Special Relativity in the 21$^{rm st}$ century

AnnalenderPhysik,18July2008SpecialRelativityinthe21stcenturyS.Cacciatori1,,V.Gorini1,,andA.Kamenshchik2,3,1DipartimentodiFisicaeMatematica,Universitàdell’Insubria,viaValleggio11,22100Como,ItalyandINFN,Sez.diMilano,Italy2DipartimentodiFisicaandINFN,viaIrnerio46,40126Bologna,Italy3L.D.LandauInstituteforTheoreticalPhysicsoftheRAS,Kosyginstreet2,119334Moscow,RussiaReceived27May2008,revised18July2008,acceptedXXJuly2008byF.W.HehlPublishedonlineXXXXKeywordsMinkowski,specialrelativity,generalrelativity,deSittergroup,Minkowskihyperboloid.PACS03.30.+p,04.20.Cv,98.80.-k,98.80.JkWededicatethispapertoAndrzejKossakowskiontheoccasionofhis70thbirthday.Thispaper,whichismeanttobeatributetoMinkowski’sgeometricalinsight,restsontheideathatthebasicobservedsymmetriesofspacetimehomogeneityandofisotropyofspace,whicharedisplayedbythespacetimemanifoldinthelimitingsituationinwhichtheeffectsofgravitycanbeneglected,leadstoaformulationofspecialrelativitybasedontheappearanceoftwouniversalconstants:alimitingspeedcandacosmologicalconstantwhichmeasuresaresidualcurvatureoftheuniverse,whichisnotascribabletothedistributionofmatter-energy.Thattheseconstantsshouldexistisanoutcomeoftheunderlyingsymmetriesandisconfirmedbyexperimentsandobservations,whichfurnishtheiractualvalues.Specifically,itturnsoutonthesefoundationsthatthekinematicalgroupofspecialrelativityisthedeSittergroupdS(c;)=SO(1;4).Onthisbasis,wedevelopatanelementaryclassicaland,hopefully,sufficientlydidacticallevelthemainaspectsofthetheoryofspecialrelativitybasedonSO(1;4)(deSitterrelativity).Asanapplication,weapplytheformalismtoanintrinsicformulationofpointparticlekinematicsdescribingbothinertialmotionandparticlecollisionsanddecays.CopyrightlinewillbeprovidedbythepublisherContents1Introduction22Kinematicalalgebras63Spacetimemanifold94Causalstructureandcoordinatesystems114.1Flatcoordinates........................................144.2Sphericalcoordinates.....................................154.3Hyperboliccoordinates....................................155Flatlimit166KinematicsintheMinkowskihyperboloid196.1Energy.............................................216.2Energyinflatcoordinates...................................226.3Momentum..........................................236.4deSittermassshell......................................256.5Collisionsanddecays.....................................25E-mail:[e-mail]E-mail:[e-mail]E-mail:[e-mail]videdbythepublisherarXiv:0807.3009v1[gr-qc]18Jul20082S.Cacciatori,V.Gorini,andA.Kamenshchik:SpecialRelativityinthe21stcentury6.6Detection...........................................277KillingvectorsinflatFriedmannanddeSitteruniverses288Conclusions31AConstructionofthedeSittermanifolddS(1;3)33BCurvaturetensor34B.1Covariantderivatives,connection...............................34B.2Levi-Civita(Christoffel)connection.............................35B.3Geodesicmotions.......................................37B.4Curvaturetensor........................................37CKillingequations38KillingvectorsoftheMinkowskihyperboloid...........................38References391IntroductionItseemsappropriatetous,ontheoccasionofthecentennialofHermannMinkowski’sgeometricalformu-lationofspecialrelativity,whichheputforwardinhiscelebratedaddress“RaumundZeit”deliveredonthe21stofSeptember1908,atthe80thAssemblyofGermanNaturalScientistsandPhysiciansatCologne[1],topaytributetothisgreatscientistandmathematicianbyprovidingacontributiontotheinterpretationandfurtherdevelopmentofhisideasandintuitionsinthelightoftheprogresswhichhasbeenachievedinourunderstandingoftheuniverseinthelasthundredyears.SpecialrelativitywasoriginallyformulatedbyEinsteininthe“annusmirabilis”1905restingon1)theprincipleofrelativity,statingtheequivalenceoftheclassofallinertialobserversasregardsthedescriptionofthelawsofnatureandon2)theprincipleofconstancyofthevelocityoflightwhichstatesthatelectro-magneticwavespropagateinemptyspacewithauniversalspeedc(=2:997924581010cms1)whichdoesnotdependonthestateofmotionofitssourceandoftheobserverwhomeasuresthisspeed[2].StrictlyspeakingEinstein’soriginalformulationdiffersfromtheabovetotheextentthathedoesreferallmotionstosomeparticularinertialobserverwhichisassumedconventionallytobe“stationary”.However,thatthisobserverdoesnotplayanyprivilegedrolecomparedtotheotheronesisclearfromhisstatementthat«theintroductionofa“luminiferousaether”willprovetobesuperfluousinasmuchastheviewheretobedevelopedwillnotrequirean“absolutelystationaryspace”providedwithspecialproperties,norassignavelocity-vectortoapointoftheemptyspaceinwhichelectromagneticprocessestakeplace»[2].Thus,Einstein’sinsightirretrievablyshatteredthenotionsofabsolutespaceandabsolutetimewhichweredeeplyrootedinthemindofscientistssincethetimeofNewton.However,itwassoonrealizedafterEinstein’sgreatbreakthroughthattheprincipleofconstancyofthevelocityoflightwasnotnecessaryinordertoobtaintheLorentztransformations.Indeed,beginningwithaworkbyIgnatowski[3],alar