes.........................984.3.1Theform#...........................994.3.2Theform#intermsofaframeeld.............994.3.3Thedenitionof !......................994.4Theconnectionforminaframeeldasapull-back........1004.5Gauss'theorems............................1034.5.1EquationsofstructureofEuclideanspace..........103

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CONTENTS74.5.2EquationsofstructureofasurfaceinR3..........1044.5.3Theoremaegregium......................1044.5.4Holonomy............................1044.5.5Gauss-Bonnet.........................1055Connectionsonprincipalbundles.1075.1Submersions,brations,andconnections..............1075.2Principalbundlesandinvariantconnections.............1115.2.1Principalbundles.......................1115.2.2Connectionsonprincipalbundles..............1135.2.3Associatedbundles......................1155.2.4Sectionsofassociatedbundles................1165.2.5Associatedvectorbundles...................1175.2.6Exteriorproductsofvectorvaluedforms..........1195.3Covariantdierentialsandcovariantderivatives..........1215.3.1Thehorizontalprojectionofforms..............1215.3.2ThecovariantdierentialofformsonP...........1225.3.3Aformulaforthecovariantdierentialofbasicforms...1225.3.4Thecurvatureisd !......................1235.3.5Bianchi'sidentity.......................1235.3.6Thecurvatureandd2.....................1236Gauss'slemma.1256.1Theexponentialmap.........................1256.2Normalcoordinates..........................1266.3TheEulereldEanditsimageP..................1276.4Thenormalframeeld........................1286.5Gauss'lemma.............................1296.6Minimizationofarclength......................1317Specialrelativity1337.1TwodimensionalLorentztransformations..............1337.1.1Additionlawforvelocities..................1357.1.2Hyperbolicangle........................1357.1.3Propertime..........................1367.1.4Timedilatation........................1377.1.5Lorentz-Fitzgeraldcontraction................1377.1.6Thereversetriangleinequality................1387.1.7PhysicalsignicanceoftheMinkowskidistance.......1387.1.8Energy-momentum......................1397.1.9Psychologicalunits......................1407.1.10TheGalileanlimit.......................1427.2Minkowskispace............................1427.2.1TheComptoneect......................1437.2.2NaturalUnits.........................1467.2.3Two-particleinvariants....................147

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8CONTENTS7.2.4Mandlestamvariables.....................1507.3Scatteringcross-sectionandmutualux...............1548DieGrundlagenderPhysik.1578.1Preliminaries..............................1578.1.1Densitiesanddivergences...................1578.1.2Divergenceofavectoreldonasemi-Riemannianmanifold.1608.1.3TheLiederivativeofofasemi-Riemannmetric......1628.1.4Thecovariantdivergenceofasymmetrictensoreld....1638.2Varyingthemetricandtheconnection...............1678.3Thestructureofphysicallaws....................1698.3.1TheLegendretransformation.................1698.3.2Thepassiveequations.....................1728.4TheHilbertfunction"........................1738.5Schrodinger'sequationasapassiveequation............1758.6Harmonicmaps............................1759Submersions.1799.1Submersions..............................1799.2Thefundamentaltensorsofasubmersion..............1819.2.1ThetensorT..........................1819.2.2ThetensorA..........................1829.2.3CovariantderivativesofTandA...............1839.2.4Thefundamentaltensorsforawarpedproduct.......1859.3Curvature................................1869.3.1Curvatureforwarpedproducts................1909.3.2Sectionalcurvature......................1939.4Reductivehomogeneousspaces....................1949.4.1Bi-invariantmetricsonaLiegroup.............1949.4.2Homogeneousspaces......................1979.4.3Normalsymmetricspaces...................1979.4.4Orthogonalgroups.......................1989.4.5DualGrassmannians.....................2009.5Schwarzschildasawarpedproduct..................2029.5.1Surfaceswithorthogonalcoordinates............2039.5.2TheSchwarzschildplane...................2049.5.3Covariantderivatives.....................2059.5.4Schwarzschildcurvature....................2069.5.5Cartancomputation......................2079.5.6Petrovtype...........................2099.5.7Kerr-Schildform........................2109.5.8Isometries...........................2119.6RobertsonWalkermetrics.......................2149.6.1Cosmogenyandeschatology..................216

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CONTENTS910Petrovtypes.21710.1Algebraicpropertiesofthecurvaturetensor............21710.2Linearandantilinearmaps......................21910.3Complexconjugationandrealforms.................22110.4Structuresontensorproducts....................22310.5SpinorsandMinkowskispace.....................22410.6Tracelesscurvatures..........................22510.7Thepolynomialalgebra........................22510.8Petrovtypes..............................22610.9Principalnulldirections........................22710.10Kerr-Schildmetrics..........................23011Star.23311.1Denitionofthestaroperator....................23311.2Does?:^kV!^n)]TJ/F10 6.974 Tf 6.226 0 Td[(kVdeterminethemetric?..........23511.3Thestaroperatoronforms......................24011.3.1ForR2.............................24011.3.2ForR3.............................24111.3.3ForR1;3............................24211.4Electromagnetism...........................24311.4.1Electrostatics..........................24311.4.2Magnetoquasistatics......................24411.4.3TheLondonequations.....................24611.4.4TheLondonequationsinrelativisticform..........24811.4.5Maxwell'sequations......................24911.4.6ComparingMaxwellandLondon...............249

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10CONTENTS

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Chapter1Theprincipalcurvatures.1.1VolumeofathickenedhypersurfaceWewanttoconsiderthefollowingproblem:LetYRnbeanorientedhyper-surface,sothereisawelldenedunitnormalvector,y,ateachpointofY.LetYhdenotethesetofallpointsoftheformy+ty;0th:WewishtocomputeVnYhwhereVndenotesthen)]TJ/F8 9.963 Tf 7.748 0 Td[(dimensionalvolume.Wewilldothiscomputationforsmallh,seethediscussionaftertheexamples.Examplesinthreedimensionalspace.1.SupposethatYisaboundedregioninaplane,ofareaA.ClearlyV3Yh=hAinthiscase.2.SupposethatYisarightcircularcylinderofradiusrandheight`withoutwardlypointingnormal.ThenYhistheregionbetweentherightcircularcylindersofheight`andradiirandr+hsoV3Yh=`[r+h2)]TJ/F11 9.963 Tf 9.963 0 Td[(r2]=2`rh+`h2=hA+h21 2rA=Ah+1 2kh2;whereA=2r`istheareaofthecylinderandwherek=1=risthecurvatureofthegeneratingcircleofthecylinder.Forsmallh,thisformulaiscorrect,infact,11

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12CHAPTER1.THEPRINCIPALCURVATURES.whetherwechoosethenormalvectortopointoutofthecylinderorintothecylinder.Ofcourse,intheinwardpointingcase,thecurvaturehastheoppositesign,k=)]TJ/F8 9.963 Tf 7.748 0 Td[(1=r.Forinwardpointingnormals,theformulabreaksdownwhenh>r,sincewegetmultiplecoverageofpointsinspacebypointsoftheformy+ty.3.YisasphereofradiusRwithoutwardnormal,soYhisasphericalshell,andV3Yh=4 3[R+h3)]TJ/F11 9.963 Tf 9.962 0 Td[(R3]=h4R2+h24R+h34 3=hA+h21 RA+h31 3R2A=1 3A3h+31 Rh2+1 R2h3;whereA=4R2istheareaofthesphere.Onceagain,forinwardpointingnormalswemustchangethesignofthecoecientofh2andtheformulathusobtainedisonlycorrectforh1 R.Soingeneral,wewishtomaketheassumptionthathissuchthatthemapY[0;h]!Rn;y;t7!y+tyisinjective.ForYcompact,therealwaysexistsanh0>0suchthatthisconditionholdsforallh
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1.2.THEGAUSSMAPANDTHEWEINGARTENMAP.131.2TheGaussmapandtheWeingartenmap.Inordertostatethegeneralformula,wemakethefollowingdenitions:LetYbeanimmersedorientedhypersurface.Ateachx2Ythereisauniquepositiveunitnormalvector,andhenceawelldenedGaussmap:Y!Sn)]TJ/F7 6.974 Tf 6.227 0 Td[(1assigningtoeachpointx2Yitsunitnormalvector,x.HereSn)]TJ/F7 6.974 Tf 6.227 0 Td[(1denotestheunitsphere,thesetofallunitvectorsinRn.Thenormalvector,xisorthogonaltothetangentspacetoYatx.WewilldenotethistangentspacebyTYx.Forourpresentpurposes,wecanregardTYxasasubspaceofRn:Ift7!tisadierentiablecurvelyingonthehypersurfaceY,thismeansthatt2Yforalltandif=x,then0belongstothetangentspaceTYx.Conversely,givenanyvectorv2TYx,wecanalwaysndadierentiablecurvewith=x;0=v.SoagoodwaytothinkofatangentvectortoYatxisasaninnitesimalcurve"onYpassingthroughx.Examples:1.SupposethatYisaportionofann)]TJ/F8 9.963 Tf 9.292 0 Td[(1dimensionallinearoranesub-spacespacesittinginRn.ForexamplesupposethatY=Rn)]TJ/F7 6.974 Tf 6.227 0 Td[(1consistingofthosepointsinRnwhoselastcoordinatevanishes.ThenthetangentspacetoYateverypointisjustthissamesubspace,andhencethenormalvectorisaconstant.TheGaussmapisthusaconstant,mappingallofYontoasinglepointinSn)]TJ/F7 6.974 Tf 6.227 0 Td[(1.2.SupposethatYisthesphereofradiusRsaycenteredattheorigin.TheGaussmapcarrieseverypointofYintothecorrespondingparallelpointofSn)]TJ/F7 6.974 Tf 6.227 0 Td[(1.Inotherwords,itismultiplicationby1=R:y=1 Ry:3.SupposethatYisarightcircularcylinderinR3whosebaseisthecircleofradiusrinthex1;x2plane.ThentheGaussmapsendsYontotheequatoroftheunitsphere,S2,sendingapointxinto1=rxwhere:R3!R2isprojectionontothex1;x2plane.AnothergoodwaytothinkofthetangentspaceisintermsofalocalparameterizationwhichmeansthatwearegivenamapX:M7!RnwhereMissomeopensubsetofRn)]TJ/F7 6.974 Tf 6.226 0 Td[(1andsuchthatXMissomeneighborhoodofxinY.Lety1;:::;yn)]TJ/F7 6.974 Tf 6.226 0 Td[(1bethestandardcoordinatesonRn)]TJ/F7 6.974 Tf 6.227 0 Td[(1.PartoftherequirementthatgoesintothedenitionofparameterizationisthatthemapXberegular,inthesensethatitsJacobianmatrixdX:=@X @y1;;@X @yn)]TJ/F7 6.974 Tf 6.227 0 Td[(1

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14CHAPTER1.THEPRINCIPALCURVATURES.whosecolumnsarethepartialderivativesofthemapXhasrankn)]TJ/F8 9.963 Tf 10.123 0 Td[(1every-where.ThematrixdXhasnrowsandn)]TJ/F8 9.963 Tf 9.37 0 Td[(1columns.Theregularityconditionamountstotheassertionthatforeachz2Mthevectors,@X @y1z;;@X @yn)]TJ/F7 6.974 Tf 6.226 0 Td[(1zspanasubspaceofdimensionn)]TJ/F8 9.963 Tf 9.554 0 Td[(1.Ifx=XythenthetangentspaceTYxispreciselythespacespannedby@X @y1y;;@X @yn)]TJ/F7 6.974 Tf 6.227 0 Td[(1y:SupposethatFisadierentiablemapfromYtoRm.Wecanthendeneitsdierential,dFx:TYx7!Rm.Itisalinearmapassigningtoeachv2TYxavaluedFxv2Rm:Intermsoftheinnitesimalcurve"description,ifv=0thendFxv=dF dt:Youmustcheckthatthisdoesnotdependonthechoiceofrepresentingcurve,.Alternatively,togivealinearmap,itisenoughtogiveitsvalueattheelementsofabasis.Intermsofthebasiscomingfromaparameterization,wehavedFx@X @yiy=@FX @yiy:HereFX:M!RmisthecompositionofthemapFwiththemapX.YoumustcheckthatthemapdFxsodetermineddoesnotdependonthechoiceofparameterization.Bothofthesevericationsproceedbythechainrule.Oneimmediateconsequenceofeithercharacterizationisthefollowingim-portantproperty.SupposethatFtakesvaluesinasubmanifoldZRm.ThendFx:TYx!TZFx:LetusapplyallthistotheGaussmap,,whichmapsYtotheunitsphere,Sn)]TJ/F7 6.974 Tf 6.227 0 Td[(1.Thendx:TYx!TSn)]TJ/F7 6.974 Tf 6.226 0 Td[(1x:ButthetangentspacetotheunitsphereatxconsistsofallvectorsperpendiculartoxandsocanbeidentiedwithTYx.WedenetheWein-gartenmaptobethedierentialoftheGaussmap,regardedasamapfromTYxtoitself:Wx:=dx;Wx:TYx!TYx:ThesecondfundamentalformisdenedtobethebilinearformonTYxgivenbyIIxv;w:=Wxv;w:

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1.2.THEGAUSSMAPANDTHEWEINGARTENMAP.15Inthenextsectionwewillshow,usinglocalcoordinates,thatthisformissymmetric,i.e.thatWxu;v=u;Wxv:Thisimplies,fromlinearalgebra,thatWxisdiagonizablewithrealeigenvalues.Theseeigenvalues,k1=k1x;;kn)]TJ/F7 6.974 Tf 6.227 0 Td[(1=kn)]TJ/F7 6.974 Tf 6.227 0 Td[(1x,oftheWeingartenmaparecalledtheprincipalcurvaturesofYatthepointx.Examples:1.Foraportionofn)]TJ/F8 9.963 Tf 10.408 0 Td[(1spacesittinginRntheGaussmapisconstantsoitsdierentialiszero.HencetheWeingartenmapandthusalltheprincipalcurvaturesarezero.2.ForthesphereofradiusRtheGaussmapconsistsofmultiplicationby1=Rwhichisalineartransformation.Thedierentialofalineartransformationisthatsametransformationregardedasactingonthetangentspaces.HencetheWeingartenmapis1=Ridandsoalltheprincipalcurvaturesareequalandareequalto1=R.3.Forthecylinder,againtheGaussmapislinear,andsotheprincipalcurvaturesare0and1=r.WeletHjdenotethejthnormalizedelementarysymmetricfunctionsoftheprincipalcurvatures.SoH0=1H1=1 n)]TJ/F8 9.963 Tf 9.962 0 Td[(1k1++kn)]TJ/F7 6.974 Tf 6.227 0 Td[(1Hn)]TJ/F7 6.974 Tf 6.227 0 Td[(1=k1k2kn)]TJ/F7 6.974 Tf 6.226 0 Td[(1and,ingeneral,Hj=n)]TJ/F8 9.963 Tf 9.963 0 Td[(1j)]TJ/F7 6.974 Tf 6.227 0 Td[(1X1i1<
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16CHAPTER1.THEPRINCIPALCURVATURES.1.3Proofofthevolumeformula.WerecallthattheGaussmap,assignstoeachpointx2Yitsunitnormalvector,andsoisamapfromYtotheunitsphere,Sn)]TJ/F7 6.974 Tf 6.226 0 Td[(1.TheWeingartenmap,Wx,isthedierentialoftheGaussmap,Wx=dx,regardedasamapofthetangentspace,TYxtoitself.WenowdescribethesemapsintermsofalocalparameterizationofY.SoletX:M!RnbeaparameterizationofclassC2ofaneighborhoodofYnearx,whereMisanopensubsetofRn)]TJ/F7 6.974 Tf 6.227 0 Td[(1.Sox=Xy;y2M,say.LetN:=XsothatN:M!Sn)]TJ/F7 6.974 Tf 6.227 0 Td[(1isamapofclassC1.ThemapdXy:Rn)]TJ/F7 6.974 Tf 6.226 0 Td[(1!TYxgivesaframeofTYx.Thewordframe"meansanisomorphismofourstan-dard"n)]TJ/F8 9.963 Tf 8.431 0 Td[(1-dimensionalspace,Rn)]TJ/F7 6.974 Tf 6.227 0 Td[(1withourgivenn)]TJ/F8 9.963 Tf 8.431 0 Td[(1-dimensionalspace,TYx.HerewehaveidentiedTRn)]TJ/F7 6.974 Tf 6.227 0 Td[(1ywithRn)]TJ/F7 6.974 Tf 6.227 0 Td[(1,sotheframedXygivesusaparticularisomorphismofRn)]TJ/F7 6.974 Tf 6.227 0 Td[(1withTYx.Givingaframeofavectorspaceisthesameasgivingabasisofthatvectorspace.Wewillusethesetwodierentwaysofusingthewordframe"inter-changeably.Lete1;:::;en)]TJ/F7 6.974 Tf 6.227 0 Td[(1denotethestandardbasisofRn)]TJ/F7 6.974 Tf 6.227 0 Td[(1,andforXandN,letthesubscriptidenotethepartialderivativewithrespecttotheithCartesiancoordinate.ThusdXyei=Xiyforexample,andsoX1y;:::;Xn)]TJ/F7 6.974 Tf 6.226 0 Td[(1yis"theframedeterminedbydXywhenweregardTYxasasubspaceofRn.Forthesakeofnotationalsim-plicitywewilldroptheargumenty.ThuswehavedXei=Xi;dNei=Ni;andsoWxXi=Ni:Recallthedenition,IIxv;w=Wxv;w,ofthesecondfundamentalform.LetLijdenotethematrixofthesecondfundamentalformwithrespecttothebasisX1;:::Xn)]TJ/F7 6.974 Tf 6.226 0 Td[(1ofTYx.SoLij=IIxXi;Xj=WxXi;Xj=Ni;XjsoLij=)]TJ/F8 9.963 Tf 7.749 0 Td[(N;@2X @yi@yj;.3

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1.3.PROOFOFTHEVOLUMEFORMULA.17thelastequalitycomingfromdierentiatingtheidentityN;Xj0intheithdirection.Inparticular,itfollowsfrom.3andtheequalityofcrossderivativesthatWxXi;Xj=Xi;WxXjandhence,bylinearitythatWxu;v=u;Wxv8u;v2TYx:Wehaveprovedthatthesecondfundamentalformissymmetric,andhencetheWeingartenmapisdiagonizablewithrealeigenvalues.Recallthattheprincipalcurvaturesare,bydenition,theeigenvaluesoftheWeingartenmap.WewillletW=WijdenotethematrixoftheWeingartenmapwithrespecttothebasisX1;:::;Xn)]TJ/F7 6.974 Tf 6.227 0 Td[(1.Explicitly,Ni=XjWjiXj:IfwewriteN1;:::;Nn)]TJ/F7 6.974 Tf 6.226 0 Td[(1;X1;:::;Xn)]TJ/F7 6.974 Tf 6.227 0 Td[(1ascolumnvectorsoflengthn,wecanwritetheprecedingequationasthematrixequationN1;:::;Nn)]TJ/F7 6.974 Tf 6.226 0 Td[(1=X1;:::;Xn)]TJ/F7 6.974 Tf 6.226 0 Td[(1W:.4Thematrixmultiplicationontherightisthatofannn)]TJ/F8 9.963 Tf 9.68 0 Td[(1matrixwithann)]TJ/F8 9.963 Tf 9.709 0 Td[(1n)]TJ/F8 9.963 Tf 9.709 0 Td[(1matrix.Tounderstandthisabbreviatednotation,letuswriteitoutinthecasen=3,sothatX1;X2;N1;N2arevectorsinR3:X1=0@X11X12X131A;X2=0@X21X22X231A;N1=0@N11N12N131A;N2=0@N21N22N231A:Then.4isthematrixequation0@N11N21N12N22N13N231A=0@X11X21X12X22X13X231AW11W12W21W22:MatrixmultiplicationshowsthatthisgivesN1=W11X1+W21X2;N2=W12X1+W22X2;andmoregenerallythat.4givesNi=PjWjiXjinalldimensions.NowconsidertheregionYh,thethickenedhypersurface,introducedintheprecedingsectionexceptthatwereplacethefullhypersurfaceYbytheportionXM.ThustheregioninspacethatweareconsideringisfXy+Ny;y2M;0
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18CHAPTER1.THEPRINCIPALCURVATURES.ItistheimageoftheregionM;h]Rn)]TJ/F7 6.974 Tf 6.227 0 Td[(1Runderthemapy;7!Xy+Ny:Weareassumingthatthismapisinjective.By1.4,ithasJacobianmatrixdierentialJ=X1+N1;:::;Xn)]TJ/F7 6.974 Tf 6.227 0 Td[(1+Nn)]TJ/F7 6.974 Tf 6.227 0 Td[(1;N=X1;:::;Xn)]TJ/F7 6.974 Tf 6.227 0 Td[(1;NIn)]TJ/F7 6.974 Tf 6.227 0 Td[(1+W001:.5Therighthandsideof.5isnowtheproductoftwonbynmatrices.ThechangeofvariablesformulainseveralvariablessaysthatVnh=ZMZh0jdetJjdhdy1dyn)]TJ/F7 6.974 Tf 6.226 0 Td[(1:.6Letustakethedeterminantoftherighthandsideof.5.ThedeterminantofthematrixX1;:::;Xn)]TJ/F7 6.974 Tf 6.227 0 Td[(1;NisjusttheorientedndimensionalvolumeoftheparallelepipedspannedbyX1;:::;Xn)]TJ/F7 6.974 Tf 6.226 0 Td[(1;N.SinceNisofunitlengthandisperpendiculartotheX0s,thisisthesameastheorientedn)]TJ/F8 9.963 Tf 9.269 0 Td[(1dimensionalvolumeoftheparallelepipedspannedbyX1;:::;Xn)]TJ/F7 6.974 Tf 6.227 0 Td[(1.Thus,bydenition",jdetX1;:::;Xn)]TJ/F7 6.974 Tf 6.227 0 Td[(1;Njdy1dyn)]TJ/F7 6.974 Tf 6.227 0 Td[(1=dn)]TJ/F7 6.974 Tf 6.226 0 Td[(1A:.7Wewillcomebackshortlytodiscusswhythisistherightdenition.Thesecondfactorontherighthandsideof.5contributesdet+W=+k1+kn)]TJ/F7 6.974 Tf 6.227 0 Td[(1:Forsucientlysmall,thisexpressionispositive,soweneednotworryabouttheabsolutevaluesignifhsmallenough.Integratingwithrespecttofrom0tohgives.2.Weproved.2ifwedenedn)]TJ/F7 6.974 Tf 6.226 0 Td[(1Atobegivenby.7.Butthenitfollowsfrom.2thatd dhVnYhjh=0=ZYdn)]TJ/F7 6.974 Tf 6.227 0 Td[(1A:.8Amoment'sthoughtshowsthatthelefthandsideof1.8isexactlywhatwewanttomeanbyarea":itisthevolumeofaninnitesimallythickenedregion".Thisjustiestaking.7asadenition.Furthermore,althoughthedenition.7isonlyvalidinacoordinateneighborhood,andseemstodependonthechoiceoflocalcoordinates,equation.8showsthatitisindependentofthelocaldescriptionbycoordinates,andhenceisawelldenedobjectonY.ThefunctionsHjhavebeendenedindependentofanychoiceoflocalcoordinates.Hence.2worksglobally:Tocomputetherighthandsideof.2wemayhavetobreakYupintopatches,anddotheintegrationineachpatch,summingthepieces.ButweknowinadvancethatthenalanswerisindependentofhowwebreakYuporwhichlocalcoordinatesweuse.

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1.4.GAUSS'STHEOREMAEGREGIUM.191.4Gauss'stheoremaegregium.Supposeweconsiderthetwosidedregionaboutthesurface,thatisVnY+h+VnY)]TJ/F10 6.974 Tf -2.214 -7.267 Td[(hcorrespondingtothetwodierentchoicesofnormals.Whenwereplacexby)]TJ/F11 9.963 Tf 7.749 0 Td[(xateachpoint,theGaussmapisreplacedby)]TJ/F11 9.963 Tf 7.749 0 Td[(,andhencetheWein-gartenmapsWxarealsoreplacedbytheirnegatives.Theprincipalcurvatureschangesign.Hence,intheabovesumthecoecientsoftheevenpowersofhcancel,sincetheyaregivenintermsofproductsoftheprincipalcurvatureswithanoddnumberoffactors.Forn=3weareleftwithasumoftwoterms,thecoecientofhwhichisthearea,andthecoecientofh3whichistheintegraloftheGaussiancurvature.ItwastheremarkablediscoveryofGaussthatthiscurvaturedependsonlyontheintrinsicgeometryofthesurface,andnotonhowthesurfaceisembeddedintothreespace.Thus,forboththecylinderandtheplanethecubictermsvanish,becauselocallythecylinderisisometrictotheplane.Wecanwraptheplanearoundthecylinderwithoutstretchingortearing.ItwasthisfundamentalobservationofGaussthatledRiemanntoinvestigatetheintrinsicmetricgeometryofhigherdimensionalspace,eventuallyleadingtoEinstein'sgeneralrelativitywhichderivesthegravitationalforcefromthecurvatureofspacetime.ArstobjectivewillbetounderstandthismajortheoremofGauss.AnimportantgeneralizationofGauss'sresultwasprovedbyHermannWeylin1939.Heshowed:ifYisanykdimensionalsubmanifoldofndimensionalspacesofork=1;n=3Yisacurveinthreespace,letYhdenotethetube"aroundYofradiush,thesetofallpointsatdistancehfromY.Then,forsmallh,VnYhisapolynomialinhwhosecoecientsareintegralsoverYofintrinsicexpressions,dependingonlyonthenotionofdistancewithinY.Letusmultiplybothsidesof.4ontheleftbythematrixX1;:::;Xn)]TJ/F7 6.974 Tf 6.226 0 Td[(1TtoobtainL=QWwhereLij=Xi;Njasbefore,andQ=Qij:=Xi;Xjiscalledthematrixoftherstfundamentalformrelativetoourchoiceoflocalcoordinates.Allthreematricesinthisequalityareofsizen)]TJ/F8 9.963 Tf 8.986 0 Td[(1n)]TJ/F8 9.963 Tf 8.986 0 Td[(1.IfwetakethedeterminantoftheequationL=QWweobtaindetW=detL detQ;.9anexpressionforthedeterminantoftheWeingartenmapageometricalprop-ertyoftheembeddedsurfaceasthequotientoftwolocalexpressions.Forthecasen)]TJ/F8 9.963 Tf 10.316 0 Td[(1=2,wethusobtainalocalexpressionfortheGaussiancurvature,K=detW.

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20CHAPTER1.THEPRINCIPALCURVATURES.Therstfundamentalformencodestheintrinsicgeometryofthehypersur-faceintermsoflocalcoordinates:itgivestheEuclideangeometryofthetangentspaceintermsofthebasisX1;:::;Xn)]TJ/F7 6.974 Tf 6.226 0 Td[(1.Ifwedescribeacurvet7!tonthesurfaceintermsofthecoordinatesy1;:::;yn)]TJ/F7 6.974 Tf 6.227 0 Td[(1bygivingthefunctionst7!yjt;=1;:::;n)]TJ/F8 9.963 Tf 9.963 0 Td[(1thenthechainrulesaysthat0t=n)]TJ/F7 6.974 Tf 6.226 0 Td[(1Xj=1Xjytdyj dttwhereyt=y1t;:::;yn)]TJ/F7 6.974 Tf 6.226 0 Td[(1t:ThereforetheEuclideansquarelengthofthetangentvector0tisk0tk2=n)]TJ/F7 6.974 Tf 6.227 0 Td[(1Xi;j=1Qijytdyi dttdyj dtt:ThusthelengthofthecurvegivenbyZk0tkdtcanbecomputedintermsofytasZvuut n)]TJ/F7 6.974 Tf 6.227 0 Td[(1Xi;j=1Qijytdyi dttdyj dttdtsolongasthecurvelieswithinthecoordinatesystem.SotwohypersurfaceshavethesamelocalintrinsicgeometryiftheyhavethesameQinanylocalcoordinatesystem.Inordertoconformwithasomewhatvariableclassicalliterature,weshallmakesomeslightchangesinournotationforthecaseofsurfacesinthreedi-mensionalspace.Wewilldenoteourlocalcoordinatesbyu;vinsteadofy1;y2andsoXuwillreplaceX1andXvwillreplaceX2,andwewilldenotethescalarproductoftwovectorsinthreedimensionalspacebyainsteadof;.WewriteQ=EFFG.10whereE:=XuXu.11F:=XuXv.12G:=XvXv.13sodetQ=EG)]TJ/F11 9.963 Tf 9.962 0 Td[(F2:.14

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1.4.GAUSS'STHEOREMAEGREGIUM.21Wecanwritetheequations1.11-.13asQ=Xu;XvyXu;Xv:.15Similarly,letussete:=NXuu.16f:=NXuv.17g:=NXvv.18soL=)]TJ/F1 9.963 Tf 9.409 14.047 Td[(effg.19anddetL=eg)]TJ/F11 9.963 Tf 9.963 0 Td[(f2:Hence.9specializestoK=eg)]TJ/F11 9.963 Tf 9.962 0 Td[(f2 EG)]TJ/F11 9.963 Tf 9.963 0 Td[(F2;.20anexpressionfortheGaussiancurvatureinlocalcoordinates.Wecanmakethisexpressionevenmoreexplicit,usingthenotionofvectorproduct.Noticethattheunitnormalvector,NisgivenbyN=1 jjXuXvjjXuXvandjjXuXvjj=p jjXujj2jjXvjj2)]TJ/F8 9.963 Tf 9.963 0 Td[(XuXv2=p EG)]TJ/F11 9.963 Tf 9.963 0 Td[(F2:Thereforee=NXuu=1 p EG)]TJ/F11 9.963 Tf 9.963 0 Td[(F2XuuXuXv=1 p EG)]TJ/F11 9.963 Tf 9.963 0 Td[(F2detXuu;Xu;Xv;Thislastdeterminant,isthethedeterminantofthethreebythreematrixwhosecolumnsarethevectorsXuu;XuandXv.ReplacingtherstcolumnbyXuvgivesacorrespondingexpressionforf,andreplacingtherstcolumnbyXvvgivestheexpressionforg.Substitutinginto1.20givesK=detXuu;Xu;XvdetXvv;Xu;Xv)]TJ/F8 9.963 Tf 9.963 0 Td[(detXuv;Xu;Xv2 [XuXuXvXv)]TJ/F8 9.963 Tf 9.962 0 Td[(XuXv2]2:.21Thisexpressionisrathercomplicatedforcomputationbyhand,sinceitinvolvesallthosedeterminants.Howeverasymbolicmanipulationprogramsuchasmapleormathematicacanhandleitwithease.Hereistheinstructionformathematica,takenfromarecentbookbyGray,intermsofafunctionX[u,v]denedinmathematica:

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22CHAPTER1.THEPRINCIPALCURVATURES.gcurvature[X ][u ,v ]:=Simplify[Det[D[X[uu,vv],uu,uu],D[X[uu,vv],uu],D[X[uu,vv],vv]]*Det[D[X[uu,vv],vv,vv],D[X[uu,vv],uu],D[X[uu,vv],vv]]-Det[D[X[uu,vv],uu,vv],D[X[uu,vv],uu],D[X[uu,vv],vv]]^2/D[X[uu,vv],uu].D[X[uu,vv],uu]*D[X[uu,vv],vv].D[X[uu,vv],vv]-D[X[uu,vv],uu].D[X[uu,vv],vv]^2^2]/.uu->u,vv->vWearenowinapositiontogivetwoproofs,bothcorrectbutbothsomewhatunsatisfactoryofGauss'sTheoremaegregiumwhichassertsthattheGaussiancurvatureisanintrinsicpropertyofthemetricalcharacterofthesurface.How-evereachproofdoeshaveitsmerits.1.4.1Firstproof,usinginertialcoordinates.Fortherstproof,weanalyzehowtherstfundamentalformchangeswhenwechangecoordinates.Supposewepassfromlocalcoordinatesu;vtolocalcoordinatesu0;v0whereu=uu0;v0;v=vu0;v0.ExpressingXasafunctionofu0;v0andusingthechainrulegives,Xu0=@u @u0Xu+@v @u0XvXv0=@u @v0Xu+@u @v0XvorXu0;Xv0=Xu;XvJwhereJ:=@u @u0@u @v0@v @u0@v @v0soQ0=Xu0;Xv0yXu0;Xv0=JyQJ:Thisgivestheruleforchangeofvariablesoftherstfundamentalformfromtheunprimedtotheprimedcoordinatesystem,andisvalidthroughouttherangewherethecoordinatesaredened.HereJisamatrixvaluedfunctionofu0;v0.Letusnowconcentrateattentiononasinglepoint,P.Therstfundamentalformisasymmetricpostivedenitematrix.Bylinearalgebra,wecanalwaysndamatrixRsuchthatRyQuP;vpR=I,thetwodimensionalidentityma-trix.HereuP;vParethecoordinatesdescribingP.Withnolossofgeneralitywemayassumethatthesecoordinatesare;0.Wecanthenmakethelin-earchangeofvariableswhoseJ;0isR,andsondcoordinatessuchthatQ;0=Iinthiscoordinatesystem.Butwecandobetter.WeclaimthatwecanchoosecoordinatessothatQ=I;@Q @u;0=@Q @v;0=0:.22

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1.4.GAUSS'STHEOREMAEGREGIUM.23Indeed,supposewestartwithacoordinatesystemwithQ=I,andlookforachangeofcoordinateswithJ=I,hopingtodeterminethesecondderivativessothat.22holds.WritingQ0=JyQJandusingLeibniz'sformulaforthederivativeofaproduct,theequationsbecome@J+Jy @u0=)]TJ/F11 9.963 Tf 8.944 6.739 Td[(@Q @u@J+Jy @v0=)]TJ/F11 9.963 Tf 8.944 6.739 Td[(@Q @v;whenwemakeuseofJ=I.Writingouttheseequationsgives2@2u @u02@2u @u0@v0+@2v @u02@2u @u0@v0+@2v @u022@2v @u0@v0!=)]TJ/F11 9.963 Tf 8.944 6.74 Td[(@Q @u2@2u @u0@v0@2u @v02+@2v @u0@v0@2u @v02+@2v @u0@v02@2v @v02!=)]TJ/F11 9.963 Tf 8.944 6.739 Td[(@Q @v:Thelowerrighthandcorneroftherstequationandtheupperlefthandcornerofthesecondequationdetermine@2v @u0@v0and@2u @u0@v0:AlloftheremainingsecondderivativesarethendeterminedconsistentlysinceQisasymmetricmatrix.Wemaynowchooseuandvasfunctionsofu0;v0:whichvanishat;0togetherwithalltheirrstpartialderivatives,andwiththesecondderivativesasabove.Forexample,wecanchoosetheuandvashomogeneouspolynomialsinu0andv0withtheabovepartialderivatives.Acoordinatesysteminwhich.22holdsatapointPhavingcoordinates;0iscalledaninertialcoordinatesystembasedatP.ObviouslythecollectionofallinertialcoordinatesystemsbasedatPisintrinsicallyassociatedtothemetric,sincethedenitiondependsonlyonpropertiesofQinthecoordinatesystem.WenowclaimthefollowingProposition1Ifu;visaninertialcoordinatesystemofanembeddedsurfacebasedatPthenthentheGaussiancurvatureisgivenbyKP=Fuv)]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 2Guu)]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 2Evv.23theexpressionontherightbeingevaluatedat;0.Asthecollectionofinertialsystemsisintrinsic,andas.23expressesthecurvatureintermsofalocalexpressionforthemetricinaninertialcoordinatesystem,thepropositionimpliestheTheoremaegregium.Toprovetheproposition,letusrstmakearotationandtranslationinthreedimensionalspaceifnecessarysothatXPisattheoriginandthetangentplanetothesurfaceatPisthex;yplane.ThefactthatQ=IimpliesthatthevectorsXu;Xvformanorthonormalbasisofthex;yplane,so

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24CHAPTER1.THEPRINCIPALCURVATURES.byafurtherrotation,ifnecessary,wemayassumethatXuistheunitvectorinthepositivex)]TJ/F8 9.963 Tf 11.431 0 Td[(directionandbyreplacingvby)]TJ/F11 9.963 Tf 7.749 0 Td[(vifnecessary,thatXvistheunitvectorinthepositiveydirection.TheseEuclideanmotionsweuseddonotchangethevalueofthedeterminantoftheWeingartenmapandsohavenoeectonthecurvature.Ifwereplacevby)]TJ/F11 9.963 Tf 7.749 0 Td[(v,EandGareunchangedandGuuorEvvarealsounchanged.Underthechangev7!)]TJ/F11 9.963 Tf 20.962 0 Td[(vFgoesto)]TJ/F11 9.963 Tf 7.748 0 Td[(F,butthecrossderivativeFuvpicksupanadditionalminussign.SoFuvisunchanged.Wehavearrangedthatweneedprove.23undertheassumptionsthatXu;v=0@u+ru;vv+su;vfu;v1A;wherer;s;andfarefunctionswhichvanishtogetherwiththeirrstderivativesattheorigininu;vspace.SofarwehaveonlyusedthepropertyQ=I,notthefullstrengthofthedenitionofaninertialcoordinatesystem.Weclaimthatifthecoordinatesystemisinertial,allthesecondpartialsofrandsalsovanishattheorigin.Toseethis,observethatE=+ru2+s2u+f2uF=rv+rurv+su+susv+fufvG=r2v++sv2+f2vsoEu=2ruuEv=2ruvFu=ruv+suuFv=rvv+suvGu=2suvGv=2svv:ThevanishingofalltherstpartialsofE;F;andGat0thusimpliesthevanishingofsecondpartialderivativesofrands.Bytheway,turningthisargumentaroundgivesusageometricallyintuitivewayofconstructinginertialcoordinatesforanembeddedsurface:AtanypointPchooseorthonormalcoordinatesinthetangentplanetoPandusethemtoparameterizethesurface.Intheprecedingnotationjustchoosex=uandy=vascoordinates.NowNisjusttheunitvectorinthepositivez)]TJ/F8 9.963 Tf 11.069 0 Td[(directionandsoe=fuuf=fuvg=fvvsoK=fuufvv)]TJ/F11 9.963 Tf 9.963 0 Td[(f2uvalltheabovemeantasvaluesattheoriginsinceEG)]TJ/F11 9.963 Tf 10.109 0 Td[(F2=1attheorigin.Ontheotherhand,takingthepartialderivativesoftheaboveexpressionsfor

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1.4.GAUSS'STHEOREMAEGREGIUM.25E;FandGandevaluatingattheorigininparticulardiscardingtermswhichvanishattheorigingivesFuv=ruvv+suuv+fuufvv+f2uvEvv=2ruvv+f2uvGuu=2suuv+f2uvwhenevaluatedat;0.So.23holdsbydirectcomputation.1.4.2Secondproof.TheBrioschiformula.SincetheGaussiancurvaturedependsonlyonthemetric,weshouldbeabletondageneralformulaexpressingtheGaussiancurvatureintermsofametric,validinanycoordinatesystem,notjustaninertialsystem.Thisweshalldobymassaging.21.Thenumeratorin.21isthedierenceofproductsoftwodeterminants.NowdetB=detBysodetAdetB=detAByandwecanwritethenumeratorof1.21asdet0@XuuXvvXuuXuXuuXvXuXvvXuXuXuXvXvXvvXvXuXvXv1A)]TJ/F8 9.963 Tf 7.749 0 Td[(det0@XuvXuvXuvXuXuvXvXuXivXuXuXuXvXvXuvXvXuXvXv1A:AllthetermsinthesematricesexceptfortheentriesintheupperlefthandcornerofeachiseitheratermoftheformE;F;orGorexpressibleasintermsofderivativesofE;FandG.Forexample,XuuXu=1 2EuandFu=XuuXv+XuXuvsoXuuXv=Fu)]TJ/F7 6.974 Tf 10.844 3.923 Td[(1 2Evandsoon.Soifnotforthetermsintheupperlefthandcorners,wewouldalreadyhaveexpressedtheGaussiancurvatureintermsofE;FandG.Soourproblemishowtodealwiththetwotermsintheupperlefthandcorner.Noticethatthelowerrighthandtwobytwoblockinthesetwomatricesarethesame.Soexpandingbothmatricesalongthetoprow,forexamplethedierenceofthetwodeterminantswouldbeunchangedifwereplacetheupperlefthandterm,XuuXvvintherstmatrixbyXuuXvv)]TJ/F11 9.963 Tf 9.471 0 Td[(XuvXuvandtheupperlefthandterminthesecondmatrixby0.WenowshowhowtoexpressXuuXvv)]TJ/F11 9.963 Tf 10.204 0 Td[(XuvXuvintermsofE;FandGandthiswillthengiveaproofoftheTheoremaegregium.WehaveXuuXvv)]TJ/F11 9.963 Tf 9.962 0 Td[(XuvXuv=XuXvvu)]TJ/F11 9.963 Tf 9.962 0 Td[(XuXvvu)]TJ/F8 9.963 Tf 7.749 0 Td[(XuXuvu+XuXuvv=XuXvvu)]TJ/F8 9.963 Tf 9.962 0 Td[(XuXuvv=XuXvu)]TJ/F11 9.963 Tf 9.962 0 Td[(XuvXvu)]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 2XuXuvv=XuXvvu)]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 2XvXvuu)]TJ/F8 9.963 Tf 11.158 6.739 Td[(1 2XuXuvv=)]TJ/F8 9.963 Tf 8.944 6.74 Td[(1 2Evv+Fuv)]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 2Guu:

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26CHAPTER1.THEPRINCIPALCURVATURES.WethusobtainBrioschi'sformulaK=detA)]TJ/F8 9.963 Tf 9.963 0 Td[(detB EG)]TJ/F11 9.963 Tf 9.962 0 Td[(F2where.24A=0@1 2Evv+Fuv)]TJ/F7 6.974 Tf 11.159 3.923 Td[(1 2Guu1 2EuFu)]TJ/F7 6.974 Tf 11.158 3.923 Td[(1 2EvFv)]TJ/F7 6.974 Tf 11.159 3.923 Td[(1 2GuEF1 2GvFG1AB=0@01 2Ev1 2Gu1 2EvEF1 2GuFG1A:Brioschi'sformulaisnottforhumanusebutcanbefedtomachineifnecessary.ItdoesgiveaproofofGauss'theorem.Noticethatifwehavecoordinateswhichareinertialatsomepoint,P,thenBrioschi'sformulareducesto.23sinceE=G=1;F=0andallrstpartialsvanishatP.WewillreproduceamathematicaprogramforBrioschi'sformulafromGrayattheendofthissection.Incasewehaveorthogonalcoordinates,acoordinatesysteminwhichF0,Brioschi'sformulasimpliesandbecomesuseful:IfwesetF=Fu=Fv=0inBrioschi'sformulaandexpandthedeterminantsweget1 EG2)]TJ/F8 9.963 Tf 8.945 6.74 Td[(1 2Evv)]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 2GuuEG+1 4EuGuG+1 4EvGvE+1 4E2vG+1 4G2uE=)]TJ/F8 9.963 Tf 8.944 6.74 Td[(1 2Evv EG+1 4E2v E2G+1 4EvGv EG2+)]TJ/F8 9.963 Tf 8.944 6.739 Td[(1 2Guu EG+1 4G2u EG2+1 4EuGu E2G:Weclaimthattherstbracketedexpressioncanbewrittenas)]TJ/F8 9.963 Tf 18.486 6.74 Td[(1 p EG@ @v1 p G@p E @v!:Indeed,1 p EG@ @v1 p G@p E @v!=1 p EG)]TJ/F11 9.963 Tf 12.208 6.74 Td[(Gv 2G3 2@p E @v+1 p G@2p E @v2!=1 p EG)]TJ/F11 9.963 Tf 14.279 6.74 Td[(EvGv 4G3 2p E+1 2p G@ @vE1 2Ev=1 p EG)]TJ/F11 9.963 Tf 14.279 6.74 Td[(EvGv 4G3 2p E+1 2p G)]TJ/F11 9.963 Tf 12.34 6.74 Td[(E2v 2E3 2+Evv p E=)]TJ/F11 9.963 Tf 9.223 6.74 Td[(GvEv 4G2E)]TJ/F11 9.963 Tf 17.565 6.74 Td[(E2v 4E2G+Evv 2EG:DoingasimilarcomputationforthesecondbracketedtermgivesK=)]TJ/F8 9.963 Tf 7.748 0 Td[(1 p EG"@ @u1 p E@p G @u!+@ @v1 p G@p E @v!#.25

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1.5.PROBLEMSET-SURFACESOFREVOLUTION.27astheexpressionfortheGaussiancurvatureinorthogonalcoordinates.WeshallgiveamoredirectproofofthisformulaandofGauss'theoremaegregiumoncewedeveloptheCartancalculus.1.5Problemset-Surfacesofrevolution.Thesimplestnon-trivialcaseiswhenn=2-thestudyofacurveintheplane.ForthecaseofacurveXt=xt;ytintheplane,wehaveX0t=x0t;y0t;Nt=1 x0t2+y0t21=2)]TJ/F11 9.963 Tf 7.749 0 Td[(y0t;x0t;wherethereectsthetwopossiblechoicesofnormals.Equation.3saysthattheonebyonematrixLisgivenbyL11=)]TJ/F8 9.963 Tf 7.749 0 Td[(N;X00=1 x02+y02)]TJ/F11 9.963 Tf 7.749 0 Td[(y0x00+x0y00:TherstfundamentalformistheonebyonematrixgivenbyQ11=kX0k2:Sothecurvatureis1 x02+y023 2x00y0)]TJ/F11 9.963 Tf 9.963 0 Td[(y00x0:Verifythatastraightlinehascurvaturezeroandthatthecurvatureofacircleofradiusris1=rwiththeplussignwhenthenormalpointsoutward.1.Whatdoesthisformulareducetointhecasethatxisusedasaparameter,i.e.xt=t;y=fx?Wewanttostudyasurfaceinthreespaceobtainedbyrotatingacurve,,inthex;zplaneaboutthez)]TJ/F8 9.963 Tf 7.749 0 Td[(axis.Suchasurfaceiscalledasurfaceofrevolution.Surfacesofrevolutionformoneofsimplestyetveryimportantclassesofsurfaces.Thesphere,torus,paraboloid,ellipsoidwithtwoequalaxesareallsurfacesofrevolution.Becauseofmodesofproductiongoingbacktothepotter'swheel,thesurfacesofmanyobjectsofdailylifearesurfacesofrevolution.WewillndthatthegeometryoffamousSchwarzschildblackholecanbeconsideredasaparticularanalogueofasurfaceofrevolutioninfourdimensionalspace-time.Letustemporarilyassumethatthecurveisgivenbyafunctionx=fz>0sothatwecanusez;ascoordinates,wherethesurfaceisgivenbyXz;=0@fzcosfzsinz1A;

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28CHAPTER1.THEPRINCIPALCURVATURES.andwechoosethenormaltopointawayfromthez)]TJ/F8 9.963 Tf 7.749 0 Td[(axis.2.Findz;andshowthattheWeingartenmapisdiagonalintheXz;Xbasis,infactNz=Xz;N=d fXwhereisthecurvatureofthecurveandwheredisthedistanceofthenormalvector,,fromthez)]TJ/F8 9.963 Tf 7.749 0 Td[(axis.ThereforetheGaussiancurvatureisgivenbyK=d f:.26CheckthattheGaussiancurvatureofacylindervanishesandthatofasphereofradiusRis1=R2.Noticethat.26makessenseevenifwecan'tusezasaparameterevery-whereon.Indeed,supposethatisacurveinthex;zplanethatdoesnotintersectthez)]TJ/F8 9.963 Tf 7.749 0 Td[(axis,andweconstructthecorrespondingsurfaceofrevolution.Atpointswherethetangenttoishorizontalparalleltothex)]TJ/F8 9.963 Tf 7.749 0 Td[(axisthenor-malvectortothesurfaceofrevolutionisvertical,sod=0.AlsotheGaussiancurvaturevanishes,sincetheGaussmaptakestheentirecircleofrevolutionintothenorthorsouthpole.So.26iscorrectatthesepoints.Atallotherpointswecanusezasaparameter.Butwemustwatchthesignof.RememberthattheGaussiancurvatureofasurfacedoesnotdependonthechoiceofnormalvector,butthecurvatureofacurveintheplanedoes.Inusing.26wemustbesurethatthesignofistheonedeterminedbythenormalpointingawayfromthez)]TJ/F8 9.963 Tf 7.749 0 Td[(axis.3.Forexample,taketobeacircleofradiusrcenteredatapointatdistanceD>rfromthez)]TJ/F8 9.963 Tf 7.749 0 Td[(axis,sayx=D+rcos;z=rsinintermsofanangularparameter,.Thecorrespondingsurfaceofrevolutionisatorus.Noticethatinusing.26wehavetotakeasnegativeonthesemicircleclosertothez)]TJ/F8 9.963 Tf 7.749 0 Td[(axis.SotheGaussiancurvatureisnegativeontheinner"halfofthetorusandpositiveontheouterhalf.Using.26and;ascoordinatesonthetorus,expressKasafunctionon;.Also,expresstheareaelementdAintermsofdd.Withoutanycomputation,showthatthetotalintegralofthecurvaturevanishes,i.e.RTKdA=0.RecallourdenitionsofE;F;andGgiveninequations.11-.13.Intheclassicalliterature,onewritetherstfundamentalformasds2=Edu2+2Fdudv+Gdv2:

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1.5.PROBLEMSET-SURFACESOFREVOLUTION.29themeaningofthisexpressionisasfollows:lett7!ut;vtdescribethecurveC:t7!Xut;vtonthesurface.Thendsgivestheelementofarclengthofthiscurveifwesubstituteu=ut;v=vtintotheexpressionfortherstfundamentalform.Sotherstfundamentalformdescribestheintrinsicmetricalpropertiesofthesurfaceintermsofthelocalcoordinates.Recallequation.25whichsaysthatifu;visanorthogonalcoordinatesystemthentheexpressionfortheGaussiancurvatureisK=)]TJ/F8 9.963 Tf 7.749 0 Td[(1 p EG"@ @u1 p E@p G @u!+@ @v1 p G@p E @v!#:4.Showthatthez;coordinatesintroducedinproblem2forasurfaceofrevolutionisanorthogonalcoordinatesystem,ndEandGandverify??forthiscase.Acurves7!Csonasurfaceiscalledageodesicifitsacceleration,C00,iseverywhereorthogonaltothesurface.Noticethatd dsC0s;C0s=2C00s;C0sandthis=0ifCisageodesic.ThetermgeodesicreferstoaparametrizedcurveandtheaboveequationshowsthattheconditiontobeageodesicimpliesthatkC0skisaconstant;i.ethatthecurveisparametrizedbyaconstantmultipleofarclength.Ifweuseadierentparameterization,says=stwithdotdenotingderivativewithrespecttot,thenthechainruleimpliesthat_C=C0_s;C=C00_s2+C0s:Soifuseaparameterotherthanarclength,theprojectionoftheaccelerationontothesurfaceisproportionaltothetangentvectorifCisageodesic.Inotherwords,theaccelerationisintheplanespannedbythetangentvectortothecurveandthenormalvectortothesurface.Conversely,supposewestartwithacurveCwhichhasthepropertythatitsaccelerationliesintheplanespannedbythetangentvectortothecurveandthenormalvectortothesurfaceatallpoints.Letusreparametrizethiscurvebyarclength.ThenC0s;C0s1andhenceC00;C00.AsweareassumingthatCliesintheplanespannedby_Candthenormalvectortothesurfaceateachpointofthecurve,andthat_sisnowhere0weconcludethatC,initsarclengthparametrizationisageodesic.Standardusagecallsacurvewhichisageodesicuptoreparametrization"apregeodesic.Idon'tlikethisterminologybutwilllivewithit.5.Showthatthecurves=constintheterminologyofproblem2areallpregeodsics.Showthatthecurvesz=const.arepregeodesicsifandonlyifzisacriticalpointoff,i.e.f0s=0.

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30CHAPTER1.THEPRINCIPALCURVATURES.ThegeneralsettingfortheconceptofsurfacesofrevolutionisthatofaRiemanniansubmersion,whichwillbethesubjectofChapter8.

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Chapter2Rulesofcalculus.2.1Superalgebras.AcommutativeassociativesuperalgebraisavectorspaceA=AevenAoddwithagivendirectsumdecompositionintoevenandoddpieces,andamapAA!Awhichisbilinear,satisestheassociativelawformultiplication,andAevenAeven!AevenAevenAodd!AoddAoddAeven!AoddAoddAodd!Aeven!=!ifeither!orareeven,!=)]TJ/F11 9.963 Tf 7.749 0 Td[(!ifboth!andareodd.Wewritetheselasttwoconditionsas!=)]TJ/F8 9.963 Tf 7.749 0 Td[(1degdeg!!:Heredeg=0ifiseven,anddeg=1mod2ifisodd.2.2Dierentialforms.Alineardierentialformonamanifold,M,isarulewhichassignstoeachp2MalinearfunctiononTMp.Soalineardierentialform,!,assignstoeachpanelementofTMp.Wewill,asusual,onlyconsiderlineardierentialformswhicharesmooth.31

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32CHAPTER2.RULESOFCALCULUS.Thesuperalgebra,MisthesuperalgebrageneratedbysmoothfunctionsonMtakenasevenandbythelineardierentialforms,takenasodd.Multiplicationofdierentialformsisusuallydenotedby^.Thenumberofdierentialfactorsiscalledthedegreeoftheform.Sofunctionshavedegreezero,lineardierentialformshavedegreeone.Intermsoflocalcoordinates,themostgenerallineardierentialformhasanexpressionasa1dx1++andxnwheretheaiarefunctions.Expressionsoftheforma12dx1^dx2+a13dx1^dx3++an)]TJ/F7 6.974 Tf 6.227 0 Td[(1;ndxn)]TJ/F7 6.974 Tf 6.227 0 Td[(1^dxnhavedegreetwoandareeven.Noticethatthemultiplicationrulesrequiredxi^dxj=)]TJ/F11 9.963 Tf 7.749 0 Td[(dxj^dxiand,inparticular,dxi^dxi=0.Sothemostgeneralsumofproductsoftwolineardierentialformsisadierentialformofdegreetwo,andcanbebroughttotheaboveform,locally,aftercollectionsofcoecients.Similarly,themostgeneraldierentialformofdegreekninndimensionalmanifoldisasum,locally,withfunctioncoecients,ofexpressionsoftheformdxi1^^dxik;i1<
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2.4.DERIVATIONS.33wehaved!=da1^dx1++dan^dxn=@a1 @x1dx1+@a1 @xndxn^dx1+@an @x1dx1++@an @xndxn^dxn=@a2 @x1)]TJ/F11 9.963 Tf 11.372 6.74 Td[(@a1 @x2dx1^dx2++@an @xn)]TJ/F7 6.974 Tf 6.226 0 Td[(1)]TJ/F11 9.963 Tf 11.158 6.74 Td[(@an)]TJ/F7 6.974 Tf 6.226 0 Td[(1 @xndxn)]TJ/F7 6.974 Tf 6.226 0 Td[(1^dxn:Inparticular,equalityofmixedderivativesshowsthatd2f=0,andhencethatd2!=0foranydierentialform.Hencetherulestorememberaboutdare:d!=d!+)]TJ/F8 9.963 Tf 7.749 0 Td[(1deg!!dd2=0df=@f @x1dx1++@f @xndxn:2.4Derivations.Alinearoperator`:A!Aiscalledanoddderivationif,liked,itsatises`:Aeven!Aodd;`:Aodd!Aevenand`!=`!+)]TJ/F8 9.963 Tf 7.749 0 Td[(1deg!!`:Alinearmap`:A!A,`:Aeven!Aeven;`:Aodd!Aoddsatisfying`!=`!+!`iscalledanevenderivation.SotheLeibnizruleforderivations,evenorodd,is`!=`!+)]TJ/F8 9.963 Tf 7.748 0 Td[(1deg`deg!!`:Knowingtheactionofaderivationonasetofgeneratorsofasuperalgebradeterminesitcompletely.Forexample,theequationsdxi=dxi;ddxi=08iimpliesthatdp=@p @x1dx1++@p @xndxnforanypolynomial,andhencedeterminesthevalueofdonanydierentialformwithpolynomialcoecients.Thelocalformulawegavefordfwherefisany

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34CHAPTER2.RULESOFCALCULUS.dierentiablefunction,wasjustthenaturalextensionbycontinuity,ifyoulikeoftheaboveformulaforpolynomials.Thesumoftwoevenderivationsisanevenderivation,andthesumoftwooddderivationsisanoddderivation.Thecompositionoftwoderivationswillnot,ingeneral,beaderivation,butaninstructivecomputationfromthedenitionsshowsthatthecommutator[`1;`2]:=`1`2)]TJ/F8 9.963 Tf 9.963 0 Td[()]TJ/F8 9.963 Tf 7.749 0 Td[(1deg`1deg`2`2`1isagainaderivationwhichisevenifbothareevenorbothareodd,andoddifoneisevenandtheotherodd.Aderivationfollowedbyamultiplicationisagainaderivation:specically,let`beaderivationevenoroddandletbeanevenoroddelementofA.Considerthemap!7!`!:Wehave`!=`!+)]TJ/F8 9.963 Tf 7.748 0 Td[(1deg`deg!!`=`!+)]TJ/F8 9.963 Tf 7.748 0 Td[(1deg`+degdeg!!`so!7!`!isaderivationwhosedegreeisdeg+deg`:2.5Pullback.Let:M!Nbeasmoothmap.ThenthepullbackmapisalinearmapthatsendsdierentialformsonNtodierentialformsonMandsatises!^=!^d!=d!f=f:Thersttwoequationsimplythatiscompletelydeterminedbywhatitdoesonfunctions.Thelastequationsaysthatonfunctions,isgivenbysubstitution":IntermsoflocalcoordinatesonMandonNisgivenbyx1;:::;xm=y1;:::;ynyi=ix1;:::;xmi=1;:::;nwheretheiaresmoothfunctions.Thelocalexpressionforthepullbackofafunctionfy1;:::;ynistosubstituteifortheyisasintotheexpressionforfsoastoobtainafunctionofthex0s.Itisimportanttoobservethatthepullbackondierentialformsisde-nedforanysmoothmap,notmerelyfordieomorphisms.Thisisthegreatadvantageofthecalculusofdierentialforms.

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2.6.CHAINRULE.352.6Chainrule.Supposethat:N!Pisasmoothmapsothatthecomposition:M!Pisagainsmooth.Thenthechainrulesays=:Onfunctionsthisisessentiallyatautology-itistheassociativityofcomposition:f=f.Butsincepull-backiscompletelydeterminedbywhatitdoesonfunctions,thechainruleappliestodierentialformsofanydegree.2.7Liederivative.LettbeaoneparametergroupoftransformationsofM.If!isadierentialform,wegetafamilyofdierentialforms,t!dependingdierentiablyont,andsowecantakethederivativeatt=0:d dtt!jt=0=limt=01 t[t!)]TJ/F11 9.963 Tf 9.963 0 Td[(!]:Sincet!^=t!^titfollowsfromtheLeibnizargumentthat`:!7!d dtt!jt=0isanevenderivation.Wewantaformulaforthisderivation.Noticethatsincetd=dtforallt,itfollowsbydierentiationthat`d=d`andhencetheformulafor`iscompletelydeterminedbyhowitactsonfunc-tions.LetXbethevectoreldgeneratingt.Recallthatthegeometricalsigni-canceofthisvectoreldisasfollows:Ifwexapointx,thent7!txisacurvewhichpassesthroughthepointxatt=0.Thetangenttothiscurveatt=0isthevectorXx.Intermsoflocalcoordinates,XhascoordinatesX=X1;:::;XnwhereXixisthederivativeofit;x1;:::;xnwithrespecttotatt=0.Thechainrulethengives,foranyfunctionf,`f=d dtf1t;x1;:::;xn;:::;nt;x1;:::;xnjt=0=X1@f @x1++Xn@f @xn:

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36CHAPTER2.RULESOFCALCULUS.ForthisreasonweusethenotationX=X1@ @x1++Xn@ @xnsothatthedierentialoperatorf7!Xfgivestheactionof`onfunctions.Aswementioned,thisactionof`onfunctionsdeterminesitcompletely.Inparticular,`dependsonlyonthevectoreldX,sowemaywrite`=LXwhereLXistheevenderivationdeterminedbyLXf=Xf;LXd=dLX:2.8Weil'sformula.ButwewantamoreexplicitformulaLX.ForthisitisusefultointroduceanoddderivationassociatedtoXcalledtheinteriorproductanddenotedbyiX.Itisdenedasfollows:FirstconsiderthecasewhereX=@ @xjanddeneitsinteriorproductbyi@ @xjf=0forallfunctionswhilei@ @xjdxk=0;k6=jandi@ @xjdxj=1:Thefactthatitisaderivationthengivesaneasyruleforcalculatingi@=@xjwhenappliedtoanydierentialform:Writethedierentialformas!+dxj^wheretheexpressionsfor!anddonotinvolvedxj.Theni@ @xj[!+dxj^]=:

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2.8.WEIL'SFORMULA.37TheoperatorXji@ @xjwhichmeansrstapplyi@=@xjandthenmultiplybythefunctionXjisagainanoddderivation,andsowecanmakethedenitioniX:=X1i@ @x1++Xni@ @xn:.1Itiseasytocheckthatthisdoesnotdependonthelocalcoordinatesystemused.NoticethatwecanwriteXf=iXdf:InparticularwehaveLXdxj=dLXxj=dXj=diXdxj:Wecancombinethesetwoformulasasfollows:SinceiXf=0foranyfunctionfwehaveLXf=diXf+iXdf:Sinceddxj=0wehaveLXdxj=diXdxj+iXddxj:HenceLX=diX+iXd=[d;iX].2whenappliedtofunctionsortotheformsdxj.Buttherighthandsideoftheprecedingequationisanevenderivation,beingthecommutatoroftwooddderivations.Soiftheleftandrighthandsideagreeonfunctionsandonthedierentialformsdxjtheyagreeeverywhere.Thisequation,2.2,knownasWeil'sformula,isabasicformulaindierentialcalculus.Wecanusetheinteriorproducttoconsiderdierentialformsofdegreekask)]TJ/F8 9.963 Tf 7.749 0 Td[(multilinearfunctionsonthetangentspaceateachpoint.Toillustrate,letbeadierentialformofdegreetwo.Thenforanyvectoreld,X,iXisalineardierentialform,andhencecanbeevaluatedonanyvectoreld,Ytoproduceafunction.SowedeneX;Y:=[iX]Y:

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38CHAPTER2.RULESOFCALCULUS.WecanusethistoexpressexteriorderivativeintermsofordinaryderivativeandLiebracket:Ifisalineardierentialform,wehavedX;Y=[iXd]YiXd=LX)]TJ/F11 9.963 Tf 9.962 0 Td[(diXdiXY=Y[X][LX]Y=LX[Y])]TJ/F11 9.963 Tf 9.963 0 Td[(LXY=X[Y])]TJ/F11 9.963 Tf 9.963 0 Td[([X;Y]wherewehaveintroducedthenotationLXY=:[X;Y]whichislegitimatesinceonfunctionswehaveLXYf=LXYf)]TJ/F11 9.963 Tf 9.962 0 Td[(YLXf=XYf)]TJ/F11 9.963 Tf 9.963 0 Td[(YXfsoLXYasanoperatoronfunctionsisexactlythecommutatorofXandY.SeebelowforamoredetailedgeometricalinterpretationofLXY.PuttingthepreviouspiecestogethergivesdX;Y=XY)]TJ/F11 9.963 Tf 9.963 0 Td[(YX)]TJ/F11 9.963 Tf 9.962 0 Td[([X;Y];.3withsimilarexpressionsfordierentialformsofhigherdegree.2.9Integration.Let!=fdx1^^dxnbeaformofdegreenonRn.Recallthatthemostgeneraldierentialformofdegreenisanexpressionofthistype.ThenitsintegralisdenedbyZM!:=ZMfdx1dxnwhereMisanymeasurablesubset.This,ofcourseissubjecttotheconditionthattherighthandsideconvergesifMisunbounded.Thereisalotofhiddensubtletybuiltintothisdenitionhavingtodowiththenotionoforientation.Butforthemomentthisisagoodworkingdenition.Thechangeofvariablesformulasaysthatif:M!RnisasmoothdierentiablemapwhichisonetoonewhoseJacobiandeterminantiseverywherepositive,thenZM!=ZM!:2.10Stokestheorem.LetUbearegioninRnwithachosenorientationandsmoothboundary.Wethenorienttheboundaryaccordingtotherulethatanoutwardpointingnormal

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2.11.LIEDERIVATIVESOFVECTORFIELDS.39vector,togetherwiththeapositiveframeontheboundarygiveapositiveframeinRn.Ifisann)]TJ/F8 9.963 Tf 9.963 0 Td[(1)]TJ/F8 9.963 Tf 7.748 0 Td[(form,thenZ@U=ZUd:AmanifoldiscalledorientableifwecanchooseanatlasconsistingofchartssuchthattheJacobianofthetransitionmaps)]TJ/F7 6.974 Tf 6.227 0 Td[(1isalwayspositive.Suchachoiceofanatlasiscalledanorientation.Notallmanifoldsareorientable.Ifwehavechosenanorientation,thenrelativetothechartsofourorientation,thetransitionlawsforann)]TJ/F8 9.963 Tf 7.749 0 Td[(formwheren=dimMandforadensityarethesame.Inotherwords,givenanorientation,wecanidentifydensitieswithn)]TJ/F8 9.963 Tf 7.749 0 Td[(formsandn)]TJ/F8 9.963 Tf 7.748 0 Td[(formwithdensities.Thuswemayintegraten)]TJ/F8 9.963 Tf 7.748 0 Td[(forms.ThechangeofvariablesformulathenholdsfororientationpreservingdieomorphismsasdoesStokestheorem.2.11Liederivativesofvectorelds.LetYbeavectoreldandtaoneparametergroupoftransformationswhoseinnitesimalgenerator"issomeothervectoreldX.Wecanconsiderthepulledback"vectoreldtYdenedbytYx=d)]TJ/F10 6.974 Tf 6.227 0 Td[(tfYtxg:Inwords,weevaluatethevectoreldYatthepointtx,obtainingatangentvectorattx,andthenapplythedierentialoftheinversemap)]TJ/F10 6.974 Tf 6.226 0 Td[(ttoobtainatangentvectoratx.IfwedierentiatetheoneparameterfamilyofvectoreldstYwithrespecttotandsett=0wegetavectoreldwhichwedenotebyLXY:LXY:=d dttYjt=0:If!isalineardierentialform,thenwemaycomputeiY!whichisafunctionwhosevalueatanypointisobtainedbyevaluatingthelinearfunction!xonthetangentvectorYx.ThusitYt!x=hdt!tx;d)]TJ/F10 6.974 Tf 6.226 0 Td[(tYtxi=fiY!gtx:Inotherwords,tfiY!g=itYt!:Wehaveveriedthiswhen!isadierentialformofdegreeone.Itistriviallytruewhen!isadierentialformofdegreezero,i.e.afunction,sincethenbothsidesarezero.Butthen,bythederivationproperty,weconcludethatitistrueforformsofalldegrees.WemayrewritetheresultinshorthandformastiY=itYt:

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40CHAPTER2.RULESOFCALCULUS.Sincetd=dtweconcludefromWeil'sformulathattLY=LtYt:Untilnowthesubscripttwassuperuous,theformulasbeingtrueforanyxeddieomorphism.Nowwedierentiatetheprecedingequationswithrespecttotandsett=0.Weobtain,usingLeibniz'srule,LXiY=iLXY+iYLXandLXLY=LLXY+LYLX:ThislastequationsaysthatLiederivativeonformswithrespecttothevectoreldLXYisjustthecommutatorofLXwithLY:LLXY=[LX;LY]:Forthisreasonwewrite[X;Y]:=LXYandcallittheLiebracketorcommutatorofthetwovectoreldsXandY.Theequationforinteriorproductcanthenbewrittenasi[X;Y]=[LX;iY]:TheLiebracketisantisymmetricinXandY.WemaymultiplyYbyafunctiongtoobtainanewvectoreldgY.FormthedenitionswehavetgY=tgtY:Dierentiatingatt=0andusingLeibniz'sruleweget[X;gY]=XgY+g[X;Y].4whereweusethealternativenotationXgforLXg.Theantisymmetrythenimpliesthatforanydierentiablefunctionfwehave[fX;Y]=)]TJ/F8 9.963 Tf 7.749 0 Td[(YfX+f[X;Y]:.5FromboththisequationandfromWeil'sformulaappliedtodierentialformsofdegreegreaterthanzeroweseethattheLiederivativewithrespecttoXatapointxdependsonmorethanthevalueofthevectoreldXatx.2.12Jacobi'sidentity.Fromthefactthat[X;Y]actsasthecommutatorofXandYitfollowsthatforanythreevectoreldsX;YandZwehave[X;[Y;Z]]+[Z;[X;Y]]+[Y;[Z;X]]=0:

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2.13.LEFTINVARIANTFORMS.41ThisisknownasJacobi'sidentity.Wecanalsoderiveitfromthefactthat[Y;Z]isanaturaloperationandhenceforanyoneparametergrouptofdif-feomorphismswehavet[Y;Z]=[tY;tZ]:IfXistheinnitesimalgeneratoroftthendierentiatingtheprecedingequa-tionwithrespecttotatt=0gives[X;[Y;Z]]=[[X;Y];Z]+[Y;[X;Z]]:Inotherwords,Xactsasaderivationofthemutliplication"givenbyLiebracket.ThisisjustJacobi'sidentitywhenweusetheantisymmetryofthebracket.InthefuturewewewillhaveoccasiontotakecyclicsumssuchasthosewhichariseontheleftofJacobi'sidentity.SoifFisafunctionofthreevectoreldsorofthreeelementsofanysetwithvaluesinsomevectorspaceforexampleinthespaceofvectoreldswewilldenethecyclicsumCycFbyCycFX;Y;Z:=FX;Y;Z+FY;Z;X+FZ;X;Y:WiththisdenitionJacobi'sidentitybecomesCyc[X;[Y;Z]]=0:.6Exercises2.13Leftinvariantforms.LetGbeagroupandMbeaset.AleftactionofGonMconsistsofamap:GM!Msatisfyingtheconditionsa;b;m=ab;manassociativitylawande;m=m;8m2Mwhereeistheidentityelementofthegroup.Whenthereisnoriskofconfusionwewillwriteamfora;m.ButinmuchofthebeginningofthefollowingexercisestherewillbeariskofconfusionsincetherewillbeseveraldierentactionsofthesamegroupGonthesetM.Wethinkofanactionasassigningtoeachelementa2Gatransformation,a,ofM:a:M!M;a:m7!a;m:Sowealsousethenotationam=a;m:

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42CHAPTER2.RULESOFCALCULUS.Forexample,wemaytakeMtobethegroupGitselfandlettheactionbeleftmultiplication,L,soLa;m=am:WewillwriteLa:G!G;Lam=am:Wemaymayalsoconsidertheleftactionofrightmultiplication:R:GG!G;Ra;m=ma)]TJ/F7 6.974 Tf 6.227 0 Td[(1:TheinverseisneededtogettheorderrightinRa;Rb;m=Rab;m.SowewillwriteRa:G!G;Ram=ma)]TJ/F7 6.974 Tf 6.227 0 Td[(1:WewillbeinterestedinthecasethatGisaLiegroup,whichmeansthatGisamanifoldandthemultiplicationmapGG!GandtheinversemapG!G;a7!a)]TJ/F7 6.974 Tf 6.227 0 Td[(1arebothsmoothmaps.Thenthedierential,dLammapsthetangentspacetoGatm,tothetangentspacetoGatam:dLa:TGm!TGamandsimilarlydRa:TGm!TGma:Inparticular,dLa)]TJ/F6 4.981 Tf 5.397 0 Td[(1:TGa!TGe:LetG=Glnbethegroupofallinvertiblennmatrices.Itisanopensubsethenceasubmanifoldofthen2dimensionalspaceMatnofallnnmatrices.WecanthinkofthetautologicalmapwhichsendseveryA2GintoitselfthoughtofasanelementofMatnasamatrixvaluedfunctiononG.Putanotherway,AisamatrixoffunctionsonG,eachofthematrixentriesAijofAisafunctiononG.HencedA=dAijisamatrixofdierentialformsor,wemaysay,amatrixvalueddierentialform.SowemayconsiderA)]TJ/F7 6.974 Tf 6.227 0 Td[(1dAwhichisalsoamatrixvalueddierentialformonG.LetBbeaxedelementofG.1.ShowthatLBA)]TJ/F7 6.974 Tf 6.226 0 Td[(1dA=A)]TJ/F7 6.974 Tf 6.227 0 Td[(1dA:.7SoeachoftheentriesofA)]TJ/F7 6.974 Tf 6.227 0 Td[(1dAisleftinvariant.2.ShowthatRBA)]TJ/F7 6.974 Tf 6.227 0 Td[(1dA=BA)]TJ/F7 6.974 Tf 6.227 0 Td[(1dAB)]TJ/F7 6.974 Tf 6.226 0 Td[(1:.8SotheentriesofA)]TJ/F7 6.974 Tf 6.227 0 Td[(1dAarenotrightinvariantingeneral,but.8showshowtheyaretransformedintooneanotherbyrightmultiplication.

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2.14.THEMAURERCARTANEQUATIONS.43ForanytwomatrixvalueddierentialformsR=RijandS=SijdenetheirmatrixexteriorproductR^Sbytheusualformulaformatrixproduct,butwithexteriormultiplicationoftheentriesinsteadofordinarymultiplication,soR^Sik:=XjRij^Sjk:Also,ifR=Rijisamatrixvalueddierentialform,denedRbyapplyingdtoeachoftheentries.SodRij:=dRij:Finally,if:X!YisasmoothmapandR=RijisamatrixvaluedformonYthenwedeneitspullbackbypullingbackeachoftheentries:Rij:=Rij:2.14TheMaurerCartanequations.3.Inelementarycalculuswehavetheformulad=x=)]TJ/F11 9.963 Tf 7.749 0 Td[(dx=x2.WhatisthegeneralizationofthisformulaforthematrixfunctionA)]TJ/F7 6.974 Tf 6.227 0 Td[(1.Inotherwords,whatistheformulafordA)]TJ/F7 6.974 Tf 6.226 0 Td[(1?4.Showthatifweset!=A)]TJ/F7 6.974 Tf 6.226 0 Td[(1dAthend!+!^!=0:.9HereisanotherwayofthinkingaboutA)]TJ/F7 6.974 Tf 6.226 0 Td[(1dA:SinceG=GlnisanopensubsetofthevectorspaceMatn,wemayidentifythetangentspaceTGAwiththevectorspaceMatn.ThatiswehaveanisomorphismbetweenTGAandMatn.Ifyouthinkaboutitforaminute,itistheformdAwhicheectsthisisomorphismateverypoint.Ontheotherhand,leftmultiplicationbyA)]TJ/F7 6.974 Tf 6.227 0 Td[(1isalinearmap.Underthisidentication,thedierentialofalinearmapLlooksjustlikeL.Sointermsofthisidentication,A)]TJ/F7 6.974 Tf 6.227 0 Td[(1dA,whenevaluatedatthetangentspaceTGAisjusttheisomorphismdL)]TJ/F7 6.974 Tf 6.227 0 Td[(1A:TGA!TGIwhereIistheidentitymatrix.2.15RestrictiontoasubgroupLetHbeaLiesubgroupofG.ThismeansthatHisasubgroupofGanditisalsoasubmanifold.Inotherwordswehaveanembedding:H!G

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44CHAPTER2.RULESOFCALCULUS.whichisaninjectivegrouphomomorphism.Leth=THIdenotethetangentspacetoHattheidentityelement.5.Concludefromtheprecedingdiscussionthatifwenowset!=A)]TJ/F7 6.974 Tf 6.227 0 Td[(1dAthen!takesvaluesinh.Inotherwords,whenweevaluate!onanytangentvectoratanypointofHwegetamatrixbelongingtothesubspaceh.6.Showthatonagroup,theonlytransformationswhichcommutewithalltherightmultiplications,Rb;b2G,aretheleftmultiplications,La.Foranyvector2THI,denethevectoreldXbyXA=dRA)]TJ/F6 4.981 Tf 5.396 0 Td[(1:RecallthatRA)]TJ/F6 4.981 Tf 5.396 0 Td[(1isrightmultiplicationbyAandsosendsIintoA.Forexample,ifwetakeHtobethefullgroupG=GlnandidentifythetangentspaceateverypointwithMatnthentheabovedenitionbecomesXA=A:Byconstruction,thevectoreldXisrightinvariant,i.e.isinvariantunderallthedieomorphismsRB.7.ConcludethattheowgeneratedbyXisleftmultiplicationbyaoneparam-etersubgroup.AlsoconcludethatinthecaseH=GlntheowgeneratedbyXisleftmultiplicationbytheoneparametergroupexpt:FinallyconcludethatforageneralsubgroupH,if2hthenalltheexptlieinH.8.WhatisthespacehinthecasethatHisthegroupofEuclideanmotionsinthreedimensionalspace,thoughtofasthesetofallfourbyfourmatricesoftheformAv01;AAy=I;v2R3?2.16Frames.LetVbeanndimensionalvectorspace.RecallthatframeonVis,bydeni-tion,anisomorphismf:Rn!V.Givingfisthesameasgivingeachofthe

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2.17.EUCLIDEANFRAMES.45vectorsfi=fiwheretheirangeoverthestandardbasisofRn.SogivingaframeisthesameasgivinganorderedbasisofVandwewillsometimeswritef=f1;:::;fn:IfA2GlnthenAisanisomorphismofRnwithitself,sofA)]TJ/F7 6.974 Tf 6.227 0 Td[(1isanotherframe.Sowegetanaction,R:GlnF!FwhereF=FVdenotesthespaceofallframes:RA;f=fA)]TJ/F7 6.974 Tf 6.227 0 Td[(1:.10Iffandgaretwoframes,theng)]TJ/F7 6.974 Tf 6.226 0 Td[(1f=MisanisomorphismofRnwithitself,i.e.amatrix.Sogivenanytwoframes,fandg,thereisauniqueM2Glnsothatg=fM)]TJ/F7 6.974 Tf 6.226 0 Td[(1.Oncewexanf,wecanusethisfacttoidentifyFwithGln,buttheidenticationdependsonthechoiceoff.Butinanyeventthenon-uniqueidenticationshowsthatFisamanifoldandthat.10denesanactionofGlnonF.Eachofthefithei)]TJ/F8 9.963 Tf 7.749 0 Td[(thbasisvectorintheframecanbethoughtofasaVvaluedfunctiononF.Sowemaywritedfj=X!ijfi.11wherethe!ijareordinarynumbervaluedlineardierentialformsonF.Wethinkofthisequationasgivingtheexpansionofaninnitesimalchangeinfjintermsofthebasisf=f1;:::;fn.Ifweusetherow"representationoffasabove,wecanwritetheseequationsasdf=f!.12where!=!ij.9.Showthatthe!denedby.12satisesRB!=B!B)]TJ/F7 6.974 Tf 6.227 0 Td[(1:.13Toseetherelationwithwhatwentonbefore,noticethatwecouldtakeV=Rnitself.Thenfisjustaninvertiblematrix,Aand.12becomesouroldequation!=A)]TJ/F7 6.974 Tf 6.226 0 Td[(1dA.So.13reducesto.8.Ifwetaketheexteriorderivativeof2.12weget0=ddf=df^!+fd!=f!^!+d!fromwhichweconcluded!+!^!=0:.142.17Euclideanframes.WespecializetothecasewhereV=Rn;n=d+1sothatthesetofframesbecomesidentiedwiththegroupGlnandrestricttothesubgroup,H,of

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46CHAPTER2.RULESOFCALCULUS.EuclideanmotionswhichconsistofallnmmatricesoftheformAv01;A2Od;v2Rd:Suchamatrix,whenappliedtoavectorw1sendsitintothevectorAw+v1andAw+vistheorthogonaltransformationAappliedtowfollowedbythetranslationbyv.ThecorrespondingEuclideanframesconsistingofthecolumnsoftheelementsofHarethusdenedtobetheframesoftheformfi=ei0;i=1;:::d;wheretheeiformanorthonormalbasisofRdandfn=v1;wherev2Rdisanarbitraryvector.Theideaisthatvrepresentsachoiceoforigininddimensionalspaceande=e1;:::;edisanorthonormalbasis.Wecanwritethisinshorthandnotationasf=ev01:IfdenotestheembeddingofHintoG,weknowfromtheexercise5that!=00;whereij=)]TJ/F8 9.963 Tf 7.748 0 Td[(ji:Sothepullbackof.12becomesdev01=ee00.15or,inmoreexpandednotation,dej=Xiijei;dv=Xiiei:

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2.18.FRAMESADAPTEDTOASUBMANIFOLD.47Let;denotetheEuclideanscalarproduct.Thenwecanwritei=dv;ei.16anddej;ei=ij:Ifweset=)]TJ/F8 9.963 Tf 7.749 0 Td[(thisbecomesdei;ej=ij:.17Then.14becomesd=^;d=^:.18Or,inmoreexpandednotation,di=Xjij^j;dik=Xjij^jk:.19Equations.16-.18or.19areknownasthestructureequationsofEuclideangeometry.2.18Framesadaptedtoasubmanifold.LetMbeakdimensionalsubmanifoldofRd.Thisdeterminesasubmanifoldofthemanifold,H,ofallEuclideanframesbythefollowingrequirements:iv2Mandiiei2TMvforik.Wewillusuallywriteminsteadofvtoemphasizetherstrequirement-thattheframesbebasedatpointsofM.ThesecondrequirementsaysthattherstkvectorsintheframebasedatmbetangenttoMandhencethatthelastn)]TJ/F11 9.963 Tf 10.154 0 Td[(kvectorsintheframearenormaltoM.WewilldenotethismanifoldbyOM.Ithasdimensionk+kk)]TJ/F8 9.963 Tf 9.963 0 Td[(1 2+d)]TJ/F11 9.963 Tf 9.962 0 Td[(k)]TJ/F8 9.963 Tf 9.963 0 Td[(1d)]TJ/F11 9.963 Tf 9.963 0 Td[(k 2:ThersttermcomesfromthepointmvaryingonM,thesecondisthedimensionoftheorthogonalgroupOkcorrespondingtothechoicesoftherstkvectorsintheframe,andthethirdtermisdimOd)]TJ/F11 9.963 Tf 9.667 0 Td[(kcorrespondtothelastn)]TJ/F11 9.963 Tf 9.667 0 Td[(kvectors.WehaveanembeddingofOMintoH,andhencetheformsandpullbacktoOM.Aswearerunningoutofletters,wewillcontinuetodenotethesepullbacksbythesameletters.Sothepulledbackformssatisfythesamestructureequations.16-.18or.19asabove,buttheyaresupplementedbyi=0;8i>k:.20

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48CHAPTER2.RULESOFCALCULUS.2.19Curvesandsurfaces-theirstructureequa-tions.WewillbeparticularlyinterestedincurvesandsurfacesinthreedimensionalEuclideanspace.Foracurve,C,themanifoldofframesistwodimensional,andwehavedC=1e1.21de1=12e2+13e3.22de2=21e1+23e3.23de3=31e1+32e2:.24Onecanvisualizethemanifoldofframesasasortoftube:abouteachpointofthecurvethereisacircleintheplanenormaltothetangentlinecorrespondingthepossiblechoicesofe2.Forthecaseofasurfacethemanifoldofframesisthreedimensional:wecanthinkofitasaunionofcircleseachcenteredatapointofSandintheplanetangenttoSatthatpoint.Thenequation.21isreplacedbydX=1e1+2e2.25butotherwisetheequationsareasabove,includingthestructureequations.19.Thesebecomed1=12^2.26d2=)]TJ/F8 9.963 Tf 7.749 0 Td[(12^1.270=31^1+32^2.28d12=13^32.29d13=12^23.30d23=21^13.31Equation.29isknownasGauss'equation,andequations.30and.31areknownastheCodazzi-Mainardiequations.2.20Thesphereasanexample.Incomputationswithlocalcoordinates,wemaynditconvenienttouseacross-section"ofthemanifoldofframes,thatisamapwhichassignstoeachpointofneighborhoodonthesurfaceapreferredframe.Ifwearegivenaparametrizationm=mu;vofthesurface,onewayofchoosingsuchacross-sectionistoapplytheGram-Schmidtorthogonalizationproceduretothetan-gentvectoreldsmuandmv,andtakeintoaccountthechosenorientation.Forexample,considerthesphereofradiusR.Wecanparameterizethespherewiththenorthandsouthpolesandonelongitudinalsemi-circleremoved

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2.20.THESPHEREASANEXAMPLE.49bytheu;v2;2;byX=Xu;vwhereXu;v=0@RcosusinvRsinusinvRcosv1A:Herevdenotestheangulardistancefromthenorthpole,sotheexcludedvaluev=0correspondstothenorthpoleandtheexcludedvaluev=correspondstothesouthpole.Eachconstantvalueofvbetween0andisacircleoflatitudewiththeequatorgivenbyv= 2.Theparameterudescribesthelongitudefromtheexcludedsemi-circle.InanyframeadaptedtoasurfaceinR3,thethirdvectore3isnormaltothesurfaceatthebasepointoftheframe.Therearetwosuchchoicesateachbasepoint.Inoursphereexampleletuschoosetheoutwardpointingnormal,whichatthepointmu;vise3mu;v=0@cosusinvsinusinvcosv1A:Wewillwritethelefthandsideofthisequationase3u;v.Thecoordinatesu;vareorthogonal,i.e.XuandXvareorthogonalateverypoint,sotheor-thonormalizationprocedureamountsonlytonormalization:Replaceeachofthesevectorsbytheunitvectorspointinginthesamedirectionateachpoint.Sowegete1u;v=0@)]TJ/F8 9.963 Tf 9.409 0 Td[(sinucosu01A;e2u;v=0@cosucosvsinucosv)]TJ/F8 9.963 Tf 9.41 0 Td[(sinv1A:Wethusobtainamapfrom;2;tothemanifoldofframes,u;v=Xu;v;e1u;v;e2;u;v;e3u;v:SinceXue1=RsinvandXve2=RwehavedXu;v=Rsinvdue1u;v+Rdve2u;v:Thusweseefrom.25that1=Rsinvdu;2=Rdvandhencethat1^2=R2sinvdu^dv:NowR2sinvdudvisjusttheareaelementofthesphereexpressedinu;vco-ordinates.Thechoiceofe1;e2determinesanorientationofthetangentspacetothesphereatthepointXu;vandso1^2isthepull-backofthecorrespondingorientedareaform.

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50CHAPTER2.RULESOFCALCULUS.10.Compute12;13,and23andverifythatd12=)]TJ/F11 9.963 Tf 7.749 0 Td[(K1^2whereK=1=R2isthecurvatureofthesphere.WewillgeneralizethisequationtoanarbitrarysurfaceinR3insection??.2.21RibbonsTheideahereistostudyacurveonasurface,orratheracurvewithaninnitesimal"neighborhoodofasurfacealongit.SoletCbeacurveandOCitsassociatedtwodimensionalmanifoldofframes.Wehaveaprojection:OC!Csendingeveryframeintoitsorigin.ByaribbonbasedonCwemeanasectionn:C!OC,sonassignsauniqueframetoeachpointofthecurveinasmoothway.Wewillonlybeconsideringcurveswithnon-vanishingtangentvectoreverywhere.Withnolossofgeneralitywemayassumethatwehaveparametrizedthecurvebyarclength,andthechoiceofe1determinesanorientationofthecurve,so=ds.Thechoiceofe2ateverypointthende-terminese3uptoasign.Soagoodwaytovisualizesistothinkofarigidmetalribbondeterminedbythecurveandthevectorse2perpendiculartothecurvedeterminedbynateachpoint.Theformsijallpullbackunderntofunctionmultiplesofds:n12=kds;n23=)]TJ/F11 9.963 Tf 7.749 0 Td[(ds;n13=wds.32wherek;andwarefunctionsofs.Wecanwriteequations2.21-.24aboveasdC ds=e1;andde1 ds=ke2+we3;de2 ds=)]TJ/F11 9.963 Tf 7.749 0 Td[(ke1)]TJ/F11 9.963 Tf 9.963 0 Td[(e3;de3 ds=)]TJ/F11 9.963 Tf 7.748 0 Td[(we1+e3:.33Forlaterapplicationswewillsometimesbesloppyandwriteijinsteadofnijforthepullbacktothecurve,soalongtheribbonwehave12=kdsetc.Alsoitwillsometimesbeconvenientincomputationsasopposedtoprovingtheoremstouseparametersotherthanarclength.11.ShowthattworibbonsdenedoverthesameintervalofsvaluesarecongruentthatisthereisaEuclideanmotioncarryingoneintotheotherifandonlyifthefunctionsk;;andwarethesame.Aribbonisreallyjustacurveinthespace,H,ofallEuclideanframes,havingthepropertythatthebasepoint,thatisthevoftheframev;e1;e2;e3hasnon-vanishingderivative.Thepreviousexercisesaysthattwocurves,i:I!Handj:I!HinHdierbyanoveralllefttranslationthatissatisfyj=Lhiifandonlyiftheforms;12;13;23pullbacktothesameformsonI.The

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2.22.DEVELOPINGARIBBON.51formiisjustthearclengthformdsaswementionedabove.Itisabsolutelycrucialfortherestofthiscoursetounderstandthemeaningoftheformi12.ConsideracircleoflatitudeonasphereofradiusR.Toxthenotation,supposethatthecircleisatangulardistancevfromthenorthpoleandthatweuseuasangularcoordinatesalongthecircle.Taketheribbonadaptedtothesphere,soe1istheunittangentvectortothecircleoflatitudeande2istheunittangentvectortothecircleoflongitudechosenasabove.Problem10thenimpliesthati12=)]TJ/F11 9.963 Tf 7.749 0 Td[(cosvdu.12.LetCbeastraightlinesayapieceofthez-axisparametrizedaccordingtoarclengthandlete2berotatingataratefsaboutCso,forexample,e2=cosfsi+sinfsjwhereiandjaretheunitvectorsinthexandydirections.Whatisi12?Tocontinueourunderstandingof12,letusconsiderwhatitmeansfortworibbons,i:I!Handj:I!Htohavethesamevalueofthepullbackof12atsomepoints02IwhereIissomeintervalontherealline.Soi12js=s0=j12js=s0:Thereisauniqueleftmultiplication,thatisauniqueEuclideanmotion,whichcarriesis0tojs0.Letassumethatwehaveappliedthismotionsoweassumethatis0=js0.Letuswriteis=Cs;e1s;e2s;e3s;js=Ds;f1s;f2s:f3sandweareassumingthatCs0=Ds0,C0s0=e1s0=f1s0=D0s0sothecurvesCandDaretangentats0,andthate2s0=f2s0sothattheplanesoftheribbonspannedbythersttwoorthonormalvectorscoincide.Thenourconditionabouttheequalityofthepullbacksof12assertsthate02)]TJ/F11 9.963 Tf 9.962 0 Td[(f02s0;e1s0=0andofcoursee02)]TJ/F11 9.963 Tf 10.712 0 Td[(f02s0;e2s0=0automaticallysincee2sandf2sareunitvectors.Sotheconditionisthattherelativechangeofe2andf2andsimilarlye1andf1ats0benormaltothecommontangentplanetotheribbon.2.22Developingaribbon.Wewillnowdroponedimension,andconsiderribbonsintheplaneor,ifyoulike,ribbonslyinginaxedplaneinthreedimensionalspace.Soallwehaveisand12.Also,theorientationofthecurveandoftheplanecompletelydeterminese2astheunitvectorintheplaneperpendiculartothecurveandsuchthate1;e2givethecorrectorientation.soaribbonintheplaneisthesameasanorientedcurve.13.Letk=ksbeanycontinuousfunctionofs.Showthatthereisaribbonintheplanewhosebasecurveisparametrizedbyarclengthandforwhich

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52CHAPTER2.RULESOFCALCULUS.j12=kds.Furthermore,showthatthisplanarribboncurveisuniquelydetermineduptoaplanarEuclideanmotion.Itfollowsfromtheprecedingexercise,thatwehaveawayofassociatingacurveintheplanedetermineduptoaplanarEuclideanmotiontoanyribboninspace.Itconsistsofrockingandrollingtheribbonalongtheplaneinsuchawaythatinnitesimalchangeinthee1ande2arealwaysnormaltotheplane.Mathematically,itconsistsinsolvingproblem13forthek=kswherei12=kdsfortheribbon.Wecallthisoperationdevelopingtheribbonontoaplane.Inparticular,ifwehaveacurveonasurface,wecanconsidertheribbonalongthecurveinducedbythesurface.Inthisway,wemaytalkofdevelopingthesurfaceonaplanealongthegivencurve.Intuitively,ifthesurfacewereconvex,thisamountstorollingthesurfaceonaplanealongthecurve.noindent14.WhatareresultsofdevelopingtheribbonsofProblem12andtheribbonweassociatedtoacircleoflatitudeonthesphere?2.23Paralleltransportalongaribbon.Recallthataribbonisacurveinthespace,H,ofallEuclideanframes,havingthepropertythatthebasepoint,thatistheCoftheframeC;e1;e2;e3hasnon-vanishingderivativeatallpoints.SoCdenesacurveinEuclideanthreespacewithnowherevanishingtangent.Wewillparameterizethiscurveandtheribbonbyarclength.Byaunitvectoreldtangenttotheribbonwewillmeanacurve,vsofunitvectorseverywheretangenttotheribbon,sovs=cosse1s+sinse2s:.34Wesaythatthevectoreldisparallelalongtheribboniftheinnitesimalchangeinvisalwaysnormaltotheribbon,i.e.ifv0s;e1sv0s;e2s0:Recalltheform12=kdsfrombefore.15.Showthatthevectoreldasgivenaboveisparallelifandonlyifthefunctionsatisesthedierentialequation0+k=0:Concludethatthenotionofparallelismdependsonlyontheform12.Alsoconcludethatgivenanyunitvector,v0atsomepoints0,thereisauniqueparallelvectoreldtakingonthevaluev0ats0.Thevaluevs1atsomesecondpointiscalledtheparalleltransportofv0alongtheribbonfroms0tos1.

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2.24.SURFACESINR3.5316.Whatistheconditiononaribbonthatthetangentvectortothecurveitself,i.e.thevectorelde1,beparallel?Whichcirclesonthespherearesuchthattheassociatedribbonhasthisproperty?Supposetheribbonisclosed,i.e.Cs+L=Cs;e1s+L=e1s;e2s+L=e2sforsomelengthL.Wecanthenstartwithavectorv0atpoints0andtransportitallthewayaroundtheribbonuntilwegetbacktothesamepoint,i.e.transportfroms0tos0+L.Thevectorv1wesoobtainwillmakesomeangle,callitwiththevectorv0.Theangleiscalledtheholonomyoftheparalleltransportoftheribbon.17.Showthatisindependentofthechoiceofs0andv0.Whatisitsexpressionintermsof12?18.Whatistheholonomyforacircleonthesphereintermsofitslatitude.19.Showthatiftheribbonisplanarsoe1ande2lieinaxedplaneavectoreldisparallelifandonlyifitisparallelintheusualsenseofEuclideangeometrysaymakesaconstantanglewiththex-axis.Butrememberthatthecurveisturning.Sotheholonomyofacircleintheplaneis2dependingontheorientation.Similarlyforthesumoftheexterioranglesofatrianglethinkofthecornersasbeingroundedout.Convinceyourselfofthefollowingfactwhichisnotsoeasyunlessyouknowthetrick:Showthatforanysmoothsimpleclosedcurvei.e.onewithnoselfintersectionsintheplanetheholonomyisalways2.Exercises15,17,and19,togetherwiththeresultsabovegiveanalternativeinterpretationofparalleltransport:developtheribbonontotheplaneandthenjusttranslatethevectorv0intheEuclideanplanesothatitsoriginliesattheimageofs1.Thenconsiderthecorrespondingvectoreldalongtheribbon.Thefunctionkin12=kdsiscalledthegeodesiccurvatureoftheribbon.TheintegralR12=Rkdsiscalledthetotalgeodesiccurvatureoftheribbon.Itgivesthetotalchangeinangleincludingmultiplesof2betweenthetangentstotheinitialandnalpointsofthedevelopedcurve.2.24SurfacesinR3.WeletMbeatwodimensionalsubmanifoldofR3andOitsbundleofadaptedframes.Wehaveaprojection"map:O!M;m;e1;e2;e37!m;whichwecanalsowrite=m:

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54CHAPTER2.RULESOFCALCULUS.Supposethatweconsiderthetruncated"versionoftheadaptedbundleofframes~Owhereweforgetaboute3.Thatis,letconsistofallm;e1:e2wherem2Mande1;e2isanorthonormalbasisofthetangentspaceTMmtoMatm.Noticethatthedenitionwejustgavewasintrinsic.TheconceptofanorthonormalbasisofTMmdependsonlyonthescalarproductonTMm.Thedierentialofthemapm:~O!Matapointm;e1;e2sendsatangentvectorto~Oatm;e1;e2;e3toatangentvectortoMatm,andthescalarproductofthisimagevectorwithe1isalinearfunctionof.Wehavejustgivenanintrinsicof1.ByabuseoflanguageIamusingthissameletter1fortheformdm;e1on~Oase3doesnotenterintoitsdenition.Similarly,weseethat2isanintrinsicallydenedform.Fromtheirverydenitions,theforms1and2arelinearlyindependentateverypointof~O.Thereforetheformsd1andd2areintrinsic,andthisprovesthattheform12isintrinsic.Indeed,ifwehadtwolineardierentialformsandonOwhichsatisedd1=^2;d1=^2d2=)]TJ/F11 9.963 Tf 7.749 0 Td[(^1d2=)]TJ/F11 9.963 Tf 7.749 0 Td[(^1thenthersttwoequationsgive)]TJ/F11 9.963 Tf 9.963 0 Td[(^20whichimpliesthat)]TJ/F11 9.963 Tf 9.98 0 Td[(isamultipleof2andthelasttwoequationsimplythat)]TJ/F11 9.963 Tf 8.84 0 Td[(isamultipleof1so=.ThenextfewproblemswillgiveathirdproofofGauss'stheoremaegregium.Theywillshowthatd12=)]TJ/F11 9.963 Tf 7.749 0 Td[(K1^2whereKistheGaussiancurvature.ThisassertionislocalinM,sowemaytemporarilymaketheassumptionthatMisorientable-thisallowsustolookatthesub-bundle OOoforientedframes,consistingofthoseframesforwhiche1;e2formanorientedbasisofTMmandwheree1;e2;e3anorientedframeonR3.LetdAdenotetheorientedareaformonthesurfaceM.Abadbutstandardnotation,sincewetheareaformisnotthedierentialofaoneform,ingeneral.Recallthatwhenevaluatedonanypairoftangentvectors,1;2atm2Mitistheorientedareaoftheparallelogramspannedby1and2,andthisisjustthedeterminantofthematrixofscalarproductsofthe'swithanyorientedorthonormalbasis.Conclude20.ExplainwhydA=1^2:Thethirdcomponent,e3ofanyframeiscompletelydeterminedbythepointonthesurfaceandtheorientationastheunitnormal,ntothesurface.Nown

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2.24.SURFACESINR3.55canbethoughtofasamapfromMtotheunitsphere,SinR3.LetdSdenotetheorientedareaformoftheunitsphere.SondSisatwoformonMandwecandenethefunctionKbyndS=KdA:21ShowthathefunctionKisGaussiancurvatureofthesurface.22.ShowthatndS=31^32and23.Concludethatd12=)]TJ/F11 9.963 Tf 7.748 0 Td[(KdA:WearegoingtowanttoapplyStokes'theoremtothisformula.Butinordertodoso,weneedtointegrateoveratwodimensionalregion.SoletUbesomeopensubsetofMandlet:U!)]TJ/F7 6.974 Tf 6.227 0 Td[(1UObeamapsatisfying=id:SoassignsaframetoeachpointofUinadierentiablemanner.LetCbeacurveonMandsupposethatCliesinU.Thenthesurfacedeterminesaribbonalongthiscurve,namelythechoiceofframesfromwhiche1istangenttothecurveandpointinginthepositivedirection.SowehaveamapR:C!Ocomingfromthegeometryofthesurface,andwithnownecessarilydierentnotationfromtheprecedingsectionR12=kdsisthegeodesiccurvatureoftheribbonasstudiedabove.SincetheribbonisdeterminedbythecurveasMisxedwecancallitthegeodesiccurvatureofthecurve.Ontheotherhand,wecanconsidertheform12pulledbacktothecurve.LetCs=Cs;f1s;f2s;nsandletsbetheanglethate1smakeswithf1ssoe1s=cossf1s+sinsf2s;e2s=)]TJ/F8 9.963 Tf 9.409 0 Td[(sinsf1s+cossf2s:24.LetC12denotethepullbackof12tothecurve.Showthatkds=d+C12:Concludethat

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56CHAPTER2.RULESOFCALCULUS.Proposition2Thetotalgeodesiccurvature=b)]TJ/F11 9.963 Tf 8.956 0 Td[(a+RC12whereb)]TJ/F11 9.963 Tf 8.955 0 Td[(adenotesthetotalchangeofanglearoundthecurve.Howcanweconstructa?Hereisonewaythatwedescribedearlier:SupposethatUisacoordinatechartandthatx1;x2arecoordinatesonthischart.Then@ @x1;@ @x2arelinearlyindependentvectorsateachpointandwecanapplyGramSchmidttoorthonomalizethem.Thisgiveaandtheangleaboveisjusttheanglethatthevectore1makeswiththex)]TJ/F8 9.963 Tf 7.749 0 Td[(axisinthiscoordinatesystem.SupposewetakeCtobetheboundaryofsomeniceregion,D,inU.Forexample,supposethatCisatriangleorsomeotherpolygonwithitsedgesroundedtomakeasmoothcurve.Thenthetotalchangeinangleis2andso25.ConcludethatforsuchacurveZZDKdA+ZCkds=2:TheintegralofKdAiscalledthetotalGaussiancurvature.26.Showthatasthecurveactuallyapproachesthepolygon,thecontributionfromtheroundedcornersapproachestheexteriorangleofthepolygon.Con-cludethatifaregioninacoordinateneighborhoodonthesurfaceisboundedbycontinuouspiecewisedierentiablearcsmakingexterioranglesatthecornersProposition3thetotalGaussiancurvature+Ptotalgeodesiccurvatures+Pexteriorangles=2.27.Supposethatwehavesubdividedacompactsurfaceintopolygonalregions,eachcontainedinacoordinateneighborhood,withffaces,eedges,andvvertices.Let=f)]TJ/F11 9.963 Tf 9.963 0 Td[(e+v.showthatZMKdA=2:

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Chapter3Levi-CivitaConnections.3.1Denitionofalinearconnectiononthetan-gentbundle.AlinearconnectionronamanifoldMisarulewhichassignsavectoreldrXYtoeachpairofvectoreldsXandYwhichisbilinearoverRsubjecttotherulesrfXY=frXY.1andrXgY=XgY+grXY:.2Whilecondition.2isthesameasthecorrespondingconditionLXgY=[X;gY]=XgY+gLXYforLiederivatives,condition.1isquitedierentfromthecorrespondingformulaLfXY=[fX;Y]=)]TJ/F8 9.963 Tf 7.749 0 Td[(YfX+fLXYforLiederivatives.IncontrasttotheLiederivative,condition.1impliesthatthevalueofrXYatx2MdependsonlyonthevalueXx.If2TMxisatangentvectoratx2M,andYisavectorelddenedinsomeneighborhoodofxweusethenotationrY:=rXYx;whereXx=:.3Bytheprecedingcomments,thisdoesnotdependonhowwechoosetoextendtoXsolongasXx=.WhiletheLiederivativeisanintrinsicnotiondependingonlyonthedier-entiablestructure,aconnectionisanadditionalpieceofgeometricstructure.57

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58CHAPTER3.LEVI-CIVITACONNECTIONS.3.2Christoelsymbols.Thesegivetheexpressionofaconnectioninlocalcoordinates:Letx1;:::;xnbeacoordinatesystem,andletuswrite@i:=@ @xiforthecorrespondingvectorelds.Thenr@i@j=Xk)]TJ/F10 6.974 Tf 6.227 4.114 Td[(kij@kwherethefunctions)]TJ/F10 6.974 Tf 93.198 3.615 Td[(kijarecalledtheChristoelsymbols.Wewillfrequentlyusetheshortenednotationri:=r@i:SothedenitionoftheChristoelsymbolsiswrittenasri@j=Xk)]TJ/F10 6.974 Tf 6.227 4.113 Td[(kij@k:.4IfY=XjYj@jisthelocalexpressionofageneralvectoreldYthen.2impliesthatriY=Xk8<:@Yk @xi+Xj)]TJ/F10 6.974 Tf 6.227 4.114 Td[(kijYj9=;@k:.53.3Paralleltransport.LetC:I!MbeasmoothmapofanintervalIintoM.WerefertoCasaparameterizedcurve.Wewillsaythatthiscurveisnon-singularifC0t6=0foranytwhereC0tdenotesthetangentvectoratt2I.ByavectoreldZalongCwemeanarulewhichsmoothlyattachestoeacht2IatangentvectorZttoMatCt.WewillletVCdenotethesetofallsmoothvectoreldsalongC.Forexample,ifVisavectoreldonM,thentherestrictionofVtoC,i.e.theruleVCt:=VCtisavectoreldalongC.SincethecurveCmightcrossitself,orbeclosed,itisclearthatnoteveryvectoreldalongCistherestrictionofavectoreld.Ontheotherhand,ifCisnon-singular,thentheimplicitfunctiontheoremsaysthatforanyt02IwecanndanintervalJcontainingt0andasystemofcoordinatesaboutCt0inMsuchthatintermsofthesecoordinatesthecurveisgivenbyx1t=t;xit=0;i>1

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3.3.PARALLELTRANSPORT.59fort2J.IfZisasmoothvectoreldalongCthenfort2JwemaywriteZt=XjZjt@jt;0;:::;0:WemaythendenethevectoreldYonthiscoordinateneighborhoodbyYx1;:::;xn=XjZjx1@janditisclearthatZistherestrictionofYtoConJ.Inotherwords,locally,everyvectoreldalonganon-singularcurveistherestrictionofavectoreldofM.IfZ=YCistherestrictionofavectoreldYtoCwecandeneitsderivative"Z0,alsoavectoreldalongCbyY0Ct:=rC0tY:.6IfgisasmoothfunctiondenedinaneighborhoodoftheimageofC,andhisthepullbackofgtoIviaC,soht=gCtthenthechainrulesaysthath0t=d dtgCt=C0tg;thederivativeofgwithrespecttothetangentvectorC0t.ThenifZ=YCforsomevectoreldYonMandh=gCtequation.2impliesthathZ0=h0Z+hZ0:.7WeclaimthatthereisauniquelinearmapZ7!Z0denedonallofVCsuchthat.7and.6hold.Indeed,toproveuniqueness,itisenoughtoproveuniquenessinacoordinateneighborhood,whereZt=XjZjt@iC:Equations.7and.6thenimplythatZ0t=XjZj0t@jC+ZjtrC0t@j:.8Inotherwords,anynotionofderivativealongC"satisfying.7and.6mustbegivenby3.8inanycoordinatesystem.Thisprovestheuniqueness.Ontheotherhand,itisimmediatetocheckthat.8satises.7and.6ifthe

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60CHAPTER3.LEVI-CIVITACONNECTIONS.curveliesentirelyinacoordinateneighborhood.Buttheuniquenessimpliesthatontheoverlapoftwoneighborhoodsthetwoformulascorrespondingto.8mustcoincide,provingtheglobalexistence.Wecanmakeformula.8evenmoreexplicitinlocalcoordinatesusingtheChristoelsymbolswhichtellusthatrC0t@j=Xk)]TJ/F10 6.974 Tf 6.227 4.113 Td[(kijdxiC dt@kC:Substitutinginto.8givesZ0=Xk0@dZk dt+Xij)]TJ/F10 6.974 Tf 6.227 4.114 Td[(kijdxiC dtZj1A@kC:.9AvectoreldZalongCissaidtobeparallelifZ0t0:LocallythisamountstotheZisatisfyingthesystemoflineardierentialequa-tionsdZk dt+Xij)]TJ/F10 6.974 Tf 6.226 4.113 Td[(kijdxiC dtZj=0:.10HencetheexistenceanduniquenesstheoremforlinearhomogeneousdierentialequationsinparticularexistenceovertheentireintervalofdenitionimpliesthatProposition4Forany2TMCthereisauniqueparallelvectoreldZalongCwithZ=.Therulet7!C0tisavectoreldalongCandhencewecancomputeitsderivative,whichwedenotebyC00andcalltheaccelerationofC.Whereasthenotionoftangentvector,C0,makessenseonanymanifold,theaccelerationonlymakessensewhenwearegivenaconnection.3.4Geodesics.Acurvewithaccelerationzeroiscalledageodesic.InlocalcoordinateswesubstituteZk=xk0into.10toobtaintheequationforgeodesicsinlocalcoordinates:d2xk dt2+Xij)]TJ/F10 6.974 Tf 6.227 4.114 Td[(kijdxi dtdxj dt=0;.11wherewehavewrittenxkinsteadofxkCin3.11tounburdenthenotation.Theexistenceanduniquenesstheoremforordinarydierentialequationsimpliesthat

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3.5.COVARIANTDIFFERENTIAL.61Proposition5Foranytangentvectoratanypointx2MthereisanintervalIabout0andauniquegeodesicCsuchthatC=xandC0=.Bytheusualargumentswecanthenextendthedomainofdenitionofthegeodesicthroughtobemaximal.Thisistherstofmanydenitionsorcharacterizations,iftakethistobethebasicdenitionthatweshallhaveofgeodesics-thenotionofbeingself-parallel.Inthecasethatallthe)]TJ/F10 6.974 Tf 162.792 3.615 Td[(kij=0wegettheequationsforstraightlines.SupposethatC:I!Misanon-constantgeodesic,andweconsiderareparametrization"ofC,i.e.considerthecurveB=Ch:J!Mwhereh:J!IisadieomorphismoftheintervalJontotheintervalI.Wewritet=hssothatdB ds=dC dtdh dsandhenced2B ds2=d2C dt2dh ds2+dC dsd2h ds2=dC dsd2h ds2sinceC00=0asCisageodesic.ThefactthatCisnotconstantandtheuniquenesstheoremfordierentialequationssaysthatC0isneverzero.HenceBisageodesicifandonlyifd2h ds20orhs=as+bwhereaandbareconstantswitha6=0.Inshort,thefactofbeinganon-constantgeodesicdeterminestheparameterizationuptoananechangeofparameter.3.5Covariantdierential.WecanextendthenotionofcovariantderivativewithrespecttoavectoreldXwhichhasbeendenedonfunctionsbyf7!XfandonvectoreldsbyY7!rXYtoalltensorelds:WerstextendtolineardierentialformsbytherulerXY=XY)]TJ/F11 9.963 Tf 9.962 0 Td[(rXY.12ReplacingYbygYhastheeectofpullingoutafactorofgsincethetwotermsontherightinvolvingXgcancel.ThisshowsthatrXisagainalineardierentialform.NoticethatrfX=frXandrXg=Xg+grX:

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62CHAPTER3.LEVI-CIVITACONNECTIONS.WenowextendrXtobeatensorderivation"requiringthatrX=rX+rXforanypairoftensoreldsand.ForexamplerXZ=rXZ+rXZ:ThisthendenesrXonalltensoreldswhicharesumsofproductsofoneformsandvectorelds.Noticethatifwedenethecontraction"C:Z7!Zthenthedenition.12ofrXimpliesthatrXCZ=rXZ=CrXZ+rXZ=CrXZ:inotherwords,rXcommuteswithcontractionrXC=CrX:.13Thiswascheckedinthespecialcasethatwehadatensoroftype,1whichwasthetensorproductofaoneformandavectoreld.Butifwehaveatensoroftyper,swhichisaproductofoneformsandvectorelds,thenwemayformthecontractionofanyone-formfactorwithanyvectoreldfactortoobtainatensoroftyper-1,s-1and.13continuestohold.Ifisageneraltensoreldoftyper,s,itiscompletelydeterminedbyevaluationonalltensoreldsoftypes,rwhichareproductsofoneformsandvectorelds.WethendenerXbyrX=X)]TJ/F11 9.963 Tf 9.963 0 Td[(rX:Inthecasethatisitselfasumofproductsofone-formsandvectoreldsthiscoincideswithourolddenition.AgainthisimpliesthatrXisatensor.Furthermore,contractioninanytwopositionsinisduallocallytoinsertionofPiEiintothecorrespondingpositionsinatensoroftypes-1,r-1wheretheEiformabasislocallyofthevectoreldsateachpointandtheiformthedualbasis.ButrXiEi=0sinceifthefunctionsaijaredenedbyrXEj=PjaijEithenrXi=)]TJ/F1 9.963 Tf 9.409 7.472 Td[(Paijjasfollowsfrom.12.Thisshowsthat.13holdsingeneral.Wecanthinkofthecovariantderivativeasassigningtoeachtensoreldoftyper,satensoreldroftyper,s+1,givenbyrX=rX:Thetensorriscalledthecovariantdierentialof.

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3.6.TORSION.633.6Torsion.Letrbeaconnection,XandYvectoreldsandfandgfunctions.Using.1,3.2,andthecorrespondingequationsforLiebracketswendrfXgY)-222(rgYfX)]TJ/F8 9.963 Tf 9.963 0 Td[([fX;gY]=fgrXY)-222(rYX)]TJ/F8 9.963 Tf 9.963 0 Td[([X;Y];InotherwordsthevalueofX;Y:=rXY)-222(rYX)]TJ/F8 9.963 Tf 9.962 0 Td[([X;Y]atanypointxdependsonlyonthevaluesXx;Yxofthevectoreldsatx.Sodenesatensoreldoftype,2inthesensethatitassignstoanypairoftangentvectorsatapoint,athirdtangentvectoratthatpoint.Thistensoreldiscalledthetorsiontensoroftheconnection.SoaconnectionhaszerotorsionifandonlyifrXY)-222(rYX=[X;Y]3.14forallpairsofvectoreldsXandY.Intermsoflocalcoordinates,[@i;@j]=0.So@i;@j=ri@j)-222(rj@i=Xk)]TJ/F8 9.963 Tf 4.566 -8.07 Td[()]TJ/F10 6.974 Tf 6.227 4.113 Td[(kij)]TJ/F8 9.963 Tf 9.963 0 Td[()]TJ/F10 6.974 Tf 6.227 4.113 Td[(kji@k:ThusaconnectionhaszerotorsionifandonlyifitsChristoelsymbolsaresymmetriciniandj.3.7Curvature.ThecurvatureR=RroftheconnectionrisdenedtobethemapVM3!VMassigningtothreevectoreldsX;Y;ZthevalueRXYZ:=[rX;rY]Z)-222(r[X;Y]Z:.15Theexpression[rX;rY]occurringontherightin.15isthecommutatorofthetwooperatorsrXandrY,thatis[rX;rY]=rXrY)-235(rYrX.WerstobservethatRisatensor,i.e.thatthevalueofRXYZatapointdependsonlyonthevaluesofX;Y,andZatthatpoint.ToseethiswemustshowthatRfXgYhZ=fghRXYZforanythreesmoothfunctionsf;gandh.Forthisitsucestocheckthisoneatatime,i.e.whentwoofthethreefunctionsareidenticallyequaltoone.Forexample,iff1hwehave)]TJ/F11 9.963 Tf 7.749 0 Td[(RX;gYZ=r[X;gY]Z)-222(rXrgYZ+rgYrXZ=XgrYZ+gr[X;Y]Z)]TJ/F8 9.963 Tf 9.963 0 Td[(XgrXrYZ)]TJ/F11 9.963 Tf 9.963 0 Td[(grXrYZ+grYrXZ=gRXYZ:

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64CHAPTER3.LEVI-CIVITACONNECTIONS.SinceRisanti-symmetricinXandYweconcludethatRfXYZ=fRXYZ.Finally,)]TJ/F11 9.963 Tf 7.749 0 Td[(RXYhZ=[X;Y]hZ+hr[X;Y]Z)-222(rXYhZ+hrYZ+rYXhZ+hrXZ=hRXYZ+[X;Y]h)]TJ/F8 9.963 Tf 9.962 0 Td[(XY)]TJ/F11 9.963 Tf 9.962 0 Td[(YXhZ)]TJ/F11 9.963 Tf 9.963 0 Td[(XhrYZ)]TJ/F11 9.963 Tf 7.749 0 Td[(YhrXZ+YhrXZ+XhrYZ=hRXYZ:Thuswegetacurvaturetensoroftype,3whichassignstoeverythreetangentvectors;;atapointxthevalueR:=RXYZxwhereX;Y;ZareanythreevectoreldswithXx=;Yx=;Zx=.Alternatively,wespeakofthecurvatureoperatoratthepointxdenedbyR:TMx!TMx;R:7!R:Aswementioned,thecurvatureoperatorisanti-symmetricinand:R=)]TJ/F11 9.963 Tf 7.748 0 Td[(R:TheclassicalexpressionofthecurvaturetensorintermsoftheChristoelsymbolsisobtainedasfollows:Since[@k;@`]=0,R@k@`@j=rkr`@j)-222(r`rk@j=r`Xm)]TJ/F10 6.974 Tf 6.226 4.114 Td[(mkj@m!+rkXm)]TJ/F10 6.974 Tf 6.226 4.114 Td[(m`j@r!=)]TJ/F1 9.963 Tf 9.41 9.464 Td[(Xm@ @x`)]TJ/F10 6.974 Tf 6.227 4.113 Td[(mkj@m+Xm;r)]TJ/F10 6.974 Tf 6.226 4.113 Td[(mkj)]TJ/F10 6.974 Tf 6.226 4.113 Td[(r`m!@r+Xm@ @xk)]TJ/F10 6.974 Tf 6.227 4.113 Td[(m`j@m)]TJ/F1 9.963 Tf 9.963 9.464 Td[(Xm;r)]TJ/F10 6.974 Tf 6.226 4.113 Td[(m`j)]TJ/F10 6.974 Tf 6.227 4.113 Td[(rkm!@r=XiRijk`@iwhereRijk`=)]TJ/F11 9.963 Tf 13.708 6.74 Td[(@ @x`)]TJ/F10 6.974 Tf 6.227 4.114 Td[(ikj+@ @xk)]TJ/F10 6.974 Tf 6.227 4.114 Td[(i`j)]TJ/F1 9.963 Tf 9.963 9.465 Td[(Xm)]TJ/F10 6.974 Tf 6.226 4.114 Td[(i`m)]TJ/F10 6.974 Tf 6.227 4.114 Td[(mkj+Xm)]TJ/F10 6.974 Tf 6.227 4.114 Td[(ikm)]TJ/F10 6.974 Tf 6.227 4.114 Td[(m`j:.16IftheconnectionhaszerotorsionweclaimthatR+R+R=0;.17or,usingthecyclicsumnotationweintroducedwiththeJacobiidentity,thatCycR=0:

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3.8.ISOMETRICCONNECTIONS.65Toprovethis,wemayextend;,andtovectoreldswhosebracketsallcommutesaybyusingvectoreldswithconstantcoecientsinacoordinateneighborhood.ThenRXYZ=rXrYZ)-222(rYrXZ:ThereforeCycRXYZ=CycrXrYZ)-222(CycrYrXZ=CycrXrZY)-222(CycrXrZYsincemakingacyclicpermutationinanexpressionCycFX;Y;Zdoesnotaectitsvalue.ButthefactthattheconnectionistorsionfreemeansthatwecanwritethelastexpressionasCycrX[Y;Z]=0byourassumptionthatallLiebracketsvanish.QED3.8Isometricconnections.SupposethatMisasemi-Riemannianmanifold,meaningthatwearegivenasmoothlyvaryingnon-degeneratescalarproducth;ixoneachtangentspaceTMx.GiventwovectoreldsXandY,welethX;YidenotethefunctionhX;Yix:=hXx;Yxix:Wesaythataconnectionrisisometricforh;iifXhY;Zi=hrXY;Zi+hY;rXZi.18foranythreevectoreldsX;Y;Z.ItisasortofLeibniz'sruleforscalarprod-ucts.Ifwegobacktothedenitionofthederivativeofavectoreldalongacurvearisingfromtheconnectionr,weseethat.18impliesthatd dthY;Zi=hY0;Zi+hY;Z0iforanypairofvectoreldsalongacurveC.Inparticular,ifYandZareparallelalongthecurve,sothatY0=Z0=0,weseethathY;Ziisconstant.Thisisthekeymeaningoftheconditionthataconnectionbeisometric:paralleltranslationalonganycurveisanisometryofthetangentspaces.3.9Levi-Civita'stheorem.Thisassertsthatonanysemi-Riemannianmanifoldthereexistsauniquecon-nectionwhichisisometricandistorsionfree.ItischaracterizedbytheKoszulformula

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66CHAPTER3.LEVI-CIVITACONNECTIONS.2hrVW;Xi=VhW;Xi+WhX;Vi)]TJ/F11 9.963 Tf 13.842 0 Td[(XhV;Wi)-111(hV;[W;X]i+hW;[X;V]i+hX;[V;W]i.19foranythreevectoreldsX;V;W.ToproveKoszul'sformula,weapplytheisometricconditiontoeachoftherstthreetermsoccurringontherighthandsideof.19.ForexamplethersttermbecomeshrVW;Xi+hW;rVXi.Weapplythetorsionfreeconditiontoeachofthelastthreeterms.ForexamplethelasttermbecomeshX;rVW)-248(rWVi.Therewillbealotofcancellationleavingthelefthandside.SincethevectoreldrVWisdeterminedbyknowingitsscalarproducthrVW;XiforallvectoreldsX,theKoszulformulaprovestheuniquenesspartofLevi-Civita'stheorem.Ontheotherhand,therighthandsideoftheKoszulformulaisfunctionlinearinX,i.e.hrVW;fXi=fhrVW;XiascanbecheckedusingthepropertiesofrandLiebracket.Soweobtainawelldenedvectoreld,rVWanditisroutinetocheckthatthissatisestheconditionsforaconnectionandistorsionfreeandisometric.WecanusetheKoszulidentitytoderiveaformulafortheChristoelsymbolsintermsofthemetric.Firstsomestandardnotations:Wewillusethesymbolgtostandforthemetric,sogisjustanothernotationforh;i.Inalocalcoordinatesystemwewritegij:=h@i;@jisog=Xijgijdxidxj:Herethegijarefunctionsonthecoordinateneighborhood,butwearesuppress-ingthefunctionaldependenceonthepointsinthenotation.Themetricgisasymmetrictensoroftype,2.Itinducesanisomorphismateachpointofthetangentspacewiththecotangentspace,eachtangentvectorgoingintothelinearfunctionh;iconsistingofscalarproductby.Bytheaboveformulathemapisgivenby@i7!Xjgijdxj:Thisisomorphisminducesascalarproductonthecotangentspaceateachpoint,andsoatensoroftype,0whichweshalldenoteby^gorsometimesbyg"".Wewritegij:=hdxi;dxjiso^g=Xijgij@i@j:

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3.10.GEODESICSINORTHOGONALCOORDINATES.67Thetransitionfromthetwolowerindicestothetwoupperindicesisthereasonfortheverticalarrowsnotation.Themetriconthecotangentspacesinducesamapintoitsdualspacewhichisthetangentspacegivenbydxi7!Xgij@jandthetwomaps-fromtangentspacestocotangentspacesandviceversa-areinversesofoneanothersoXkgikgkj=ij;thematrices"gijandgk`areinverses.NowletussubstituteX=@m;V=@i;W=@jintotheKoszulformula.19.Allbracketsontherightvanishandweget2hri@j;@mi=@igjm+@jgim)]TJ/F11 9.963 Tf 9.963 0 Td[(@mgij:Sinceri@j=Xk)]TJ/F10 6.974 Tf 6.227 4.114 Td[(kij@kisthedenitionoftheChristoelsymbols,theprecedingequationbecomes2Xa)]TJ/F10 6.974 Tf 6.227 4.113 Td[(aijgam=@igjm+@jgim)]TJ/F11 9.963 Tf 9.963 0 Td[(@mgij:Multiplyingthisequationbygmkandsummingovermgives)]TJ/F10 6.974 Tf 6.226 4.114 Td[(kij=1 2Xmgkm@gjm @xi+@gim @xj)]TJ/F11 9.963 Tf 11.905 6.74 Td[(@gij @xm:.20Inprinciple,weshouldsubstitutethisformulainto.11andsolvetoobtainthegeodesics.Inpracticethisisamessforageneralcoordinatesystemandsowewillspendagoodbitoftimedevelopingothermeansusuallygrouptheoreticalforndinggeodesics.Howevertheequationsaremanageableinorthogonalcoordinates.3.10Geodesicsinorthogonalcoordinates.Acoordinatesystemiscalledorthogonalifgij=0;i6=j:Ifweareluckyenoughtohaveanorthogonalcoordinatesystemtheequationsforgeodesicstakeonasomewhatsimplerform.Firstnoticethat.20becomes)]TJ/F10 6.974 Tf 6.226 4.114 Td[(kij=1 2gkk@gjk @xi+@gik @xj)]TJ/F11 9.963 Tf 11.158 6.74 Td[(@gij @xk

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68CHAPTER3.LEVI-CIVITACONNECTIONS.So.11becomesd2xk dt2+gkkXi@gkk @xidxk dtdxi dt)]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 2gkkXi@gii @xkdxi dtdxi dt=0:Ifwemultiplythisequationbygkkandbringthenegativetermtotheothersideweobtaind dtgkkdxk dt=1 2Xi@gii @xkdxi dt2.21astheequationsforgeodesicsinorthogonalcoordinates.3.11Curvatureidentities.ThecurvatureoftheLevi-Civitaconnectionsatisesseveraladditionalidentitiesbeyondthetwocurvatureidentitiesthatwehavealreadydiscussed.LetuschoosevectoreldsX;Y;Vwithvanishingbrackets.WehavehRXYV;Vi=hrXrYV;Vi+hrYrXV;Vi=YhrXV;Vi)-222(hrXV;rYVi)]TJ/F11 9.963 Tf 16.051 0 Td[(XhrYV;Vi+hrYV;rXVi=1 2YXhV;Vi)]TJ/F8 9.963 Tf 17.247 6.74 Td[(1 2XYhV;Vi=1 2[X;Y]hV;Vi=0:ThisimpliesthatforanythreetangentvectorswehavehR;i=0andhencebypolarizationthatforanyfourtangentvectorswehavehR;i=h;Ri:.22ThisequationsaysthatthecurvatureoperatorRactsasaninnitesimalorthogonaltransformationonthetangentspace.ThelastidentitywewanttodiscussisthesymmetrypropertyhR;i=hR;i:.23TheproofconsistsofstartingwiththeidentityCycR;=0andtakingthescalarproductwithtoobtainhCycR;;i=0:

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3.12.SECTIONALCURVATURE.69Thisisanequationinvolvingthreeterms.Takethecyclicpermutationofthefourvectorstoobtainfourequationslikethisinvolvingtwelvetermsinall.Whenweaddthefourequationseightofthetermscancelinpairsandtheremainingtermsgive.23.WesummarizethesymmetrypropertiesoftheRiemanncurvature:R=)]TJ/F11 9.963 Tf 7.749 0 Td[(RhR;i=hRiR+R+R=0hR;i=hR;i:3.12Sectionalcurvature.LetVisavectorspacewithanon-degeneratesymmetricbilinearformh;i.Asubspaceiscallednon-degenerateiftherestrictionoftheh;itothissub-spaceisnon-degenerate.Ifh;iispositivedenite,thenallsubspacesarenon-degenerate.Atwodimensionalsubspaceisnon-degenerateifandonlyifforanybasisv;wofthearea"Qv;w:=hv;vihw;wi)-222(hv;wi2doesnotvanish.Letbeanon-degenerateplane=twodimensionalsubspaceofthetangentspaceTMxofasemi-Riemannianmanifold.ThenitssectionalcurvatureisdenedasK:=hRvwv;wi Qv;w:.24Itiseasytocheckthatthisisindependentofthechoiceofbasisv;w.3.13Riccicurvature.Ifwehold2TMxand2TMxxedinRvthenthemapv7!Rvv2TMxisalinearmapofTMxintoitself.ItstracewhichisbiinearinandisknownastheRiccicurvaturetensor.Ric;:=tr[v7!Rx;v]:.25RiccicurvatureplaysakeyroleingeneralrelativitybecauseitistheRiccicurvatureratherthanthanthefullRiemanncurvaturewhichentersintotheEinsteineldequations.

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70CHAPTER3.LEVI-CIVITACONNECTIONS.3.14Bi-invariantmetricsonaLiegroup.ThesimplestexampleofaRiemmanmanifoldisEuclideanspace,wherethegeodesicsarestraightlinesandallcurvaturesvanish.WemaythinkofEuclideanspaceasacommutativeLiegroupunderaddition,andviewthestraightlinesastranslatesofoneparametersubgroupslinesthroughtheorigin.Aneasybutimportantgeneralizationofthisiswhenweconsiderbi-invariantmetricsonaLiegroup,aconceptweshallexplainbelow.Inthiscasealso,thegeodesicsarethetranslatesofoneparametersubgroups.3.14.1TheLiealgebraofaLiegroup.LetGbeaLiegroup.ThismeansthatGisagroup,andisasmoothmanifoldsuchthatthemultiplicationmapGG!Gissmooth,asisthemapinv:G!Gsendingeveryelementintoitsinverse:inv:a7!a)]TJ/F7 6.974 Tf 6.227 0 Td[(1;a2G:UntilnowtheLiegroupswestudiedweregivenassubgroupsofGln.Wecancontinueinthisvein,orworkwiththemoregeneraldenitionjustgiven.WehavetheleftactionofGonitselfLa:G!G;b7!abandtherightactionRa:G!G;b7!ba)]TJ/F7 6.974 Tf 6.226 0 Td[(1:WeletgdenotethetangentspacetoGattheidentity:g=TGe:WeidentifygwiththespaceofallleftinvariantvectoreldsonG,so2gisidentiedwiththevectoreldXwhichassignstoeverya2GthetangentvectordLae2TGa:WewillalternativelyusethenotationX;Yor;forelementsofg.TheleftinvariantvectoreldXgeneratesaoneparametergroupoftrans-formationswhichcommuteswithallleftmultiplicationsandsomustconsistofaoneparametergroupofrightmultiplications.InthecaseofasubgroupofGln,wheregwasidentiedwithasubspaceofofthespaceofallnnmatrices,wesawthatthiswastheoneparametergroupoftransformationsA7!AexptX;i.e.theoneparametergroupRexp)]TJ/F10 6.974 Tf 6.227 0 Td[(tX:Sowemightaswellusethisnotationingeneral:exptXdenotestheoneparame-tersubgroupofGobtainedbylookingatthesolutioncurvethrougheoftheleft

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3.14.BI-INVARIANTMETRICSONALIEGROUP.71invariantvectoreldX,andthentheoneparametergroupoftransformationsgeneratedbythevectoreldXisRexp)]TJ/F10 6.974 Tf 6.227 0 Td[(tX.LetXandYbeelementsofgthoughtofasleftinvariantvectorelds,andletuscomputetheirLiebracketasvectorelds.Solett=Rexp)]TJ/F10 6.974 Tf 6.227 0 Td[(tXbetheoneparametergroupoftransformationsgeneratedbyX.Accordingtothegeneraldenition,theLiebracket[X;Y]isobtainedbydierentiatingthetimedependentvectoreldtYatt=0.Bydenition,thepull-backtYisthevectoreldwhichassignstothepointathetangentvectordt)]TJ/F7 6.974 Tf 6.226 0 Td[(1aYta=dRexptXaYaexptX:.26InthecasethatGisasubgroupofthegenerallineargroup,thisispreciselytheleftinvariantvectorelda7!aeexptXYexp)]TJ/F11 9.963 Tf 7.748 0 Td[(tX:Dierentiatingwithrespecttotandsettingt=0showsthatthevectoreld[X;Y]ispreciselytheleftinvariantvectoreldcorrespondingtothecommutatorofthetwomatricesXandY.WecanmimicthiscomputationforageneralLiegroup,notnecessarilygivenasasubgroupofGln:Firstletusrecordthespecialcaseof.26whenwetakea=e:dt)]TJ/F7 6.974 Tf 6.226 0 Td[(1eYte=dRexptXYexptX:.27Foranya2GweletAadenoteconjugationbytheelementa2G,soAa:G!G;Aab=aba)]TJ/F7 6.974 Tf 6.227 0 Td[(1:WehaveAae=eandAacarriesone-parametersubgroupsintooneparam-etersubgroups.InparticularthedierentialofAaatTGe=gisalineartransformationofgwhichweshalldenotebyAda:dAae=:Ada:TGe!TGe:WehaveAa=LaRa=RaLa:SoifYistheleftinvariantvectoreldonGcorrespondingto2TGe=g,wehavedLa=YaandsodAae=dRaadLae=dRaaYa:Seta=exptX,andcomparethiswith3.27.Dierentiatewithrespecttotandsett=0.Weseethattheleftinvariantvectoreld[X;Y]correspondstotheelementofTGeobtainedbydierentiatingAdexptXwithrespecttotandsettingt=0.Insymbols,wecanwritethisasd dtAdexptXjt=0=adXwhereadX:g!g;adXY=[X;Y]:.28

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72CHAPTER3.LEVI-CIVITACONNECTIONS.NowadXasdenedaboveisalineartransformationofg.SowecanconsiderthecorrespondingoneparametergroupexptadXoflineartransformationsofgusingtheusualformulafortheexponentialofamatrix.But.28saysthatAdexptXisaoneparametergroupoflineartransformationswiththesamederivative,adXatt=0.TheuniquenesstheoremforlineardierentialequationsthenimpliestheimportantformulaexptadX=AdexptX:.293.14.2ThegeneralMaurer-Cartanform.Ifv2TGaistangentvectoratthepointa2G,therewillbeauniqueleftinvariantvectoreldXsuchthatXa=v.Inotherwords,thereisalinearmap!a:TGa!gsendingthetangentvectorvtotheelement=!av2gwheretheleftinvariantvectoreldXcorrespondingtosatisesXa=v.Sowehavedenedagvaluedlineardierentialform!identiedthetangentspaceatanya2Gwithg.IfdLbv=w2TGbathenXba=wsinceXv=vandXisleftinvariant.Inotherwords,!LbadLb=!a;or,whatamountstothesamethingLb!=!forallb2G.Theform!isleftinvariant.WhenweprovedthisforasubgroupofGlnthiswasacomputation.Butinthegeneralcase,aswehavejustseen,itisatautology.WenowwanttoestablishthegeneralizationoftheMaurer-Cartanequation.9whichsaidthatforsubgroupsofGlnwehaved!+!^!=0:Sincewenolongerhave,ingeneral,thenotionofmatrixmultiplicationwhichentersintothedenitionof!^!,wemustrstmustrewrite!^!inaformwhichgeneralizestoanarbitraryLiegroup.SoletustemporarilyconsiderthecaseofasubgroupofGln.RecallthatforanytwoformandapairofvectoreldsXandYwewriteX;Y=iYiX.Thus!^!X;Y=!X!Y)]TJ/F11 9.963 Tf 9.962 0 Td[(!Y!X;thecommutatorofthetwomatrixvaluedfunctions,!Xand!Y.Considerthecommutatoroftwomatrixvaluedoneforms,!and,!^+^!

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3.14.BI-INVARIANTMETRICSONALIEGROUP.73accordingtoourusualrulesofsuperalgebra.Wedenotethisby[!^;]:Inparticularwemaytake!=toobtain[!^;!]=2!^!:SowecanrewritetheMaurer-CartanequationforasubgroupofGlnasd!+1 2[!^;!]=0:.30NowforageneralLiegroupwedohavetheLiebracketmapgg!g:Sowecandenethetwoform[!^;!].ItisagvaluedtwoformwhichsatisesiX[!^;!]=[X;!])]TJ/F8 9.963 Tf 9.962 0 Td[([!;X]foranyleftinvariantvectoreldX.Hence[!^;!]X;Y:=iYiX[!^;!]=iY[X;!])]TJ/F8 9.963 Tf 9.963 0 Td[([!;X]=[X;Y])]TJ/F8 9.963 Tf 9.963 0 Td[([Y;X]=2[X;Y]foranypairofleftinvariantvectoreldsXandY.Sotoprove.30ingeneral,wemustverifythatforanypairofleftinvariantvectoreldswehaved!X;Y=)]TJ/F11 9.963 Tf 7.748 0 Td[(![X;Y]:Butthisisaconsequenceofourgeneralformula.3fortheexteriorderivativewhichinourcasesaysthatd!X;Y=X!Y)]TJ/F11 9.963 Tf 9.963 0 Td[(Y!X)]TJ/F11 9.963 Tf 9.962 0 Td[(![X;Y]:Inoursituationthersttwotermsontherightvanishsince,forexample,!Y=Y=aconstantelementofgsothatX!Y=0andsimilarlyY!X=0.3.14.3Leftinvariantandbi-invariantmetrics.Anynon-degeneratescalarproduct,h;i,ongdeterminesandisequivalenttoaleftinvariantsemi-RiemannmetriconGviatheleft-identicationdLa:g=TGe!TGa;8a2G,SinceAa=LaRa,theleftinvariantmetric,h;iisrightinvariantifandonlyifitisAainvariantforalla2G,whichisthesameassayingthath;iisinvariantundertheadjointrepresentationofGong,i.e.thathAdaY;AdaZi=hY;Zi;8Y;Z2g;a2G:

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74CHAPTER3.LEVI-CIVITACONNECTIONS.Settinga=exptX;X2g,dierentiatingwithrespecttotandsettingt=0givesh[X;Y];Zi+hY;[X;Z]i=0;8X;Y;Z2g:.31IfeveryelementofGcanbewrittenasaproductofelementsoftheformexp;2gwhichwillbethecaseifGisconnected,thisconditionimpliesthath;iisinvariantunderAdandhenceisinvariantunderrightandleftmultiplication.Suchametriciscalledbi-invariant.Letinvdenotethemapsendingeveryelementintoitsinverse:inv:a7!a)]TJ/F7 6.974 Tf 6.227 0 Td[(1;a2G:SinceinvexptX=exp)]TJ/F11 9.963 Tf 7.748 0 Td[(tXweseethatdinve=)]TJ/F8 9.963 Tf 7.749 0 Td[(id:Alsoinv=Ra)]TJ/F6 4.981 Tf 5.396 0 Td[(1invLa)]TJ/F6 4.981 Tf 5.396 0 Td[(1sincetherighthandsidesendsb2Gintob7!a)]TJ/F7 6.974 Tf 6.227 0 Td[(1b7!b)]TJ/F7 6.974 Tf 6.226 0 Td[(1a7!b)]TJ/F7 6.974 Tf 6.226 0 Td[(1:Hencedinva:TGa!TGa)]TJ/F6 4.981 Tf 5.397 0 Td[(1isgiven,bythechainrule,asdRa)]TJ/F6 4.981 Tf 5.396 0 Td[(1dinvedLa)]TJ/F6 4.981 Tf 5.396 0 Td[(1=)]TJ/F11 9.963 Tf 7.749 0 Td[(dRa)]TJ/F6 4.981 Tf 5.397 0 Td[(1dLa)]TJ/F6 4.981 Tf 5.397 0 Td[(1implyingthatabi-invariantmetricisinvariantunderthemapinv.Conversely,ifaleftinvariantmetricisinvariantunderinvthenitisalsorightinvariant,hencebi-invariantsinceRa=invL)]TJ/F7 6.974 Tf 6.227 0 Td[(1ainv:3.14.4Geodesicsarecosetsofoneparametersubgroups.TheKoszulformulasimpliesconsiderablywhenappliedtoleftinvariantvectoreldsandbi-invariantmetricssinceallscalarproductsareconstant,sotheirderivativesvanish,andweareleftwith2hrXY;Zi=hX;[Y;Z]i)-222(hY;[X;Z]i+hZ;[X;Y]iandthersttwotermscancelby.50.WeareleftwithrXY=1 2[X;Y]:.32Conversely,ifh;iisaleftinvariantmetricforwhich.51holds,thenhX;[Y;Z]i=2hX;rYZi=)]TJ/F8 9.963 Tf 7.749 0 Td[(2hrYX;Zi=h[Y;X];Zi=h[X;Y];Zi

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3.14.BI-INVARIANTMETRICSONALIEGROUP.75sothemetricisbi-invariant.LetbeanintegralcurveoftheleftinvariantvectoreldX.Equation.51impliesthat00=rXX=0soisageodesic.Thustheone-parametergroupsarethegeodesicsthroughtheidentity,andallgeodesicsareleftcosetsofoneparametergroups.ThisisthereasonforthenameexponentialmapinRiemanniangeometrywhichweshallstudyinChapterV.InChapterVIIIwewillstudyRiemanniansubmersions.ItwillemergefromthisstudythatifawehaveaquotientspaceB=G=Hofagroupwithabi-invariantmetricsatisfyingsomemildconditions,thenthegeodesicsonBintheinducedmetricareorbitsofcertainoneparametersubgroups.Forexample,thegeodesicsonspheresarethegreatcircles.3.14.5TheRiemanncurvatureofabi-invariantmetric.WecomputetheRiemanncurvatureofabi-invariantmetricbyapplyingthedenition.15toleftinvariantvectorelds:RXYZ=1 4[X;[Y;Z]])]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 4[Y;[X;Z]])]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 2[[X;Y];Z]Jacobi'sidentityimpliesthersttwotermsaddupto1 4[[X;Y];Z]andsoRXYZ=)]TJ/F8 9.963 Tf 8.945 6.74 Td[(1 4[[X;Y];Z]:.333.14.6Sectionalcurvatures.InparticularhRXYX;Yi=)]TJ/F8 9.963 Tf 8.944 6.739 Td[(1 4h[[X;Y];X];Yi=)]TJ/F8 9.963 Tf 8.944 6.739 Td[(1 4h[X;Y];[X;Y]isoKX;Y=1 4jj[X;Y]jj2 jjX^Yjj2:.343.14.7TheRiccicurvatureandtheKillingform.RecallthatforeachX2gthelineartransformationofgconsistingofbracketingontheleftbyXiscalledadX.SoadX:g!g;adXV:=[X;V]:WecanthuswriteourformulaforthecurvatureasRXVY=1 4adYadXV:NowtheRiccicurvaturewasdenedasRicX;Y=tr[V7!RXVY]:

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76CHAPTER3.LEVI-CIVITACONNECTIONS.Wethusseethatforanybi-invariantmetric,theRiccicurvatureisalwaysgivenbyRic=1 4B.35whereB,theKillingform,isdenedbyBX;Y:=tradXadY:.36TheKillingformissymmetric,sincetrAC=trCAforanypairoflinearoperators.Itisalsoinvariant.Indeed,let:g!gbeanyautomorphismofg,so[X;Y]=[X;Y]forallX;Y2g.WecanreadthisequationassayingadXY=adXYoradX=adX)]TJ/F7 6.974 Tf 6.227 0 Td[(1:HenceadXadY=adXadY)]TJ/F7 6.974 Tf 6.227 0 Td[(1:Sincetraceisinvariantunderconjugation,itfollowsthatBX;Y=BX;Y:Appliedto=exptadZanddierentiatingatt=0showsthatB[Z;X];Y+BX;[Z;Y]=0.SotheKillingformdenesabi-invariantsymmetricbilinearformonG.Ofcourseitneednot,ingeneral,benon-degenerate.Forexample,ifthegroupiscommutative,itvanishesidentically.AgroupGiscalledsemi-simpleifitsKillingformisnon-degenerate.Soonasemi-simpleLiegroup,wecanalwayschoosetheKillingformasthebi-invariantmetric.Forsuchachoice,ourformulaabovefortheRiccicurvaturethenshowsthatthegroupmanifoldwiththismetricisEinstein,i.e.theRiccicurvatureisamultipleofthescalarproduct.SupposethattheadjointrepresentationofGongisirreducible.Thengcannothavetwoinvariantnon-degeneratescalarproductsunlessoneisamultipleoftheother.Inthiscase,wecanalsoconcludefromourformulathatthegroupmanifoldisEinstein.3.14.8Bi-invariantformsfromrepresentations.HereisawaytoconstructinvariantscalarproductsonaLiealgebragofaLiegroupG.LetbearepresentationofG.ThismeansthatisasmoothhomomorphismofGintoGln;RorGln;C.Thisinducesarepresentation_ofgby_X:=d dtexptXjt=0:So_:g!gln

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3.14.BI-INVARIANTMETRICSONALIEGROUP.77whereglnistheLiealgebraofGln,and_X:Y=[_X;_Y]wherethebracketontherightisingln.Moregenerally,alinearmap_:g!gln;Corgln;RsatisfyingtheaboveidentityiscalledarepresentationoftheLiealgebrag.EveryrepresentationofGgivesrisetoarepresentationofgbutnoteveryrepresentationofgneedcomefromarepresentationofGingeneral.If_isarepresentationofg,withvaluesingln;R,wemaydenehX;Yig:=tr_X_Y:Thisisrealvalued,symmetricinXandY,andh[X;Y];Zig+hY;X;Zig=tr_X_Y_Z)]TJ/F8 9.963 Tf 11.984 0 Td[(_Y_X_Z+_Y_X_Z)]TJ/F8 9.963 Tf 11.985 0 Td[(_Y_Z_X=0:Sothisisinvariant.Ofcourseitneednotbenon-degenerate.Acaseofparticularinterestiswhentherepresentation_takesvaluesinun,theLiealgebraoftheunitarygroup.Anelementofunisaskewadjointmatrix,i.e.amatrixoftheformiAwhereA=Aisselfadjoint.IfA=AandA=aijthentrA2=trAA=Xi:jaijaji=Xi:jaij aij=Xijjaijj2whichispositiveunlessA=0.So)]TJ/F8 9.963 Tf 9.409 0 Td[(triAiAispositiveunlessA=0.Thisimpliesthatif_:g!unisinjective,thentheformhX;Yi=)]TJ/F8 9.963 Tf 9.41 0 Td[(tr_X_Yisapositivedeniteinvariantscalarproductong.Forexample,letusconsidertheLiealgebrag=uandtherepresentation_ofgontheexterioralgebraofC2.Wemaydecompose^C2=^0C2^1C2^2C2andeachofthesummandsisinvariantunderourrepresentation.Everyelementofuactstriviallyon^0C2andactsinitsstandardfashionon^1C2=C2.Everyelementofuactsviamultiplicationbyitstraceon^2C2soinparticularallelementsofsuacttriviallythere.Thusrestrictedtosu,theinducedscalarproductisjusthX;Yi=)]TJ/F8 9.963 Tf 9.409 0 Td[(trXY;X;Y2su;whileonscalarmatrices,i.e.matricesoftheformS=riIwehavehS;Si=)]TJ/F8 9.963 Tf 9.409 0 Td[(tr_S2=2r2+r2=6r2=)]TJ/F8 9.963 Tf 7.749 0 Td[(3trS2=)]TJ/F8 9.963 Tf 8.944 6.74 Td[(3 2trS2:

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78CHAPTER3.LEVI-CIVITACONNECTIONS.3.14.9TheWeinbergangle.Theprecedingexampleillustratesthefactthatiftheadjointrepresentationofgisnotirreducible,theremaybemorethanaoneparameterfamilyofinvariantscalarproductsong.Indeedthealgebraudecomposesasasumu=suuofsubalgebras,whereuconsistsofthescalarmatriceswhichcommutewithallelementsofu.Itfollowsfromtheinvarianceconditionthatumustbeorthogonaltosuunderanyinvariantscalarproduct.Eachofthesesummandsisirreducibleundertheadjointrepresentation,sotherestrictionofanyinvariantscalarproducttoeachsummandisdetermineduptopositivescalarmultiple,butthesemultiplescanbechosenindependentlyforeachsummand.Sothereisatwoparameterfamilyofchoices.InthephysicsliteratureitisconventionaltowritethemostgeneralinvariantscalarproductonuashA;Bi=)]TJ/F8 9.963 Tf 11.243 6.74 Td[(2 g22trA)]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 2trAIB)]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 2trBI+)]TJ/F8 9.963 Tf 11.243 6.74 Td[(1 g21trAtrBwhereg1andg2aresometimescalledcouplingstrengths".Therstsummandvanishesonuandthesecondsummandvanishesonsu.TheWeinbergangleWisdenedbysinW:=g1 p g21+g22andplaysakeyroleinElectro-Weaktheorywhichuniestheelectromagneticandweakinteractions.Inthecurrentstateofknowledge,thereisnobroadlyagreedtheorythatpredictstheWeinbergangle.Itisaninputderivedfromexperiment.ThedataasofJuly2002fromtheParticleDataGroupgivesavalueofsin2W=0:23113::::Noticethatthecomputationthatwedidfromtheexterioralgebrahasg21=2 3andg22=2sosin2W=2 3 2 3+2=:25:OfcourseseveralquitedierentrepresentationswillgivethesamemetricorWeinbergangle.3.15Frameelds.Byaframeeldwemeanann-tupletE=E1;:::;Enofvectoreldsdenedonsomeneighborhoodwhosevaluesateverypointformabasisofthetangent

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3.16.CURVATURETENSORSINAFRAMEFIELD.79spaceatthatpoint.Thesethendeneadualcollectionofdierentialforms=0B@1...n1CAwhosevaluesateverypointformthedualbasis.Forexample,acoordinatesys-temx1;:::;xnprovidestheframeeld@1;:::;@nwithdualformsdx1;;dxn.Buttheuseofmoregeneralframeeldsallowsforexibilityincomputation.Aframeeldiscalledorthonormal"ifhEi;Eji0fori6=jandhEi;Eiiiwherei=1.Forexample,applyingtheGram-Schmidtproceduretoanarbitraryframeeldforapositivedenitemetricyieldsanorthonormalone.3.16Curvaturetensorsinaframeeld.IntermsofaframeeldthecurvaturetensorisgivenasXRijk`Eik`jwhereRijk`=iREkE`Ej:TheRiccitensor,whichaswementioned,playsakeyroleingeneralrelativity,takestheformRic=XRijijwhereRij=RicEi;Ej:=XRmimj:IftheframeisorthonormalthenforanypairofvectoreldsV;WwehaveRicV;W=XmhRVEmEm;Wi:AmanifoldiscalledRicciatifitsRiccicurvaturevanishes.ThescalarcurvatureSisdenedasS:=XgijRij:3.17Frameeldsandcurvatureforms.LetMbeasemi-Riemannianmanifold.LetE1;:::;Enbeanorthonormal"frameelddenedonsomeopensubsetofM.Inordernottoclutterupthenotationwewillnotintroduceaspecicnameforthedomainofdenitionofourframeeld.ThismeanstheEiarevectoreldsandhEi;Eji0;i6=jwhilehEi;Eiii;i=1:

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80CHAPTER3.LEVI-CIVITACONNECTIONS.ThusE1p;:::;Enpformanorthonormal"basisofthetangentspaceTMpateachpointpinthedomainofdenition.Thedualbasisofthecotangentspacethenprovidesafamilyoflineardierentialforms,1;:::;n.Itfollowsfromthedenition,thatifv2TMpthenhv;vi=1)]TJ/F11 9.963 Tf 4.566 -8.069 Td[(1v2++nnv2:Thisequation,trueatallpointsinthedomainofdenitionoftheframeeldisusuallywrittenasds2=112++nn2:.37Conversely,if1;:::;nisacollectionoflineardierentialformssatisfying.37denedonsomeopensetthenthedualvectoreldsconstitutean"orthonormal"frameeld.Onanymanifold,wehavethetautologicaltensoreldoftype,1whichassignstoeachtangentspacetheidentitylineartransformation.Wewilldenotethistautologicaltensoreldbyid.Thusforanyp2Mandanyv2TMp,idv=v:Intermsofaframeeldwehaveid=E11+Enninthesensethatbothsidesyieldvwhenappliedtoanytangentvectorvinthedomainofdenitionoftheframeeld.Wecansaythattheigivetheexpressionforidintermsoftheframeeldandalsointroducethevectorofdierentialforms":=0B@1...n1CAasashorthandforthecollectionofthei.ForeachitheLevi-CivitaconnectionyieldsatensoreldrEi,thecovariantdierentialofEiwithrespecttotheconnection,andhencelineardierentialforms!ijdenedby!ij=irEj:.38SorEj=Xm!mjEm:TherststructureequationofCartanassertsthatdi=)]TJ/F1 9.963 Tf 9.409 9.464 Td[(Xm!im^m:.39Toprovethis,weapplytheformula2.3whichsaysthatdX;Y=XY)]TJ/F11 9.963 Tf 9.963 0 Td[(YX)]TJ/F11 9.963 Tf 9.963 0 Td[([X;Y]

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3.17.FRAMEFIELDSANDCURVATUREFORMS.81holdsforanylineardierentialformandvectoreldsXandY.Weapplythistoi;Ea;EbtoobtaindiEa;Eb=EaiEb)]TJ/F11 9.963 Tf 9.962 0 Td[(EbiEa)]TJ/F11 9.963 Tf 9.963 0 Td[(i[Ea;Eb]:SinceiEbandiEa=0or1areconstants,thersttwotermsvanishandsothelefthandsideof.39whenevaluatedonEa;Ebbecomes)]TJ/F11 9.963 Tf 7.748 0 Td[(i[Ea;Eb]:Astotherighthandsidewehaveh)]TJ/F1 9.963 Tf 9.409 9.464 Td[(X!im^miEa;Eb=iEah)]TJ/F1 9.963 Tf 9.409 9.464 Td[(X!jm^miEb=h)]TJ/F1 9.963 Tf 9.409 9.464 Td[(X!imEam+XmEa!imiEb=)]TJ/F11 9.963 Tf 7.748 0 Td[(!ibEa+!iaEb=)]TJ/F11 9.963 Tf 7.748 0 Td[(irEaEb)-222(rEbEa=)]TJ/F11 9.963 Tf 7.748 0 Td[(i[Ea;Eb]:QEDNoticethat!ij=irEj=ihrEj;Eii:Since0=dhEi;Ejiwehavej!ji=)]TJ/F11 9.963 Tf 7.749 0 Td[(i!ij:.40Inparticular!ii=0.Ifweintroducethematrixoflineardierentialforms"!:=!ijwecanwritetherststructuralequationsasd+!^=0:Fortangentvectors;2TMplet)]TJ/F8 9.963 Tf 4.566 -8.07 Td[(ij;bethematrixofthecurvatureoperatorRwithrespecttothebasisE1p;:::;Enp.SoREjp=Xiij;Ei:SinceR;=)]TJ/F11 9.963 Tf 7.748 0 Td[(R;,ij;=)]TJ/F8 9.963 Tf 7.748 0 Td[(ij;sotheijareexteriordierentialformsofdegreetwo.Cartan'ssecondstructuralequationassertsthatij=d!ij+Xm!im^!mj:.41

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82CHAPTER3.LEVI-CIVITACONNECTIONS.WehaveREaEbEj=XiijEa;EbEibydenition.Wemustshowthattherighthandsideof.41yieldsthesameresultwhenwesubstituteEa;Ebintothedierentialforms,multiplybyEiandsumoveri.WriteREaEbEj=rEarEbEj)-222(rEbrEaEj)-222(r[Ea;Eb]Ej:SincerEbEj=Pi!ijEbEiwegetrEarEbEj=XEa[!ijEb]Ei+X!mjEbrEaEm=XEa[!ijEb]Ei+Xi;m!mjEb!imEaEirEbrEaEj=XEb[!ijEa]Ei+Xi;m!mjEa!imEbEir[Ea;Eb]Ej=X!ij[Ea;Eb]EjsoREaEbEj=XiEa!ijEb)]TJ/F11 9.963 Tf 9.962 0 Td[(Eb!ijEa)]TJ/F11 9.963 Tf 9.963 0 Td[(!ij[Ea;Eb]Ei+Xm;i!imEa!mjEb)]TJ/F11 9.963 Tf 9.962 0 Td[(!imEb!mjEaEi:TherstexpressioninsquarebracketsisthevalueonEa;Ebofd!ijby2.3whilethesecondexpressioninsquarebracketsisthevalueonEa;EbofP!im^!mj.ThisprovesCartan'ssecondstructuralequation.Wecanwritethetwostructuralequationsasd+!^=03.42d!+!^!=3.433.18Cartan'slemma.Wewillshowthattheequations.42and.40determinethe!ij.Firstaresultinexterioralgebra:Lemma1Letx1;:::;xpbelinearlyindependentelementsofavectorspace,V,andsupposethaty1;:::yp2Vsatisfyx1^y1+xp^yp=0:Thenyj=pXk=1AjkxkwithAjk=Akj:

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3.19.ORTHOGONALCOORDINATESONASURFACE.83Proof.Choosexp+1;:::;xnifp
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84CHAPTER3.LEVI-CIVITACONNECTIONS.isorthonormal"withdualframegivenby1=edu;2=gdv:Takingexteriorderivativesyieldsd1=evdv^du=)]TJ/F8 9.963 Tf 7.749 0 Td[(ev=gdu^2d2=gudu^dv=)]TJ/F8 9.963 Tf 7.749 0 Td[(gu=edu^1:Hence!12=ev=gdu)]TJ/F11 9.963 Tf 9.963 0 Td[(12gu=edvbytheuniquenessofthesolutiontotheCartanequations.Intwodimensionsthesecondstructuralequationreducesto12=d!12andwecomputed!12=)]TJ/F8 9.963 Tf 7.749 0 Td[([ev=gv+12gu=eu]du^dv=)]TJ/F8 9.963 Tf 10.958 6.74 Td[(1 eg[ev=gv+12gu=eu]1^2:Thesectionalcurvature=theGaussiancurvatureisthengivenbyK=112E1;E2=)]TJ/F8 9.963 Tf 10.958 6.74 Td[(1 eg[ev=gv+12gu=eu]:.44Weobtainedthisformulainthepositivedenitecasebymuchmorecomplicatedmeansintherstchapter.Exercises.1.3.20ThecurvatureoftheSchwartzschildmetricWeusepolarcoordinatesonspaceandtfortimesocoordinatest;r;#;andintroducetheshorthandnotationS:=sin#;C:=cos#:WexapositiverealnumberMandassumethatr>2M:TheSchwartzschildmetricisgivenasds2=)]TJ/F8 9.963 Tf 7.749 0 Td[(02+12+22+32where0=p hdt;h:=1)]TJ/F8 9.963 Tf 11.158 6.74 Td[(2M r1=1 p hdr2=rd#3=rSd

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3.21.GEODESICSOFTHESCHWARTZSCHILDMETRIC.851.Computedp handtheneachofthedi;i=0;1;2;3.2.Findtheconnectionformmatrix!.3.Findthecurvatureformmatrix=d!+!^!.4.ShowthattheSchwarzschildmetricisRicciat.5.Findthesectionalcurvaturesofthecoordinateplanes",i.e.theplanesspannedbyanytwoof@t;@r;@#;@.6.ThespaceoftheSchartzschildmetricisthetwistedproduct"oftheSchwartzschildplane"Bspannedbyr;twiththemetricgivenby)]TJ/F8 9.963 Tf 7.748 0 Td[(02+120andtheunitsphereSinthesensethatthemetrichastheformg==gB+r2gS:Fromthisfactalonei.e.usingnonoftheprecedingcomputationstogetherwithKoszul'sformulashowthathrXY;Zi=0ifXandYarevectoreldsonBandZisavectoreldonSallthoughtofasvectoreldsonthefullspace.Exercises2.3.21GeodesicsoftheSchwartzschildmetric.Thepurposeofthisproblemsetistogothroughthedetailsoftwoofthefamousresultsgeneralrelativity,theexplanationoftheadvanceoftheperihelionofMer-curyandthedeectionoflightpassingnearthesun.Einstein,1915.Inordertogetresultsinusefulform,weshallexplicitlyincludeNewton'sgravitationalconstantGTheequationsforgeodesicsinalocalcoordinatesystemonasemi-Riemannianmanifoldared2xk ds2+Xi;j)]TJ/F10 6.974 Tf 6.226 4.113 Td[(kijdxi dsdxj ds=0.45where)]TJ/F10 6.974 Tf 6.227 4.113 Td[(kij:=1 2Xmgkm@gjm @xi+@gim @xj)]TJ/F11 9.963 Tf 11.905 6.739 Td[(@gij @xm:.46Oneofthepostulatesofgeneralrelativityisthatasmall"particlewillmovealongageodesicinafourdimensionalLorentzianmanifoldwhosemet-ricisdeterminedbythematterdistributionoverthemanifold.Heretheword

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86CHAPTER3.LEVI-CIVITACONNECTIONS.small"istakentomeanthattheeectofthemassoftheparticleonthemetricitselfcanbeignored.Wecanignorethemassofaplanetwhenthemetricisdeterminedbythemassdistributionofthestars.Thisnotionofsmall"orpassive"issimilartothatinvolvedintheequationsofmotionofachargedparticleinanelectromagneticeld.Generalelectromagnetictheorysaysthattheparticleitselfaectstheelectromagneticeld,butforsmall"particlesweignorethisandtreattheparticlesaspassivelyrespondingtotheeld.Similarlyhere.Wewillhavealottosayaboutthephilosophicalunderpinningsofthepostulatesmallparticlesmovealonggeodesics"whenwehaveenoughmathe-maticalmachinery.Thetheoryalsospeciesthatiftheparticleismassivethenthegeodesicistimelike,whileiftheparticlehasmasszerothenthegeodesicisanullgeodesic,i.e.lightlike.Thesecondcomponentofthetheoryishowthedistributionofmatterde-terminesthemetric.ThisisgivenbytheEinsteineldequations:Matterdistributionisdescribedbyapossiblydegeneratesymmetricbilinearformonthetangentspaceateachpointcalledthestressenergytensor,T.TheEinsteinequationstaketheformG=8TwhereGisrelatedtotheRiccicurvature.Inparticular,inemptyspace,theEinsteinequationsbecomeG=0.Althoughthestudyoftheseequationsisahugeenterprise,thesolutionfortheequationsG=0intheexteriorofastarofmassMwhichissphericallysymmetric",stationary"andtendstotheMinkowskimetricatlargedistanceswasfoundalmostimmediatelybySchwarzschild.Thewordsinquotesneedtobemorecarefullydened.Thisisthemetricds2:=)]TJ/F11 9.963 Tf 7.748 0 Td[(hdt2+h)]TJ/F7 6.974 Tf 6.227 0 Td[(1dr2+r2d2.47wherehr:=1)]TJ/F8 9.963 Tf 11.158 6.739 Td[(2GM r.48whereGisNewton'sgravitationalconstantandd2istheinvariantmetricontheordinaryunitsphere,d2=d2+sin2d2:.49Tobemoreprecise,letPIR2consistofthosepairs,t;rwithr>2GM:LetN=PIS2;thesetofallt;r;q;r>2GM;q2S2.Thecoordinates;canbeusedonthespherewiththenorthandsouthpoleremoved,and.49isthelocalex-pressionfortheinvariantmetricoftheunitsphereintermsofthesecoordinates.ThenthemetricweareconsideringonNisgivenby.47asabove.NoticethatthestructureofNislikethatofasurfaceofrevolution,withtheintervalonthez)]TJ/F8 9.963 Tf 7.749 0 Td[(axisreplacedbythetwodimensionalregion,N,thecirclereplacedbythesphere,andtheradiusofrevolution,f,replacedbyr2.I

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3.21.GEODESICSOFTHESCHWARTZSCHILDMETRIC.87Ifwesetx0:=t;x1:=r;x2:=;x3:=thengij=0;i6=j.50whileg00=)]TJ/F11 9.963 Tf 7.749 0 Td[(h;g11=h)]TJ/F7 6.974 Tf 6.227 0 Td[(1;g22=r2;g33=r2sin2:Recallthatasystemofcoordinatesinwhichametricsatises.50iscalledanorthogonalcoordinatesystem.Insuchacoordinatesystemwehaveseenthatthegeodesicequationsared dsgkkdxk ds=1 2Xj@gjj @xkdxj ds2:.511.ShowthatfortheSchwarzschildmetric,.47,theequationinvolvingg22ontheleftisd dsr2d ds=r2sincosd ds2:Concludefromtheuniquenesstheoremforsolutionsofdierentialequationsthatif==2;_=0thens=2alongthewholegeodesic.Concludefromrotationalinvariancethatallgeodesicsmustlieinaplane,i.e.bysuitablechoiceofpolesofthespherewecanarrangethat=2.2.Withtheabovechoiceofsphericalcoordinatesalongthegeodesic,showthattheg00andg33equationsbecomehdt ds=Er2d ds=LwhereEandLareconstants.Theseconstantsarecalledtheenergy"andtheangularmomentum".NoticethatforL>0,asweshallassume,d=ds>0,sowecanuseasaparameterontheorbitifwelike.Generalprinciplesofmechanicsimplythatthereisaconstantofmotion"associatedtoeveryoneparametergroupofsymmetriesofthesystem.TheSchwarzschildmetricisinvariantundertimetranslationst7!t+candunderrotations7!+.Underthegeneralprinciplesmentionedabove,itturnsoutthatEcorrespondstotimetranslationandthatLcorrespondsto7!+.Wenowconsiderseparatelythecaseofamassiveparticlewherewecanchoosetheparameterssothath0s;0si1andmasslessparticlesforwhichh0s;0si0.

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88CHAPTER3.LEVI-CIVITACONNECTIONS.3.21.1Massiveparticles.Wecanwritethetangentvector,0stothegeodesicatthepointsas0s=_x0s@ @x0s+_x1s@ @x1s+_x2s@ @x2s+_x3s@ @x3s:Letusassumethatweusepropertimeastheparameterizationofourgeodesicsothath0s;0sis)]TJ/F8 9.963 Tf 18.265 0 Td[(1:vskip.2in3.Usingthislastequationandtheresultsofproblem2,showthatE2=dr ds2++L2 r2hr.52alonganygeodesic.OrbitTypes.Wecanwrite.52asE2=dr ds2+Vr.53wheretheeectivepotentialVisgivenasVr:=1)]TJ/F8 9.963 Tf 11.158 6.739 Td[(2GM r+L2 r2)]TJ/F8 9.963 Tf 11.158 6.739 Td[(2GML2 r3:ThebehavioroftheorbitdependsonthetherelativesizeofLandGM.Inparticular,.53impliesthatonanyorbit,risrestrictedtoanintervalIfr:VrE2gsuchthatr2I:Ifwedierentiate.53weget2d2r ds2dr ds=)]TJ/F11 9.963 Tf 7.749 0 Td[(V0rdr ds:.54Inparticular,acriticalpointofV,i.e.apointr0forwhichV0r0=0,givesrisetoacircularorbitrr0.IfRisanon-criticalpointofVforwhichVR=E2,thenRisaturningpoint-theorbitreachestheendpointRoftheintervalIandthenturnsaroundtomovealongIintheoppositedirection.ObservethatVGM=0andVr!1asr!1.TodeterminehowVgoesfrom0to1on[2GM:1wecomputeV0r=2 r4)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(GMr2)]TJ/F11 9.963 Tf 9.963 0 Td[(L2r+3GML2.55

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3.21.GEODESICSOFTHESCHWARTZSCHILDMETRIC.89andthequadraticpolynomialinrgivenbytheexpressioninparenthesishasdiscriminantL2L2)]TJ/F8 9.963 Tf 9.963 0 Td[(12G2M2:Ifthisdiscriminantisnegative,therearenocriticalpoints,soVincreasemono-tonicallyform0to1.Ifthisdiscriminantispositivetherearetwocriticalpoints,r10,weseethatr1alocalmaximumandr2alocalminimum.Wewillignoretheexceptionalcaseofdiscriminantzero.Inthepositivediscriminantcasewemustdistinguishbetweenthecaseswherethelocalmaximumatr1isnotaglobalmaximum,andwhenitis.SinceVr!1asr!1thesetwocasesaredistinguishedbyVr1<1andVr1>1.Ignoringnon-genericcaseswethuscanclassifythebehaviorofrsas:L2<12G2M2soVhasnocriticalpointsandhenceismonotoneincreasingontheinterval[2GM;1.Thebehaviorofrsfors0subdividesintofourcases,allleadingtocrashing"i.e.reachingtheSchwartzschildboundary2GMinnitesorescapetoinnity.Thefourpossibilitieshavetodowiththesignof_randwhetherE2<1orE2>1.1.E2<1;_r<0.SinceVdecreasesasrdecreases,.53impliesthat_rs;_r<0foralls>0whereitisdened.Theparticlecrashesintothebarrierat2GMinnitetime.2.E2<1;_r>0:Theorbitinitiallymovesinthedirectionofincreasingr,reachesitsmaximumvaluewhereVr=E2,turnsaroundandcrashes.3.E2>1;_r>0:Theparticleescapestoinnity.4.E2>1;_r<0.Theparticlecrashes.12G2M20.2.E2r1.theintervalInowliesinawelltotherightofr1,andsothevalueofrhastwoturningpointscorrespondingtotheendpointsofthisinterval.Inotherwordsthevalueofrisboundedalongtheentireorbit.Wecallthisaboundorbit.Inthenon-relativistic"approximation,thiscorrespondstoKepler'sellipses.Inproblems4and5belowwewillexaminemorecloselyhowthisapproximationworksandderiveEinstein'sfamouscalculationoftheadvanceoftheperihelionofMercury.3.Vr1
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90CHAPTER3.LEVI-CIVITACONNECTIONS.4.E2>1.Nowthepossiblebehaviorsarecrash"if_r<0ofescapetoinnityif_r>0.L>4M.NowVr1>1.Againtherewillbefourpossibleintervals:1.E20orturnaroundandthenescapeif_r<0.3.E2>1.TheintervalIextendsfrom2GMtoinnityandtheorbitiseithercrashorescapedependingonthesignof_r.4.Vr20.Accordingtoproblem2wecanuseasaparameteronsuchanorbitandbythesecondequationinthatproblemwehave_r:=dr ds=dr=d ds=d=L r2dr d:Substitutingthisandthedenitionofhinto.52wegetE2=L2 r4dr d2+1+L2 r21)]TJ/F8 9.963 Tf 11.158 6.74 Td[(2GM r:Itisnowconvenienttointroducethevariableu:=1 rinsteadofr.Wehavedu d=)]TJ/F8 9.963 Tf 11.074 6.74 Td[(1 r2dr d=)]TJ/F11 9.963 Tf 7.749 0 Td[(u2dr dsoE2=L2du d2++L2u2)]TJ/F8 9.963 Tf 9.962 0 Td[(2GMu:.56Wecanrewritethisasdu d2=2GMQ;Q:=u3)]TJ/F8 9.963 Tf 20.451 6.74 Td[(1 2GMu2+1u+0.57where0and1areconstants,combinationsofE;L;andGM:1=1 L2;0=1)]TJ/F11 9.963 Tf 9.962 0 Td[(E2 2GML:

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3.21.GEODESICSOFTHESCHWARTZSCHILDMETRIC.91Perihelionadvance.Wewillbeinterestedinthecaseofboundorbits.Inthiscase,amaximumvalue,u1alongtheorbitmustbearootofthecubicpolynomial,Q,asmustbeaminimum,u2,sincetheseareturningpointswherethelefthandsideof.57vanishes.Noticethatthesevaluesdonotdependon,beingrootsofagivenpolynomialwithconstantcoecients.SincetwooftherootsofQarereal,soisthethird,andallthreerootsmustaddupto1 2GM,thenegativeofthecoecientofu2.Thusthethirdrootis1 2GM)]TJ/F11 9.963 Tf 9.963 0 Td[(u1)]TJ/F11 9.963 Tf 9.963 0 Td[(u2:Wethushavedu d2=2GMu)]TJ/F11 9.963 Tf 9.963 0 Td[(u1u)]TJ/F11 9.963 Tf 9.963 0 Td[(u2u)]TJ/F8 9.963 Tf 20.45 6.74 Td[(1 2GM+u1+u2:Sincetherstfactorontherightisnon-positiveandthesecondnon-negative,thethirdisnon-positiveastheproductmustequalthenon-negativeexpressionontheleft.Furthermore,wewillbeinterestedintheregionwherer2GMso2GMu+u1+u2<6GMu11:Wethereforehavethefollowingexpressionsforjd=duj:d du=1 p u1)]TJ/F11 9.963 Tf 9.963 0 Td[(uu)]TJ/F11 9.963 Tf 9.962 0 Td[(u2[1)]TJ/F8 9.963 Tf 9.962 0 Td[(2GMu+u1+u2])]TJ/F6 4.981 Tf 7.423 2.677 Td[(1 2.58:=1+GMu+u1+u2 p u1)]TJ/F11 9.963 Tf 9.963 0 Td[(uu)]TJ/F11 9.963 Tf 9.962 0 Td[(u2.59:=1 p u1)]TJ/F11 9.963 Tf 9.963 0 Td[(uu)]TJ/F11 9.963 Tf 9.962 0 Td[(u2.60Here.59isobtainedfrom.58byignoringtermswhicharequadraticorhigherin2GMu+u1+u2and3.60isobtainedfrom.58byignoringtermswhicharelinearin2GMu+u1+u2.Thestrategynowistoobservethat.60isreallytheequationofanel-lipse,whoseAppolonianparameters,thelatusrectumandtheeccentricity,areexpressedintermsofu1andu2.Then.59isusedtoapproximatetheadvanceintheperihelionofKeplerianmotionassociatetothisellipse.4.Showthattheellipseu=1 `+ecosisasolutionof.60whereeand`aredeterminedfromu1=1 `+e;u2=1 `)]TJ/F11 9.963 Tf 9.962 0 Td[(e

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92CHAPTER3.LEVI-CIVITACONNECTIONS.sothatthemeandistancea:=1 21 u1+1 u2=` 1)]TJ/F11 9.963 Tf 9.962 0 Td[(e2:Thisistheapproximatingellipsewiththesamemaximumandminimumdis-tancetothesunasthetrueorbit,ifwechooseourangularcoordinatesothatthex)]TJ/F8 9.963 Tf 11.07 0 Td[(axisisalignedwiththeaxisoftheellipse.Inprinciple3.58isinsolvedform;ifweintegratetherighthandsidefromu1tou2andthenbackagain,wewillgetthetotalchangeinacrossacompletecycle.Instead,wewillapproximatethisintegralbyreplacing.58by.59andthenalsomaketheapproximatechangeofvariablesu=`)]TJ/F7 6.974 Tf 6.226 0 Td[(1+ecos.5.BymakingtheseapproximationsandsubstitutionsshowthattheintegralbecomesZ20[1+GM`)]TJ/F7 6.974 Tf 6.227 0 Td[(1+cos]d=2+6GM `sotheperihelionadvanceinonerevolutionis6GM a)]TJ/F11 9.963 Tf 9.962 0 Td[(e2:Wehavedonethesecomputationsinunitswherethespeedoflightisone.Ifwearegiventhevariousconstantsinconventionalunits,sayG=6:6710)]TJ/F7 6.974 Tf 6.227 0 Td[(11m3=kgsec;andthemassofthesuninkilogramsM=1:991030kgwemustreplaceGbyG=c2wherecisthespeedoflight,c=3108m/sec.Then2GM=c2:=1:5km.Wemaydividebytheperiodoftheplanettogettherateofadvanceas6GM ca)]TJ/F11 9.963 Tf 9.963 0 Td[(e2T:Ifwesubstitute,forMercury,themeandistancea=5:7681010m,eccentricitye=0:206andperiodT=88days,andusetheconversionscentury=36524daysradian=[360=2]degreesdegree=360000wegetthefamousvalueof43.1"/centuryfortheadvanceoftheperihelionofMercury.Thisadvancehadbeenobservedinthemiddleofthelastcentury.

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3.21.GEODESICSOFTHESCHWARTZSCHILDMETRIC.93Upuntilrecently,thisobservationalvericationofgeneralrelativitywasnotconclusive.ThereasonisthatNewton'stheoryisbasedontheassumptionthatthemassofthesunisconcentratedatapoint.AfamoustheoremofNewtonsaysthattheattractionduetoahomogeneousballonaparticleoutsideisthesameasifallthemassisconcentratedatapoint.Butifthesunisnotaperfectsphere,orifitsmassisnotuniformlydistributed,onewouldexpectsomedeviationfromKepler'slaws.ThesmalleectoftheadvanceoftheperihelionofMercurymighthaveanexplanationintermsofNewtonianmechanics.Intherecentyears,measurementsfrompulsarsindicatelargeperihelionadvancesoftheorderofdegreesperyearinsteadofarcsecondspercenturyyieldingastrikingconrmationofEinstein'stheory.3.21.2Masslessparticles.Wenowhave0s=_x0s@ @x0s+_x1s@ @x1s+_x2s@ @x2s+_x3s@ @x3s:h0s;0sis0:6.Usingproblem2verifythatE2=dr ds2+L2 r2handthend2u d2+u=3GMu2:.61Wewillbeinterestedinorbitswhichgoouttoinnityinbothdirections.Forlargevaluesofr,therighthandsideisnegligiblysmall,soweshouldcompare.61withd2u0 d2+u0=0whosesolutionsareu0=acos+bsinor1=ax+by;x=rcos;y=rsin;inotherwordsstraightlines.Wemightaswellchooseourangularcoordinatesothatthisstraightlineisparalleltothey)]TJ/F8 9.963 Tf 11.069 0 Td[(axis,i.e.u0=r)]TJ/F7 6.974 Tf 6.226 0 Td[(10cos

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94CHAPTER3.LEVI-CIVITACONNECTIONS.wherer0isthedistanceofclosestapproachtotheorigin.Supposeweareinterestedinlightrayspassingthesun.Theradiusofthesunisabout7105kmwhile2GMisabout1:5km.Henceinunitswherer0isoforder1theexpression3GMisaverysmallquantity,callit.Sowriteourequationasu00+u=u2;=3GM:.62Wesolvethisbythemethodofperturbationtheory:lookforasolutionoftheformu=u0+v+wheretheerrorisoforder2.Wechooseu0asabovetosolvetheequationobtainedbyequatingthezero-thordertermsin.7.Comparecoecientsoftoobtaintheequationv00+v=1 2r20+cos2andtryasolutionoftheformv=a+bcos2tondthesolutionofthisequationandsoobtaintherstorderapproximationu=1 r0cos)]TJ/F11 9.963 Tf 16.248 6.74 Td[( 3r20cos2+2 3r20.63to.62.Theasymptotesasr!1oru!0willbestraightlineswithanglesobtainedbysettingu=0in.63.Thisgivesaquadraticequationforcos.8.Rememberingthatcosinemustbe1showthatupthroughorderwehavecos=)]TJ/F8 9.963 Tf 11.404 6.74 Td[(2 3r0=)]TJ/F8 9.963 Tf 8.944 6.74 Td[(2GM r0:Writing==2+thisgivessin=2GM=r0orapproximately=2GM=r0.Thiswasforoneasymptote.Thesamecalculationgivesthesameresultfortheotherasymptote.Addingthetwoandpassingtoconventionalunitsgives=4GM c2r0.64forthedeection.Forlightjustgrazingthesunthispredictsadeectionof1.75".Thiswasapproximatelyobservedintheexpeditiontothesolareclipseof1919.Recent,remarkable,photographsfromtheHubblespacetelescopehavegivenstrongconrmationtoEinstein'stheoryfromthedeectionoflightbydarkmatter.

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Chapter4Thebundleofframes.4.1Connectionandcurvatureformsinaframeeld.LetE=E1;:::;Enbeanorthonormalframeeldand=0B@1...n1CAthedualframeeldsoid=E11++Ennorid=E1;:::;En0B@1...n1CAwhereidisthetautologicaltensoreldwhichassignstheidentitymaptoeachtangentspace.Wewritethismoresuccinctlyasid=E:Thematrixofconnectionformsintermsoftheframeeldisthendeterminedbyd+!^=0andthecurvaturebyd!+!^!=:WenowrepeatanargumentthatwegavewhendiscussingthegeneralMaurerCartanform:RecallthatforanytwoformandapairofvectoreldsXandYwewriteX;Y=iYiX.Thus!^!X;Y=!X!Y)]TJ/F11 9.963 Tf 9.963 0 Td[(!Y!X;95

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96CHAPTER4.THEBUNDLEOFFRAMES.thecommutatorofthetwomatrixvaluedfunctions,!Xand!Y.Considerthecommutatoroftwomatrixvaluedoneforms,!and,!^+^!accordingtoourusualrulesofsuperalgebra.Wedenotethisby[!^;]:Inparticularwemaytake!=toobtain[!^;!]=2!^!:Wecanthusalsowritethecurvatureas=d!+1 2[!^;!]:Thiswayofwritingthecurvaturehasusefulgeneralizationswhenwewanttostudyconnectionsonprincipalbundleslateroninthischapter.4.2Changeofframeeld.SupposethatE0isasecondframeeldwhosedomainofdenitionoverlapswiththedomainofdenitionofE.OntheintersectionoftheirdomainsofdenitionwemusthaveE0=ECisanotherframeeldwhereCisanorthogonalmatrixvaluedfunction.Let0bethedualframeeldofE0.OnthecommondomainofdenitionwehaveEC0=E00=id=Eso=C0:Let!0betheconnectionformassociatedto0,so!0isdeterminedusingCartan'slemmabytheanti-symmetryconditionandd0+!0^0=0:Thend=dC0=dC^0+Cd0=dCC)]TJ/F7 6.974 Tf 6.227 0 Td[(1^)]TJ/F11 9.963 Tf 9.963 0 Td[(C!0C)]TJ/F7 6.974 Tf 6.227 0 Td[(1^implyingthat!=)]TJ/F11 9.963 Tf 7.748 0 Td[(dCC)]TJ/F7 6.974 Tf 6.227 0 Td[(1+C!0C)]TJ/F7 6.974 Tf 6.226 0 Td[(1or!0=C)]TJ/F7 6.974 Tf 6.227 0 Td[(1!C+C)]TJ/F7 6.974 Tf 6.226 0 Td[(1dC:.1

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4.2.CHANGEOFFRAMEFIELD.97Wehave!0^!0=C)]TJ/F7 6.974 Tf 6.227 0 Td[(1!^!C+C)]TJ/F7 6.974 Tf 6.227 0 Td[(1!^dC+C)]TJ/F7 6.974 Tf 6.227 0 Td[(1dCC)]TJ/F7 6.974 Tf 6.227 0 Td[(1^!C+C)]TJ/F7 6.974 Tf 6.227 0 Td[(1dC^C)]TJ/F7 6.974 Tf 6.227 0 Td[(1dCwhiled!0=dC)]TJ/F7 6.974 Tf 6.227 0 Td[(1^!C+C)]TJ/F7 6.974 Tf 6.227 0 Td[(1d!C)]TJ/F11 9.963 Tf 9.962 0 Td[(C)]TJ/F7 6.974 Tf 6.227 0 Td[(1!^dC+dC)]TJ/F7 6.974 Tf 6.227 0 Td[(1^dC:NowitfollowsfromC)]TJ/F7 6.974 Tf 6.227 0 Td[(1CIthatdC)]TJ/F7 6.974 Tf 6.227 0 Td[(1=)]TJ/F11 9.963 Tf 7.749 0 Td[(C)]TJ/F7 6.974 Tf 6.227 0 Td[(1dCC)]TJ/F7 6.974 Tf 6.226 0 Td[(1andhencefromtheexpression0=!0^!0+d!0weget0=C)]TJ/F7 6.974 Tf 6.227 0 Td[(1C:.2Noticethatthisequationcontainstheassertionthatthecurvatureisatensor.Indeed,recallthatforanypairoftangentvectors;2TMpthematrix;givesthematrixoftheoperatorR:TMp!TMprelativetotheorthonormalbasisE1p;:::;Enp.Let2TMpbeatangentvectoratpandletzibethecoordinatesofrelativetothisbasisso=z1E1+znEnwhichwecanwriteas=Epzwherez=0B@z1...zn1CA:ThenR=Ep;z:IfweuseadierentframeeldE0=ECthen=E0pz0wherez0=C)]TJ/F7 6.974 Tf 6.227 0 Td[(1pz.Equation.2impliesthat0;z0=C)]TJ/F7 6.974 Tf 6.227 0 Td[(1p;zwhichshowsthatE0p0;z0=Ep;z:Thusthetransformation7!Ep;zisawelldenedlineartransforma-tion.SoifwedidnotyetknowthatRisawelldenedlineartransformation,wecouldconcludethisfactfrom.2.

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98CHAPTER4.THEBUNDLEOFFRAMES.4.3Thebundleofframes.Wewillnowmakeareinterpretationoftheargumentsoftheprecedingsectionwhichwillhavefarreachingconsequences.LetOMdenotethesetofallorthonormal"basesofallTMp.Soapoint,E,ofOMisanorthonormal"basisofTMpforsomepointp2M,andwewilldenotethispointbyE.So:OM!M;Eisano.n.basisofTMEassignstoeachEthepointatwhichitistheorthonormalbasis.SupposethatEisaframeelddenedonanopensetUM.Ifp2U,andE=p,thenthereisauniqueorthogonal"matrixAsuchthatE=EpA:WewilldenotethismatrixAbyE.Ifwewanttomakethedependenceontheframeeldexplicit,wewillwriteEinsteadof.ThusE=EEE:Thisgivesanidentication:)]TJ/F7 6.974 Tf 6.227 0 Td[(1U!UG;E=E;E.3whereGdenotesthegroupofallorthogonal"matrices.ItfollowsfromthedenitionthatEB=EB;8B2G:.4LetE0beasecondframeelddenedonanopensetU0.WehaveamapC:UU0!GsuchthatE0=ECasinthelastsection.ThusE=EEE=ECEE0EsoE)]TJ/F7 6.974 Tf 6.227 0 Td[(1E0=C:.5Thisshowsthattheidenticationsgivenby.3dene,inaconsistentway,amanifoldstructureonOM.ThemanifoldOMtogetherwiththeactionoftheorthogonalgroup"Gbymultiplicationontheright"RA:E7!EA)]TJ/F7 6.974 Tf 6.227 0 Td[(1andthedierentiablemap:OM!Miscalledthebundleoforthonor-malframes.Wewillnowdeneforms#=0B@#1...#n1CAand != !ijonOM:

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4.3.THEBUNDLEOFFRAMES.994.3.1Theform#.Let2TOMEbeatangentvectoratthepointE2OM.ThendEisatangentvectortoMatthepointE:dE2TME:Assuch,thevectordEhascoordinatesrelativetothebasis,EofTMEandthesecoordinatesdependlinearlyon.SowemaywritedE=#1E1+#nEndeningtheforms#i.Asusual,wewritethismoresuccinctlyasd=E#:4.3.2Theform#intermsofaframeeld.Letv2TOMEbeatangentvectoratthepointE2OM.AssumethatEliesinthedomainofdenitionofaframeeldEandthatE=EpAwherep=E.LetuswritedvinsteadofdEvsoasnottooverburdenthenotation.Wehavedv=Epdv=E#v=EpA#vsoA#v=dv:Wecanwritethisas#=A)]TJ/F7 6.974 Tf 6.226 0 Td[(1.6whereA)]TJ/F7 6.974 Tf 6.227 0 Td[(1istheoneformdenedonUGbyA)]TJ/F7 6.974 Tf 6.227 0 Td[(1+=A)]TJ/F7 6.974 Tf 6.226 0 Td[(1;2TMx;2TGA:HerewehavemadethestandardidenticationofTUGx;Aasadirectsum,TUGx;ATMxTGA;validonanyproductspace.4.3.3Thedenitionof !.Nextwewilldene !intermsoftheidentication:)]TJ/F7 6.974 Tf 6.227 0 Td[(1U!UG

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100CHAPTER4.THEBUNDLEOFFRAMES.givenbyalocalframeeld,andcheckthatitsatisesd#+ !^#=0;i !ij=)]TJ/F11 9.963 Tf 7.749 0 Td[(j !ji:ByCartan'slemma,thisuniquelydetermines !,sothedenitionmustbeinde-pendentofthechoiceofframeeld,andso !isgloballydenedonOM.Let!betheconnectionformoftheLevi-CivitaconnectionoftheframeeldE.Dene !:=A)]TJ/F7 6.974 Tf 6.227 0 Td[(1!A+A)]TJ/F7 6.974 Tf 6.227 0 Td[(1dA.7wheretheexpressioninbracketsontherightisamatrixvaluedoneformdenedonUG.ThenonUGwehaved[A)]TJ/F7 6.974 Tf 6.226 0 Td[(1]=)]TJ/F11 9.963 Tf 7.748 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1dA^A)]TJ/F7 6.974 Tf 6.227 0 Td[(1+A)]TJ/F7 6.974 Tf 6.227 0 Td[(1d=)]TJ/F11 9.963 Tf 7.748 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1dA^A)]TJ/F7 6.974 Tf 6.227 0 Td[(1)]TJ/F11 9.963 Tf 9.962 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1!A^A)]TJ/F7 6.974 Tf 6.226 0 Td[(1so0=d[A)]TJ/F7 6.974 Tf 6.227 0 Td[(1]+A)]TJ/F7 6.974 Tf 6.227 0 Td[(1!A+A)]TJ/F7 6.974 Tf 6.227 0 Td[(1dA^A)]TJ/F7 6.974 Tf 6.227 0 Td[(1:Applyingyieldsd#+ !^#=0:asdesired.Theantisymmetryconditionsaysthat!takesvaluesintheLiealgebraofG.HencesodoesA!A)]TJ/F7 6.974 Tf 6.226 0 Td[(1foranyA2G.WealsoknowthatA)]TJ/F7 6.974 Tf 6.227 0 Td[(1dAtakesvaluesintheLiealgebraofG.Hencesodoes !.4.4Theconnectionforminaframeeldasapull-back.Wenowhaveareinterpretationoftheconnectionform,!,associatedtoaframeeld.Indeed,theform !isamatrixvaluedlineardierentialformdenedonallofOM.Aframeeld,E,denedonanopensetU,canbethoughtofasamap,x7!ExfromUtoOM:E:U!OM;x7!Ex:Thenthepull-backof !underthismapisexactly!,theconnectionformasso-ciatedtotheframeeld!InsymbolsE !=!:Toseethis,observethatunderthemap:)]TJ/F7 6.974 Tf 6.227 0 Td[(1U!UG,wehaveEx=x;IwhereIistheidentitymatrix.ThusE=id;Iwhereid:U!UistheidentitymapandImeanstheconstantmapsendingeverypointxintotheidentitymatrix.BythechainruleE !=EA)]TJ/F7 6.974 Tf 6.227 0 Td[(1!A+A)]TJ/F7 6.974 Tf 6.226 0 Td[(1dA=EA)]TJ/F7 6.974 Tf 6.227 0 Td[(1!A+A)]TJ/F7 6.974 Tf 6.227 0 Td[(1dA=!:

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4.4.THECONNECTIONFORMINAFRAMEFIELDASAPULL-BACK.101Thus,forexample,theframeeldEisparallelrelativetoavectoreld,XonMifandonlyifrXE0whichisthesameasiX!0where!istheconnectionformoftheframeeld.Inviewoftheprecedingresultthisisthesameasi[dEX] !0:HeredEXdenotesthevectoreldalongthemapE:U!OMwhichassignstoeachx2UthevectordExXx.Letmerepeatthisimportantpointinaslightlydierentversion.SupposethatC:[0;1]!MisacurveonM,andwestartwithaninitialframeEatC.Weknowthatthereisauniquecurvet7!EtinOMwhichgivestheparalleltransportofEalongthecurveC.Wehavelifted"thecurveConMtothecurve:t7!EtonEM.Thecurveiscompletelydeterminedbyitsinitialvalue,thefactthatitisaliftofC,i.e.thatt=Ctforallt,andi0t !=0:.8Wenowwanttodescribetwoimportantpropertiesoftheform !.ForB2G,recallthatrBdenotesthetransformationrB:OM!OM;E7!EB)]TJ/F7 6.974 Tf 6.227 0 Td[(1:Wewillusethesameletter,rBtodenotethetransformationrB:UG!UG;x;A7!x;AB)]TJ/F7 6.974 Tf 6.227 0 Td[(1:Becauseof.4,wemayusethisambiguousnotationsincerB=rB:Itthenfollowsfromthelocaldenition4.7thatrB !=B !B)]TJ/F7 6.974 Tf 6.227 0 Td[(1:.9Indeedrb !=rbA)]TJ/F7 6.974 Tf 6.227 0 Td[(1!A+A)]TJ/F7 6.974 Tf 6.227 0 Td[(1dA=rBA)]TJ/F7 6.974 Tf 6.227 0 Td[(1!A+A)]TJ/F7 6.974 Tf 6.227 0 Td[(1dAandrBA)]TJ/F7 6.974 Tf 6.227 0 Td[(1!A=BA)]TJ/F7 6.974 Tf 6.226 0 Td[(1!AB)]TJ/F7 6.974 Tf 6.226 0 Td[(1

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102CHAPTER4.THEBUNDLEOFFRAMES.since!doesnotdependonGandrBA)]TJ/F7 6.974 Tf 6.227 0 Td[(1dA=BA)]TJ/F7 6.974 Tf 6.227 0 Td[(1dAB)]TJ/F7 6.974 Tf 6.226 0 Td[(1:Wecanwrite.9asrB !=AdB !:.10Forthesecondproperty,weintroducesomenotation.Letbeamatrixwhichisantisymmetric"inthesensethatiij=)]TJ/F11 9.963 Tf 7.749 0 Td[(jji:Thisimpliesthattheoneparametergroupt7!exp)]TJ/F11 9.963 Tf 7.748 0 Td[(t=I)]TJ/F11 9.963 Tf 9.963 0 Td[(t+1 2t22)]TJ/F8 9.963 Tf 12.542 6.74 Td[(1 3!t33+liesinourgroupGforallt.Thentheoneparametergroupoftransformationsrexp)]TJ/F10 6.974 Tf 6.227 0 Td[(t:OM!OMhasasitsinnitesimalgeneratoravectoreld,whichweshalldenotebyX.Theoneparametergroupoftransformationsrexp)]TJ/F10 6.974 Tf 6.227 0 Td[(t:UG!UGalsohasaninnitesimalgenerator:Identifyingthetangentspacetothespaceofmatriceswiththespaceofmatrices,weseethatthevectoreldgeneratingthisoneparametergroupoftransformationsofUGisY:x;A7!A:SothevectoreldYtakesvaluesateachpointintheTGcomponentofthetangentspacetoUGandassignstoeachpointx;AthematrixA.Inparticular!Y=0since!isonlysensitivetotheTUcomponent.AlsodAisbydenitionthetautologicalmatrixvalueddierentialformwhichassignstoanytangentvectorZthematrixZ.HenceA)]TJ/F7 6.974 Tf 6.227 0 Td[(1dAY=:FromrB=rBitfollowsthatY=Xandhencethat !X:.11Finally,thecurvatureformfromthepointofviewofthebundleofframesisgivenasusualas :=d !+1 2[ !^; !]:.12

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4.5.GAUSS'THEOREMS.1034.5Gauss'theorems.Wepausewiththissectiontogobacktoclassicaldierentialgeometryusingthelanguagewehavedevelopedsofar.4.5.1EquationsofstructureofEuclideanspace.SupposewetakeM=RnwithitsstandardEuclideanscalarproduct.TheLevi-CivitaconnectionisthenderivedfromtheidenticationofthetangentspaceateverypointwithRnitself-avectoreldbecomesidentiedwithanRnvaluedfunctionwhichwecanthendierentiate.ApointofORncanthenbewrittenasx;E1;:::;Enwherex2RnandEi2RnwithE1;:::;EnforminganorthonormalbasisofRn.Wethenhavex;E1;:::;En=xand#i=hdx;Eii;therighthandsidebeingthescalarproductofthevectorvalueddierentialform,dxandthevectorvaluedfunctionEi.Wehavedx=E#:Dierentiatingthisequationgives0=dE^#+Ed#:WehavedEj=X !ijEiwhere !ij:=hdEj;EiiordE=E !:Wethusgetd#+ !^#=0showingthat !isindeedtheconnectionform.TakingtheexteriorderivativeoftheequationdE=E^ !givesd !+ !^ !=0showingthatthecurvaturedoesindeedvanish.Tosummarize,theequationsofstructureofEuclideangeometryare#i:=hdx;Eii.13 !ij:=hdEj;Eii.14 !ij=)]TJET1 0 0 1 48.947 5.485 cmq[]0 d0 J0.398 w0 0.199 m6.559 0.199 lSQ1 0 0 1 0 -5.485 cmBT/F11 9.963 Tf 0 0 Td[(!ji.15dx=E#.16dE=E !.17d#+ !^#=0.18d !+ !^ !=0:.19

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104CHAPTER4.THEBUNDLEOFFRAMES.4.5.2EquationsofstructureofasurfaceinR3.Wespecializeton=3.LetSbeasurfaceinR3.ThispicksoutathreedimensionalsubmanifoldofthesixdimensionalOR3,callitFSconsistingofallx;E1;E2;E3wherex2SandE1andE2aretangenttoS:OfcoursethisimpliesthatE3isnormaltoS.WewilluseasubscriptStodenotethepullbackofallfunctionsandformsfromOR3toFS.Forexample,thevectorvalueddierentialformdxStakesvaluesinthetangentspaceTSxregardedasasubspaceofR3.Hence#3S0.ThesetofallxS;E1S;E2SconstitutesOS,thebundleofframesofSthoughtofasatwodimensionalRiemannmanifold.SinceE3SisdetermineduptoasignbythepointxS,wecanthinkofFSasatwofoldcoverofOS.[Fromalocalpointofviewwecanalwaysmayachoiceofthesign,andalsofromtheglobalpointofviewifthesurfaceisorientable.]FromtheequationsofstructureofEuclideanspaceweobtaindxS=#1SE1S+#2SE2S.20dE3S= !13SE1S+ !23SE2S.21d#1S+ !12S^#2S=04.22d#2S)]TJET1 0 0 1 0 -9.459 cmq[]0 d0 J0.398 w0 0.199 m6.559 0.199 lSQ1 0 0 1 0 -5.485 cmBT/F11 9.963 Tf 0 0 Td[(!12S^#1S=04.23d !12S+ !13S^ !32S=04.24thelastequationfollowingfrom !12=)]TJET1 0 0 1 32.06 5.484 cmq[]0 d0 J0.398 w0 0.199 m6.559 0.199 lSQ1 0 0 1 0 -5.484 cmBT/F11 9.963 Tf 0 0 Td[(!21and !11=0.Equations.22and.23showthat !12Sand !21S=)]TJET1 0 0 1 37.333 5.485 cmq[]0 d0 J0.398 w0 0.199 m6.559 0.199 lSQ1 0 0 1 0 -5.485 cmBT/F11 9.963 Tf 0 0 Td[(!12Saretheconnec-tionformsofOSifwelocallyidentifyitwithFS.Inparticular, !12Sisintrinsicallydened-itgivestheLevi-CivitaconnectionoftheinducedRiemannmetriconS.4.5.3Theoremaegregium.Equation.21showsthat !13S^ !23S=K#1S^#2S.25whereKistheGaussiancurvature.Gauss'stheoremaegregiumnowfollowsimmediatelyfrom.24.4.5.4Holonomy.LetSbeanytwodimensionalRiemannmanifoldnotnecessarilyembeddedinthreespace.Theconnectionmatrixisatwobytwoantisymmetricmatrix !=0 !12 !210=0 !12)]TJET1 0 0 1 -20.991 -6.47 cmq[]0 d0 J0.398 w0 0.199 m6.559 0.199 lSQ1 0 0 1 0 -5.485 cmBT/F11 9.963 Tf 0 0 Td[(!120:

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4.5.GAUSS'THEOREMS.105LetE=E1;E2beaframeeldonS,andlet!12=E !12SbethecorrespondingformonS.LetbeacurveonSlyingentirelyinthedomainofdenitionoftheframeeld,andlett7!vt2TStbeaeldofunitvectorsalongthecurve.Wecanwritevt=costE1t+sintE2twheretistheanglethattheunitvectorvtmakeswiththerstbasisvector,E1t,oftheframeatt.Thenv0=)]TJ/F11 9.963 Tf 7.749 0 Td[(0sinE1+!210cosE2+0cosE2+!120E1=0)]TJ/F11 9.963 Tf 9.962 0 Td[(!120[)]TJ/F8 9.963 Tf 9.41 0 Td[(sinE1+cosE2]:Inparticular,visparallelalongifandonlyif0t!120t:.26So[]=Z!12.27givesthechangeinofaparallelvectoreldalong.Ofcoursetheangleisrelativetoachoiceofframeeld,andsohasnointrinsicmeaning.Butsupposethatisaclosedcurve,so[]measurestherotationinvolvedintransportingatangentvectoralltheawayaroundthecurvebacktothestartingpoint.Thisiswelldened,independentoftheframeeld,and4.27isvalidforanyclosedcurveonthesurface.Inparticular,supposethat=@Di.e.supposethatistheboundarycurveofsomeorientedtwodimensionalre-gion.WethenmayapplyStokes'theoremand.25toconcludethat[]=ZDKdA:.284.5.5Gauss-Bonnet.SupposethatDiscontainedinthedomainofaframeeld,sayaframeeldobtainedbyorthonormalizingthebasiceldsofacoordinatepatch,toxtheideas.Letdenotetheanglethatthevectoreldmakeswith0ratherthanwithE1.Thetangentvector0turnsthroughanangleof2relativetotheframeeldaswetraversethecurve.thisrequiressomeproofingeneral,butisobviousifDisconvexinsomecoordinatechart,sincethentheanglethat0makeswiththex1)]TJ/F8 9.963 Tf 11.353 0 Td[(axisissteadilyincreasing.Sowecanrestricttothiscasetoavoidcallinginadditionalarguments.Thus[]=[])]TJ/F8 9.963 Tf 9.962 0 Td[(2:

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106CHAPTER4.THEBUNDLEOFFRAMES.Ifisonlypiecewisedierentiable,liketheboundaryofapolygon,thenthechangeinwillcomefromtwosources,thecontributionofthesmoothportionsandtheexterioranglesatthecorners.Sowecanwrite[]=[edgecontributions])]TJ/F1 9.963 Tf 9.963 9.465 Td[(Xexteriorangles:Weget2=ZDKdA+Xexteriorangles)]TJ/F8 9.963 Tf 9.963 0 Td[([edgecontributions]:Nowsupposewesubdividethesurfaceintosuchpolygonal"regions,andsumtheprecedingequationoverallregions.Theedgecontributionswillcancel,sinceeachedgewillcontributetwice,traversedinoppositedirections.Thus2f=ZSKdA+Xexteriorangleswherefisthenumberofregions,D,orfaces".Nowwecanwriteeachexteriorangleas)]TJ/F8 9.963 Tf 9.963 0 Td[(interiorangle:Thesumofalltheinterioranglesateachcornerfromtheregionsimpingingonitaddupto2.Eachedgecontributestotwocorners.Soifweletedenotethenumberofedgesandvthenumberofvertices"orcornersweobtaintheGauss-Bonnetformulaf)]TJ/F11 9.963 Tf 9.962 0 Td[(e+v=1 2ZSKdA:.29Theamazingpropertyofthisformulaisthatthelefthandsidedoesnotdependonthechoiceofmetric,whiletherighthandsidedoesnotdependonthechoiceofsubdivisionandisnotobviouslyanintegeronthefaceofit.SoweobtainEuler'stheoremthatf)]TJ/F11 9.963 Tf 9.918 0 Td[(e+visindependentofthechoiceofsubdivision,andalsothattheintegralofthecurvatureisindependentofthechoiceofmetric,andisanintegerequaltotheEulernumberf)]TJ/F11 9.963 Tf 9.963 0 Td[(e+v.

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Chapter5Connectionsonprincipalbundles.Accordingtothecurrentstandardmodel"ofelementaryparticlephysics,everyfundamentalforceisassociatedwithakindofcurvature.Butthecurvaturesinvolvedarenotonlythegeometriccurvaturesofspace-time,butcurvaturesassociatedwiththenotionofaconnectiononageometricalobjectaprinci-palbundle"whichisageneralizationofthebundleofframesstudiedintheprecedingchapter.Wedevelopthenecessarygeometricalfactsinthischapter.5.1Submersions,brations,andconnections.Asmoothmap:Y!Xiscalledasubmersionifdy:TYy!TXyissurjectiveforeveryy2Y.SupposethatXisn-dimensionalandthatYisn+kdimensional.Theimplicitfunctiontheoremimpliesthefollowingforasubmersion:If:Y!Xisasubmersion,thenaboutanyy2Ythereexistcoordinatesz1;:::;zn;y1;:::;yksuchthatyhascoordinates;:::;0;0:::;0andcoor-dinatesx1;:::;xnaboutysuchthatintermsofthesecoordinatesisgivenbyz1;:::;zn;y1;:::;yk=z1;:::;zn:Inotherwords,locallyinY,asubmersionlookslikethestandardprojectionfromRn+ktoRnneartheorigin.Fortherestofthissectionwewilllet:Y!Xdenoteasubmersion.Foreachy2YwedenetheverticalsubspaceVertyofthetangentspaceTYytoconsistofthose2TYysuchthatdy=0:107

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108CHAPTER5.CONNECTIONSONPRINCIPALBUNDLES.Intermsofthelocaldescription,theverticalsubspaceatanypointinthecoordinateneighborhoodofygivenaboveisspannedbythevaluesofthevectorelds@ @y1;:::;@ @ykatthepointinquestion.ThisshowsthattheVertyttogethertoformasmoothsub-bundle,callitVert,ofthetangentbundleTY.Ageneralconnectiononthegivensubmersionisachoiceofcomplemen-tarysubbundleHortoVert.Thismeansthatateachy2YwearegivenasubspaceHoryTYysuchthatVertyHory=TYyandthattheHoryttogethersmoothlytoformasub-bundleofTY.ItfollowsfromthedenitionthatHoryhasthesamedimensionasTXyand,infact,thattherestrictionofdytoHoryisanisomorphismofHorywithTXy.WeshouldemphasizethattheverticalbundleVertcomesalongwiththenotionofthesubmersion.AconnectionHor,ontheotherhand,isanadditionalpieceofgeometricaldataaboveandbeyondthesubmersionitself.Letusdescribeaconnectionintermsofthelocalcoordinatesgivenabove.Thelocalcoordinatesx1;:::;xnonXgiverisetothevectorelds@ @x1;:::;@ @xnwhichformabasisofthetangentspacestoXateverypointinthecoordinateneighborhoodonX.SincedrestrictedtoHorisabijectionateverypointofY,weconcludethattherearefunctionsari;r=1;:::k;i=1;:::nonthecoordinateneighborhoodonYsuchthat@ @z1+kXr=1ar1@ @yr;:::;@ @zn+kXr=1arn@ @yrspanHorateverypointoftheneighborhood.LetC:[0;1]!XbeasmoothcurveonX.WesaythatasmoothcurveonYisahorizontalliftofCif=Cand0t2Hortforallt.Fortherstconditiontohold,eachpointCtmustlieintheimageof.Theconditionofbeingasubmersiondoesnotimply,withoutsomeadditionalhypotheses,thatissurjective.Letusexaminethesecondconditionintermsofourlocalcoordinatedescription.Supposethatx=C,thatx=y,andwelookforahorizontalliftwith=y.WecanwriteCt=x1t;:::;xnt

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5.1.SUBMERSIONS,FIBRATIONS,ANDCONNECTIONS.109intermsofthelocalcoordinatesystemonX.SoifisanylifthorizontalornotofC,wehavet=x1t;:::;xnt;y1t;:::;yktintermsofthelocalcoordinatesystem.Fortobehorizontal,wemusthave0t=nXi=1xi0t@ @zi+XrXiaritxi0t@ @yr:Thustheconditionthatbeahorizontalliftamounttothesystemofordinarydierentialequationsdyr dt=Xrarix1t;:::;xnt;y1t;:::;yktxi0twherethexiandxi0aregivenfunctionsoft.Thisisasystemofpossiblynon-linearordinarydierentialequations.Theexistenceanduniquenesstheoremforordinarydierentialequationssaysthatforagiveninitialconditionthereissome>0forwhichthereexistsauniquesolutionofthissystemofdierentialequationsfor0t<.Standardexamplesinthetheoryofdierentialequationsshowthatthesolutionscanblowup"inaniteamountoftime;thatingeneralonecannotconcludetheexistenceofthehorizontalliftovertheentireintervalofdenitionofthecurveC.Inthecaseoflineardierentialequations,wedohaveexistenceforalltime,andthereforeinthecaseoflinearconnections,ortheconnectionthatwestudiedonthebundleoforthogonalframes,therewasgloballifting.Wewillnowimposesomerestrictiveconditions.Wewillsaythatthemap:Y!XisalocallytrivialbrationifthereexistsamanifoldFsuchthateveryx2XhasaneighborhoodUsuchthatthereexistsadieomorphismU)]TJ/F7 6.974 Tf 6.227 0 Td[(1U!UFsuchthat1=where1:UF!Uisprojectionontotherstfactor.Theimplicitfunctiontheoremassertsthatasubmersion:Y!XlookslikeaprojectionontoarstfactorlocallyinY.ThemorerestrictiveconditionofbeingabrationrequiresthatlooklikeprojectionontotherstfactorlocallyonX,withasecondfactorFwhichisxeduptoadieomorphism.Ifthemap:Y!XisasurjectivesubmersionandispropermeaningthattheinverseimageofacompactsetiscompactthenweshallprovebelowthatisabrationifXisconnected.AsecondconditionthatwewillimposeisontheconnectionHor.WewillassumethateverysmoothcurveChasaglobalhorizontallift.Wesawthat

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110CHAPTER5.CONNECTIONSONPRINCIPALBUNDLES.thisisthecasewhenlocalcoordinatescanbechosensothattheequationsfortheliftingarelinear,weshallseethatitisalsotruewhenisproper.Butletustakethisgloballiftingconditionasahypothesisforthemoment.LetC:[a;b]:!Xbyasmoothcurve.Foranyy2)]TJ/F7 6.974 Tf 6.227 0 Td[(1Cawehaveauniquelifting:[a;b]!Ywitha=y,andthisliftingdependssmoothlyonybythesmoothdependenceofsolutionsofdierentialequationsoninitialconditions.WethushaveasmoothdieomorphismassociatedwithanysmoothcurveC:[a;b]!Xsending)]TJ/F7 6.974 Tf 6.227 0 Td[(1Ca!)]TJ/F7 6.974 Tf 6.226 0 Td[(1Cb:Ifc2[a;b]iffollowsfromthedenitionandtheexistenceanduniquenesstheoremfordierentialequationsthatthecompositeofthemap)]TJ/F7 6.974 Tf 6.227 0 Td[(1Ca!)]TJ/F7 6.974 Tf 6.227 0 Td[(1CcassociatedwiththerestrictionofCto[a;c]withthemap)]TJ/F7 6.974 Tf 6.227 0 Td[(1Cc!)]TJ/F7 6.974 Tf 6.226 0 Td[(1CbassociatedwiththerestrictionofthecurveCto[c;b]isexactlythemap)]TJ/F7 6.974 Tf 6.226 0 Td[(1Ca!)]TJ/F7 6.974 Tf 6.227 0 Td[(1Cbabove.Thisthenallowsustodeneamap)]TJ/F7 6.974 Tf 6.226 0 Td[(1Ca!)]TJ/F7 6.974 Tf 6.227 0 Td[(1Cbassociatedtoanypiecewisedierentiablecurve,andthedieomorphismassociatedtotheconcatenationoftwocurveswhichformapiecewisedierentiablecurveisthecompositedieomorphism.SupposethatXhasasmoothretractiontoapoint.Thismeansthatthereisasmoothmap:[0;1]X!Xsatisfyingthefollowingconditionswheret:X!Xdenotesthemaptx=t;xasusual.Herearetheconditions:0=id.1x=x0;axedpointofX.tx0=x0forallt2[0;1].Supposealsothatthesubmersion:Y!Xissurjectiveandhasaconnectionwithgloballifting.Weclaimthatthisimpliesthatthatthesubmersionisatrivialbration;thatthereisamanifoldFandadieomorphism:Y!XFwith1=

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5.2.PRINCIPALBUNDLESANDINVARIANTCONNECTIONS.111where1isprojectionontotherstfactor.Indeed,takeF=)]TJ/F7 6.974 Tf 6.227 0 Td[(1x0:Foreachx2Xdenex:)]TJ/F7 6.974 Tf 6.227 0 Td[(1x!Ftobegivenbytheliftingofthecurvet7!tx:Thendeney=y;yy:Thefactthatisadieomorphismfollowsfromthefactthatwecanconstructtheinverseofbydoingtheliftingintheoppositedirectiononeachoftheabovecurves.EverypointonamanifoldhasaneighborhoodwhichisdieomorphictoaballaroundtheorigininEuclideanspace.Suchaballisretractibletotheoriginbyshrinkingalongradiallines.Thisprovesthatanysurjectivesubmersionwhichhasaconnectionwithagloballiftingislocallytrivial,i.e.isabration.Foranysubmersionwecanalwaysconstructaconnection.SimplyputaRiemannmetriconYandletHorbetheorthogonalcomplementtoVertrelativetothismetric.Sotoprovethatif:Y!Xisasurjectivesubmersionwhichisproperthenitisabration,itismorethanenoughtoprovethateveryconnectionhasthegloballiftingpropertyinthiscase.SoletC:[0;1]!Xbeasmoothcurve.ExtendCsoitisdenedonsomeslightlylargerinterval,say[)]TJ/F11 9.963 Tf 7.749 0 Td[(a;1+a];a>0.Foranyy2)]TJ/F7 6.974 Tf 6.226 0 Td[(1Ct;t2[0;1]wecanndaneighborhoodUyandan>0suchthattheliftngofCsexistsforallz2Uyandt)]TJ/F11 9.963 Tf 8.157 0 Td[(0thatwillworkforally2)]TJ/F7 6.974 Tf 6.227 0 Td[(1C[0;1].Butthisclearlyimpliesthatwehavegloballifting,sincewecandotheliftingpiecemealoverintervalsoflengthlessthanandpatchthelocalliftingstogether.5.2Principalbundlesandinvariantconnections.5.2.1Principalbundles.LetGbeaLiegroupwithLiealgebrag.LetPbeaspaceonwhichGacts.Totieinwithourearliernotation,andalsoforlaterconvenience,wewilldenotethisactionbyp;a7!pa)]TJ/F7 6.974 Tf 6.226 0 Td[(1;p2P;a2Gsoa2GactsonPbyadieomorphismthatwewilldenotebyra:ra:P!P;rap=pa)]TJ/F7 6.974 Tf 6.227 0 Td[(1:

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112CHAPTER5.CONNECTIONSONPRINCIPALBUNDLES.If2g,thenexp)]TJ/F11 9.963 Tf 7.749 0 Td[(tisaoneparametersubgroupofG,andhencerexp)]TJ/F10 6.974 Tf 6.226 0 Td[(tisaoneparametergroupofdieomorphismsofP,andforeachp2P,thecurverexp)]TJ/F10 6.974 Tf 6.227 0 Td[(tp=pexptisasmoothcurvestartingattatt=0.Thetangentvectortothiscurveatt=0isatangentvectortoPatp.Inthiswaywegetalinearmapup:g!TPp;up=d dtpexptjt=0:.1Forexample,ifwetakeP=GwithGactingonitselfbyrightmultiplication,andifweassumedthatGisasubgroupofGln,sothatwemayidentifyTPpasasubspaceofthethespaceofallnnmatrices,thenwehaveseenthatup=pwherethemeaningofpontherighthandsideistheproductofthematrixpwiththematrix.Forthiscase,ifrap=pforsomep2P,thea=e,theidentityelement.Ingeneral,wesaythatthegroupactionofGonPisfreeifnopointofPisxedbyanyelementofGotherthantheidentity.Sofree"meansthatifrap=pforsomep2Gthena=e.Clearly,iftheactionisfree,thenthemapupisinjectiveforallp2P.IfwehaveanactionofGonPandonQ,thenweautomaticallygetanactionofGdiagonallyonPQ,andiftheactionofPisfreethensoistheactiononPQ.Forexampletochangethenotationslightly,ifXisaspaceonwhichGactstrivially,andifweletGactonitselfbyrightmultiplication,thenwegetafreeactionofGonXG.Thisiswhatweencounteredwhenwebegantoconstructthemanifoldstructureonthebundleoforthogonalframesoutofalocalframeeld.Wenowgeneralizethisconstruction:IfwearegivenanactionofGonPwehaveaprojection:P!P=Gwhichsendseachp2PtoitsG-orbit.Wemakethefollowingassumptions:TheactionofGonPisfree.ThespaceP=GisadierentiablemanifoldMandtheprojection:P!Misasmoothbration.ThebrationislocallytrivialconsistentwiththeGactioninthesensethateverym2MhasaneighborhoodUsuchthatthereexistsadieo-morphismU)]TJ/F7 6.974 Tf 6.227 0 Td[(1U!UGsuchthat1=

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5.2.PRINCIPALBUNDLESANDINVARIANTCONNECTIONS.113where1:UF!Uisprojectionontotherstfactorandifp=m;bthenrap=m;ba)]TJ/F7 6.974 Tf 6.227 0 Td[(1:Whenallthishappens,wesaythat:P!MisaprincipalberbundleoverMwithstructuregroupG.Supposethat:P!MisaprincipalberbundlewithstructuregroupG.Sinceisasubmersion,wehavethesub-bundleVertofthetangentbundleTP,andfromitsconstruction,thesubspaceVertpTPpisspannedbythetangentstothecurvespexpt;2g.Inotherwords,upisasurjectivemapfromgtoVertp.SincetheactionofGonPisfree,weknowthatupisinjective.PuttingthesetwofactstogetherweconcludethatProposition6If:P!MisaprincipalberbundlewithstructuregroupGthenupisanisomorphismofgwithVertpforeveryp2P.Letuscomparetheisomorphismupwiththeisomorphismurbp=upb)]TJ/F6 4.981 Tf 5.396 0 Td[(1.Theactionofb2GonPpreservesthebrationandhencedrbp:Vertp!Vertpb)]TJ/F6 4.981 Tf 5.397 0 Td[(1:Letv=up2Vertp.Thismeansthatv=d dtpexptt=0:Bydenitiondrbpv=d dtrbpexptjt=0=d dtpexptb)]TJ/F7 6.974 Tf 6.226 0 Td[(1jt=0:Wehavepexptb)]TJ/F7 6.974 Tf 6.226 0 Td[(1=pb)]TJ/F7 6.974 Tf 6.227 0 Td[(1bexptb)]TJ/F7 6.974 Tf 6.227 0 Td[(1=pb)]TJ/F7 6.974 Tf 6.227 0 Td[(1exptAdbwhereAdistheconjugation,oradjoint,actionofGonitsLiealgebra.Wehavethusshownthatdrbpup=urbpAdb:.25.2.2Connectionsonprincipalbundles.Let:P!MbeaprincipalbundlewithstructuregroupG.Recallthatinthegeneralsetting,wedenedageneralconnectiontobeasub-bundleHorofthetangentbundleTPwhichiscomplementarytotheverticalsub-bundleVert.GiventhegroupactionofG,wecandemandthatHorbeinvariantunder

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114CHAPTER5.CONNECTIONSONPRINCIPALBUNDLES.G.Sobyaconnectiononaprincipalbundlewewillmeanasub-bundleHorofthetangentbundlesuchthatTPp=VertpHorpatallp2PanddrbpHorp=Horrbp8b2G;p2P:.3AtanypwecandenetheprojectionVp:TPp!VertpalongHorp,i.e.VpistheidentityonVertpandsendsallelementsofHorpto0.GivingHorpisthesameasgivingVpandcondition.3isthesameastheconditiondrbpVp=Vrbpdrbp8b2G;p2P:.4Letuscomposeu)]TJ/F7 6.974 Tf 6.226 0 Td[(1p:Vertp!gwithVp.Sowedenethegvaluedform !by !p:=u)]TJ/F7 6.974 Tf 6.227 0 Td[(1pVp:.5Thenitfollowsfrom.2and.4thatrb !=Adb !:.6LetPbethevectoreldonPwhichistheinnitesimalgeneratorofrexpt.Inviewofdenitionofupasidentifyingwiththetangentvectortothecurvet7!pexpt=rexp)]TJ/F10 6.974 Tf 6.227 0 Td[(tpatt=0,weseethatiP !=)]TJ/F11 9.963 Tf 7.749 0 Td[(:.7Theinnitesimalversionof.6isDP !=[; !]:.8Denethecurvaturebyourformula :=d !+1 2[ !; !]:.9Itfollowsfrom.6thatrb =Adb 8b2G:.10NowiPd !=DP !)]TJ/F11 9.963 Tf 9.963 0 Td[(diP !byWeil'sformulafortheLiederivative.By.7thesecondtermontherightvanishesbecauseitisthedierentialoftheconstant)]TJ/F11 9.963 Tf 7.749 0 Td[(.SoiPd !=[; !]:

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5.2.PRINCIPALBUNDLESANDINVARIANTCONNECTIONS.115OntheotherhandiP[ !; !]=[iP !; !])]TJ/F8 9.963 Tf 9.963 0 Td[([ !;iP !]=)]TJ/F8 9.963 Tf 7.749 0 Td[(2[; !]whereweused.7again.Soiv =0ifv2Vertp:.11Tounderstandthemeaningof whenevaluatedonapairofhorizontalvectors,letXandYbepairofhorizontalvectorelds,thatisvectoreldswhosevaluesateverypointareelementsofHor.TheniX !=0andiY !=0:So X;Y=iYiX =iYiXd !=d !X;Y:Butbyourgeneralformulafortheexteriorderivativewehaved !X;Y=XiY !)]TJ/F11 9.963 Tf 9.963 0 Td[(YiX !)]TJET1 0 0 1 22.609 5.484 cmq[]0 d0 J0.398 w0 0.199 m6.559 0.199 lSQ1 0 0 1 0 -5.484 cmBT/F11 9.963 Tf 0 0 Td[(![X;Y]:Thersttwotermsvanishandso =)]TJET1 0 0 1 28.228 5.485 cmq[]0 d0 J0.398 w0 0.199 m6.559 0.199 lSQ1 0 0 1 0 -5.485 cmBT/F11 9.963 Tf 0 0 Td[(![X;Y]:.12Thisshowshowthecurvaturemeasuresthefailureofthebracketoftwohori-zontalvectoreldstobehorizontal.5.2.3Associatedbundles.Let:P!MbeaprincipalbundlewithstructuregroupG,andletFbesomemanifoldonwhichGacts.Wewillwritethisactionasmultiplicationontheleft;i.e.wewilldenotetheactionofanelementa2Gonanelementf2Fasaf.WethenhavethediagonalactionofGonPF:Fora2Gwedenediaga:PF!PF;diagap;f=pa)]TJ/F7 6.974 Tf 6.226 0 Td[(1;af:SincetheactionofGonPisfree,soisitsdiagonalactiononPF.Wecanformthequotientspaceofthisaction,i.e.identifyallelementsofPFwhichlieonthesameorbit;soweidentifythepointsp;fandpa)]TJ/F7 6.974 Tf 6.227 0 Td[(1;af.ThequotientspaceunderthisidenticationwillbedenotedbyPGForbyFP:Itisamanifoldandtheprojectionmap:P!MdescendstoaprojectionofFP!MwhichwewilldenotebyForsimplybywhenthereisnodangerofconfusion.ThemapF:FP!Misabration.ThebundleFPiscalledthebundleassociatedtoPbytheG-actiononF.Let:PF!PF

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116CHAPTER5.CONNECTIONSONPRINCIPALBUNDLES.bethemapwhichsendp;fintoitsequivalenceclass.SupposethatwearegivenaconnectionontheprincipalbundleP.Recallthatthismeansthatateachp2PwearegivenasubspaceHorpTPpwhichiscomplementarytothevertical,andthatthisassignmentisinvariantundertheactionofGinthesensethatHorpa)]TJ/F6 4.981 Tf 5.396 0 Td[(1=draHorp:Givenanf2F,wecanconsiderHorpasthesubspaceHorpf0gTPFp;f=TPpTFfandthenformdp;fHorpTFPp;fwhichiscomplementarytotheverticalsubspaceVFPp;fTFPp;f:TheinvarianceconditionofHorimpliesthatdp;fHorpisindependentofthechoiceofp;finitsequivalenceclass.Soaconnectiononaprincipalbundleinducesaconnectiononeachofitsassociatedbundles.5.2.4Sectionsofassociatedbundles.If:Y!Xisasubmersion,thenasectionofthissubmersionisamaps:X!Ysuchthats=id:Inotherwords,sisamapwhichassociatestoeachx2Xanelementsx2Yx=)]TJ/F7 6.974 Tf 6.226 0 Td[(1x:Naturally,wewillbeprimarilyinterestedinsectionswhicharesmooth.Forexample,wemightconsiderthetangentbundleTM.Asectionofthetangentbundlethenassociatestoeachx2Matangentvectorsx2TMx.Inotherwords,sisavectoreld.Similarly,alineardierentialformonMisasectionofthecotangentbundleTM.Supposethat=F:FP!MisanassociatedbundleofaprincipalbundleP,andthats:M!FPisasectionofthisbundle.LetxbeapointofM,andletp2Px=)]TJ/F7 6.974 Tf 6.227 0 Td[(1xbeapointintheberoftheprincipalbundleP!Mlyingintheberoverx.Thenthereisauniquef2Fsuchthatp;f=sx:Wethusgetafunctions:P!Fbyassigningtopthiselementf2F.Inotherwords,sisuniquelydeterminedbyp;sp=sp:.13

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5.2.PRINCIPALBUNDLESANDINVARIANTCONNECTIONS.117Supposewereplacepbyrap=pa)]TJ/F7 6.974 Tf 6.226 0 Td[(1.Sincepa)]TJ/F7 6.974 Tf 6.227 0 Td[(1;af=p;fweseethatssatisestheconditionra=a8a2G:.14Conversely,supposethat:P!Fsatises.14.Thenp;p=pa)]TJ/F7 6.974 Tf 6.227 0 Td[(1;pa)]TJ/F7 6.974 Tf 6.227 0 Td[(1andsodenesanelementsx;x=p.Soa:P!Fsatisfying.14determinesasections:M!FPwith=s.Itisroutinetocheckthatsissmoothifandonlyifissmooth.WehavethusprovedProposition7Thereisaonetoonecorrespondencebetweensmoothsectionss:M!FPandsmoothfunctions:P!Fsatisfying.14.Thecorrespondenceisgivenby.13.AnextremelyspecialcaseofthispropositioniswherewetakeFtobetherealnumberswiththetrivialactionofGonR.ThenRP=MRsincethemapdoesnotidentifytwodistinctelementsofRbutmerelyidentiesallelementsofPx.AsectionsofMRisoftheformsx=x;fxwherefisarealvaluedfunction.thepropositionthenassertsthatwecanidentifyrealvaluedfunctionsonMwithrealvaluedfunctionsonPwhichareconstantonthebersPx.5.2.5Associatedvectorbundles.WenowspecializetothecasethatFisavectorspace,andtheactionofGonFislinear.Inotherwords,wearegivenalinearrepresentationofGonthevectorspaceF.Ifx2Mwecanaddtwoelementsv1andv2ofFPxbychoosingp2Pxwhichthendeterminesf1andf2inFsuchthatp;f1=v1andp;f2=v2:Wethendenev1+v2:=p;f1+f2:ThefactthattheactionofGonFislinearguaranteesthatthisdenitionisindependentofthechoiceofp.Inasimilarway,wedenemultiplicationofanelementofFPxbyascalarandverifythatalltheconditionsforFPxtobeavectorspacearesatised.LetV!Mbeavectorbundle.SoV!MisabrationforwhicheachVxhasthestructureofavectorspace.AsaclassofexamplesofvectorbundleswecanconsidertheassociatedvectorbundlesFPjustconsidered.WecanthenconsiderVvalueddierentialformsonM.Forexample,aVvaluedlineardierentialformwillbearulewhichassignsalinearmapx:TMx!Vx

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118CHAPTER5.CONNECTIONSONPRINCIPALBUNDLES.foreachx2M,andsimilarlywecantalkofVvaluedk-forms.ForthecasethatV=FPisanassociatedvectorbundlewehaveagen-eralizationofProposition7tothecaseofdierentialforms.Thatis,wecandescribeFPvalueddierentialformsascertainkindsofF-valuedformsonP.Toseehowthisworks,supposethatisanFP-valuedk-formonM.Letx2Mandletp2Px.Nowx:^kTMx!FPxandpgivesanidenticationmapwhichwewilldenotebyidentpofFPxwithF-theelementf2Fbeingidentiedwithp;f2FPx.Also,dp:TPp!TMxandsoinducesmapwhichweshallalsodenotebydpdp:^kTPp!^kTMx:Sop:=identpxdpmaps^kTPp!F.ThuswehavedenedanF-valuedk-formonP.IfvisaverticaltangentvectoratanypointpofPwehavedpv=0,soiv=0ifv2VertP:.15Letusseewhathappenswhenwereplacepbyrap=pa)]TJ/F7 6.974 Tf 6.227 0 Td[(1intheexpressionfor.sincera=,weconcludethatdpa)]TJ/F6 4.981 Tf 5.397 0 Td[(1drap=dp:Also,identpa)]TJ/F6 4.981 Tf 5.396 0 Td[(1=aidentpwheretheaontherightdenotestheactionofaonF.Wethusconcludethatra=a:.16Conversely,supposethatisanF-valuedk-formonPwhichsatises5.15and.16.ItdenesanFPvaluedk-formonMasfollows:Ateachx2Mchooseap2Px.Foranyktangentvectorsv1;:::;vk2TMxchoosetangentvectorsw1;:::;wk2TPpsuchthatdpwj=vj;j=1;:::;k:Thenconsiderpw1^^wk2F:

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5.2.PRINCIPALBUNDLESANDINVARIANTCONNECTIONS.119Condition.15guaranteesthatthisvalueisindependentofthechoiceofthewiwithdpwj=vj.Inthiswaywedeneamap^kTMx!F:Ifwenowapplyp;totheimage,wegetamap^kTMx!FPxandcondition.16guaranteesthatthismapisindependentofthechoiceofp2Px.Fromtheconstructionitisclearthattheassignments!and!areinversesofoneanother.Wehavethusproved:Proposition8ThereisonetoonecorrespondencebetweenFPvaluedformsonMandFvaluedformsonPwhichsatisfy.15and.16.FormsonPwhichsatisfy.15and.16arecalledbasicformsbecauseaccordingtothepropositionF-valuedformsonPformsonPwhich.15and.16correspondtoformsonthebasemanifoldMwithvaluesintheassociatedbundleFP.Forexample,equations.10and5.11saythatthecurvatureofaconnec-tiononaprincipalbundleisabasicgvaluedformrelativetotheadjointactionofGong.Accordingtotheproposition,wecanconsiderthiscurvatureasatwoformonthebaseMwithvaluesingP,thevectorbundleassociatedtoPbytheadjointactionofGonitsLiealgebra.Hereisanotherimportantillustrationoftheconcept.Equation.6saysthataconnectionform !satises.16,butitcertainlydoesnotsatisfy.15.Indeed,theinteriorproductofaverticalvectorwiththelineardierentialform !isgivenby.7.However,supposethatwearegiventwoconnectionforms !1and !2.Thentheirdierence !1)]TJET1 0 0 1 24.47 5.485 cmq[]0 d0 J0.398 w0 0.199 m6.559 0.199 lSQ1 0 0 1 0 -5.485 cmBT/F11 9.963 Tf 0 0 Td[(!2doessatisfy.15and,ofcourse,.16.Wecanphrasethisbysayingthatthedierenceoftwoconnectionsisabasicgvaluedone-form.5.2.6Exteriorproductsofvectorvaluedforms.SupposethatF1andF1aretwovectorspacesonwhichGacts,andsupposethatwearegivenabilinearmapb:F1F2!F3intoathirdvectorsspaceF3onwhichGacts,andsupposethatbisconsistentwiththeactionsofGinthesensethatbaf1;af2=abf1;f2:Examplesofsuchasituationthatwehavecomeacrossbeforeare:1.GisasubgroupofGlnandF1;F2andF3areallthevectorspaceofnnmatrices,andbismatrixmultiplication.

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120CHAPTER5.CONNECTIONSONPRINCIPALBUNDLES.2.GisasubgroupofGln,F1isthespaceofallnnmatrices,F2andF3areRnandbismultiplicationofamatrixtimesavector.3.GisageneralLiegroup,F1=F2=F3=g,theLiealgebraofGandbisLiebracket.IneachofthesecaseswehavehadoccasiontoformtheexteriorproductofanF1valueddierentialformwithanF2valueddierentialformtoobtainanF3valuedform.Wecandothisconstructioningeneral:formtheexteriorproductofanF1valuedk-formwithanF2-valued`formtogetandF3valuedk+`form.Forexample,iff11;:::;f1misabasisofF1andf21;:::;f2nisaabasisofF2thenthemostgeneralF1-valuedk-formcanbewrittenas=Xif1iwheretheiarerealvaluedk-forms,andthemostgeneralF2-valued`-formcanbewrittenas=Xjf2jwherethejarerealvalued`forms.Letf31;:::;f3qbeabasisofF3anddenethenumbersBkijbybf1i;f2j=XkBkijf3k:Thenyoucancheckthat^denedby^:=XBkiji^jf3kisindependentofthechoiceofbases.Inasimilarwaywecandenetheexteriorderivativeofavectorvaluedform,theinteriorproductofavectorvaluedformwithavectoreld,thepullbackofavectorvaluedformunderamapetc.Thereshouldbelittleprobleminunderstandingtheconceptinvolved.Thereisabitofanotationalproblem-howexplicitdowewanttomakethemapbinwritingdownasymbolforthisexteriorproduct.Inexample1wesimplywrote^fortheexteriorproductoftwomatrixvaluedforms.Thisforcedustousetheratherugly[;^]fortheexteriorproductoftwoLiealgebravaluedforms,wherethebwascommutatororLiebracket.Weshallretainthisuglynotationforthesakeoftheclarityitgives.Asituationthatwewillwanttodiscussinthenextsectionis:wearegivenanactionofGonavectorspaceF,andunlessforcedtobemoreexplicit,wehavechosentodenotetheactionofanelementa2Gonanelementf2Fsimplybyaf.Thisdeterminesabilinearmapb:gF!Fbyb;f:=d dtexptfjt=0:

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5.3.COVARIANTDIFFERENTIALSANDCOVARIANTDERIVATIVES.121Wethereforegetanexteriormultiplicationofag-valuedformwithanF-valuedform.Weshalldenotethisparticulartypeofexteriormutliplicationby.Soifisag-valuedk-formandisanF-valued`formthenisanF-valuedk+`-form.Wepointoutthatconditions.15and.16makeperfectlygoodsenseforvectorvaluedforms,andsowecantalkofbasicvectorvaluedformsonP,andtheexteriorproductoftwobasicvectorvaluedformsisagainbasic.5.3Covariantdierentialsandcovariantderiva-tives.InthissectionweconsideraxedconnectiononaprincipalbundleP.ThismeansthatwearegivenaprojectionVofTPontotheverticalbundleandthereforeaconnectionform !.Ofcoursewealsohaveaprojectionid)]TJ/F48 9.963 Tf 9.962 0 Td[(VontothehorizontalbundleHoroftheconnection,whereidistheidentityop-erator.Thisprojectionkillsallverticalvectors.5.3.1Thehorizontalprojectionofforms.Ifisapossiblyvectorvaluedk-formonP,wewilldenethehorizontalprojectionHofbyHv1;:::;vk=id)]TJ/F48 9.963 Tf 9.963 0 Td[(Vv1;:::;id)]TJ/F48 9.963 Tf 9.963 0 Td[(Vvk:.17ThefollowingpropertiesofHfollowimmediatelyfromitsdenitionandtheinvarianceofthehorizontalbundleundertheactionofG:1.H^=H^H.2.raH=Hra8a2G.3.Ifhasthepropertythatiw=0foranyhorizontalvectorwthenH=0.Inparticular,4.H !=0.5.Ifhasthepropertythativ=0foranyverticalvectorvthenH=.Inparticular,6.Histheidentityonbasicforms.In1andcouldbevectorvaluedformsifwehavethebilinearmapbwhichallowsustomultiplythem.ThemapHisclearlyaprojectioninthesensethatH2=H:

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122CHAPTER5.CONNECTIONSONPRINCIPALBUNDLES.5.3.2ThecovariantdierentialofformsonP.Denedmappingk-formsintok+1formsbyd:=Hd:.18Thefollowingfactsareimmediate:d^=d^H+)]TJ/F8 9.963 Tf 7.749 0 Td[(1kH^disisak-form.ivd=0foranyverticalvectorv.rad=dra8a2G.Itfollowsfromthesecondandthirditemsthatdcarriesbasicformsintobasicforms.IfFisavectorspaceonwhichGactslinearly,wecanformtheassociatedvectorbundleFP,andweknowfromProposition8thatk-formsonMwithvaluesinFParethesameasbasick-formsonPwithvaluesinF.SogivingaconnectiononPinducesanoperatordmappingk-formsonMwithvaluesinFPtok+1-formsonMwithvaluesinFP.Forexample,asectionsofFPisjustazeroformonMwithvaluesinthevectorbundleFM.GivingtheconnectiononPallowsustoconstructtheoneformdswithvaluesinFP.IfXisavectoreldonM,thenwecandenerXs:=iXds;thecovariantderivativeofsinthedirectionX.5.3.3Aformulaforthecovariantdierentialofbasicforms.LetbeabasicformonPwithvaluesinthevectorspaceFonwhichGactslinearly.Letdbethecovariantdierentialassociatedwiththeconnectionform !.Weclaimthatd=d+ !:.19Inordertoprovethisformula,itisenoughtoprovethatwhenweapplyivtotherighthandsidewegetzero,ifvisvertical.ForthenapplyingHdoesnotchangetherighthandside.ButapplyingHtotherighthandsideyieldsdsinced:=HdandH !=0soH !=0:Soitisenoughtoshowthatforany2gwehaveiPd=)]TJ/F11 9.963 Tf 7.749 0 Td[(iP !:Sinceisbasic,wehaveiP=0,sobyWeil'sidentitywehaveiP=DP=P

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5.3.COVARIANTDIFFERENTIALSANDCOVARIANTDERIVATIVES.123bytheinnitesimalversionoftheinvariancecondition5.16.Ontheotherhand,sinceiP=0andiP!=)]TJ/F11 9.963 Tf 7.749 0 Td[(,wehaveprovedourformula.Thereareacoupleofspecialcasesof.19worthmentioning.IfFisRwiththetrivialrepresentationthen.19saysthatd=d.Thisimplies,thatifsisasectionofanassociatedvectorbundleFP,andifisafunctiononM,sothatsisagainasectionofFPthends=d^s+sdsimplyingthatforanyvectoreldXonMwehaverXs=Xs+rXs:AnotherimportantspecialcaseiswherewetakeF=gwiththeadjointaction.Then.19saysthatd=d+[ !^;]:5.3.4Thecurvatureisd !.Wewishtoprovethatd !=d!+1 2[!^;!]:.20Bothsidesvanishwhenweapplyivwherevisaverticalvector-thisistrueforthelefthandsidebydenition,andwehavealreadyveriedthisfortherighthandside,seeequation.11.ButifweapplyHtobothsides,wegetd !ontheleft,andalsoontherightsinceH !=0.5.3.5Bianchi'sidentity.Inoursettingthissaysthatd =0:.21Proof.Wehaved =dd !+d1 2[ !^; !]=[d !^; !]:ApplyingHyieldszerobecauseH !=0.5.3.6Thecurvatureandd2.Wewishtoshowthatd2= :.22InthisequationisabasicformonPwithvaluesinthevectorspaceFwhereGacts,andweknowthat isabasicformwithvaluesing,sotherighthandsidemakessenseandisabasicFvaluedform.Toprovethisweuseourformulad=d+ !

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124CHAPTER5.CONNECTIONSONPRINCIPALBUNDLES.andapplyitagaintogetd2=dd+ !+ !d+ !:Wehaved2=0sotherstexpressionunderthedbecomesd !=d !)]TJET1 0 0 1 34.554 5.485 cmq[]0 d0 J0.398 w0 0.199 m6.559 0.199 lSQ1 0 0 1 0 -5.485 cmBT/F11 9.963 Tf 0 0 Td[(!d:Thesecondtermontherightherecancelstheterm !dsowegetd2=d !+ !:Sotocompletetheproofwemustcheckthat1 2[ !^; !]= !:Thisisavariantofacomputationwehavedoneseveraltimesbefore.Sinceinteriorproductwithverticalvectorssendstozero,whileinteriorproductwithhorizontalvectorssends !tozero,itsucestoverifythattheaboveequationistrueafterwetaketheinteriorproductofbothsideswithtwoverticalvectors,sayPandP.NowiP=[ !^; !]=)]TJ/F8 9.963 Tf 7.749 0 Td[([; !]+[ !;]=)]TJ/F8 9.963 Tf 7.749 0 Td[(2[; !]andsoipiP1 2[ !^; !]=[;]:AsimilarcomputationshowsthatiPiP !=)]TJ/F11 9.963 Tf 9.962 0 Td[(:ButtheequalityofthesetwoexpressionsfollowsfromthefactthatwehaveanactionofGonFwhichimpliesthatforany;2gandanyf2Fwehave[;]f=f)]TJ/F11 9.963 Tf 9.963 0 Td[(f:

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Chapter6Gauss'slemma.Wehavedenedgeodesicsasbeingcurveswhichareselfparallel.Butthereareseveralothercharacterizationsofgeodesicswhicharejustasimportant:forexample,inaRiemannmanifoldgeodesicslocallyminimizearclength:astraightlineistheshortestdistancebetweentwopoints".Wewanttogiveoneexplanationofthisfacthere,usingtheexponentialmap,"aconceptintroducedbyalBiruni-1048butunappreciatedforabout1000years.Thekeyresult,knownasGauss'lemmaassertsthatradialgeodesicsareorthogonaltotheimagesofspheresundertheexponentialmap,andthiswillallowustorelategeodesicstoextremalpropertiesofarclength.6.1Theexponentialmap.SupposethatMisamanifoldwithaconnectionr.Letm0beapointofMand2TMm0.Thenthereisauniquemaximalgeodesicwith=m0;0=.Itisfoundbysolvingasystemofsecondorderordinarydier-entialequations.Theexistenceanduniquenesstheoremforsolutionsofsuchequationsimpliesthatthesolutionsdependsmoothlyon.Inotherwords,thereexistsaneighborhoodNofinthetangentbundleTMandanintervalIabout0inRsuchthat;s7!sissmoothonNI.Ifwetake=0,thezerotangentvector,thecorrespondinggeodesic".denedforalltistheconstantcurve0tm0.ThecontinuitythusimpliesthatforinsomeneighborhoodoftheorigininTMm0,thegeodesicisdenedfort2[0;1].LetD0bethesetofvectorsinTMm0suchthatthemaximalgeodesicthroughisdenedon[0;1].Bytheprecedingremarksthiscontainssomeneighborhoodoftheorigin.Denetheexponentialmapexp=expm0:D0!M;exp=:.1For2TMm0andxedt2Rthecurves7!ts125

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126CHAPTER6.GAUSS'SLEMMA.isageodesicwhosetangentvectorats=0ist.SotheexponentialmapcarriesstraightlinesthroughtheorigininTMm0intogeodesicsthroughminM:exp:t7!t:Nowthetangentvectortothelinet7!tatt=0isjustunderthestandardidenticationofthetangentspacetoavectorspacewiththevectorspaceitself.Also,thetangentvectortothecurvet7!tatt=0is,bythedenitionof.Sotakingthederivativesofbothsidesshowsthatthedierentialoftheexponentialmapistheidentity:dexp0:TTMm00!TMm0=idunderthestandardidenticationofthetangentspaceTTMm00withTMm0.Fromtheinversefunctiontheoremitfollowsthattheexponentialmapisadieomorphisminsomeneighborhoodoftheorigin.LetUbeastarshapedneighborhoodoftheorigininTMm0onwhichexpisadieomorphism,andletU:=expUbeitsimageinMundertheexponentialmap.ThenUiscalledanormalneighborhoodofm0.Byconstructionandtheuniquenesstheoremfordierentialequationsforeverym2Uthereexistsauniquegeodesicwhichjoinsm0tomandliesentirelyinU.6.2Normalcoordinates.Supposethatwechooseabasise=e1;:::;enofTMm0andlet`1;:::;`nbethedualbasis.WethengetacoordinatesystemonUdenedbyexp)]TJ/F7 6.974 Tf 6.226 0 Td[(1m=Xximeior,whatisthesame,xi=`iexp)]TJ/F7 6.974 Tf 6.227 0 Td[(1:Thesecoordinatesareknownasnormalcoordinates,orsometimesasinertialcoordinatesforthefollowingreason:Let=PaieibeanelementofUTMm0.Sinceexpt=tthecoordinatesoftaregivenbyxit=`it=t`i=tai:ThusthesecondderivativeofxitwithrespecttotvanishesandthegeodesicequationssatisedbytbecomesXij)]TJ/F10 6.974 Tf 6.227 4.113 Td[(kijtaiaj=0;8k:Inparticular,evaluatingatt=0wegetXij)]TJ/F10 6.974 Tf 6.227 4.114 Td[(kijaiaj=0;8k:

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6.3.THEEULERFIELDEANDITSIMAGEP.127Butthismustholdforallsucientlysmallvaluesoftheaiandhenceforallvaluesoftheai.Iftheconnectionhaszerotorsion,sothatthe)]TJ/F10 6.974 Tf 208.836 3.616 Td[(kijaresymmetriciniandj,thisimpliesthat)]TJ/F10 6.974 Tf 6.227 4.113 Td[(kij=0:.2Inanormalcoordinatesystem,theChristoelsymbolsofatorsionlessconnec-tionvanishattheorigin.Henceatthisonepoint,theequationsforageodesiclookliketheequationsofastraightlineintermsofthesecoordinates.ThiswasEinstein'sresolutionofMach'sproblem:Howcanthelawsofphysics-particu-larlymechanics-involverectilinearmotioninabsenceofforces,asthisdependsonthecoordinatesystem.AccordingtoEinsteinthedistributionofmatterintheuniversedeterminesthemetricwhichthendeterminestheconnectionwhichpicksouttheinertialframe.6.3TheEulereldEanditsimageP.ThemultiplicativegroupR+actsonanyvectorspace:r2R+sendsanyvectorvintorv.Wesetr=et:ThevectoreldcorrespondingEcorrespondingtotheoneparametergroupv7!etvisknownastheEuleroperator.Fromitsdenition,ifqisahomogeneouspolynomialofdegreek,thenEq=kq;anequationwhichisknownasEuler'sequation.Alsofromitsdenition,dier-entiatingthecurvet7!etvatt=0showsthatthevalueofEatanyvectorvisEv=vunderthenaturalidenticationofthetangentspaceatvofthevectorspacewiththevectorspaceitself.WewanttoconsidertheEulereldonthetangentspaceTMm0anditsrestrictiontothestarshapedneighborhoodUanditsimageundertheexpo-nentialmap,callitP.SoPisavectorelddenedonU.Sinceexpr=rwehavePexp=d dt)]TJ/F8 9.963 Tf 4.567 -8.069 Td[(expetjt=0=d dtetjt=0

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128CHAPTER6.GAUSS'SLEMMA.soPexp=_whereweareusingthedottodenotedierentiationofthegeodesicr7!rwithrespecttor.AppliedtothevectorsweobtainPs=s_s:WeclaimthatrPP=P:.3Indeed,rPPt=tr_tt_t=tdt dt_t+t2r_t_t=t_tsinceisageodesic=Pt:Sincethepointsoftheformtlloutthenormalneighborhood,weconcludethat.3holds.Supposethatwehavechosenabasise=e1;:::;enofTMm0andsothecorrespondingnormalcoordinatesx1;:::;xnonU.Eachvectoreideterminestheconstant"vectoreldonTMm0whichassignstoeachvectorthevalueeiundertheidenticationofTMm0withTMm0.Letustemporarilyintroducethenotation~eitodenotethisvectoreld.As`1;:::;`nformthedualbasis,theneachofthe`jisalinearfunctiononTMm0,andthederivativeofthefunction`jwithrespecttothevectoreld~eiisgivenby~ei`j=0;i6=j;~ei`i=1:Nowxj=`jexp)]TJ/F7 6.974 Tf 6.227 0 Td[(1soweconcludethatundertheexponentialmapthevectoreld~eiiscarriedoverinto@iintermsofthenormalcoordinates.NowE==X`iei=X`i~eiand`i=ximif=exp)]TJ/F7 6.974 Tf 6.226 0 Td[(1m.WeconcludethattheexpressionforPinnormalcoordinatesisgivenbyP=Xxi@i:.4Thusinnormalcoordinates,theexpressionforPisthesameastheexpressionfortheEuleroperatorEinlinearcoordinates.6.4Thenormalframeeld.LetEibethevectoreldobtainedfrom@im0=eibyparalleltranslationalongthet.Bytheexistenceanduniquenesstheoremfordierentialequations,

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6.5.GAUSS'LEMMA.129weknowthatEiisasmoothvectoreldonournormalneighborhoodN.BythedenitionofPwehaverPEi=0:.5Noticethatatthesinglepointm0wehaveEim0=@im0butthisequalityneednotholdatanyotherpoint.ButE=E1;:::;EmisaframeeldwhichiscovariantconstantwithrespecttoP.Wecallitthenormalframeeldassociatedtothebasise1;:::;en.WethenalsoconstructthedualframeeldwhichisalsocovariantlyconstantwithrespecttoP.Weclaimthat,remarkably,P=XixiEi:.6Indeed,thecoecientsiPofPwithrespecttotheEiaresmoothfunctionsonournormalneighborhood.Ourrstclaimisthatthesefunctionsareintermsofournormalcoordinateshomogeneousfunctionsoforderone.Toshowthisitisenough,byEuler'stheorem,toshowthattheysatisfytheequationPf=f.ButwehavePiP=rPi+irPP=iPsincerPi=0andrPP=P.SoeachoftheiPisahomogenouslinearfunctionintermsofthenormalcoordinates.ThismeansthatwecanwriteiP=PaijxjforsomeconstantsaijThusP=XijaijxjEi=Xkxk@k;Wewanttoshowthataij=ij.Bydenition,Ei=@i.Ifwewritejxj2=Xixi2wehaveXijaijxjEj=Xijxj@i+Ojxj2andalsoXijaijxjEi=P=Xjxj@j:Theonlywaythattwolinearexpressionscanagreeuptotermsquadraticorhigherisiftheyareequal.sowehaveprovedthat.6holds.6.5Gauss'lemma.NowsupposethatMisasemi-Riemannianmanifold,andristhecorrespondingLevi-Civitaconnection.Wechooseourbasise=e1;:::;enofTMm0tobeorthonormal",sothatE=E1;:::;Enisanorthonormal"frameeld.

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130CHAPTER6.GAUSS'SLEMMA.SincetheEiformanorthonormalframeateachpoint,itfollowsfrom.6thathP;Pi=Xiix2i:.7WeclaimthatwealsohavehP;@ii=ixi:.8ToprovethisobservethathP;@ii=Xjxjh@j;@ii=ixi+Ojxj2:SoitisenoughtoshowthatPhP;@ii=ixiinordertoconclude.8.Now[P;@i]=)]TJ/F11 9.963 Tf 7.749 0 Td[(@ifromtheformula.4forP,andhencerP@i=r@iP)]TJ/F11 9.963 Tf 19.925 0 Td[(@i;sincethetorsionoftheLevi-Civitaconnectionvanishes.HencePhP;@ii=hrPP;@ii+hP;rP@ii=hP;@ii+hP;r@iPi)-222(hP;@ii=1 2@ihP;Pi=1 2@iXiixi2=ixi:Inparticular,itfollowsfrom.8thathP;jxi@j)]TJ/F11 9.963 Tf 9.963 0 Td[(ixj@ii=0:.9Nowthevectoreldsjxi@j)]TJ/F11 9.963 Tf 9.963 0 Td[(ixj@icorrespond,undertheexponentialmap,tothevectoreldsj`i~ej)]TJ/F11 9.963 Tf 9.963 0 Td[(i`j~eiwhichgeneratetheoneparametergroupofrotations"intheei;ejplaneinTMm0.Theserotations,actinginthetangentspace,whenappliedtoanypoint,sweepoutthepseudo-sphere"centeredattheoriginandpassingthroughthatpoint.LetSbethepseudo-sphereinthetangentspaceTMm0passingthroughthepoint2TMm0andletp=expSbeitsimageundertheexponentialmap.Thenwecanrestateequation.9asProposition9Theradialgeodesicthroughthepointp=expisorthogonalintheRiemannmetrictothehypersurfacep.ThisresultisknownasGauss'lemma.

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6.6.MINIMIZATIONOFARCLENGTH.1316.6Minimizationofarclength.WenowspecializetotheRiemanniancasesothatSisanactualsphereintheEuclideansense.Let:[0;1]!Mbeanycurvejoiningm0toapointminthenormalneighborhood.Inparticular,forsmallvaluesoftthepointstalllieinthenormalneighborhood.Letd=jxmji.e.d2=Pixi2intermsofthenormalcoordinates.Weknowthatdisthelengthofthegeodesicemanatingfromm0andendingatmbythedenitionoftheexponentialmapandnormalcoordinates.Wewishtoshowthatlengthofd:Inotherwords,thatthegeodesicjoiningm0tomistheshortestcurvejoiningm0tom.Since=mwehavejj=d.LetTbethersttimethatjxtjd.Thatis,Tisthegreatestlowerboundofthesetofalltforwhichtdoesnotliestrictlyinsidethesphereofradiusdinnormalcoordinates.ThenTmustlieonthesurface,theimageofthesphereofradiusdundertheexponentialmap.Itisenoughtoprovethatcurve:[0;T]!Mhaslengthd,wherenowxtliesinsidethesphereofradiusdforall0t0forallt>0.Thend=ZT0djxtj dtdt:Letudenotetheunitvectoreldintheradialdirection,denedoutsidetheorigininthenormalcoordinates.Soux=1 jxjPx.Decomposethetangentvector,0tintoitscomponentalonguanditscomponent,alongtheplanespannedbythevectoreldsxi@j)]TJ/F11 9.963 Tf 9.962 0 Td[(xj@i.So0t=ctut+t:ThenZT0djxtj dtdt=ZT0ctdtZT0jctjdt:Ontheotherhand,utandtareorthogonalrelativetotheRiemannmetric,andhencek0tk2=jctj2+ktk2soj0tjjctjandhencelength=ZT0k0tkdtdaswastobeproved.

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132CHAPTER6.GAUSS'SLEMMA.

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Chapter7Specialrelativity7.1TwodimensionalLorentztransformations.Westudyatwodimensionalvectorspacewithscalarproducth;iofsignature+)]TJ/F8 9.963 Tf 7.749 0 Td[(.ALorentztransformationisalineartransformationwhichpreservesthescalarproduct.Inparticularitpreservesjjujj2:=hu;uiwherewiththeusualabuseofnotationthisexpressioncanbepositivenegativeorzero.Inparticular,everysuchtransformationmustpreservethelightcone"consistingofalluwithjjujj2=0.Allsuchtwodimensionalspacesareisomorphic.Inparticular,wecanchooseourvectorspacetobeR2withmetricgivenbyjjuvjj2=uv:Thelightconeconsistsofthecoordinateaxes,soeveryLorentztransformationmustcarrytheaxesintothemselvesorinterchangetheaxes.Atransforma-tionwhichpreservestheaxesisjustadiagonalmatrix.HencetheconnectedcomponentoftheLorentzgroupconsistsofallmatricesoftheformr00r)]TJ/F7 6.974 Tf 6.226 0 Td[(1;r>0:Sothegroupisisomorphictothemultiplicativegroupofthepositiverealnum-133

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134CHAPTER7.SPECIALRELATIVITYbers.Weintroducet;xcoordinatesbyu=t+xv=t)]TJ/F11 9.963 Tf 9.963 0 Td[(xoruv=111)]TJ/F8 9.963 Tf 7.749 0 Td[(1txsojjtxjj2=t2)]TJ/F11 9.963 Tf 9.963 0 Td[(x2:Noticethat111)]TJ/F8 9.963 Tf 7.749 0 Td[(12=21001;soifuv=111)]TJ/F8 9.963 Tf 7.749 0 Td[(1txu0v0=r00r)]TJ/F7 6.974 Tf 6.226 0 Td[(1uvu0v0=111)]TJ/F8 9.963 Tf 7.749 0 Td[(1t0x0thent0x0=1 2111)]TJ/F8 9.963 Tf 7.749 0 Td[(1r00r)]TJ/F7 6.974 Tf 6.227 0 Td[(1111)]TJ/F8 9.963 Tf 7.749 0 Td[(1txMultiplyingoutthematricesgivest0x0=1ww1tx.1where:=r+r)]TJ/F7 6.974 Tf 6.227 0 Td[(1 2.2w:=r)]TJ/F11 9.963 Tf 9.963 0 Td[(r)]TJ/F7 6.974 Tf 6.227 0 Td[(1 r+r)]TJ/F7 6.974 Tf 6.227 0 Td[(1:.3Theparameterwiscalledthevelocity"andis,ofcourse,restrictedbyjwj<1:.4Wehave1)]TJ/F11 9.963 Tf 9.963 0 Td[(w2=r2+2+r)]TJ/F7 6.974 Tf 6.226 0 Td[(2)]TJ/F11 9.963 Tf 9.962 0 Td[(r2+2)]TJ/F11 9.963 Tf 9.962 0 Td[(r)]TJ/F7 6.974 Tf 6.226 0 Td[(2 r+r)]TJ/F7 6.974 Tf 6.227 0 Td[(12=4 r+r)]TJ/F7 6.974 Tf 6.226 0 Td[(12

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7.1.TWODIMENSIONALLORENTZTRANSFORMATIONS.135so=1 p 1)]TJ/F11 9.963 Tf 9.963 0 Td[(w2:.5Thuswdetermines.Similarly,wecanrecoverrfromw:r=r 1+w 1)]TJ/F11 9.963 Tf 9.963 0 Td[(w:SowecanusewtoparameterizetheLorentztransformations.WewriteLw:=1ww17.1.1Additionlawforvelocities.Itisusefultoexpressthemultiplicationlawintermsofthevelocityparameter.Ifw1=r)]TJ/F11 9.963 Tf 9.963 0 Td[(r)]TJ/F7 6.974 Tf 6.227 0 Td[(1 r+r)]TJ/F7 6.974 Tf 6.227 0 Td[(1w2=s)]TJ/F11 9.963 Tf 9.962 0 Td[(s)]TJ/F7 6.974 Tf 6.227 0 Td[(1 s+s)]TJ/F7 6.974 Tf 6.227 0 Td[(1thenrs)]TJ/F8 9.963 Tf 9.962 0 Td[(rs)]TJ/F7 6.974 Tf 6.226 0 Td[(1 rs+rs)]TJ/F7 6.974 Tf 6.226 0 Td[(1=r)]TJ/F10 6.974 Tf 6.227 0 Td[(r)]TJ/F6 4.981 Tf 5.397 0 Td[(1 r+r)]TJ/F6 4.981 Tf 5.396 0 Td[(1+s)]TJ/F10 6.974 Tf 6.226 0 Td[(s)]TJ/F6 4.981 Tf 5.396 0 Td[(1 s+s)]TJ/F6 4.981 Tf 5.397 0 Td[(1 1+s)]TJ/F10 6.974 Tf 6.227 0 Td[(s)]TJ/F6 4.981 Tf 5.397 0 Td[(1 s+s)]TJ/F6 4.981 Tf 5.396 0 Td[(1r)]TJ/F10 6.974 Tf 6.226 0 Td[(r)]TJ/F6 4.981 Tf 5.396 0 Td[(1 r+r)]TJ/F6 4.981 Tf 5.396 0 Td[(1soweobtainLw1Lw2=Lwwherew=w1+w2 1+w1w2:.6Thisisknowastheadditionlawforvelocities".7.1.2Hyperbolicangle.Onealsointroducesthehyperbolicangle",actuallyarealnumber,byr=eso=cosh=1 p 1)]TJ/F11 9.963 Tf 9.963 0 Td[(w2andLw=coshsinhsinhcosh:Herew=tanh:

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136CHAPTER7.SPECIALRELATIVITYForanytxwitht>0andt2)]TJ/F11 9.963 Tf 10.912 0 Td[(x2=1,wemusthavet>xandt)]TJ/F11 9.963 Tf 9.963 0 Td[(x=t+x)]TJ/F7 6.974 Tf 6.226 0 Td[(1sot+xt)]TJ/F11 9.963 Tf 9.963 0 Td[(x=r00r)]TJ/F7 6.974 Tf 6.226 0 Td[(111r=t+x:ThisshowsthatthegroupofallonedimensionalproperLorentztransfor-mations,fLwg,actssimplytransitivelyonthehyperbolajjtxjj2=1;t>0:Thismeansthatiftxandt0x0aretwopointsonthishyperbola,thereisauniqueLwwithLwtx=t0x0::Iftx=Lz10thismeansthatt0x0=LwLz10=LzLw10andsohtx;t0x0i=tt0)]TJ/F11 9.963 Tf 9.963 0 Td[(xx0=h10;Lw10i:Writingw=tanhasabovewehavehu;u0i=cosh;u=txu0=t0x0;andiscalledthehyperbolicanglebetweenuandu0.Moregenerally,ifwedon'trequirejjujj=jju0jj=1butmerelyjjujj>0;jju0jj>0;t>0;t0>0wedenethehyperbolicanglebetweenthemtobethehyperbolicanglebetweenthecorrespondingunitvectorssohu;u0i=jjujjjju0jjcosh:7.1.3Propertime.Amaterialparticleisacurve::7!whosetangentvector0haspositivetcoordinateeverywhereandsatisesjj0jj1:

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7.1.TWODIMENSIONALLORENTZTRANSFORMATIONS.137Ofcourse,thisxestheparameteruptoanadditiveconstant.iscalledthepropertimeofthematerialparticle.Itistobethoughtofasastheinternalclock"ofthematerialparticle.Foranunstableparticle,forexample,itisthisinternalclockwhichtellstheparticlethatitstimeisup.Let@0denoteunitvectorinthetdirection,@0:=10:7.1.4Timedilatation.Letuswritetforthetcoordinateofandxforitsxcoordinatesothat=tx0:=d d=dt=ddxd:Wehavedt d=h@0;0i;=cosh=1 p 1)]TJ/F11 9.963 Tf 9.963 0 Td[(w21wherew:=dx dt=dx=d dt=d.7isthevelocity"oftheparticlemeasuredinthet;xcoordinatesystem.Thustheinternalclockofamovingparticleappearstorunslowinanycoordinatesystemwhereitisnotatrest.Thisphenomenon,knownastimedilatation"isobservedallthetimeinelementaryparticlephysics.Forexample,fastmovingmuonsmakeitfromtheupperatmospheretothegroundbeforedecayingduetothiseect.7.1.5Lorentz-Fitzgeraldcontraction.Letandbematerialparticleswhosetrajectoriesareparallelstraightlines.OncewehavechosenaMinkowskibasis,wehaveanotionofsimultaneity"relativetothatbasis,meaningthatwecanadjustthearbitraryadditiveconstantinthedenitionofthepropertimeofeachparticlesothatthetwoparallelstraightlinesaregivenby7!ab+c;and7!ab+c+`:Wecanthenthinkofthecongurationasthemotionoftheendpointsofarigidrod"oflength`.Thelength`dependsonournotionofsimultaneity.Forexample,supposeweapplyaLorentztransformationLwtoobtaina=1;b=0

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138CHAPTER7.SPECIALRELATIVITYandreadjusttheadditiveconstantsintheclockstoachievesimultaneity.Thecorrespondingframeiscalledtherestframeoftherodandtheitslength,`rest,calledtherestlengthoftherodisrelatedtoourlaboratoryframe"by`rest=cosh`labor`lab=p 1)]TJ/F11 9.963 Tf 9.963 0 Td[(w2`rest;.8amovingobjectcontracts"inthedirectionofitsmotion.ThisistheLorentz-Fitzgeraldcontractionwhichwasdiscoveredbeforespecialrelativityinthecon-textofelectromagnetictheory,andcanbeconsideredasaforerunnerofspecialrelativity.Asaneectinthelaboratory,itisnotnearlyasimportantastimedilatation.7.1.6Thereversetriangleinequality.Consideranyinterval,say[0;T],onthetaxis,andlet0
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7.1.TWODIMENSIONALLORENTZTRANSFORMATIONS.139t20onthetaxiswhicharejoinedtothepointtxbylightrayslinesparalleltot=xort=)]TJ/F11 9.963 Tf 7.749 0 Td[(x.Thenassumingt2>t>t1t)]TJ/F11 9.963 Tf 9.962 0 Td[(t1=xsot1=t)]TJ/F11 9.963 Tf 9.963 0 Td[(xandt2)]TJ/F11 9.963 Tf 9.963 0 Td[(t=xsot2=t+xhencet1t2=t2)]TJ/F11 9.963 Tf 9.963 0 Td[(x2:.10Thisequationhasthefollowingsignicance:PointP=00atrestorinuniformmotionwishestocommunicatewithpointQ=tx.Itrecordsthetime,t1onitsclockwhenalightsignalwassenttoQandthetimet2whentheanswerwasreceivedassuminganinstantaneousresponse.Eventhoughtheindividualtimesdependonthecoordinates,theirproduct,t1t2givesthesquareoftheMinkowskinormofthevectorjoiningPtoQ.7.1.8Energy-momentumInclassicalmechanics,amomentumvectorisusuallyconsideredtobeanele-mentofthecotangentspace,i.ethedualspacetothetangentspace.Thusinoursituation,whereweidentifyalltangentspaceswiththeMinkowskiplaneitself,amomentum"vectorwillbearowvectoroftheform=E;p.Foramaterialparticletheassociatedmomentumvector,calledtheenergymo-mentumvector"inspecialrelativity,isarowvectorwiththepropertythattheevaluationmapv7!vforanyvectorvisapositivemultipleofthescalarproductevaluationv7!hv;0i:Inotherwords,evaluationunderisthesameasscalarproductwithm0wherem,isaninvariantofthematerialparticleknownastherestmass.Therestmassisaninvariantoftheparticleinquestion,constantthroughoutitsmotion.Sointherestframeoftheparticle,where0=@0,theenergymomentumvectorhastheformm;0.Heremisidentieduptoachoiceofunits,andwewillhavemoretosayaboutunitslaterwiththeusualnotionofmass,asdeterminedbycollisionexperiments,forexample.Inageneralframewewillhave=E;p;E2)]TJ/F11 9.963 Tf 9.962 0 Td[(p2=m2:.11Inthisframewehavep=E=w.12

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140CHAPTER7.SPECIALRELATIVITYwherewisthevelocityasdenedin.7.Wecansolveequations.11and.12toobtainE=m p 1)]TJ/F11 9.963 Tf 9.963 0 Td[(w2.13p=mw p 1)]TJ/F11 9.963 Tf 9.963 0 Td[(w2:.14ForsmallvaluesofwwehavetheTaylorexpansion1 p 1)]TJ/F11 9.963 Tf 9.962 0 Td[(w2=1+1 2w2+andsowehaveE:=m+1 2mw2+.15p:=mw+1 2mw3+:.16Thersttermin.16looksliketheclassicalexpressionp=mwforthemomentumintermsofthevelocityifwethinkofmastheclassicalmass,andthesecondtermin.15looksliketheclassicalexpressionforthekineticenergy.Wearethusledtothefollowingmodicationoftheclassicaldenitionsofenergyandmomentum.Associatedtoanyobjectthereisadenitevalueofmcalleditsrestmass.Iftheobjectisatrestinagivenframe,itsrestmasscoincideswiththeclassicalnotionofmass;whenitisinmotionrelativetoagivenframe,itsenergymomentumvectorisoftheformE;pwhereEandparedeterminedbyequations7.13and7.14.Wehavebeenimplicitlyassumingthatm>0whichimpliesthatjwj<1.Wecansupplementtheseparticlesbyparticlesofrestmass0whoseenergymomentumvectorsatisfy7.11,sohavetheformE;E.Thesecorrespondtoparticleswhichmovealonglightraysx=t.Thelawofconservationofenergymomentumsaysthatinanycollisionthetotalenergymomentumvectorisconserved.7.1.9Psychologicalunits.OurdescriptionoftwodimensionalMinkowskigeometryhasbeenintermsofnaturalunits"wherethespeedoflightisone.Pointsinourtwodimensionalspacetimearecalledevents.Theyrecordwhenandwheresomethinghappens.Ifwerecordthetotaleventsofasinglehumanconsciousnesssayroughly70yearsmeasuredinsecondsandseveralthousandmetersmeasuredinseconds,wegetasetofeventswhichisenormouslystretchedoutinoneparticulartimedirectioncomparedtospacedirection,byafactorofsomethinglike1018.Beingveryskinnyinthespacedirectionasopposedtothetimedirectionwetendtohaveapreferredsplittingofspacetimewithspaceandtimedirectionspickedout,andtomeasuredistancesinspacewithmuchsmallerunits,suchasmeters,thantheunitsweusesuchassecondstomeasuretime.Ofcourse,ifweusea

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7.1.TWODIMENSIONALLORENTZTRANSFORMATIONS.141smallunit,thecorrespondingnumericalvalueofthemeasurementwillbelarge;intermsofhumanorordinaryunits"spacedistanceswillbegreatlymagniedincomparisonwithtimedierences.ThissuggeststhatweconsidervariablesTandXrelatedtothenaturalunitstandxbyT=c)]TJ/F7 6.974 Tf 6.227 0 Td[(1t;X=xorTX=c)]TJ/F7 6.974 Tf 6.227 0 Td[(1001tx:Thelightconejtj=jxjgoesovertojXj=cjTjandwesaythatthespeedoflightiscinordinaryunits".Similarly,thetime-likehyperbolast2)]TJ/F11 9.963 Tf 9.527 0 Td[(x2=k>0becomeveryattenedoutandarealmosttheverticallinesT=const.,linesofsimultaneity".TondtheexpressionfortheLorentztransformationsinordinaryunits,wemustconjugatetheLorentztransformation,L,bythematrixc)]TJ/F7 6.974 Tf 6.226 0 Td[(1001soM=c)]TJ/F7 6.974 Tf 6.227 0 Td[(1001Lc001=coshc)]TJ/F7 6.974 Tf 6.227 0 Td[(1sinhcsinhcosh=1c)]TJ/F7 6.974 Tf 6.226 0 Td[(1wcw1;whereL=Lw.Ofcoursewisapurenumberinnaturalunits.Inpsychologicalunitswemustwritew=v=c,theratioofavelocityinunitslikemeterspersecondtothespeedoflight.ThenM=Mv=1v c2v1;=1 )]TJ/F10 6.974 Tf 11.158 3.923 Td[(v2 c21=2:.17SincewehavepassedtonewcoordinatesinwhichjjTXjj2=c2T2)]TJ/F11 9.963 Tf 9.963 0 Td[(X2;thecorrespondingmetricinthedualspacewillhavetheenergycomponentdividedbyc.Aswehaveusedcapforenergyandlowercaseformomentum,weshallcontinuetodenotetheenergymomentumvectorinpsychologicalunitsbyE;pandwehavejjE;pjj2=E2 c2)]TJ/F11 9.963 Tf 9.963 0 Td[(p2:Westillmustseehowtheseunitsrelatetoourconventionalunitsofmass.Forthis,observethatwewantthesecondtermin7.15tolooklikekineticenergywhenEisreplacedbyE=c,sowemustrescalebym7!mc.ThuswegetjjE;pjj2=E2 c2)]TJ/F11 9.963 Tf 9.962 0 Td[(p2=m2c2:.18

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142CHAPTER7.SPECIALRELATIVITYSoinpsychologicalcoordinateswerewrite7.11-.15as.18togetherwithp E=v c2.19E=mc2 )]TJ/F11 9.963 Tf 9.962 0 Td[(v2=c21=2.20p=mv )]TJ/F11 9.963 Tf 9.962 0 Td[(v2=c21=2.21E:=mc2+1 2mv2+.22p:=mv+1 2mv3 c2+:.23OfcourseatvelocityzerowegetthefamousEinsteinformulaE=mc2.7.1.10TheGalileanlimit.Inthelimit"c!1thetransformationsMvbecomeGv=10v1whichpreserveTandsendX7!X+vT.TheseareknownasGalileantrans-formations.Theysatisfythemorefamiliaradditionruleforvelocities:Gv1Gv2=Gv1+v2:7.2Minkowskispace.Sinceoureverydayspaceisthreedimensional,thecorrectspaceforspecialrelativityisafourdimensionalLorentzianvectorspace.ThiskeyideaisduetoMinkowski.InafamouslectureatCologneinSeptember1908hesaysHenceforthspacebyitself,andtimebyitselfaredoomedtofadeawayintomereshadows,andonlyakindofunionofthetwowillpreserveanindependentreality.Muchofwhatwedidinthetwodimensionalcasegoesoverunchangedtofourdimensions.Ofcourse,velocity,worv,becomevectors,wandvasdoesmomentum,pinsteadofp.Soinanyexpressionatermsuchasv2mustbereplacedbyjjvjj2,thethreedimensionalnormsquared,etc..Withthismodicationthekeyformulasoftheprecedingsectiongothrough.Wewillnotrewritethem.Thereversetriangleinequalityandsothetwineectgothroughunchanged.Ofcoursethereareimportantdierences:thelightconeisreallyacone,andnottwolightrays,thespace-likevectorsformaconnectedset,theLorentzgroupistendimensionalinsteadofonedimensional.WewillstudytheLorentz

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7.2.MINKOWSKISPACE.143groupinfourdimensionsinalatersection.Inthissectionwewillconcentrateontwo-particlecollisions,wheretherelativeanglebetweenthemomentagivesanadditionalingredientinfourdimensions.7.2.1TheComptoneect.Weconsideraphotonaparticle"ofmasszeroimpingingonamassiveparticlesayanelectronatrest.Afterthecollisionthethephotonmovesatanangle,,toitsoriginalpath.Thefrequencyofthelightischangedasafunctionoftheangle:Ifistheincomingwavelengthand0thewavelengthofthescatteredlightthen0=+h mc)]TJ/F8 9.963 Tf 9.962 0 Td[(cos;.24wherehisPlanck'sconstantandmisthemassofthetargetparticle.Theexpressionh mcisknownastheComptonwavelengthofaparticleofmassm.Comptonderived.24fromtheconservationofenergymomentumasfol-lows:Wewillworkinnaturalunitswherec=1.AssumeEinstein'sformulaEphoton=h.25fortheenergyofthephoton,whereisthefrequency,orequivalently,Ephoton=h .26whereisthewavelength.Workintherestframeofthetargetparticle,soitsenergymomentumvectorism;0;0;0.Takethex)]TJ/F8 9.963 Tf 7.749 0 Td[(axistobethedirectionoftheincomingphoton,soitsenergymomentumvectorish ;h ;0;0.Assumethatthecollisioniselasticsothattheoutgoingphotonstillhasmasszeroandtherecoilingparticlestillhasmassm.Choosethey)]TJ/F8 9.963 Tf 7.749 0 Td[(axissothattheoutgoingphotonandtherecoilingparticlemoveinthex;yplane.Thentheoutgoingphotonhasenergymomentumh 0;h 0cos;h 0sin;0whiletherecoilingparti-clehasenergymomentumE;px;py;0andconservationofenergymomentumtogetherwiththeassumedelasticityofthecollisionyieldh +m=h 0+Eh =h 0cos+px0=h 0sin+pym2=E2)]TJ/F11 9.963 Tf 9.962 0 Td[(p2x)]TJ/F11 9.963 Tf 9.962 0 Td[(p2y:SubstitutingthesecondandthirdequationsintothelastgivesE2=m2+h2 2+h2 02)]TJ/F8 9.963 Tf 9.963 0 Td[(2h2 0cos

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144CHAPTER7.SPECIALRELATIVITYwhiletherstequationyieldsE2=m2+h2 2+h2 02+2mh )]TJ/F11 9.963 Tf 9.963 0 Td[(mh 0)]TJ/F11 9.963 Tf 13.262 6.739 Td[(h2 0:ComparingthesetwoequationsgivesCompton'sformula,.24.NoticethatCompton'sformulamakesthreestartlingpredictions:thattheshiftinwavelengthisindependentofthewavelengthoftheincomingradiation,theexplicitnatureofthedependenceofthisshiftonthescatteringangle,andanexperimentaldeterminationofh=mc,inparticular,ifhandcareknown,ofthemass,m,ofthescatteringparticle.TheseweretheresultsofCompton'sexperiment.ItisworthrecallingthehistoricalimportanceofCompton'sexperiment.Attheendofthenineteenthcentury,statisticalmechanics,whichhadbeenenormouslysuccessfulinexplainingmanyaspectsofthermodynamics,yieldedwrong,andevennon-sensical,predictionswhenitcametothestudyoftheelectromagneticradiationemittedbyahotbody-thestudyofblackbodyradiation".In1900Planckshowedthattheparadoxescouldberesolvedandaanexcellentttotheexperimentaldataachievedifoneassumedthattheelectromagneticradiationisemittedinpacketsofenergygivenby.25wherehisaconstant,nowcalledPlanck'sconstant,withvalueh=6:2610)]TJ/F7 6.974 Tf 6.226 0 Td[(27ergs:ForPlanck,thisquantizationoftheenergyofradiationwasapropertyoftheemissionprocessinblackbodyradiation.In1905Einsteinproposedtheradicalviewthat.25wasapropertyoftheelectromagneticelditself,andnotofanyparticularemissionprocess.Light,accordingtoEinstein,isquantizedaccordingto.25.Heusedthistoexplainthephotoelectriceect:Whenlightstrikesametallicsurface,electronsareemitted.AccordingtoEinstein,anincominglightquantumofenergyhstrikesanelectroninthemetal,givingupallitsenergytotheelectron,whichthenusesupacertainamountofenergy,w,toescapefromthesurface.Theelectronmayalsouseupsomeenergytoreachthesurface.Inanyevent,theescapingelectronhasenergyEh)]TJ/F11 9.963 Tf 9.963 0 Td[(wwherewisanempiricalpropertyofthematerial.Thestartlingconsequencehereisthatthemaximumenergyoftheemittedelectrondependsonlyonthefrequencyoftheradiation,butnotontheintensityofthelightbeam.Increas-ingtheintensitywillincreasethenumberofelectronsemitted,butnottheirmaximumenergy.Einstein'stheorywasrejectedbytheentirephysicscom-munity.WiththetemporaryexceptionofStarkwholaterbecameaviciousnaziandattackedthetheoryofrelativityasaJewishplotphysicistscouldnotaccepttheideaofacorpuscularnaturetolight,forthisseemedtocontradictthewellestablishedinterferencephenomenawhichimpliedawavetheory,andalsocontradictedMaxwell'sequations,whichwerethecornerstoneofallofthe-oreticalphysics.Foratypicalview,letusquoteatlengthfromMillikanofoil

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7.2.MINKOWSKISPACE.145dropfamewhoseexperimentalresultgavethebestconrmationofEinstein'spredictionsforthephotoelectriceect.InhisNobellecturehewritesAftertenyearsoftestingandchangingandlearningandsometimesblundering,alleortsbeingdirectedfromthersttowardtheaccu-rateexperimentalmeasurementoftheenergiesofemissionofphoto-electrons,nowasafunctionoftemperature,nowofwavelength,nowofmaterialcontacte.m.f.relations,thisworkresulted,contrarytomyownexpectation,intherstdirectexperimentalproofin1914oftheexactvalidity,withinnarrowlimitsofexperimentalerror,oftheEinsteinequation,andtherstdirectphotoelectricdeterminationofPlanck'sh.ButdespiteMillikan'sownexperimentalvericationofEinstein'sformulaforthephotoelectriceect,hedidnotregardthisasconrmationofEinstein'stheoryofquantizedradiation.Onthecontrary,inhispaper,AdirectPhotoelectricDeterminationofPlanck'sh"Phy.Rev.7355-388wherehepresentshisexperimentalresultshewrites:...thesemi-corpusculartheorybywhichEinsteinarrivedathisequationseemsatpresenttowhollyuntenable....[Einstein's]bold,nottosayreckless[hypothesis]seemsaviolationoftheverycon-ceptionofelectromagneticdisturbance...[it]iesinthefaceofthethoroughlyestablishedfactsofinterference....Despite...theappar-entlycompletesuccessoftheEinsteinequation,thephysicaltheoryofwhichitwasdesignedtobethesymbolicexpressionisfoundsountenablethatEinsteinhimself,Ibelieve,nolongerholdstoit,andweareinthepositionofhavingbuiltaperfectstructureandthenknockedoutentirelytheunderpinningwithoutcausingthebuildingtofall.Itstandscompleteandapparentlywelltested,butwithoutanyvisiblemeansofsupport.Thesesupportsmustobviouslyexist,andthemostfascinatingproblemofmodernphysicsistondthem.Experimenthasoutruntheory,or,better,guidedbyanerroneoustheory,ithasdiscoveredrelationshipswhichseemtobeofthegreat-estinterestandimportance,butthereasonsforthemareasyetnotatallunderstood.Ofcourse,MillikanwasmistakenwhenhewrotethatEinsteinhimselfhadabandonedhisowntheory.Infact,Einsteinextendedhistheoryin1916toincludethequantizationofthemomentumofthephoton.ButforMillikan,asformostphysicists,Einstein'shypothesisofthelightquantumwasclearlyanerroneoustheory".Bytheway,itisamusingtocompareMillikan'sactualstateofmindin1916whichwastheacceptedviewoftheentirephysicscommunityoutsideofEinsteinwithhisfallaciousaccountofitinhisautobiography1950pp.100-101,wherehewritesabouthisexperimentalvericationofEinstein'sequationforthephotoelectriceect:

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146CHAPTER7.SPECIALRELATIVITYThisseemedtome,asitdidtomanyothers,amatterofverygreatimportance,foritrenderedwhatIwillcallPlanck's1912explosiveortriggerapproachtotheproblemofquantacompletelyuntenableandprovedsimplyandirrefutedly,Ithought,thattheemittedelectronthatescapeswiththeenergyhgetsthatenergybythedirecttransferofhunitsofenergyfromthelighttotheelectronandhencescarcelypermitsofanyinterpretationthanthatwhichEinsteinhadoriginallysuggested,namelythatofthesemi-corpuscularorphotontheoryoflightitself.Self-delusionoroutrightmendacity?IngeneralIhavefoundthatonecannottrusttheaccountsgivenbyscientistsoftheirownthoughtprocesses,especiallythosegivenmanyyearsaftertheevents.Inanyevent,itwasonlywiththeComptonexperiment,thatEinstein'sformula,.25wasacceptedasapropertyoflightitself.ForadetailedhistoryseethebookTheComptonEectbyRogerH.Stuewer,ScienceHistoryPublications,NewYork1975,fromwhichIhavetakentheabovequotes.7.2.2NaturalUnits.InthissectionIwillmaketheparadoxicalargumentthatPlanck'sconstantand.25haveapurelyclassicalinterpretation:Likec,Planck'sconstant,h,maybeviewedasaconversionfactorfromnaturalunitstoconventionalunits.ForthisIwillagainbrieycallonahighertheory,symplecticgeometry.Inthattheory,conservedquantitiesareassociatedtocontinuoussymmetries.Moreprecisely,ifGisaLiegroupofsymmetrieswithLiealgebrag,themomentmap,foraHamiltonianactiontakesvaluesing,thedualspaceoftheLiealgebra.Abasisofgdeterminesadualbasisofg.Inthecaseathand,theLiealgebrainquestionisthealgebraoftranslation,andthemomentmapyieldsthetotalenergy-momentumvector.Henceifwemeasuretranslationsinunitsoflength,thenthecorrespondingunitsforenergymomentumshouldbeinverselength.InthissensetheroleofPlanck'sconstantin.26isaconversionfactorfromnaturalunitsofinverselengthtotheconventionalunitsofenergy.Soweinterpreth=6.62610)]TJ/F7 6.974 Tf 6.227 0 Td[(27ergsastheconversionfactorfromthenaturalunitsofinversesecondstotheconventionalunitsofergs.Inordertoemphasizethispoint,letusengageinsomehistoricalsciencection:Supposethatmechanicshaddevelopedbeforetheinventionofclocks.Sowecouldobservetrajectoriesofparticles,theircollisionsanddeections,butnottheirvelocities.Forinstance,wemightbeabletoobservetracksinabubblechamberoronaphotographicplate.Ifourtheoryisinvariantunderthegroupoftranslationsinspace,thenlinearmomentumwouldbeaninvariantoftheparticle;ifourtheoryisinvariantunderthegroupofthreedimensionalEuclideanmotions,thesymplecticgeometrytellsthatjjpjj,thelengthofthelinearmomentumisaninvariantoftheparticle.Intheabsenceofanotionofvelocity,wemightnotbeabletodistinguishbetweenaheavyparticlemoving

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7.2.MINKOWSKISPACE.147slowlyoralightparticlemovingfast.Withoutsomewayofrelatingmomentumtolength,wewouldintroduceindependentunits"ofmomentum,perhapsbycombiningparticlesinvariouswaysandbyperformingcollisionexperiments.Butsymplecticgeometrytellsusthatthenatural"unitsofmomentumshouldbeinverselength,andthatdeBroglie'sequationjjpjj=h .27givesPlanck'sconstantasaconversionfactorfromnaturalunitstoconventionalunits.Infact,thecrucialexperimentwasthephoto-electriceect,carriedoutindetailbyMillikan.Theabovediscussiondoesnotdiminish,eveninretrospect,fromtheradicalcharacterofEinstein's1905proposal.Evenintermsofnaturalunits"thestartlingproposalisthatitisasingleparticle,thephoton,whichinteractswithasingleparticle,theelectrontoproducethephotoelectriceect.Itisthiscorpuscular"picturewhichwassodiculttoaccept.Furthermore,itisaboldhypothesistoidentifythenaturalunits"ofthephotonmomentumwiththeinversewavelength.Forreasonsofconveniencephysicistsfrequentlyprefertouse~:=h=2astheconversionfactor.Onewayofchoosingnaturalunitsistopicksomeparticularparticleanduseitsmassasthemassunit.Supposewepicktheproton.ThenmP,themassoftheprotonisthebasicunitofmass,and`P,theComptonwavelengthoftheprotonisthebasicunitoflength.AlsotP,thetimeittakesforlighttotravelthedistanceofoneComptonwavelength,isthebasicunitoftime.Theconversionfactorstothecgssystemusing~are:mP=1:67210)]TJ/F7 6.974 Tf 6.227 0 Td[(24g`P=:21110)]TJ/F7 6.974 Tf 6.226 0 Td[(13cmtP=0:0710)]TJ/F7 6.974 Tf 6.226 0 Td[(23sec:Wewilloscillatebetweenusingnaturalunitsandfamiliarunits.Usually,wewillderivetheformulaswewantinnaturalunits,wherethecomputationsarecleanerandthenstatetheresultsinconventionalunitswhichareusedinthelaboratory.7.2.3Two-particleinvariants.SupposethatAandBareparticleswithenergymomentumvectorspAandpB.InanyparticularframetheyhavetheexpressionpA=EA;pAandpB=EB;pB.Wehavethethreeinvariantsp2A=m2A=E2A)]TJ/F8 9.963 Tf 9.962 0 Td[(pA;pAp2B=m2B=E2B)]TJ/F8 9.963 Tf 9.963 0 Td[(pB;pBpapB=EAEB)]TJ/F8 9.963 Tf 9.963 0 Td[(pA;pB:

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148CHAPTER7.SPECIALRELATIVITYForthepurposeofthissectionournotationisthat;referstothethreedimensionalscalarproduct,asymbolsuchaspAdenotestheenergymomentumfourvectorofparticleA,pApBdenotesthefourdimensionalscalarproductandwewritep2AforpApA.Theseareallstandardnotations.Thelefthandsidesareallinvariantsinthesensethattheircomputationdoesnotdependonthechoiceofframe.Manycomputationsbecometransparentbychoosingaframeinwhichsomeoftheexpressionsontherighttakeonaparticularlysimpleform.Itisintuitivelyobviousandalsoatheoremthatthesearetheonlyinvariants-thatanyotherinvariantexpressioninvolvingthetwomomentavectorsmustbeafunctionofthesethree.Forexample,pA+pB2=p2A+2pApB+p2BandpA)]TJ/F11 9.963 Tf 9.963 0 Td[(pB2=p2A)]TJ/F8 9.963 Tf 9.962 0 Td[(2pApB+p2B:Herearesomeexamples:Decayatrest.ParticleA,atrest,decaysintoparticlesBandC,symbolicallyA!B+C.Findtheenergies,andthemagnitudesofthemomentaandvelocitiesoftheoutgoingparticlesintherestframeofparticleA.ConservationofenergymomentumgivespA=pB+pCorpC=pA)]TJ/F11 9.963 Tf 9.963 0 Td[(pB;sop2C=p2A+p2B)]TJ/F8 9.963 Tf 9.963 0 Td[(2pApBorm2C=m2A+m2B)]TJ/F8 9.963 Tf 9.963 0 Td[(2mAEBsincepA=mA;0;0;0:SolvinggivesEB=m2A+m2B)]TJ/F11 9.963 Tf 9.963 0 Td[(m2C 2mA:InterchangingBandCgivestheformulaforEC.WehavepB=)]TJ/F48 9.963 Tf 7.749 0 Td[(pCandE2B)-222(jjpBjj2=m2B.SubstitutingintotheaboveexpressionforEBgivesjjpBjj2=1 4m2A)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(m4A+m4B+m4C)]TJ/F8 9.963 Tf 9.963 0 Td[(2m2Am2C)]TJ/F8 9.963 Tf 9.963 0 Td[(2m2Bm2C+2m2Am2B)]TJ/F8 9.963 Tf 9.963 0 Td[(4m2Am2BsojjpBjj=jjpCjj=p m2A;m2B;m2C 2mA.28whereisthetrianglefunction"x;y;z:=x2+y2+z2)]TJ/F8 9.963 Tf 9.963 0 Td[(2xy)]TJ/F8 9.963 Tf 9.963 0 Td[(2xz)]TJ/F8 9.963 Tf 9.963 0 Td[(2yz:.29

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7.2.MINKOWSKISPACE.149Ifwenowredothecomputationinordinaryunitskeepingtrackofdividingbyc2intheenergypartofthescalarproductandmultiplyingallm0sbycwegetEB=m2A+m2B)]TJ/F11 9.963 Tf 9.963 0 Td[(m2C 2mAc2.30jjpBjj=jjpCjj=p m2A;m2B;m2C 2mAc:.31Equation.12becomesv=c2 Ep:.32Takingthemagnitudesandusingtheaboveformulasfortheenergiesandmag-nitudesofthemomentagivethemagnitudesofthevelocitiesoftheoutgoingparticles.Sinceourenergiesareallnon-negative,.30showsthatdecayfromrestcannotoccurunlessmAmB+mC.SimilarlytheexpressionforjjpBjjwouldbecomeimaginaryifmA
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150CHAPTER7.SPECIALRELATIVITYatrest.Thetotalenergymomentumvector,callitq,forthefourparticleswillthenbeq=m;0;0;0inthecenterofmomentumsystem,henceq2=m2andsop2tot=q2impliesE+m2)-222(jjpjj2=4m2:ButE2)-222(jjpjj2=m2soweget2mE+m2=15m2orE=7m:InordinaryunitswewouldwritethisasE=7mc2:NowE=mc2+kineticenergy+soapproximately6mc2ofkineticenergymustbesupplied.Ontheotherhand,ifweshoottwobeamsofparticlesoftypeAagainstoneanother,thenforthecollision,thelaboratoryframeandthecenterofmomentumframecoincide,andtheincomingtotalenergymomentumvectorisE;0;0;0andourconservationequationbecomes4E=4m.Wethusmustsupplykineticenergyequaltoaboutmtoeachparticle,oratotalenergyofabout2mc2inordinaryunits.Comparingthetwoexperimentsweseethatthecollidingbeamexperimentismoreenergyecientbyafactorofthree.Todayvirtuallyallnewmachinesforcollisionexperimentsarecollidersforthisreason.7.2.4Mandlestamvariables.Weconsideratwobodyscatteringeventwithatwobodyoutcome,soA+B!C+D:Boththeincomingandtheoutgoingparticlescanexistinvariousstates,anditistheroleofanyquantummechanicaltheorytoyieldaprobabilityampli-tudeforapairofincomingstatestoscatterintoapairofoutgoingstates.Ingeneral,thestatesarecharacterizedbyvariousinternal"parameterssuchasspin,isospinetc.,inadditiontotheirmomentum.Howeverweshallconsiderthesituationwheretheonlyimportantparametersdescribingthestatesaretheirmomenta.Sothequantummechanicaltheoryistoprovidethetransitionamplitude,TpA;pB;pC;pD,acomplexnumbersuchthatjTj2givestherela-tiveprobabilityoftwoenteringstateswithenergymomentumvectorspAandpBtoscattertotheoutgoingstateswithenergymomentapCandpD.Thislookslikeafunctionoffourvectors,i.e.ofsixteenvariables,butLorentzin-varianceandconservationofenergymomentumimpliesthatthereareonlytwofreevariables.Indeed,LorentzinvarianceimpliesthatTshouldbeafunctionofvariousscalarproductsp2A;pApBetc.,ofwhichthereareteninall.Of

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7.2.MINKOWSKISPACE.151these,p2A=m2A;p2B=m2B;p2C=m2C;andp2D=m2Dareparametersoftheparticles,andhencedonotvary,leavingthesixproductsoftheformpApBetc.asvariables.Buttheseareconstrainedbyconservationofenergymomen-tum,pA+pB=pC+pDwhichprovidesfourequations,leavingonlytwooftheproductsindependent.Itturnsout,forreasonsofcrossingsymmetry"thatisconvenienttousetwoofthethreeMandelstamvariablesdenedbys:=c)]TJ/F7 6.974 Tf 6.227 0 Td[(2pA+pB2.33t:=c)]TJ/F7 6.974 Tf 6.227 0 Td[(2pA)]TJ/F11 9.963 Tf 9.963 0 Td[(pC2.34u:=c)]TJ/F7 6.974 Tf 6.227 0 Td[(2pA)]TJ/F11 9.963 Tf 9.963 0 Td[(pD2.35asindependentvariables.NowconservationofenergymomentumimpliesthatpApC+pD=pApA+pB.Hences+t+u=m2A+m2B+m2C+m2D.36givestherelationbetweenthethreeMandelstamvariables.AlthoughtheMan-delstamvariablesareimportantfortheoreticalwork,theparametersthataremeasuredinthelaboratoryareincomingandoutgoingenergiesandscatteringangle.ItthereforebecomesusefultoexpresstheselaboratoryparametersintermsoftheMandlestamvariables.Energiesintermsofs.Bythedenitionofs,weseethatthetotalenergyinthecenterofmomentumsystemisgivenbyECMA+ECMB=ECMC+ECMD=c2p s:.37TondtheenergyofAinthecenterofmomentumsystem,weagainemploythetrickofthinkingofactitiousparticleofenergymomentumpC+pDhenceatrestintheCMsystemandwithmassp sdecayingintoparticlesAandBwithenergymomentapAandpBandapply.30toobtainECMA=s+m2A)]TJ/F11 9.963 Tf 9.962 0 Td[(m2Bc2 2p s:.38TondthelaboratoryenergyofparticleAwhereparticleBisatrestwegothroughanargumentsimilartothatusedinderiving7.30.Asusualwewilldothederivationinasystemofwherec=1.WehavepB=mB;0;0;0andpA=pC+pD)]TJ/F11 9.963 Tf 9.963 0 Td[(pBsom2A=pC+pD2+m2B)]TJ/F8 9.963 Tf 9.963 0 Td[(2pBpC+pD=s+m2B)]TJ/F8 9.963 Tf 9.963 0 Td[(2pBpA+pB=s)]TJ/F11 9.963 Tf 9.963 0 Td[(m2B)]TJ/F8 9.963 Tf 9.963 0 Td[(2mBEASolvinggivesEA=s)]TJ/F11 9.963 Tf 9.963 0 Td[(m2B)]TJ/F11 9.963 Tf 9.962 0 Td[(m2A 2mB:

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152CHAPTER7.SPECIALRELATIVITYRevertingtogeneralcoordinatesgivesELabA=s)]TJ/F11 9.963 Tf 9.963 0 Td[(m2A)]TJ/F11 9.963 Tf 9.963 0 Td[(m2Bc2 2mB:.39AnglesintermsofMandelstamvariables.WewillstudythespecialcasewheremC=mAandmD=mB,forexamplewhentheoutgoingandincomingparticlesarethesame.Thevariabletiscalledthemomentumtransfer.Conservationofenergymomentumsaysthatq:=pA)]TJ/F11 9.963 Tf 9.963 0 Td[(pC=pD)]TJ/F11 9.963 Tf 9.962 0 Td[(pBandthedenitionthatt=q2:Weworkinunitswherec=1.SquaringbothsidesofpD=pB+qandusingtheassumptionthatmB=mDgivest=)]TJ/F8 9.963 Tf 7.749 0 Td[(2pBq:IntheLaboratoryframewhereBisatrestsopB=mB;0;0;0thisbecomest=)]TJ/F8 9.963 Tf 7.749 0 Td[(2mBEA)]TJ/F11 9.963 Tf 9.963 0 Td[(EC:SupposethatAisaverylightparticle,practicallyofmasszero,sothatpA:=jjkAjj;kAandpC:=jjkCjj;kC.Thent=q2=jjkAjj)-222(jjkCjj2)-222(jjkA)]TJ/F48 9.963 Tf 9.963 0 Td[(kCjj2=)]TJ/F8 9.963 Tf 7.749 0 Td[(2jjkAjjjjkCjj)]TJ/F8 9.963 Tf 9.963 0 Td[(cos=)]TJ/F8 9.963 Tf 7.749 0 Td[(4jjkAjjjjkCjjsin2=2;where,thescatteringangle,istheanglebetweenkAandkC.Substitutingt=)]TJ/F8 9.963 Tf 7.749 0 Td[(2mBjjkAjj)-222(jjkCjjintotheaboveexpressiongives2mBjjkAjj)-222(jjkCjj=4jjkAjjjjkCjjsin2=2orjjkAjj jjkCjj=1+2jjkAjj mBsin2=2:IfweassumethatjjkAjjissmallincomparisontomBthenjjkAjj:=jjkCjjandwegett:=)]TJ/F8 9.963 Tf 7.749 0 Td[(4jjkAjj2sin2=2:.40Thisformulaisforalightparticleofmoderateenergyscatteringoamassiveparticle.

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7.2.MINKOWSKISPACE.153Thegeneralexpressionisabitmoremessy,butnotmuch.Wewishtondtheangle,,betweentheincomingmomentumpAandtheoutgoingmomentumpCintherestframeofBintermsoftheMandelstamvariablesandthemasses.InthisframewehavepApC=EAEC)-222(jjpAjjjjpCjjcos:Sowewillproceedintwosteps:rsttoexpressEA;EC;jjpAjj;jjpCjjintermsofthefourdimensionalscalarproductsandthemassesandthentoexpressthescalarproductsintermsoftheMandlestamvariables.WehaveEA=pApB mBEC=pCpB mBjjpAjj=q E2A)]TJ/F11 9.963 Tf 9.963 0 Td[(m2AjjpCjj=q E2C)]TJ/F11 9.963 Tf 9.963 0 Td[(m2Csocos=EAEC)]TJ/F11 9.963 Tf 9.962 0 Td[(pApC p E2A)]TJ/F11 9.963 Tf 9.963 0 Td[(m2AE2C)]TJ/F11 9.963 Tf 9.963 0 Td[(m2C:IfwedenotethecommonvalueofmAandmCbymwehavecos=pApBpCpB)]TJ/F11 9.963 Tf 9.962 0 Td[(m2BpApC p [pApB2)]TJ/F11 9.963 Tf 9.962 0 Td[(m2Bm2][pCpB2)]TJ/F11 9.963 Tf 9.962 0 Td[(m2Bm2]:.41Tocompletetheprogramobservethatitfollowsfromthedenitionsthat2pApB=s)]TJ/F11 9.963 Tf 9.962 0 Td[(m2A)]TJ/F11 9.963 Tf 9.963 0 Td[(m2B2pApC=m2A+m2C)]TJ/F11 9.963 Tf 9.963 0 Td[(tand2pCpB=mA+m2B)]TJ/F11 9.963 Tf 9.962 0 Td[(u:Substitutingthesevaluesinto.41givesusourdesiredexpression.Bytheway,equation.41hasaniceinterpretationintermsofthescalarproductinducedonthespaceofexteriortwovectors.WehavepA^pBpC^pB=pApCpBpB)]TJ/F8 9.963 Tf 9.962 0 Td[(pApBpCpB=)]TJ/F8 9.963 Tf 7.749 0 Td[([pApBpCpB)]TJ/F11 9.963 Tf 9.963 0 Td[(m2BpApC]whilejjpA^pBjj2=m2Bm2A)]TJ/F8 9.963 Tf 9.962 0 Td[(pApB2andjjpC^pBjj2=m2Bm2C)]TJ/F8 9.963 Tf 9.963 0 Td[(pCpB2:Thetwolastexpressionsarenegative,sincethetwoplanespannedbytwotimelikevectorshassignature+-.Wecanthuswrite.41ascos=)]TJ/F8 9.963 Tf 18.224 6.739 Td[(pA^pBpA^pB p jjpA^pBjj2jjpC^pBjj2:.42

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154CHAPTER7.SPECIALRELATIVITY7.3Scatteringcross-sectionandmutualux.LetusgobacktotheexpressionjjpA^pBjj2=m2Bm2A)]TJ/F8 9.963 Tf 9.108 0 Td[(pApB2fromtheendofthelastsection.Intermsofagivenspacetimesplittingwithunittimelikevector@0,wecanwritepA=EA@0+pA;pB=EB@0+pBsopA^pB=pA^pB+@0^EApB)]TJ/F11 9.963 Tf 9.962 0 Td[(EBpAandhencejjpA^pBjj2=jjpApBjj2R3)-222(jjEApB)]TJ/F11 9.963 Tf 9.963 0 Td[(EBpAjj2R3wheredenotesthecrossproductinR3andthenormsinthelastexpressionarethethreedimensionalnorms.Inaframewherethemomentaarealigned,suchasatheCMframewherepA=)]TJ/F48 9.963 Tf 7.749 0 Td[(pBorthelaboratoryframewherepB=0,wehavepApB=0.Recallourrelativisticdenitionofvelocityasp=E.SoinaframewherethemomentaarealignedwehavejjEApB)]TJ/F11 9.963 Tf 9.962 0 Td[(EBpAjjR3=EAEBjjvjjR3wherev=1 EApA)]TJ/F8 9.963 Tf 15.774 6.74 Td[(1 EBpBisthemutualvelocity.SoinsuchaframewehavejjpA^pBjj2=E2AE2Bjjvjj2R3:.43Wewanttoapplythistothefollowingsituationwhichwerststudyinaxedframewherethemomentaarealigned.AbeamofparticlesoftypeAimpactsonatargetofparticlesoftypeBandsomeeventsoftypefareobserved.Letnfdenotethenumberofeventsoftypefperunittime,sonfhasdimensionstime)]TJ/F7 6.974 Tf 6.227 0 Td[(1.WeassumethatthetargetdensityisBwithdimensionsvol)]TJ/F7 6.974 Tf 6.227 0 Td[(1andthebeamdensityisA.Weassumethatthebeamiswellcollimatedandthatallofitsparticleshaveapproximatelythesamemomentum,pA.ThemutualuxperunittimeattimetisvZAt;xBt;xd3xwherev:=jjvjjR3andvisthemutualvelocity.Sothemutualuxhasdimensionsdistance timevol.=1 timearea:Thusnf vRAt;xBt;xd3x

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7.3.SCATTERINGCROSS-SECTIONANDMUTUALFLUX.155hasthedimensionsofarea.Similarly,ifweintegratethenumeratoranddenomi-natorwithrespecttotime,thecorrespondingquotientwillhavethedimensionsofarea.LetNf:=Rnfdtbethetotalnumberofeventsoftypef.Then,integratingthedenominatoraswell,f:=Nf vRAxBxd4x.44iscalledthetotalcross-sectionforeventsoftypef.Soithasthedimensionsofarea.Theconvenientunitisthebarnasinhecan'thitthesideofabarn"where1barn=10)]TJ/F7 6.974 Tf 6.227 0 Td[(24cm2:Thedenominatorintheexpressionforthetotalcross-sectioniscalledthemutualux.Ithasamoreinvariantexpressionasfollows:Intheframewherethetargetparticlesareatrest,thecurrent"ofthetargetparticlesathreeformhastheexpressionJB=Bdx^dy^dzwhilethecurrentforthebeamwillhaveanexpressionoftheformJA=Adx^dy^dz+dt^jAzdx^dy+jAydz^dx+jAxdy^dz:SoAB=)]TJ/F11 9.963 Tf 7.749 0 Td[(JAJB:Also,inthisframe,EAEB=pApB.Soby.43wehavemutualux=jjpA^pBjj pApBZJAJBd4x:.45Itisthefunctionofanydynamicaltheoryinquantummechanicstomakesomepredictionsabouttheexpectednumberofeventsoftypef.

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156CHAPTER7.SPECIALRELATIVITY

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Chapter8DieGrundlagenderPhysik.ThiswasthetitleofHilbert's1915paper.Itsoundsabitaudacious,butletustrytoputtheideasinageneralcontext.Weneedtodoafewcomputationsinadvance,soasnottodisrupttheowoftheargument.8.1Preliminaries.8.1.1Densitiesanddivergences.IfweregardRnasadierentiablemanifold,thelawforthechangeofvariablesforanintegralinvolvestheabsolutevalueoftheJacobiandeterminant.Thisisdierentfromthelawofchangeofvariablesofafunctionwhichisjustsubstitution.[Butitisclosetothetransitionlawforann-formwhichinvolvestheJacobiandeterminantnotitsabsolutevalue.]Forthisreasonwecannotexpecttointegratefunctionsonamanifold.Theobjectsthatwecanintegrateareknownasdensities.Webrieyrecalltwoequivalentwaysofdeningtheseobjects:Coordinatechartdescription.AdensityisarulewhichassignstoeachcoordinatechartU;onMwhereUisanopensubsetofMand:U!RnafunctiondenedonUsubjecttothefollowingtransitionlaw:IfW;isasecondchartthenv=)]TJ/F7 6.974 Tf 6.227 0 Td[(1vjdetJ)]TJ/F6 4.981 Tf 5.396 0 Td[(1jvforv2UW8.1whereJ)]TJ/F6 4.981 Tf 5.396 0 Td[(1denotestheJacobianmatrixofthedieomorphism)]TJ/F7 6.974 Tf 6.227 0 Td[(1:UV!UV:Ofcourse8.1isjustthechangeofvariablesformulaforanintegrandinRn.Tangentspacedescription.IfVisann-dimensionalvectorspace,letj^Vjdenotethespaceofrealorcomplexvaluedfunctionsofn-tuplets157

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158CHAPTER8.DIEGRUNDLAGENDERPHYSIK.ofvectorswhichsatisfyAv1;:::;Avn=jdetAjv1;:::;vn:.2Thespacej^Vjisclearlyaone-dimensionalvectorspace.Adensityisthenarulewhichassignstoeachx2Manelementofj^TMxj.Therelationbetweenthesetwodescriptionsisthefollowing:Letbeadensityaccordingtothetangentspacedescription.Thusx2j^TMxjforeveryx2M.LetU;beacoordinatechartwithcoordinatesx1;:::;xn.ThenonUwehavethevectorelds@ @x1;:::;@ @xn:Wecanthenevaluatexonthevaluesofthesevectoreldsatanyx2U,andsodenex=x@ @x1x;:::;@ @xnx:IfW;isasecondcoordinatechartwithcoordinatesy1;:::;ynthenonUWwehave@ @xj=X@yi @xj@ @yiandJ)]TJ/F6 4.981 Tf 5.397 0 Td[(1=@yi @xjso.1followsfrom8.2.IfU;isacoordinatechartwithcoordinatesx1;:::;xnthenthedensitydenedonUby1,thatisbyx@ @x1x;:::;@ @xnx=18x2Uisdenotedbydx.EveryotherdensitythenhasthelocaldescriptionGdxonUwhereGisafunction.If:N!MisadieomorphismandifisadensityonM,thenthepullbackisthedensityonNdenedbyzv1;:::;vn:=zdzv1;:::;dzvnz2N;v1;:::;vn2TNz:Itiseasytocheckthatthisisindeedadensity,i.e.that.2holdsateachz2N.Inparticular,ifXisavectoreldonMgeneratingaoneparametergroupt7!t=exptXofdieomorphisms,wecanformtheLiederivativeDX:=d dttjt=0:

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8.1.PRELIMINARIES.159WewillneedalocaldescriptionofthisLiederivative.WecanderivesuchalocaldescriptionfromWeil'sformulafortheLiederivativeofadierentialformbythefollowingdevice:SupposethatthemanifoldMisorientableandthatwehavechosenanorientationofM.ThismeansthatwehavechosenasystemofcoordinatechartssuchthatalltheJacobiandeterminantsdetJ)]TJ/F6 4.981 Tf 5.397 0 Td[(1arepositive.Relativetothissystemofcharts,wecandroptheabsolutevaluesignin.1sincedetJ)]TJ/F6 4.981 Tf 5.396 0 Td[(1>0.But.1withouttheabsolutevaluesignsisjustthetransitionlawforann-formonthen-dimensionalmanifoldM.Inotherwords,oncewehavechosenanorientationonanorientablemanifoldMwecanidentifydensitieswithn-forms.AxedchartU;carriestheorientationcomingfromRnandouridenticationamountstoidentifyingthedensitydxwiththen-formdx1^^dxn.Ifisann-formonann-dimensionalmanifoldthenWeil'sformulaDX=iXd+diXreducestoDX=diXsinced=0astherearenonon-zeron+1formsonann-dimensionalmanifold.IfX=X1@ @x1++Xn@ @xnand=Gdx1^^dxnintermsoflocalcoordinatesthenanimmediatecomputationgivesdiX=nXi=1@iGXi!dx1^^dxn.3where@i:=@ @xi:Itisusefultoexpressthisformulasomewhatdierently.Itmakesnosensetotalkaboutanumericalvalueofadensityatapointxsinceisnotafunction.Butitdoesmakesensetosaythatdoesnotvanishatx,sinceifx6=0then.1impliesthatx6=0.Supposethatisadensitywhichdoesnotvanishanywhere.ThenanyotherdensityonMisoftheformfwherefisafunction.IfXisavectoreld,sothatDXisanotherdensity,thenDXisoftheformfwherefisafunction,calledthedivergenceofthevectoreldXrelativetothenon-vanishingdensityanddenotedbydivX.Insymbols,DX=divX:Wecanthenrephrase.3assayingthatdivX=1 GnXi=1@iGXi.4

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160CHAPTER8.DIEGRUNDLAGENDERPHYSIK.inalocalcoordinatesystemwhereX=X1@1++Xn@nisthelocalexpressionforXandGdxisthelocalexpressionfor.8.1.2Divergenceofavectoreldonasemi-Riemannianmanifold.Supposethatgisasemi-Riemannmetriconann)]TJ/F8 9.963 Tf 11.27 0 Td[(dimensionalmanifold,M.Thengdeterminesadensity,callitg,whichassignstoeveryntangentvectors,1;:::;natapointxthevolume"oftheparallelepipedthattheyspan:g:1;:::;n7!jdethi;jij1 2:.5IfwereplacetheibyAiwhereA:TMx!TMxthedeterminantisreplacedbydetAi;Aj=detAi;jA=detA2deti;jsoweseethat.2issatised.Sogisindeedadensity,andsincethemetricisnon-singular,thedensitygdoesnotvanishatanypoint.SoifXisavectoreldonM,wecanconsideritsdivergencedivgXwithrespecttog.Sincegwillbexedfortherestofthissubsection,wemaydropthesubscriptgandsimplywritedivX.SodivXg=DXg:.6Ontheotherhand,wecanformthecovariantdierentialofXwithrespecttotheconnectiondeterminedbyg,rX:ItassignsanelementofHomTMp;TMptoeachp2Maccordingtotherule7!rX:Thetraceofthisoperatorisanumber,assignedtoeachpoint,p,i.e.afunctionknownasthecontraction"ofrX,soCrX:=f;fp:=tr7!rX:WewishtoprovethefollowingformuladivX=CrX:.7Wewillprovethisbycomputingbothsidesinacoordinatechartwithcoordi-nates,say,x1;:::;xn.Letdx=dx1dx2dxndenotethestandarddensitytheonewhichassignsconstantvalueonetothe@1;:::;@n;@i:=@=@xi.Theng=Gdx;G=jdeth@i;@jij1 2=deth@i;@ji1 2

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8.1.PRELIMINARIES.161where:=sgndeth@i;@ji:Recallthelocalformula.4forthedivergence:divX=1 GXi@iXiG:Write:=deth@i;@jiso1 G@iG=1 p 1 2p @ @xi=1 21 @ @xiindependentofwhether=1or)]TJ/F8 9.963 Tf 7.748 0 Td[(1.Tocomputethispartialderivative,letususethestandardnotationgij:=h@i;@jiso=detgij=Xjgijijwherewehaveexpandedthedeterminantalongthei)]TJ/F8 9.963 Tf 7.749 0 Td[(throwandtheijarethecorrespondingcofactors.Ifwethinkofasafunctionofthen2variables,gijthen,sincenoneoftheikforaxediinvolvegij,weconcludefromtheabovecofactorexpansionthat@ @gij=ij.8andhencebythechainrulethat@ @xk=Xijij@gij @xk:But1 ij=gij)]TJ/F7 6.974 Tf 6.226 0 Td[(1;theinversematrixofgij,whichisusuallydenotedbygklsowehave@ @xk=Xijgij@gij @xk

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162CHAPTER8.DIEGRUNDLAGENDERPHYSIK.or1 G@kG=1 2Xijgij@gij @xk:Recallthat)]TJ/F10 6.974 Tf 6.226 4.113 Td[(abc:=1 2Xrgar@grb @xc+@grc @xb)]TJ/F11 9.963 Tf 11.158 6.74 Td[(@gbc @xrsoX)]TJ/F10 6.974 Tf 6.227 4.113 Td[(aba=1 2Xargar@gar @xborXa)]TJ/F10 6.974 Tf 6.227 4.113 Td[(aka=1 G@G @xk:.9Ontheotherhand,wehaver@iX=X@Xj @xi@j+X)]TJ/F10 6.974 Tf 6.227 4.788 Td[(jikXk@ksoCrX=X@Xj @xj+1 GXXj@G @xj;proving.7.Forlateruseletusgooveronestepofthisproof.From.8wecanconclude,asabove,that@G @gij=1 2Ggij:.108.1.3TheLiederivativeofofasemi-Riemannmetric.WewishtoproveLVg=SrV#:.11ThelefthandsideofthisequationistheLiederivativeofthemetricgwithrespecttothevectoreldV.Itisarulewhichassignsasymmetricbilinearformtoeachtangentspace.Bydenition,itistherulewhichassignstoanypairofvectorelds,XandY,thevalueLVgX;Y=VhX;Yi)-222(h[V;X];Yi)-222(hX;[V;Y]i:Therighthandsideof.11meansthefollowing:V#denotesthelineardierentialformwhosevalueatanyvectoreldYisV#Y:=hV;Yi:Intensorcalculusterminology,#istheloweringoperator",anditcommuteswithcovariantdierential.Since#commuteswithr,wehaverV#X;Y=rXV#Y=hrXV;Yi:

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8.1.PRELIMINARIES.163ThesymbolSin.11denotessymmetricsum,sothattherighthandsideof.11whenappliedtoX;YishrXV;Yi+hrYV;Xi:Butnow.11followsfromtheidentitiesLVhX;Yi=VhX;Yi=hrVX;Yi+hX;rVYirVX)]TJ/F8 9.963 Tf 9.962 0 Td[([V;X]=rXVrVY)]TJ/F8 9.963 Tf 9.963 0 Td[([V;Y]=rYV:8.1.4Thecovariantdivergenceofasymmetrictensoreld.LetTbeasymmetriccontravariant"tensoreldofsecondorder,sothatinanylocalcoordinatesystemThastheexpressionT=XTij@i@j;Tij=Tji:Ifisalineardierentialform,thenwecancontract"Twithtoobtainavectoreld,T:Inlocalcoordinates,if=XaidxithenT=XTijaj@i:Wecanformthecovariantdierential,rTwhichthenassignstoeverylineardierentialformalineartransformationofthetangentspaceateachpoint,andthenformthecontraction,CrT.SinceTissymmetric,wedon'thavetospecifyonwhichoftheupperindices"wearecontracting.WedenedivT:=CrT;calledthecovariantdivergenceofT.Itisavectoreld.ThepurposeofthissectionistoexplainthegeometricalsignicanceoftheconditiondivT=0:.12IfSisacovariant"symmetrictensoreldsothatS=XSijdxidxjinlocalcoordinates,letSTdenotethedoublecontraction.Itisafunction,giveninlocalcoordinatesbyST=XSijTij:

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164CHAPTER8.DIEGRUNDLAGENDERPHYSIK.ThusTcanberegardedasalinearfunctiononthespaceofallcovariantsym-metrictensorsofcompactsupportbytheruleS7!ZMSTg;wheregisthevolumedensityassociatedtog.LetVbeavectoreldofcompactsupport.ThenLVgisasymmetrictensorofcompactsupport.WeclaimProposition10Equation.12isequivalenttoZMLVgTg=0.13forallvectoreldsVofcompactsupport.Proof.Let:=V#soTisavectoreldofcompactsupport,andsoZMCrTg=ZMLTg=0bythedivergencetheorem.Recallournotation:thesymboldenotesasingle"contraction,sothatTisavectoreld.Ontheotherhand,rT=rT+Tr:Applythecontraction,C:2CTr=2Tr=TLVg;usingthefactthatTissymmetricand.11.So2TrV#=TLVgandandhenceRMTLVgg=0forallVofcompactsupportifandonlyifRMdivTg=0forallofcompactsupport.IfdivT60,wecanndapointpandalineardierentialformsuchthatdivTp>0atsomepoint,p.Multiplyingbyablipfunctionifnecessary,wecanarrangethathascompactsupportanddivT0sothatRMdivTg>0.Letuswrite`Tforthelinearfunctiononthespaceofsmoothcovarianttensorsofcompactsupportgivenby`TS:=ZMSTg:Wecanrewrite.13as`LVg=08Vofcompactsupport.14

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8.1.PRELIMINARIES.165when`=`T.Wecanaskaboutcondition8.14fordierenttypesoflinearfunctions,`.Forexample,consideradeltatensorconcentratedatapoint",thatisalinearfunctionoftheform`S=Sptwheretisacontravariant"symmetrictensordenedatthepointp2M.Weclaimthatnonon-zerolinearfunctionofthisformcansatisfy.14.Indeed,letWbeavectoreldofcompactsupportandletbeasmoothfunctionwhichvanishesatp.SetV=W.ThenrV#=dW#+rW#andthesecondtermvanishesatp.Thereforecondition.14saysthat0=tdpW#p=[tW#p]dp:ThissaysthatthetangentvectortW#pyieldszerowhenappliedtothefunction:tW#p=0:Thisistoholdforallvanishingatp,whichimpliesthattW#p=0:Nowgivenanytangentvector,w2TMpwecanalwaysndavectoreldWofcompactsupportsuchthatWp=w.Hencetheprecedingequationimpliesthattw#=08w2TMpwhichimpliesthatt=0.Letusturntothenextsimplestcase,adeltatensorconcentratedonacurve".Thatis,let:I!Mbeasmoothcurveandletbeacontinuousfunctionwhichassignstoeachs2Iasymmetriccontravarianttensor,satthepoints.Denethelinearfunction`onthespaceofcovariantsymmetrictensoreldsofcompactsupportby`S=ZISssds:Letusexaminetheimplicationsof.14for`=`.Onceagain,letuschooseV=W,thistimewith=0on.WethengetthatZI[sW#s]sds=0forallvectoreldsWandallfunctionsofcompactsupportvanishingon.Thisimpliesthatforeachs,thetangentvectorsw#istangenttothecurveforanytangentvectorwats.Forotherwisewecouldndafunctionwhichvanishedonandforwhich[sw]6=0.ByextendingwtoavectoreldWwithWs=wandmodifyingifnecessarysoastovanishoutsideasmallneighborhoodofswecouldthenarrangethattheintegralonthelefthandsideoftheprecedingequationwouldnotvanish.

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166CHAPTER8.DIEGRUNDLAGENDERPHYSIK.Thesymmetryofsthenimpliesthats=cs0s0sforsomescalarfunction,c.Indeed,inlocalcoordinatessupposethat0s=Pvi@isands=Ptij@is@js.Appliedsuccessivelytothebasisvectorsw=@isweconcludethattij=civjandhencefromtij=tjithattij=ctji.Letusassumethats6=0socs6=0.Changingtheparameterizationmeansmultiplying0sbyascalarfactor,andhencemultiplyingbyapositivefactor.Sobyreparametrizingthecurvewecanarrangethat=00.Toavoidcarryingaroundthesign,letusassumethat=00.Sincemultiplyingby)]TJ/F8 9.963 Tf 7.748 0 Td[(1doesnotchangethevalidityof.14,wemaymakethischoicewithoutlossofgenerality.AgainletuschooseV=W,butthistimewithnorestrictionon,butletususethefactthats=0s0s.WegetrV#=[dW#+rW#]=0h0;Wi+hr0W;0i=0h0;Wi)]TJ/F11 9.963 Tf 9.962 0 Td[(hW;r00i:Theintegralofthisexpressionmustvanishforeveryvectoreldandeveryfunctionofcompactsupport.Weclaimthatthisimpliesthatr000,thatisageodesic!Indeed,supposethatr0s0s6=0forsomevalue,s0,ofs.Wecouldthenndatangentvectorwats0suchthathw;r0s00s0i=1andthenextendwtoavectoreldW,andsohWr0s0si>0forallsnears0.Nowchoose0withs0=1andofcompactsupport.Indeed,choosetohavesupportcontainedinasmallneighborhoodofs0,sothatZI0h0;Wi=h0;Wib)]TJ/F8 9.963 Tf 9.963 0 Td[(h0;Wia=0whereabor
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8.2.VARYINGTHEMETRICANDTHECONNECTION.1678.2Varyingthemetricandtheconnection.WewillregardthespaceofsmoothcovariantsymmetrictensoreldsSsuchasthoseweconsideredintheprecedingsectionasthecompactlysupportedpiece"ofthetangentspacetoagivenmetricg.Thisistobeinterpretedinthefollowingsense:LetMdenotethespaceofallsemi-Riemannmetricsonamanifold,M,sayallwithaxedsignature.Ifg2Misaparticularmetric,andifSisacompactlysupportedsymmetrictensoreld,theng+tSisagainametricofthesamesignatureforsucientlysmalljtj.SowecanregardSastheinnitesimalvariationingalongthislinesegment"ofmetrics.Ontheotherhand,ifgtisanycurveofmetricsdependingsmoothlyont,andwiththepropertythatgt=goutsidesomexedcompactset,K,thenS:=dgt dtjt=0isasymmetrictensoreldofcompactsupport.SowewilldenotethespaceofallcompactlysupportedsmootheldsofsymmetriccovarianttwotensorsbyTMcompact:Noticethatwehaveidentiedthisxedvectorspaceasthecompactlysup-portedtangentspaceateverypoint,ginthespaceofmetrics.Wehavetriv-ialized"thetangentbundletoM.Thespaceofallsymmetricconnectionsalsohasanaturaltrivialization.Indeed,letrandr0betwoconnections.ThenrfXY)-222(r0fXY=frXY)]TJ/F11 9.963 Tf 9.963 0 Td[(fr0XY:Inotherwords,themapA:X;Y=rXY)-222(r0XYisatensor;itsvalueatanypointpdependsonlyonthevaluesofXandYatp.WecansaythatA=r)-222(r0isatensoreld,oftypeTTTonewhichassignstoeverytangentvectoratp2ManelementofHomTMp;TMp.Conversely,ifAisanysuchtensoreldandifr0isanyconnectionthenr=r0+Aisanotherconnection.ThusthespaceofallconnectionsisananespacewhoseassociatedlinearspaceisthespaceofallA0s.Wewillbeinterestedinsymmetricconnections,inwhichcasetheA0sarerestrictedtobeingsymmetric:AXZ=AZX.Checkthisasanexercise.LetAdenotethespaceofallsuchsmoothsymmetricAandletCdenotethespaceofallsymmetric

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168CHAPTER8.DIEGRUNDLAGENDERPHYSIK.connections.Thenwecanidentifythetangentspace"toCatanyconnectionrwiththespaceA,becauseinanyanespacewecanidentifythetangentspaceatanypointwiththeassociatedlinearspace.Insymbols,wemaywriteTCr=A;independentoftheparticularr.Onceagainwewillbeinterestedinvariationsofcompactsupportintheconnection,sowewillwanttoconsiderthespaceAcompactconsistingoftensoreldsofourgiventypeofcompactsupport.TheLevi-CivitamapassignstoeveryRiemannmetricasymmetricconnec-tion.Soitcanbeconsideredasamap,callitL:C:,frommetricstoconnections:L:C::M!C:ThevalueofL:C:gatanypointdependsonlyongijanditsrstderivativesatthepoint,andhencethedierentialoftheLevi-CivitamapcanbeconsideredasalinearmapdL:C:g:TMcompact!Acompact:Thespacesonbothsidesareindependentofgbutthedierentialdenitelydependsong.Inwhatfollows,wewillletAdenotethevalueofthisdierentialatagivengandS2TMcompact:A:=dL:C:g[S]:Asanexercise,youshouldcomputetheexpressionforAintermsofrSwherer=L:C:gistheLevi-Civitaconnectionassociatedtog.ThemapRassociatestoeverymetricitsRiemanncurvaturetensor.ThemapRicassociatestoeverymetricitsRiccicurvature.Forreasonsthatwillsoonbecomeapparent,weneedtocomputethedierentialsofthesemaps.Thecurvatureisexpressedintermsoftheconnection:RX;Y=r[X;Y])]TJ/F8 9.963 Tf 9.963 0 Td[([rX;rY]:Sowemaythinkoftherighthandsideofthisequationasdeningamap,curv,fromthespaceofconnectionstothespaceoftensorsofcurvaturetype.DierentiatingthisexpressionusingLeibniz'srulegives,foranyA2A,dcurvr[A]X;Y=A[X;Y])]TJ/F11 9.963 Tf 9.963 0 Td[(AXrY)-222(rXAY+AYrX+rYAX:Wehave[X;Y]=rXY)-222(rYXsoA[X;Y]=ArXY)]TJ/F11 9.963 Tf 9.962 0 Td[(ArYX:

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8.3.THESTRUCTUREOFPHYSICALLAWS.169Ontheotherhand,thecovariantdierential,rAofthetensoreldAwithrespecttotheconnection,risgivenbyrAX;YZ=rXAYZ)]TJ/F11 9.963 Tf 9.962 0 Td[(ArXYZ)]TJ/F11 9.963 Tf 9.963 0 Td[(AYrXZor,moresuccinctly,rAX;Y=rXAY)]TJ/F11 9.963 Tf 9.962 0 Td[(ArXY)]TJ/F11 9.963 Tf 9.962 0 Td[(AYrX:Fromthisweseethatdcurvr[A]X;Y=rAY;X)]TJ/F8 9.963 Tf 9.963 0 Td[(rAX;Y:Ifwelet~rAdenotethetensorobtainedfromrAby~rAX;Y=rAY;Xwecanwritethisequationevenmoresuccinctlyasdcurvr[A]=~rA)-222(rA:IfwesubstituteA=dL:C:g[S]intothisequationweget,bythechainrule,thevalueofdRg[S].Takingthecontraction,C,whichyieldstheRiccitensorfromtheRiemanntensor,weobtaindRicg[S]=C~rA)-222(rA:Let^gdenotethecontravariantsymmetrictensorcorrespondingtog,thescalarproductinducedbygonthecotangentspaceateachpoint.Thus,forexample,thescalarcurvature,S,isobtainedfromtheRiccicurvaturebycon-tractionwith^g:S=^gRic:Contractingtheprecedingequationwith^gandusingthefactthatrcommuteswithcontractionwith^gandwithCweobtain^gdRicg[S]=CrVwhereVisthevectoreldV:=CA")]TJ/F8 9.963 Tf 15.95 0.139 Td[(^gA:WehaveCrV=divV.AlsoVhascompactsupportsinceSdoes.Henceweobtain,fromthedivergencetheorem,thefollowingimportantresult:ZM^gdRicg[S]g=0:.158.3Thestructureofphysicallaws.8.3.1TheLegendretransformation.Letfbeafunctionofonerealvariable.Wecanconsiderthemapt7!f0twhichisknownastheLegendretransformation,orthepointslopetransformation",

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170CHAPTER8.DIEGRUNDLAGENDERPHYSIK.Lf,associatedtof.Forexample,iff=1 2kt2thentheassociatedLegendretransformationisthelinearmapt7!kt.Asforanytransformation,wemightbeinterestedincomputingitsinverse.Thatis,ndtheorapointtwithagivenvalueoff0t.Forafunction,f,oftwovariableswecanmakethesamedenitionandposethesamequestion:DeneLfasthemapxy7!@f=@x;@f=@y:Givena;bwemayasktosolvetheequations@f=@x=a@f=@y=bforxy.Thegeneralsituationisasfollows:SupposethatMisamanifoldwhosetangentbundleistrivialized,i.e.thatwearegivenasmoothidenticationofTMwithMV,allthetangentspacesareidentiedwithaxedvectorspace,V.OfcoursethisalsogivesanidenticationofallthecotangentspaceswiththexedvectorspaceV.Inthissituation,ifFisafunctiononM,theassociatedLegendretransformationisthemapLF:M!V;x7!dFx:Inparticular,given`2V,wemayasktondx2MwhichsolvestheequationdFx=`:.16Thisisthesourceequation"ofphysics,withthecaveatthatthefunctionFneednotbecompletelydened.Nevertheless,itsdierentialmightbedened,providedthatwerestricttovariationswithcompactsupport"asisillustratedbythefollowingexample:InNewtonianphysics,thebackgroundisEuclideangeometryandtheobjectsareconservativeforceeldswhicharelineardierentialformsthatareclosed.Withamildlossofgeneralityletusconsiderpotentials"insteadofforceelds,sotheobjectsarefunctions,onEuclideanthreespace.OurspaceMcon-sistsofallsmoothfunctions.SinceMisavectorspace,itstangentspaceisautomaticallyidentiedwithMitself,soV=M.Theforceeldassociatedwiththepotentialis)]TJ/F11 9.963 Tf 7.749 0 Td[(d,anditsenergydensity"atapointisonehalftheEuclideanlength2.Thatis,thedensityisgivenby1 22x+2y+2zwheresubscriptdenotespartialderivative.WewouldliketodenethefunctionFtobethetotalenergy"F=1 2ZR32x+2y+2zdxdydz

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8.3.THESTRUCTUREOFPHYSICALLAWS.171butthereisnoreasontobelievethatthisintegralneedconverge.However,supposethatsisasmoothfunctionofcompactsupport,K.Thussvanishesoutsidetheclosedboundedset,K.Foranyboundedset,B,theintegralFB:=1 2ZB2x+2y+2zdxdydzconverges,andthederivativedFB[+ts] dtjt=0=ZR3xsx+ysy+zszdxdydzexistsandisindependentofBsolongasBK.SoitisreasonabletodenetherighthandsideofthisequationasdFevaluatedats:dF[s]:=ZR3xsx+ysy+zszdxdydzeventhoughthefunctionFitselfisnotwelldened.Ofcoursetodoso,wemustnottakeV=MbuttakeVtobeofthesubspaceconsistingoffunctionsofcompactsupport.AlinearfunctiononVisjustadensity,butinEuclideanspace,withEu-clideanvolumedensitydxdydzwemayidentifydensitieswithfunctions.Sup-posethatisasmoothfunction,andwelet`bethecorrespondingelementofV,`s=ZR3sdxdydz:Equation.16with`=`becomesZR3xsx+ysy+zszdxdydz=ZR3sdxdydz8s2V;whichistoberegardedasanequationforwhereisgiven.Wehavexsx+ysy+zsz=xsx+ysy+zsz)]TJ/F11 9.963 Tf 9.963 0 Td[(swhereistheEuclideanLaplacian,=xx+yy+zz:Thus,sincethetotalderivativessxxetc.contributezerototheintegral,equation.16isthePoissonequation=:Asweknow,asolutiontothisequationisgivenbyconvolutionwiththe1=rpotential:x;y;z=1 4Z;; p x)]TJ/F11 9.963 Tf 9.962 0 Td[(2+y)]TJ/F11 9.963 Tf 9.962 0 Td[(2+z)]TJ/F11 9.963 Tf 9.963 0 Td[(2dddifhascompactsupport,forexample,sothatthisintegralconverges.InthissenseEuclideangeometrydeterminesthe1=rpotential.

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172CHAPTER8.DIEGRUNDLAGENDERPHYSIK.8.3.2Thepassiveequations.SymmetriesofthefunctionFmayleadtoconstraintsontherighthandsideof.16.Inourexampleofafunctionoftwovariables,supposethatthefunctionfontheplaneisinvariantunderrotations.Thusfwouldhavetobeafunctionoftheradius,r,andhencetherighthandsideof.16wouldhavetobeproportionaltodr,andinparticular,vanishonvectorstangenttothecirclethroughthepointxy.Moregenerally,supposethatGisgroupactingonM,andthatthefunctionFisinvariantundertheactionofthisgroup,i.e.Fax=Fx8a2G:LetO=Gxdenotetheorbitthroughx,soGxconsistsofallpointsoftheformax;a2G.ThenthefunctionFisconstantonOandsodFxmustvanishwhenevaluatedonthetangentspacetoO.Wemaywritethissymbolicallyas`2TO0x.17if.16holds.Ofcourse,intheinnitedimensionalsituationswherewewanttoapplythisequation,wemustusesomeimaginationtounderstandwhatismeantbythetangentspacetotheorbit.Wewanttoconsiderwhathappenswhenwemodify`byaddingtoitasmall"element,2V.Presumablythesolutionxtooursourceequation".16wouldthenbemodiedbyasmallamountandsothetangentspacetotheorbitwouldchange.Wewouldthenhavetoapply.17to`+usingthemodiedtangentspace.[Onesituationwheredisregardingthischangeinxcouldbejustiediswhen`=0.Presumablythemodicationofxwillbeofrstorderin,andhencethechangein.17willbeasecondordereectwhichcanbeignoredifissmall.]Apassiveequationofphysicsiswhereweapply.17butdisregardthechangeinthetangentspaceandsoobtaintheequation2TM0x:.18Thejusticationforignoringthenon-lineareectofofxmaybeproblematicalfromourabstractpointofview,buttheequationwehavejustobtainedforthepassivereactionoftothepresenceofxisapowerfulprincipleofphysics.AbouthalfthelawsofphysicsareofthisformWehaveenunciatedtwoprinciplesofphysics,asourceequation.16whichamountstoinvertingaLegendretransformation,andthepassiveequation.18whichisaconsequenceofsymmetry.WenowturntohowHilbertandEinsteinimplementedtheseprinciplesforgravity.

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8.4.THEHILBERTFUNCTION".1738.4TheHilbertfunction".ThespaceMisthespaceofLorentzianmetricsonagivenmanifold,M.HilbertchoosesashisfunctionFG=)]TJ/F1 9.963 Tf 9.409 13.56 Td[(ZMSg;S=^gRicg:Asdiscussedabove,thisfunction"neednotbedenedsincetheintegralinquestionneednotconverge.ButthedierentialdFg[S]willbedenedwhenevaluatedonavariationofcompactsupport.TheintegraldeningFinvolvesgatthreelocations:inthedenitionofthedensityg,inthedualmetric^gandinRic.Thus,byLeibniz'srule)]TJ/F11 9.963 Tf 7.749 0 Td[(dFg[S]=ZM^gRicgdg[S]+ZMd^g[S]Ricg+ZM^gdRicg[S]:Wehavealreadydonethehardworkinvolvedinshowingthatthethirdintegralvanishes,equation.15.Soweareleftwiththersttwoterms.Astotherstterm,thecoordinatefreewayofrewriting.10isdgg[S]=1 2^gSg:Astothesecondterm,recallthatinlocalcoordinates,^gisgivenbyPgij@i@jwheregijistheinversematrixofgij.Sowerecallaformulawederivedforthedierentialoftheinversefunctionofamatrix.Ifinvdenotestheinversefunction,soinvB=B)]TJ/F7 6.974 Tf 6.226 0 Td[(1;thenitfollowsfromdierentiatingtheidentityBB)]TJ/F7 6.974 Tf 6.227 0 Td[(1IusingLeibniz'srulethatdinvB[C]=)]TJ/F11 9.963 Tf 7.749 0 Td[(B)]TJ/F7 6.974 Tf 6.227 0 Td[(1CB)]TJ/F7 6.974 Tf 6.227 0 Td[(1:Itfollowsthatthedierentialofthefunctiong7!^gwhenevaluatedatSisS"",thecontravariantsymmetrictensorobtainedfromSbyapplyingtheraisingoperatorcomingfromgtwice.NowS""Ric=SRic"":SoifwedeneRIC:=Ric""

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174CHAPTER8.DIEGRUNDLAGENDERPHYSIK.tobethecontravariantformoftheRiccitensorweobtaindFg[S]=ZMRIC)]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 2S^gSg:.19Thisislefthandsideofthesourceequation8.16.TherighthandsideisalinearfunctiononthespaceTMcompact.WeknowthatifTisasmoothsymmetrictensoreld,thenitdenesalinearfunctiononTMcompactgivenby`TS=ZMSTg:Thusfor`=`Tequation.16becomesthecelebratedEinsteineldequationsRIC)]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 2S^g=T.20SoifweregardthephysicalobjectsassemiRiemannmetrics,andifwebelievethatmatterdeterminesthemetric,byasourcetypeequation,thenmattershouldbeconsideredasalinearfunctiononTMcompact.Inparticularasmooth"matterdistributionisacontravariantsymmetrictensoreld.IfwebelievethatthelawsofphysicsaredescribedbythefunctiongivenbyHilbert,wegettheEinsteineldequations.Modifyingthefunctionwouldchangethesourceequations.Forexample,ifwereplaceSbyS+cwherecisaconstant,thiswouldhavetheeectofaddingaterm1 2c^gtothelefthandsideoftheeldequations.Thisisthenotoriouscosmologicalconstant"term.WewilltakeourgroupofsymmetriestobethegroupofdieomorphismsofMofcompactsupport-dieomorphismswhicharetheidentityoutsidesomecompactset.SuchtransformationspreservethefunctionF.IfVisavectoreldofcompactsupportwhichgeneratesaoneparametergroup,softransformations,thenthesetransformationshavecompactsup-port,andthefactthatthefunctionFisinvariantunderthesetransformationstranslatesintotheassertionthatdFg[LVg]=0.Inotherwords,thetangentspacetotheorbitthroughg"isthesubspaceofTMcompactconsistingofallLVgwhereVisavectoreldofcompactsup-port.FromtheresultsobtainedabovewenowknowthatthepassiveequationtranslatesintodivT=0forasmoothtensoreldandintoT=00;ageodesicforacontinuoustensoreldconcentratedalongacurve.TheseresultsareindependentofthechoiceofF.

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8.5.SCHRODINGER'SEQUATIONASAPASSIVEEQUATION.1758.5Schrodinger'sequationasapassiveequa-tion.Inquantummechanics,thebackgroundisacomplexHilbertspace.Inordertoavoidtechnicalities,letusassumethatHisanitedimensionalcomplexvectorspacewithanHermitianscalarproduct.LetMdenotespaceofallselfadjointoperatorsonH.LetGbethegroupofallunitaryoperators,andletGactonMbyconjugation:U2GactsonMbysendingA7!UAU)]TJ/F7 6.974 Tf 6.227 0 Td[(1:SinceMisavectorspace,itstangentbundleisautomaticallytrivialized.WemayalsoidentifythespaceoflinearfunctionsonMwithMbyassigningtoB2Mthelinearfunction`Bdenedby`BA=trAB:IfCisaselfadjointmatrix,thetangenttothecurveexpitCAexp)]TJ/F11 9.963 Tf 7.748 0 Td[(itCatt=0isi[C;A].SothetangentspacetotheorbitthroughA"consistsofalli[C;A]Showthatthepassiveequation.18becomes[A;B]=0for=`B.Alinearfunctioniscalledapurestateifitisoftheform`BwhereBisprojectionontoaonedimensionalsubspace.Thismeansthatthereisaunitvector2HdetermineduptophasesothatBu=u;8u2Hwhere;denotesthescalarproductonH.Showthatapurestatesatises.18ifandonlyifisaneigenvectorofH:H=forsomerealnumber.ThisisthetimeindependentSchrodingerequation.8.6Harmonicmaps.Letusreturntoequation.18inthesettingofthegroupofdieomorphismsofcompactsupportofamanifoldMactingonthesemi-Riemannianmetrics.Inthecasethatweourlinearfunctionwasgivenbyadeltafunctiontensoreldsupportedalongacurve"wesawthatcondition8.18impliesthatthecurveisageodesicandthetensoreldis00undersuitablereprametrizationofthecurveandassumingthatthetensorelddoesnotvanishanywhereonthe

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176CHAPTER8.DIEGRUNDLAGENDERPHYSIK.curve.Wenowexaminewhatcondition.18saysforadeltafunctiontensoreld"onamoregeneralsubmanifold.SoweareinterestedintheconditionSrV#=0forallVofcompactsupportwhereisprovidedbythefollowingdata:1.AkdimensionalmanifoldQandapropermapf:Q!M,2.Asmoothsectiontoff]S2TM,sotassignstoeachq2Qanelementtq2S2TMfq,and3.Adensity!onQ.ForanysectionsofS2TMandanyq2Qwecanformthedoublecontraction"sqtqsincesqandtqtakevaluesindualvectorspaces,andsincefsproper,ifshascompactsupportthensodoesthefunctionq7!sqtqonQ.Wecanthenformtheintegral[s]:=ZQst!:.21Weobserveandthiswillbeimportantinwhatfollowsthatdependsonthetensorproductt!asasectionoff]S2TMDwhereDdenotesthelinebundleofdensitiesofQratherthanontheindividualfactors.WeapplytheequationSrV#=0tothisandtov=WwhereisafunctionofcompactsupportandWavectoreldofcompactsupportonM.SincerW=dW+rWandtissymmetric,thisbecomesZQtdW#+rW#!=0:.22WerstapplythistoawhichvanishesonfQ,sothatthetermrWvanisheswhenrestrictedtoQ.Weconcludethatthesinglecontraction"tmustbetangenttofQatallpointsforalllineardierentialformsandhencethatt=dfhforsomesectionhofS2TQ.Again,letusapplycondition.22,butnolongerassumethatvanishesonfQ.ForanyvectoreldZonQletus,byabuseoflanguage,writeZforZf;foranyfunctiononM,writehZ;WiforhdfZ;WiM

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8.6.HARMONICMAPS.177whereWisavectoreldonM,andrZWforrdfZW:Writeh=Xhijeiejintermsofalocalframeelde1;:::;ekonQ.ThentrV#=Xhij[eihej;Wi+hreiW;eji]:NowhreiW;eji=eihW;eji)-222(hW;reiejisotrV#=Xijhijeihej;Wi)]TJ/F11 9.963 Tf 9.962 0 Td[(hW;hijreieji:Also,ZQXhijeihej;Wi!=)]TJ/F1 9.963 Tf 9.41 13.561 Td[(ZQhej;WiLPihijei!:Letuswritezj=div!XhijeisoLPihijei!=wj!:IfwesetZ:=Xzjejthencondition.22becomesXijhijMreiej=)]TJ/F11 9.963 Tf 7.749 0 Td[(Z;.23wherewehaveusedMrtoemphasizethatweareusingthecovariantderivativewithrespecttotheLevi-CivitaconnectiononM,i.e.Mreiej:=rdfeidfej:Tounderstand.23supposethatweassumethathisnon-degenerate,andsoinducesasemi-RiemannianmetrichonQ,andletusassumethat!isthevolumeformassociatedwithh.Inalldimensionsexceptk=2thissecondassumptionisharmless,sincewecanrescalehtoarrangeittobetrue.Lethrdenotecovariantdierentialwithrespecttoh.Letuschoosetheframeelde1;:::;ektobeorthonormal"withrespecttoh,i.e.hij=jij;wherej=1sothatXihijei=je)]TJ/F11 9.963 Tf 9.962 0 Td[(j:

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178CHAPTER8.DIEGRUNDLAGENDERPHYSIK.ThenLej!=Chrej!andChrej=Xiihhreiej;eiih=)]TJ/F1 9.963 Tf 9.409 9.465 Td[(Xihej;hireieiih;soZ=)]TJ/F1 9.963 Tf 9.41 9.465 Td[(XjXijhej;ihreieiihej=)]TJ/F1 9.963 Tf 7.749 9.465 Td[(Xiihreiei=)]TJ/F1 9.963 Tf 9.409 9.465 Td[(Xijhreiej:GivenametrichonQ,ametricgonM,thesecondfundamentalformofamapf:Q!M,isdenedasBfX;Y:=grdfXdfY)]TJ/F11 9.963 Tf 9.963 0 Td[(dfhrXY:.24HereXandYarevectoreldsonQanddfXdenotesthevectoreldalongf"whichassignstoeachq2QthevectordfqXq2TMfq.Thetensioneldfofthemapfrelativetoagivengandhisthetraceofthesecondfundamentalformsof=Xijhijgrdfeidfej)]TJ/F11 9.963 Tf 9.963 0 Td[(dfhreiejintermsoflocalframeeld.Amapfsuchthatf0iscalledharmonic.Wethusseethatundertheaboveassumptionsabouthand!Theorem2Condition.22saysthatfisharmonicrelativetogandh.Supposethatwemakethefurtherassumptionthathisthemetricinducedfromgbythemapf.ThendfhrXY=grdfXdfYtan;thetangentialcomponentofgrdfXdfYandhenceBfX;Y=grdfXdfYnor;thenormalcomponentofgrdfXdfY.Thisisjusttheclassicalsecondfunda-mentalformvectorofQregardedasanimmersedsubmanifoldofM.TakingitstracegiveskHwhereHisthemeancurvaturevectoroftheimmersion.Thusifinadditiontotheaboveassumptionswemaketheassumptionthatthemetrichisinducedbythemapf,thenweconcludethat.18saysthatH=0,i.e.thattheimmersionfmustbeaminimalimmersion.

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Chapter9Submersions.Thetreatmenthereisthatofa1966paperbyO'NeillMichiganJournalofMath.followingearlierbasicworkbyHermann.Inasense,thesubjectcanberegardedastheappropriategeneralizationofthenotionofasurfaceofrevolution"9.1Submersions.LetMandBbedierentiablemanifolds,and:M!Bbeasubmersion,whichmeansthatdm:TMm!TBmissurjectiveforallm2M.Theimplicitfunctiontheoremthenguaranteesthat)]TJ/F7 6.974 Tf 6.226 0 Td[(1bisasubmanifoldofMforallb2B.Thesesubmanifoldsarecalledthebersofthesubmersion.Bytheimplicitfunctiontheorem,thetangentspacetotheberthroughm2Mjustthekernelofthedierentialoftheprojection,.CallthisspaceVMm.SoVMm:=kerdm:Thesetofsuchtangentvectorsatmiscalledthesetofverticalvectors,andavectoreldonMwhosevaluesateverypointareverticalwillbecalledaverticalvectoreld.WewilldenotethesetofverticalvectoreldsbyVM.IfisasmoothfunctiononB,andVisaverticalvectoreld,thenV=0.Conversely,ifV=0forallsmoothfunctions,onB,thenVisvertical.Inparticular,ifUandVareverticalvectorelds,thensois[U;V].NowsupposethatbothMandBaresemi-Riemannmanifolds.LetHMm:=VM?m:Weassumethefollowing:dm:HMm!TBmisanisometricisomorphism,i.e.isbijectiveandpreservesthescalarproductoftangentvectors.NoticethatthisimpliesthatVMmHMm=f0gsothat179

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180CHAPTER9.SUBMERSIONS.therestrictionofthescalarproducttoVMmisnon-singular.OfcourseintheRiemanniancasethisisautomatic.WeletH:TMm!HMmdenotetheorthogonalprojectionateachpointandalsoletHMdenotethesetofhorizontal"vectoreldsvectoreldswhichbelongtoHMmateachpoint.Similarly,weletVdenoteorthogonalprojectionontoVMmateachpoint.SoifEisavectoreldonM,thenVEisaverticalvectoreldandHEisahorizontalvectoreld.WewillreservethelettersU;V;Wforverticalvectorelds,andX;Y;Zforhorizontalvectorelds.Amongthehorizontalvectorelds,thereisasubclass,thebasicvectorelds.Theyaredenedasfollows:LetXBbeavectoreldonB.Ifm2M,thereisauniquetangentvector,callitXm2HMmsuchthatdmXm=XBm.Thisdenesthethebasicvectoreld,X,correspondingtoXB.NoticethatifXisthebasicvectoreldcorrespondingtoXB,andifisasmoothfunctiononB,thenX=XB:Also,bydenition,hX;YiM=hXB;YBiBforbasicvectoreldsXandY.Ingeneral,ifXandYarehorizontal,orevenbasicvectorelds,theirLiebracket,[X;Y]neednotbehorizontal.ButifXandYarebasic,thenwecancomputethehorizontalcomponentof[X;Y]asfollows:IfisanyfunctiononBandifXandYarebasicvectorelds,thenH[X;Y]=[X;Y]=XY)]TJ/F11 9.963 Tf 9.963 0 Td[(YX=XBYB)]TJ/F11 9.963 Tf 9.963 0 Td[(YBXB=[XB;YB]soH[X;Y]isthebasicvectoreldcorrespondingto[XB;YB].WeclaimthatHrXYisthebasicvectoreldcorrespondingtorBXBYB.1whererBdenotestheLevi-CivitacovariantderivativeonBandrdenotesthecovariantderivativeonM.Indeed,letXB;YB;ZBbevectoreldsonBandX;Y;ZthecorrespondingbasicvectoreldsonM.ThenXhY;ZiM=XhXB;YBiB=XBhYB;ZBiBwhilehX;[Y;Z]i=hX;H[Y;Z]i=hXB;[YB;ZB]iBsinceH[Y;Z]isthebasicvectoreldcorrespondingto[YB;ZB].FromtheKoszulformulaitthenfollowsthathrXY;ZiM=hrBXBYB;ZBiB:

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9.2.THEFUNDAMENTALTENSORSOFASUBMERSION.181ThereforedmrXYm=rBXBYBmforallpointsmwhichimplies.1.Supposethatisahorizontalgeodesic,sothatisaregularcurve,soanintegralcurveofavectoreldXBonB.LetXbethecorrespondingbasicvectoreld,soisanintegralcurveofX.ThefactthatisageodesicimpliesthatrXX=0along,andhenceby9.1rXBXB=0alongsoisageodesic.Wehaveprovedisageodesicifisahorizontalgeodesic..2IfVandWareverticalvectorelds,thenwemayconsidertheirrestrictiontoeachberasavectoreldalongthatber,andmayalsoconsidertheLevi-Civitaconnectionontheberconsideredasasemi-Riemannmanifoldinitsownright.WewilldenotethecovariantderivativeofWwithrespecttoVrelativetotheconnectioninducedbythemetriconeachberbyrVVW.ItfollowsfromtheKoszulformula,andthefactthat[V;W]isverticalifVandWarethatrVVW=VrVW.3forverticalvectorelds.HereristheLevi-CivitacovariantderivativeonM,sothatrVWhasbothahorizontalandaverticalcomponent.9.2Thefundamentaltensorsofasubmersion.9.2.1ThetensorT.ForarbitraryvectoreldsEandFonMdeneTEF:=H[rVEVF]+V[rVEHF];where,inthisequation,rdenotestheLevi-Civitacovariantderivativedeter-minedbythemetriconM.IffisanydierentiablefunctiononM,thenVfF=fVFandrVEfVF=[VEf]VF+frVEVFsoH[rVEVfF]=fH[rVEVF]:SimilarlyfpullsoutofthesecondterminthedenitionofT.AlsoVfE=fVEandrfVE=frVEbyadeningpropertyofr.ThisprovesthatTisatensoroftype;2:TfEF=TEfF=fTEF.Bydenition,TE=TVEdependsonlyontheverticalcomponent,VEofE.IfUandVareverticalvectorelds,thenTUV=HrUV=HrVU+H[U;V]=HrVUsince[U;V]isvertical.ThusTUV=TVU.4

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182CHAPTER9.SUBMERSIONS.forverticalvectorelds.AlsonoticethatifUisaverticalvectoreldthenTEUishorizontal,whileifXisahorizontalvectoreld,thenTEXisvertical.1.ShowthathTEF1;F2i=hF1;TEF2i.5foranypairofvectoreldsF1;F2.9.2.2ThetensorA.ThisisdenedbyinterchangingtheroleofhorizontalandverticalinT,soAEF:=VrHEHF+HrHEVF:ThesameproofasaboveshowsthatAisatensor,thatAEsendshorizontalvectoreldsintoverticalvectoreldsandviceversa,andtheyoursolutionofproblem1willalsoshowthathAEF1;F2i=hF1;AEF2iforanypairofvectoreldsF1;F2.Noticethattheanyhorizontalvectoreldcanbewrittenlocallyasafunctioncombinationofbasicvectorelds,andifVisverticalandXbasic,then[V;X]=VXB)]TJ/F11 9.963 Tf 9.963 0 Td[(XV=0;sotheLiebracketofaverticalvectoreldandabasicvectoreldisvertical.2.ShowthatAXX=0foranyhorizontalvectoreld,X,andhencethatAXY=)]TJ/F11 9.963 Tf 7.748 0 Td[(AYX.6foranypairofhorizontalvectoreldsX;Y.SinceV[X;Y]=VrXY)-222(rYX=AXY)]TJ/F11 9.963 Tf 9.962 0 Td[(AYXitthenfollowsthatAXY=1 2V[X;Y]:.7Hint,itsucestoshowthatAXX=0forbasicvectorelds,andforthis,thathV;AXXi=0forallverticalvectoreldssinceAXXisvertical.UseKoszul'sformula.Wecanexpresstherelationsbetweencovariantderivativesofhorizontaland

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9.2.THEFUNDAMENTALTENSORSOFASUBMERSION.183verticalvectoreldsandthetensorsTandA:rVW=TVW+rVVW.8rVX=HrVX+TVX.9rXV=AXV+VrXV.10rXY=HrXY+AXY.11IfXisabasicvectoreld,thenrVX=rXV+[V;X]and[V;X]isvertical.HenceHrVX=AXVifXisbasic:.129.2.3CovariantderivativesofTandA.ThedenitionofcovariantderivativeofatensoreldgivesrE1AE2E3=rE1AE2E3)]TJ/F11 9.963 Tf 9.963 0 Td[(ArE1E2E3)]TJ/F11 9.963 Tf 9.963 0 Td[(AE2rE1E3foranythreevectoreldsE1;E2;E3.Suppose,inthisequation,wetakeE1=VandE2=Wtobevertical,andE3=Etobeageneralvectoreld.ThenAE2=AW=0sotherstandthirdtermsontherightvanish.InthemiddletermwehaveArVW=AHrVW=ATVWsothatwegetrVAW=)]TJ/F11 9.963 Tf 7.748 0 Td[(ATVW:.13IfwetakeE1=XtobehorizontalandE2=Wtobevertical,againonlythemiddletermsurvivesandwegetrXAW=)]TJ/F11 9.963 Tf 7.749 0 Td[(AAXW:.14Similarly,rXTY=)]TJ/F11 9.963 Tf 7.749 0 Td[(TAXY.15rVTY=)]TJ/F11 9.963 Tf 7.749 0 Td[(TTVY:.163.ShowthathrUAXV;Wi=hTUV;AXWi)-222(hTUW;AXVi.17hrEAXY;Vi=hrEAYX;Vi.18hrETVW;Xi=hrETWV;Xi.19whereU;V;Warevertical,X;YarehorizontalandEisageneralvectoreld.WealsoclaimthatShrZAXY;Vi=ShAXY;TVZi.20

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184CHAPTER9.SUBMERSIONS.whereVisvertical,X;Y;ZhorizontalandSdenotescyclicsumoverthehori-zontalvectors.Proof.Thisisatensorequation,sowemayassumethatX;Y;ZarebasicandthatthecorrespondingvectoreldsXB;YB;ZBhavealltheirLiebracketsvanishatb=mwheremisthepointatwhichwewanttochecktheequation.ThusallLiebracketsofX;Y;Zareverticalatm.Wehave1 2[X;Y]=AXYby.7,so1 2[[X;Y];Z]=[AXY;Z]=rAXYZ)-222(rZAXYandthecyclicsumoftheleftmostsidevanishesbytheJacobiidentity.SoS[rAXYZ]=S[rZAXY]:.21TakingscalarproductwiththeverticalvectorVm,wehaveatthepointmbyrepeateduseof.4and.5hrAXYZ;Vi=hTAXYZ;Vi=hZ;TAXYVi=hZ;TVAXYi=hTVZ;AXYiWerecordthisfactforlateruseashTAXYZ;Vi=hTVZ;AXYi:.22Using.21weobtainShrZAXY;Vi=ShTVZ;AXYi:.23NowhrZAXY;Vi)-222(hrZAXY;Vi=hArZXY;Vi+hAXrZY;ViwhileArZXY=)]TJ/F11 9.963 Tf 7.748 0 Td[(AYHrZX=)]TJ/F11 9.963 Tf 7.749 0 Td[(AYHrXZusing.6fortherstequationsandthefactthat[X;Z]isverticalforthesecondequation.TakingscalarproductwithVgiveshAYHrXZ;Vi=hAYrXZ;VisinceAYUishorizontalforanyverticalvector,andhencehAYVrXZ;Vi=0:WethusobtainhrZAXY;Vi)-222(hrZAXY;Vi=hAXrZY;Vi)-222(hAYrXZ;Vi:Thecyclicsumoftherighthandsidevanishes.So,takingcyclicsumandapplying.23establishes.20.

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9.2.THEFUNDAMENTALTENSORSOFASUBMERSION.1859.2.4Thefundamentaltensorsforawarpedproduct.Averyspecialcaseofasemi-Riemannsubmersionisthatofawarpedproduct"followingO'Neill'sterminology.HereM=BFasamanifold,soisjustprojectionontotherstfactor.Wearegivenapositivefunction,fonBandmetricsh;iBandh;iFoneachfactor.Ateachpointm=b;q;b2B;q2FwehavethedirectsumdecompositionTMm=TBbTFqasvectorspaces,andthewarpedproductmetricisdenedasthedirectsumh;i=h;iBf2h;iF:O'NeillwritesM=BfFforthewarpedproduct,themetricsonBandFbeingunderstood.Thenotionofwarpedproductcanitselfbeconsideredasageneralizationofasurfaceofrevolution,whereBisaplanecurvenotmeetingtheaxisofrevolution,wherefisthedistancetotheaxis,andwhereF=S1,theunitcirclewithitsstandardmetric.Onawarpedproduct,thebasicvectoreldsarejustthevectoreldsofBconsideredasvectoreldsofBFintheobviousway,havingnoFcomponent.Inparticular,theLiebracketoftwobasicvectorelds,XandYonMisjusttheLiebracketofthecorrespondingvectoreldsXBandYBonB,consideredasavectoreldonMviathedirectproduct.Inparticular,[X;Y]hasnoverticalcomponent,soAXY=0.Infact,wecanbemoreprecise.Foreachxedq2F,theprojectionrestrictedtoBfqgisanisometryofBfqgwithB.ThusrXY=thebasicvectoreldcorrespondingtorBXBYB:Onawarpedproduct,thereisaspecialclassofverticalvectorelds,thosethatarevectoreldsonFconsideredasvectoreldsonBFviathedirectproductdecomposition.LetusdenotethecollectionofthesevectoreldsbyLF,thelifts"ofvectoreldsonFtouseO'Neill'sterminology.IfV2LFandXisabasicvectoreld,then[X;V]=0sincetheydependondierentvariables"andhencerXV=rVX.ThevectoreldrXVisvertical,sincehrXV;Yi=hV;rXYi=0foranybasicvectoreld,Y,asrXYishorizontal.ThisshowsthatAXV=0aswell,soA=0.Weclaimthatonceagainwecanbemoreprecise:rXV=rVX=Xf fV8basicX;and8V2LF:.24Indeed,theonlytermthatsurvivesintheKoszulformulafor2hrXV;Wi;W2LFisXhV;Wi.WehavehV;Wi=f2hVF;WFiFwherewehavewrittenfinsteadoffbytheusualabuseoflanguageforadirectproduct.NowhVF;WFiFisafunctiononFpulledbacktoBFand

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186CHAPTER9.SUBMERSIONS.soisannihilatedbyX.HenceXhV;Wi=2fXfhVF;WFiF=2Xf fhV;Wi;proving.24.Noticethat.24givesusapieceofT,namelyTVX=Xf fV:Wecanalsoderivethehorizontal"pieceofT,namelyTVW=)]TJ 8.944 6.74 Td[(hV;Wi fgradf:.25IndeedhrVW;Xi=hW;rVXi=)]TJ/F11 9.963 Tf 8.944 6.74 Td[(Xf fhV;WiandXf=hgradf;Xi:Inthisformula,itdoesn'tmatterwhetherweconsiderfasafunctiononMandcomputeitsgradientthere,orthinkoffasafunctiononBandcomputeitsgradientrelativetoBandthentakethehorizontallift.TheansweristhesamesincefhasnoFdependence.Finally,theverticalcomponentofrVW;V;W2LFisjustthesameastheextensiontoMofrFVFWFsincethemetriconeachberdiersfromthatofFbyaconstantfactor,whichhasnoinuenceonthecovariantderivative.9.3Curvature.WewantequationsrelatingthecurvatureofthebaseandthecurvatureoftheberstothecurvatureofMandthetensorsT,A,andtheircovariantderivatives.SowewillbeconsideringexpressionsoftheformhRE1E2E3;E4iwhereRisthecurvatureofMandtheE0sareeitherhorizontalorvertical.Weletn=0;1;2;3;or4denotethenumberofhorizontalvectors,theremainingbeingvertical.Thisgivesvecases.Sowewillgetveequationsforcurvature.Forexample,n=0correspondstoallvectorsvertical,soweareaskingfortherelationbetweenthecurvatureoftheberandthefullcurvature.LetRVdenotethecurvaturetensoroftheberasasemi-Riemannsubmanifold.Thecasen=0istheGaussequationofeachber:hRUVW;Fi=

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9.3.CURVATURE.187hRVUVW;Fi)-222(hTUW;TVFi+hTVW;TUFi;U;V;W;F2VM:.26WerecalltheproofO'Neill100.Wemayassume[U;V]=0soRUV=rUrV+rVrUand,using.3andthedenitionofT,ifwehavehrUrVW;Fi=hVrUrVVW;Fi+hrUTVW;Fi=hrVUrVVW;Fi+UhTVW;Fi)-222(hTVW;rUFi=hrVUrVVW;Fi)-222(hTVW;TUFi:SubstitutingtheaboveexpressionforRUVintohRUVW;Fithenproves9.26.Thecasen=1istheCodazziequationforeachber:LetU;V;WbeverticalvectoreldsandXahorizontalvectoreld.ThenhRUVW;Xi=hrVTUW;Xi)-222(hrUTVW;Xi.27ThisisalsoinO'Neill,page115.Werecalltheproof.Weassumethat[U;V]=0soRUV=rUrV+rVrUasbefore.WehavehrUrVW;Xi=hrUrVVW;Xi+hrUTVW;Xi=hTUrVVW;Xi+hrUTVW;Xi:WewriterUTVW=rUTVW+TrUVW+TVrUWandhTVrUW;Xi=hTVrVUW;XisohrUrVW;Xi=hrUTVW;Xi+hTUrVVW;Xi+hTVrUW;Xi+hTrUVW;Xi:InterchangingUandVandsubtracting,usingrUV=rVUproves9.27.Wenowturntotheoppositeextreme,n=4andn=3butrstsomenotation.WeletRHdenotethehorizontalliftofthecurvaturetensorofB:Ifhi2HMmwithvi:=dmhideneRHh1h2h3tobetheuniquehorizontalvectorsuchthatdm)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(RHh1h2h3=RBv1v2v3:Thecasen=4isgivenby

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188CHAPTER9.SUBMERSIONS.hRXYZ;Hi=hRHXYZ;Hi)]TJ/F8 9.963 Tf 16.051 0 Td[(2hAXY;AZHi+hAYZ;AXHi+hAZX;AYHi.28foranyfourhorizontalvectoreldsX;Y;Z;H.Asusual,wemayassumeX;Y;Zarebasicandalltheirbracketsarevertical.WewillmassageeachtermontherightofRXYZ=r[X;Y]Z)-222(rXrYZ+rYrXZ:Since[X;Y]isvertical,[X;Y]=2AXY.Sor[X;Y]Z=2HrAXYZ+2TAXYZ:SinceZisbasicwecanapply.12tothersttermgivingr[X;Y]Z=2AZAXY+2TAXYZ:LetuswriterYZ=HrYZ+AYZandapplyequation9.1whichwewrite,byabuseoflanguageasHrYZ=rBYZ:ThenrXrYZ=rBXrBYZ+AXrBYZ+AXAYZ+VrXAYZ:SeparatingthehorizontalandverticalcomponentsinthedenitionofRgivesHRXYZ=)]TJ/F8 9.963 Tf 7.749 0 Td[([rBX;rBY]Z+2AZAXY)]TJ/F11 9.963 Tf 9.963 0 Td[(AXAYZ+AYAXZ.29VRXYZ=2TAXYZ)-222(VrXAYZ++VrYAXZ)]TJ/F11 9.963 Tf 9.963 0 Td[(AXrBYZ+AYrBXZ.30AswehavechosenX;Ysuchthat[XB;YB]=0,thersttermontherightof.29isjustRHXYZ.Takingthescalarproductof.29withahorizontalvectoreldandusingthefactthatAEisskewadjointrelativetothemetricandAXZ=)]TJ/F11 9.963 Tf 7.748 0 Td[(AZXproves.28.Ifwetakethescalarproductof.30withaverticalvectoreld,VwegetanexpressionforhRXYZ;ViandwecandroptheprojectionsV.Letusexaminewhathappenswhenwetakethescalarproductofthevarioustermsontherightof.30withV.ThersttermgiveshTAXYZ;Vi=hTVZ;AXYiby.22.ThenexttwotermsgivehrYAXZ;Vi)-222(hrXAYZ;Vi=hrYAXZ;Vi)-222(hrXAYZ;Vi+hAXrYZ;Vi)-222(hAYrXZ;Vi

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9.3.CURVATURE.189sincerXY)-297(rYX=[X;Y]isverticalbyassumption.Thelasttwotermscancelthetermsobtainedbytakingthescalarproductofthelasttwotermsin.30withVandweobtainhRXYZ;Vi=2hAXY;TVZi+hrYAXZ;Vi)-222(hrXAYZ;Vi:.31Wecansimplifythisabitusing.18and.20.Indeed,by.18wecanreplacethesecondtermontherightbyhrXAYZ;Viandthenapply.20toget,forn=3,hRXYZ;Vi=hrZAXY;Vi+hAXY;TZVi)-222(hAYZ;TVXi)-222(hAZX;TVYi:.32Finallywegiveanexpressionforthecasen=2:hRXVY;Wi=hrXTVW;Yi+hrVAXY;Wi)-69(hTVX;TWYi+hAXV;AYWi:.33Toprovethis,writeRXV=rrXV)-222(rrVX)-222(rXrV+rVrXandhrrXVY;Wi=hY;TrXVWi+hArXVY;WihrrVXY;Wi=hTrVXY;Wi)-222(hArVXY;WihrXrVY;Wi=hrXTVY;Wi+hrVY;AXWihrVrXY;Wi=hrVAXY;Wi)-222(hrXY;TVWiwhere,forexample,inthelastequationwehavewrittenrXY=AXY+HrXYandhrVHrXY;Wi=hHrXY;rVWi=hHrXY;TVWi=hrXY;TVWi:WehavehrXTVW;Yi=hW;rXTVYi=hW;rXTVYi+hW;TrXVYi+hW;TVrXYihrVAXY;Wi=hrVAXY;Wi)-222(hArVXY;Wi)-222(hAXrVY;Wi:

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190CHAPTER9.SUBMERSIONS.Thesixtermsontherightofthelasttwoequationsequalsixoftheeighttermsontherightoftheprecedingfourleavingtworemainingterms,hArXVY;Wi)-222(hTrVXY;Wi:ButhArXVY;Wi=hAYHrXV;Wi=hAYAXV;Wi=hAXV;AYWiandasimilarargumentdealswiththesecondterm.Werepeatourequations.IntermsofincreasingvaluesonnwehavehRUVW;Fi=hRVUVW;Fi)-222(hTUW;TVFi+hTVW;TUFi;hRUVW;Xi=hrVTUW;Xi)-222(hrUTVW;XihRXVY;Wi=hrXTVW;Yi+hrVAX;Wi)-222(hTVX;TWYi+hAXV;AYWi;hRXYZ;Vi=hrZAXY;Vi+hAXY;TZVi)-222(hAYZ;TVXi)-222(hAZX;TVYi;hRXYZ;Hi=hRHXYZ;Hi)]TJ/F8 9.963 Tf 16.051 0 Td[(2hAXY;AZHi+hAYZ;AXHi+hAZX;AYHi:Wehavestatedtheformulaforn=2,i.e.twoverticalandtwohorizontaleldsforthecasehRXVY;Wi,i.e.whereonehorizontalandoneverticalvectoroccurinthesubscriptRE1E2.Butitiseasytocheckthatallotherarrangementsoftwohorizontalandtwoverticaleldscanbereducedtothisonebycurvatureidentities.Similarlyforn=1andn=3.9.3.1Curvatureforwarpedproducts.ThecurvatureformulassimplifyconsiderablyinthecaseofawarpedproductwhereA=0andTVX=Xf fV;TVW=)]TJ 8.944 6.739 Td[(hV;Wi fgradf:WewillgivetheformulaswhereX;Y;Z;HarebasicandU;V;W;F2LF.WehaveVf=0andhrVgradf;Xi=VXf)-251(hgradf;rVXi=0.Weconcludethattherighthandsideof.27vanishes,soRUVWisverticalandweconcludefrom.26thatRUVW=RFUVW)]TJ 11.158 6.74 Td[(hgradf;gradfi f2hU;WiV)-222(hV;WiU.34TheHessianofafunctionfonasemi-RiemannmanifoldisdenedtobethebilinearformonthetangentspaceateachpointdenedbyHfX;Y=hrXgradf;Yi:

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9.3.CURVATURE.191Infact,wehavehrXgradf;Yi=XYf)-222(hgradf;rXYi=rXdfY)]TJ/F11 9.963 Tf 9.963 0 Td[(dfrXY=[rrf]X;YwhichgivesanalternativedenitionoftheHessianasHf=rrfandshowsthatitisindeeda,2typetensoreld.AlsoHfX;Y=XYf)]TJ/F8 9.963 Tf 9.963 0 Td[(rXYf=[X;Y]f+YXf)]TJ/F8 9.963 Tf 9.962 0 Td[([rXY)-222(rYX+rYX]f=YXF)]TJ/F8 9.963 Tf 9.963 0 Td[(rYXf=HfY;XshowingthatHfisasymmetrictensoreld.WehaverXV=Xf fV=TVX;TVW=)]TJ 8.944 6.739 Td[(hV;Wi fgradf;ifXisbasicandV;W2LF.SorXTVW=rXTVW)]TJ/F11 9.963 Tf 9.963 0 Td[(TrXV)]TJ/F11 9.963 Tf 9.962 0 Td[(TVrXW=hV;WiXf f2gradf+1 frXgradf+2hV;WiXf f2andhgradf;Yi=Yf.Thereforethecasen=2aboveyieldshRXVY;Wi=)]TJ/F11 9.963 Tf 8.945 6.74 Td[(HfX;Y fhV;Wi:.35Thecasen=3giveshRXYZ;Vi=0andhencebyasymmetrypropertyofthecurvaturetensor,hRXYZ;Vi=hRZVX;Yi=0,or,changingnotation,hRXVY;Zi=0:ThusRVXY=HfX;Y fV:.36WehavehRUVX;Wi=hRUVX;Wi=0andby.36andtherstBianchiidentityhRUVX;Yi=HfX;Y=fhU;Vi)-222(hV;Ui=0soRUVX=0:.37

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192CHAPTER9.SUBMERSIONS.Ifweusethisfact,thesymmetryofthecurvaturetensorand.35weseethatRXVW=hV;Wi frXgradf:.38Itfollowsfromthecasen=3andn=4thatRXYZ=RHXYZ;.39thebasicvectoreldcorrespondingtothevectoreldRBXBYBZB.HencehRXYV;Zi=0.WealsohavehRXYV;Wi=hRVWX;Yi=0soRXYV=0:.40Riccicurvatureofawarpedproduct.RecallthattheRiccicurvature,RicX;YdenedasthetraceofthemapV7!RXVYisgivenintermsofanorthonormal"frameeldE1;:::;EnbyRicX;Y=XihRXEiY;Eii;i=hEi;Eii:WewillapplythistoaframeeldwhoserstdimBvectorslieinVectBandwhoselastd=dimFvectorslieinVectF.Wewillassumethatd>1andthatX;Y2VectBandU;V2VectF.WegetRicX;Y=RicBX;Y)]TJ/F11 9.963 Tf 11.541 6.74 Td[(d fHessBfX;Y.41RicX;V=0.42RicV;W=RicFV;W)-222(hV;Wif#where.43f#:=f f+d)]TJ/F8 9.963 Tf 9.963 0 Td[(1hgradf;gradfi f2.44wherefistheLaplacianoffwhichisthesameasthecontractionoftheHessianoff.GeodesicsforawarpedproductWenowcomputetheequationsforageodesiconBfF.Lets=s;sbeacurveonBfFandsupposetemporarilytheneither0s=0nor0s=0inanintervalwearestudying.Sowecanembedthetangentvectorsalongbothprojectedcurvesinvectorelds,XonBandVonF,sothatisasolutioncurvetoX+VonBfF.TheconditionthatbeageodesicisthenthatrX+VX+V=0along.ButrX+VX+V=rXX+rXV+rVX+rVV=rBXX+2Xf fV)]TJ 11.158 6.739 Td[(hV;Vi fgradf+rFVV:

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9.3.CURVATURE.193Separatingtheverticalandhorizontalcomponents,andusingthefactthatrBXX=00alongand0=V;rFVV=00alongshowsthatthegeodesicequationstaketheform00=h0;0iFfgradfonB.4500=)]TJ/F8 9.963 Tf 17.338 6.74 Td[(2 fdf ds0onF.46Alimitingargument[O-208]showsthattheseequationsholdforallgeodesics.Werepeatalltheimportantequationsofthissubsection:rXY=rBXYrXV=Xf fV=rVXHrVW=TUV=)]TJ/F7 6.974 Tf 9.292 3.923 Td[(1 fhV;WigradfvertrVW=rFVWgeodesiceqns00=h0;0iFfgradfonB00=)]TJ/F7 6.974 Tf 13.923 3.923 Td[(2 fdf ds0onFcurvatureRXYZ=RHXYZRVXY=HessBfX;Y fVRXVW=hV;Wi frXgradfRUVW=RFUVW)]TJ/F13 6.974 Tf 11.158 5.026 Td[(hgradf;gradfi f2hU;WiV)-222(hV;WiURiccicurvRicX;Y=RicBX;Y)]TJ/F10 6.974 Tf 11.417 3.922 Td[(d fHessBfX;YRicX;V=0RicV;W=RicFV;W)-222(hV;Wif#wheref#:=f f+d)]TJ/F8 9.963 Tf 9.963 0 Td[(1hgradf;gradfi f2:9.3.2Sectionalcurvature.Wereturntothegeneralcaseofasubmersion,andrecallthatthesectionalcurvatureoftheplane,PabTMm,spannedbytwoindependentvectors,a;b2TMmisdenedasKPab:=hRaba;bi ha;aihb;bi)-222(ha;bi2:

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194CHAPTER9.SUBMERSIONS.Wecanwritethedenominatormoresimplyasjja^bjj2.Inthefollowingformulas,allpairsofvectorsareassumedtobeindependent,withu;vverticalandx;yhorizontal,andwherexBdenotesdmxandyB:=dmy.SubstitutingintoourformulasforthecurvaturegivesKPvw=KVPvw)]TJ 11.158 6.74 Td[(hTvv;Twwi)-222(jjTvwjj2 jjv^wjj2.47KPxv=hrxTvv;xi+jjAxvjj2)-222(jjTvxjj2 jjxjj2jjvjj2.48KPxy=KBPxByB)]TJ/F8 9.963 Tf 11.158 6.74 Td[(3jjAxyjj2 jjx^yjj2:.499.4Reductivehomogeneousspaces.9.4.1Bi-invariantmetricsonaLiegroup.LetGbeaLiegroupwithLiealgebra,g,whichweidentifywiththeleftin-variantvectoreldsonG.Anynon-degeneratescalarproduct,h;i,ongthusdeterminesandisequivalenttoaleftinvariantsemi-RiemannmetriconG.WeletAadenoteconjugationbytheelementa2G,soAa:G!G;Aab=aba)]TJ/F7 6.974 Tf 6.227 0 Td[(1:WehaveAae=eanddAa=Ada:TGe!TGe:SinceAa=LaRa)]TJ/F6 4.981 Tf 5.397 0 Td[(1,theleftinvariantmetric,h;iisrightinvariantifandonlyifitisAainvariantforalla2G,whichisthesameassayingthath;iisinvariantundertheadjointrepresentationofGong,i.e.thathAdaY;AdaZi=hY;Zi;8Y;Z2g;a2G:Settinga=exptX;X2g,dierentiatingwithrespecttotandsettingt=0givesh[X;Y];Zi+hY;[X;Z]i=0;8X;Y;Z2g:.50IfGisconnected,thisconditionimpliesthath;iisinvariantunderAdandhenceisinvariantundertightandleftmultiplication.Suchametriciscalledbi-invariant.Letinvdenotethemapsendingeveryelementintoitsinverse:inv:a7!a)]TJ/F7 6.974 Tf 6.227 0 Td[(1;a2G:SinceinvexptX=exp)]TJ/F11 9.963 Tf 7.748 0 Td[(tXweseethatdinve=)]TJ/F8 9.963 Tf 7.749 0 Td[(id:

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9.4.REDUCTIVEHOMOGENEOUSSPACES.195Alsoinv=Ra)]TJ/F6 4.981 Tf 5.396 0 Td[(1invLa)]TJ/F6 4.981 Tf 5.396 0 Td[(1sincetherighthandsidesendsb2Gintob7!a)]TJ/F7 6.974 Tf 6.226 0 Td[(1b7!b)]TJ/F7 6.974 Tf 6.227 0 Td[(1a7!b)]TJ/F7 6.974 Tf 6.226 0 Td[(1:Hencedinva:TGa!TGa)]TJ/F6 4.981 Tf 5.396 0 Td[(1isgiven,bythechainrule,asdRa)]TJ/F6 4.981 Tf 5.397 0 Td[(1dinvedLa)]TJ/F6 4.981 Tf 5.397 0 Td[(1=)]TJ/F11 9.963 Tf 7.748 0 Td[(dRa)]TJ/F6 4.981 Tf 5.396 0 Td[(1dLa)]TJ/F6 4.981 Tf 5.396 0 Td[(1implyingthatabi-invariantmetricisinvariantunderthemapinv.Conversely,ifaleftinvariantmetricisinvariantunderinvthenitisalsorightinvariant,hencebi-invariantsinceRa=invL)]TJ/F7 6.974 Tf 6.227 0 Td[(1ainv:TheKoszulformulasimpliesconsiderablywhenappliedtoleftinvariantvectoreldsandbi-invariantmetricssinceallscalarproductsareconstant,sotheirderivativesvanish,andweareleftwith2hrXY;Zi=hX;[Y;Z]i)-222(hY;[X;Z]i+hZ;[X;Y]iandthersttwotermscancelby.50.WeareleftwithrXY=1 2[X;Y]:.51Conversely,ifh;iisaleftinvariantbracketforwhich.51holds,thenhX;[Y;Z]i=2hX;rYZi=)]TJ/F8 9.963 Tf 7.749 0 Td[(2hrYX;Zi=h[Y;X];Zi=h[X;Y];Zisothemetricisbi-invariant.LetbeanintegralcurveoftheleftinvariantvectoreldX.Condition.51impliesthat00=rXX=0soisageodesic.Thustheone-parametergroupsarethegeodesicsthroughtheidentity,andallgeodesicsareleftcosetsofoneparametergroups.ThisisthereasonforthenameexponentialmapinRiemanniangeometry.Wecomputethecurvatureofabi-invariantmetricbyapplyingthedenitiontoleftinvariantvectorelds:RXYZ=1 2[[X;Y];Z])]TJ/F8 9.963 Tf 11.159 6.739 Td[(1 4[X;[Y;Z]]+1 4[Y;[X;Z]]:Jacobi'sidentityimpliesthelasttwotermsaddupto)]TJ/F7 6.974 Tf 8.944 3.923 Td[(1 4[[X;Y];Z]andsoRXYZ=1 4[[X;Y];Z]:.52

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196CHAPTER9.SUBMERSIONS.InparticularhRXYX;Yi=1 4h[[X;Y];X];Yi=1 4h[X;Y];[X;Y]isoKX;Y=1 4jj[X;Y]jj2 jjX^Yjj2:.53ForeachX2gthelineartransformationofgconsistingofbracketingontheleftbyXiscalledadX.SoadX:g!g;adXV:=[X;V]:WecanthuswriteourformulaforthecurvatureasRXVY=)]TJ/F8 9.963 Tf 8.944 6.74 Td[(1 4adYadXV:NowtheRiccicurvaturewasdenedasRicX;Y=tr[V7!RXVY]:Wethusseethatforanybi-invariantmetric,theRiccicurvatureisalwaysgivenbyRic=)]TJ/F8 9.963 Tf 8.944 6.739 Td[(1 4B.54whereB,theKillingform,isdenedbyBX;Y:=tradXadY:.55TheKillingformissymmetric,sincetrAB=trBAforanypairoflinearoperators.Itisalsoinvariant.Indeed,let:g!gbeanyautomorphismofg,so[X;Y]=[X;Y]forallX;Y2g.WecanreadthisequationassayingadXY=adXYoradX=adX)]TJ/F7 6.974 Tf 6.227 0 Td[(1:HenceadXadY=adXadY)]TJ/F7 6.974 Tf 6.227 0 Td[(1:Sincetraceisinvariantunderconjugation,itfollowsthatBX;Y=BX;Y:Appliedto=exptadZanddierentiatingatt=0showsthatB[Z;X];Y+BX;[Z;Y]=0.SotheKillingformdenesabi-invariantscalarproductonG.Ofcourseitneednot,ingeneral,benon-degenerate.Forexample,ifthegroupiscommuta-tive,itvanishesidentically.AgroupGiscalledsemi-simpleifitsKillingform

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9.4.REDUCTIVEHOMOGENEOUSSPACES.197isnon-degenerate.Soonasemi-simpleLiegroup,wecanalwayschoosetheKillingformasthebi-invariantmetric.Forsuchachoice,ourformulaabovefortheRiccicurvaturethenshowsthatthegroupmanifoldwiththismetricisEinstein,i.e.theRiccicurvatureisamultipleofthescalarproduct.SupposethattheadjointrepresentationofGongisirreducible.Thengcannothavetwoinvariantnon-degeneratescalarproductsunlessoneisamultipleoftheother.Inthiscase,wecanalsoconcludefromourformulathatthegroupmanifoldisEinstein.9.4.2Homogeneousspaces.NowsupposethatB=G=HwhereHisasubgroupwithLiealgebra,hsuchthathhasanHinvariantcomplementarysubspace,mg.Infact,forsimplicity,letusassumethatghasanon-degeneratebi-invariantscalarproduct,whoserestrictiontohisnon-degenerate,andletm=h?.ThisdenesaGinvariantmetriconB,andtheprojection!G=H=Bisasubmersion.TheleftinvarianthorizontalvectoreldsareexactlythevectoreldsX2m,andsoAXY=1 2V[X;Y];X;Y2m:Ontheotherhand,thebersarecosetsofH,hencetotallygeodesicsincethegeodesicsareoneparametersubgroups.HenceT=0.Wecanread9.49backwardstodetermineKBPBasKBPXBYB=KPXY+3 4jjV[X;Y]jj2 jjX^Yjj2orKBPXBYB=1 4jjH[X;Y]jj2+jjV[X;Y]jj2 jjX^Yjj2;X;Y2m:.56SeeO'Neillpp.313-15foraslightlymoregeneralformulationofthisresult.Itfollowsfrom.2thatthegeodesicsemanatingfromthepointH2B=G=HarejustthecurvesexptXH;X2m.9.4.3Normalsymmetricspaces.Formula.56simpliesifallbracketsofbasicvectoreldsarevertical.Soweassumethat[m;m]h.ThenwegetKBPXBYB=jj[X;Y]jj2 jjX^Yjj2=h[[X;Y];X];Yi jjX^Yjj2:.57Forexampleswherethisholds,weneedtosearchforaLiegroupGwhoseLiealgebraghasanAd-invariantnon-degeneratescalarproduct,h;ianda

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198CHAPTER9.SUBMERSIONS.decompositiong=h+msuchthath?m[h;h]h[h;m]m[m;m]h:Let:g!gbethelinearmapdeterminedbyX=)]TJ/F11 9.963 Tf 7.749 0 Td[(X;X2m;U=U;U2h:Thenisanisometryofh;i[E;F]=[E;F]8E;F2g2=id.Conversely,supposewestartwithasatisfyingtheseconditions.Since2=id,wecanwritegasthelineardirectsumofthe+1and)]TJ/F8 9.963 Tf 7.749 0 Td[(1eigenspacesof,i.e.deneh:=fUjU=Ugandm:=fXjX=)]TJ/F11 9.963 Tf 7.748 0 Td[(Xg.Sincepreservesthescalarproduct,eigenspacescorrespondingtodierenteigenvaluesmustbeorthogonal,andthebracketconditionsonhandmfollowautomaticallyfromtheirdenition.Onewayofndingsuchaistondadieomorphism:G!GsuchthatGhasabi-invariantmetricwhichisalsopreservedby,isanautomorphismofG,i.e.ab=ab,2=id.Ifwehavesucha,then:=desatisesourrequirements.Furthermore,thesetofxedpointsof,F:=fa2Gja=agisclearlyasubgroup,whichwecouldtakeasoursubgroup,H.Infact,letF0denotetheconnectedcomponentoftheidentityinF,andletHbeanysubgroupsatisfyingF0HF.ThenM=G=Hsatisesallourrequirements.Suchaspaceiscalledanormalsymmetricspace.Weconstructalargecollectionofexamplesofsuchspacesinthenexttwosubsections.9.4.4Orthogonalgroups.WebeginbyconstructinganexplicitmodelforthespacesRp;qandtheorthog-onalgroupsOp;q.WeletdenotethestandardEuclideanpositivedenitescalarproductonRn.Foranymatrix,M,squareorrectangular,welettMde-noteitstranspose.Foragivenchoiceofp;qwithp+q=nweletdenotethe

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9.4.REDUCTIVEHOMOGENEOUSSPACES.199diagonalmatrixwith+1intherstppositionsand)]TJ/F8 9.963 Tf 7.749 0 Td[(1inthelastqpositions.Thenhu;vi:=uv=uvisascalarproductonRnoftypep;q.TheconditionthatamatrixAbelongtoOp;qisthenthatAvAw=v;w;8v;w2RnwhichisthesameastAAvw=vw8v;w2RnwhichistheastheconditiontAAv=v8v2Rn:SotAA=orA2Op;q,tA=A)]TJ/F7 6.974 Tf 6.226 0 Td[(1:.58NowsupposethatA=exptM;M2g:=op;q.Then,sincetheex-ponentialofthetransposeofamatrixisthetransposeofitsexponential,wehaveexpstM=exp)]TJ/F11 9.963 Tf 7.748 0 Td[(sM=exp)]TJ/F11 9.963 Tf 7.748 0 Td[(sMsince)]TJ/F7 6.974 Tf 6.226 0 Td[(1=.Dierentiationats=0givestM=)]TJ/F11 9.963 Tf 7.749 0 Td[(M.59astheconditionforamatrixtobelongtotheLiealgebraop;q.IfwewriteMinblock"formM=axybthentM=tatytxtbandtheconditiontobelongtoop;qisthatta=)]TJ/F11 9.963 Tf 7.749 0 Td[(a;tb=)]TJ/F11 9.963 Tf 7.748 0 Td[(b;y=txsothemostgeneralmatrixinop;qhastheformM=axtxb;ta=)]TJ/F11 9.963 Tf 7.749 0 Td[(a;tb=)]TJ/F11 9.963 Tf 7.749 0 Td[(b:.60ConsiderthesymmetricbilinearformX;Y7!trXY,calledthetraceform".Itisclearlyinvariantunderconjugation,hence,restrictedtoX;Ybothbelongingtoop;q,itisaninvariantbilinearform.Letusshowthatisnon-degenerate.Indeed,supposethatX=axtxb;Y=cytyd

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200CHAPTER9.SUBMERSIONS.areelementsofop;q.ThentrXY=tr)]TJ/F11 9.963 Tf 4.566 -8.069 Td[(ac+bd+2xty:thisshowsthatthesubalgebrah:=opoqconsistingofallblockdiagonal"matricesisorthogonaltothesubspacemconsistingofallmatriceswithzeroentriesonthediagonal,i.e.oftheformM=0xtx0:Formatricesofthelatterform,wehavetrxtx=Xijx2ijandsoispositivedenite.Ontheotherhand,sinceta=)]TJ/F11 9.963 Tf 7.749 0 Td[(aandtb=)]TJ/F11 9.963 Tf 7.749 0 Td[(bwehavea7!tra2=)]TJ/F1 9.963 Tf 9.409 7.472 Td[(Pija2ijisnegativedenite,andsimilarlyforb.Hencetherestrictionofthetraceformtohisnegativedenite.9.4.5DualGrassmannians.SupposeweconsiderthespaceRp+q,thepositivedeniteEuclideanspace,withorthogonalgroupOp+q.ItsLiealgebraconsistsofallanti-symmetricmatricesofsizep+qandtherestrictionofthetraceformtoop+qisnegativedenite.Sowecanchooseapositivedeniteinvariantscalarproductong=op+qbysettinghX;Yi:=)]TJ/F8 9.963 Tf 8.944 6.74 Td[(1 2trXY:Letbeasintheprecedingsubsection,soisdiagonalwithpplus1'sandqminus1'sonthediagonal.NoticethatisitselfanorthogonaltransformationforthepositivedenitescalarproductonRp+q,andhenceconjugationbyisanautomorphismofOp+qandalsoofSOp+qthesubgroupconsistingoforthogonalmatriceswithdeterminantone.LetustakeG=SOp+qandtobeconjugationby.Soabcd=a)]TJ/F11 9.963 Tf 7.749 0 Td[(b)]TJ/F11 9.963 Tf 7.749 0 Td[(cdandhencethexedpointsubgroupisF=SOpOq.WewilltakeH=SOpSq.ThesubspacemconsistsofallmatricesoftheformX=0)]TJ/F10 6.974 Tf 7.748 3.615 Td[(txx0andtrX2=)]TJ/F8 9.963 Tf 7.749 0 Td[(2trtxx

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9.4.REDUCTIVEHOMOGENEOUSSPACES.201sohX;Xi=trtxx;whichwasthereasonforthe1 2inourdenitionofh;i.Ourformulaforthesectionalcurvatureofanormalsymmetricspaceshowsthatthesectionalcurvatureof~Gp;qisnon-negative.Thespecialcasep=1thequotientspaceistheq)]TJ/F8 9.963 Tf 10.442 0 Td[(dimensionalsphere,~G1;q=Sqandthexoccurringintheaboveformulaisacolumnvector.Hence[X;Y]whereYcorrespondstothecolumnvectoryistheoperator=ytx)]TJ/F11 9.963 Tf 9.909 0 Td[(xty2oq,andjj[X;Y]jj2=jjX^Yjj2;provingthattheunitspherehasconstantcurvature+1.NextletGbetheconnectedcomponentofOp;q,asdescribedinthepre-cedingsubsection,andagaintaketobeconjugationby.ThistimetakehX;Yi=1 2trXY:The-1eigenspace,mofconsistsofallmatricesoftheform0txx0andtherestrictionofh;itomispositivedenite,whiletherestrictiontoH:=SOpSOqisnegativedenite.ThecorrespondingsymmetricspaceG=HisdenotedbyGpq.Ithasnegativesectionalcurvature.Inparticular,thecasep=1ishyperbolicspace,andthesamecomputationasaboveshowsthatithasconstantsectionalcurvatureequalto)]TJ/F8 9.963 Tf 7.749 0 Td[(1.ThisrealizeshyperbolicspaceasthespaceoftimelikelinesthroughtheorigininaLorentzspaceofonehigherdimension.Thesetwoclassesofsymmetricspacesaredualinthefollowingsense:Sup-posethath;mandh;maretheLiealgebradataofsymmetricspacesG=HandG=H.SupposewehaveaLiealgebraisomorphism`:h!hsuchthath`U;`Vi=hU;Vi;8U;V2handalinearisometryi:m!mwhichreversesthebracket:[iX;iY]=)]TJ/F8 9.963 Tf 7.749 0 Td[([X;Y]8X;Y2m:

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202CHAPTER9.SUBMERSIONS.ThenitisimmediatefromourformulaforthesectionalcurvaturethatKiX;iY=)]TJ/F11 9.963 Tf 7.748 0 Td[(KX;YforanyX;Y2mspanninganon-degenerateplane.WesaythatthesymmetricspacesG=HandG=Hareinduality.Inourcase,H=SOpSOqforboth~Gp;qandGp;qsowetake`=id.Wedeneibyi:0)]TJ/F10 6.974 Tf 7.748 3.615 Td[(txx07!0txx0:Itiseasytocheckthatthesesatisfyouraxiomsandso~Gp;qandGp;qaredual.Forexample,thesphereandhyperbolicspacearedualinthissense.9.5Schwarzschildasawarpedproduct.IntheSchwarzschildmodelwedeneP:=ft;rjr>0;r6=2Mgwithmetric)]TJ/F11 9.963 Tf 7.749 0 Td[(hdt2+1 hdr2;h=hr=1)]TJ/F8 9.963 Tf 11.158 6.74 Td[(2M r:ThenconstructthewarpedproductPrS2whereS2istheordinaryunitspherewithitsstandardpositivedenitemetric,callitd2.So,followingO'Neill'sconventions,thetotalmetricisoftype;1timelike=negativesquarelengthgivenby)]TJ/F11 9.963 Tf 7.748 0 Td[(hdt2+1 hdr2+r2d2:WewriteP=PI[PIIwherePI=ft;rjr>2Mg;PII=ft;rjr<2MgandN=PIrS2;B=PIIrS2:NiscalledtheSchwarzschildexteriorandBiscalledtheblackhole.Intheexterior,@tistimelike.InB,@tisspacelikeand@ristimelike.Ineither,thevectorelds@t;@rareorthogonalandbasic.Sothebaseisasurfacewithorthogonalcoordinates.Toapplytheformulasforwarpedproductsweneedsomepreliminarycomputationsonconnectionsandcurvatureofsurfaceswithorthogonalcoordinates.

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9.5.SCHWARZSCHILDASAWARPEDPRODUCT.2039.5.1Surfaceswithorthogonalcoordinates.Weconsiderasurfacewithcoordinatesu;vandmetricEdu2+Gdv2andset1:=sgnE;2:=sgnGandwriteEdu2+Gdv2=112+222where1:=edu;e:=p 1E;e>0;and2:=gdv;g:=p 2G;g>0:ThedualorthonormalframeeldisgivenbyF1=1 e@u;F2=1 g@v:Theconnectionforms,!12and!21=)]TJ/F11 9.963 Tf 7.749 0 Td[(12!12aredenedby!12X=1rXF2;!21X=2rXF1foranyvectoreld,X,andaredeterminedbytheCartanequationsd1+!12^2=0;d2+!21^1=0:Thecurvatureformisthengivenbyd!12.Wendtheconnectionformsbystraightforwardcomputation:d1=evdv^du=)]TJ/F10 6.974 Tf 8.944 4.085 Td[(ev gdu^2d2=gudu^dv=)]TJ/F10 6.974 Tf 8.945 4.444 Td[(gu edv^1wheresubscriptsdenotepartialderivatives.Thus!12=ev gdu)]TJ/F11 9.963 Tf 9.963 0 Td[(12gu edvsatisesbothstructureequationswith!21=)]TJ/F11 9.963 Tf 7.748 0 Td[(12!12andisuniquelydeter-minedbythem.Wecomputed!12=ev gvdv^du)]TJ/F11 9.963 Tf 9.963 0 Td[(12gu eudu^dv=)]TJ/F1 9.963 Tf 9.41 14.048 Td[(ev gv+12gu eudu^dv:

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204CHAPTER9.SUBMERSIONS.Thisisthecurvatureform12.Ingeneral,theRiemanncurvatureisrelatedtotheconnectionformbyRvwFj=)]TJ/F1 9.963 Tf 9.409 9.464 Td[(Xijv;wFi:Inourcasethereisonlyoneterminthesumandthesectionalcurvature,whichequalstheGausscurvatureisgivenbyhRF1F2F1;F2i=hRF1F2F2;F1i=h12F1;F2F1;F1i=112F1;F2=1 eg12@u;@v=)]TJ/F11 9.963 Tf 9.562 6.74 Td[(1 egev gv+12gu eu=)]TJ/F8 9.963 Tf 11.328 6.74 Td[(1 eg1ev gv+2gu eu:SoK=)]TJ/F8 9.963 Tf 11.327 6.739 Td[(1 eg1ev gv+2gu eu.61istheformulaforthecurvatureofasurfaceintermsoforthogonalcoordinates.9.5.2TheSchwarzschildplane.InthecaseoftheSchwarzschildplane,P,wehaveeg=1soer=g=eer=11 2Er,andthepartialderivativeswithrespecttotvanish.TheformulasimpliestoK=1 2ErrorK=2M r3:.62TheconnectionformintheSchwarzchildplaneisgivenby!12=M r2dt;!21=M r2dtbythesamecomputationsince12=)]TJ/F8 9.963 Tf 7.749 0 Td[(1.Sor@t@t=r@th1 2F1=h1 2r@tF1=h1 2!21@tF2=h1 2M r2F2=Mh r2@r:

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9.5.SCHWARZSCHILDASAWARPEDPRODUCT.205Similarly,r@r@t=@rh1 2F1=h)]TJ/F6 4.981 Tf 7.423 2.678 Td[(1 2M r2F1=M r2h@t;andr@r@r=@rh)]TJ/F6 4.981 Tf 7.422 2.677 Td[(1 2F2=)]TJ/F11 9.963 Tf 11.06 6.74 Td[(M r2h@r:.WewillalsoneedtheHessianofthefunctionr.Wehave,bydenition,HrX;Y=hrXgradr;Yi:Nowgradr=h@randr@th@r=hr@t@r=M r2@tr@rh@r=hr@r+hr@r@r=M r2@r:ThusHr=M r2h;i:.639.5.3Covariantderivatives.Wewishtoapplytheformulasforcovariantderivativesinwarpedproductstothebasicvectorelds,@t;@randtovectoreldsV;WtangenttothesphereconsideredasvectoreldsonN[B,thewarpedproduct.Thecovariantderivativesofbasicvectoreldsaretheliftsofthecorrespond-ingvectoreldsonthebase,andsofromtheprevioussubsectionwegetr@t@t=Mh r2@r.64r@t@r=r@r@t=M r2h@t.65r@r@r=)]TJ/F11 9.963 Tf 11.06 6.74 Td[(M r2h@r.66

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206CHAPTER9.SUBMERSIONS.FromtheformularXV=rVX=Xf fVforawarpedproductweget,takingf=r,r@tV=rV@t=0;.67andr@rV=rV@r=1 rV:.68ApplyingtheformulaforTVWforawarpedproductgivesTVW=)]TJ/F11 9.963 Tf 8.944 6.74 Td[(h rhV;Wi@r.69sincegradr=h@r.ThisisthehorizontalcomponentofrVW.TheverticalcomponentisjusttheliftofrSVW,thecovariantderivativeonthesphere.9.5.4Schwarzschildcurvature.Fromformulas.39and.40forwarpedproductsandourformula.62forthecurvatureintheSchwarzschildplanewegetR@t@r@t=)]TJ/F8 9.963 Tf 7.749 0 Td[(2Mh=r3@r.70R@r@t@r=2M=r3h@t.71R@t@rV=0:.72From.36and.63weobtainRXVY=)]TJ/F11 9.963 Tf 7.749 0 Td[(RVXY=)]TJ/F11 9.963 Tf 8.944 6.74 Td[(M r3hX;YiVsoR@tV@t=Mh=r3V.73R@tV@r=0.74R@rV@t=0.75R@rV@r=M=hr3V:.76Weapply.34tocomputeRUV.Wehavehgradh;gradhi=h2h@r;@ri=handtheberovert;risthesphereofradiusrwhosecurvatureisr)]TJ/F7 6.974 Tf 6.226 0 Td[(2.wegetRVWU=M=r3hU;ViW)-222(hU;WiV.77RVW@t=0.78RVW@r=0.79

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9.5.SCHWARZSCHILDASAWARPEDPRODUCT.207Toapply.38wecomputer@tgradr=hr@t@r=M r2@tr@rh@r=2M=r2@r+hr@r@r=M r2@rsoR@tVW=R@tWV=M=r3hV;Wi@t.80R@rVW=R@rWV=M=r3hV;Wi@r:.81WeshowthattheRiccicurvaturevanishesbyapplyingourformulasfortheRiccicurvatureofawarpedproduct,.41-.43.Forasurface,RicX;Y=KhX;YiandthisisM=r3hX;YiforvectorsintheSchwarzschildplane.Ontheotherhand,d=2;f=r;Hf=M=r2h;i.ThisshowsthatRicX;Y=0.Forverticalvectors,wehaveRicFV;W=r)]TJ/F7 6.974 Tf 6.227 0 Td[(2hV;Wiwhiler=CHessr=2M=r2hgradf;gradfi=hsof#=r2showingthatRicV;W=0.9.5.5Cartancomputation.WehaveusedthetechniquesofwarpedproducttocomputetheSchwarzschildconnectionandcurvature.However,theCartanmethodismoredirect:

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208CHAPTER9.SUBMERSIONS.ds2=02)]TJ/F8 9.963 Tf 9.963 0 Td[(12)]TJ/F8 9.963 Tf 9.962 0 Td[(22)]TJ/F8 9.963 Tf 9.963 0 Td[(32where0=p hdt;h:=1)]TJ/F8 9.963 Tf 11.158 6.74 Td[(2M r1=1 p hdr2=rd#3=rSdS=sin#C=cos#dp h=1 2p h2M r2drsod0=M r2p hdr^dt=)]TJ/F11 9.963 Tf 15.21 6.739 Td[(M r2p h0^1d1=0d2=)]TJ 8.944 15.231 Td[(p h r2^1d3=)]TJ 8.944 15.231 Td[(p h r3^1)]TJ/F11 9.963 Tf 12.968 6.74 Td[(C rS3^2ord=)]TJ/F11 9.963 Tf 7.749 0 Td[(!^where=0BB@01231CCA!=0BBB@0M r2p h000M r2p h00)]TJ/F13 6.974 Tf 8.945 9.873 Td[(p h r2)]TJ/F13 6.974 Tf 8.944 9.873 Td[(p h r30p h r20)]TJ/F10 6.974 Tf 10.436 3.923 Td[(C Sr30p h r3C Sr301CCCA:

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9.5.SCHWARZSCHILDASAWARPEDPRODUCT.209NowM r2p h0=M r2dtsodM r2p h0=2M r30^1p h r2=p hd#sodp h r2!=M r2p hdr^d#=M r31^2dp h r3!=dp hSd=M r2p hdr^Sd+p hCd^d=M r31^3+Cp h Sr22^3dC Sr3=dCd=)]TJ/F11 9.963 Tf 7.749 0 Td[(Sd^d=)]TJ/F8 9.963 Tf 11.074 6.74 Td[(1 r22^3:Thisthengivesthecurvaturematrixinthisframeas:=d!+^!=0BBBBBBBB@02M r30^1)]TJ/F10 6.974 Tf 8.944 3.922 Td[(M r30^2)]TJ/F10 6.974 Tf 8.944 3.922 Td[(M r30^32M r30^10)]TJ/F10 6.974 Tf 8.944 3.922 Td[(M r31^2M r33^1)]TJ/F10 6.974 Tf 8.945 3.922 Td[(M r30^2M r31^202M r32^3)]TJ/F10 6.974 Tf 8.945 3.923 Td[(M r30^3)]TJ/F10 6.974 Tf 8.945 3.923 Td[(M r33^1)]TJ/F7 6.974 Tf 8.945 3.923 Td[(2M r32^301CCCCCCCCA:ThecurvaturetensorisgivenintermsofasRvwEj=Xijv;wEiorRijk`=ijEk;E`:Noticefromtheformofgivenabove,thatRijk`=0ifj6=k;`.HenceRmjm`=0ifj6=`.LookingatthecolumnsweseethatPRmimi=I)]TJ/F11 9.963 Tf 9.067 0 Td[(I)]TJ/F11 9.963 Tf 9.066 0 Td[(I=0.ThustheSchwarzschildmetricisRicciat.9.5.6Petrovtype.ThetensorRabcdisobtainedfromthetensorRabcd=abEc;Edbyraising"thesecondindex.Wewanttoconsiderthisasthematrixoftheoperator[R]

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210CHAPTER9.SUBMERSIONS.relativetothebasisEi^Ej.IfweuseijtostandforEi^Ejandomitthezeroentriesweseethatthematrixof[R]is[R]=010203233112012M=r302)]TJ/F11 9.963 Tf 7.749 0 Td[(M=r303)]TJ/F11 9.963 Tf 7.749 0 Td[(M=r3232M=r331)]TJ/F11 9.963 Tf 7.749 0 Td[(M=r312)]TJ/F11 9.963 Tf 7.749 0 Td[(M=r3.82Wecanwritethisinblockthreebythreeformas[R]=A00AwhereA=0@2M=r3000)]TJ/F11 9.963 Tf 7.749 0 Td[(M=r3000)]TJ/F11 9.963 Tf 7.749 0 Td[(M=r31A;:Ontheotherhandwehave?E0^E1=E2^E3?E0^E2=E3^E1?E0^E3=E1^E2and?2=)]TJ/F8 9.963 Tf 7.748 0 Td[(id:Thusthematrixof?relativetothesamebasisis[?]=0)]TJ/F11 9.963 Tf 7.749 0 Td[(II0whereIisthethreebythreeidentitymatrix.ClearlytheoperatorgivenbyRon^2TMcommuteswiththestaroperator,aspredictedbythegeneraltheoryforanyRicciatcurvature,andweseefromtheformofthematrixAthatitisofPetrovtypeD,withrealeigenvalues2M=r3;)]TJ/F11 9.963 Tf 7.749 0 Td[(M=r3)]TJ/F11 9.963 Tf 9.963 0 Td[(M=r3.9.5.7Kerr-Schildform.Wewillshowthatbymakingachangeofvariablesthatthemetricisthesumofaatmetric,andamultipleofthesquareofalineardierentialform,,wherejjjj2=0intheatmetric.Thegeneralizationofthisconstructionwillbeimportantinthecaseofrotatingblackholes.Wemakethechangeofvariablesintwostages:Letu=t+TrwhereTisanyfunctionofrdetermineduptoadditiveconstantsuchthatT0=1 h:

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9.5.SCHWARZSCHILDASAWARPEDPRODUCT.211Thendu=dt+1 hdrsohdt2=hdu2)]TJ/F8 9.963 Tf 9.962 0 Td[(2dudr+1 hdr2)]TJ/F11 9.963 Tf 7.748 0 Td[(hdt2+1 hdr2=)]TJ/F11 9.963 Tf 7.748 0 Td[(hdu2+2dudr=)]TJ/F8 9.963 Tf 7.748 0 Td[(du)]TJ/F11 9.963 Tf 9.963 0 Td[(dr2+dr2+2M rdu2:Soifwesetx0:=u)]TJ/F11 9.963 Tf 9.963 0 Td[(rthisbecomes)]TJ/F11 9.963 Tf 7.748 0 Td[(dx02+dr2)]TJ/F8 9.963 Tf 11.158 6.74 Td[(2M rdx0+dr2:Theformdx0+drhassquarelengthzerointheatmetric)]TJ/F11 9.963 Tf 7.749 0 Td[(dx02+dx2+y2+dz2;r2=x2+y2+z2andtheSchwarzschildmetricisgivenbydx02+dr2+r2d2)]TJ/F8 9.963 Tf 11.159 6.74 Td[(2M rdx0+dr2whichisthedesiredKerr-Schildform.9.5.8Isometries.AvectoreldXisaninnitesimalisometryoraKillingeldifitsowpreservesthemetric.ThisequivalenttotheassertionthatLXhY;Zi=h[X;Y];Zi+hY;[X;Z]i.83forallvectoreldsYandZ.NowXhY;Zi=hrXY;Zi+hY;rXZiandrXY=rYX+[X;Y]withasimilarequationforZ.So.83isequivalenttohrYX;Zi+hrZX;Yi=0:.84LetSbeasubmanifold,NanormalvectoreldtoS,andY;V;WtangentialvectoreldstoS,allextendedtovectoreldsintheambientmanifold.ThenalongSwehavethedecompositionrVY=rSVY+IIV;Y

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212CHAPTER9.SUBMERSIONS.intotangentialandnormalcomponents.So0=VhN;Yi=hrVN;Yi+hN;IIV;Yi:Ifthesubmanifoldistotallygeodesic,soIIV;Y0,weseethathrVN;Yi=0:Soforanyvectoreld,hrVX;Wi=hrVtanX;WiandhenceifXisaKillingvectoreldandSatotallygeodesicsubmanifoldthenthetangentialcomponentofXalongSisaKillingeldforS.ThecurvatureoftheSchwarzschildplaneis2M=r3.Soanyisometrymustpreserversinceitpreservescurvature.Henceitmustbeoftheformt;r7!t;r;randsocarriesthevectorelds@t7!@ @t@t;@r7!@ @r@t+@r:Comparingthelengthsof@randitsimageweseethat@ @r=0andcomparingthelengthsof@tanditsimageshowsthat@ @t=1:SotheonlyisometriesoftheSchwarzschildplanearetranslationsint,i.e.t;r7!t+c;r.Sincetheplanesatxedsphericalanglearetotallygeodesic,thismeansthatthetangentialcomponentofanyKillingvector,Ymustbeamultipleof@t.SothemostgeneralKillingeldisoftheformY=f@t+VwhereVisverticalandfisafunctiononS2.TheclaimisthatfisaconstantandVdoesnotdependont;randisaKillingvectorforthesphere.Inthefollowing,Udenotesanyvectoreldonthespherelifteduptobeaverticalvectoreld,andudenotesthevalueofthisvectoreldatsomepointq2S2.

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9.5.SCHWARZSCHILDASAWARPEDPRODUCT.213Wehaver@r@t=M r2h@tr@rV=1 rVr@rU=1 rUso@rhV;Ui=hr@rV;Ui+hV;r@rUi=2 rhV;Ui:Solvingthisequationforaxedpointq2S2andxedtangentvectoruatq,weseethatatthesexedvalueshV;Ui=gtr2:Nowr@tU=0sohr@tY;Ui=@thY;UirUY=Uf@t+@r+verticalsohrUY;@ti=)]TJ/F11 9.963 Tf 7.749 0 Td[(hUfsohr@tY;Ui+hrUY;@ti=0Killingimplies@thV;Ui=0:Again,xingu,thisgivesg0tr2=hrUf:Butnomultipleofr2canequalanymultipleofhr=1)]TJ/F7 6.974 Tf 11.839 3.922 Td[(2M runlessbothmultiplesarezero.Sog0=0whichimpliesthathV;Ui=kUr2:Butthefactorr2iswhatwemultiplythesphericalmetricbyintheSchwarzschildmetric.HencethislastequationshowsthattheprojectionofVontothespheredoesnotdependonrort.ThenitmustbeaKillingeldonS2.TheconditionUf0impliesthatfisaconstant.Conclusion:theconnectedgroupofisometriesoftheSchwarzschildsolutionisRSOconsistingoftimetranslationsandrotationsofthesphere.

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214CHAPTER9.SUBMERSIONS.9.6RobertsonWalkermetrics.ThesearewarpedproductsofanintervalIRwithanegativedenitemetricandathreedimensionalspacelikemanifold,S,ofconstantcurvature.SoSiseitherthethreesphere,Euclideanthreedimensionalspace,orhyperbolicthreespace.Weusearclength,tasthecoordinateonI,sothetotalmetrichastheform)]TJ/F11 9.963 Tf 7.749 0 Td[(dt2+f2d2whered2istheconstantcurvaturemetriconSandf=ft.Wewrite@tf=f0;gradf=)]TJ/F11 9.963 Tf 7.748 0 Td[(f0@tsocovariantderivativesaregivenbyr@t@t=0.85r@tV=rV@t=f0=fV.86TVW=hV;Wif0=f@t.87rVVW=rFVW:.88WehaveHf@t;@t:=f00andsoRV@t@t=f00=fV.89RVW@t=0.90R@tVW=f00=fhV;Wi@t.91RUVW=f0=f2+k=f2[hU;WiV)-222(hV;WiU]:.92wherek=1;0or)]TJ/F8 9.963 Tf 7.749 0 Td[(1istheconstantcurvatureofS.Theberdimensionisd=3soRic@t;@t=)]TJ/F8 9.963 Tf 8.944 6.739 Td[(3f00 f.93whileRic@t;V=0asalwaysinawarpedproductandRicV;W=2f0 f2+2k f+f00 f!hV;Wi:.94TakingthecontractionoftheRiccitensorgivesthescalarcurvatureasS=6f0 f2+k f2+f00 f!.95andhencetheEinsteintensorT=Ric)]TJ/F7 6.974 Tf 11.158 3.922 Td[(1 2Sh;iisgivenbyTV;W=}hV;Wi;}:=)]TJ/F1 9.963 Tf 9.409 17.036 Td[("2f00 f+f0 f2+k f2#:

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9.6.ROBERTSONWALKERMETRICS.215AlsoT@t;@t=)]TJ/F8 9.963 Tf 8.944 6.74 Td[(3f00 f+1 2S=3f0 f2+3k f2:=:Withthesedenitionsof}andwecanwriteT=+}dtdt+}h;i:.96AnenergymomentumtensoroftypeT=+}+}gwhereXisaforwardtimelikevectorand=hX;iandwhereand}arefunctions,iscalledaperfectuidforreasonsexplainedinO'Neill.Thefunction}iscalledthepressure.Theuidiscalledadustif}=0.ARobertsonWalkermodelwhichisadustiscalledaFriedmanmodel.LetuscomputethecovariantdivergenceoftheTgivenby9.96.Wecom-puterelativetoaframeeldwhoserstcomponentis@tandwhoselastthreecomponentsU1;U2;U3arethereforvertical.ThecovariantdivergenceisdenedtobeXirEiTEi;:Inallsituations,thecovariantdivergentofhgisjustdhsincerEg=0andXdhEiihEi;i=dh:Henceweobtain,fordivT,theexpression)]TJ/F8 9.963 Tf 7.749 0 Td[(0+}0dt++}XirEidtEidt+}0dt:Nowr@tdt@t=dtr@t@t=0,whilerUdtU=)]TJ/F11 9.963 Tf 7.749 0 Td[(dtrUU=)]TJ/F11 9.963 Tf 8.944 6.74 Td[(f0 fforanyunitvectororthogonalto@t.Thusweobtain)]TJ/F1 9.963 Tf 9.409 14.047 Td[(0+3+}f0 fdtforthecovariantdivergence.

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216CHAPTER9.SUBMERSIONS.9.6.1Cosmogenyandeschatology.ThefunctionH:=f0 fiscalledtheHubbleexpansionrateforobviousreasons.ThevanishingofthecovariantdivergenceofTyieldstheequation0=)]TJ/F8 9.963 Tf 7.749 0 Td[(3+}H:.97ifwegobacktothedenitionsofand}weseethat)]TJ/F8 9.963 Tf 7.749 0 Td[(6f00 f=+3}:Nowintheknownuniverse,}>0,sof00<0.Sothegraphoffisconvexdown,i.e.itliesbelowitstangentlineatanypoint.LetH0=Ht0denotetheHubbleconstantatthepresenttime,t0.Thetangentlinetothegraphoffatt0hasslopeH0ft0andhenceisgivenbytheequation`t=ft0+H0ft0t)]TJ/F11 9.963 Tf 9.963 0 Td[(t0:Att0)]TJ/F11 9.963 Tf 10.272 0 Td[(H)]TJ/F7 6.974 Tf 6.226 0 Td[(10theline`crossestheaxis.Sincef>0bydenition,thisshowsthatthemodelmustfailatsometimetinthepast,nomorethanH)]TJ/F7 6.974 Tf 6.226 0 Td[(10unitsoftimeago.ThecurrentestimatesonHubble'sconstantgivethisvalueassomewherebetweentenandtwentybillionyears.Noticealsothatiff0T<0atsometimeinthefuturethentheconvexdownwardpropertyimpliesthatthemodelwillalsofailatsomefuturetimeT.Foradiscussionoffurtherdetailsofthebigbang"andbigcrunch"andmorespecicallyFriedmanmodelswhereitisassumedthat}=0seeO'Neill.

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Chapter10Petrovtypes.10.1Algebraicpropertiesofthecurvatureten-sorTheRiemanncurvaturetensorhRXYZ;Wiisanti-symmetricinX;YandinZ;Wsocanbethoughtofasabilinearformon^2TMmatanypointmofasemi-RiemannmanifoldM.ItisalsoinvariantundersimultaneousinterchangeofX;YwithZ;Wsothisbilinearformissymmetric.Inaddition,itsatisesthecyclicityconditionhRXYZ;Wi+hRXZW;Yi+hRXWY;Zi=0:Wewanttoconsiderthealgebraicpossibilitiesandpropertiesofthistensor,sowillreplaceTMxbyageneralvectorspaceVwithnon-degeneratescalarproductandwanttoconsidersymmetricbilinearformsRon^2VwhichsatisfyRv^x;y^z+Rv^y;z^x+Rv^z;x^y=0:.1Forexample,ifVisfourdimensional,then^2Vissixdimensional,andthespaceofsymmetricbilinearformson^2Vis21dimensional.ThecyclicityconditioninthiscaseimposesnoconstraintonRifvisequaltoandhencelinearlydependentonx;yorz.HencethereisonlyoneequationonRimpliedby10.1inthiscase.Thusthespaceofpossiblecurvaturetensorsatanypointinafourdimensionalsemi-Riemannianmanifoldis20dimensional.TheRiccitensoristhecontractionsaywithrespecttothe,3positionoftheRiemanncurvature:RicR=C13R;RicRx;y:=XaRea^x;ea^ywherethesumisoveranyorthonormal"basis.ItisasymmetrictensoronV.SowecanthinkofRicasamapfromthespaceofpossiblecurvaturestopossibleRiccicurvatures.IfweletCurvVS2^2V217

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218CHAPTER10.PETROVTYPES.denotethesubspaceofthespaceofsymmetricbilinearformson^2Vsatisfying.1.ThenRic:CurvV!S2V:LetusshowthatifdimV>2thismapissurjective.Indeed,supposethatA2S2V.LetA^Adenotetheinducedsymmetricformon^2VsothatA^Au^v;x^y:=Av;xAw;y)]TJ/F11 9.963 Tf 9.963 0 Td[(Av;yAw;x:Holdingvxedandcyclicallysummingoverw;x;ywegetAv;x[Aw;y)]TJ/F11 9.963 Tf 7.749 0 Td[(Aw;y]+Av;y[Ax;w)]TJ/F11 9.963 Tf 7.749 0 Td[(Aw;x]+Av;w[Ay;x)]TJ/F11 9.963 Tf 7.749 0 Td[(Ax;y]=0:ThusA^Asatises.1.IfAandBaretwoelementsofS2VweseethatA^B+B^A:=A+B^A+B)]TJ/F11 9.963 Tf 9.963 0 Td[(A^A)]TJ/F11 9.963 Tf 9.963 0 Td[(B^Balsosatises.1.Letg2S2Vdenotethescalarproductitself.WeclaimthatRicg^g=n)]TJ/F8 9.963 Tf 9.963 0 Td[(1g:IndeedRicg^gv;w=Xahea;eaihv;wi)-222(hea;wihv;eai=nhv;wi)]TJ/F1 9.963 Tf 16.051 9.465 Td[(Xahv;eaihea;wi=n)]TJ/F8 9.963 Tf 9.962 0 Td[(1hv;wi:ForanyR2CurvVon^2Vdeneitsscalarcurvature"S=SRbyS:=XaRicRea;ea=CRicR:Also,foranyA2S2V,wehaveCA:=XaRicRea;easoSR=CRicR:ThenXaAea;eahv;wi)]TJ/F11 9.963 Tf 16.051 0 Td[(Aea;vhea;wi=CAhv;wi)]TJ/F11 9.963 Tf 16.051 0 Td[(Av;wXahea;eaiAv;w)]TJ/F11 9.963 Tf 9.962 0 Td[(Aea;whea;vi=n)]TJ/F8 9.963 Tf 9.962 0 Td[(1Av;wsoRicg^A+A^g=n)]TJ/F8 9.963 Tf 9.962 0 Td[(2A+CAg.2wheren=dimV.SinceRicg^g=n)]TJ/F8 9.963 Tf 9.322 0 Td[(1gthisshowsthatRic:CurvV!S2Vissurjection.WesaythatRisRicciatifRicR=0.Thusinfour

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10.2.LINEARANDANTILINEARMAPS.219dimensions,thespaceofRicciatcurvaturetensorsatanypointistendi-mensional.ThepurposeofthischapteristoexplainhowthecomplexgeometryofspinorsleadstoaclassicationofallpossibleRicciatcurvaturesintovetypes,thePetrovclassicationpublishedin1954intherelativelyobscurejour-nalSci.Nat.KazanStateUniversity.InanalyzingPetrovtypeD,KerrwasledtohisdiscoveryoftherotatingblackholesolutionsoftheEinsteinequations,whichgeneralizetheSchwartzschildsolution,in1963.Unfortunatelywewillnothavetimetostudytheremarkablepropertiesofthissolution.Itwouldtakeawholesemester.LetusbrieygobacktothegeneralsituationwheredimV>2.LetR2CurvV.ThenWdenedbyR=W+1 n)]TJ/F8 9.963 Tf 9.963 0 Td[(2g^RicR+RicR^g)]TJ/F11 9.963 Tf 31.008 6.74 Td[(SR n)]TJ/F8 9.963 Tf 9.962 0 Td[(1n)]TJ/F8 9.963 Tf 9.963 0 Td[(2g^gsatisesRicW=0:ItiscalledtheWeylcurvatureortheWeylcomponentoftheRiemanncurva-ture.Itis,aswasdiscoveredbyHermannWeyl,aconformalinvariantofthemetric.Inthreedimensionswehavedim^2=3andhencekerRic=0,therearenoWeyltensors.Theyexistinfourormoredimensions.Wenotturntothespecialpropertiesofthecurvaturetensorsingeneralrelativity.Inwhatfollows,allvectorspacesandtensorproductsareoverthecomplexnumbersunlessotherwisespecied.Allvectorspacesareassumedtobenitedimensional.10.2Linearandantilinearmaps.Amap:U!Vbetweenvectorspacesiscalledantilinearifa1u1+a2u2= a1u1+ a2u28u1;U22U;a1;a22C:Thecompositionoftwoantilinearmapsislinear,andthecompositionofalinearmapwithanantilinearmapineitherorderisantilinear.WeletU#denotethespaceofallantilinearfunctionsonU,thatisthesetofallantilinearmaps:U!C.Asusual,weletU,thecomplexdualspaceofUdenotethespaceoflinearmapsofU!C.WehaveacanonicallinearisomorphismU!U##whereu2Uissenttotheantilinearfunctionoff2U#givenbyf7! fu:Noticethat fau= afu=a fu;

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220CHAPTER10.PETROVTYPES.sothismapofU!U##islinear.Itisinjectiveandhencebijectivesinceourspacesarenitedimensional.Wedene U:=U#so UconsistsofantilinearfunctionsonU.Givenalinearfunction,`,onanvectorspace,W,wegetanantilinearfunctionbycomposingwiththestandardconjugationonthecomplexnumbers,so `= `; :C!Cor `w= `w8w2W:Also,startingwithanantilinearfunctionweproducealinearfunctionbycom-positionwithcomplexconjugation.Thus,forexample,themostgenerallinearfunctiononUisoftheform`7!`uu2U;andhencethemostgeneralantilinearfunctiononUisoftheform`7! `u:Butifwewrite`= m= mwherem2U#,then,consideredasafunctionofmthisistheassignmentm7!muwhichisalinearfunctionofm.Thuswehaveacanonicalidentication U:=U#=U#:Also U=U##=U:Wehaveanantilinearmapu7! u;U! Ugivenbycompositionwithconju-gationonCasabove,wherewethinkofUasU.So u`= `u;8`2Uor um=mu;m= `2U#:So um= mu=`u=u`andthus u=uundertheidenticationof UwithU.Wealsohave UV= U V

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10.3.COMPLEXCONJUGATIONANDREALFORMS.221asacanonicalidentication,with uv= u vasthemap :UV! U V:IfbisabilinearformonU,thenwecanthinkof basabilinearformon Uaccordingtotherule b u; v:= bu;v:Indeed, ba u; v= b au; v= b au;v= abu;v=a b u; v;andsimilarlyfor b u;a v.Ifbissymmetricorantisymmetricthensois b.10.3Complexconjugationandrealforms.Acomplexconjugationofacomplexvectorspace,V,isanantilinearmapofVtoitselfwhosesquareistheidentity.Supposethaty:v7!vyissuchacomplexconjugation.Thenthesetofvectorsxedbyy,fvjvy=vgisarealvectorspace.Itiscalledtherealformofthecomplexvectorspace,V,relativetotheconjugation,y.WedenotethisvectorspacebyVy;realorsimplybyVrealwhenyisunderstood.Ifv2Vrealthenivsatisestheequationwy=)]TJ/F11 9.963 Tf 7.748 0 Td[(wandwemightwanttocallsuchvectorsimaginary".Everyvectoru2Vcanbewritteninauniquewayasu=v+iw;v;w2Vreal;indeedv=1 2v+vy;w=)]TJ/F11 9.963 Tf 7.749 0 Td[(i 2v)]TJ/F11 9.963 Tf 9.963 0 Td[(vy:Familiarexamplesare:Visthesetofallnncomplexmatricesandyisconjugatetranspose.Therealvectorsarethentheselfadjointmatrices.Anotherexampleistostartwitharealvectorspace,E,andthencomplexifyitbytensoringwiththecomplexnumbers:V=ERC

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222CHAPTER10.PETROVTYPES.withxRcy=xR c:Thecorrespondingrealsubspaceisthenidentiedwithourstartingspace,E.Theaboveremarksabouteveryvectorbeingwrittenasu=v+iwshowsthatanycomplexvectorspacewithconjugationcanbyidentiedwiththisexample,i.e.asV=ERCwhereE=Vreal.Weshallbeinterestedintwoothertypesofexamples.SupposewestartwithavectorspaceUandconstructV=U U.Denecomplexconjugationbyu vy:=v u:Soy=s wheres: UU7!U Uswitchestheorderofthefactors.Therealsubspaceisspannedbytheelementsoftheformu u.AsecondexampleisV=U U UUwithx+ yy=y+ x:Therealsubspaceconsistsofallx+ xandhencecanbeidentiedwithUasarealvectorspace.ThatiswecanconsiderUasavectorspaceovertherealnumbersforgettingaboutmultiplicationbyi,andthiscanbeidentiedasarealvectorspacewiththerealsubspaceofU+ U.Forexample,supposethatgisasymmetriccomplexbilinearformonU.Wethenobtainacomplexsymmetricbilinearform, gon Uandhenceacomplexsymmetricbilinearform,g gonU UbydeclaringUand Utobeorthogonal:g gx+ u;y+ v:=gx;y+ gu;v:Thisrestrictstoarealbilinearformontherealsubspace:g gx+ x;y+ y=2Regx;y:SoundertheidenticationoftherealsubspaceofU UwithU,themetricg gbecomesidentiedwiththerealquadraticform2Reg.Supposethatgisnon-degenerate,andwechooseacomplexorthonormalbasis,e1;:::;enforg.Sogei;ei=1andgei;ej=0fori6=j.Thisisalwayspos-siblefornon-degeneratesymmetricformsoncomplexvectorspaces.Thene1;:::;en;ie1;:::;ienisanorthogonalbasisforUasarealvectorspacewithscalarproductRegandRegiek;iek=)]TJ/F8 9.963 Tf 7.749 0 Td[(1.Sothemetric2Regisoftypen;nonthespaceUthoughtofasa2ndimensionalrealvectorspace.

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10.4.STRUCTURESONTENSORPRODUCTS.22310.4Structuresontensorproducts.IfUandVarecomplexvectorspaces,then^2UV=S2U^2V^2US2V:Exteriormultiplicationisgivenbyu1v1^u2v2=u1u2v1^v2+u1^u2v1v2whereu1u2denotestheproductofu1andu2inthesymmetricalgebra,andsimilarlyforv1v2.WewillwanttoapplythisconstructiontothecaseV= U.IfUhasanantisymmetricbilinearform,!,andVhasanantisymmetricform,,thenthisinducesasymmetricbilinearformonUVbyhu1v1;u2v2i=!u1;u2v1;v2:WewillwanttoapplythisconstructiontoV= Uand= !.ThesymmetricbilinearinducedonUVinturninducesascalarproducton^2UV=S2U^2V^2US2Vaccordingtotheusualrulehu1v1^u2v2;u3v3^u4v4i==hu1v1;u3v3ihu2v2;u4v4i)-222(hu1v1;u4v4ihu2v2;u3v3i=!u1;u3!u2;u4v1;v3v2;v4)]TJ/F11 9.963 Tf 9.963 0 Td[(!u1;u4!u1;u3v1;v4v2;v3:Wecaninterpretthisscalarproductasfollows,putscalarproductsonthespacesS2Uand^2Uaccordingtotheruleshu1u2;u3u4i:=1 2!u1;u3!u2;u4+!u1;u4!u2;u3andhu1^u2;u3^u4i:=!u1;u3!u2;u4)]TJ/F11 9.963 Tf 9.963 0 Td[(!u1;u4!u2;u3:MakesimilardenitionsforS2V;^2V.Putthetensorproductscalarprod-uctonS2U^2Vand^2US2V.DeclarethespacesS2U^2Vand^2US2Vinthedirectsum,^2UV=S2U^2V^2US2V:Thisdirectsumscalarproductthencoincideswiththescalarproductdescribedabove.Inparticular,whenV= Uand= !,andwhenwethinkofconjugationasmappingS2U^2 U7!^2US2 U,weareinthesituationdescribedabove,ofg g,wheregisthetensorproductmetriconS2U^2 U.

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224CHAPTER10.PETROVTYPES.10.5SpinorsandMinkowskispace.LetUbeatwodimensionalcomplexvectorspacewithanantisymmetricnon-degeneratebilinearform,!.ThenwegetasymmetricbilinearformonU U.Letuscheckthattherestrictionofthissymmetricformtotherealsubspaceisreal,andisoftype1;3.Toseethis,letubeanynon-zeroelementofU,andletvbesomeothervectorwith!u;v=1 p 2:Thenu wisanullvectorofU Uforanyw,since!u;u=0.Thenhu u+v v;u u+v vi=2!u;v2=1:Alsohu u)]TJ/F11 9.963 Tf 9.963 0 Td[(v v;u u)]TJ/F11 9.963 Tf 9.962 0 Td[(v vi=)]TJ/F8 9.963 Tf 7.749 0 Td[(1hu v+v u;u v+v ui=)]TJ/F8 9.963 Tf 7.749 0 Td[(1hiu v)]TJ/F11 9.963 Tf 9.963 0 Td[(v u;iu v)]TJ/F11 9.963 Tf 9.962 0 Td[(v ui=)]TJ/F8 9.963 Tf 7.749 0 Td[(1andthevectorsu u+v v;u u)]TJ/F11 9.963 Tf 9.924 0 Td[(v v;u v+v u;iu v)]TJ/F11 9.963 Tf 9.925 0 Td[(v uaremutuallyorthogonal,andspantherealsubspace.Let:=u^v.Socanbecharacterizedastheuniqueelementof^2Usatisfying!=1 2.Thenu u^u v+v u=u2 + u2:Thiselementof^2T,whereTistherealsubspaceofU Uisthewedgeproductofanullvector,u uandaspacelikevectororthogonaltothenullvector.Henceitcorrespondstoanullplane"containingthenullvectoru u.Thuseachnon-zerou2Udeterminesanullvector,u u,andanullplane",Qu,correspondingtothedecomposableelementu2 + u2.Multi-plyingubyaphasefactor,eimultiplies ubye)]TJ/F10 6.974 Tf 6.227 0 Td[(iandhencedoesnotchangethenullvectoru u.Butitchangesthenullplanesinceu27!e2iu2.Geometri-cally,thisamountstoreplacingvbye)]TJ/F10 6.974 Tf 6.227 0 Td[(ivandsorotatesthevectoru v+v uby2.SoQeiuisobtainedfromQubyrotationthroughangle2.Wecancomputethestaroperatorintermsoftheorthonormalbasiscon-structedabovefromuandv,andndbydirectcomputationthat?u2 =iu2 thesamechoiceofsignforallu.Sincethesignofthestaroper-atorisdeterminedbytheorientation,wecanchoosetheorientationsothat?u2 =iu2 8u2U,andhencethedecomposition^2U U=S2U^2 U^2US2 Uisthedecompositionintothe+iand)]TJ/F11 9.963 Tf 7.749 0 Td[(ieigenspacesofthecomplexicationofstaron^2U U=^2TRC.

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10.6.TRACELESSCURVATURES.22510.6Tracelesscurvatures.Ifweuseand toidentify^2Uand^2 UwithC,wethencanwrite^2U U=S2US2 U;asthedecompositionintotheieigenvaluesofthestaroperator.ThenS2^2U U)]TJ/F8 9.963 Tf 9.492 1.494 Td[(=S2S2US2S2 Uisthe)]TJ/F8 9.963 Tf 7.749 0 Td[(1eigenspaceoftheinducedactionof?onS2^2U U.ThecomplexconjugationistheobviousonecomingfromthecomplexconjugationU! U.Thuswemayidentifythespaceofreal)]TJ/F8 9.963 Tf 7.748 0 Td[(1eigenvectorsof?onS2^2TwithS2S2Uconsideredasarealvectorspace.ThespaceS2S2Uissixdimensionaloverthecomplexnumbers.Ithasaninvariantvedimensionalsubspace,S4U,thespaceofquarticpolynomialsinelementsofU.Wecanalsodescribethissubspaceasfollows:wecanusethequadraticformonS2UandonS2S2Utodeneamap:S2S2U!EndS2U;hts1;s2i=ht;s1s2i;t2S2S2U;s1;s22S2U;andwheres1s22S2S2U.ThisidentiesS2S2UwiththespaceofallsymmetricoperatorsonS2U,symmetricwithrespecttothequadraticformonS2U.Themapt7!trtisalinearformwhichisinvariantlydened.SinceSlUactsirreduciblyonS4U,therestrictionofthislinearformtoS4Umustbezero,sowecanthinkofS4Uasconsistingoftracelessoperators.Uptoaninessentialscalar,wecanconsidertherestrictionoftoS4U,callit,characterizedbyhts1;s2i=ht;s1s2i;wheres1s22S4Uistheproductofs1ands2inthesymmetricalgebra,andthescalarproductontherightisthescalarproductinS4U.10.7Thepolynomialalgebra.Itwillbeconvenienttodealwiththeentiresymmetricalgebra,S:=SU,whereSkdenotethehomogeneouspolynomialsofdegreek.Foranyu6=02U,letusnowchoosewsuchthat!u;w=1,anddenethederivationonSu:Sk!Sk)]TJ/F7 6.974 Tf 6.227 0 Td[(1byuz=!u;z8z2UwhichdenesitongeneratorsandhencedeterminesitonallofS.Thecom-mutatorofanytwoderivationsisaderivation,andthecommutator[iu;iu0]

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226CHAPTER10.PETROVTYPES.vanishesonS1andhenceonSforanypairofvectorsuandu0.Thusallderivationsucommute,andhenceu7!uextendstoahomomorphism:S!EndS:Thisallowustoextend!toabilinearformonSbyhs;ti:=[ist]0wherethesubscript0denotesthecomponentindegreezero.SothespacesSkandS`areorthogonalwithrespecttothisbilinearform,andtherestrictiontoSkSkissymmetricwhenkiseven,andantisymmetricwhenkisodd.Wecanwritetheoperatort;t2S4ashts1;s2i=ts1s2;s1;s22S2:Sinceeveryquartichomogeneouspolynomialintwovariablesisaproductoffourlinearpolynomials,t=u1u2u3u4,wecanusethisformulaandthederivationpropertytodescribetheoperatort.10.8Petrovtypes.Forexample,supposethatt=u4,i.e.allfourfactorsareidentical.Thenu4ukw4)]TJ/F10 6.974 Tf 6.226 0 Td[(k=0,fork6=0andu4w4=12.Henceu4u2=u4uw=0;u4w2=6u2:Thusforanynonzerou2U,theoperatoru4isarankonenilpotentoperatorwithimageCu2.Supposethatthreeofthefactorsoftarethesame,andthefourthlinearlyindependent.Sowemayassumethatt=u3wforu;w2Uwith!u;w=1.Thenu3wukwn)]TJ/F10 6.974 Tf 6.226 0 Td[(k=0;k6=1andu3wuw3=)]TJ/F8 9.963 Tf 7.749 0 Td[(1:Sou3wu2=0;u3wuw2Cu2;u3ww22Cuw:Thusu3whaskernelCu2andimagetheplanespannedbyu2anduwinS2.Theimageofthisplaneisthekernel,sou3wisatwostepnilpotentoperator.Nextconsiderthecasewhereu1=u2;u3=u4;u26=u3,allnotzero.Thenon-zerovalueof!u1;u3isaninvariant.Butwecanalwaysmultiplyourelementtbyascalarfactor,toarrangethatthisvalueisone.Souptoscalarmultiplewehavet=u2w2for06=u2U;!u;w=1.Thenu2w2ukwk=0;k6=2;u2w2u2w2=4:

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10.9.PRINCIPALNULLDIRECTIONS.227OurcurrentchoiceofthenormalizationofthescalarproductonS2yieldshu2;w2i=hw2;u2i=2;huw;uwi=)]TJ/F8 9.963 Tf 7.748 0 Td[(1allotherscalarproductsequalzeroforthebasisu2;uw;w2ofS2.Hencefollowsthatu2w2isdiagonizablewitheigenvalues)]TJ/F8 9.963 Tf 7.749 0 Td[(4;2;2:u2w2u2=2u2;u2w2w2=2w2;u2w2uw=)]TJ/F8 9.963 Tf 7.749 0 Td[(4uw:Supposethatexactlytwofactorsareequal.Wecanassumethatthetwoequalfactorsareu.Multiplyingbyscalarsifnecessary,wecanarrangethat!u;u3=!u;u4=1.Sou3=w+au;u4=w+bu;a6=b:Replacingwbyw)]TJ/F7 6.974 Tf 11.159 3.923 Td[(1 2a+buwemaywritet=u2w+ruw)]TJ/F11 9.963 Tf 9.963 0 Td[(ru=u2w2)]TJ/F11 9.963 Tf 9.962 0 Td[(r2u4;r6=0;wherewehaver=1 2a)]TJ/F11 9.963 Tf 10.256 0 Td[(b.Nowthesemisimpleelementu2w2commuteswiththerankonenilpotentelementu4sinceu2w2=0@2000)]TJ/F8 9.963 Tf 7.749 0 Td[(400021A;u4=0@0060000001Aintermsofthebasisu2;uw;w2ofS2UInfact,theformofthesetwomatricesshowsthattheoperatoru2w2)]TJ/F11 9.963 Tf 9.962 0 Td[(r2u4isnotdiagonizable.Finally,thegenericcaseoffourdistinctlinearfactorscorrespondstothegenericcaseofthreedistincteigenvalues.WethushavethevariousPetrovtypesfornon-zeroelements:name#linearfactorsstructureoft I4distinctdistincteigenvalues,diagonalizableII3distinct2;)]TJ/F11 9.963 Tf 7.749 0 Td[(;)]TJ/F11 9.963 Tf 7.749 0 Td[(;non-diagonalizableDu1=u26=u3=u42;)]TJ/F11 9.963 Tf 7.749 0 Td[(;)]TJ/F11 9.963 Tf 7.749 0 Td[(;diagonalizableIIIu1=u2=u36=u4nilpotent,rank2Nu1=u2=u3=u4nilpotent,rankone10.9Principalnulldirections.WehaveidentiedthespaceS2=S2Uwiththe+ieigenspaceof?actingon^2TC.Themap7!)]TJ/F11 9.963 Tf 9.518 0 Td[(i?isareallinearidenticationof^2Twiththiseigenspace,underwhichmultiplicationbyiispulledbacktothestaroperator.SoanelementcorrespondstoanullvectorinS2ifandonlyifitsatises^=0andh;i=0,andsodeterminesanullplanewhichisdegenerateundertherestrictionoftheLorentzscalarproduct.Suchanullplanecontainsauniquenullline.WecandescribethisnullplaneandnulllineintermsofS2asfollows.ThenullelementsofS2arethoseelementswhicharesquaresof

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228CHAPTER10.PETROVTYPES.linearelements.Indeed,everyelementofS2canbefactoredintotheproductoftwolinearfactors,sayasuv,andifvisnotamultipleofuthenhuv;uvi6=0.Sothenullbivectorsin^2Tcorrespondtoelementsoftheformu2,andthecorrespondingnulllineinTisthelinespannedbyu u.Ifu2satiseshtu2;u2i=0theniscalledaprincipalnullbivectoranditscorrespondingnulllineiscalledaprincipalnullline,andanon-zerovectorinaprincipalnulllineiscalledaprincipalnullvector.If=u2thenwesaythatuisaprincipalspinor.Projectively,thetwoquadriccurvesh;i=0andht;i=0willintersectatfourpoints,butthesepointsmaycoalescetogivemultiplepointsofintersection.Themultiplicity,m,ofaprincipalnullvector`=u uisdenedtobem=1ifu2isnotaneigenvectoroft,m=2ifu2isaneigenvectoroftwithnon-zeroeigenvalue,m=3iftu2=0;dimkert=1;m=4iftu2=0anddimkert=2.Theconditionforutobeaprincipalnullspinorcanbewrittenastu4=0:Ifwewritetasaproductoflinearfactors,t=u1u2u3u3weseethatthisisequivalenttosayingthatu=uiuptoaconstantfactor,i.e.thatubeafactoroft.Ifwenowgobacktotheprevioussectionandexamineeachofthenormalformsweconstructedforeachtype,weseethatthefactorizationpropertiesdeningthetypeoftalsogivethemultiplicitiesoftheprincipalnullvectors.SotypeIhasfourdistinctprincipalnullvectorseachofmultiplicity1,typeIIhasoneprincipalnullvectorofmultiplicity2andtwoofmultiplicity1,typeDhastwoprincipalnullspinorseachofmultiplicitytwo,typeIIIhasoneofmultiplicity3andoneofmultiplicity1,andtypeNhasoneprincipalnullvectorofmultiplicity4.Insymbols:I,;1;1;1II,;1;1D,;2III,;1N,4:Hereisanotherdescriptionofthemultiplicityofanullvector,k=u u.Theelementu2correspondstoabivector=k^xwherexissomespacelikevectorperpendiculartok.Tosaythatkisprincipalisthesameastosaythatgt;=0wheregisthecomplexscalarproductpulledbackto^2T.The

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10.9.PRINCIPALNULLDIRECTIONS.229realpartofgisjusttheoriginalscalarproductsoht;i=0.Sincewecanmultiplyuandbyanarbitraryphasefactor,theconditionofbeingprincipalisthathtk^x;k^xi=08x?k:Writinghtk^x;k^xi=hRkxk;xiandthenpolarizing,weseethatthisisthesameassayingthathRkxk;yi=08x;y?k:.3Weclaimthatthenullvectorisprincipalwithmultiplicity2ifandonlyifhRkxk;yi=0;8x?kand8y:.4Proof.Supposethatk=u uisafactoroforderatleasttwoint.Thishappensifandonlyiftu3v=tu4=0.Thisisthesameassayingthattu2isorthogonaltothecomplextwodimensionalspaceu2?relativetothecomplexmetric.Thiscomplextwodimensionalspacecorrespondstoarealfourdimensionalspace,theorthogonalcomplementofthetwodimensionalsubspaceof^2Tspannedbyk^x=u2andk^z=iu2.Herexandzarespacelikevectorsorthogonaltokandtoeachotherasabove.Souisarepeatedfactoroftifandonlyifh[R]k^x;i=0forallinthisfourdimensionalsubspaceof^2Tandsimilarlyforz.Thefourdimensionalspaceinquestionisspannedbythethreedimensionalspaceofelementsoftheformk^y;y2Tandtheelementx^z.Inparticular,appliedtoelementsoftheformx^ywegetcondition10.4forthexwehavechosenandalsoforxreplacedbyz.ItisautomaticwithxreplacedwithksinceRkk=0.Thisprovesthat.4holdsifuisarepeatedfactor.Toprovetheconverse,wemustshowthat[R]k^xisorthogonaltothefourdimensionalsubspaceof^2Tspannedbyallk^yandx^z.Condition.4guaranteestheorthogonalityfortheelementsoftheformk^y.SowemustprovethathRkxx;zi=0:ThiswillfollowfromtheRicciatnesscondition.Indeed,chooseanullvector`orthogonaltok;xandzwithhk;`i=0.Then0=Ric[R]k;z=XijgijhRkyiz;yjasyi;yjrangeovertheelementsk;`;x;z.ThissumreducestohRk`z;k+hRkxz;xiallothertermsvanishing.Thersttermvanishesby.4andthisimpliesthevanishingofthesecond.

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230CHAPTER10.PETROVTYPES.10.10Kerr-Schildmetrics.Wewanttouse.4toconcludethatifaRicciatmetricisobtainedfromaatmetricbyaddingthesquareofanullformtoaatmetric,thenthetracefreecurvaturehasarepeatedfactor.Moreprecisely,ifthenullformis,andthecorrespondingvectoreldisN,thenwewillshowthatthisrepeatedfactorisuwhereu u=N.Forthiswerecallthefollowingfacts,AnecessaryconditionforthenewmetrictobeRicciatisthatrNN=Nforsomefunction.ThustheintegralcurvesofNarenullgeodesicsintheoldandnewmetricbutwithapossiblynon-aneparametrization..ThenewaneconnectionisdiersfromtheoldaneconnectionbyaddingtheatensorA2TS2Twhichcanbeexpressedintermsofthenullform.Thatis,thenewconnectionisrXY+AXYwhereristheoldconnectionandwecanwritedownaformulaforAXYinvolvingthenullformanditscovariantderivatives.Inparticular,AN=N.5i.e.ANX=XN:AlsoA=i.e.AXY=XY:.6IftheaneconnectionismodiedbytheadditionofatensorA,thenthenewcurvaturediersfromtheoldcurvaturebyR0XY=RXY+[AX;AY]+rAX;Y)]TJ/F8 9.963 Tf 9.962 0 Td[(rAY;X:HererAX;Y2HomT;TisdenedbyrAX;YZ=rXAYZ)]TJ/F11 9.963 Tf 9.963 0 Td[(ArXYZ)]TJ/F11 9.963 Tf 9.962 0 Td[(AYrXZwhereristheconnectionrelativetotheoldmetric.

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10.10.KERR-SCHILDMETRICS.231InourcasetheoldcurvatureiszeroandweareinterestedincomputinghR0NXN;Yi=)]TJ/F11 9.963 Tf 7.749 0 Td[(RNXY:WehaveANN=0andANAXN=ANANX=0sothebrackettermmakesnocontribution.SinceNisanullvectoreld,wehaverXN?NandsoArXNN=ANrXN=0foranyX,andANN0sorXANN=0foranyX.SoweareleftwiththeformulaR0NXY=rNAXY:NowrNA=rNA)]TJ/F8 9.963 Tf 9.962 0 Td[(rNA=)]TJ/F8 9.963 Tf 7.749 0 Td[(rN+2orRNXY=)]TJ/F8 9.963 Tf 7.749 0 Td[(rN+2XY:.7Inparticular,ifX?NsoX=0,theprecedingexpressionvanishesforanyYprovingthatNisaprincipalnullvectorofmultiplicityatleasttwo.QED

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232CHAPTER10.PETROVTYPES.

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Chapter11Star.11.1Denitionofthestaroperator.WestartwithanitedimensionalvectorspaceVovertherealnumberswhichcarriestwoadditionalpiecesofstructure:anorientationandanon-degeneratescalarproduct.Thescalarproduct,h;ideterminesascalarproductoneachofthespaces^kVwhichisxedbytherequirementthatittakeonthevalueshx1^^xk;y1^^yki=dethxi;yjiondecomposableelements.Thisscalarproductisnon-degenerate.Indeed,startingfromanorthonormal"basise1;:::;enofV,thebasisei1^^eik;i1<
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234CHAPTER11.STAR.Thewedgeproductallowsustoassigntoeachelementof2^kVthelinearfunction,`on^n)]TJ/F10 6.974 Tf 6.227 0 Td[(kVgivenby^!=`!8!2^n)]TJ/F10 6.974 Tf 6.226 0 Td[(kV:Butsincetheinducedscalarproducton^n)]TJ/F10 6.974 Tf 6.226 0 Td[(kVisnon-degenerate,anylinearfunction`isgivenas`!=h;!iforaunique=`.Sothereisauniqueelement?2^n)]TJ/F10 6.974 Tf 6.227 0 Td[(kVdeterminedby^!=h?;!i:.1Thisisourconventionwithregardtothestaroperator.Inshort,wehavedenedalinearmap?:^kV!^n)]TJ/F10 6.974 Tf 6.227 0 Td[(kVforeach0knwhichisdeterminedby.1.LetuschooseanorthonormalbasisofVasabove,butbeingsuretochooseourorthonormalbasistobeoriented,whichmeansthat=e1^en:LetI=i1;:::;ikbeak)]TJ/F8 9.963 Tf 11.335 0 Td[(subsetoff1;:::;ngwithitselementsarrangedinorder,i1<
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11.2.DOES?:^KV!^N)]TJ/F10 6.974 Tf 6.226 0 Td[(KVDETERMINETHEMETRIC?235andthescalarproductonRisthestandardoneassigningtoeachrealnumberitssquare.Takingthenumber1asabasisforRthoughtofasaonedimen-sionalvectorspaceoveritself,thismeansthath1;1i=1.Wedgeproductbyanelementof^0V=Risjustordinarymultiplicationbyofavectorbyarealnumber.So,^1=1^=andthedenition^1=h?;1irequiresthat?=1.4nomatterwhatthesignatureofthescalarproductonVis.Ontheotherhand,?1=.Wedeterminethesignfrom1^==h?1;iso?1=h;i=)]TJ/F8 9.963 Tf 7.749 0 Td[(1q.5inaccordancewithourgeneralrule.Applying?twicegivesalinearmapof^kVintoitselfforeachk.Weclaimthat?2=)]TJ/F8 9.963 Tf 7.749 0 Td[(1kn)]TJ/F10 6.974 Tf 6.227 0 Td[(k+qid:.6Indeed,sincebothsidesarelinearoperatorsitsucestoverifythisequationonbasiselements,e.g.onelementsoftheformeI,andbyrelabelingifnecessarywemayassume,withoutlossofgenerality,thatI=f1;:::;kg.Then?e1^^ek=)]TJ/F8 9.963 Tf 7.749 0 Td[(1rIcek+1^^en;while?ek+1^^en=)]TJ/F8 9.963 Tf 7.749 0 Td[(1kn)]TJ/F10 6.974 Tf 6.226 0 Td[(k+rIe1^^eksincetherearen)]TJ/F11 9.963 Tf 10.015 0 Td[(ktranspositionsneededtobringeachoftheei;ik,pastek+1^^en.SincerI+rIc=q,.6follows.11.2Does?:^kV!^n)]TJ/F11 9.963 Tf 7.749 0 Td[(kVdeterminethemet-ric?Thestaroperatordependsonthemetricandontheorientation.Clearly,chang-ingtheorientationchangesthesignofthestaroperator.Letusdiscussthequestionofwhenthestaroperatordeterminesthescalarproduct.Weclaim,asapreliminary,thatitfollowsfromthedenitionthat^?!=)]TJ/F8 9.963 Tf 7.749 0 Td[(1qh;!i8;!2^k.7forany0kn.Indeed,wehavereallyalreadyveriedthisformulaforthecasek=0ork=n.Foranyintermediatek,weobservethatbothsidesare

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236CHAPTER11.STAR.bilinearinand!,soitsucestoverifythisequationonbasiselements,i.ewhen=eIand!=eKwhereIandKarek)]TJ/F8 9.963 Tf 7.748 0 Td[(subsetsoff1;:::;ng.IfK6=IthenheI;eKi=0,whileKcandIhaveatleastoneelementincommon,soeI^?eK=0.Hencebothsidesequalzero.SowemustonlychecktheequationforI=K,andwithoutlossofgeneralitywemayassumebyrelabelingtheindicesthatI=f1;2;:::;kg.Thenthelefthandsideof11.7is)]TJ/F8 9.963 Tf 7.748 0 Td[(1rIcwhiletherighthandsideis)]TJ/F8 9.963 Tf 7.748 0 Td[(1q+rIby.2.Sinceq=rI+rIctheresultfollows.Onemightthinkthat.7impliesthat?actingon^kV;k6=0;ndeter-minesthescalarproduct,butthisisnotquitetrue.Hereisthesimplestandveryimportantcounterexample.TakeV=R2withthestandardpositivedef-initescalarproductandk=1.So?:^1V=V!V.Intermsofanorientedorthonormalbasiswehave?e1=e2;?e2=)]TJ/F11 9.963 Tf 7.748 0 Td[(e1,thus?iscounterclockwiserotationthroughninetydegrees.Anynon-zeromultipleofthestandardscalarproductwilldeterminethesamenotionofangle,andhencethesame?operator.Thus,intwodimensions,the?operatoronlydeterminesthemetricuptoscale.Thereasonforthebreakdownintheargumentisthattheoccurringontherighthandsideof1.7dependsonthechoiceofmetric.Itisclearfrom.7thatthestaroperatoractingon^kVdeterminestheinducedscalarproducton^kVuptoscale.Indeed,leth;i0denoteasecondscalarproductonV.Let0denotetheelementof^nVdeterminedbythescalarproducth;i0,so0=aforsomenon-zeroconstant,a>0.Finally,forpurposesofthepresentargument,letususemoreprecisenotationanddenotethescalarproductsinducedon^kVbyh;ikandh;i0k.Then.7impliesthath;i0k=1 ah;ik:.8Forexample,supposethatweknowthattheoriginalscalarproductsonVdierbyapositivescalarfactor,sayh;i0=ch;i;c>0:Thenh;i0k=ckh;iwhile0=1 cn=2sinceh;i0n=cnh;i.Hencethefactthethestaroperatorsarethesameon^kVimpliesthatc=1foranykotherthank=n 2.Thiswasexactlythepointofbreakdowninourtwodimensionalexamplewheren=2;k=1.

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11.2.DOES?:^KV!^N)]TJ/F10 6.974 Tf 6.226 0 Td[(KVDETERMINETHEMETRIC?237Ingeneral,ifh;iispositivedenite,andh;i0isanyothernon-degeneratescalarproduct,thentheprincipalaxistheoremthediagonalizationtheoremforsymmetricmatricesfromlinearalgebrasaysthatwecanndabasise1;:::;enwhichisorthonormalforh;iandorthogonalwithrespecttoh;i0withhei;eii0=si;si6=0:ThenheI;eIi0=si1sikheI;eIi;I=fi1;:::;ikg:Theonlywaythat.8canholdforagiven00ifweusethesignature;1.Conversely,ifnisanynon-zeronullvectorandwisanyspacelikevectorperpendiculartonthentheplanespannedbynandwisadegenerateplanesothathu^v;u^vi2=0foranypairofvectorsspanningthisplane.

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238CHAPTER11.STAR.Noticethatifu0;v0issomeotherpairofvectorsspanningtheplanePfu;vg,thenu0^v0=bu^vforsomescalarb6=0.Conversely,ifu0^v0=bu^v;b6=0,thenu0;v0spanthesameplane,Pfu;vgasdouandv.Soeverylineofnulldecomposablebivectorsi.e.alineoftheformfru^vg;hu^v;u^vi2=0determinesalineofofnullvectors,fcng.Conversely,ifwestartwiththelinefcngofnullvectors,letQn:=n?betheorthogonalcomplementofn.ItisathreedimensionalsubspaceofVcontainingn;allelementsofQnnotlyingonthelinefcngbeingspacelike.Thechoiceofanyspacelikevector,w,inQn,saywithjhw;wij=1thendeterminesadegenerateplanecontainingnandlyinginQn.Wethusgetawholecircle"ofnullplanesPwithf0gfcngPQnV:Ingeneral,achainofincreasingsubspacesiscalledaag"inthemathematicalliterature.Ifthedimensionsincreasebyoneateachstepitiscalledacompleteag".Whatwehavehereisthateachu^vwithhu^v;u^vi2=0determinesaspecialkindofcompleteag,startingwithalineofnullvectors.Penroseusesthefollowingpicturesquelanguage:hecallsfcngtheagpoleaboutwhichtheplaneProtates.Allthisisoverkillforourpresentpurpose,butwillbeneededlateron.Whatwedoconcludeforourcurrentneedsisthattheconeofnullbivectors,f!2^2Vjh!;!i2=0gdeterminestheconeofnullvectors,N:=fw2Vjhw;wi=0g.Sowecanconcludetheproofwiththefollowing:3.LetWbeanyvectorspacewithanon-degeneratescalarproducth;ioftypep;qwithp6=0;q6=0andletN:=fw2Wjhw;wi=0gbeitsnullcone.Ifh;i0isanyothernondegeneratescalarproductwiththesamenullconethenh;i0=sh;iforsomenon-zeroscalar,s.SowenowknowthatinourfourdimensionalMinkowskispace,aknowledgeof?:^2V!^2Vdeterminesthemetricuptoscale.Herearesomemorespecialfactswewillneedlater.4.Showthat?:^2V!^2Visselfadjointrelativetoh;i2,i.e.h?;!i=h;?!i8;!2^2V:ThenextthreeproblemsrelatetothediscussioninChapterIX.Itfollows

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11.2.DOES?:^KV!^N)]TJ/F10 6.974 Tf 6.226 0 Td[(KVDETERMINETHEMETRIC?239fromourgeneralformulathat?:^2V!^2Vsatises?2=)]TJ/F8 9.963 Tf 11.069 0 Td[(id:Thismeansthat?:^2V!^2Vhaseigenvaluesiand)]TJ/F11 9.963 Tf 7.749 0 Td[(i.Inordertohaveactualeigen-vectors,wemustcomplexify.Soweintroducethespace^2VC:=^2VC:Anelementof^2VCisanexpressionoftheform+i!;;!2^2V.Anylinearoperatoron^2Vautomaticallyextendstobecomeacomplexlinearoperatoron^2VC.Forexample?+i!:=?+i?!.Similarly,everyrealbilinearformon^2Vextendstoacomplexbilinearformon^2VC.Forexample,h;i=h;i2wewillnowreverttotheimprecisenotationanddropthesubscript2extendsash+i!;+ii:=h;i+ih!;i+ih;i)-222(h!;i:Thesubspaces^2V+C:=f)]TJ/F11 9.963 Tf 9.962 0 Td[(i?g^2V)]TJ/F49 6.974 Tf -2.213 -7.209 Td[(C:=f+i?g2^2Varecomplexlinearsubspaceswhicharethe+iand)]TJ/F11 9.963 Tf 7.749 0 Td[(ieigenspacesof?on^2VC.Theyareeachofthreecomplexdimensionsand^2VC=^2V+C^2V)]TJ/F49 6.974 Tf -2.214 -7.209 Td[(C:Inthephysicsliteraturetheyhavetheunfortunatenamesofthespaceofselfdual"andanti-selfdual"bivectors.5.Showthatthesetwosubspacesareorthogonalunderthecomplexextensionofh;i.Therealvectorspace^2Vhasdimension6.Hencethespaceofsymmetrictwotensorsover^2V,thespaceS2^2Vhasdimension67=2=21.Theoperator?:^2V!^2Vinducesanoperatorshallwealsodenoteitby??ofS2^2V!S2^2V.Theeigenvaluesofthisinducedoperatorwillbeallpos-sibleproductsoftwofactorsofeitherior)]TJ/F11 9.963 Tf 7.749 0 Td[(i,sotheeigenvaluesoftheinducedoperator?:S2^2V!S2^2Vare1.Thecorrespondingeigenspacesarenowreal.6.Showthatthedimensionofthe)]TJ/F8 9.963 Tf 7.748 0 Td[(1eigenspaceis12andthedimensionofthe+1eigenspaceis9.Hint:Thedimensionsofrealeigenspacesdonotchangeifwecomplexifyandthenconsiderdimensionsoverthecomplexnumberswiththesamerealeigenvaluesofthecomplexiedoperator.DescribethespaceS2W1W2,thesymmetrictwotensorsoveradirectsumoftwovectorspaces.

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240CHAPTER11.STAR.ThereasonthattheprecedingproblemwillbeofimportancetousisthatthecurvaturetensorRatanypointofaLorentzianmanifoldcanbethoughtofaslyinginS2^2VwhereV=TMx,thecotangentspaceatapoint.Actually,oneoftheBianchiidentitiesthecyclicsumconditionimposesoneadditionalalgebraicconstraintonthecurvaturetensorsothatRliesina20dimensionalsubspaceofthe21dimensionalspaceS2^2V.TheEinsteinconditioninfreespacewillturnouttofurtherrestrictRtolieinanelevendimensionalsubspaceofthetwelvedimensionalspaceof)]TJ/F8 9.963 Tf 7.749 0 Td[(1eigenvectors,andthemorestringentconditionofbeingRicciatwillrestrictRtolieinatendimensionalsubspaceofthiselevendimensionalspace.Wewillspendagoodbitoftimestudyingthistendimensionalspace.11.3Thestaroperatoronforms.IfMisanorientedsemi-Rimannianmanifold,wecanconsiderthestaroperatorassociatedtoeachcotangentspace.Thus,operatingpointwise,wegetastaroperatormappingk)]TJ/F8 9.963 Tf 7.749 0 Td[(formsinton)]TJ/F11 9.963 Tf 9.962 0 Td[(kforms,wheren=dimM:?:kM!n)]TJ/F10 6.974 Tf 6.227 0 Td[(kM:Manyoftheimportantequationsofphysicshavesimpleexpressionsintermsofthestaroperatoronforms.thepurposeoftherestoftheseexercisesistodescribesomeofthem.Infact,alloftheequationsweshallwritedownwillbeforvariousstaroperatorsofatspaceoftwo,threeandfourdimensions.Butthegeneralformulationgoesoverunchangedforcurvedspacesorspacetimes.11.3.1ForR2.Wetakeasourorthonormalframeofformstobedx;dyandtheorientationtwoformtobe:=dx^dy.Then?dx=dy;?dy=)]TJ/F11 9.963 Tf 7.748 0 Td[(dxaswehavealreadyseen.7.Foranypairofsmoothrealvaluedfunctionsuandv,let!:=udx)]TJ/F11 9.963 Tf 9.963 0 Td[(vdy:Writeoutthepairofequationsd?!=0;d!=0.9asasystemoftwopartialdierentialequationsforuandv.WewillndlateronthatMaxwell'sequationsintheabsenceofsourceshasexactlythissame

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11.3.THESTAROPERATORONFORMS.241expression,exceptthatforMaxwell'sequations!isatwoformonMinkowskispaceinsteadofbeingaoneformontheplane.Ifweallowcomplexvaluedforms,writef=u+ivanddz=dx+idythentheabovepairofequationscanbewrittenasd[fdz]=0:ItthenfollowsfromStokes'theoremthattheintegraloffdzaroundthebound-aryofanyregionwherefisdenedandsmoothmustbezero.ThisisknownastheCauchyintegraltheorem.Noticethatfdz=!+i?!istheanti-selfdualformcorrespondingto!intheterminologyoftheprecedingsection.11.3.2ForR3.Wehavetheorthonormalcoframeelddx;dy;dz,with=dx^dy^dz,so?1=;?dx=dy^dz?dy=)]TJ/F11 9.963 Tf 7.749 0 Td[(dx^dz?dz=dx^dywith?2=1inalldegrees.Let:=@2 @x2+@2 @y2+@2 @z2:Let=adx+bdy+cdz=Adx^dy+Bdx^dz+Cdy^dz:8.Showthat?d?d)]TJ/F11 9.963 Tf 9.963 0 Td[(d?d?=)]TJ/F8 9.963 Tf 7.749 0 Td[(adx)]TJ/F8 9.963 Tf 9.963 0 Td[(bdy)]TJ/F8 9.963 Tf 9.963 0 Td[(cdz.10and)]TJ/F11 9.963 Tf 9.693 0 Td[(?d?d+d?d?=)]TJ/F8 9.963 Tf 7.749 0 Td[(Adx^dy)]TJ/F8 9.963 Tf 9.694 0 Td[(Bdx^dz)]TJ/F8 9.963 Tf 9.693 0 Td[(Cdy^dz:.11

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242CHAPTER11.STAR.11.3.3ForR1;3.Wewillchoosethemetrictobeoftype;3sothatwehavetheorthonormal"coframeeldcdt;dx;dy;dzwithhcdt;cdti=1andhdx;dxi=hdy;dyi=hdz;dzi=)]TJ/F8 9.963 Tf 7.748 0 Td[(1:Wewillchoose=cdt^dx^dy^dz:Thisxesthestaroperator.ButIamfacedwithanawkwardnotationalprob-leminthenextsectionwhenwewilldiscusstheMaxwellequationsandtherelativisticLondonequations:WewillwanttodealwiththestaroperatoronR3andR1;3simultaneously,infactinthesameequation.Icoulduseasub-script,say?3todenotethethreedimensionalstaroperatorand?4todenotethefourdimensionalstaroperator.Thiswouldclutteruptheequations.SoIhaveoptedtokeepthesymbol?forthestaroperatorinthreedimensionsandforthepurposeoftherestofthissectiononly,useadierentsymbol,|,forthestaroperatorinfourdimensions.So|cdt^dx^dy^dz=1;|1=)]TJ/F11 9.963 Tf 7.749 0 Td[(cdt^dx^dy^dzwhile|cdt=)]TJ/F11 9.963 Tf 7.749 0 Td[(dx^dy^dz|dx=)]TJ/F11 9.963 Tf 7.749 0 Td[(cdt^dy^dz|dy=cdt^dx^dz|dz=)]TJ/F11 9.963 Tf 7.749 0 Td[(cdt^dx^dywhichwecansummarizeas|cdt=)]TJ/F11 9.963 Tf 9.963 0 Td[(?1|=)]TJ/F11 9.963 Tf 7.749 0 Td[(cdt^?for=adx+bdy+cdzand|cdt^dx=dy^dz|cdt^dy=)]TJ/F11 9.963 Tf 7.749 0 Td[(dx^dz|cdt^dz=dx^dy|dx^dy=)]TJ/F11 9.963 Tf 7.749 0 Td[(cdt^dz|dx^dz=cdt^dy|dy^dz=)]TJ/F11 9.963 Tf 7.749 0 Td[(cdt^dx:

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11.4.ELECTROMAGNETISM.243Noticethatthelastthreeequationsfollowfromtheprecedingthreebecause|2=)]TJ/F8 9.963 Tf 7.749 0 Td[(idasamapontwoformsinR1;3.Wecansummarizetheselastsixequationsas|cdt^=?;|=)]TJ/F11 9.963 Tf 7.749 0 Td[(cdt^?:Iwanttomakeitclearthatintheseequations=adx+bdy+cdzwherethefunctionsa;b;andccandependonallfourvariables,t;x;yandz.Similarlyisalinearcombinationofdx^dy;dx^dzanddx^dzwhosecoecientscandependonallfourvariables.Sowemaythinkofandasformsonthreespacewhichdependontime.Wehave|2=idononeformsandonthreeformswhichcheckswith|cdt^dx^dy=)]TJ/F11 9.963 Tf 7.748 0 Td[(dzor,moregenerally,|cdt^=)]TJ/F11 9.963 Tf 9.962 0 Td[(?:11.4Electromagnetism.WebeginwithtworegimesinwhichwesolelyusethestaroperatoronR3.Thenwewillpasstothefullrelativistictheory.11.4.1Electrostatics.Theobjectsofthetheoryare:alineardierentialform,E,calledtheelectriceldstrength.ApointchargeeexperiencestheforceeE.TheintegralofEalonganypathgivesthevoltagedropalongthatpath.TheunitsofEarevoltage length=energy chargelength:Thedielectricdisplacement,D,whichisatwoform.Inprinciple,wecouldmeasureDv1;v2wherev1;v22TR3xR3areapairofvectorsasfollows:constructaparallel-platecapacitorwhoseplatesaremetalparallelogramsde-terminedbyhv1;hv2wherehisasmallpositivenumber.Placetheseplateswiththecorneratxtouchthemtogether,thenseparatethem.TheyacquirechargesQ.TheorientationofR3picksoutoneofthesetwoplateswhichwecallthetopplate.ThenDv1;v2=limh!0chargeontopplate h2:TheunitsofDarecharge area:

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244CHAPTER11.STAR.Thechargedensitywhichisathreeform,.Weidentifydensitieswiththreeformssincewehaveanorientation.Thekeyequationsinthetheoryare:dE=0which,inasimplyconnectedregionimpliesthatthatE=)]TJ/F11 9.963 Tf 7.748 0 Td[(duforsomefunc-tion,ucalledthepotential.TheintegralofDovertheboundarysurfaceofsomethreedimensionalregionisthetotalchargeintheregionGauss'law:Z@UD=ZUwhich,byStokes,canbewrittendierentiallyasdD=:Iwilluseunitswhichabsorbthetraditional4into.FinallythereisaconstituitiveequationrelatingEandD.InanisotropicmediumitisgivenbyD=?Ewhereiscalledthedielectricfactor.Inahomogeneousmediumitisacon-stant,calledthedielectricconstant.Inparticular,thedielectricconstantofthevacuumisdenotedby0.Theunitsof0arecharge areachargelength energy=charge2 energylength:Thelawsofelectrostatics,sincetheyinvolvethestaroperator,determinethethreedimensionalEuclideangeometryofspace.11.4.2Magnetoquasistatics.Inthisregime,itisassumedthattherearenostaticcharges,so=0,andthatMaxwell'sterm@D=@tcanbeignored;energyisstoredinthemagneticeldratherthanincapacitors.Thefundamentalobjectsare:aoneformEgivingtheelectricforceeld.TheforceonachargeeiseE,asbefore.atwoformBgivingthemagneticinductionorthemagneticuxdensity.TheforceonacurrentelementIwhichisavectorisiIBwhereidenotesinteriorproduct.

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11.4.ELECTROMAGNETISM.245Thecurrentux,Jwhichistwoform[measuredinamps/area].aoneform,Hcalledthemagneticexcitationorthemagneticeld.TheintegralofHovertheboundary,CofasurfaceSisequaltotheuxofcurrentthroughthesurface.ThisisAmpere'slaw.ZCH=ZSJ.12Faraday'slawofinductionsaysthat)]TJ/F11 9.963 Tf 10.743 6.74 Td[(d dtZSB=ZCE:.13ByStokes'theorem,thedierentialformofAmpere'slawisdH=J;.14andofFaraday'slawis@B @t=)]TJ/F11 9.963 Tf 7.748 0 Td[(dE:.15Faraday'slawimpliesthatthetimederivativeofdBvanishes.ButinfactwehavethestrongerassertionHertz'slawdB=0:.16Equations.14,.15,and.16arethestructurallawsofelectrody-namicsinthemagnetoquasistaticapproximation.Wemustsupplementthembyconstituitiveequations.OneoftheseisB=?H;.17where?denotesthestaroperatorinthreedimensions.AccordingtoAmpere'slaw,Hhasunitscharge timelengthwhileaccordingtoFaraday'slawBhasunitsenergytime chargelength2sothathasunitsenergytime2 charge2length:Thushasunitstime2 length2=velocity)]TJ/F7 6.974 Tf 6.226 0 Td[(2

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246CHAPTER11.STAR.anditwasMaxwell'sgreatdiscovery,thefoundationstoneofallthathashap-penedsinceinphysics,that1 00=c2wherecisthespeedoflight.ThisdiscussionisabitprematureinourpresentregimeofquasimagnetostaticswhereDplaysnorole.Weneedonemoreconstituitiveequation,torelatethecurrenttotheelec-tromagneticeld.Inordinaryconductivity,onemimicstheequationV=RIforaresistorinanetworkbyOhm'slaw:J=?E:.18AccordingtotheDrudetheoryasmodiedbySommerfeldthechargecarri-ersarefreeelectronsandcanbedeterminedsemi-empiricallyfromamodelinvolvingthemeanfreetimebetweencollisionsasaparameter.Noticethatinordinaryconductivitythechargecarrierissomethingexternaltotheelec-tromagneticeld,andisnotregardedasafundamentalconstantofnaturelikec,saybutisanempiricalparametertobederivedfromanothertheory,saystatisticalmechanics.Infact,Drudeproposedthetheoryofthefreeelec-trongasin1900,somethreeyearsafterthediscoveryoftheelectron,byJ.J.Thompson,andithadamajorsuccessinexplainingthelawofWiedemannandFranz,relatingthermalconductivitytoelectricalconductivity.However,ifyoulookatthelengthyarticleonconductivityinthe1911editionoftheEncyclo-pediaBritannica,writtenbyJ.J.Thompsonhimself,youwillndnomentionofelectronsinthesectiononconductivityinsolids.ThereasonisthatDrude'stheorygaveabsolutelythewronganswerforthespecicheatofmetals,andthiswasonlyrectiedin1925inthebrilliantpaperbySommerfeldwherehereplacesMaxwellBoltzmannstatisticsbytheFermi-Diracstatistics.Allthisisexplainedinasolidstatephysicscourse.Irepeatmymainpoint-isnotafundamentalconstantandthesourceofJisexternaltotheelectromagenticelds.11.4.3TheLondonequations.Inthesuperconductingdomain,itisnaturaltomimicanetworkinductorwhichsatisestheequationV=LdI dt:SotheLondonsintroducedtheequationE=?@J @t;.19

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11.4.ELECTROMAGNETISM.247whereisanempiricalparametersimilartotheconductance,buttheanalogueofinductanceofacircuitelement.Equation.19isknownastherstLondonequation.Ifweassumethatisaconstant,wehave@ @t?dH=?@J @t=1 E:SettingH=)]TJ/F7 6.974 Tf 6.227 0 Td[(1?B,applyingd,andusing.15weget@ @td?d?B+ B=0:.20Fromthisonecandeducethatanappliedexternaleldwillnotpenetrate,butnotthefullMeissnereectexpellingallmagneticeldsinanysuperconduct-ingregion.Hereisasampleargumentaboutthenon-penetrationofimposedmagneticeldsintoasuperconductingdomain:SincedB=0wecanwrited?d?B=d?d?+?d?dB=B;where4istheusualthreedimensionalLaplacianappliedtothecoecientsofBusing.11.Supposewehaveasituationwhichisinvariantundertranslationinthexandzdirection.Forexampleaninniteslabofwidth2awithsidesaty=aparalleltothey=0plane.Thenassumingthesolutionalsoinvariant,.20becomes )]TJ/F11 9.963 Tf 13.779 6.74 Td[(@2 @y2@B @t=0:Ifweassumesymmetrywithrespecttoy=0intheproblem,weget@B @t=Ctcoshy ;where=s iscalledthepenetrationdepthofthesuperconductingmaterial.Itistypicallyoforder.1m.Supposeweimposesometimedependentexternaleldwhichtakesonthethevaluebtdx^dy;forexample,onthesurfaceoftheslab.Continuitythengives@B @t=b0tcoshy= cosha=:Thequotientontherightdecaysexponentiallywithpenetrationy=.Soexternallyappliedmagneticeldsdonotpenetrate,inthesensethatthetimederivativeofthemagneticuxvanishesexponentiallywithinafewmultiplesofthepenetrationdepth.ButthefullMeissnereectsaysthatallmagneticeldsintheinteriorareexpelled.

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248CHAPTER11.STAR.SotheLondonsproposedstrengthening.20byrequiringthattheexpres-sioninparenthesisin.20beactuallyzero,insteadofmerelyassumingthatitisaconstant.Sinced?B=dH=Jassumingthatisaconstantwegetd?J=)]TJ/F8 9.963 Tf 9.913 6.74 Td[(1 B:.21Equation.21isknownasthesecondLondonequation.11.4.4TheLondonequationsinrelativisticform.WecanwritethetwoLondonequationsinrelativisticform,bylettingj=)]TJ/F11 9.963 Tf 7.749 0 Td[(J^dtbethethreeformrepresentingthecurrentinspacetime.Ingeneral,wewritej:=)]TJ/F11 9.963 Tf 9.963 0 Td[(J^dt.22asthethreeforminspacetimegivingtherelativisticcurrent",butinthequasistaticregime=0.Wehave|j=1 c?J;aoneformonspacetimewithnodtcomponentunderourassumptionoftheabsenceofstaticchargeinourspacetimesplitting.Socd|j=dspace?J)]TJ/F11 9.963 Tf 11.159 6.739 Td[(@?J @t^dt;wherethedontheleftisthefulldoperatoronspacetime.Fromnowon,untiltheendofthishandout,wewillbeinspace-time,andsousedtodenotethefulldoperatorinfourdimensions,andusedspacetodenotethethreedimensionaldoperator.WerecallthatintherelativistictreatmentofMaxwell'sequations,theelec-triceldandthemagneticinductionarecombinedtogivetheelectromagneticeldF=B+E^dtsothatFaraday'slaw,.15,andHertz'slaw,.16arecombinedintothesingleequation,dF=0;.23knownastherstMaxwellequation.WeseethatthetwoLondonequationscanalsobecombinedtogived|cj=)]TJ/F11 9.963 Tf 7.749 0 Td[(F;.24whichimplies.23.ThissuggeststhatsuperconductivityinvolvesmodifyingMaxwell'sequations,incontrasttoordinaryconductivitywhichissupplemen-tarytoMaxwell'sequations.

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11.4.ELECTROMAGNETISM.24911.4.5Maxwell'sequations.Toseethenatureofthismodication,werecallthesecondMaxwellequationwhichinvolvesthetwoformG=D)]TJ/F11 9.963 Tf 9.963 0 Td[(H^dtwhereDisthedielectricdisplacement",asabove.RecallthatdspaceDgivesthedensityofchargeaccordingtoGauss'law.ThesecondMaxwellequa-tioncombinesGauss'lawandMaxwell'smodicationofAmpere'slawintothesingleequationdG=j;.25wherethethreecurrent,jisgivenby.22.Theproduct)]TJ/F7 6.974 Tf 6.227 0 Td[(1=2hastheunitsofvelocity,aswehaveseen,andletususassumethatweareinthevacuumorinamediumforwhichthatthisvelocity,isc,thesamevalueasthevacuum.SousingthecorrespondingLorentzmetriconspacetimetodeneour|operatorthecombinedconstituitiverelationscanbewrittenasG=)]TJ/F8 9.963 Tf 11.611 6.74 Td[(1 c|F;orusingunitswherec=1moresimplyasG=)]TJ/F8 9.963 Tf 9.455 6.74 Td[(1 |F:.26Fromnowon,wewillusenatural"unitsinwhichc=1andinwhichenergyandmasshaveunitslength)]TJ/F7 6.974 Tf 6.226 0 Td[(1.11.4.6ComparingMaxwellandLondon.Thematerialinthissubsection,especiallythecommentsattheend,mightbeacceptableinthemathematicsdepartment.Youshouldbewarnedthattheydonotreectthecurrentlyacceptedphysicaltheoriesofsuperconductivity,andhencemightencountersometroubleinthephysicsdepartment.Inclassicalelectromagnetictheory,jisregardedasasourceterminthesensethatoneintroducesaoneform,A,thefourpotential,withF=)]TJ/F11 9.963 Tf 7.748 0 Td[(dAandMaxwell'sequationsbecomethevariationalequationsfortheLagrangianwithLagrangedensityLMA;j=1 2dA^|dA)]TJ/F11 9.963 Tf 9.963 0 Td[(A^j:.27

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250CHAPTER11.STAR.Thismeansthefollowing:LMA;jisafourformonR1;3andwecanimaginethefunction"LMA;j:="ZR1;3LMA;j:Itisofcoursenotdenedbecausetheintegralneednotconverge.ButifCisanysmoothoneformwithcompactsupport,thevariationdLMA;j[C]:=d dsLMA+sC;jjs=0iswelldened,andtheconditionthatthisvariationvanishforallsuchCgivesMaxwell'sequations.9.ShowthatthesevariationalequationsdoindeedgiveMaxwell'sequations.UsedC^|A=dC^|dA)]TJ/F11 9.963 Tf 9.428 0 Td[(C^d|dAandthefactthat^|!=!^|fortwoforms.Inparticular,onehasgaugeinvariance:Aisonlydetermineduptotheadditionofaclosedoneform,andtheMaxwellequationsbecomed|dA=j:.28FortheLondonequations,ifweapply|to.24anduse.26weget|d|j= G;andsobythesecondMaxwellequation,.25wehaved|d|j=1 2j:.29Wenolongerrestrictjbyrequiringtheabsenceofstationarycharge,butdoobservethatconservationofcharge".i.e.dj=0isaconsequenceof.29.Ifweset|j=A;.30weseethattheMaxwellLagrangedensity.27ismodiedtobecometheProca"LagrangedensityLLA=1 2dA^|dA)]TJ/F8 9.963 Tf 13.808 6.74 Td[(1 2A^|A:.3110.Verifythis.

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11.4.ELECTROMAGNETISM.251Anumberofremarksareinorder:1.TheLondonequationshavenogaugefreedom.2.TheMaxwellequationsinfreespacethatiswithj=0areconformallyinvariant.Thisisageneralpropertyofthestaroperatoronmiddledegrees,inourcasefrom^2to^2,aswehaveseen.ButtheLondonequationsinvolvethestaroperatorfrom^1to^3andhencedependon,anddetermine,theactualmetricandnotjustontheconformalclass.ThisistobeexpectedinthattheMeissnereectinvolvesthepenetrationdepth,.3.Sincetheunitsofarelength,theunitsof1=2aremass2asistobeexpected.SotheLondonmodicationofMaxwell'sequationscanbeexpressedastheadditionofamassliketermtothemasslessphotons.Infact,substitutingaplanewavewithfourmomentumkdirectlyinto.29showsthatkmustlieonthemassshellk2=1=2.4.SincetheMaxwellequationsarethemasszerolimitoftheProcaequations,onemightsaythattheLondonequationsrepresentthemoregenericsituationfromthemathematicalpointofview.Perhapsthetrueworld"isalwayssuperconductingandweexistinsomelimitingcasewherethephotoncanbeconsideredtohavemasszero.5.Ontheotherhand,ifonestartsfromarmbeliefingaugetheories,thenonewouldregardthemassacquisitionastheresultofspontaneoussymmetrybreakingviatheHiggsmechanism.InthestandardtreatmentonegetstheHiggseldasthespinzeroeldgivenbyaCooperpair.Butsincetheelectronsarenotneededforchargetransport,asnoexternalsourcetermoccursin11.29,onemightimagineanentirelydierentoriginfortheHiggseld.Doweneedelectronsforsuperconductivity?Wedon'tusethemtogivemasstoquarksorleptonsinthestandardmodel.