Special Relativity and Theory of Gravity via Maxim

arXiv:0707.3855v2[gr-qc]15Mar2008SpecialRelativityandTheoryofGravityviaMaximumSymmetryandLocalization—InHonorofthe80thBirthdayofProfessorQikengLu∗Han-YingGuoInstituteofTheoreticalPhysics,ChineseAcademyofSciences,Beijing100080,China.LikeEuclid,RiemannandLobachevskygeometriesareonanalmostequalfoot-ing,basedontheprincipleofrelativityofmaximumsymmetryproposedbyPro-fessorQikengLuandthepostulateoninvariantuniversalconstantscandR,thedeSitter/anti-deSitter(dS/AdS)specialrelativityondS/AdS-spacewithradiusRcanbesetuponanalmostequalfootingwithEinstein’sspecialrelativityonMinkowski-spaceasthecaseofR→∞.ThusthedS-spaceiscoin-like:AlawofinertiainBeltramiatlaswithBeltramitimesimultaneityfortheprincipleofrelativityononeside.Theproper-timesi-multaneityandaRobertson-Walker-likedS-spacewithentropyandanacceleratedexpandingS3fittingthecosmologicalprincipleonanother.IfouruniverseisasymptotictotheRobertson-Walker-likedS-spaceofR⋍(3/Λ)1/2,itshouldbeslightlyclosedinO(Λ)withentropyboundS≃3πc3kB/ΛG~.Contrarily,viaitsasymptoticbehavior,itcanfixonBeltramiinertialframeswithout‘anargumentinacircle’andactsastheoriginofinertia.Thereisatrialityofconformalextensionsofthreekindsofspecialrelativityandtheirnullphysicsontheprojectiveboundaryofa5-dAdS-space,anullconemoduloprojectiveequivalence[N]∼=∂P(AdS5).ThusthereshouldbeadS-spaceontheboundaryofS5×AdS5asavacuumofsupergravity.InthelightofEinstein’s‘Galileanregions’,gravityshouldbebasedonthelocalizedprincipleofrelativityoffullmaximumsymmetrywithagauge-likedynamics.Thus,thismayleadtotheoryofgravityofcorrespondinglocalsymmetry.AsimplemodelofdS-gravitycharacterizedbyadimensionlessconstantg≃(ΛG~/3c3)1/2∼10−61showsthefeaturesonumbilicalmanifoldsoflocaldS-invariance.Somegravitationaleffectsoutofgeneralrelativitymayplayaroleasdarkmatter.ThedarkuniverseanditsasymptoticbehaviormayalreadyindicatethatthedSspecialrelativityanddS-gravitybethefoundationoflargescalephysics.PACSnumbers:03.30.+p,02.40.Dr,04.90.+e,04.50+h,∗Invitedtalkatthe2006InternationalConferenceonSeveralComplexVariables,June5-9,2006;Beijing,China.2ContentsI.Introduction2II.ThreeKindsofSpecialRelativitywithMaximumSymmetry6A.BeltramimodelofRiemannsphere6B.BeltramimodelofdS-spacetime81.dS-hyperboloidHR+⊂M1,4anduniform‘greatcircular’motion92.BeltramimodelofdS-spacetimeandinertialmotions10C.Klein’sErlangenprogramversusprincipleofrelativityinallpossiblekinematics11D.Historicalremarks12III.PrincipleofRelativityandDeSitterSpecialRelativity13A.Transformationsamonginertialsystemsandprincipleofrelativity13B.Lawofinertia,generalizedEinsteinformula,lightconeandhorizon15C.Twokindsofsimultaneity,principleofrelativityandcosmologicalprinciple161.Beltramisimultaneityforprincipleofrelativity172.Proper-timesimultaneityandRobertson-Walker-likedS-space17D.CosmologicalsignificanceofdeSitterspecialrelativity19IV.ConformalExtensionsofThreeKindsofSpecialRelativity21A.ConformalextensionsofMink/dS/AdS-spacesonanullcone21B.TrialityofnullphysicsonconformalMink/dS/AdS-spaces241.Motionoffreemasslessparticlesandlightsignals242.OnCFTandAdS/CFTcorrespondence25V.TheoryofGravitywithLocalizationofMaximumSymmetry27A.Fromtheequivalenceprincipletotheprincipleoflocalization27B.Principleoflocalizationandgravitationaldynamics29C.LocalizationofdS-hyperboloidandumbilicalmanifold31D.AsimplemodelofdS-gravity34VI.ConcludingRemarks35Acknowledgments37References37I.INTRODUCTIONAsafamousmathematician,ProfessorQikengLu’scontributionstophysicsconcernvariousfields,suchasdispersionrelations,specialandgeneralrelativity,theoryofgravity,gaugetheory,integrablesystems,conformalfieldtheory,soonandsoforth.Asoneofhisgreatcontributions,he3hassuggestedthattheprincipleofrelativityshouldbegeneralizedtoconstantcurvaturespace-timeswithradiusR,i.e.deSitter(dS)andanti-deSitter(AdS)spacetimes[1]andbegantoresearchthespecialrelativityonthesemaximallysymmetricspacetimesin1970s[1,2].Actually,basedontheprincipleofrelativityofmaximumsymmetryandthepostulateoninvariantuniversalconstantsofcandR,thedS/AdS-invariantspecialrelativity,thedS/AdSspecialrelativityforshort,canbesetup[1,2,3,4,5,6,7,8,9,10,11,12,13,14]onanalmostequalfootingwithEinstein’sspecialrelativityonMinkowski(Mink)-spacetimeasthecaseofR→∞.Asiswellknown,Einstein’stheoryofrelativityincludingspecialrelativity,generalrelativityandcosmologyprovidessomeofmostimportantbreakthroughsinthelastcentury.Togetherwithquantumtheory,theyconstitutethefoundationofmodernphysicsalthoughhowtoquantizegravityformulatedingeneralrelativityisstillopen.Recentobservations[15,16]show,however,thatouruniverseisalmostcompletelydarkandacceleratedexpanding.ItisnotasymptotictoaMink-spacetime,butpossiblyadS-spacetimewithatinypositivecosmologicalconstantΛ0.WithplentyofdS-puzzles,thesegreatlychallengeEinstein’stheoryofrelativityasafoundationofphysicsinlargescale.Thecosmologicalconstantisregardedassomequantum‘vacuum’energyinordinaryapproach.Thisleadstoahugedifferenceof10−122asrealΛ0isextremelytiny.AccordingtoProfessorLu’sproposal[1]andthedSspecialrelativity[1,2,3,4,5,6,7,8,9,10,11,12,13,14],however,itshould