The Spin Holonomy Group In General Relativity

arXiv:gr-qc/9207006v123Jul1992May1992UMDGR-92-208THESPINHOLONOMYGROUPINGENERALRELATIVITYTedJacobson1andJosephD.Romano2DepartmentofPhysicsUniversityofMaryland,CollegePark,MD20742ABSTRACTIthasrecentlybeenshownbyGoldbergetalthattheholonomygroupofthechiralspin-connectionispreservedundertimeevolutioninvacuumgeneralrelativity.Here,theunderlyingreasonforthetime-independenceoftheholonomygroupistracedtotheself-dualityofthecurvature2-formforanEinsteinspace.Thisobservationrevealsthattheholonomygroupistime-independentnotonlyinvacuum,butalsointhepresenceofacosmologicalconstant.Italsoshowsthatoncematteriscoupledtogravity,the“conservationofholonomy”islost.Whenthefundamentalgroupofspaceisnon-trivial,theholonomygroupneednotbeconnected.Foreachhomotopyclassofloops,theholonomiescompriseacosetofthefullholonomygroupmoduloitsconnectedcomponent.Thesecosetsarealsotime-independent.Allpossibleholonomygroupsthatcanariseareclassified,andexamplesaregivenofconnectionswiththeseholonomygroups.Theclassificationoflocalandglobalsolutionswithgivenholonomygroupsisdiscussed.PACS:04.20.Cv,04.60.+n,[e-mail][e-mail]suchadearthofknownobservablesingeneralrelativity,anyobservableisworthstudying.Thisisespeciallytrueinviewofissuesraisedbyquantumtheory.Forexample,itisonlywhentrueobservablesareknownthatthephysicalinnerproductinHilbertspacecanbeconstrainedbyrealityconditions,andmeaningfulphysicalstatementscanbeextractedfromthetheory.Moreover,anobservableconstructedentirelyfromthechiralspin-connectionisparticularlyinterestingbecause,asrealizedbyAshtekar,thecomponentsofthisconnectionformacompletesetofcoordinateshavingvanishingPoissonbracketsonthephasespaceofcomplexifiedgeneralrelativity.Thecorrespondingquantumoperatorsthereforecommute,soanobservablebuiltpurelyfromthemmaybefreeofoperator-orderingambiguities.Itisthereforenoteworthythattheholonomygroupofthechiralspin-connectionisanobservableinvacuumgeneralrelativity(GR).Moreprecisely,asshowninarecentpaperbyGoldberg,Lewandowski,andStornaiolo[1],thecomplexificationoftheholonomygroupbasedatapoint∗ispreservedundertimeevolution.Inaddi-tion,thisholonomygroupisinvariantunderspatialdiffeomorphismsandSL(2,C)spin-transformationsthataretheidentityat∗.Thus,theholonomygroupqualifiesasanobservableoncethebasepoint∗andspin-frameat∗arefixed.Thisobserv-ableisdeterminedbythespin-connectiononaninitialvaluehypersurface.Thus,inahamiltonianformulationofthetheory,itisdeterminedbythephasespacecoordinates,withoutimplicitlyorexplicitlysolvingthedynamics.Inthispaper,theresultof[1]isextendedinseveraldirections.First,theunderlyingreasonforthetime-independenceoftheholonomygroupistracedtotheself-dualityofthecurvature2-formintheabsenceofmatter.Thisobservationrevealsthattheholonomygroupistime-independentnotonlyinvacuum,butalsointhepresenceofacosmologicalconstant.Italsoshowsthatoncematteriscoupledtogravity,the“conservationofholonomy”islost.Foragenericpointinphasespace,theholonomygroupwillbeallofSL(2,C).Thus,theholonomygroupobservabledoesnotcontainverymuchinformationaboutthegravitationalfieldingeneral,andforthisreason,itwouldappearnottobeaveryinterestingobservable.Itis,however,betterthannothing.More-over,whenthefundamentalgroupofspaceisnon-trivial,thereisarefinementoftheholonomygroupobservable.Foreachhomotopyclassofloops,theholonomiescompriseacosetofthefullholonomygroup.Thiscosetisalsotime-independent.1Butsincethefundamentalgroupisnot,ingeneral,invariantunderlargediffeomor-phisms,thehomotopy-classholonomycosetsarenotquiteobservables.Toobtainobservables,theactionofthesediffeomorphismsmustbefactoredout.Thisconstructionmirrorsthatof2+1dimensionalgravity,wherethevacuumequationsimplylocalflatnessoftheSO(2,1)frame-connection,andthemapfromhomotopyclassestoholonomyelementsyieldsanobservableafterfactoringbythemappingclassgroup[2,3].Remarkably,the3+1-dimensionalhomotopyobserv-ablesexisteventhoughtheconnectionisnotnecessarilyflat.Theyarenon-trivial,however,onlyiftheholonomygroupisnotallofSL(2,C).Therestofthispaperisorganizedasfollows:First,insection2,weprovideouralternateproofoftheconservationofholonomy,showingthattheresultextendstothecaseofacosmologicalconstant,butnottoarbitrarymattercoupling.Next,insection3,wespellouttherelationbetweenthislocalresultandtheglobalstatementthattheholonomygroupisanobservable.Thisdiscussionisintendedtoprovideanexplicittreatmentofsomepointsthatwereimplicitin[1].Inordertohavethemostgeneralresult,wewilltakecaretoallowforarbitraryspatialtopologyandarbitraryconnections.Insection4,thehomotopyobservableisintroduced,andinsection5,weclassifythecasesthatcanpotentiallyariseforthisobservable.Insection6,thelocalclassificationofsolutionswithrestrictedholonomyalgebrasisgiven,andtheglobalclassificationproblemisdiscussedbutnotsolved.Finally,section7containsabriefdiscussionoftheresults,theirpossibleuses,andopenquestions.Anappendixcontainsaproofofthereductiontheoremusedinsection3.2.LocalResult:Time-independenceoftheconnectionalgebraThelocalresultwasprovedin[1]usingAshtekar’shamiltonianformulationo