Relativity as the quantum mechanics of space-time

arXiv:physics/0005034v2[physics.gen-ph]16Jun2000Relativityasthequantummechanicsofspace-timemeasurementsRichardLieuDepartmentofPhysics,UniversityofAlabama,Huntsville,AL35899,U.S.A.email:[e-mail]1AbstractCanasimplemicroscopicmodelofspaceandtimedemonstrateSpecialRelativityasthemacroscopic(aggregate)behaviorofanensemble?Thequestionwillbeinvestigatedinthreeparts.First,itisshownthattheLorentztransformationformallystemsfromtheFirstRelativityPostulate(FRP)aloneifspace-timequantizationisafundamentallawofphysicswhichmustbeincludedaspartofthePostulate.Animportantcorollary,however,isthatwhenmeasuringdeviceswhichcarrythebasicunitsoflengthsandtime(e.g.aclocktickingeverytimequantum)are‘moving’uniformly,theyappeartobemeasuringwithlargerunits.Secondly,suchanapparentincreaseinthesizesofthequantacanbeattributedtoextrafluctuationsassociatedwithmotion,whicharepreciselydescribedintermsofathermallyagitatedharmonicoscillatorbyusingatemperatureparameter.Thisprovidesastringentconstraintonthemicroscopicpropertiesofflatspace-time:itisanarrayofquantizedoscillators.Thirdly,sincetheforegoingdevelopmentwouldsuggestthatthespace-timearrayofanacceleratedframecannotbeinthermalequilibrium,(i.e.itwillhaveadistributionoftemperatures),theapproachisappliedtothecaseofaccelerationbythefieldofanypointobject,whichcorrespondstoatemperature‘spike’inthearray.Itisshownthattheoutwardtransportofenergybyphononconductionimpliesaninverse-squarelawofforceatlowspeeds,andthefullSchwarzschildmetricathighspeeds.Apredictionofthenewtheoryisthatwhentwoinertialobserversmovetoofastrelativetoeachother,orwhenfieldsaretoostrong,anharmoniccorrectionswillmodifyeffectsliketimedilation,andwillleadtoasymmetrieswhichimpliesthattheFRPmaynotbesustainableinthisextremelimit.2IncontemporaryphysicstheFRPisusuallyviewedasaportraitofthecompletesymmetrybetweeninertialobservers.Specificallyallsuchobserversexperiencethesamelawsofphysics[1].ThisPostulate,thoughsimplystated,ispowerfulbecauseapplicationofittonewphysicalprinciplescouldleadtoimportantconsequences.ThesupremeexampleiswhenMaxwell’sequationofelectromagneticwave(i.e.light)propagationfinallybecamerecognizedasauniversallawofnature[2].Inorderthattheequationbeinvariantwithrespecttoallinertialframes,theircoordinatescannotberelatedbytheGalileantransformation(whichassumesspaceandtimeareabsolute),asfirstrealizedbyLorentz[3].Awayofarrivingatthecorrecttransformationsimplyandlogically,assuggestedbyEinstein[4],istointroducetheSecondRelativityPostulate(hereaftertheSRP).Itstatesthatthespeedoflightisauniversalconstant,independentof(a)inertialframes,and(b)themotionoftheemissionsourceineachframe.Sincethesetwoindependences,especially(b),donotariseunambiguouslyfromtheassertionthatMaxwell’sequationsmustbeformallyinvariantinallframes,theneedforanexplicitdeclarationasapostulateisinevitable.Thus,althoughEinsteindidfruitfullyintegrateMaxwell’stheorywiththeFRP,itwasnecessaryforhimtoenlistaseparatepostulate.DoesthismeanthestatementofsymmetryprovidedbytheFRP,thoughveryelegant,onlyplaystheroleofcompellingandguidingustoinvokefurtherpostulateseverytimewehavetodealwithanewphysicallaw?HereIsuggestanotherpossibility,viz.thatthereasonfortheproblemsoftheFRPisbecause,unlikeNewton’slaws,neitherMaxwell’stheorynorEinstein’sSRPoperateonasufficientlyfundamentallevel.Themissingpieceissimplythatmicroscopically3spaceandtimearequantized:thereexistsnaturallyimposedunitsofdistancesandtimeintervalsascharacterizedbytheparameters(xo,to),theminimumuncertaintiesinone’sabilitytomeasurethecoordinates(x,t)ofanevent,whichcannotbesurpassedirrespectiveofaninstrument’saccuracy.WhenincorporatedwiththeFRP,thePostulatenowreads:thelawsofphysicsasobservedthroughasystemofquantizedspace-time,thelattera‘grid’whichinevitablycontrolstheoutcomesofmeasurements,isthesameforallinertialobservers.Thus,ifaframetransformationarbitrarilychangesxoandto,thereference‘grid’willingeneralbedistorted;butifatleasttheratioofthetwoquantitiesremainsconstant,the‘grid’willbeenlargedorreducedproportionately,andsolongasnophysicsexistswhichdependsonabsolutescales,suchatransformationwillstillpreservethesymmetrybetweeninertialframes.ItisworthinvestigatingwhetherthisgeneralizedFRPalonewillleadtoauniquesetofequationswhichconnectthecoordinatesofdifferentframes,Beforedoingso,however,Idesignatetheratioxo/toauniversalconstantvowhichhasthedimensionofaspeed.LetusfirstexaminetheGalileantransformationG.TheeventdataofaninertialobserverΣare(x,t)withaccuracies(xo,to).AccordingtoG,suchdatawouldappeardifferentlytoanotherobserverΣ′whomovesrelativetoΣwithvelocityv=(v,0,0).MorepreciselythisobservernoticesthatifheweretorepeatthemeasurementfollowingeverystepadoptedbyΣ,hisresultswouldbe(x′,t′)withaccuracies(x′o,t′o),wherex′=x−vt,t′=t;andx′o=xoq1+v2/v2o,t′o=to.Basedontheaboveperception,Σ′constructsaphysicalmodelofhisworldview.Wouldthisbethesamemodelastheonederivedfromexperimentsperformeddirectlybyhimself?Suchaquestionisindeedvalidandpertinent4ifweinterprettheforementionedaccuraciesaslimitingoneswhichformbasicunitsofmeasurements.Theansweris‘no’becau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