Proper Time Dynamics in General Relativity and Con

arXiv:gr-qc/9805083v121May1998ProperTimeDynamicsinGeneralRelativityandConformalUnifiedTheoryL.N.Gyngazov∗,M.Pawlowski†,V.N.Pervushin‡,V.I.Smirichinski§AbstractThepaperisdevotedtothedescriptionameasurabletime-interval(“propertime”)intheHamiltonianversionofgeneralrelativitywiththeDirac-ADMmetric.ToseparatethedynamicalparameterofevolutionfromthespacemetricweusetheLichnerowiczconformallyinvariantvari-ables.IntermsofthesevariablesGRisequivalenttotheconformallyinvariantPenrose-Chernicov-TagirovtheoryofascalarfieldtheroleofwhichisplayedbythescalefactormultipliedonthePlanckconstant.IdentificationofthisscalarfieldwiththemodulusoftheHiggsfieldinthestandardmodelofelectroweakandstronginteractionsallowsustoformulateanexampleofconformallyinvariantunifiedtheorywherethevacuumaveragingofthescalarfieldisdeterminedbycosmologicalintegralsofmotionoftheUniverseevolution.Keywords:Hamiltonianreduction,conformaltheory,unificationoffundamentalinteractions∗ParticlePhysicsLaboratory,JointInstituteforNuclearResearch,Dubna,Russia†SoltanInstituteforNuclearStudies,Warsaw,Poland.;e-mail:[e-mail]‡BogolubovLaboratoryonTheoreticalPhysics,JointInstituteforNuclearResearch,Dubna,Russia§BogolubovLaboratoryonTheoreticalPhysics,JointInstituteforNuclearResearch,Dubna,Russia;e-mail:[e-mail]01IntroductionThenotionof“time”,ingeneralrelativity,ismany-sided[1,2,3].Generalrelativityisinvariantwithrespecttogeneralcoordinatetransformationsin-cludingthereparametrizationsofthe“initialtime-coordinate”t7→t′=t′(t).TheEinsteinobserver,inGR,measuresthepropertimeastheinvariantgeometricalinterval.TheHamiltonianreduction[1]ofcosmologicalmodelsinspiredbyGR[1,2,3]revealstheinternaldynamical“parameterofevolution”oftheDiracinvariantsectorofphysicalvariables[4,5,6,7,8].Incosmologicalmodelsthis“evolutionparameter”isthecosmicscalevariable,andtherelationbetweenaninvariantgeometricalintervalanddynamical“evolutionparameter”(the“propertime”dynamics)describesdataoftheobservationalcosmology(theredshiftandHubblelaw).InthispaperwewouldliketogeneralizetheHamiltonianreductionwithinternalevolutionparametertothecaseoffieldtheoriesofgravity.Forresearchingtheproblemof“time”inatheorywiththegeneralcoordinatetrans-formations[1],oneconventionallyuses[9,10]theDirac-ADMparametrizationofthemetric[11]andtheLichnerowiczconformallyinvariantvariables[12]constructedbyhelpofthescalefactor(i.e.thedeterminantofthespacemetric).TheDirac-ADMparametrizationistheinvariantunderthegroupofkinemetrictrans-formations.Thelattercontainstheglobalsubgroupofthereparameterizationoftimet7→t′=t′(t).TheHamiltonianreductionofsuchthetime-reparametrizationinvari-antmechanicalsystemsisaccompaniedbytheconversionofoneoftheinitialdynamicalvariablesintoparameterofevolutionofthecorrespondingreducedsystems.YorkandKuchar[9,10]pointedoutthatsuchvariableinGR(whichisconvertedintheevolutionparameter)canbeproportionaltothetraceofthesecondform.Inthecontrastwith[9,10],wesupposethatthesecondformcanbedecomposedonbothglobalexcitationandlocalone.TheADM-metricandtheLichnerowiczconformallyinvariantvariablesallowsus[13,14]toextractthisevolutionparameterofthereducedsystem,inGR,astheglobalcom-ponentofthescalefactor.ThemaindifficultyoftheHamiltonianreductioninGRisthenecessityofseparationofparametersofgeneralcoordinatetransformationsfrominvariantphysicalvariablesandquantitiesincludingtheparameterofevolutionandpropertime.Recently,thisseparationwasfulfilledinthecosmologicalFriedmannmodels[7,8]withtheuseoftheLevi-Civitacanonicaltransformation[15,16,17],whichallowsonetoestablishdirectrelationsbetweentheDiracobservablesofthegeneralizedHamiltonianapproachandtheFriedmannonesintheobservationalcosmology(theredshiftandtheHubblelaw)expressedintermsofthepropertime.IthasbeenshownthatinthiswayonecanconstructthenormalizablewavefunctionoftheUniversesothatthevariationofthisfunctionunderthepropertimeleadstothe“redshift”measuredinobservationalcosmology[8].WeshowthattheHamiltonianreductionofGRdistinguishestheconformaltimeasmorepreferablethanthepropertimefromthepointofviewofthecorrespondenceprincipleandcausality[18].Theusageoftheconformaltime(insteadoftheproperone)asameasurableintervalcanbearguedintheconformalunifiedtheory(CUT)[19,20]based1onthestandardmodeloffundamentalinteractionswheretheHiggspotentialischangedbythePenrose-Chernicov-TagirovLagrangianforascalarfield[21].Thecontentofthepaperisthefollowing.InSection2,weuseamodelofclassicalmechanicswiththetimereparametrizationinvariancetointroducedefinitionsofalltimesusedintheextendedandreducedHamiltoniansystems.Section3isdevotedtospecialrelativitytoemphasizethemainfeaturesofrelativisticsystemswiththeframeofreferenceofanobserver.InSection4,weconsidertheFriedmanncosmologicalmodelsofexpandingUniversetofindtherelationbetweentheevolutionparameterinthereducedHamiltoniansystemandthepropertimeoftheEinstein-Friedmannobserver.InSection5,adynamicalparameterofevolutionisintroducedinGRastheglobalcomponentofthespacemetric,andanequationforthepropertimeintermsofthisdynamicalparameterisderived.Section6isdevotedtotheconstructionof