IndependentprojectreportTheNielsBohrInstituteUniversityofCopenhagenKil - PDF document

1 Independentprojectreport.TheNielsBohrIns
Independentprojectreport.TheNielsBohrInstituteUniversityofCopenhagen....Killing-YanotensorsAugust13,2014Author:DennisHansen,Mail:xnw909@alumni.ku.dk.Supervisor:NielsObers.AbstractInthisproject,weinvestigatethetheoryofKilling-Yanotensorsandtheirap-plicationtothestudyofhigherdimensionalblackholes.Wewillrstdevelopthegeneraltheoryofexplicitandhiddensymmetriesforparticlesandelds,andrelatethemtoconservedquantitiesandintegrabilityofphysicaltheories.Themostgen-eralspacetimeallowingtheprincipalconformalKilling-Yanotensorisconstructed,andwederiveasuitablecoordinatebasisforthiscanonicalmetric.ThisweidentifywiththeKerr-NUT-(A)dSmetricuponimposingtheEinsteineldequations,whichdescribesaverygeneralclassofspacetimeswithblackholeswithsphericaltopologyoftheeventhorizon.Focusisthenturnedontothosehigherdimensionalstation-aryblackholes,whichcanallbeconsideredspecialcasesoftheKerr-NUT-(A)dSmetric.ThistreatmentincludesespeciallytheKerr,Schwarzschild-TangherliniandMyers-Perryblackholes. CONTENTSCONTENTS Contents1Introduction12Killingtensors22.1IsometriesandtheLiederivative.......................22.2Killingvectorsandtensors...........................42.3TheLieandsynmmetricSchoutenNijenhuisbrackets............62.4Conservationlawsforparticles.........................92.5Conservationlawsforelds..........................122.6Classicalintegrabilityandseparability....................163Killing-Yanotensors193.1Basicresults...................................193.2AntisymmetricSchouten-Nijenhuisbracket..................213.3ClosedconformalKilling-Yanotensors....................233.4TheprincipalconformalKilling-Yanotensor.................243.5Killing-Yanotowers...............................254Thecanonicalmetric284.1Darbouxbasis..................................284.2TheKilling-YanotowersinDarbouxbasis..................324.3Eigenvectorsof !h...............................334.4ThecanonicalandKerr-NUT-(A)dSmetric............

2 ......374.5Proofofexistenceanduniqueness
......374.5Proofofexistenceanduniqueness.......................384.6ImposingEinsteinsequations.........................494.7Wickrotatedmetric..............................504.8Integrabilityandseparability..........................505Stationaryblackholes515.1Aspectsofblackholespacetimes........................515.2Energyandangularmomentum........................525.3ThePCKYTandblackholes.........................546SpecialcasesofKerr-NUT-(A)dS546.1TheD=4Kerrsolution............................556.2Schwarzschild-Tangherlini...........................586.3Myers-Perryspacetimes.............................597Discussionandfurtherdevelopments618Summary62References63AAlternativeproofoftheorem466ii BMechanicsongeneralmanifolds68B.1Lagrangianformalism..............................68B.2Hamiltonianformalism.............................69CClassicaleldtheoryongeneralmanifolds73DNon-coordinate(vielbein)bases74EUsefultheoremsandidentities78E.1SecondordercovariantderivativesforKVsandKYTs............78iii 1INTRODUCTION 1IntroductionSymmetrieshavealwaysbeenanimportantaspectofphysics.Usually,theysimplifycalculationsdirectlybygivingconstraintsthatthemodelunderconsiderationmustobey.Thereisaintimateconnectionbetweensymmetriesandconservedquantities.Forthecaseofcontinuoussymmetries,thefamousNoethertheoremstatesthatforeverycontinuoussymmetry,thereisacorrespondingconservationlawwhentheequationsofmotionofthesystemareobeyed[31].ThisalsogeneralizesoutsideofclassicalphysicsregimetothecaseofquantumphysicsbytheWardidentitiesforquantumeldtheory.Withintheclassofcontinuousspacetimesymmetries,physicistshaveintherecentyearsbeenmadeawareofthefactthattherearetwokindsofsuchsymmetries,whichwecouldboldandnameexplicitandhidden.Explicitsymmetriesareisometries,dieo-morphismsofthemetric,generatedbyaKillingvector,andtheirphysicalinterpretationareingeneralratherintuitive-couldforexamplebetranslationorrotationalsymme-tries.On

3 theotherhand,hiddensymmetriesaresymmetri
theotherhand,hiddensymmetriesaresymmetriesthatcannotbeidentiedwithisometriesdirectly.Hiddensymmetriesarealwaysdescribedbytensorsofranktwoorgreater,whichwewillcallKillingtensor.TheystillgiveconservedquantitiesusingNoetherstheorem,andthisiswhywecallthemhidden,buttheirphysicalinterpreta-tionmayinsomecasesbeobscure.Inothercases,asfortheKerrmetric,whichhasahiddensymmetrytheyhaveagoodphysicalmeaning,hereitcanbeinterpretedasthetotalangularmomentumofsomeparticleoreldintheasymptoticallyatregion.ItturnsoutthatinaHamiltonianformulationongeneralmanifoldswecannaturallymakesenseofthesehiddensymmetriesinphasespaceasweshallsee.TherstoccurrenceofsuchahiddensymmetrywasindeeddiscoveredfortheKerrmetric.In1968Carter[8]discoveredthattheKerrmetrichadsomeratherunexpectedconservedquantity,anditwasrstin1970thatWalkerandPenrose[38]guredoutthatitoriginatedfromaKillingtensorofranktwo.Asaconsequenceofthis,therewerenowatotalof4constantsofmotion,2fromKillingvectorsandtwofromrank2Killingtensors(theotheronebeingthemetricitself),whichrendersthegeodesicequationintegrable-aremarkableresult.However,WalkerandPenrosewerenotabletodevelopafulltheorythatwouldexplainthereasonwhysomespacetimesadmittedKillingtensorsinthispaper.Arathercompletesolutiontothisproblemhasonlyemergedwithinthelast10yearsorliterature,culminatingwiththepaperbyKrtousetal.[27].ItwasunderstoodthatforalargesubclassofthesolutionstotheEinsteineldequationsadmittinghiddensymmetries,theintegrabilityofthegeodesicequationisveryfundamentallyrelatedtothat.Itwasrealizedthatforthisclassofspacetimes,afurtherdecompositionoftheKillingtensorsintocontractedproductsofKilling-Yanotensorswasindeedpossible.TheKilling-YanotensorscanbethoughtofasthesquarerootofKillingtensors,andtheyhavemanyinterestingproperties.TheexistenceofaspecialKilling-Yanotensor,theso-calledprincipalconformalKilling-Yanotensorturnouttoexactlydenethelargeclassofintegrablespacet

4 imes.TheprincipalconformalKilling-Yanote
imes.TheprincipalconformalKilling-YanotensorwillgenerateawholetowerofKillingtensorsandKillingvectors,thatwillagaingivesconservedquantitiesandsecureintegrability.Thesewillalsosecureseparabilityofeldequationsofinterest,suchastheKlein-GordonandDiracequations[29],whichisaremarkableresultaswell.InthisprojectweshallstudythetheoryofKilling-Yanotensorsindetailandtheir1 2KILLINGTENSORS relationtointegrablespacetimesandsymmetries.OurendgoalwillbetoconstructthemostgeneralmetricsolvingEinsteinseldequationsallowingsuchaprincipalconformalKilling-Yanotensor,whichturnsouttobetheverygeneralKerr-NUT-(A)dSmetricinD�2spacetimesdimensions,rstconstructedbyChenetal.[9]in2006.Therewillbegivenfullproofsofmoststatements,buttocontaintheprojecttoareasonablesize,welimitourselvestogivereferencestoasomeresultsandjustdescribethemainideasoftheproofs.WewillstartwithdevelopmentofthemathematicalbackgroundofKillingvectorsandtensors,andtheirconformalgeneralizations.Therelationofthesetosymmetriesoftheoriesofparticlesandelds,areconstructedusingbothaLagrangianandHamiltonianformalismofmechanics,andtwogeneralversionsofNoether'stheoremareproven.Integrabilityandseparabilitytheoryforparticlesandeldsarethendiscussed,andwegiveasimplecriterionforintegrabilityofHamiltonians,includingthegeodesicequation,alongwithastatementofatheoremontheexistenceofseparabilitystructures.WethenmoveontothetheoryofKilling-Yanotensorsandtheirconformalgeneralizations,provingvariousnumberoftheoremsandlemmas.TheseareusedforconstructingthemostgeneralmetricallowingaprincipalconformalKilling-Yanotensor,thecanonicalmetric,usingthetheory.Thedetailedproofthatwewillbegivingonthisissomewhatsimplerinsomeaspectsthanwhatcanbefoundintheliterature,andthisisthemainresultofthisproject.WethenshowthatthisisinfacttheKerr-NUT-(A)dSmetricwhenimposingtheEinsteineldequations.Wethenleavethemostgeneralcase,andtakeaquickreviewofstationaryblackholesspacetimesandth

5 eirrelationtotheKerr-NUT-(A)dSmetric,and
eirrelationtotheKerr-NUT-(A)dSmetric,andtakealookatimportantspecialcasesofit.Atlastwediscussbrieyhowthetheorythatwehavedevelopedcanbeputintoagreaterperspective,applicationsandgeneralizations.NotationandconventionsUnlessotherwisestated,wealwaysusetheChristoelconnectionforthecovariantderivative.Thesignatureofthemetricisingeneralarbitraryunlessotherwisestated.WeworkinnaturalunitswithG=c=~=1.Boldsymbolslikeg;hreferstotensorsthatarewrittenincomponent-freenotation,i.e.includingthebasis.2Killingtensors2.1IsometriesandtheLiederivativeInthefollowing,letusworkwithamanifoldMofdimensionD=2n+","=0;1sowehaveexplicitlyodd("=1)oreven("=0)dimensions,towhichwecanassociateametricg,thatissingularatanitenumberofpointsatmost.Thesimplestkindofsymmetryofsuchamanifoldareisometries,activecoordinatetransformationsthatleavesthemetricginvariant[7].Tomakethesestatementsmoreprecise,letusrstassumethatwehavesomedieomorphism':M!M,i.e.adier-entiablebijectivemappingwithandierentiableinverse.Wethendenethepullback1ofthemetric'gas: 1Foradieomorphism,wecanpullbackageneraltensor,butcontravariantindicestransformoppo-sitely,theyarepushedforward.2 2.1IsometriesandtheLiederivative2KILLINGTENSORS Denition1(Pullback).Foradieomorphism':M!Mandachartwithcoordinatepointxofthemanifoldandwithy'(x),wedenethepullbackofthemetric'ginlocalcoordinatesas('g)00(x)@x @y(x)0@x @y(x)0g(y(x)):(2.1)Inwords,thisistosaythatthevalueofgaty='(x)ismappedback(pulledback)tobethevalueofg0'gatx,whichbothdescribesthesamespacetimepointofthemanifold.Nowitisalsoeasytomakeaprecisestatement(therstoftwoequivalentones)ofwhatwemeanbyanisometryofthemetric:Denition2(Isometry1).Wesaythatadieomorphism'isanisometryif'g=g.Thisiswhatweshouldunderstandbyanactivecoordinatetransformation,thatleavesthemetricinvariant.Saynowthatwehaveaone-

6 parameterfamilyofdieomorphisms(notn
parameterfamilyofdieomorphisms(notnecessarilyisometries)':MR!M,whereistheparameter2takensuchthat'0=I(theidentitymap),andthefamilyisdierentiablewithrespectto.Thepullbackofsuchafamilyisgeneratedbytheowofsomevectoreldbecauseitdenesa(dierentiable)vectoreldasthetangentvectorateachspacetimepointforeachvalueoftheparameter.Conversely,givenavectoreldK,theintegralcurvesofitsowisthesolutionoftheODEgivenbysettingthedirectionalderivativeofacurvex()equaltothevectoreld:dx() d=K(x());(2.2)withboundaryconditionsx(=0)=xanddx() d=0=K(x)ateachpointofthemanifoldinsomechosenchartx.TheexistenceanduniquenessofthesolutionisguaranteedbythePicard-Lindelöftheorem[35],giventhatKiscontinuous-wewillassumethatitisasdierentiableasneededinthefollowing.WewillcallKthegeneratoroftheone-parameterfamilyofdieomorphisms'.Wemaythenconcludethatavectorelddenes(orgenerates)aone-parameterfamilyofdieomorphisms,andtheexistenceofoneimpliestheother.GivensuchageneratorKofafamily',wecandenetheLiederivativeLKalongKofthemetric(similarlyforgeneraltensors)asLKg(x)lim!0"('g)(x)�g(x) #;(2.3)whichisatensoritself,sinceitisjustthedierenceoftwotensors.TheLiederivativemaybedescribedinwordsasrateofchangeofthetensoraswemovealongtheowateachspacetimepoint.Ithasalotofgoodpropertiesandisoneofthesimplestconstructionsofadierentialoperatoronamanifold,see[35]. 2Itiseasytoseethatunderfunctionalcomposition,theyformaone-parametergroupwiththisasthegroupproduct,becausefortwoparameters;0('0')(x)='0('(x))='0+(x),andtheremainingaxiomsmayalsoeasilybeproventohold.3 2.2Killingvectorsandtensors2KILLINGTENSORS Givennowaone-parameterfamilyofisometries',wecallthegeneratorKaKillingvector(KV),

7 whichwillturnouttobeveryfundamentalquant
whichwillturnouttobeveryfundamentalquantity.Wehave'g=gforany2R,andusingthedenitionoftheLiederivative,thisisequivalenttothevanishingofLKg,whichgivesusatheoremthatconnectsisometries,KillingvectorsandtheLiederivative,whichwecanformulateasbelow:Theorem3(Isometry2).KisaKillingvectorifandonlyifLKg=0.2.2KillingvectorsandtensorsAswehavenowestablishedtheroleofKillingvectorsinconnectiontoisometries,thenaturalquestionishowtondthem.Thefollowingfamousequationsinglesthemoutasaspecialclassoftensors.Theorem4(TheKillingequation).KisaKillingvectorifandonlyifitsatisesr(K)=0:(2.4)Proof.ThisismosteasilydoneexpressingtheLiederivativeLKgthroughthecovariantderivative.AssumingthatKgeneratesanisometry,wehavebytheorem4that0=LKg=Krg+rKg+rKg=2r(K):(2.5)Thisprovestheclaim.SeeappendixAforanalternativeversionoftheproof. TheKillingvectorequationmakesiteasytotestifonehasageneratorofsymmetryinagivenspacetime,andinprinciplewecouldalsodetermineKillingvectorsbysolvingthecorrespondingPDE.DoingthisinatMinkowskispace,wewillnd10Killingvectorscorrespondingthe4translational,3rotationaland3boostisometries[36].Unfortunatelyitisnotsosimpleingeneralgeometries,andwemustarguedierentlytondtheKillingvectors.TheconceptofaKillingvectorcanbegeneralizedfurther.WesaythatatotallysymmetricrankptensorKisaKillingtensor(KT),ifitsatisestheKillingtensorequationr(K1


p)=0:(2.6)TheinterpretationofthegeneralKillingtensorisnotasstraight-forwardasfortheKillingvector.Kisnotassociatedwithanisometryforp�1,andweusetheterminologythatitgeneratesahiddensymmetryinthemeaningintroducedearlier,thatwillimplythatsomequantitiesareconservedasweshallseesooninsection2.4.Herewewillalsondadierentwaytoderive(2.6).4 2.2Killingvectorsandtensors2KILLINGTENSORS 2.2.1ConformalKillingvectorsandtensorsA

8 gain,wemaygeneralizetheconceptofaKilling
gain,wemaygeneralizetheconceptofaKillingtensor.TherstgeneralizationofthisistheconformalKillingtensor(CKT),whichisconstructedbydoingalocalchangeofscaleofthegeometrysothatthemetricchangesasg0(x)=!(x)2g(x);(2.7)where!(x)issomearbitraryrealandnon-vanishingsmoothfunction[7].Thisisclearlyadieomorphism,astheinverseconformaltransformationisgivenbymultiplyingthemetricby!�2.Wesaythatg0istheconformalmetricoftheconformalframe,andgtheoriginalmetric.Inthelanguageofdieomorphisms,wesaythatthedieomorphism'!that(2.7)denes,iscalledtheconformaltransformation.Thenwemaywriteg0'!g=!2g:(2.8)Wecantheningeneralconsiderconformaltransformedtensors,andrelatethembacktotheoriginaltensorsandtheoriginalmetric.TheconformalKillingtensorequationisobtainedinthiswayfrom(2.6),aswecanformulatemorepreciselyasadenition:Denition5(ConformalKillingtensor).AconformalKillingtensorofrankpisatotallysymmetrictensorK1


p,thatintheconformalframeobeysconformalKillingtensorequationr(K1


p)=pg(1 K2


p);(2.9)wherethetensor K2


pissometotallysymmetrictensorofrankp�1,foundbytracingbothsides.ToseehowthisrelatedtoanordinaryKillingtensor,letusperformtheconformaltransformationexplicitly.Thecovariantderivativeintheconformalframerisrelatedtotheordinarycovariantderivative~rbyachangeoftheconnection,�=~�+C;(2.10)whereCisatensor,symmetricinlowerindices,whoseexplicitformmaybecalcu-latedintermsof!.AssumethatK1


pisaKillingtensor.WethenhaverK1


p=~rK1


p+C1K


p+C2K1


p+:::~rK1


p+Q1


p:(2.11)NotethatQ1


pC1K&

9 #1;

p+C2K
#1;

p+C2K1


p+:::istotallysymmetricaswell,becauseCissymmetricinthelowerindices.Ifwedenepg(1 K2)


pQ1


p,thisisconsistentwiththedenitionofQ1


p,when K2


pisatotallysymmetrictensorofrankp�1.Usingthisandsymmetrizing(2.11)usingthatK1


pisaKillingtensor,wend5 2.3TheLieandsynmmetricSchoutenNijenhuisbrackets2KILLINGTENSORS r(K1


p)=~r(K1


p)+pg((1 K2)


p)=pg(1 K2


p):ThuswehavefoundtheconformalKillingtensorequationaspromised.If K2


pvanishes,thenK1


pisanormalKillingtensor.ForanyconformalKillingtensor,wecanalsoconcludethattherealwaysexistsaninverseconformaltransformation,thattakesitbacktoanormalKillingtensor.2.3TheLieandsynmmetricSchoutenNijenhuisbracketsTheLiebracket[
;
]isdenedastheactionoftheLiederivativealongthevectoreldXactingonavectoreldYLXY[X;Y];(2.12)andithasanumberofgoodpropertieslistedbelowsuchasbilinearityandtheJacobiidentitythatcanallbeveried[35]:[X;Y]=�[Y;X][X;Y+Z]=[X;Y]+[X;Z](2.13)0=[[X;Y];Z]+[[Y;Z];X]+[[Z;X];Y]where;arescalars,andX;Y;Zarevectorelds.[X;Y]shouldbethoughtofastherateofchangeofYalongtheowofX,orequivalentlyminustherateofchangeof�XalongY.Wemayalsondusingthepartialorcovariant3derivativethatitcanbeexpressedas[X;Y]=(X
r)Y�(Y
r)X;(2.14)andinthisformtheinterpretationgivenaboveismoreclear.IfwehavesomeLiegroupofdieomorphisms4Gofdimensionn,thenthegeneratingvectoreldsfX1;:::;XngwillformaLiealgebrag,whicharealgebraswithaproductthatexactlyfullls(2.13).TheLiealgebrafullydescribestheLiegroupthroughthestructureconstantsCijkthatdoesn'tdepend

10 onspacetime.Itisbasicallytheinnites
onspacetime.Itisbasicallytheinnitesimalowsgeneratedaroundtheidentitytransformation.Ingeneralwehave[Xi;Xj]=nXk=1CijkXk:(2.15)Noticethatthisexpressionisindependentofanycoordinateswemightintroduce,the 3Becauseoftheantisymmetry[X;Y]=�[Y;X],theconnectiontermsdropsoutwhentheconnectionistorsionfree-ifthemetricisnottorsionfree,thenwemustusepartialderivativesin(2.14).4ForsuchagrouptheLiealgebragwillautomaticallyclose,becauseapplyingtwodierentdieomor-phismsofGmustalsobeadieomorphism.6 2.3TheLieandsynmmetricSchoutenNijenhuisbrackets2KILLINGTENSORS onlyindicesaregroupindicesrelatedtoG.Ingeneralwecouldcomparethecalculatedstructureconstantsforonemanifoldtothosecalculatedforothermanifolds,andiftheyarethesame,thenthetwodieomorphismgroupsareisomorphic.Thiscanbeputtogooduse,aswethenhaveacoordinateindependentwayofdeterminingwhatkindofsymmetrieswehave.ThiswaywecouldforexampledeterminethatwewouldhavesphericalsymmetryifwefoundthreevectorsthathadcommutationrelationsisomorphictothatofSO(3),i.e.Cijk=ijk.ForagivenspacetimeofdimensionDandagivenbasisfX1;:::;XDgofthetangentspaceT(M),thatisnotnecessarilyacoordinatebasis,theLiebracketcanactuallytellussomethingaboutthepossibilityofintroducingacoordinatebasis:Theorem6(Coordinatebasis).LetfX1;:::;XDgbeabasisforT(M).Wehave[Xi;Xj]=0ifandonlyifXi=@xi,i.e.thereexistsalocalcoordinatebasisforT(M)denedbytheowofthebasisvectors.Theproofcanbefoundinanyreferencebookondierentialgeometry,seeforexample[35].Theintuitionbehindthisisthatifanytwovectoreldscommute,thenthereisnochangeinthegeneratorsgoingalongeitherow,andtheowequation(A.1)thatwoulddenethecoordinatesismucheasiertosolve,asthereisonlychangeinonedirection.Particularlyrelevantforusistherestrictiontoisometries.ThesetofalltheKillingvectorsthatwouldgeneratetheisometriesofaspacetimeiscalledtheisometrygroup.Themaximalnumberofgeneratorsthatwecanhaveforagivenspacetimeisnmax=D+ D2!=D

11 2(D+1);(2.16)becausethiswouldcorrespondt
2(D+1);(2.16)becausethiswouldcorrespondtotheDtranslationalKillingvectors,andD2rota-tionalKillingvectorswehaveinatspace.ThemaximalnumberofrotationalKillingvectorscanbecalculatedasfollows:wecanchoosetherstcoordinateaxisxiofthehyperplanethatwerotatein,inDwaysandtheotherxjinD�1ways,butasthehyperplaneof(xi;xj)isthesameas(xj;xi),weonlyhavehalfoftheproduct,whichisD2.SpacetimeswithmaximalnumberofKillingvectorsarecalledmaximalsymmetricspacetimes,forexample(A)dSspacetime.Forsuchmaximallysymmetricspacetimes,therotationalpartoftheisometrygroupisSO(p;q),wherep+q=D,and(p;q)isthesignature,andthetranslationalpartisRp;q,andthefullisometrygroupisSO(p;q) Rp;q.Fornon-maximalsymmetricspaces,theisometrygroupisasubgroupofthis,assomeoftheKillingvectorsmaynotbepresent.Forexample,theRobertson-Walkerspacetimesdoesn'thavetime-translationalKillingvectorwhenthescalefactordependsonthetimecoordinate.Likewise,foraxialsymmetricspacetimesasareofinteresttousinthefol-lowing,thereisn'tfullrotationalsymmetry,onlyalongonerotationalcoordinateaxis,soonlyoneofthethreeSO(3)generatorsexists.Thisgeneralizestohigherdimensionalhypersphericalandaxisymmetricalsymmetriesaswellaswediscussinsection5.2.WecannaturallygeneralizetheabovediscussiontoKillingtensorsbyintroducingamoregeneralbracket.SuchaconstructionisgivenbythesymmetricSchoutenNijenhuisbracket[32],whichisdened(incomponentform)as:7 2.3TheLieandsynmmetricSchoutenNijenhuisbrackets2KILLINGTENSORS [X;Y]1


p+q�1pX(1


p�1rYp


p+q�1)�qY(1


q�1rXq


p+q�1)(2.17)wheretheinputsaretotallysymmetrictensorsX1


p=X(1


p)andY1


q=Y(1


q).Thisdenitionholdswhentheconnectionistorsionfree,butifthisisnotthecase,wemustreplacethemwithpartialderivatives5.[X;Y

12 ]1


p+q�1isitself
]1


p+q�1isitselfasymmetrictensorofrankp+q�1.WeseethatwehavetheSchoutenNijenhuisbracketreducestotheLiebracket(2.14)whenbothX;Yarevectors,p=q=1.ThesymmetricSchoutenNijenhuis(SSN)bracketdenesaLiealgebraonthevectorspaceofsymmetrictensors,becauseonecaneasilycheckthatitsatisestheLiealgebraaxioms(2.13)ofwithZbeeingarankrsymmetrictensor.Thevectorspaceofsymmetrictensorsisingeneralinnitedimensional,becausewecancontinuetogeneratesymmetrictensorsofhigherandhigherrankbysymmetrizingtensorproductsoflowerranktensors.WehavethatifXisavectorandYisageneraltensor,thenLXY=[X;Y];(2.18)whichonecancheckusingthecoordinateexpressionoftheLiederivativeactingonatensor.TheLeibnizpropertyoftheLiederivativeactingonmultivectorsisaspecialcaseofthisequationand2.17,aswecanrewritetheexpressionasLX(Y +Z)=(LXY) +Z+Y +(LXZ);(2.19)where +isthesymmetricpartofthetensorproduct.Now,restrictingthediscussiontotoKillingtensors,wemayshowthefollowingresultTheorem7(SSNbracketandKillingtensor).KisaKillingtensorofrankpifandonlyif[g;K]=0,whentheconnectionismetriccompatible.Proof.Wedoadirectcalculationtoshowthatthetwonotionsareequivalentbyusing(2.17)withcovariantderivatives:[g;K]1


p+1=2g(1rK2


p+1)�pK(1


p�1rgpp+1)()=2g(1rK2


p+1)=2r(1K2


p+1);wherewein(*)usedthattheconnectionwasassumedtobemetriccompatibleandthenraisedtheindexofthecovariantindex.Uponloweringalloftheindices,weobtainexactlytheKillingtensorequation(2.6)andprovesthetheorem. Ofcourse,aspecialcaseofthistheorem,isthecaseofaKillingvectorstatedintheorem3.Also,wemightbegintothinkaboutwhetherthesetofKillingtensorsisa 5Whentorsionfree,theconnectiontermswillcancelanyway,butitisjustmoreconvenientusingcovariantderivatives.8 2.4Conservationlawsforparticles2KILLINGTENSORS LiealgebraonitsownwiththeSSNbr

13 acket[
;
].Wehavethefollowingnicer
acket[
;
].Wehavethefollowingniceresult:Theorem8(SSNbracketofKTs).LetKandQbeaKillingtensorsofrankpandq.Then[K;Q]isaKillingtensorofrankp+q�1.Proof.WejusthavetoverifythattheKillingtensorequation(2.6)holdsfor[K;Q].ThisismosteasilydoneusingtheJacobiidentityon[[K;Q];g]andusingtheorem7.Wesee0=[[K;Q];g]+[[Q;g];K]+[[g;K];Q]=[[K;Q];g];whereweusedthat[g;Q]=[g;K]=0bytheorem7.Thisshowsthat[K;Q]isaKillingtensor. ThismeansthattheSSNbracketofKillingtensorscloses,andthusthesetofKillingtensorsofagivenspacetimewiththeSSNbracketisaLiealgebra.Ingeneralitisinnitedimensional,anditisasubalgebraoftheLiealgebraofallsymmetrictensors.ThisconcludesthetreatmentofthealgebraicaspectsofKillingtensorsfornow.2.4Conservationlawsforparticles2.4.1ConservationofKillingtensorsThegreatinterestinndingsuchKillingtensorsforaphysicistisbecausetheygiveusconservedquantities,thatputsconstraintsontheequationsofmotionthatwewanttosolve.Ourrstexampleofthisisconservationofquantitiesalonggeodesics.Wemayprovethefollowingresult:Theorem9(Conservationandgeodesics).LetK1


pbeaKillingtensorandx()ageodesic.ThenthescalarJ=K1


pp1


ppisconstantalongageodesic,wherep_x()isthetangentvectorofthegeodesic.Proof.WewouldliketoshowthatDJ d=D dK1


pp1


pp=0,whereD dpristhecovariantdirectionalderivativealongthegeodesic,whichfulllsthegeodesicequationDp d=0.WedoadirectcalculationusingthisandthetotalsymmetryoftheKillingtensorDJ d=prK1


pp1


pp()=r(K1


p)pp1


pp()=0:In(*)weusedthattheproductpp1


ppistotallysymmetricintheindices,soonlythetotallysymmetricpartofrK1


pcontributes.ThisgivesustheKillingtensorequat

14 ionthatwecaninvokein(**)toconcludewhatwe
ionthatwecaninvokein(**)toconcludewhatwewantedtoshow. OnecouldtakethisasthemotivationfortheKillingtensorequation(2.6).9 2.4Conservationlawsforparticles2KILLINGTENSORS 2.4.2Noether'stheoremWhatwehaveconsideredsofarhavesolelybeenspacetimesymmetries.Wecanhoweverrelatethesetotheoriesdenedonthespacetime,givenbysomeLagrangian[density],whichgivesanotherviewontheroleoftheconservationlawoftheorem9.Theformercanbeseenasaspecialcaseofamoregeneraltheorem,thefamousNoethertheorem,whichwewillformulateshortly[35].ForashortreviewofLagrangianandHamiltonianmechanicsongeneralmanifolds,seeappendixB.ImaginenowthattheLagrangianLhassomecontinuoussymmetrydescribedbysomerepresentationofaLiegroupG,inthemeaningthatatransformationofLbysomeT2Gleavesitinvariant,formallyTL=L.ThesymmetrytransformationwillchangethecoordinatesanddirectionalderivativesinexactlysuchawaythattheLagrangianremainsunchangedafterthetransformation6.Workinginphasespace7�,themostgeneralinnitesimaltransformation^xinlocalcoordinatesisgivenbythevertical(momentum)derivativeof(B.5)[36]:^xR(x;p)=K(x)+K(x)p+1 2!K(x)pp+:::;(2.20)whereisinnitesimalandthefamilyoftensorsK


aretotallysymmetricwithallindiceslowered,andareindependentofmomenta.ItdependsonthestructureofGhowmanytermsthereare.Forordinarydieomorphisms,whichisrestrictedtothecongurationspacepartof�,theseriestruncatesaftertherstterm.Formoregeneraltransformationstherearemoretermsastherecouldbesomemomentumdependentsymmetrytransformation.Wecouldgobacktocongurationspacebyimposing(B.16)ifwelike.Thenotationforthetensorsintheexpansionisnotarbitrary-wewillseesoonthattheyareexactlyKillingtensorsforspecialLagrangians.TheNoethertheoremthensaysthatwhenimposingtheclassicalequationsofmotions,thereareconservedquantities,givenacontinuoussymmetry.Theorem10(Noether'stheorem

15 forparticles).LetaLagrangianL:T(M)!Rhave
forparticles).LetaLagrangianL:T(M)!Rhaveacontinuoussymmetryunderatransformationgivenby(2.20).ThenthereexistsacorrespondingconservedchargegivenbyJ=R@L @x;(2.21)whenxobeystheequationsofmotion.Proof.AsymmetrytransformationmustgiveusLTL�L=0evenwhenweare 6Uptoboundaryterms,thatweassumearezero,sotheydoesn'tcontributetotheEOMs.7Forsimplicitywestudythecaseofasingleparticlehere,butitiseasilygeneralized.10 2.4Conservationlawsforparticles2KILLINGTENSORS o-shell.Usingthis,wecandoanexpansionin^xand^_x,anddosomerewriting:0=L=Lx+^x;_x+^_x�L(x;_x)=Lx+R;_x+_R�L(x;_x)=L(x;_x)+@L(x;_x) @xR+@L(x;_x) @_x_R�L(x;_x)=@L(x;_x) @xR+@L(x;_x) @_x_R:Applyingtheequationsofmotion@L(x;_x) @x=d d@L(x;_x) @_x,thiscanbewrittenasatotalderivative:0=@L(x;_x) @xR+@L(x;_x) @_x_R=d d@L(x;_x) @_xR+@L(x;_x) @_x_R=d d @L(x;_x) @_xR!:ThuswehavethatJ@L(x;_x) @_xR;(2.22)isconservedalongtheequationsofmotion,aswewantedtoshow. OneshouldespeciallynoticethattheLagrangianwhichhasthegeodesicequationasEOMisthesingle-particleLagrangianL(x;_x)=1 2g_x_x;(2.23)whichcanbeseenasafreeparticleLagrangian,withcorrespondingHamiltonianH(x;p)=1 2gpp:(2.24)Inthiscaseequation(2.22)tellsusthattheconservedquantityisJ=@L(x;_x) @_xR=pR=Kp+Kpp+1 2!Kppp+:::(2.25)ToseethattheyareactuallyourKillingtensors,wenoticethat11 2.5Conservationlawsforelds2KILLINGTENSORS DJ d=prKp+Kpp+1 2!Kppp+:::=r(K)pp+r(K)ppp+1 2!r(K)pppp&#

16 27;+:::(2.26)=0:Becauseeachtermisdi
27;+:::(2.26)=0:Becauseeachtermisdierentfromanother8,wehavethateachtensormustsatisfytheKillingtensorequation(2.6).Thuswehavesuccessfullyconnectedatheoryoffreeparticlestothespacetimesymmetries.Interactionsbetweenparticlesofanon-quantumtheoryisgivenbyapotentialfunctionV:T(M)!R,sotheLagrangiantakestheformL=1 2g_x_x�V,andtheHamiltonianH=1 2gpp+V,ifthereisnovelocitydependence.ThesymmetriesofLcouldthenbethoughttobeconstrainedinnumber.Thisishowevernotphysical-goodphysicalpotentialsshouldasaminimumrespectthespacetimesymmetriesifwearelookingatatheoryshouldbeapplicableanywhere.Otherwisetheconsequencewouldbethattheresultofanexperimentcoulddependonthelocation,rotation,etc.intheuniverse.Letusformulatethismoreconcisely:Principle1(SymmetriesofaLagrangian):ThespacetimesymmetriesisasubsetofallthesymmetriesofaLagrangian.Thisshouldserveasaguidingprincipleforconstructingtheories.2.5ConservationlawsforeldsForeldswehaveadditionaldegreesoffreedom,astheybothdependonspacetime,andmaycarryinternalindices.Wecanwriteageneraleldas`,where`isshortforbothspacetimeandinternalindices,seeappendixCforashortreviewofclassicaleldtheoryongeneralmanifolds.TherepresentationofcontinuoussymmetrytransformationbyT2Gmustnowbeextended,becausealongwithspacetimesymmetries,wecouldalsohaveinternalsymmetries,wherethetensorcomponentschanges:T`(x)=`(Tx)=U`(Tx);(2.27)whereUisarepresentationoftheinternalsymmetryonthecomponents,andTx 8Proof.Assumerstthattherearetwonon-zerocontractedtermsthatwouldcancelin(2.26).Wecanfactoroutthep'ssuchthatwehavesomethinglike�K


+cK





pp



pp


,c2R,whichwouldhavetobezero,soK


=�cK





pp.Butaccordingtoourinitialdenitionof(2.20),K


isin

17 dependentofp's,andsowehaveacontradiction
dependentofp's,andsowehaveacontradiction.Thisargumentcanberepeatedwithanynumberofterms,factoringouttermshigherorderinp's,andthuswehaveproventhattheKillingtensorequationholdsforallmembersofthefamily. 12 2.5Conservationlawsforelds2KILLINGTENSORS representsthespacetimetransformation.Likewisewecoulddoaninnitesimalvariation`(x)`x+^x�`(x);(2.28)where^xisgivenby(2.20),and`isaninnitesimalchangeofthecomponents,whichwecouldwritelike`(x)!(x)`(x);(2.29)whereisinnitesimaland!(x)`isatransformationofthecomponents.Forthemetric,whichcanbethoughtofasaclassicaleldonitsown,wewouldhavethatTg(x)=g0(x)carriessomerepresentationofthesymmetry.Notice,thatiftherepresentationofthetransformationisgeneratedbyaKillingvector,thenwesimplyhaveTg='g=gbydenitionoftheKillingvector,andthereisnovariationing.AgainwesaythatTisasymmetryif9TL=LandNoether'stheoremwillthengeneralizewithessentiallythesamecontentandsimilarproof.Theorem11(Noether'stheoremforelds).LetaLagrangiandensityL:Tpq(M)!Rhaveacontinuoussymmetryunderatransformationgivenby(2.27).ThenthereexistsacorrespondingconservedcurrentgivenbyJ@L @[r`]!`0`^`0�r`R+LR;(2.30)when`obeystheequationsofmotion,and^`0isthevalueoftheeldsatthetrans-formedspacetime.Proof.Wecanrstsplit`(x)`x+^x�`(x)intotwotermsusingdenitions(2.28)and(2.29),aninnitesimalvariationofthespacetimeargumentandaninnitesimalvariationofthecomponentsbyaexpansionintorstorder`(x)=`x+^x�`(x)=`(x)+`(x)+r`(x)^x�`(x)+O2`(x)+^`(x);(2.31)whereisthepurelyinternalvariation,and^isthepurespa

18 cetimevariation.Asymmetrytransformationm
cetimevariation.AsymmetrytransformationmustgiveusS=0evenwhenweareo-shell.Wecandoanexpansionofthistorstorderininthespacetimevariationtodoarewritingoftheintegrationstowritethemasasingleintegraloverthenon-transformedcoordinates: 9ThetheoremcanactuallyeasilybeextendedbyallowingLtobeatotaldivergence.13 2.5Conservationlawsforelds2KILLINGTENSORS 0=S=TML(T`;Tr`)dDTx�'(M)L(`;r`)dDx=TML(`+`;r`+r`)dDTx�'(M)L(`;r`)dDx()=ML`+`;r`+r`det @Tx @x!dDx�ML(`;r`)dDx=ML`+`;r`+r`1�r^xdDx�L(`;r`)dDx+O2()=ML`+`;r`+r`�L(`;r`)+rL(`;r`)^xdDx+O2=M@L(`;r`) @[`]`+@L(`;r`) @[r`]r`+rL(`;r`)^xdDx+O2;wherewein(*)didachangeofvariablesfromTxtox,whichgivesdDTx=det@Tx @xdDx=1�r^x+O(2)dDxintherstintegral,soonlytheinternalvari-ationsremains.In(**)wedidanexpansionandkeptonlyO(2)terms,andapartialintegrationoftheonlysurvivingtermfromthedeterminantfactorgaveusthetermrL(`;r`)^x.InthelastlinedidanexpansionofL`+`;r`+r`torstorderintheinternaleldvariationsonly.Wehaver`=r`,exactlybecausetheyareevaluatedatthesamepoint(theyactondierentspaces),andusingthiswecandoafurtherrewritingofthevariationoftheaction.Noticethatgoingon-shellusingr@L(`;r`) @[r`]=@L(`;r`) @[`]givesusr"@L(`;r`) @[r`]`#= r@L(`;r`) @[r`]!`+@L

19 (`;r`) @[r`]r
(`;r`) @[r`]r`=@L(`;r`) @[`]`+@L(`;r`) @[r`]r`;andwecanexactlyrecognizethersttwotermsinourvariation.Wemaythenwrite0=Mr"@L(`;r`) @[r`]`#+rL(`;r`)^xdDx=Mr"@L(`;r`) @[r`]`+L(`;r`)^x#dDx:14 2.5Conservationlawsforelds2KILLINGTENSORS Nowwenoticethatfrom(2.31)wehave`(x)=`(x)�^`(x)=`x+^x�r`^x=!(x)`0``0x+^x�r`R;=!(x)`0`^`0(x)�Rr`wherewealsousedthat^x=R,anddened^`0`0x+^x,whichisthevalueof`0atthevariationspacetimepoint.Usingthis,wecanfactorout,andwrite0=Mr"@L(`;r`) @[r`]!`0`^`0�Rr`+L(`;r`)R#dDx:WecanthendeneaconservedquantitybythecurrentJ@L(`;r`) @[r`]!`0`^`0�Rr`+L(`;r`)R;(2.32)thatfulllsthecovariantconservationlawrJ=0. AspecialcaseofgreatimportanceforourambitionsisofcoursetheEinstein-Hilbertlagrangianforjustthemetricandsomematterelds I,givenbyS[g; I]=1 16Rq jgj| {z }Lvac+LmatdDx=Svac[g]+Smat[g; I];(2.33)whereRistheRicciscalar,thatdependsonlyonthemetricgthroughtheChristoelsymbols,andLmatisthelagrangiandensityforinteractionwithmatter,thatmightdependonotherelds I,forexamplegaugeelds.Variationwrt.tothemetricyieldsEinsteinsequationsforgeneralrelativity[7]with(Hilbert)energy-momentumtensorgivenbyT�21 q jgjSmat[g; I] g:(2.34)Theenergy-momentumtensorisastrangebeastinclassicaleldtheory[16].Wecouldalsod

20 eneatheso-calledcanonicalenergy-mom
eneatheso-calledcanonicalenergy-momentumtensorTcan:associatedwithspacetimetranslationinvariantlagrangians,i.e.inlocalcoordinatesthefulllagrangianisinvariantunderx!x+a,whereaissomeconstant.Inthiscasewehave!I0I=0,nochangeofthetensorcomponentsofthematterelds I,and(2.30)willgiveusJ=@L @[r I]ar I�aL="@L @[r I]r I�Lg#aTcan:a:15 2.6Classicalintegrabilityandseparability2KILLINGTENSORS IngeneralTcan:isnotsymmetricasrequiredbytheEinsteinequations,andneitherisitgaugeinvariant,whichisrequiredif Iaregaugeelds.Thisisabitawkwardandcansometimesbexedbyaddingextratermstothecurrent.Onereasonwhyitdoesn'talwayshold,isbecausewedon'talwayshasfulltranslationinvariance,asisthecasefortheblackholespacetimesaswewilllookatlater.Afurtherdiscussionoftheproblemcanbefoundin[16].Itisnotreallyofconcernforourfuturepurposes,whereweareeitherinthevacuumorhaveacosmologicalconstant.AscorollarybenetofthisversionofNoether'stheorem,weseethatifwejustconsiderthespacetimesymmetries,wewillhaveexactlythesamenumberofconservedquantitiesforeldsaswehadforparticles.Thesymmetriesthatwehave,mustatleasthavethesymmetriesofthefreetheorywithnomatterelds,equivalenttothediscussionthatledtoprinciple1.Theextrainternalsymmetriescangivesomeextraconservedquantities,asforexampleifwehaveanon-abeliangaugetheorydeningthematterelds.2.6ClassicalintegrabilityandseparabilityWemayputtheconservationofcertainquantitiesintogooduse.Sometimeswewillbeabletoprovethatifthereareenoughconservedquantities,wemayalwaysinprinciplesolvethegeodesicequationandotherclassicalequationsofmotionforparticlesandelds,likethecurvedspaceversionsoftheKlein-GordonorDiracequations.Inthiscasewesaythattheequationsofmotionareintegrable.Ofcourse,saywehaveNdegreesoffreedomintheequationunderconsideration,thenitisclearthatwewouldneedatleastNcon

21 servedquantitiestosolvethem.Itmightnotbe
servedquantitiestosolvethem.ItmightnotbethatNisenough;theywouldallhavetobeindependentanditmightalsobethatincertaincasessomeoftheconservedquantitiesarenotuseful.Sometimeswemaynotprovefullintegrability,butitmightbethatwecouldprovesomethinglesspowerfulsuchasseparability.Toputalloftheseaspectsonmoresolidground,wewillhavetodevelopamathematicaltheory.GivensomeobservableAandaHamiltonianH,wehavethatthederivativeofAwrt.thecurveparameter,dA d_A;canberelatedtoPoissonbracketfA;Hg,whenimposingtheHamiltonianequationsofmotion(B.14).Expandingthisinthe1-formbasisdxiofT(�),wend_A=@A @xidxi d=@A @xdx d+@A @pdp d()=@A @x@H @p�@A @p@H @x=fA;Hg:(2.35)IfwehavethatAisaconservedquantity,_A=0,thenitsimplycommuteswiththeHamiltonian,andthetwostatementsareequivalent.WewilluseaslightlydierentlanguageandsaythattheobservableAcommuteswithH,thenitisarstintegral[12].BecauseA(x;p)=constantwhenx;pobeystheHamiltonequations,itdenesa16 2.6Classicalintegrabilityandseparability2KILLINGTENSORS hyperplaneinphasespace�ofdimension2D�2(insomenon-emptyregionof�atleast).ThesolutionoftheEOMsmustbelongtothishyperplane,andthuseectivelywehavereducedthedegreesoffreedomofthemby2(1forpositionand1formomentum),sotheequivalentHamiltoniansystemisofdimension2D�2.BythevaguestatementthattwoobservablesA;Bshouldbeindependent,wemeanthattheyshouldbefunctionallyindependent:Denition12(Functionalindependence).TwoobservablesA;Barefunctionallyinde-pendent,iftheyarenotrelatedtoeachotherbyaconstantfactor,orequivalently,thattheirverticalderivativesarelinearlyindependentalongtheEOMs.GiventwofunctionallyindependentobservablesA;B,thenwewouldliketoseewhichconditionstheymustobeyfortheretobe2dierentrstintegrals,sowehaveafurtherrestrictionofthesystem.UsingthepropertiesofthePoissonbracketoftheorem37,especiallytheJacobiidentity,wecanlookatf

22 fA;Bg;Hg:0=ffA;Bg;Hg+ffH;Ag;Bg+ffB;Hg;Ag
fA;Bg;Hg:0=ffA;Bg;Hg+ffH;Ag;Bg+ffB;Hg;Ag)ffA;Bg;Hg=�ffB;Hg;Ag+ffA;Hg;Bg=nA;_Bo+n_A;Bo=d dfA;Bg=0;andthusiftheyarereallyfunctionallyindependentwedon'thaved dfA;Bg=0ingeneral.WecanthenconcludethatwemusthavefA;Bg=0.Bythediscussionabovewecancharacterizeasystemwherewecanndauniquesolution,acompletelyintegrablesystem,bythetheorembelow,aresultcalledLioville'stheorem.Theorem13(Lioville).WesaythataHamiltoniansystemisintegrableifwehaveDfunctionallyindependentobservablesAithatallPoissoncommute,fAi;Ajg=0foralli;j.Ifthisisthecase,wethensaythattheobservablesareininvolution.Forsuchasystemthereare2D�2D=0degreesoffreedomleft,andweshouldbeabletosolvetheequationsofmotioncompletely,atleastinsomeregionwheretherearenodegeneraciesbetweentheobservables.WealwayshavethatHitselfisarstintegral,andthiscorrespondsto(classical)conservationofenergy10.ForthegeodesicequationthistranslatestothattheHamiltonianH=1 2gppshouldPoissoncommutewithDfunctionallyindependentobservablesAithatareininvolution,forittobeintegrable.ThereisadeepconnectionbetweenthePoissonbracketsofconservedobservablesandKillingtensorsthatgeneratesthem.Thefollowingtheoremholds:Theorem14(SSNandPoissonbrackets).WehavefA;Bg=0ifandonlyif[A;B]=0forthecorrespondingKillingtensors. 10Asoursystemisdenedon�,whichisagainisastructuredenedonageneralmanifoldM,itisnotcertainthatHcanreallybeinterpretedasanenergyfunction.17 2.6Classicalintegrabilityandseparability2KILLINGTENSORS Proof.WerstshowthatthisholdsfortherestrictedcaseA=A1


p(x)p1


ppandB=B1


q(x)p1


pq.Wedoadirectcalculation:fA;Bg=rA@B�@ArB=(rA1


p)p1


ppB1


q@p1


pq�A1


p@p1


pp(rB1


q)p1&#

23 1;

pp()=24qXi=1
1;

pp()=24qXi=1iBi1


q�1(rAq


p+q�1)�pXj=1jAj1


p�1(rBp


p+q�1)35p1


pp+q�1()=hqB(1


q�1rAq


p+q�1)�pA(1


p�1rBp


p+q�1)ip1


pp+q�1=�[A;B]1


p+q�1p1


pp+q�1In(*)werewrotetheqandptermsresultingfromthemomentaderivativesrela-beledthesummation,andin(**)weusedthattheproductofmomentaistotallysym-metricandthenidentiedtheSSNbracket.ThisshowsfortherestrictedcasethatfA;Bg=0,[A;B]=0.ThegeneralcaseofA=P1n=01 n!A1


n(x)p1


pnandB=P1n=01 n!B1


n(x)p1


pnthenfollowsfromlinearityofthePoissonbracket. Foreldtheoriesdescribedbyalagrangiandensity,weshouldstillthinkaboutcon-servedquantitiesofNoetherstheoremasputtingconstraintsontheequationsofmotionfortheelds.However,astheseequationsarepartialdierentialequationsincontrasttotheordinarydierentialequationsoftheparticlemechanics,andtheremightbeinternalindicesaswell,therearemanymoredegreesoffreedom.IfwehaveanumberofdIinternaldegreesoffreedominDspacetimedimensionswheretheeldsarerankptensors,thenthereareapriori~N=dI+pDdegreesoffreedomateachspacetimepoint,butthisnumberisclearlyreducedbyboundaryconditionsalongwithadditionalpropertiesoftheinternalindices.Ingeneral,wedonothaveenoughconservedquantitiestoconstraintheequationsenoughsowewouldhavetheeldtheoreticanalogofintegrabilityofdenition13.Wemaysometimesusethesymmetriestoprovethatcertainequationsofmotionsareseparableinspecialcoordinates.Onecandevelopatheoryofseparabilitystructures,whichhelpsputtingthe

24 senotionsonamorerigorousground[3].Achart
senotionsonamorerigorousground[3].AchartissaidtobearseparabilitystructureofthemanifoldiftheHamilton-Jacobiequationallowsaadditiveseparationofvariables,whererofthecoordinatesareignorable,i.e.wehaverindependentKillingvectorsinsomechart.Onecanprovethefollowingresult:Theorem15(Separability).AmanifoldMofdimensionDwithmetricgadmitsarseparabilitystructureifandonlyifthereexistsrfunctionallyindependentKillingvectors ( j), j=0;:::;r�1andD�rfunctionallyindependentrank2KillingtensorsK(i),j=1;:::;D�rsuchthathK(i);K(j)i=0;(2.36)K(i); ( j)=0;(2.37)18 3KILLING-YANOTENSORS  ( i); ( j)=0;(2.38)allwithrespecttothesymmetricSchouten-Nijenhuisbracket,andtheKillingtensorsK(i)hasD�rcommoneigenvectorsx(i)suchthathx(i);x(j)i=x(i); ( j)=0;(2.39)x(i)
( j)=0:(2.40)Therelationtoeldequationsisabitsubtle,buttheexistenceofaseparabilitystructureandtheorem15impliesthattheKlein-Gordonequationisseparable[29].OnecanalsoshowthatasimilarresultfortheDiracequationholds[6].WewillndthattheprincipalconformalKilling-YanotensordictatestheexistenceofatowerofKillingvectorsandatowerofKillingtensorsthatfulllstherequirementoftheorem15.3Killing-Yanotensors3.1BasicresultsThemotivationtoconstructtheKillingtensorsofbeforewasthattheygaveusconservedquantities.WemayintroducetheKilling-Yanotensor(KYT)class,whichcanbethoughtofasthesquarerootofaKillingtensor.TheyareevenmorefundamentalthantheKillingtensors,buttheirrelationtosymmetriesisabitobscure.Denition16(Killing-Yanotensor).ArankpKYTfisdenedasatotallyantisym-metrictensorf1


p=f[1


p]thatfulllsrf1


p=r[f1


p];(3.1)i.e.thecovariantderivativeactingonitbecomestotallyantisymmetric.TheKYTshaveanumberofnicepropertiesthatareallowsustodofurtherdevelop-ments.ThereasonwhyonecansaythatKYTsarethesquarerootofKillingtensors,isbecauseoftheirr

25 elationgivenbythefollowingtheorem:Theore
elationgivenbythefollowingtheorem:Theorem17(PropertiesofKYTs).(1):ForaKYTf1


p,wehavethatK=f2


pf2


pisaKillingtensorofrank2.(2):f2


pppisparalleltransportedalongageodesicwithtangentvectorpp.Proof.(1):ItiseasytoseethatKasdenedissymmetric,aswecanjustchangetheorderofthecontractionandinterchangethelowered/risedindices.WethendoadirectcalculationofrKrK=rf2


pf2


p=rf2


pf2


p+f2


prf2


p=f2


pr[f2


p]+f2


pr[f2


p]:19 3.1Basicresults3KILLING-YANOTENSORS Symmetrizingthis(antisymmetrizingwasdonerst)wendr(K)=f2


p(jr[f)2


p]+f2


p(jr[f)2


p]=2f2


p(jr[f)2


p]($)=�2f2


p(jr[f)2


p]=0;Thelastconclusionfollowsbecauser(K)issymmetricin;,soweshouldnotgetaminus,butwedogetonefromtheantisymmetrizationthatwasdonerst,andthusitmustvanish.(2):WewanttoshowthatD df2


ppp=0.Toprovethis,wedoadirectcalculationusingtheantisymmetryoftheKilling-Yanotensorequation(3.2):D df2


ppp=prf2


ppp=r[f2


p]ppp=0;whereweinthelastlineusedthatr[f2


p]istotallyantisymmetric,whilepppistotallysymmetric,andtheymustthenvanish. The&#

26 28;rstpartofthetheoremisunfortunatelynot
28;rstpartofthetheoremisunfortunatelynotanifandonlyiftheorem;itmightverywellbethatwecannottakethesquarerootofaKillingtensoranddecomposeitintoaKYT.Wecanagaindoaconformaltransformationofthem,anddoingthis,theirdenitionbecomes:Denition18(ConformalKilling-Yanotensor).AconformalKilling-Yanotensor(CKYT)ofrankpistotallyantisymmetrictensork1


p,p-form,thatfulllsrk1


p=r[k1


p]+pg[1 k2


p];(3.2)where k2


pisaantisymmetrictensorofrankp�1.Wendexplicitlybydoingacontractionofand1that11 k2


p=1 D�p+1rk2


p;(3.3)If k2


pvanishes,thenwesaythatk1


p=f1


pisaregularKilling-Yanotensor.ForgeneralrankCKYTs,wedonotgetthatthecontractionabovegivesarank2CKT,butitisactuallythecaseforarank2CKYT[29],aswecanprove: 11Firstwendthatg1rk1


p=g1r[k1


p]+pg1g[1 k2


p]=pg1g[1 k2


p].Now,usingthat kisantisymmetric,wecanwriteg1g[1 k2


p]=1 p!g1�g1(p�1)! k23


p�g2(p�1)! k13


p
=1 pD k2


p�1 pg1g2 k13


p=1 pD k2


p�1 p(p�1) k2


p=1 p(D�p+1) k2


p,whichthenwheninvertedgivesus(3.3).20 3.2AntisymmetricSchouten-Nijenhuisbracket3KILLING-YANOTENSORS Lemma19(CKTsandCKYTs).Letkbearank2CKYT.ThenKkkisaCKT.Proof.Weverifydirectlythat(2.9)isfullledbytakingthecovariantderivativeanduse(3.2)rK=(rk)k

27 +k(rk
+k(rk)=r[k]+2g[ k]k+kr[k]+2g[ k]:Symmetrizingthis,makestherstandthirdtermvanishastheyareantisymmetricin;.r(K)=2g([ k]k)+k(g[ k])=g( kjjk)+g( kjjk)=2g( kjjk)2g( K);whereweinthelastlineidentied kk= KastheRHSof(2.9),whichisseentohold,becauseithasthecorrectformandacontractionofbothsideswouldthengiveus K.Thisprovesthelemma. 3.2AntisymmetricSchouten-NijenhuisbracketWecandeneabracketthatworksontotallyantisymmetrictensoreldsaswell.ThisisgivenbytheantisymmetricSchoutenNijenhuis(ASN)bracket[11],whichisdened(incomponentform)as[X;Yg1


p+q�1pX[1


p�1rYp


p+q�1]+q(�1)pqY[1


q�1rXq


p+q�1](3.4)wherebothoftheinputsareformswithallindicesraised(multivectors)X1


p=X[1


p]andY1


q=Y[1


q],andwehaveassumedthattheconnectionistorsionfree.Ifthisisnotthecase,thenwemustusepartialderivativesinstead,theconnectiontermswillcancelifitistorsionfree.TheantisymmetricSchoutenNijenhuisbracketdenesaZ2-gradedLiealgebraonthevectorspaceof(anti)symmetricmultivectors,becauseonecancheckthatitsatisesthefollowinggradedLiealgebraaxiomswithZbeeingarankr(anti)symmetricmultivector[X;Yg=(�1)pq[Y;X][X;Y+Zg=[X;Y]+[X;Z](3.5)0=(�1)p(r+1)[[X;Yg;Zg+(�1)q(p+1)[[Y;Zg;Xg+(�1)r(q+1)[[Z;Xg;Yg:21 3.2AntisymmetricSchouten-Nijenhuisbracket3KILLING-YANOTENSORS Thevectorspaceofantisymmetricmultivectorsisnitedimensional,becausethereca

28 nbenoformsofranklargerthanD,sothegradedL
nbenoformsofranklargerthanD,sothegradedLiealgebramustbenitedimensionalaswell.Wemayshowthatthefollowingproductruleholds[X;Y �Zg=[X;Yg �Z+(�1)q(p+1)[X;Zg �Y(3.6)whereY �Zistheantisymmetricpartofthetensorproduct.WehavethatifXisavectorandYisamultivector,thenLXY=[X;Yg;(3.7)whichonecancheckusingthecoordinateexpressionoftheLiederivativeactingonatensor.TheLeibnizpropertyoftheLiederivativeactingonmultivectorsisaspecialcaseofthisequationand(3.6),aswecanrewritetheexpressionasLX(Y �Z)=(LXY) �Z+Y �(LXZ):(3.8)LetusnowdiscusswhattheASNbracketimpliesonthesetofallKilling-Yanotensors(withallindicesraised).AgoodquestionwouldthenbeiftheywouldformagradedLiealgebra,i.e.istheASNbracketoftwoKYTsagainaKYT.Ingeneraltheanswerisnegative,ashasbeeninvestigatedinKastoretal.[26].However,wemayprovealessgeneralresult:Theorem20(ASNbracketofrank1and2KYTs).LetbeaKilling-Yanotensorofrank1(aKillingvector),andfaKYTofrank2.Thenq[;fgisaKYTofrank2.Proof.Incomponentform,wehaveq[;fggivenby(3.4)withindicesloweredq=rf[]+2f[rjj]=rf+fr�fr;(3.9)forwhichwewanttoshowthatrq=r[q].Doingadirectcalculationofrqwemayndusing(3.1)andtherelationsofforthesecondorderderivativesfromappendixE,thatwehaverq=�3r[rf]+3 2R[f]:(3.10)Thisismanifestlyantisymmetricinallfreeindices,sorq=r[q],andthusitisaKilling-Yanotensor.Thisprovesthetheorem. Ingeneral,theSchoutenNijenhuisbracketdoesn'tdenegradedLiealgebraofKYTs,becausethebracketfailstocloseonsomethingthatisaKYT.Counter-examplesoftheclosurecanbefoundin[26],andincludesimportantclassesofspacetimessuchasthegeneralKerr-NUT-(A)dSone.However,onecanprovethatformaximallysymmetric

29 spacetimes,theydoformagradedLiealgebra[2
spacetimes,theydoformagradedLiealgebra[26].22 3.3ClosedconformalKilling-Yanotensors3KILLING-YANOTENSORS 3.3ClosedconformalKilling-YanotensorsNextlogicalstepistogoonandclassifydierentCKYTs.Wehavethreedierentclassesthatwecandivide(3.2)into,thathavedierentproperties:KYTs:Herewehaveg[1 k2


p]=0,whichmakesitaKilling-Yanotensorbydenition.Closed:Herewehaver[k1


p]=0.Thisimpliesthatk=db,wherebissomep�1form-thisholdsgloballywhenthespacetimeissimplyconnectedandlocallyifthesingularitiesaremild.SuchclosedCKYTs(CCKYTs)areveryimportantforthetheoryweareabouttobuild.Both:Inthecasebothtermsvanishes,andwesimplyhaverk1


p=0,whichmeansthatkiscovariantlyconstantandisalsobothaCCKYTandaKYT.ItturnsoutthattheHodgedualitytransformationtakesaCKYTthatisnotaKYT,intoaHodgedualthatisaKYT[5].Letusprovethis:Theorem21(KYTsandCCKYTs).TheHodgedual?kofarankpCCKYTkisaKYTf?kofrankD�pandviceversa.Proof.Assumethatk1


pisaCCKYT.Wedoadirectcalculationbytakingthecovariantderivativeoffp+1


D(?k)p+1


D=1 p!1


pp+1


Dk1


pandsimplifytheexpressionbyrelatingittotheCCKYTanditsproperties:rfp+1


D=r(?k)p+1


D=1 p!1


pp+1


Drk1


p()=1 p!1


pp+1


Dg[1 k2


p]=1 p!1


pp+1


D1 pg1 k2


p�(p�1)g2 k13


p()=1 p!p2


pp+1


D k2


p+(p�1) p!p13


pp+1


D k13


p=1

30 p!2


p
p!2


pp+1


D k2


p;wherewein(*)usedthedenitionoftheCCKY,andthenexpandedtheantisym-metrization.In(**)weinterchanged1$2togetaminus.Inthisform,wecanseeexplicitlythatrfp+1


D=r[fp+1


D]becausetheLevi-Civitatensoristotallyantisymmetricinitslowerindices.Thusf?kisaKYT,andtheconversefollowsfromthebijectivepropertiesoftheHodgedualtransformationandwehaveproventhetheorem. OneoftheimportantpropertiesofCCKYTsisthattheirwedgeproductisagainaCCKYtensorofhigherrank[28].23 3.4TheprincipalconformalKilling-Yanotensor3KILLING-YANOTENSORS Theorem22(CCKYTsandwedgeproduct).LetwbeaCCKYTofrankpandvbeaCCKYTofrankq.Thentheirwedgeproductkw^visalsoaCCKYTofrankp+q.Proof.WethenneedtoshowthatkisaCKYT,i.e.thatitobeys(3.2).Noticerstthatkw^visclosed,becauseofthepropertiesoftheexteriorderivative:dk=dw^v+(�1)pw^dv=0+0:(3.11)Wethenonlyneedtoshowthatrk1


p+qhastherightform.Todothis,weusetheproductruleofthecovariantderivative.rk1


p+q=r(w^v)1


p+q=r (p+q)! p!q!w[1


pvp+1


p+q]!=(p+q)! p!q!hrw[1


pvp+1


p+q]+w[1


prjjvp+1


p+q]i()=(p+q)! p!q!hpg[1 w2


pvp+1


p+q]+qw[1


pg[p+1 vp+2


p+qi(p+q)g[1 k2


p+q]:In(*)weusedtheactionofacovariantderivativeonaCCKYT.(*)alsoshowsthatrk1


p+qhasthecorrectformrequiredbyaCCKYTwithametricfactorthatisantisymmetrized.Theexpressionwehavefoundmustthereforebeequaltowhatwewouldndbyacontraction,(p+q)g[1 k2


p+q].Thisconcludestheproof. 3.4Theprincipal

31 conformalKilling-YanotensorWhenitexists,
conformalKilling-YanotensorWhenitexists,evenmorefundamentalistheprincipalconformalKilling-Yanotensor(PCKYT)h,whichisaspecialCCKYTofrank2.Webettergiveaproperdenition:Denition23(PCKYT).hiscalledaPCKYT,ifitisaCCKYTofrank2,i.e.from(3.2)wehavethatitisantisymmetricandsatisesrh=2g[];1 D�1rh:(3.12)Further,hmustbenon-degenerate,inthemeaningthatitasamatrixhaverank2n.iscalledtheprimaryvector.TheprimaryvectorisactuallyaKillingvectorforthemetricifweareinanEinsteinspace.Toshowthis,wecandoasmallcalculationon24 3.5Killing-Yanotowers3KILLING-YANOTENSORS r=1 D�1rrh=1 D�1grrh;whereweusedthedenitionofandfactoredoutametric.UsingtheresultsforthesecondordercovariantderivativeofthePCKYTgiveninappendixE,wendbyinsertingthisandsymmetrizingthatr()=3 21 D�1ghR(j[hjjj)]i=�3 21 D�1R(jhj);whereweusedthatRistotallyantisymmetricinthelastthreeindices,andwecouldidentifytheRiccitensorRgR.NowusingtheEinsteinspaceconditionR=gwendr()=�3 21 D�1g(jhj)=�3 41 D�1gh+gh=�3 41 D�1(h+h)=0;(3.13)usingtheantisymmetryofhtomakethenalconclusion.Actually,thisalsoholdso-shellwithoutimposingtheEinsteinconditionforthecanonicalmetric,whichwewillshowlater.ThisishowevermorecomplicatedtoshowandintimatelyrelatedtothestructureofthePCKYTandthekindofmetricsthatallowssuchone.Again,sincethathisclosed,wehavethatthereexistsaKYpotential1-formbsuchthath=db;(3.14)giventhesameassumptionsasstatedpreviously.3.5Killing-Yanotowers3.5.1Thetensortowers

32 Bytheorem22,wecanconstructamanyCCKYtenso
Bytheorem22,wecanconstructamanyCCKYtensorsfromthePCKYTinparticularbytakingthewedgeproductofitwithitself.ThisisknownastheKYtensortower-therewillbeavectortoweraswell[28].Letusdeneitproperly:25 3.5Killing-Yanotowers3KILLING-YANOTENSORS Denition24(KYtensortower).Wedenethej'thCCKYToftheKYtensortowerash(j)h^j=Vjn=1h.ThereareanumberofpropertiesconnectedtotheKYtensortower,whichwesum-marizeandproofusingmostlyearlierobtainedresultsinthefollowingtheorem:Theorem25(PropertiesofKYtower).(1):h(j)isaCCKYofrank2j.(2):Forb(j)b^h(j�1),wehaveh(j)=db(j).(3):f(j)?h(j)isaD�2jformthatisaKYtensor.(4):Tothej'thstepofthetower,thereisassociatedastepoftheKillingtensortowergivenbyK(j)f(j)2


D�2jf(j)2


D�2j:(3.15)Proof.(1):Thisissimplyaconsequenceoftheorem22.(2):BythedenitionoftheKYpotentialb,weseebydirectcalculationanduseoftheproductlawoftheexteriorderivativedthatdb(j)=db^h(j�1)=(db)^h(j�1)+(�1)2(j�1)b^dh(j�1)=(db)^h(j�1)=h(j);whereweusedthatdh(j�1)=0sinceitisclosed.(3)and(4):Thisfollowsfromtheorem17. SinceweknowthatthePCKYisnon-degenerate,wecannothavethatanyoftheh(j)stepsinthetowervanishes,becausethepotentialzeroeigenvaluecanatmostgivenzerocolumns/rows.Thelaststep,theh(n)isinevendimensionsD=2nexactlyproportionaltotheLevi-Civitatensor,theonlypossibility,whileinodddimensionsD=2n+1,wehavethath(n)isa2nformthatisdualtoaKillingvectorbytheorem25.h(n)isnotveryinteresting,andwewillexcludeitfromourtower,becausetheKillingtensorsthatwewillgenerateareeitherjustconstantorjustaproductofKillingvectors.WethereforetaketheallowedstepsoftheKillingtensortowerinanycasetobe1jn�1.Thuswegenerateatotalofn�1Killingtensorsthatgivesusconservedquantities.Ifweincludethemetrictensor(whichistriviallyaKillingtensor)asthej=0step,K(0)g,thenwehavenKillingtensorsofthe(extended)KillingtensortowerK(i),0

33 in�1.TheexplicitformoftheseKilli
in�1.TheexplicitformoftheseKillingtensorscanbeobtainedusingtheidentitiesforcontractedproductsoftheLevi-CivitatensorinappendixE.Wethenndthat(3.15)canbewrittenas26 3.5Killing-Yanotowers3KILLING-YANOTENSORS K(j)f(j)2


D�2jf(j)2


D�2j=1 (2j!)21


2j2


D�2jh(j)1


2j01


02j2


D�2jh(j)01


02j()=(2j!)(D�2j)! (2j!)2[101


2j]02jh(j)1


2jh(j)01


02j()=(2j!)(D�2j)! (2j!)2[[101


2j]02j]h(j)1


2jh(j)01


02j=(2j+1)! (2j(j!))2[[h101


h2j02j]h101


h2j02j]()=(2j)! (2jj!)2h[101


h2j02j]h[101


h2j02j]�2jh[01


h2j02j]h[01


h2j02j]A(j)�~K(j);(3.16)wherewein(*)used(E.1),in(*)thattheh(j)saretotallyantisymmetricandproductsofhsforuseinthefollowingline.In(***)weuse(E.2)tosplituptheexpression.InthelastlinewehavedenedtheconvenientquantitiesA(j)(2j)! (2jj!)2h[101


h2j02j]h[101


h2j02j];(3.17)~K(j)=2j(2j)! (2jj!)2h[01


h2j02j]h[01


h2j02j]:(3.18)Atthepresentstage,itisnotclearwhetherallofthecorrespondingconservedquanti-tiesarefunctionallyindependent.Wewilleventuallyshowthattheyinfactareinsection4.8,buthavehavealotofworktodobeforewecanmaketheconclusion.3.5.2ThevectortowerWehadawholetoweroftensorsthatgaverisetoncons

34 ervedcharges.Wealsondthattherearen+
ervedcharges.Wealsondthattherearen+"(Killing)vectors,thatgivesaKillingvectortowerwithsteps ( j),0 jn�1+".Therststepofthetoweristheprimary(Killing)vector(3.12)ofthePCKY12, (0)=K(j):(3.19)TheotherstepsofthetowerisgeneratedbycontractionofwiththeremainingstepsoftheKillingtensortower,whichgivesusn�1stepsfromcontractionoftheprimary(Killing)vectorwithtensorsfromthenon-extendedKillingtower(3.15): 12WemuststressthefactthatatpresentwehaveonlyshownthatitisaKillingvectorwhentheEinsteinequationsareimposed.WearegoingtoshowthatitisactuallyaKillingvectorforthecanonicalmetricwithouttheEinsteinequationsimposed.27 4THECANONICALMETRIC ( j)K( j):(3.20)Inevendimensions,therearenomoresteps,andtherearenowintotalD=2nstepsfromthetwotowers.Forodddimensions,thereisanextrastep,exactlytheoneKillingvectorfromtheHodgedualofh(n): ( n)f(n)=(?h)(n):(3.21)SowehaveforodddimensionsthattherearealsoD=2n+"stepsfromthetwotowerscombined.Again,atthepresentstageitisnotclearthattheyareindependent,andthusyielddierentconservedquantities.Weshowthisinsection4.8.4ThecanonicalmetricWenowrestrictourselvestospacetimesofeuclideansignatures.Thecaseswithindenitesignature,especially(�+:::+)areobtainedfromtheresultsbelowbyaWickrotation,whichwewilldiscusslater.Thisismainlybecauseofthecomputationaladvantagesinthisform,butithasnoinuenceontheexistenceanduniquenessofthePCKYT,astheformofitdoesn'tchangebythisprocedure.4.1DarbouxbasisTostudythePCKYTmoreclosely,itisusefultoseeifwecanbringagivenPCKYThtoasimplerformbyachangeofbasis(toanon-coordinateone).Taking !hh=gh(4.1)asanormalmatrixofdimensionsDD,anoperatoronthetangentspaceT(M),whichhasrealvectorspacestructure.Wecanalsoendowitwithaninnerproduct
byusingthemetricg,sowedeneX
YXY=

35 gXY;X;Y2T(M):(4.2)On
gXY;X;Y2T(M):(4.2)Onthetangentspace !histhenanantisymmetricoperator,aswehaveX
 !h
Y=gXghY=XhY=�YhX=�Y
 !h
X:(4.3)TheassociatedCKTHfromlemma19withoneindexraiseddenesanotheroperator28 4.1Darbouxbasis4THECANONICALMETRIC onT(M)whichwecanwriteas !HH�hh=� !h
!h:(4.4)Nowthinkingof !HasanoperatoronT(M),thisisclearlyasymmetricoperator,anditisalsopositivedeniteasX
 !H
Y=�X
 !h
!h
Y= !h
X
 !h
Y0:Sinceitissymmetric,itmaybediagonalizedbythespectraltheoremoflinearalgebra,andwethenknowthatthereexistsanorthonormalbasiswhereitisdiagonal.Saythatweareinthisorthonormalbasis,whichwillbeanon-coordinate(vielbein)basisingeneralasconsideredinappendixD,andconsiderthegenericeigenvalueproblem !H
n=An;wherewehavenormalizedthevectornsuchthatn
n=gnn=+1:Wecanthenprovethefollowinglemma:Lemma26(Conjugateeigenvectorsproperties).Ifnisnormalizedandaneigenvectorof !H,witheigenvalueA6=0,thenwedenetheconjugatevectoras n1 q jAj !h
n;(4.5)andthisisalsoaneigenvectorof !Hwithsameeigenvalue.Furtherwehavethatthattheyareorthogonal,i.e.n
n=0andthat n
n=1.IfA=0,thenwesetn= nandsaythatnisself-conjugate.Proof.WedoadirectcalculationofH nusingH�hh=�hh:H n=�hh0@1 q jAjhn1A=1 q jAjh(�hh)n=1 q jAjhHn=A0@1 q jAjhn1A=A n;29 4.1Darbouxbasis4THECANONICALMETRIC asclaimed.Toshowthattheconjugateandregulareigenvectorsareorthogonal,wedoasmallcalculationusingthathisantisymmetricgn n

36 =1 q jAjghnn
=1 q jAjghnn=1 q jAjhnn=0;becausetheproductnnissymmetric.Toshowthenormalizationoftheconjugateeigenvectors,weuse(4.5)g n n=1 jAjghnhn=1 jAjhhnn=1 jAjHnn=A jAjnn=sgn(A)nn=1;wheresgn(A)=1because !Hispositivedenite13.Thisnishestheproof. Theconclusiontodrawfromlemma26isthateacheigenvaluehasamultiplicityofatleasttwoifnon-zero,andthusthedimensionofthecorrespondingeigenspaceisatleast2.AstheeigenvaluesAiarepositive,wecanwritethemasAi=x2i.Now,as !hisanantisymmetricoperator,toget !H=� !h
!hdiagonalinthisbasis,wemusthavethatitisoftheform !h=0BBBBBBBBBBBB@::::::0xi�xi0:::0::::::1CCCCCCCCCCCCA:(4.6) 13Thisargumentobviouslydoesn'tholdforLorentziansignatures.Forsuchmetricswehavethatoneeigenvaluewillbenegative,andgiveusatime-likevectorinsteadofallspacelike.30 4.1Darbouxbasis4THECANONICALMETRIC Becauseoftherankofhassumedtobe2n,thereisatmostoneeigenvalueAi=0,whichcanthenonlybepresentinodddimensions,andtheremainingnxismustbefunctionallyindependent,andhcannotbecovariantlyconstant.Thusthenumberof22submatricesisncorrespondingtoAi=x2iwitheigenvectorsni; ni,andinodddimensionswehaveadditionallya11submatrixcorrespondingtotheAi=0eigenvaluewitheigenvectorn0= n0.Theeigenvaluesof !histhenbeseentobeixi.Fromthespectraltheoremwealsoknowthateigenvectorsof !Hcorrespondingtodierenteigenvaluesareorthogonal,andthusthatthesetofD=2n+"vectorsfni; nig.Inthisnotation,wemaysummarizetheresultsoflemma26asni
nj=ij; ni
nj=ij;ni
nj=0; !H
ni=Aini; !H
ni=Ai ni(4.7)ThisiscalledtheDarbouxbasis,andwedeneaorderingofthevectornianditsconjugate niintheorthonormalbasis,suchthatwehaveni=0BBBBBB@:::10:::1CCCCCCA; ni=0BBBBBB@:::01:::1CCCCCCA:(4.8)Wecanalsocollecttheeigenvectorsintoasinglesetofvectors^ni=^ni@

37 ,denedas^n0=n0(inodddimensionsonly)
,denedas^n0=n0(inodddimensionsonly),^n2i�1=ni,^n2i= ni.Withthisnotationwehave^ni
^nj=ij,i.e.thecomponentsofthemetricisdiagonal.Wecandeneadualbasisofcovectors^ni=^niasusualbyrequiring^ni(^nj)=^ni
^nj=ij[35].Theadvantageofthisisthatthemetrichasbeendiagonalized.Writingthecoordinatebasisindicesexplicitlywehavefrom(4.7)thatij=^ni
^nj=g^ni^nj:AcontractionwithtwocovectorsontheRHSonthisgivesg^ni^ni^nj^nj=g=g;andontheLHSwesimplyhaveij^ni^nj.Equatingthetwosidesyieldsg=ij^ni^nj;(4.9)andwithoutthecoordinateindicesbycontractingwiththecoordinateone-formbasis:g=ij^ni^nj(4.10)Inthesamewaywecouldderivethattheinversemetricisg�1=ij^ni^nj:(4.11)NowweturntothePCKYThandseehowthislooksintheDarbouxbasis.Ifwenowactwith(4.9)on !hweobtainthePCKYT2-formh.Wendusinghowthecovectors^ni(whichcanbethoughtofasrow-vectors)workson !hthelastdesiredresultofthissection:31 4.2TheKilling-YanotowersinDarbouxbasis4THECANONICALMETRIC h=gh=ij^ni^njh=nXi=1ninih+ ni nih+"n0n0h=nXi=1nixi ni+ ni�xini+"n0(0)=nXi=1xini^ ni:(4.12)Incoordinatefreenotationwehaveh=nXi=1xini^ ni:(4.13)4.2TheKilling-YanotowersinDarbouxbasisIntheDarbouxbasis,thestepsoftheKillingtensorandvectortowershaveverysimpleexpressionsbecauseofthesimplestructure[29].Wendbysomesimplecombinatoricsthatwecanexpressh(j)ofdenition24ash(j)=j!Xi1


jxi1


xijni1^ ni1^


^nik^ nij;(4.14)becauseoftheantisymmetricpropertiesofthewedgeproduct.OnecanalsoseebyusingtheabovethatthescalarfunctionA(j)givenby(3.17)canbeexpressedasA(j)=Xi1


jx2i1

38 ;x2ij;(4.15)whenonedoesthecombinatoricso
;x2ij;(4.15)whenonedoesthecombinatoricsoftheantisymmetricproduct.IfwedeneA(j)iXi1


ji6=iix2i1


x2ij;(4.16)wecanlowerindicesandwrite~K(k)denedin(3.18)as~K(j)=nXi=1x2iA(j�1)ini ni+ ni ni:(4.17)Noticethatfromtheequations(4.15)and(4.16),wehavetherelationA(j)=A(j)i+x2iA(j�1)i;(4.18)becausex2iA(j�1)iistheexactlythetermsthatareexcludedinthesummationofA(j)i.ThisallowsustoexplicitlywritedownthecorrespondingnstepsoftheKillingtensortower(3.16)forj=0;:::;n�1inasimpleformas32 4.3Eigenvectorsof !h4THECANONICALMETRIC K(j)=nXi=1A(j)ini ni+ ni ni+"A(j)n0 n0;(4.19)wherethelasttermisonlypresentinodddimensions.Inthisform,theKillingtensorsofthetowerareallsimultaneouslydiagonalizedandthushavecommoneigenvectors.Wemayobtainamoreexplicitformulaforthen+"stepsoftheKillingvectortower ( j)for j=0;:::;n�1+",whichinthisnotationarecovectors,givenbyequations(3.19)-(3.21)usingtheaboveresultsifwelike.Forthenrststeps j=0;:::;n�1wehave ( j)=
K( j);(4.20)whilethelastKillingvector14presentinodddimensions ( n)isgivenby(3.21).4.3Eigenvectorsof !hWewanttobuildthemostgeneralmetricfrom !Hand !husingtheireigenvectors,eigenvalues,andotherobjects.NoticerstthattheydidspecifythemetricgaswellasthePCKYThinaparticularsimpleform.WecanthereforereversetheconstructionandstartoutwithaPCKYThandtheDarbouxbasisdeterminedbyit,alongwiththemetricg,givenbyequations(4.9)and(4.12).TheresultisthennaturallygoingtobethemostgeneralmetricwithaPCKYT,wherewewillsoondetermineapropercoordinatebasissothestructureofg;haremoreclear.Theeigenvaluesof !hareixi,andwecaneasilydenethecorrespondingly(complex)eigenvectorsasalinearcombinationoftheDarbouxbasisvectorsmi1 p 2( ni+ini); mi(mi)y=1 p 2( ni�ini):(4.21)Thiswewillproveshortly.Theyarestilleigenvectorsof !H,astheyarejustalinearcombinationofeigenvectorsofthesameeigenspacewitheigenvalueAi=x2i.Thecomplexeigenvectors

39 of !hcomplexiesthegeometryifwewantt
of !hcomplexiesthegeometryifwewanttousethemasabasisinstead(forodddimensionsm0 m0^n0,whichisnotnullbutspacelike)-whichwewillbecausethen !H; !haresimultaneouslydiagonalized,whilegisnolongerdiagonal.Thisisjustaconvenienttrick,becauseweusethemasindependentvariables,butrelatedbycomplexconjugation,sothemanifoldisstillreal,astherearenomoredegreesoffreedominthisbasis.Lemma27(Propertiesof mi;mi).(1):Wehave !h
mi=�iximiand !h
mi=+ixi mi.(2):Theyarecomplexnullvectors,i.e.mi
mj= mi
mj=0andmi
mj=ijforevendimensions,andforodddimensionswehaveadditionally m0
m0=m0
m0= m0
m0=+1.(3):Forthedualvectorswehave= mimi+mi mi.Proof.(1):Wedoadirectcalculationusingthepropertiesofthe2nDarbouxbasisvectorsandmi1 p 2( ni+ini)intheeigenspacecorrespondingtothex2ieigenvalueof !H: 14YetagainwemuststressthatwehavenotyetproventhattheyareKillingvectors.33 4.3Eigenvectorsof !h4THECANONICALMETRIC !h
mi=1 p 2 !h
( ni+ini)=1 p 2 !h
ni+i !h
ni=1 p 2(+xini�ixi ni)=�ixi1 p 2(ini+ ni)=�iximi:(4.22)Wehavethat !h
mi=+ixi mifollowsfromcomplexconjugationofthisresult.(2):Thisweproveusingthepropertiesofni; njasgivenby(4.7).2mi
mj=( ni+ini)
( nj+inj)= ni
nj�ni
nj=ij�ij=0:(4.23)Wehave mi
mj=0bycomplexconjugationoftheabove.Toshowmi
mj=0wedoasimilarcalculation:2mi
mj=( ni+ini)
( nj�inj)= ni
nj�ni
nj=ij+ij=2ij;whichwaswhatwewantedforevendimensions.Fortheodddimensions,theadditionalbasisvectorm0 m0isnotnullbutspacelike,andtherelationsfollowssimplyfromtheorthonormalityof^n0.(3):ThiscanbeshowndirectlyusingthedualDarbouxbasisanditsproperties: mimi+mi mi=1 2( ni�ini) ni+ini+( ni+ini) ni�ini=1 2 ni ni+nini+ ni ni+nini()=^n

40 ;i^ni=;wherewein(*)u
;i^ni=;wherewein(*)used(4.10)withoneindexofthemetricraised.Thisconcludestheproof. 34 4.3Eigenvectorsof !h4THECANONICALMETRIC Inwhatistocome,wearegoingtoneedacovariantderivativeinthenon-coordinateDarbouxbasis.Deningthecomplexiedcovariantderivativealongmi=mi@and mi= mi@asDirmimir; Dir mi mir;D0 D0rm0m0r;(4.24)wherethespinconnectionabandfurtherdetailscanbefoundinappendixD.Wearegoingtousesomelemmas,thatgivesususefulrelationsforthe2neigenvectors mi;miandm0 m0.Lemma28.WehaveforthenulleigenvectorsthatDj !h
mi=(mi
)mj+j0i0:(4.25)whereistheprimaryvectorof(3.12)inthenullbasis,andthelasttermcanonlybepresentinodddimensions.Proof.WeprovethisusingthedenitionofthePCKYT(3.12),rh=2g[].Wecancontractthiswiththevielbeinmjg,forwhichwendmjgrh=2mjgg[],Djh=2mjgg[]=mjg(g�g)=mj�g=mj�mjContractingwithmi,i6=0,givesDjhmi=mj�mjmi=mjmi�mjmi;andtranslatingthenotationwehaveusinglemma27thatDj !h
mi=(mi
)mj+(mi
mj)| {z }=j0i0(4.26)=(mi
)mj+j0;wherethelasttermisonlypresentforodddimensions,wherewehavetheextrabasisvectorm0(whichisspacelike).Thisprovesthelemma. 35 4.3Eigenvectorsof !h4THECANONICALMETRIC Lemma29.Wehaveforthenrelationsforthenulleigenvectorsmithatfullls !h+ixiI
Djmi+i(Djxi)mi+(mi
)mj+j0i0=0;(4.27)whereIistheidentitymatrix.Proof.Ourstar

41 tingpointistheeigenvectorequationoflemma
tingpointistheeigenvectorequationoflemma27forgeneraldimensionalspacetimes, !h
mi+iximi=0.ActingwithDjonthisandusingtheLeibnizpropertiesofthecovariantderivativegivesus0=Dj !h
mi+iximi=Dj !h
mi+i(Djxi)mi+ !h
(Djmi)+ixi(Djmi)=Dj !h
mi+i(Djxi)mi+ !h+ixiI
Djmi;whereIistheidentitymatrix.Now,using(4.25)wecanrewritetherstterm,andwethennd0=(mi
)mj+j0i0+i(Djxi)mi+ !h+ixiI
Djmi:(4.28)Thisprovesthelemma. Asimilarrelationwith miisobtainedbycomplexconjugation.Aconsequenceofthislastlemmaisthatwecansaysomethingabouttheeigenvaluesxi.Thisweformulateinyetanotherlemma.Lemma30.Wehavefori6=jthatDjxi=0;Dixi=imi
;D0xi=0(nosum);(4.29)wherethelastequationisonlypresentinodddimensions.Proof.Toprovethis,werststartwiththecasei6=0contract(4.27)with mi(nosumoverrepeatedindices)andusetheresultsoflemma27,especiallymi
mj=ij:0= !h+ixiI
Djmi+i(Djxi)mi+(mi
)mj+j0i0
mi= mi
!h+ixi mi
Djmi+i(Djxi)+(mi
)ij()=(�ixi+ixi) mi
Djmi+i(Djxi)+(mi
)ij=(0) mi
Djmi+i(Djxi)+(mi
)ij=i(Djxi)+(mi
)ij;wherewein(*)usedthetransposeoftheeigenvalueequationoflemma27for miandthefactthat !hisantisymmetric,intherstterm.ThisyieldsafterasmallrewritingDjxi=i(mi
)ij;(4.30)36 4.4ThecanonicalandKerr-NUT-(A)dSmetric4THECANONICALMETRIC andespeciallyD0xi=i(mi
)i0=0;whichconcludesthei6=0case.Forodddimensionsandi=0noticethatx0=0,andthisautomaticallyimpliesD0xi=0andthisconcludestheproofofthelemma. 4.4ThecanonicalandKerr-NUT-(A)dSmetricBeforendingthemostgeneral(euclidean)metricthatallowsanPCKYT,letusrststateitsform[27]:Theorem31(Canonicalmetric).ThemostgeneralmetricindimensionsD=2n+"�2thatallowsaPCKYTisuniquelygivenbyg=nXi=1264(dxi)2 Qi+Qi0@n�1X k=0A( k)id k1A2375�"c A(n)0@nX k=0A(

42 k)d k1A2;(4.31)incoordinatesnx1;:::;xn
k)d k1A2;(4.31)incoordinatesnx1;:::;xn; 0;:::; D�n�1o,withquantitiesA( k)iXi1


ki6=iix2i1


x2i k;(4.32)A( k)Xi1


kx2i1


x2i k(4.33)UinYj=1j6=ix2j�x2i(4.34)XiXi(xi)(4.35)QiXi Ui(4.36)Theequation(4.35)forXiholdso-shell,whereitisunderstoodthatthereisonlyasinglecoordinatedependence.WhenvacuumEinsteinseldequationswithacosmologicalconstantareimposed,wendthatthemetricistheKerr-NUT-(A)dSwithXi=�2bix1�"i+"c x2i+nXk="ckx2ki;(4.37)wherewehavenparametersbi,n+1�"parametersck�0,and"parametersc,wherethetotalnumberofparametersis2n+1,andD�"areindependent.Thestructureofthemetricisquiteregularandsimple.Itisabitsimplerinevendi-mensions,wherethelasttermisnotthere.TheDcoordinatesnx1;:::;xn; 0;:::; D�n�1oarethennon-zeroxisrelatedtotheeigenvaluesof !h,alongwithD�n=n+"extracoordinates,whichwearegoingtoshowareexactlyKillingcoordinatesgeneratedbyh37 4.5Proofofexistenceanduniqueness4THECANONICALMETRIC also.Thexicoordinatescanbethoughtofasdirectionalcosinecoordinates,while kscanbethoughtofasazimuthal-likecoordinates.Itisexactlytheassumptionoffunc-tionalindependenceofthexisthatallowsustousethemascoordinates.Thesignaturehereiseuclidean,butaWickrotationwillgiveusthetheLorentziansignaturethatweareinterestedin.Aprescriptionforthiswillbediscussedlateron.WhenimposingtheEinsteinequation,oneseebycomparisontotheKerr-NUT-(A)dSmetricfoundinforexampleChenetal.[9],thatthecanonicalmetricisidenticaltoit.Asacorollaryofthetheorem,wehavethenproventhatthisverygeneralmetrichasaPCKYTalongwithallofitsgoodproperties.Alloftheparameters,whichareconservedNoetherchargesarerelatedtothecosmologicalconstant,angularmomentum,massandNUTcharges.Therelationsareabitsubtle,astheyarenotallindependent,anddependsonwhetherthedimensionisoddoreven.TheresultscanbefoundinChenetal.[9],butwewillconsiderthequestionoftheinterpretationofthemlater,anddeduce

43 specialcasesofrelevancetous.4.5Proofofex
specialcasesofrelevancetous.4.5ProofofexistenceanduniquenessTheproofwilltakeuptheremainderofthissection.Firstwedotheevendimensionscase,D=2n,andthenwegeneralizeintheendtoodddimensions.Thestrategyistodenen+n=Dnaturalorcanonicalcoordinatesnx1;:::;xn; 0; 1:::; n�1odirectlyfromwhatwehaveprovensofar.ItisessentiallyenoughtoprovethatLh=0andLg=0;(4.38)whereistheprimaryvectorgivenbythePCKYTequation(3.12).Theseequationstellsusthatgeneratesafamilyofdieomorphismsthatdoesn'tchangethePCKYThandthemetricg,andespeciallythatitisaKillingvector.Theproofisthensubdividedintothecorrespondingparts,followedbyadditionalpartswhereconstructthecoordinatebasisandimposetheEinsteinequations.Lh=0:Theresultoflemma30forthexiscanbeputtogooduseintheevendimensionscase.Ifwenowsimplydenetheabsolutesquareofthei=jcaseDixi=imi
anditscomplexconjugateasQi2jDixij2;(4.39)andwecanalsoinvertthisequationtowriteDixi=i p 2q Qi;(4.40)uptoacomplexphase,whichwesuchthatDixiisalwaysimaginary.Withthechoiceofphasein(4.40),wehavethatmi
=1 p 2p Qiand mi
=1 p 2p Qi.Wecanwriteintermsofthenullbasis:38 4.5Proofofexistenceanduniqueness4THECANONICALMETRIC =nXi=1(mi
)mi+( mi
) mi=nXi=11 i(Diximi+Dixi mi)=nXi=11 ii p 2q Qi(mi+ mi)=nXi=1q Qi1 2(( ni+ini)+( ni�ini))=nXi=1q Qi ni;(4.41)whichshowsthatisaprojectionontothesubspacespannedbythe nis.Thisshowsalreadynowthatisnotarbitrary.Asourstrategyistousethexisascoordinates,wetaketheexteriorderivativedxiofthemtoobtainasetof1-forms.Wedothisforthecomponentsrst:(dxi)=rxi()=mj mj+ mjmjrxi=mj Djxi+ mjDjxi()=iji p 2q Qi�mj+ mj()=i 2q Qi� nj+inj+ nj�inj=i 2q Qi�2ini=q Qini(nosum)(4.42)wherewein(*)usedproperty3oflemma27.In(**)weusedlemma30and(4.40),a

44 ndin(***)weusedthedenitionofthecomp
ndin(***)weusedthedenitionofthecomplexnullcovectors.Incoordinatefreenotationwehavedxi=q Qini(nosum):(4.43)Noticethatthisequationgivesusarelationfornofbasiscovectorsniinrelationtothecanonicalcoordinates.Ausefulexpressionfor
hisobtainedusingtheaboveresultsand(4.13)39 4.5Proofofexistenceanduniqueness4THECANONICALMETRIC 
h=0@nXj=1q Qj nj1A
nXi=1xini^ ni!()=nXi;j=1xjq Qj nj
nini=nXi=1xiq Qini()=d 1 2nXi=1x2i!(4.44)wherewein(*)usedtheorthonormalrelationsoftheconjugatevectors,andin(**)weused4.43.Thisshowsthatthevector
hisclosed.WecanthennallytaketheLiederivativealongusingthisandtheclosednessofh(3.12)intheCartanformula(E.3),whichgivesusLh=
dh+d(
h)=0+0;(4.45)whichwaswhatwewanted.Lg=0:Theclaimiseasytoverifyon-shellaswehavealreadydone.Itdoesalsoholdo-shell,whichwewillprovenow.TheprocessofprovingthiswillalsoallowustointroducetheremainingD�n=ncoordinatesinevendimensions.Firstnoticetheveryusefulrelation
dxi=0@nXj=1q Qj nj1A
q Qini=0;(4.46)bytheorthonormalityrelationsoftheDarbouxbasis(4.7).Ifwedeneqi
dlnq Qi;(4.47)thenwehavethatusing(4.46)wecanexpresstheLiederivativesofniand nialong,whicharegoingtobeuseful,intermsofthis.WerstdoLniusing(E.3):40 4.5Proofofexistenceanduniqueness4THECANONICALMETRIC Lni=
dni+d(ni
)(i)=
dni(ii)=
d 1 p Qidxi!(iii)=
" d1 p Qi!^dxi#=
" 1 Qidq Qi!^dxi#(iv)= 
1 Qidq Qi!^dxi+(�1)1 1 Qidq Qi!
dxi(v)= 
1 p Qidq Qi!1 p Qidxi=
dlnq Qi1 p Qidxi=+qini:(4.48)In(i)weusedthatisalinearcombinationof njsby(4.41)andtheorthonormalityoftheDarbouxbasis.In(ii)weused(4.43),andin(iii)theclosednessandLeibnizpropertyofd,in(iv)weusetheinteriorproductformula,whichcanbefoundinappendixE.In(v)weused(4.46)sothatwecouldrewritethewedgeproduct,andthenwecouldnallyidentifyqi

45 ni.Likewise,butalittlemoreinvolvedwecane
ni.Likewise,butalittlemoreinvolvedwecanexpressL niintermsofqi.WendL ni=
d ni+d( ni
)(i)=
d1 xih
ni+dq Qi(ii)=(�1)1+1 xi
(h
dni)+dq Qi(iii)=1 xi
h
d 1 p Qidxi!!+dq Qi(iv)=
d 1 p Qi!^1 xih
dxi!+dq Qi= 
d 1 p Qi!!1 xih
dxi+d 1 p Qi!
1 xih
dxi+dq Qi41 4.5Proofofexistenceanduniqueness4THECANONICALMETRIC (v)= 
d 1 p Qi!!1 xih
dxi= �
1 p Qidq Qi!1 xih
ni(vi)= �
1 p Qidq Qi! ni=�qi ni(4.49)wherewein(i)usedthat ni=1 xih
nibythedenitiongiveninlemma26andthat ni
=p Qibyuseof(4.41)andorthonormalityofDarbouxbasis.In(ii)weusedthechain-ruletoconcludethat
d1 xi/
dxi=0by(4.46),dh=0becauseitisclosed,soweonlygettermswiththeexteriorderivativeactingonni(usingthatitsatisesamodiedLeibnizruleandcommuteswithcontraction,thereare2wedgeproductsandtwocontractionshere,whichwillgiveaplussign),andhisunderstoodtobecontractedwiththenipart.In(iii)weusedthatdxi=p Qinicanbeinvertedtogiveni=ijnj=ij1 p Qjdxj=1 p Qidxi.In(iv)weusedthatdxiisclosedtorewritetheexteriorderivativeandthendothecontractionusingtheinteriorproductformulaofappendixE.Togettheresultin(v)werstidentieddxi=p Qini,then ni=1 xih
ni,andthennoticedthatd 1 p Qi!
1 xih
dxi=�1 Qidq Qi
1 xih
q Qini=�1 p Qidq Qi(
ni)=�1 p Qidq Qiq Qi=�dq Qi;sothatthelasttermintheexpressionwouldcancelthis.In(vi)wethenidentied ni=1 xih
niandqitonishtheproofoftheclaim.Bysomeadditionalthinking,wecanactuallyshowthatwehaveQi=Qi(x1;:::;xn).OnecouldthinkthatQiasdenedby(4.39)ingeneraldependsonthespacetimepointinsomecoordinatechartthathasfx1;:::;xngashalfofthecoordinatesbecauseoft

46 hecovariantderivativethatentersinthede&#
hecovariantderivativethatentersinthedenition,butthisisnotthecase.As nisaretheonlyotherobjectbesidestheeigenvaluesixiandnisdeterminedbyh,theremainingnyetundeterminedcoordinatesthatitmustdene,mustbeconstructedfromjustthe nisinsomeway,wherethespecicconstructionisnotimportantfortheargument.Asweknowthat ni
ni=0by(4.7),thisshowsthatthetangentspaceofthespacetimedenedbyhisthedirectsumoftwoorthogonalsubspaces,T(M)=f nigfnig,f nig?fnig.WehavethatQiisdenedentirelyfromquantitiesthatbelongstothesubspacespannedbynis,asweusing(4.43)canwriteQi=(ni
dxi)2,andthusitcannotdependon42 4.5Proofofexistenceanduniqueness4THECANONICALMETRIC anythingbutfx1;:::;xngcoordinates.Thisisaveryusefulresult.Aswehavedlnp Qi/Picidxibythechain-rule,whereciaresomefunctionsdeterminedbythechain-rule,wecanthenuse(4.46)tosimplyconcludethatwehaveqi=
dlnq Qi=0;(4.50)whichallowsustotoconcludethatL ni=Lni=0:(4.51)Thisprovestheclaiminevendimensions,becausebyraisingindiceswendforthedualbasisthatL ni=Lni=0,andbytheLeibnizpropertyoftheLiederivativewesimplyhaveLg=0:(4.52)KillingvectorsandtensorsofthemetricIfwetakealookattheKillingtowersgeneratedbythePCKYTforthecanonicalmetricandstatedinsection4.2inaconvenientform:K(j)=nXi=1A(j)ini ni+ ni ni;(4.53) ( j)=
K( j);(4.54)wecanwiththeaboveresultsprovethat ( j)areKillingvectors.Todothisweremem-berthatK( j)isconstructedfromHodgedualsofthewedgeproductsofthePCKYT.AstheLiederivativecommuteswiththeHodgedual15andwecanusetheLeibnizpropertyoftheLiederivativeonthewedgeproducts,weconcluderstbyusing(4.45)thatwehaveLK(j)=0:(4.55)UsingthattheSSNbracketcommuteswiththecontractionandfulllstheLeibnizrule,wehaveL ( j)g= ( j);g=[;g]
K(j)+
hK(j);gi=0+0;(4.56)whichshowsthatwehavenKillingvectorsgivenby ( j).UsingonlythatLg=Lh=0,onecanprovebythemethodofintroducingaspeciallych

47 osengenerating 15BecausetheLevi-Civitate
osengenerating 15BecausetheLevi-Civitatensoriscovariantlyconstant.43 4.5Proofofexistenceanduniqueness4THECANONICALMETRIC function16forthestepsK(j)[23,28],thatwehaveTheorem32.ForthenKillingvectorsoftheKillingvectortower ( j)andthenKillingtensorsoftheKillingtensorstowerK(j),wehavehK(i);K(j)i=0;(4.57)hK(i);i=0;(4.58)K(i); ( j)=0:(4.59)Theseresultsaresomeofthemostcrucialtowhatwewanttoprovenow,butalsointegrabilityandseparabilityasdiscussedpreviously.ThenKillingvectorsandnKillingtensorsareallclearlyindependent,becausetheyhavedierentpowersofxisduetoA(j)i,butthiscanalsobeshowndirectlyusingotherarguments[29].TheRiemanntensorandthespinconnectionThePCKYTequation(3.12)rh=g�gcangiveusvaluableinformationaboutthetheChristoelsymbolsandthecovariantderivative,asitisover-constrained[39].TorelateittotheRiemanntensorR,wecanusethatthecommutatorofcovariantderivativesiscontractedsumsofhwiththeRiemanntensorinatorsion-freeconnection.Wehaverrh=gr�gr;(4.60)andthus[r;r]h=gr�gr�gr+gr;(4.61)butwealsohavebythepropertiesofthecommutator[7][r;r]h=�Rh�Rh=Rh�Rh;(4.62)whereweinthelastlineusedthesymmetrypropertiesofRandh.WecanthenwritethecombinedequationasRh�Rh=gr�gr�gr+gr:(4.63)ContractingtheindiceswiththeDarbouxvielbeins^niandtheinverses^ni,wecanexpressthisequationequivalentlyintheDarbouxba

48 sisasRabdfhfc�Rabcfhfd=bcra
sisasRabdfhfc�Rabcfhfd=bcrad�acrbd�bdrac+adrbc:(4.64) 16Theorem32holdsfortheKillingtowersofodddimensionalspacetimesaswell.44 4.5Proofofexistenceanduniqueness4THECANONICALMETRIC Usingthepropertiesofthevielbeinsandthesymmetrypropertiesofthenon-vielbeinindicesoftheRiemanntensorthatwecanextractsomeinformationaboutthecovariantderivativesof.Takingforexamplec=2i�1andd=2i,sothattheycorrespondtothevectors^n2i�1=ni;^n2i= ni,wendthattheLHSof(4.64)vanishesbythesimpleformofhfa= !hinthisbasisandtheantisymmetryoftheRiemanntensorinthelasttwoindicies,andwehavetheidentity0=b(2i�1)ra(2i)�a(2i�1)rb(2i)�b(2i)ra(2i�1)+a(2i)rb(2i�1):(4.65)Takingdierentprojectionsofthisequality,wegetthefollowingresults:r(2i)(2j)=r(2i)(2j�1)=r(2i�1)(2j)=r(2i�1)(2j�1)=0fori6=j;(4.66)r(2i)(2i)+r(2i�1)(2i�1)=0:(4.67)Additionalnon-trivialrelationscanbealsobederivedusingtheothersymmetryrela-tionsoftheRiemanntensor.Withwhatweknowsofar,wecanalsocalculatethespinconnectioncoecientsforthetorsion-freecondition.Afteralongcalculation,onearrivesatsomerathersimplecoecients,thatcanbefoundin[25,39].ConstructingthecoordinateframesUsingtheaboveresults,wecancalculatetheLiebracketoftheDarbouxbasis.DoingacalculationofL ni,wewillndthat[ nj; ni]=0,butasimilarcalculationshowsthatweingeneralcanconcludethat[ni; nj]6=0and[ni;nj]6=0.Thismeansthattheydonotgenerateusefulcoordinatesbytheirow,becausethentheywoulddependimplicitlyonothercoordinatesdenedbytheotherowsasdiscussedinsection2.3.Tohaveausefulcoordinatebasis,wemustthenndonewhereallofthebasisvectordoescommuteasstatedintheorem6.Therightchoiceofanewbasisfei; e kgisdenedasthenvectors@xi,whichby(4.43)isproportionaltoni,alongwithnvectorswhicharethecorrespondingKillingvectorsoftheKillingvectortowergivenby(4.20)[24]:ei

49 1 p Qini=@xi;(4.68) e k (
1 p Qini=@xi;(4.68) e k ( k)=
K( k)=nXi=1A( k)iq Qi ni;(4.69)whereagaini=1;:::;nand k=0;:::;n�1.Itreallyisabasis,becausethevectorsarealllinearlyindependentastheA( k)ifactorgivesdierentpowersofxis[39].Oneshouldnoticethatei
e k=0foranychoiceofi; kbecauseoftheorthonormalityrelationsoftheDarbouxbasis.The e ksareexactlyKillingvectorsasweprovedpreviously,whileeiareactuallyeigenvectorsoftheKillingtensorsK(j)witheigenvalueA(j)i,as45 4.5Proofofexistenceanduniqueness4THECANONICALMETRIC K(j)
ei=nXk=1A(j)knk nk+ nk nk
ei=nXk=1A(j)k1 p Qini
nk nk+ni
nk nk=nXk=1A(j)k1 p Qikink=A(j)iei:(4.70)Usingtheorem32andusingthattheLiebracketsatisestheLeibnizruleandcom-muteswithcontractionsalongwithwhatweknowaboutthecovariantderivativesofandtheJacobiidentity,wecanshowthatthesevectoreldsallcommute[27]:h e i; e ji=[ei;ej]=hei; e ji=0:(4.71)Thisshowsthatfei; e kgistherightbasisthatallowsustodeneacoordinatebasisbytheorem6,withcoordinatesfxi; igsuchthatei=@xi; e i=@ i:(4.72)Wendthecobasisbyinverting(4.69)usingtheorthonormalityoftheDarbouxbasisandthedenitionofthedualbasisei(ej)=ijand e i e j= i j.Theresultaftersomemanipulationisei=q Qini=dxi;(4.73) e j=nXi=1(�x2i)n�1� j Uip Qi ni;(4.74)wherewehavedenedUinYj=1j6=ix2j�x2i;(4.75)andexpressedQiasQiXi Ui;(4.76)whereXi=Xi(xi)isafunctionofjustxi.TheformofXiisaconsequenceofalemmafromalgebra,whichcanbefoundintheappendixof[29].WecanthinkofA( k)iasannmatrix,andthenBi( j)(�x2i)n�1� j Uicanbethoughtofastheinversematrixthatweneedtoinvert(4.69),aswemayshowthat46 4.5Proofofexistenceanduniqueness4THECANONICALMETRIC nXi=1Bi( j)A( k)i= k j;n�1X k=0Bi(k)A( k)j=ij:(4.77)ThecorrespondingnKillingcoordinates kgeneratedbytheowofthesecovectorseldsnei; e jo.OnecanshowthattheowofKillingvectorsgivesclosed

50 curves,andthatwehave k2[0;2][9].Th
curves,andthatwehave k2[0;2][9].Thuswehaveobtainedacoordinatebasisforthemetricasstatedintheorem31.Wecanalsoinverttheequations(4.73)and(4.74)andobtainexpressionsforfni; nigintermsofnei; e ko.Wendintermsofthecoordinatefunctions:ni=1 p Qidxi; ni=q Qin�1X k=0A( k)id k:(4.78)Thisresultisveryuseful,asitisnowjustamatterofsubstitutiontondthemetric.InsertingintothecanonicalmetricIfweinserttheaboveresultsinthevielbeinmetricof(4.10)thenweobtain(4.73)g=nXi=1ni ni+ ni ni=nXi=1 1 p Qidxi!2+0@q Qin�1X k=0A( k)id k1A2=nXi=1(dxi)2 Qi+Qi0@n�1X k=0A( k)id k1A2Thisprovestheo-shellresultasthismetricisexactlytheformofthecanonicalmetric(4.31),andthecontentoftheorem31.Thisnishestheproofoftheexistenceanduniquenesspartforevendimensions.GeneralizingtoodddimensionsForodddimensions,wecangiveaheuristicalproofbydimensionalreductionfromD0=2(n+1)toD=2n+1bysettingxn+1=0andthenviewtheDdimensionalcanonicalmetricgasapullbackofthemetrictothishyperplanefromtheD0dimensionalspacetimewithmetricg0.The22blocksonthediagonalof !h0allspanindependentsubspaces,andthuseverythingthatwegeneratefromitwillsplitupnicely,sothexn+1coordinateisextrainthissense,as !hisasubmatrixof !h0(anextrarowandcolumnwithzeroes).Letusshowthatwecanputthemetricinthecorrectform.Firstwedoarewriting:47 4.5Proofofexistenceanduniqueness4THECANONICALMETRIC g0=n+1Xi=1264(dxi)2 Qi+Qi0@nX k=0A( k)id k1A2375=nXi=1264(dxi)2 Qi+Qi0@nX k=0A( k)id k1A2375+(dxn+1)2 Qn+1+Qn+10@nX k=0A( k)n+1d k1A2(4.79)=nXi=1264(dxi)2 Qi+Qi0@n�1X k=0A( k)id k1A2375+nXi=1QiA( n)id k2+(dxn+1)2 Qn+1+Qn+10@nX k=0A( k)n+1d k1A2Nowsettingxn+1=0andthusdxn+1=0,wehavethatthemetricgonthishyperplanereducestog=nXi=1264(dxi)2 Qi+Qi0@n�1X k=0A( k)id k1A2375+nXi=1QiA( n)id k2+Qn+10@nX k=0A( k)n+1d k1A2=nXi=1264(dxi)2 Qi+Qi0@n�1X k=0A( k)id k1A2375�c A(n)0@nX k=0A( k)d k1A2;(4.80)wherec=Xn+1(xn+1=0)issomeconstant.TheclaimnXi=1&#

51 20;QiA( n)id k2+Qn+10@nX
20;QiA( n)id k2+Qn+10@nX k=0A( k)n+1d k1A2=�c A(n)0@nX k=0A( k)d k1A2;(4.81)followswhensettingxn+1=0oftheLHS,aswethenhaveA( k)n+1=A( k)(becausetheexclusionofproductswithxn+1inthesumisthenjustA( k)),A( n)i=0(becausethereisaxn+1factorinallterms),andUn+1/A( n)(constantofproportionalityisabsorbedinc).Weknowthat ( n)isaKillingvectorfromtheorem21,sothecoordinateitgeneratesisconsistenttouse.Thisshowsthatthetheoremholdsfortheodddimensionalcaseaswell.ThePCKYTandpotentialInthecanonicalcoordinatesintroducedingeneralDdimensions,thePCKYTanditsKYpotentialgivenby(3.12)andintheDarbouxbasisas(4.13),canberewrittentotakethefollowingform:48 4.6ImposingEinsteinsequations4THECANONICALMETRIC h=nXi=1xini^ ni=nXi=1xi 1 p Qidxi!^0@q Qin�1X k=0A( k)id k1A=1 2nXi=1dx2i^0@n�1X k=0A( k)id k1A()=1 2n�1X k=0dA( k+1)i^d k:(4.82)In(*)weusedthatthestructureofA( k)iwillreduceittosumsofoneorderlower,whentakingderivativeswrt.x2is.Inthisformitisalsoeasytoseethatthepotentialshouldbetakenasb=1 2n�1X k=0A( k+1)id k:(4.83)4.6ImposingEinsteinsequationsHavingcalculatedthespinconnectioncoecients,onecanimposetheEinsteinequation,mosteasilydoneintheCartanformalism.Thiscanthenbeusedtocalculatethecom-ponentsoftheRiemanntensorasdiscussedearlier,andthentheEinsteinequationscanbeimposed,whichwasdonein[20].ThisismosteasilydoneintheDarbouxbasis,andtheresultofthisisthatXiisnolongerarbitraryandtakestheform:Xi=�2bix1�"i+"c x2i+nXk="ckx2ki;(4.84)ThiswasrstperformedbyHamamotoetal.[20]in2006,andmoreextensivelyworkedthroughinHourietal.[24].Theparametersbi;c;ckfoundintheprocesshavedierentinterpretation.Onendsthatcnisproportionaltothecosmologicalconstant,asfortheRiccitensoronendsR=(�1)n(D�1)cng;whichsatisestheEinsteinspaceconditionwithcosmologicalconstantgivenby=(�1)n(D�1)cn.Theremainingnparametersck;c(lastoneonlypresentinodddimen-sions)plusnparametersbiarerot

52 ationalandNUTparametersofthespacetime.Th
ationalandNUTparametersofthespacetime.TheNUTchargescanbethoughtofashigher-dimensionalgeneralizationoftheparameterintroducedbyNewmanetal.[30]forageneralizationoftheD=4Schwarschildmet-ric,whichcouldbeinterpretedasbeingproportionaltomagneticchargeineuclideansignature[4].Bothinoddandevendimensions,thereisascalingsymmetryofthemetriccoordinates49 4.7Wickrotatedmetric4THECANONICALMETRIC thatcanbeusedtomakeoneparameter(notcn)redundant.ThiseectivelyreducesthenumberofindependentparameterstoD�",withthecorrectnumberofrotationalparameters17asdiscussedinsection5.2,whiletherestaretheninterpretedasNUTparameters.4.7WickrotatedmetricIfweintroducearadialcoordinater�ixninthemetric(4.31),thenwewillchangethesignatureofthemetricsoitbecomesLorentzian[9].Aconsequenceofthisisthatwegetsignchangesinthequantitiesthatentersinthemetric.Forthosedependingonx2n,i.e.A( k)i;A( k);Ui,weshouldsubstitutexn=�r2intotheexpressions.ForXnwehaveXn=�2bnx1�"n+"c x2n+nXk="ckx2kn=�2bn(�i)1�"(r)1�"�"c r2+nXk="ck�r2k�2Mr1�"�"c r2+nXk="ck�r2k;(4.85)wherewehavedenedthemassparameterM(�i)1+"bn,asthecorrespondingWickrotatedNUTparameter.Whatbecomesthetimecoordinate,isthecoordinategeneratedbygeneratedbytheKillingvector, 0.ThisisbecauseasQnchangessign,wedon'tgetachangeofsignforthetermswithr,butthe 0withA( 0)i=1wegetachangeofsign.As 02[0;2]wecandoarescalingsoitbecomesmorephysicalifwelike.Theremainingcoordinatesarestillinterpretedasdirectionalcosinesandazimuthalcoordinates.TheWickrotationaspresentedaboveisahighlyformalprocedure,whichintroducescoordinatesingularitiesinthenewformofthemetric,whileitwaswell-denedintheoldcoordinates[39].Regardingtheparameters,thediscussionofsection5.2wehaveonelessindependentrotationthanfortheeuclideansignature,i.e.n�1+",whenthespatialnumberofdimensionsiseven(Dodd),astherearethesamenumberofpossiblerotationsastheformercasewithanoddn

53 umberofspatialdimensions.Inevenspacetime
umberofspatialdimensions.Inevenspacetimedimensions,wewillthenhaveanextraNUTparameter,whereoneofthemisthemass.Wethenhaven�1+"rotationalparameters,n�1�"non-massNUTparameters,themassandthecosmologicalconstantthatparametrizesthesolutiontoEinsteinsequations.4.8IntegrabilityandseparabilityAswehaveproventhattheKillingtensorsandvectorsoftheKYtowersallareindepen-dentandcommutewrt.theSSNbracketby(4.57)-(4.59),itisnowjustasimplematterofapplyingtheorem14tothisresult.Wethenhavethefollowingresultforthecanonicalmetric 17ThisdiscussionappliestoWickrotatedmetric,butherewejusthaveanextraspatialdimension.50 5STATIONARYBLACKHOLES Theorem33(Conservedcharges).ThenstepsoftheKillingtensortowerK(j),0jn�1,areallindependentandgiverisetondierentconservedcharges.Then+"stepsof( j)oftheKillingvectortower,0 jn�1+",areallindependentandgiveriseton+"dierentconservedcharges.ThetotalnumberofconservedquantitiesisD.ThetotalnumberofconservedquantitiesbeingDitisexactlyenoughtorenderthegeodesicequationintegrablebydenition13.Wealsoseeexplicitlythattheorem15holdsandwehavea(D�n)-separabilitystructure.5Stationaryblackholes5.1AspectsofblackholespacetimesBlackholesareobjectsthatonsomechosenlength-scaleareveryenergetic(tobemademoreprecise),andthustendtocurvespacetime.Farawayfromtheblackhole,spacetimeshouldbeatifwerestrictourselvestotheclassofso-calledvacuumblackholeswithcosmologicalconstantidenticalzero,whichcanbethoughtofasapproximatingthespacetimeofthewholeuniversebyjustthissingleobject.Fornon-zerocosmologicalconstant,thesituationismoreinvolved,butonecouldalsoconsiderblackholesthatareasymptotically(A)dSspacetimes,whichroughlymeansthattakingthelimitofsizeoftheblackholegoingtozero,oneshouldhavethatthegeometryapproaches(A)dS,seeAshtekarandDas[2]formoreprecisedenitions.Inanycase,thespecialthingaboutblackholesisthattheycurvespacetimesomuch,thatifyouarecloseenoughtooneandgoalongthegeode

54 sic,thenyouwillatsomepointbetrappedinsid
sic,thenyouwillatsomepointbetrappedinsideahypersurface,wherenophysics(followinganarbitraryfuture-directedcurve)canhelpyouescapeagain[7].Wenowgiveamoreprecisedenitionofwhatwemeanbythisstatement:Denition34(Blackhole).ByablackholespacetimewemeanaspacetimewherethereexistaclosedhypersurfaceE,theeventhorizon,atwhichpassingtime-likecurvesbecomesconnedinside.Thisiscommontospacetimesofarbitrarydimension,andforanyvalueofthecos-mologicalconstant.Theconsequenceofthisisthattheeventhorizonisaclosednullhypersurface,andthisfactinparticularmakeshelpsndequationsusingthemetricthatdeterminesit.Moreprecisedenitionsoftheeventhorizoncanbefoundin[21].Wewillrstconsideramorerestrictiveclassofblackholes,thestationaryones.Forblackholesthatareasymptotically(A)dS,havethesamesymmetriesastheasymptoticallyatspacetimes,because(A)dSaremaximallysymmetricspacetimes.Bystationarywemeanthefollowing:Denition35(Stationaryspacetime).Wesaythatanasymptoticallyator(A)dSspace-timeisstationary,ifthereexistsaKillingvectorTthatistime-likeintheasymptoticallyator(A)dSregion.Thisdenitionsmeansthatweinastationaryspacetimecanalwayscanndcoor-dinateswherethemetricisindependentofthetimecoordinate.Thesearetheblack51 5.2Energyandangularmomentum5STATIONARYBLACKHOLES holesthatarephysicallyofinterest,becauseafteranyphysicalprocessesthatwouldcarryenergy-momentumawayfromaconnedspatialregionhasdiminished,spacetimeshouldbeindependentoftime.Theendstateofanysuchprocessshouldthusbeablackhole(ifwearestillinavacuumregion).Ifwehavethemetricinsuchaformthatitisindepen-dentofthetimecoordinatetgeneratedbytheowofT,thenwehavethatT=@t,sincethisisthetangentvectorofcurvethatonlymovesinthetime-direction.Therearesomesubtletiesanddierenciesbetweenasymptoticallyatand(A)dSblackholes,forexamplethepossibilityofancosmologicalhorizonindSspacetime,andinAdSspacetimespatialinniteisatime-likeKillingvector[13,7].Stationar

55 ityimpliesstaticity,theexistenceofatime-
ityimpliesstaticity,theexistenceofatime-likeKillingvectorthatfurtheratanypointofspacetimeisorthogonaltoaspatialhypersurface(constanttimecoordinate),butnottheotherwayaround.Infact,astationaryspacetimedoesn'tneedtobeorthogonaltoanyspatialhypersurface,asisthecasefortherotatingblackholespacetimeswewillconsiderlateron.QuestionsaboutthetopologyoftheeventhorizonEaresubtle.Ithasbeenknownforalongtimethatinfourspacetimedimensions,itisveryrestrictedforstationaryspacetimesundersometechnicalbutreasonableassumptions[34]:Theorem36(Hawking).ForastationaryspacetimeinD=4dimensions,theeventhorizonEisa2-sphereS2.Likewiseinhigherdimensions,whenrigorousresultwerestilllacking,itwasthoughtforsometimethatthetopologyweretobeveryrestricted.In2002EmparanandReall[14]demonstratedthatthereexistedaslightgeneralizationoftheD=5Myers-PerrymetricallowedaS2S1topologyoftheeventhorizon,whichwastobecalleda(rotating)blackring.ThisgeneralizationisnotaspecialcaseofthegeneralKerr-NUT-(A)dSmetric.Atheoremonthetopologiesofofeventhorizons[18]thatgeneralizestheHawkingtheorem36showsthatingeneraleventhorizonscanindeedbeproducttopologiesinhigherdimensions,andevenconsistofseveraldisconnectedcomponents.SincetheEmparanandReallsolution,severalnon-trivialtopologieshasbeendeterminedforD=5since,forexampleaS3[(S2S1)blackSaturneventhorizoninagreementwiththemoregeneraltheorem,alongwithotherD�4dimensionalblackholeshavebeenfound[13],butnocompleteclassicationexistsatthepresenttime.5.2EnergyandangularmomentumSaynowthatweareinaD=2n+"dimensionalspacetimewithonetimedirectionandD�1spatialdirections.Aeuclideanrotationisanactiveorpassiveprocessthathappensinthehyperplanesoftwospatialcoordinatesfxi;xjg,keepingthedistancetorotationaxisandspacialdistanceaswellxed,butotherwisemovingthepointsofthehyperplanearound.InDdimensions,themaximalisometrygroupofrotationsinMinkowskispaceisthenSO(D�1),andthenumberof(notnecessarilyindependent)rotationhyperpla

56 nes,whichisexactlythenumberofgeneratorso
nes,whichisexactlythenumberofgeneratorsofthegroupisN= D�12!=1 2(D�1)(D�2);(5.1)52 5.2Energyandangularmomentum5STATIONARYBLACKHOLES becausewecanchoosetherstcoordinateaxisxiofthehyperplaneinD�1ways,andtheotherxjinD�2ways,butasthehyperplaneof(xi;xj)isthesameas(xj;xi),weonlyhavehalfoftheproduct.TherankofSO(D�1)isHn�1+",sotheCartansubalgebrahso(D�1)isofdimensionH[35].FromthetheoryoftheCartansubalgebra,weknowthatthesegeneratorswillallcommutewrt.theLiebracketwiththemselves,andarethereforeexactlythenumberofindependentrotationswecanhave.WecanthereforealwayschoseaclevercoordinatebasiswhereeachoftheHindependenthyperplanes18areassociatedwithasinglerotationalKillingvectorRi(@'i)thatgeneratesacoordinatebyitsow,plustheremainingKillingvectorsoftheisometrygroup.Inthisbasiswehavethatthemetricgisindependentofthecoordinates'i,1iH.Inthecasethatwedon'thavemaximallyrotationalsymmetry,itisasubgroupofSO(D�1).IftheonlyrotationalsymmetryleftisexactlytheonegeneratedbyCartansubgroupofSO(D�1),thenwesaythatwehaveaxialsymmetry.TheCartansubgroupwouldisthensimplyisomorphictoU(1)


U(1)=U(1)N.Inanycase,givenarotationalKillingvectorR,wehavebytheeld-theoreticversionofNoether'stheorem11,thattherecorrespondaconservedchargetotheLagrangiandensityforgeneralrelativity,theEinstein-Hilbertlagrangian[22].Forasymptoticallyatstationaryspacetimes,thisconservedquantityiswhatwewillcallangularmomentumJ'andisgivenbyJ'�1 8@nrRp gj@dD�2x;(5.2)whereissomeD�1dimensionalspacelikehypersurfaceatspatialinnitywherespacetimeisasymptoticallyat,and@istheboundaryofit,aD�2dimensionalhyper-surface.gj@isthedeterminantoftheinducedmetric-themetricon@isthepullbackofgfromMto@.Alsonistheunitnormalvectorto,andisanoutwardspointingunitnormalvectorof@.Astheseareconservedfortheg

57 ivenspacetime,theyplaytheroleofaparametr
ivenspacetime,theyplaytheroleofaparametrizationawholesetofsolutionstotheEinsteinequations,asdiscussedfortheKerr-NUT-(A)dSmetric.Sinceweknowwhatangularmomentumisinatspacetimes,wherewecoulddeneitaswehavedoneintheabove,andthespacetimeisasymptoticallyat,theconclusionisthatJ'shouldreallybethoughtofasangularmomentum.Forasymptotically(A)dSspacetimes,thecorrespondinglyconservedquan-titymustbecalculateddierently,becausetheaboveintegral(5.2)ingeneraldiverges,see[2]foradiscussionofthis.Likewisewendthattheexistenceofanasymptoticallytime-likeKillingvectorTforstationaryspacetimesgivesussomesortofenergy-conservation.ThecorrespondingconservedchargeMisgivenbyM1 4@nrTp gj@dD�2x;(5.3)whichnormallycanbethoughtofthetotalmassorenergyoftheblackholeincludinggravitationalbindingenergy.Againwehavetouseadierentdenitionoftheconserved 18Ofcourse,thereNconservedquantitiesforeachofthegeneratorsthatareexactlytheKillingvectors,butthepointisthattheyarenotallindependentastheyingeneraldoesn'tcommutewrt.theLiebracket.TheCasimirsareexactlytheindependentones.53 5.3ThePCKYTandblackholes6SPECIALCASESOFKERR-NUT-(A)DS chargein(A)dSspacetimes.5.3ThePCKYTandblackholesTheexistenceofthePCKYTgaverisetothecanonicalandtheKerr-NUT-(A)dSmetricbytheorem31.TheKerr-NUT-(A)dSblackholesarethemostgeneral(higherdimensional)blackholespacetimesolutionsofthevacuumEinsteinseldequations(withacosmologicalconstant)knownatthepresenttime.Actuallywealsohavethattheonlypossibilityfortheeventhorizontopology,becauseintheWickrotatedversionof(4.31),theeventhorizonislocatedataxedvalueoftheradialcoordinater=rH[10].Ifthereisasolution,thenitdenesaSD�2hypersphere,whichistheeventhorizon.OnecanthentakethepointofviewthattheexistenceofthePCKYTisexactlywhatcharacterizesthe(higherdimensional)blackholespacetimeswithsphericaltopologyoftheeventhorizon[17].Thisisalsothethingthatsolutionsofdier

58 entdimensionshaveincommon,alongwiththego
entdimensionshaveincommon,alongwiththegoodintegrabilityandseparabilityproperties,andinthiswaythePCKYTensuresthattheseblackholesaresimilarinanydimension.6SpecialcasesofKerr-NUT-(A)dS Name\Parameters NUT Rotational Mass Cosm.const. Dimensions Kerr-NUT-(A)dS X X X X D Myers-Perry-(A)dS X X X D Myers-Perry X X D Schwarzschild-AdS X X D Schwarzschild-Tangherlini X D Kerr X X 4 TaubNUT X X 4 Schwarzschild X 4 Table1:SomespecialcasesoftheKerr-NUT-(A)dSmetricclassiedaccordingtowhichparametersthatarepresent,andthenameofthesemetricsintheliterature.BasicallyallstationaryblackholesofinterestcanbederivedfromtheKerr-NUT-(A)dSmetricasclaimedbefore.Itisamatterofchoosingtheparameterscorrectly,butthesolutionsoneobtainsfromthegeneralformofthemetric(4.31),willingeneralnotbeinthestandardcoordinatesthatoneoftenderivesthesespecialcasesin.Itisthenofcoursenecessarytoperformachangeofcoordinatestomakeapositiveidentication.WecangetanoverviewofthepossibilitiesandtheircorrespondingnamesinrelationtotheNUTandrotationparameters,alongwiththemassandcosmologicalconstantasoutlinedinthetable1inincreasingorderofspecialization.54 6.1TheD=4Kerrsolution6SPECIALCASESOFKERR-NUT-(A)DS 6.1TheD=4KerrsolutionAsawarm-up,weshouldconsidertheD=4solutionforaelectromagneticallyneutralrotatingblackhole,theKerrsolution[37].Thiswillbeusefulsowehaveabaselineofcomparisontodierencesinhigherdimensionsonbothaquantitativelyandqualitativelybasis.TheKerrmetricisincomponentforminBoyerLindquistcoordinates19(t;r;;')givenbyg=�
dt�asin2d'2+sin2 hr2+a2id'�adt2+
dr2+d2;(6.1)wherethetensorproductinsquaresisunderstood,andwhere
r2�2Mr+a2;(6.2)r2+a2cos2;(6.3)aJ=M:(6.4)Weseethatgisindependentof'andt,sothereatleasttwoKillingvectors,thosethatareassociatedwithinvarianceundertranslationofthese.TheKillingvectorR(@')thatgenerates'-translation

59 sisarotationalKillingvector,because'!'+2
sisarotationalKillingvector,because'!'+2isthesamepointinspacetime.LikewisetheKillingvectorT(@t)thatgeneratest-translationsisaasymptoticallytime-likeKillingvector,aswemayverify.HereJistheconservedcharge(5.2)associatedwiththerotationalKillingvector,andweshouldthusthinkofitasangularmomentumandMislikewisetheconservedcharge(5.3)associatedwiththetime-likeKillingvector,thetotalmass.Bothcanbeveriedbydirectcalculation.Thequantitya=J=Misthusangularmomentumperunitmass.Ifwetakethelimitr!1,itiseasytoseethatg!insphericalcoordinates,soitisasymptoticallyat.Itisnotstatic,becausetakingt!�tgivesachangeofsignofthetwocrossterms(dt�asin2d')2and([r2+a2]d'�adt)2.Neitherisitfullysphericallysymmetric,becausetranslatingdoesn'tleavethemetricinvariant.Theconclusionisthatitisastationaryspacetime,thatdescribesarotatingblackholewithmassMandangularmomentumJrotatingthe=0hyperplane.TheBoyerLindquistcoordinatesarealreadyinasphericalcoordinatesformwherewecaneasilyndtheeventhorizon(s),whicharethengivenbygrr=
=0)
=r2H�2MrH+a2=0:(6.5)ThenumberofsolutionsforrHof(6.5)dependsontherelationshipbetweenthemassandangularmomentum.M2�Jcorrespondstothemostphysicalrelevantsituationwherethetotalenergyislargerthantheangularmomentum,andthisgivestwosolutionsrH=Mp M2�a2:(6.6) 19Rangest2R,r2R+,2[0;),'2[0;2).55 6.1TheD=4Kerrsolution6SPECIALCASESOFKERR-NUT-(A)DS IntheBoyerLindquistcoordinates,thiswilldeneahypersurfacethatisofS2topol-ogy.AnotherinterestingfeatureoftheKerrmetric,istheexistenceofanergosphere(ergohypersurface).Thisisahypersurface,thatisdenedbyTbecomingnull,i.e.0=gTT=gtt:Forgeneralstationaryspacetimestheergosphereisnotthesameastheeventhorizon,becauseingeneraltheremightbelinearcombinationsofothertangentvectorsthatwillbetime-likeattheergosurface,itisjustthestationarycurvesgeneratedbyTthatwillbecomeunph

60 ysical.Theconsequenceisthatinsideoftheer
ysical.Theconsequenceisthatinsideoftheergosphere,objectsareforcedtomove.Thiswillfurtherleadtoparticularitiessuchasframe-draggingandthePenroseprocess[7].6.1.1GettingKerrfromtheKerr-NUT-(A)dSWecangettheKerrmetricfromthegeneral(euclidean)Kerr-NUT-(A)dSmetric(4.31)bytakingD=4.ThefunctionsthatenterhavetheexpressionsA(0)1=A(0)2=1;A(1)1=x22;A(1)2=x21;(6.7)U1=�U2=x22�x21:(6.8)Themetricthenbecomesg=2Xi=1264(dxi)2 Qi+Qi0@1X k=0A( k)id k1A2375=(dx1)2 Q1+Q1A(0)1d 0+A(1)1d 12+(dx2)2 Q2+Q2A(0)2d 0+A(1)2d 12=x22�x21 X1(dx1)2+X1 x22�x21d 0+x22d 12�x22�x21 X2(dx2)2�X2 x22�x21d 0+x21d 12:IfwenowdotheWickrotationx2!ir,andalsodene 0; 1 ;x1y;X1Y;X2Rforsimplicityweobtaing=�r2+y2 X1dy2�X1 r2+y2d�r2d 2�r2+y2 X2dr2+X2 r2+y2d+y2d 2=1 r2+y2Rd+y2d 2�Yd�r2d 2�r2+y2"dy2 Y+dr2 R#:(6.9)WehavethatimposingtheEinsteinequationsspeciesX1Y;X2Rby(4.37),whichthen(withM�ib2)takestheformY=�2b1y+c0+c1y2+c2y4;(6.10)56 6.1TheD=4Kerrsolution6SPECIALCASESOFKERR-NUT-(A)DS R=�2Mr+c0�c1r2+c2r4:(6.11)Thenumberofconstantshereis5,buttheyarenotallindependentasweknowfromthegeneraltheory.Thescalingsymmetrygivenbyr!r;y!y;!�1; !�3 ;R!4R;Y!4Y;(6.12)asiseasytoverify.Wethendenenewparametersthatobeysthisscalinglawdirectly,andwecanwriteY=�2b1y+c0+c1y2+c2y4!�2b1y+c0+c12y2+c24y4=c20|{z}4a2�2b1|{z}�3Ny+c0|{z}2(a2 3�1)2y2+c2|{z} 34y4=4 a2+2Ny+ a2 3�1!y2� 3y4!=4"a2�y2 1+y2 3!+2Ny#;wherethe3independentparametersfa;N;gwehavedenedareexactlyangularmomentumpr.mass,NUTcharge,andthecosmologicalconstant.Theirphysicallyidenticationcanberelatedtothesescalingproperties,

61 asa!2a,N!3N,!0sca
asa!2a,N!3N,!0scalesaswewouldexpecttheirphysicalquantitiesdoesbyarescalingofthecoordinates.LikewisewiththesedenitionsandasimilaroneforthemassMwendthatwecanR=�2Mr+c0�c1r2+c2r4!�2b1y+c0+c12y2+c24y44"a2+r2 1�r2 3!�2Mr#;(6.13)whichwouldalsoscaleasM!3M.TheKerrmetricisthenrecoveredasaspecialcasebysettingN==0,albeitnotintheBoyerLindquistcoordinates,whichisrelatedbyachangeofvariableswhichcanbefoundinKubiznak[29].6.1.2TowerofKillingtensorsWehavealreadyfoundtwoKillingvectorsT;R,andconcludedthattherecannotbeanyfurther20bythegeneraltheory.IfthereexistsaPCKYfortheKerrspacetime,whichweeventuallywillshowthattheredoes,weknowthatthereshouldbetworank2KillingtensorsoftheextendedKillingtensortower.Oneisthemetricitself,andthelastone 20Inprinciple,therecouldbemore,buttheycouldnotbegeneratedbythePCKY.57 6.2Schwarzschild-Tangherlini6SPECIALCASESOFKERR-NUT-(A)DS givesrisetoaconservedchargerstfoundbyCarterin1968.ItwasnotunderstooduntilWalkerandPenrosein1970showedthatitoriginatedfromarank2Killingtensor[38].Itisgivenby22l(n)+r2g;(6.14)wherel1
r2+a2;
;0;a;n1 22r2+a2;�
;0;a(6.15)Wecaninterpretasthetotalangularmomentumofaparticle,asintheasymp-toticallyatregion
r2,r2,andwehavel(1;1;0;0);n1 2(1;�1;0;0):(6.16)then2r2l(n)+r2=0BBBB@+r2�r2�r2+r20+r20+r21CCCCA=0BBBB@00r2r21CCCCA:Contractingthiswithgeodesicsoftheatregioninsphericalcoordinates,wendthatppr224r2 d d!2+r2sin2 d' d!235;whichiswhatwewouldcalltotalangularmomentumperunitmasssquaredofaparticle.6.2Schwarzschild-TangherliniThersthigherdimensionalsolutiontotheEinsteineldequationswasthegeneralizationofthespher

62 icallysymmetricSchwarzschildsolution,the
icallysymmetricSchwarzschildsolution,theTangherlinisolution[13]givenbyg=�1� rD�3dt2+1� rD�3�1dr2+r2d 2D�2;(6.17)where58 6.3Myers-Perryspacetimes6SPECIALCASESOFKERR-NUT-(A)DS 16M (D�2) D�2;(6.18)whereMisthetotalmassoftheblackhole,and D�2isthehyperareaofa(D�2)-sphere.Thiswasaratherstraightforwardgeneralization,moreorlessdonebyusingthehigher-dimensionalgeneralizationofNewtonslawofuniversalgravitationtheweak-eldlimit,forwhichthepropagatorofthePoissonequationG(r)/1=rD�3changeswiththenumberofspatialdimensions,andthensolvingtheEinsteinequation,whichgreatlysimpliesusingtheSO(D�1)symmetryofspacetime.WemosteasilygettheSchwarzschild-TangherlinisolutionfromtheMyers-Perrysolutionthatweconsiderinthenextsection.TheSchwarzschild-TangherlinisolutionisalsoaspecialcaseoftheWick-rotatedKerr-NUT-(A)dSmetric(4.31),withonlyonenon-zeroparameter,themassM.ThismeansthatalloftheXifunctions(4.37)areverysimple.WeusethescalingsymmetrytonormalizeXi=1fori6=n,astherearenodegreeoffreedomlefthere,andweconcludethatXi=1(i6=n);Xn=�2Mr1�":(6.19)ThetopologyoftheeventhorizonisofcourseSD�2,anditislocatedatthecoordinatesingularity rD�3H=1)rH=D�3p ,andthusalwaysexistsforM�0.ThegeneralizationoftheSchwarzschildsolutiontohigherdimensionsthusdoesn'tgiveanyrealsurprises,whichisduetothehighdegreeof(SO(D�1))symmetryofthespacetime,thatrestrictsthephysicsverymuchbecauseofthenumberofKillingvectors,thatcanbeenhancedfromthenumbergeneratedfromtheKerr-NUT-(A)dSmetric.6.3Myers-PerryspacetimesOnceuponatime,peoplestartedwonderinghowoneshouldgeneralizetheKerrsolutiontohigherdimensionsinawaythatwasnatural.ThefactthatthereisnotfullsphericalsymmetryevenrenderedtheKerrsolutionhadtondintherstplace,anditwasrstin1986byMyersandPerrythatageneralizationwasachieved.Ofcourse,inhigherdimensionstherearemoredegreesoffreedomforarotationtotakeplace

63 ,aswefoundinsection5.2,thereareHn&#
,aswefoundinsection5.2,thereareHn�1+"independentplanesofrotationandHconservedangularmomenta.Soforanaxialsymmetrichigherdimensionalblackhole,thatrotatesinthemaximalnumberofindependenthyperplanes,wehavetheminimalnumberofrotationalKillingvectors,justspannedbytheCartansubalgebraofSO(D�1).ThesolutionfoundbyMyersandPerryhasthegeneralsolutiongivenbysuitablecoordinatesg=�dt2+U V�2Mdr2+2M U dt�HXi=1ai2id'i!2+HXi=1r2+a2id2i+2id'2i+(1�")r2d2;(6.20)wherewedene59 6.3Myers-Perryspacetimes6SPECIALCASESOFKERR-NUT-(A)DS Vr�("+1)HYi=1r2+a2i;(6.21)UV 1�HXi=1a2i2i r2+a2i!;(6.22)aiJi=M:(6.23)whereitisunderstoodthatinodddimensionswhen"=1,thecoordinateisnotpresent.Theft;r;'i;igcoordinates21areavariantofhypersphericalcoordinates,wherewehavedenediandasthedirectionalcosines,whichintheasymptoticallyatregionincartesiancoordinatescanbewrittenasixi=r.ThecoordinatesarethusnotallindependentandsatisestheconstraintHXi=12i+(1�")2=1;andoneshouldeliminateoneofthecoordinatesusingthis,butthiswillspoilthesomewhatsimplestructureofthemetric.Themetricisindependentoftand'i,soweshouldexpectsomeisometries.Therotationalcoordinates'iarethecoordinatesofrotationintheHindependenthyperplanes,andtherearethusHrotationalKillingvectorsK(i)(@'i)correspondingtothem,withconservedangularmomentumJiassociatedwitheachone.Alsothetime-likeKillingvectorT=(@t)denestheconservedquantity,themassoftheblackholeM.Again,theaiparametersthatareinterpretedasangularmomentumpr.massineachindependenthyperplaneofrotation.Aswetaker!1wehaveVr�("+1)r2H=r2H�("+1);UV=r2H�("+1);andthemetricreducestog�dt2+dr2+r2HXi=1d2i+2id'2i+(1�")r2d2;(6.24)theseareindeedjustatspaceinthevariantofhypersphericalcoordinates.ThuswecanreallyconcludethattheMyers-Perryspacetimeissta

64 tionaryandasymptoticallyat.6.3.1Eve
tionaryandasymptoticallyat.6.3.1EventhorizonThecoordinateswehaveusedaresuitableforndingtheeventhorizonsthatmayappear.Bydenitionoftheeventhorizon,weshouldsolve0=grr(rH)=V�2M U) 21Rangest2R,r2R+,i2[�1;1),'i2[0;2).60 7DISCUSSIONANDFURTHERDEVELOPMENTS 2Mr"+1H=HYi=1r2H+a2i:(6.25)Thisisacomplicatedpolynomialequationthatmayhavemanydierentrootsde-pendingontheparametersai.Wecanseefromtheequationthatthecosinedirectionsdoesn'tenter,anditisonlyanequationfortheradialcoordinaterH,andwhenithasasolution,thetopologyoftheeventhorizonisthenSD�2asexpected.6.3.2GettingMyers-PerryfromKerr-NUT-(A)dSLookingattheKerr-NUT-(A)dSmetric(4.31),weseethattheformsofthemetricslooksimilar,butthedenitionofthefunctionsthatentersthemetricsarenotthesame.WeobtaintheMyers-PerryspacetimesfromtheKerr-NUT-(A)dSbysettingallNUTchargesandthecosmologicalconstantequaltozero(keepingmass),whichreducesthedegreesoffreedomintheparametersbyn�".Thiswillcorrespondtothemass,andHrotationalparameters.TheprocedureforobtainingtheMyers-PerrymetricfromtheKerr-NUT-(A)dScanbeobtainedfrom[9],whichalsogivesusthecoordinatetransformationtogetthemetricintheform(6.20).InthispapertheauthorsstartswiththeKerr-(A)dSmetric,whichwasfoundasageneralizationoftheMyers-PerrymetricbyGibbonsetal.[19],andintroducesNUTparameters.Workingintheoppositedirection,onethenobtainstheMyers-Perrymetricbychoosingtheextraparametersinaspecicwayanddoingtheinversecoordinatetransformation,asaspecialcaseoftheKerr-NUT-(A)dS.Thisisinprinciplestraight-forward,butonemustpayattentiontothedomainofthecoordinatetransformation,andonethenreachesthedesiredconclusion.7DiscussionandfurtherdevelopmentsInsummary,whatwehaveshownintheformersections,isthatwecancontributemanyofthegood,desirableandremarkablepropertiesofalargeclassofthehigherdimensionalblackholesolutionstotheexistenceofthePCKYT.Toputthetheoryintoabiggerpicture,onecanaskhowitisconnectedtootheraspectsofphysi

65 cs.Itisawell-knownfactthatgeneralrelativ
cs.Itisawell-knownfactthatgeneralrelativitycannotbethecorrecttheoryofgravity,asitisaclassicaleldtheory.Thecorrecttheoryhastobedescribedbysomequantumtheory,ifgravityasexpectedfollowsthehistoricaldevelopmentoftheoriesoftheotherfundamentalforces,thathasbeenverysuccessfuluponquantizationofthecorrespondingclassicaleldtheory.ThusthekindofblackholesdescribedbytheKerr-NUT-(A)dSmetriccanbethoughtofasaclassicalsolution,subjecttoquantumcorrectionsthatbecomesmoreimportantathighenergiesorsmalllengthscales.Stringtheoriesaredierentcandidatesforatheoryofquantumgravity,andthoseofinterestthesedaysarehigher-dimensionalones,soitisimportanttounderstandwhatclassicalblackholeslookslikeinhigherdimensions,toimprovetheunderstandingofthequantumversions.AnopenprobleminthetheoryoftheKilling-Yanotensors,istogiveseparabilityresultsforothereldequationsthantheKlein-GordonandDiracequations[6]denedonthecanonicalspacetime.Forexample,separationofvariablesoftheminimallycoupled61 8SUMMARY Maxwellequationshaveneverbeenproven,butisconjecturedtobepossible[39].Ageneralresultabouttheseparabilityofeldequationsislackingatpresenttime.Likewiseitisalsoanopenproblemwhattherelationtoblackholeswithnon-sphericaleventhorizontopologiesis.Ithasbeenshownthateven-thoughtheblackringsolutiondoesn'tadmitaKilling-Yanotensor,adimensionallyKaluza-KleinreducedversionofthisadmitsaCKYT,whichisnotingeneralclosed[1].Thisexplainssomeofthegoodpropertiesofthe5Dblackringmetric,butfurtherexplorationhasn'tbeendoneatpresenttime.OnecanalsoconsiderproductmanifoldsM=LK,whereKisthecanonicalspacetimewithaPCKYT,andLissomeothermanifold.MwilltheninheritthegoodpropertiesofthePCKYTonK,butofcoursefullintegrabilityandseparabilityetc.dependsonthestructureofL.InparticularifwehaveL=Rp,i.e.weaddtoKatotalofpatdirectionssothemetriconMbecomesg=gK+pXi=1dz2i;(7.1)thenwegetap-branesolution,whereallofthegoodpropertiesofKcarriesoverinasimplewaybecauseLismaximallysymmetri

66 c.Onecouldthenforexampleapplythetheoryof
c.OnecouldthenforexampleapplythetheoryofblackfoldsdevelopedbyEmparanetal.[15]tostudysuchspacetimesinthelimitoftwowidelydierentlengthscalesofthespacetime.Itisalsopossibletogeneralizetheorem31forthecanonicalmetricbeginningfromthelessrestrictivecaseofjustaCKYTandnotnecessarilyaPCKYT.ThexisthatoccurredsquaredinthecorrespondingdiagonalizedKillingtensoraretheningeneralfewerinnumber,i=1;:::;`n,butitisstillpossibletoconstructanorthonormalbasisbythespectraltheoremoflinearalgebra.ThemetricthatoccursafterasimilartreatmenttowhatleadtotheKerr-NUT-(A)dSmetriciscalledthegeneralizedKerr-NUT-(A)dSmetric,withasetofparametersthatwillingeneralhaveadierentinterpretationbecauseofthedegeneracy[39].Resultsabouttheintegrabilityandseparabilitypropertiesofthesespacetimesappearsnottohavebeenstudiedsofarintheliterature.8SummaryInthisprojectwehaveinvestigatedthetheoryofKillingandKilling-Yanotensors,andtheirrelationstoexplicitandhiddensymmetries.Wehavesuccessfullyrelatedtheex-plicitsymmetriestoisometries,andhiddensymmetriestosymmetriesofthephasespaceoftheHamiltonianthatgivesthegeodesicequation.Theexistenceoftheprincipalcon-formalKilling-Yanotensorhasbeenshowntojustthethingthatsecuresintegrabilityandseparabilityofthemostimportantequationsofmotiondenedonalargeclassofspacetimes,denedbythecanonicalmetric.ImposingtheEinsteineldequationsonthisgivesustheKerr-NUT-(A)dSmetric,whichisaverygeneralblackholesolutionthathasanumberofwell-knownspacetimesasspecialcases.Allofthesespecialcaseshaveasphericaleventhorizontopology,whichisageneralfeatureofspacetimeswithaPCKYT.Variousgeneralizationsofthetheorywasbrieydiscussed,butitisclearthatasomewhatcompletetheoryatpresenttimewithfewopenquestionstobeanswered.62 REFERENCESREFERENCES References[1]JayArmas.Maximalanalyticextensionandhiddensymmetriesofthedipoleblackring.November2010.[2]AbhayAshtekarandSauryaDas.Asymptoticallyanti-desitterspace-times:Con-servedquantities.Class.Quant.G

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69 9]DavidKubiznak.HiddenSymmetriesofHigher
9]DavidKubiznak.HiddenSymmetriesofHigher-DimensionalRotatingBlackHoles.PhDthesis,UniversityofAlberta,2008.64 REFERENCESREFERENCES [30]E.Newman,L.Tamburino,andT.Unti.Empty-spacegeneralizationoftheschwarzschildmetric.JournalofMathematicalPhysics,4(7):915,1963.ISSN0022-2488.doi:10.1063/1.1704018.URLhttp://dx.doi.org/10.1063/1.1704018.[31]EmmyNoether.Invariantvariationproblems.TransportTheoryandStatisticalPhysics,1(3):186207,Jan1971.ISSN1532-2424.doi:10.1080/00411457108231446.URLhttp://dx.doi.org/10.1080/00411457108231446.Translationoforiginal1918article.[32]L.K.Norris.Schouten-nijenhuisbrackets.JournalofMathematicalPhysics,38(5):2694,1997.ISSN0022-2488.doi:10.1063/1.531981.URLhttp://dx.doi.org/10.1063/1.531981.[33]GSardanashvily.GeneralisedHamiltonianFormalismforFieldTheory:ConstraintSystems.WorldScienticPublishingCoPteLtd,1994.[34]MasaruSiino.Topologyofeventhorizons.Phys.Rev.D,58(10),Oct1998.ISSN1089-4918.doi:10.1103/physrevd.58.104016.URLhttp://dx.doi.org/10.1103/PhysRevD.58.104016.[35]PeterSzekeres.ACourseinModernMathematicalPhysics:Groups,HilbertSpaceandDierentialGeometry.CambridgeUniversityPress,2004.ISBN0521829607.[36]J.W.vanHoltenandR.H.Rietdijk.Symmetriesandmotionsinmanifolds.J.Geom.Phys.,11,1993.[37]MattVisser.Thekerrspacetime:Abriefintroduction.arXiv.org,June2007.[38]MartinWalkerandRogerPenrose.Onquadraticrstintegralsofthegeodesicequa-tionsfortypef22gspacetimes.CommunicationsinMathematicalPhysics,18(4):265274,1970.URLhttp://projecteuclid.org/euclid.cmp/1103842577.[39]YukinoriYasuiandTsuyoshiHouri.Hiddensymmetryandexactsolutionsineinsteingravity.April2011.65 AALTERNATIVEPROOFOFTHEOREM4 AAlternativeproofoftheorem4Analternativeproofoftheorem4canbegivenbyinnitesimalarguments,wheretheintuitiveaspectsareabitmoretransparent.Equation(2.2)isequivalenttotheintegralequationx()=x(=0)+0K(x())d:(A.1)Integratingto=innitesimal,wehav

70 ethattheinnitesimalowgenerated
ethattheinnitesimalowgeneratedatx()yby(A.1)torstorderinisequivalenttotheequationy=y(x)=x+K(x);(A.2)WehavetorstorderinthatK(x)=K(y)�@ @K(y�K)=02+:::=K(y)+O2:Wewillalsohavetousethat@K(y) @y=@K(x) @y+O2=@K(x) @x0@x0 @y+O2=@K(x) @x+O2:Wecanwritethetransformedmetric'gg0atxgivenby(2.1)torstorderin:g0(x)=@x @y@x @yg(y)=@(y�K(y)) @y@(y�K(y)) @yg(y)=g(y)� @K(y) @y+@K(y) @y!g(y)=g(y)� @K(x) @x+@K(x) @x!g(x)+O2:(A.3)Ifwenowalsodoanexpansionofthemetricatyintorstorder,wend66 AALTERNATIVEPROOFOFTHEOREM4 g(y)=g(y)j=0+@g(y) @=0+O2=g(x)+@x @@g(y) @x=0+O2=g(x)�K(x)@g(x) @x+O2:(A.4)Insertingthisinto(A.3),wendg0(x)=g(x)�" @K(x) @x+@K(x) @x!g(x)+K(x)@g(x) @x#+O2:(A.5)Now,therequirementthatthisgeneratesanisometry,istosaythatg0(x)=g(x)bydenition2,andwemustthenhavethattheordertermmustbezero,whichgivesustheequationg@K+g@K+K@g=0;(A.6)whereallpartialderivativesarewrt.thecoordinatex.Wewant(A.6)tobeequivalenttotheKillinge

71 quationr(K)=0,whichwewillshowt
quationr(K)=0,whichwewillshowthatitis.IfwetakealookatrKandexpanditusingthedenitionoftheChristoelsymbolrK=grK=g@K+�K=g@K+1 2g'(@g'+@g'�@'g)K=g@K+1 2'(@g'+@g'�@'g)K=g@K+1 2(@g�@g+@g)K=g@K+@[gjj]+1 2@gKSymmetrizingthis,wend67 BMECHANICSONGENERALMANIFOLDS r(K)=1 2(rK+rK)=1 2g@K+@[gjj]+1 2@gK+g@K+@[gjj]+1 2@gK=1 2g@K+g@K+1 2(@g+@g)K=1 2g@K+g@K+K@g=1 2(0)=0;whereweused(A.6)inthethirdlastlinetoconnecttheKillingequationtoisometries,aswewantedto.BMechanicsongeneralmanifoldsB.1LagrangianformalismSaythatwehavesometheorydenedonourspacetimebysomeactionS,afunctionalofcurves.ThecurvesweconsiderallmapsintoncopiesofthemanifoldMnSni=1Mi,eachcorrespondingtothepositionofagivenparticle,nintotal.WethinkofMnasthecongurationspaceofLagrangianmechanics.Hencesuchacurve maybedenedas :IR!Mn,andwecantakethemtobeasdierentiableaswewish.TogiveaproperdenitionoftheLagrangianfunctionL,werstdeneTn(M)Sni=1T(Mi),whereT(Mi)isthetangentbundleofthei'thcopyofthemanifold.WethenconsidertheactiondenedasanordinaryintegraloversomefunctionL:Tn(M)!R,theLagrangian,ofdierentiablecurves :IR!Mnandthetangentvectorstothecurve_ :IR!Tn,whereTnSni=1Tiisthecorrespondingntangentspaces.Then

72 ~ ( ;_ ):IR!Tn(M),theliftofthe
~ ( ;_ ):IR!Tn(M),theliftofthecurve ,tracesoutacurveinphasespace,whichmaythenbecomposedwiththeLagrangian.WemaythenwriteS[ ]=IL(~ ())d=IL( ();_ ())d=IL(x1();:::;xn();_x1();:::_xn())d;(B.1)whereweinthelastequalityhaveinsertedachartmappingfromthetangentbundletoparametrizedlocalcoordinatesxi()anddirectionalderivatives(velocities)_xi()dxi() dofthei'thparticle.Theclassicalequationsofmotionisgivenbystationarypointsoftheactionfunctional,anddoingavariationoftheintegral,weobtainthenEuler-Lagrangeequations68 B.2HamiltonianformalismBMECHANICSONGENERALMANIFOLDS Figure1:Phasespaceillustration,verticalandhorizontaldirections@L(xi;_xi) @xi�d d@L(xi;_xi) @_xi=0:(B.2)B.2HamiltonianformalismTheEuler-Lagrangeequationsareallsecond-orderdierentialequations.Itismoreneattohaverstorderequations,andtodothiswewouldhavetorewritetheEuler-Lagrangeequationsastwo(coupled)rst-orderequations.ThisistheHamiltonianformalism,andherewewilltaketakepositionxiandcovariantmomentumpiasindependentvariablesonthecotangentbundleTn(M)Sni=1T(Mi),asonMiwehavethat!Mi=pidxiisanarbitraryone-form22,(xi;pi)arecoordinatesonTn(M).Weshouldthinkof�Tn(M)asthedieomorphicinvariantphasespaceofnparticles,consistingofnsetsofpointsandmomentaofthemanifold.�isitselfamanifoldofdimensiondim(�)=2nDthathassomenicepropertiesasweshallsee.Wenowneedtodeneequationsom�thatwillgivethesameequationsofmotionas(B.2).FunctionsF:�!Rarecalledobservables,andwemaywritethenasF(xi;pi).Wewouldliketohavesomekindofcovariantderivativeon�.AswehavethecovariantderivativeridenedonMi,wecanextenditto�asadirectionalderivativealongthexicoordinates.GivenavectorXi2T(Mi),wehavethatitdenesaone-parameterfamilyofowbytheequationdxi() d=X

73 i(xi()),wherexi(=0)=x
i(xi()),wherexi(=0)=xi.Forthisow,pi2T(Mi)shouldbekeptconstant,soitmustbeparalleltransportedbytheequationDpi() d=dxi() dripi()=0,withpi(=0)=pi.ThedirectionalderivativealongXiofF(whichwewillcallthehorizontaldirection)isdenedas[Xri]F(xi;pi)dF(xi();pi()) d=0:(B.3) 22Inphasespaceitself,thisiscalledthecanonicalone-form.69 B.2HamiltonianformalismBMECHANICSONGENERALMANIFOLDS Forthemomentumdirection,wedenefora1-formatxi2Mi,!i2Txi(Mi),thattheverticalderivativealong!i,!i@icanbedenedbytheactiononFas[!i@i]F(xi;pi)dF(xi;pi+!i) d=0:(B.4)AsTx(Mi)isalinearvectorspaceofdimensionD,wedon'tneedtodeneanyactiononxi.Letusnowdroptheparticleindexifornotationalconvenience;itcanbethoughtofapartofifonelikes,withnDcoordinates.WithoutthedirectionsX;!,thehorizontalandverticalderivativesmaybewrittenasrFand@Finthemeaningthattheyhavetheactionof(B.3)and(B.4)ineachcoordinateand1-formcomponent.ForageneralobservableF,wemayexpanditasapower-seriesinpasF(x;p)=1Xn=01 n!f1


n(x)p1


pn;(B.5)wheref1


n(x)arefullysymmetricandonlydependsonx.Inthisform,theactionofthehorizontalderivativeareeasy,aswesimplyhaverF(x;p)=1Xn=01 n!(rf1


n(x))p1


pn;(B.6)becausetherpi'sarecovariantlyconstantwhenparalleltransportedalongofM.Likewise,fortheverticalderivative,wendusingtheproductrulethatF(x;p)=1Xn=01 n!f1


n(x)@(p1


pn)=1Xn=01 n!f2


n(xi)(1
p2


pn)+1Xn=01 n!f1


n(x)(p1p3


pn)+::::(B.7)Wecanusethecoordin

74 atesxifx;pgtoforma1-formb
atesxifx;pgtoforma1-formbasisforcovectorsofT(�),bydeningthebasiselementsasdxifdx;dpg,andwemaylikewisealsodeneabasisforvectoreldsofT(�)asgin@ @x@x;@ @p@po.AgeneralvectorX2T(�)canthenbewrittenincomponentasX=Xigi=(XM)@x+(XTM)@p;(B.8)whereXMisavectoronM,thehorizontalpart,andXTMisacovectoronTM,theverticalpart.Ageneralcovectoreld!2T(�)withcomponents!icanbewrittenas!=!idxi=(!M)dx+(!TM)dp;(B.9)where!MisacovectoronM,andXTMisavectoronTM(andacovectoronM).Aswesee,thisdenesnaturalmappingsfromT(�)toT(M)andfromT(�)toT(M).Thecovariantderivativeone-formoperatoron�canthenbedenedwithcomponentsri=fr;@g,andwecanwrite70 B.2HamiltonianformalismBMECHANICSONGENERALMANIFOLDS rridxi=rdx+@dp:(B.10)Wecandenea2-formon�,calledthesymplecticstructure ,by dx^dp;(B.11)whichisascalaronM.Thesymplecticstructureisobviouslyclosed,anditisinvariantunderachangeofcoordinates,becausedx;dptransformsoppositelyonM.Becauseofthisstructure,wesaythat�isasymplecticmanifold.ActingonvectoreldsA;igi=@A @x@x+@A @p@pandB;igi=@B @x@x+@B @p@p,whereA;Bareobservables,wendthat A;idxi;B;idxi=dxA;igi^dpB;igi=dx @A @x@x!dp @B @p@p!�dp @A @p@p!dx @B @x@x!=@A @x@B @p�@A @p@B @x=rA@B�@ArB:(B.12)WecandenenowthePoissonbracketfA;BgoftwoobservablesA;BfromthesymplecticstructuredirectlyasfA;BgrA@B�@ArB:(B.13)WecanshowthatthePoissonbrackethasanumberofniceproperties:Theorem37(PropertiesofthePoissonbracket).(1):fA+B;Cg=fA;Cg+fB;CgandfA;Bg=fA;Bg,2R(Linearity).(2):fA;Bg=�fB;Ag(Antis

75 ymme-try).(3):fAB;Cg=AfB;Cg+fA;CgB(Itisa
ymme-try).(3):fAB;Cg=AfB;Cg+fA;CgB(Itisaderivation).(4):ffA;Bg;Cg+ffC;Ag;Bg+ffB;Cg;Ag=0(Jacobiidentity),allforarbitraryobservablesA;B;C.(5):ThePoissonbracketisinvariantunderachangeofcoordinates.Proof.(1):Thisiseasilyprovenusingthelinearityofr;@.(2):Alsoeasilyproven:fA;Bg=rA@B�@ArB=�(rB@A�@BrA)=�fB;Ag(3):WendbyusingtheproductruleofcovariantdierentiationthatfAB;Cg=r(AB)@C�@(AB)rC=(rA@C)B�(@ArC)B+A(rB@C)�A(@BrC)=AfB;Cg+fA;CgB:(4):Thiscanbedoneexplicitlybywritingoutallterms,butonecanalsojustnotethatthisfollowsbecauseofassociativityofcompositionsofobservables.(5):Thisfollowsdirectlyfromthat isascalaronM. 71 B.2HamiltonianformalismBMECHANICSONGENERALMANIFOLDS Inotherwords,wehaveproventhatthePoissonbracketon�isaLiealgebra,calledthePoissonalgebra.Letusfurthershowthatitdoesn'tmatterwhichconnectionweuseforthecovariantderivative,aslongasitistorsion-free:Lemma38(Independenceofconnection).Let~randrbetwocovariantderivativeswitharbitrarytorsion-freeconnections,andletthetensorCbethedierencebetweentheconnections,i.e.~�=C+�.ThePoissonbrackettakesthesamevaluewitheitherone.Proof.Let^fA;Bgbethevalueusing~r,andfA;Bgthevalueusingr.Adirectcalculationof~rAusingtheexpansion(B.5)gives~rA=1Xn=01 n!~ra1


np1


pn=1Xn=01 n!ra1


n+C1a2


n+C2a1


n+:::p1


pn=rA+1Xn=01 n!C1a2


n+C2a1


n+:::p1


pn=rA+pC@1Xn=01 n!a1


np1


pn=rA+p

76 C@A:Usingthiswecando
C@A:Usingthiswecandoacalculationof^fA;Bg:^fA;Bg=~rA@B�@A~rB=rA@B�@ArB+pC@A@B�@ApCB()=rA@B�@ArB+pC(@A@B�@ApB)=fA;Bg;wherewein(*)usedtheassumptionabouttorsion-freeconnectionstofactoroutC,whichisthensymmetricinlowerindices. EspeciallythisalsoshowsthatthePoissonbrackethasthesamevalueinanyconformalframe,astheextraterminthecovariantderivativewillvanishundertheantisymmetricbehaviorofthePoissonbracket.WewillnowproceedtoformulateclassicalmechanicsintheHamiltonianformalism.LetusrstdenetheHamiltonianasanobservableH:�!R,andclaimthattheequationsofmotionforacurve :R!�parametrizedbyaregivenbyHamiltonsequations_xdx d=fx;Hg=@H @p;_pdp d=fp;Hg=�@H @x:(B.14)72 CCLASSICALFIELDTHEORYONGENERALMANIFOLDS Theorem39(Hamilton).LettheLagrangianLbegiven.ThenoneobtainstheHamil-tonianH,whichobeysHamiltonsequationsandyieldsthesameequationsofmotion,byaLegendretransformation.Proof.ToseethattheseareequivalenttotheLagrangianformulationwewilldoaLeg-endretransformationanddeneH(x;p)sup_x(p_x�L(x;_x));(B.15)wherethesupremumisfoundbysolving@ @_x(p_x�L(x;_x))=0,yieldingp=@L(x;_x) @_x:(B.16)TheEuler-Lagrangeequationscanthenbewrittenas_p=d d@L(x;_x) @_x=@L(x;_x) @x:(B.17)TakingtheexteriorderivativeofHincoordinateexpansion,wehavethatdH=@H @xidxi=@H @xdx+@H @pdp()=�_pdx+_xdp()=�@L(x;_x) @xdx+d(_xp)�pd_x()=d(_xp)�"@L(x;_x) @xdx+@L(x;_x) @_xd_x#=d(_xp�L);whichshowsthatthetwoformalismsareequivalent,uptosomeconstantofnoim-portance.In(*)weinvokedbothofHamilt

77 onsequations(B.14),in(**)weinvokedtheEul
onsequations(B.14),in(**)weinvokedtheEuler-Lagrangeequation(B.17)andusedthatd(_xp)�pd_x=_xdpusingtheprod-uctruleoftheexteriorderivative.Finallyin(***)weinvokedthesupremumcondition(B.16). Asaconsequenceofthis,wecanalwaysLegendretransformbacktoaLagrangian,givenaHamiltonian.CClassicaleldtheoryongeneralmanifoldsOurresultsintheabovemaybegeneralizedtogeneraltensorelds I:M!Tpq(M)denedonthemanifold,whereTpq(M)isthe(p;q)tensorbundle,andI=f;;:::gisshorthandforallthetensorcomponentsandotherindices.Inthiscasewewouldconsider73 DNON-COORDINATE(VIELBEIN)BASES theoriesdenedbytheLagrangiandensityL:Tpq(M)!R,andtheeldactionisgivenbycompositionoftheliftof IandL:S[ I]ML( I;r I;g)='(M)^L( I(x);r I(x);g)q jgjdDx;(C.1)='(M)L( I(x);r I(x);g)dDx(C.2)whereweinthesecondlinehaveinsertedlocalcoordinates.Wehavedenedtwolagrangians^L;Lhere;thedierencebetweenthemisthat^Lisascalar,andLisapseudo-scalar,thatdoesn'ttransformcorrectly,butwithcorrectmeasure.If^L=^L( I(x);r I(x))isindependentofthemetricinthemeaningthatitisalreadydenedonthemanifold( Iisdecoupledfromit),andwedovariationsthatdoesn'tchangethemetric,theequationsofmotionisasalwaysgivenbytheEuler-Lagrangeequations[21]:@^L @[ I]�r@^L @[r I]=0:(C.3)Thisderivedisundertheassumptionthatthevariationattheboundaryissettozero.Ingeneralwemayhavethattheeld Iwouldcoupletothemetric,asisforexamplethecasefortheEinstein-Hilbertactionforgeneralrelativity,andinthiscaseweshoulduseL=L( I(x);r I(x);g)asourstartingpoint.Inthiscasetheequationsofmotionare@L @[ I]�r@L @[r I]=0;(C.4)@L @[g]�r@L @[rg]=0;(C.5)aswewouldhavetovarythemetricitselfaswell,andeverythingthatdependsonthemetric,includingtheChristoelsymbols.WemayalsobeabletodeneaHamiltonianformalismandaHamiltoniandensity.Theremightbeproblemswiththeinterpretationanddieomorphicinvariance,a

78 stimeandspacewouldbe
stimeandspacewouldbetreateddierentlyinthemostnaivetreatment,wherewewouldsimplydoaLegendretransformation.Therearecuresforthis,seeforexample[33],butthisiswelloutsidethemainlineofthisproject.DNon-coordinate(vielbein)basesConsideradierentiablemanifoldMofdimensionDwithsignaturep+q.GivenvectoreldX=X@,wecandoachangeofbasisfromthecoordinatebasistoaspecialchoiceofvectoreldsX=~Xa^naXa^na,thatsatisesthattheyspananpseudo-orthonormalbasis,i.e.g(^na;^nb)=ab,whenweareworkingwithaLorentzianmanifoldM.Thevectorsf^nag,calledthevielbeins,areeverywherepseudo-orthonormalasdened,and74 DNON-COORDINATE(VIELBEIN)BASES theexistenceofsuchabasisforthevectorbundleT(M)comesfromregularLinearalgebra,becauseateachpointp2MthetangentspaceTp(M)isavectorspaceofdimensiond=dimM1,andsowecanalwayschooseapseudo-orthonormalbasis(weneedanon-degeneratemetric,toensurethis).Wecandeneapseudo-orthonormalbasisn^nboforthecovectorbundleT(M)bytherequirementthat^nb(^na)=ba,whichalsoprovestheirexistenceastheyaredualbasisvectors.Thetransformationbetweencoordinatebasisandthevielbeinbasisisgivenbylineartransformationofthecoordinatebasis(becausethereisageneralberbundleatpointofthemanifold,whichisavectorspace),ateachpointofthemanifold,sowehave^na=na@;(D.1)wherena=na(p),p2M,ofcourseisainvertiblematrixateachpoint,becausewearejustdoingachangeofbasis.ThecomponentsthentransformsasX=Xa^na=Xana@)X=Xana,Xa=naX;(D.2)wherenaistheinverseofthematrixna,whichthensatisesthatnbna=ba;nana=:(D.3)Thecovectors^nbwillhavetotransformoppositelybecausetheyaredenedbythedualbasisrequirement^nb(^na)=ba,sochangingbasisforthevielbeinsba=^nb(^na)=^nb(na@)=na^nb(@)naMb^nb=(nb)�1dxnbdx;(D.4)wherewehavedenedMb^n

79 b(@),becauseitisalineartransformati
b(@),becauseitisalineartransformation,butthenweseethatitisexactlytheinversematrix,andhencewegetthewantedtransformationlaw.Ingeneralwecannotexpectthatavielbeinisabasisforallofthemanifold,butwecanmakeasmoothchangeofvariablesateachoverlap,aswecanalwaysdenethemfromacoordinatebasis.SuchatransformationbetweenvielbeinsisthencalledalocalLorentztransformationaa0(p),LLT,whichdependsonmanifold,andwehave^na0=aa0(p)^na;^nb0=b0b(p)^nb;(D.5)wherea0a(p)istheinverseofaa0(p).Bydenitiontheymustleavethemetricun-changed(theyaresymmetrytransformations):g=ab^na ^nb=a0b0^na0 ^nb0=a0b0a0a(p)^na b0b(p)^nb=a0a(p)b0b(p)a0b0^na ^nb)75 DNON-COORDINATE(VIELBEIN)BASES a0b0=aa0(p)bb0(p)ab:(D.6)Thenumberofisometriesateachpointisthen6in1+3dimensions(3boostsand3rotations),asthesewillleavethemetricunchanged.IngeneralwehavethattheisometrygroupinDdimensionsislocallySO(p;q).Wecanriseandlowerindiceswithabandgandtheinversemetrics,becausethisisthesameinanybasis.Wecanwriteageneralvector=@inthevielbeinbasisas=DXi=1(g^ni
)^ni=DXi=1(^ni
)^ni(D.7)Ingeneralwecantakeatensorinamixedbasis,andwecanstilldocontractions,risingandloweringofindices,andtransformvielbeinsbyLLTandcoordinatebasistrans-formationsbyageneralcoordinatetransformationGCT.ThecomponentsofthetensorA=Aab^na @ ^nb dx=Aa00b00^na0 @0 ^nb0 dx0istheneasilyseentotransformasAb00a00=a0a@x0 @xbb0@x @x0Aab:(D.8)Aspecialtensoristhe(1;1)tensore=nadx ^na:(D.9)Thisisactuallythetheidentitymap,becauseitjusttakesavectorinthecoordinatebasisandchangesittothevielbeinbasis,ortakesacovectorinvielbeinbasisandtransformsittothecoordinatebasis-thecomponentschange,butthetensorsdoesn't.Hereweseethepoweroftheformalism,namelythatthemetricisalwayssimpleandconstant.Whatch

80 angesthenisthecovariantderivativeandothe
angesthenisthecovariantderivativeandotherkindsofdierentiation.Weknowhowtotakecovariantderivativesoftensorsincoordinatebasis,andthiswecanusetoderivethetransformationpropertiesforvielbeins.Firstwenotice,thatifwetakethecovariantderivativerofavectorXab,itneedstobesomethingofapartialderivativealongwithlineartransformationab,thespinconnection,termstoagreewiththedeningaxiomsofthecovariantderivative.Wecanndthatfora(1;1)tensorwehavethatthecovariantderivativehascomponentsgivenby[7]:rXab=@Xab+acXcb�cbXac;(D.10)wherethecovariantindexmusttransformwithasametransformationexceptforaminustoensurethatthecovariantderivativeofascalarcontractedfromvielbeinsisjustthepartialderivative.IfwenowtakeacovariantderivativeofthevectorXaandtransformbacktocoordinatebasis,whereweknowwhatisgoingon,wehavethatwecanndanexpressionforthespinconnectionabintermsoftheconnection�(whichwedon'tassumeismetriccompatibleortorsionfree):76 DNON-COORDINATE(VIELBEIN)BASES rXa=@Xa+abXb=@(naX)+abnbX=(@X)na+X(@na)+abnbX(D.11)ComparingwithnarX=na@X+na�X,whichisthecomponentsofthesametensorrX,wendthat�=na@na+nanbab;(D.12)whichcanbeinvertedtoyieldthespinconnectionab=nanb��nb@na:(D.13)Thespinconnectionabisnotatensor,butitdoestransformastensorinthecoor-dinatebasisindex.Underthevielbeinindicesittransformsnon-tensoriallyasa0b0=a0abb0ab�cb0@a0c:(D.14)Thecovariantderivativeofe=nadx ^naisofcoursezero,asthisisjusttheidentitymap,whichwecanverifydirectlyusingtheabove,whichgivesusre=rnadx ^na=rnadx dx&#

81 22; ^na=0)rna=0:(D.15)Thisisal
22; ^na=0)rna=0:(D.15)Thisisalsocalledthetetradpostulate.Thecovariantderivativealongavector^nb(theb-directioninthevielbeinbasis)canbewrittenasrbr^nbnbr;(D.16)andusing(D.11),wemayndthatthecovariantderivativeofavectorXacanbewrittenasrbXa=nbrXa=nb(@X)na+nbX(@na)+nbacncX:Wecanalsondthatasaspecialcase,thecovariantderivativeofoneofthebasisvectorsisrb^na=cba^nc:(D.17)Wecanviewtensorswithvielbeinindicesaseachoftheseindicestakesavectororcovectorasaninput.Thisviewisveryconvenient.ThisCartanformalismallowsustoreformulategeneralrelativityinthevielbeinformalism,whichiscomputationallysimpler.77 EUSEFULTHEOREMSANDIDENTITIES Wewillhowevernotpursuethisfurther.EUsefultheoremsandidentitiesWeuseanumberofidentitiestosimplifythecalculations,whichcanbefoundin[35].FirstaidentityforcontractedproductsofLevi-CivitatensorsinDdimensions:i1


irnr+1


nDj1


jrnr+1


nD=r!(D�r)![i1j1


ir]jr(E.1)Secondly,from[29]wehave(r+1)[i[ji1j1


ir]jr]=ij[i1[j1


ir]jr]�ri[j1[i1jjj


ir]jr](E.2)WehavealsousedCartan'smagicformulaseveraltimes,whichgivesusasimpleexpressionfortheLiederivativeofforms:Theorem40(Cartan'sidentity).ForavectorXandan-formk,wehavethatLXk=X
dk+d(X
k);(E.3)whereby"a
b"wemeantheinteriorproduct,i.e.thecontractionofbwithainrstvariable.Anotherusefulrelationistheinteriorproductformulaofwedgeproductsofap-formwandaq-formv:
(w^v)=(
w)^v+(�1)pw^(
v):(E.4)E.1SecondordercovariantderivativesforKVsandKYTsForaKillingvectorwehave[26]:2rr=�R+R=�2R:(E.5)Forarank-2CCKYThwehave[29]:2rrh=3R&