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Chapter13

CosmologyI–TheSpacetimeTransformationModel

The Big Bang theory is one of only a few theories in science that enjoys virtually universalacceptance.Beforeabout1929the“SteadyStatetheory”wasfavoredbutsincethentherehavebeennumerousexperimentalobservationsthatsupporttheBigBangtheoryanddisprovetheSteadyStatetheory.IamgoingtoproposeanalternativetothestandardBigBangmodel.Thismightbeconsideredbysomeastheequivalentofscientificheresy.However,wewillfaithfullyfollowourstartingassumption theuniverseisonlyspacetime toitslogicalconclusion.Asanintroductiontothecosmologicalimplicationsofourstartingassumption,wewillusealineoutofthechildren’sstoryofAliceinWonderland.InthisstoryAliceasksthequestion:“IstheroomgettingbiggeroramIgettingsmaller?”Scientistsdonotasktheequivalentquestionabouttheuniverse. Theyalwaysassumethattheuniverseisgettingbiggerwhiletheirmetersticksandclocksstaythesame. However, thisassumptionleadstonumerousmysteries. Howis itpossiblethatdistantgalaxiesarerecedingat fasterthanthespeedof light?Isnewspacetimecontinuously being created? Ismysterious dark energy required to explain the acceleratingexpansionoftheuniverse?Allofcosmologylooksdifferentwhenweassumethattheuniverseisonlyspacetimeandthepropertiesofspacetimearechanging.BeforeCopernicus,itwasassumedthattheearthwasthecenteroftheuniverse.Nowweknowthatwedonotliveinaspecialplaceintheuniverse,butweatleastassumethatbothoursizeandourrateof timeareconstant. Iamproposing that the final indignity tomankind is thatspacetime isgoing througha transformation that contractsallphysicalobjects, including theearthandourbodies.Furthermore,therateoftimeisalsoslowingdown.Itisproposedthatthespacetimefieldisundergoingatransformationwhichisresponsibleforwhatweperceivetobetheexpansionofthevolumeoftheuniverse.Supposethatweimaginegoingbackwardsintimefromtoday.Thedistancebetweengalaxieswoulddecreaseand the cosmicmicrowavebackground CMB photonswould increase theirenergy.Todayonlyasmallpercentageoftheobservableenergyintheuniverseisintheformofelectromagnetic EM radiation.However,goingbackwardsintimewewouldreachatimewhenthedominantformofobservableenergyintheuniversewasEMradiation.Whathashappenedtothisenergythatoncedominatedtheuniverse?Individualphotonsarenotdisappearing.Evenasphotonsloseenergytheyretainalltheirangularmomentum.Electronsandprotonsappeartobe immune from this energy loss, butwe have to questionwhether our standard of energyremainsconstantovertime.


Whenweassumethattheuniverseisonlyspacetimebothtodayandthroughoutitshistory,weget an entirely different perspective on cosmology. There still is something that can bedesignatedthe“BigBang”butthereisnosingularitywithinfiniteenergydensity. Infact,theenergydensityoftheuniverseneverexceedsPlanckenergydensity∿10113J/m3.Allofthiswillbeexplained,butwewillstartbyreviewingthecurrentcosmologicalprinciples.The Cosmic Microwave Background (CMB):TheprecisemeasurementoftheCMBhashadaprofoundimpactoncosmology.ThePlanckSpacecrafthasmappedthefullskydistributionoftheCMBtoaresolutionbetterthan0.1°.Theageoftheuniversehasbeendeterminedtobe13.8billion years 0.4%. Also the universe has been determined to not have any large scalecurvature.Itis“flat”toameasurementaccuracyofabout0.1%.1,2TheCMBwasalsomappedbytheWilkinsonMicrowaveAnisotropyProbe WMAP whichhadlowerresolution,butsomeofthedataismoreaccessibleandstillquiteaccurate.Therefore,numbersquotedherewillbeamixtureofthesetwosources.AnalysisofthedatafromWMAPhasalsodeterminedthatabout380,000yearsaftertheBigBang,theenergyintheuniversewas10%neutrinos,15%photons,12%ordinarymatter baryonicmatter , and 63% dark matter. For comparison the percentages quoted for today are:4.9%ordinarymatter,26.8%darkmatterand68.3%darkenergy.The15%energythatwasphotonenergyat380,000yearshastodaybeenredshiftedbyafactorofabout1080.ThishasreducedthecurrentenergycontentoftheseCMBphotonstoanalmostinsignificantpercentage ~0.014% ofthetotalenergyintheuniverse.Similarly,neutrinosandother relativistic particles have also lost momentum kinetic energy due to the redshift.Therefore,over20%ofthetotalobservableenergypresentintheuniverseatanageof380,000has disappeared. Ifwe extrapolate back further, the universewas radiation dominated andalmost all the observable energy present in the early universe has disappeared observableenergyexcludesvacuumenergy . Thequestionaboutwhathashappenedto thisobservableenergywillbediscussedlaterItcanbeshownthattheenergythatwaspresentinordinarymatteranddarkmatterat380,000years is still approximately present in the universe today. The percentages look differentbecause of the addition of dark energy and the subtraction of the energy in photons andneutrinos.Darkenergyhasneverbeenobserved,but it isrequiredbythecurrentlyacceptedcosmologicalmodel.Theneedfordarkenergywillalsobediscussedlater.ThemeasurementoftheCMBhasalsoshownthattheuniversedoeshaveapreferredframeofreference.Fromanygivenlocation,thecosmicmicrowavebackgroundonlyappearsisotropic

1 http://lambda.gsfc.nasa.gov/product/map/dr4/pub_papers/sevenyear/cosmology/wmap_7yr_cosmology.pdf 2 Astrophysical Journal Supplement 180: 225-245 February 2009


fromaparticularframeofreferencecalledthe“CMBrestframe”.Forexample,thesunappearstobemovingatabout369km/sinthedirectionoftheVirgoconstellationrelativetotheCMBasseen from the sun’s location. This relativemotionproducesaDoppler shift in theCMB thatshowsupasaredshiftintheCMBinonedirectionandablueshiftintheCMBintheoppositedirection dipoleanisotropy .ThisanisotropyissubtractedfromCMBpicturestoproducethespeckled but uniform CMB pictures commonly exhibited showing only small temperaturevariations. Each location in the universe has a unique frame of reference that is stationaryrelativetotheCMB.Hubble Parameter: The expansion rate of the universe will be referred to as the “Hubbleparameter ”.Itisnotatrueconstantsinceitchangesovertime.Theexpansionoftheuniversequoted by astrophysicists usually is given in units of kilometers/second/mega‐parseckm/s/Mpc .Thismightbeconvenientunitsforastrophysicists,butsincewearestandardizingonSIunits,wewillconvertthisintounitsofm/s/m.Theconversionis:km/s/Mpc 3.24 10‐20m/s/m. TheHubbleparametermeasuredby thePlanck spacecraft hasbeen combinedwithothermeasurementstoyieldcompositemeasurementsrangingfromabout67to68km/s/Mpc.Thisbookwilluse 67.8km/s/Mpc 2.2 10‐18m/s/m.Toputthisinperspective,twopointsatrestrelativetothelocalCMBrestframeandseparatedby4.5 1017m ∿49lightyears wouldbemovingawayfromeachotherat1m/s.TheMilkyWaygalaxyandevengalaxyclustersaregravitationallyboundanddonotexpandwiththeexpansionofspacetime.Galaxiesseparatedbymorethanabout300millionlightyearsexhibitthisexpansion.TheHubbleparameterhasunitsofinverseseconds.IftheuniverseexpandedlinearlyfromtheBigBangtotoday,then1/ wouldequaltheageoftheuniverseinseconds.Usingthemeasuredvalue,1/ 4.5 1017seconds 14.4billionyearscomparedtothe13.8billionyearsageoftheuniversemeasuredotherwaysbythePlanckspacecraft.Comoving Coordinates:TheperspectiveofauniqueCMBrestframeateachlocationhasledtotheconceptthattheuniversecanbemodeledashavinga“comovingcoordinatesystem”.ThisisacoordinategridthatexpandswiththeHubbleexpansionoftheuniverse.Eachpointonthisgrid is in theCBM rest frame for that location. This is also called the “comoving frame”. Acomoving observer is the only observer that will see the universe including the CMB asisotropic.Galaxiesarenearlyinthecomovingframe,soanyvelocitytheyhaverelativetothecomovingframeistheir“peculiarvelocity”.Atanygiveninstant,allpointsintheCMBrestframeareexperiencingthesamerateoftimeiflocal gravitational disturbances are ignored. Anotherwayof saying this is that on the scalewheretheuniverseishomogeneous about300millionlightyears ,allpointsinthecomovingframeareexperiencingthecomovingcoordinate’scosmologicaltime.


Comovingdistanceχistheproperdistancebetweentwopoints bothinthecomovingframe atthepresentinstantofcomovingtime.Whilecomovingdistancecorrespondstoproperdistanceatthepresentinstant,comovingdistanceisimaginedtoremainatafixedvalueofχovertime.InthepastorfuturetheproperdistancebetweenthesetwopointschangeswiththeHubbleflowbutthecomovingdistanceisafixeddesignationonanexpandingcoordinatesystem.Thecosmic“scalefactor a ”isafunctionoftimeandusuallydesignatedas:a t .Thisscalefactorquantifiesthe relative expansionof theuniversebetween twomoments in time. The term lt is properdistance between the two points at a different time different ages of the universe . Therelationshipbetweenthesetermsis:χ lτa t Thecurrentscaleofthecomovingcoordinatesystemisdesignated“ao”andusuallysettoequalone ao 1 .The Λ-CDM Cosmological Model: Λ‐CDMisanabbreviationforLambda‐ColdDarkMatter.ThisiscurrentlyconsideredtobethestandardBigBangmodel.In1929whenEdwinHubblediscoveredtheredshiftofgalaxies hecalledthemnebula ,theinitial interpretationwasthatthesewereDopplershiftsduetorelativemotionofgalaxiesexpandingintoapreexistingvoid.Hubbleproposedthattherewasalinearrelationshipbetweenvelocityanddistance.Thecurrentinterpretationisthattheredshiftisduetoacosmologicalexpansionoftheuniversewherespaceitself isbeingcontinuouslycreatedeverywhere. OnedifferencebetweenHubble’sconcept apreexisting void and cosmological expansion is that cosmological expansion is currentlybelievedtobeabletoproduceseparationvelocitiesthatexceedthespeedoflight.Observationsstrongly support the general relativistic interpretation of a cosmological expansion over thespecialrelativityinterpretationofmassexpandingintoapreexistingvoid.AdetaileddescriptionoftheΛ‐CDMmodelwillnotbegivenhere.However,thereareseveralphysical interpretationsandpredictionsof thismodel thatwill bedescribed toestablish thecurrent perspective of cosmologists. The favored Λ‐CDM model is usually designated as:ΩM 0.31andΩΛ 0.69withtheHubbleparameter 67.8km/s/Mpc 2.2 10‐18m/s/m inMKSunits

The matter density parameter ΩM 0.31 represents that all forms of matter makes upapproximately31%ofthecriticaldensityoftheuniverseandthecosmologicalconstantdensityparameterΩΛ 0.69indicatesthatdarkenergymakesupabout69%ofthecriticaldensityoftheuniverse.DarkenergyisthehypotheticalenergythatexpandstheuniverseandisassociatedwithEinstein’scosmologicalconstantΛ.


Figure 13‐1 is taken from a very good article titled “Expanding Confusion: CommonMisconceptionsofCosmologicalHorizonsandSuperluminalExpansionoftheUniverse”3Figure13‐1istwospacetimediagramsthatplottimeversusproperdistanceDonthetoppanelandtimeversuscomovingdistanceonthebottompanel.Thepanelsaredrawnfromtheperspectiveofanobserverlocatedattheintersectionofthe“now’line 13.8billionyearsaftertheBigBang andthedistance 0line. Thelowerpanelhasverticalworldlinesforobjectscurrentlywithvariousredshifts z 0,1,3etc. becausethecomovingdistanceχdoesnotchangeovertime.Theproperdistancechangesovertimebutthecomovingdistanceisacoordinatedistancethatexpandswiththecomovingcoordinatesystem.Thetwopanelsofferdifferentperspectivesandthefollowingdescriptionappliestobothpanels.Thedashedlinelabeled“particlehorizon”crossesthe13.8billionyear“now”lineatabout46billionlightyears.Itistobeunderstoodthateventhoughasignalfromthemostdistantsourcecurrentlyreachingus,calledtheparticlehorizon,hasbeentravelingatthespeedoflightfor13.8billionyears,thecurrentdistancetothatsourceislargerthan13.8billionlightyears.Thisisbecause the cosmic expansion of the universe has continued to increase the volume of theuniverseandthedistancebetweenpointsafterthespeedoflightsignalhaspassedanylocation.Nextwewill lookat the line labeled“Hubblesphere”. Currently theboundaryof theHubblesphereis13.8billionlightyearsfromus.Thisisthedistancewherespaceitselfissupposedly

3 T. M. Davis; C. H. Lineweaver; Publications of the Astronomical Society of Australia 2004, 21, 97-109.


recedingfromusatavelocityequaltothespeedoflight conceptexaminedlater .AccordingtotheΛ‐CDMmodel,thespacebeyondtheHubblesphereisrecedingfromusfasterthanthespeedoflight.Thisgivesrisetoan“eventhorizon”.The line labeled“eventhorizon”representsthe furthestdistancethatwecanreceivecurrentinformation.Forexample,accordingtotheΛ‐CDMmodel,galaxiesthatwecurrentlyobserveashavingaredshiftofz 1.8arecurrentlycrossingoureventhorizon.Thismeansthatlightbeingemittedbythesegalaxiestodaywillneverreachusbecauseoftheacceleratingexpansionoftheuniverse.Allgalaxieswithredshiftsgreaterthan1.8havealreadycrossedoureventhorizonandwilldisappearfromviewsometimewithinthenext14billionyears largerredshiftsdisappearsooner .According to the Λ‐CDMmodel eventually the Hubble sphere and the event horizonwill beapproximatelythesamedistance.Inthedistantfuturetheuniversewillhaveaneventhorizonataconstantproperdistanceofabout17billionlightyears.Galaxieswillcontinuetodisappearfromviewastheyarecarriedbycosmologicalexpansionbeyondthisdistance.Eventuallyonlythegravitationallyboundgalaxieswillremainvisible.Iftheconceptofacceleratingexpansionof the universe is carried to its logical conclusion, we would end with the “Big Rip” whereexpandingspacetimeeventually tearsapartourgalaxy, thenoursolarsystemandeventuallyevenatoms.ProblemswiththeBigBangSingularityandInflation:Themostwidelyacceptedmodelforthestartoftheuniverseisa“singularity”withnovolumeandinfiniteenergydensity.Thatstartingconditionrequiresthatthelawsofphysics“breakdown”.Italsocreatesaproblemforexplaininghowtodaytheenergydensityoftheuniverseishomogeneousonthelargescalelarger.Startingwithasingularityresults in inhomogeneousdistribution,evenon the largescale. Therefore,physicistspostulateaperiodof “inflation”whentheuniverseexpandedmuch faster thanthespeedoflight.Thishypothesiswouldresultintheportionoftheuniversethatwecanobservebeinghomogeneous.However,itisgenerallyagreedthattheuniverseextendsfarbeyondourcurrentparticlehorizon.Somecosmologists even suggest that the “extendeduniverse” is infinite. Wedonotneed todecidethatquestioninordertodothefollowingthoughtexperiment. Supposethatwecouldobtainamapoftheuniverseextendingfarbeyondourparticlehorizon.Wouldthemaprevealthattheuniverseisinhomogeneousontheextendedscale?Inflationsupposedlydidnotreallyeliminate the inhomogeneties causedby starting froma singularity. Itmerely expanded thevolumesomuchthatanysingleobserverdoesnotseeobviouslargescaleinhomogeneties.Theimplication is that today there are parts of the extended universe that have substantiallydifferentdensitythanthedensityofourobservableportionoftheuniverse.


It has been established byWMAP space probe that the portion of the universe thatwe canobservehasflatspacetimetowithinthe0.4%marginoferror.Doweliveinaspecialpartoftheuniversethathappenstohaveachievedthisspecialcondition?Surelyifthereareunobservablepartsof theextendeduniverse thathavesubstantiallydifferentdensity, then theywouldnothaveflatspacetime.Animportantcosmologicalprincipleisthatwedonotliveinaspecialpartoftheuniverse.Thisshouldapplyeventotheextendeduniversebeyondourparticlehorizon.Theimplicationisthat13.8billionyearsaftertheBigBang,allpartsoftheextendeduniverseshould have the same density. Inflation does not explain how the extended universe can behomogeneous.Spacetime Transformation Model: We are about to start the description of an alternativecosmologicalmodelthatwillbereferredtoasthe“spacetimetransformationmodel”.Thismodelwillbemorefullyexplainedinthenextchapter,butkeypartsofthemodelwillbepresentedinthis chapter. Initially, these key parts will be explained using the familiar concepts andterminology of the current Big Bang model. When explaining the spacetime transformationmodel,theinitiationofthetransformation thebeginningoftime willbereferredtoasthe“BigBang”. This is a highly descriptive term that is being adapted to express the start of atransformationofoneformofspacetimetoanother. Thiswasaccompaniedbyatremendousincreaseinpropervolume,sotheterm“BigBang”isstillapplicable.Iftheuniverseisonlyspacetime,howdoesthespacetimefieldcreatenewvolume?Today,alotofattentionisbeingpaidtotheapparentaccelerationoftheexpansionoftheuniverse.Whilethisisanimportantquestion,itisnotpossibletoanswerthisquestionuntilwefirstunderstandhow any new proper volume is created in the universe. If the expansion of the universe isimaginedasmass/energyexpandingintoapreexistingvoid,thenmomentumwouldcontinuethisexpansionandtherewouldbenomystery.However,theΛ‐CDMmodelofcosmologysaysthattheuniverseisundergoingacosmologicalexpansion.Thisexpansionistheresultofnewvolume continuously being created everywhere in the universe. This new volume isindistinguishablefrompreviouslyexistingvolume.Therefore,thenewvolumemustalsocontainvacuumenergywithenergydensityofmorethan10112J/m3.Supposethatweattempttoputthis in perspective. TheHubble parameter is 2.2 10‐18m/s/m. Thismeans that eachsecond,eachcubicmeterofspacetimeisexpandingby 2.2 10‐18 3m3 10‐53m3.Thissmallvolume seems insignificant until it is realized that the energy density of this new volume isapproximatelyPlanckenergydensityUp 10113J/m3.Therefore,10‐53m3requiresabout1060Jofvacuumenergy. Themassof theMilkyWayisabout1042kgequivalenttoanannihilationenergyofabout2 1059J.Therefore,eachcubicmeterintheuniverseseemstobeaddingtheequivalenttheMilkyWay’senergyeachsecond.Thisseemsridiculous,butitappearsthatthereareonlythreechoicestoexplainthisdiscrepancy.Either 1 vacuum energy is canceled by some other equally vast offsetting effect; 2 a vastamountofnewvacuumenergyisbeingcontinuouslyaddedtotheuniversetomaintainaspecific


vacuumenergy density or 3 a fixed amount of vacuumenergy is being distributed over anincreasingvolumewhichvastlydecreasestheenergydensityofvacuumenergyastheuniverseexpandsfromtheBigBangtoday. Allofthesealternativesseemunappealing.Thespacetimetransformationmodel of the universe will propose a fourth explanation which involves thetransformationofthepropertiesofspacetime.Theadditionofnewvolumetotheuniverseisnotsomesubtleeffectthatistakingplaceinremotepartsoftheuniversebetweengalaxies.Itispossibletoillustratethescaleofcosmicexpansionusingourownsolarsystemasanexample.SupposethatweconsiderasphericalvolumewiththeradiusofNeptune'sorbit radius 4.5 1012m .Thissolarsystemsizesphericalvolumewouldcontainallthemajorplanetsandhaveavolumeofabout3.8 1038m3.CalculatingfromtheHubbleparameter 2.2 10‐18s‐1 ,thisvolumewouldincreasebyabout1021m3eachsecondifthesphericalvolumeexpandedproportionaltotheHubbleexpansionoftheuniverse.Toputthisexpansioninperspective,thissolarsystemsizevolumeisaddingtheequivalentofaboutearth’svolume 1.08 1021m3 everysecond.WedonotnoticeanychangeinthevolumeofthesolarsystemortheMilkyWaygalaxybecausetheseobjectsaregravitationallybound.Theadditionofnewvolumebecomesobviouswhendistance properlighttraveltime ismeasuredbetweengalaxies thatarenotgravitationallyboundtogether. Onlyafterwehaveaplausibleexplanationforthecreationofthisnewvolumeeverywhereintheuniverse,canweseriouslyaddressthequestionaboutthenonlinearityinthiscreationprocess acceleratingexpansion .The new volume is also spacetime so a rephrasing of the question is: How does spacetimerearrange itself to creatingnewpropervolume? If theuniverse isonly spacetime, and if theuniverseappearstobeexpanding,thenthepropertiesofspacetimemustbechangingwithtime.Somethingmustbechangingbecausetodayittakesalongertimeforlighttotravelbetweentwogalaxiesthanitdidabillionyearsago assumingeachgalaxyisatrestrelativetotheCMB .Proper Volume:Whatexamplesdowehaveofspacetimeadjustingitselfinawaythatchangesproper volume? In the example of the Shapiro experiment, the sun’s gravity changed thespacetimebetweentheearthandthesuninawaythatincreasedtheproperradialdistancebyabout7.5kmcomparedtotheradialdistancethatwouldbeexpectedfromEuclidiangeometry.Thisincreaseinradialdistanceincreasedthevolumewithinaspherethatis1astronomicalunit1AU inradiusbyΔV 3.46 1026m3 previouslycalculated .Thisnon‐Euclidianincreaseinproper volume is more than 300,000 times the earth’s volume. This volume increase wasaccompaniedwithadecreaseintherateoftime.Also,therotarmodelhastwolobeswherethepropertiesofspacetimeareslightlydistorted.Theslowtimelobehasarateoftimethatloses1unitofPlancktimeinatimeperiodof1/ωc.ThatslowtimelobehasavolumethatislargerthanwhatwouldbeexpectedfromEuclidiangeometry.The fast time lobehas lesspropervolumethanwouldbeexpected fromEuclidiangeometry.


Againitappearsas if thereisaninverserelationshipbetweenpropervolumeandtherateoftime.Neartheendofchapter2itwasshownthatthestandardsolutiontotheSchwarzschildequationhasa4dimensionalvolumethatisindependentofthegravitationalgammaГbecausethechangeinthetimedimensionoffsetsthechangeintheradialspatialdimension.Inalltheseexamples,thereisaninterconnectionbetweentherateoftime,thecoordinatespeedoflightandpropervolume.Wewillexplorethefollowingidea:New proper volume is created when the rate of time decreases. Itisproposedthatspacetimeisabletoexchangetheabsoluterateoftimeforpropervolume.Thisconceptisimpliedinthefollowingequation:

ГnoteinverserelationshipbetweendτanddLR

WhiledLR/dRappliestotheradialdirectionintheSchwarzschildcoordinatesystem,theaboveequationalsocanbeinterpretedasshowinganinverserelationshipbetweentherateofpropertimeandpropervolumeevenwhenappliedtotheentireuniverse.Weseethepropervolumeoftheuniverseincreasing,butprovingthatthisiscoupledwithadecreasingrateoftimeismoredifficult.Anabsoluteproof that therateof time isslowing in theuniversewouldrequire thecomparisontoahypotheticalcoordinateclockoutsideof theuniverse. This isan impossibleexperiment,soinsteadwewillstartwithseveralthoughtexperiments.Cavity Thought Experiments:Firstwewillimagineasphericalshellwithmassmandinternalradiusr.Wewillplacethissphericalshellinspacewhereitisinaninertialframeofreference.If the spherical shell hasmass, then on the outside surface of this spherical shellwewouldexperience a gravitational acceleration accompanied by the standard time dilation andgravitational effect on volume. Inside this spherical shell, there would be no gravitationalaccelerationandwecouldconsiderthisflatspacetime.However,theescapevelocityfromtheshellisgreaterifwestartfrominsidetheshellcomparedtostartingfromtheoutsidesurface.Even though there is flat spacetime, inside the spherical shellwe still have the gravitationaleffectsthatareafunctionofescapevelocityandscalewithΓ.Thispointwasalsomadeinchapter2 with examples using a cavity at the center of the earth and the Andromeda galaxy. Thegravitationaleffectontherateoftime,thecoordinatevelocityof lightandonpropervolumeremainsevenwhenthereisnogravitationalacceleration.However,onlythegravitationaleffecton the rate of time is easy to experimentally see so wewill temporally concentrate on thegravitationaleffectontherateoftime.Background Gravitational Gamma Г:PreviouslywedefinedthegravitationalgammaГandacloselyrelatedconcept,thegravitationalmagnitudeβ.Theseweredefinedasfollows:


Г≡ 1/ 1 / –

Г gravitationalgamma;ve escapevelocity

β≡1– 1– 1– 1–Гβ gravitationalmagnitude

Г 1 βandβ weakgravityapproximations

Inside a cavity uniformly surrounded by mass m, there is a gravitational effect, but the

gravitationalgammadefinitionГ needstobereinterpreted.Thisequationpresumes

anisolatedmassinanotherwiseemptyuniverse.Theinteriorofasphericalshelldoesnotmeetthis assumption. However, the definition of Г that incorporates escape velocity ve is easily

definable:Г 1/ 1 / . Forexample,acavityatthecenteroftheearthwouldhaveanescapevelocity about22%greater than theescapevelocity starting from theearth’s surfaceassumingauniformdensityapproximationandignoringairfriction .Theotheralternativedefinitionforthegravitationalgamma stationaryframe alsoworksforacavitysurroundedbymass:Г dt/dτ.Inthiscase,wedefineacoordinateclockfarfromthesourceofgravitywitharateoftimedtandaclockinthecavitywithrateoftimedτ.Actually,thedefinitionГ dt/dτispreferablebecauseitisalsoapplicabletotheentireuniverse.Thereasonfordiscussingcavitiessurroundedbymassisthatourlocationintheuniversehassimilarities to a cavity surrounded by mass. In fact, the universe is something like beingsurroundedbyashellthatisincreasinginmasseverysecond.Whatwoulditbeliketobeinsideashellthatisincreasingitsmass? assuminginertialframeofreference Therateoftimewouldbeslowingdownandthecoordinatespeedoflightwouldbeslowingdown,butwewouldnotbeable to directly detect these changes unlesswe had communication to an outside standard.However, it is proposed thatwewould notice something strangewas happening. A carefulexperimentwouldrevealthat lightwasundergoingaslightredshift. Thewavelengthof lightpropagatinginsidethecavitywouldincreasewithpropagationtime.Aproofofthiscontentionwillbegiveninthenextchaptertoexplaintheredshiftobservedintheuniverse.We often make the assumption that a location in space far from the earth and sun can beconsidered to be a location with zero gravity. It is true that there might be negligiblegravitationalacceleration,butwearestillsurroundedbyallthemass/energywithintheparticlehorizonforourlocationintheuniverse.WeknowfromobservationsoftheCosmicMicrowaveBackground Planckspaceprobe thattheuniverseisflatspacetimetowithina0.1%marginoferror.Thissuggeststhattheuniverseextendsinfinitelybeyondourparticlehorizon.However,only the mass/energy within our particle horizon can have any influence on us. Themass/energywithinourparticlehorizonexertsabackgroundgravitationaleffectwhichmust


havegreatlysloweddownboththerateoftimeandthecoordinatespeedoflightcomparedtoahypotheticaluniversewithoutthisinfluence.Wewilluseanewnametodiscussthissubject.Background Gravitational Gamma of the Universe Гu:Thename“backgroundgravitationalgammaoftheuniverse”andthesymbolГuwillbeusedtorepresenttheconceptthattheuniversepossessesagravitationaleffectthatproducesnoaccelerationbutaffectstherateoftime,propervolume, thecoordinatespeedof lightandmanyotherphysicalproperties. Withbackgroundgammathereisdistributedmass/energyinalldirectionssowedonotmeettheSchwarzschildassumptionofasinglemassinanotherwiseemptyuniverse.However,thedefinitionofГbasedontheeffectontherateoftimeisthesame:Гu dt/dτuwheredτuistherateoftimeinaframeofreferencethatisstationaryrelativetothecosmicmicrowavebackground CMB andfarfromlocalizedsourcesofgravity.InthiscasedtisthecoordinaterateoftimewhichiseithertherateoftimeinahypotheticaluniversewithoutgravitationalinfluencewhereГu 1ortherateoftimeattheinstanttheBigBangstarted whenГu 1 .Thiswillbeexplainedlater.Toillustratetheconceptofabackgroundgravitationalgamma,anexamplewillbegivenusingtheMilkyWaygalaxy.Thisgalaxyhasavisibleradiusofabout50,000lightyears,butdarkmatterisbelievedtoextendtoaradiusofatleast150,000lightyears.Thesun’sorbitisabout26,000lightyearsinradiusaroundthegalacticcenter.Thegalacticgravitationalaccelerationfeltbythesunisdueentirelytomatterthatisinsidethesun’sorbit.MatteranddarkmatterintheMilkyWaygalaxythatliesexternaltothesun’sorbitdoesnotproduceanygravitationalaccelerationontheSun.However,thisexternalmatterdoesstillproduceagravitationaleffectinthesensethat itaffects therateof time, thecoordinatevelocityof light, thegravitationalpotential, theescapevelocityfromthegalaxy,thestandardofenergy,etc.Inotherwords,thisexternalmatterisproducingasignificantbackgroundГatthesun’sorbitaldistance.It is interestingtoestimatethebackgroundГof theMilkyWaygalaxycausedbythegalaxy’smassexternaltothesun’s26,000lightyearorbit.Thetotalmassofthegalaxy,includingdarkmatter,isestimatedatabout1.2 1042kg.Themassinsidethesun’sorbitrequiredtoproducetherequiredgravitationalaccelerationonthesunisonlyabout15%ofthistotalmassof thegalaxy.ThereforethemassoftheMilkyWaygalaxyexternaltothesun’sorbitisabout1042kg.AlmostallofthismassisdarkmatterthatissphericallydistributedaroundtheMilkyWaygalaxy.Forthiscalculationwewillestimatethatallthemassexternaltothesun’sorbitcanbesimulatedby a spherical shell 110,000 light years in radius with mass of 1042 kg. Substituting intoβ Gm/c2rweobtainβ 7 10‐7 orГ 1 7 10‐7 .Theearth’sgravitationalmagnitudeattheearth’ssurfaceisβ 7 10‐10 orГ 1 7 10‐10 .WhileneithertheearthnortheMilkyWaygalaxyhavealargeeffectontheabsoluterateoftime,theMilkyWay’suniformbackgroundГ slows down the rate of time about 1,000 times more than the earth’s own gravity7 10‐7/7 10‐10 .


ExtendingtheconcepttothebackgroundgammatotheuniverseГu,allthemass/energywithintheparticlehorizoniscreatingaverylargevalueofГu.IndividualstarsandgalaxiesrepresentonlyarelativelysmallperturbationofcurvedspacetimeintheflatandverylargebackgroundГuoftheuniverse. Blackholeswillbediscussedlater. Forexample,iftheuniversehad99%ofthem/Rratiorequiredtoformablackhole,thebackgroundgravitationalgammaoftheuniversewould be: Гu 10. Laterwewill attempt to calculate the actual current value of Гu of theuniverse. General relativity calculations normally ignores the background gamma of theuniversewhichisthesameasassumingthatthisbackgroundisГu 1.Thisisacceptableforallcalculations involvingdiscretemassor evengalactic clusters. However,whenattempting toexplaintheexpansionoftheuniverse,thiscannotbeignored.Besides Гu, there is a related concept that is the background gravitationalmagnitude of theuniverse:βu.TherelationshipbetweenГuandβuisthesameastherelationshipbetweenГandβ. SomeequationsaresimplerwhenexpressedintermsofgravitationalmagnitudeβratherthangravitationalgammaГ.TheexactconversionbetweenГandβis:β 1–1/Г 1–dτ/dt.Theserelationshipsalsoholdforβu.Thegravitationalmagnitudeβhasascalethatrangesfrom0to1whilethegravitationalГhasascalethatrangesfrom1toinfinityIsotropic and Homogeneous Universe:Allpointsonthecomovingcoordinatesystem CMBrestframe perceivetheuniversetobeisotropicandhomogeneous.ThisincludestherateoftimewhichisthesameeverywhereintheCMBrestframe.Sincepointsonthecomovingcoordinatesareexpandingawayfromeachother,whatdoesitmeanforthesepointstohavethesamerateoftimeatagiveninstant?Thiscanbeillustratedwithathoughtexperiment.Supposewehave3 spaceships located far from sources of gravity in intergalactic space. Each of the threespaceships isexactlystationaryrelative to theCMBfor its location.Thethreespaceshipsarewidelyseparatedformingastraightlinewiththemiddlespaceshipexactlyhalfwaybetweenthetwooutsidespaceships.Supposethetwooutsidespaceshipssentoutastablemicrowavesignalatthe9.19gigahertzcorrespondingtothefrequencyoftheircesiumatomicclocks.Whenthetwo signals reach themiddle spaceship, the frequency would be slightly lower redshifted comparedtotheatomicclockonthemiddlespaceshipduetocosmicexpansion.However,theimportantpointisthatboththefrequenciesfromthetwooutsidespaceshipswouldbeexactlythesame.Themiddlespaceshipobservesthatboththeoutsidespaceshipswereexperiencingthesamerateoftimeattheinstantthesignalsleftthespaceships.Withrelativitytherecanbeconfusionaboutdifferentdefinitionsof the term“simultaneous”. Therefore, theconceptofa“midpointobserver”specifiesonewayofdefiningsimultaneous.Thetwooutsidespaceshipsaresimultaneouslyexperiencingthesamerateoftimeaccordingtothemidpointobserver.Thismeansthatallpointsonthecomovinggridareexperiencingthesamerateoftimeaccordingtomidpointobservers. Therefore, this rateof time isusedas coordinate rateof time in theRobertson‐Walkermetric.Nowsupposethatallthreespaceshipsarestationaryrelativetoeachother.Anextrapolationofthethoughtexperimentshowsthattheevenifthethreespaceships


aremovingrelativetotheCMBbutstationaryrelativetoeachother,themidpointobserverwillstillseethattheoutsidespaceshipsareexperiencingthesamerateoftime.Thisconceptcanbeextendedtoanytwopointsintheuniverseinthesameframeofreference.On the scale where the universe is homogeneous, any two points in the same frame of reference experience the same rate of time according to a midpoint observer.No Large Scale Gravitational Acceleration:Thisthoughtexperimentwillbeexplainedusingtherotarmodelofanelectronandtheexplanationforgravitypreviouslydeveloped.Supposethatweplaceanelectron rotar inintergalacticspaceandignoreanylocallygeneratedforces.We are only interested in examining gravity on the scale larger than300million light yearswheretheuniverseishomogeneous.Wearegoingtotestthelargescalegravityoftheuniverseusingtherotarmodelofanelectron.Recallthatarotarexperiencesgravitationalaccelerationwhenthereisarateoftimegradientacrosstherotar.Thiscausesthevacuumenergytoexertanunbalancedpressureonopposite sidesof therotar. Thisproducesanet force that is thegravitationalforce.Theaccelerationofgravity g waspreviouslyshowntobe:

g c2 –c2 g gravitationalacceleration

Itwaspreviouslyshownthatthedistancedesignatedinthedenominator dr isproperlength,notcoordinatelength circumferentialradius fromgeneralrelativity.Ifthereisnogradientintherateoftime,thend dτ/dt /dr 0andg 0.Thereisnogravitationalaccelerationwhenthereisnogradientintherateoftime.Twopointsontheoppositesidesofthequantumvolumeofarotarcanbeconsideredtobetwopointsinthesameframeofreference.Inahomogeneousandisotropicuniverse,thesamerateoftimeispresentonoppositesidesoftherotar accordingtothemidpointobserver .Thereisnogravitationalaccelerationonthehomogeneousscaleof300millionlightyearsorlarger.Allthismightseemobvious,butitimpliesthattheuniverseisnotstrugglingtoexpandagainstagravitationalforcethatisattemptingtocollapsetheuniverse.Thegravityoflocalizedobjectslikestarsandgalaxiescancurvespacetimeonalocalscale,butthere is not a rate of time gradient in the universe on the scale where the universe ishomogeneous.If there is no large scale rate of time gradient in the universe, then there is no tendency for there to be gravitational deceleration or acceleration at this scale.Itmightbearguedthatdarkenergyiscurrentlyprovidingsomethinglikeanti‐gravity.Todefendthispositionitwouldhavetobearguedthatgravityiscurrentlyproducingarateoftimegradientthatisattemptingtocollapsetheuniversebutdarkenergyisproducinganoppositerateoftimegradient.Thecombinationoffseteachotherandeliminatesanylargescalerateoftimegradienttherebyeliminatinglargescalegravitationalacceleration.However,eventhisargumenthasan


obvious flaw. Dark energy only became a significant fraction of the energy in the universeroughly7billionyearsago.Itwouldhavetobearguedthatpriortothistimetherewasalargescalerateoftimegradientintheuniverse.ThisiscountertotheassumptionscontainedintheRobertson‐Walkermetric.Itisnotpossibletohaveanisotropicuniverseifthereisalargescalerateoftimegradient.Infact,iftherewasalargescalerateoftimegradientpresentwhentheCMB radiation was emitted, evidence of this rate of time gradient would be preserved asanisotropy intheCMBunlesswehappentobe locatedat theexactcenterof therateof timegradient.Rate of Time Gradient in a Dust Cloud:Itisanaxiomofgeneralrelativitythatadistributedcloudofdust nopressure shouldexperienceauniformgravitationalcollapse.Ifthecloudisdividedintosphericalvolumesofsuccessivelylargerradii,thenaparticulardustparticle thetestparticle inthecloudexperiencesgravitationalaccelerationonlyfromthegravityofdustparticleswithinthesmallerradiispheres.Massthatliesoutsidethesphericalvolumecontainingthetestparticledoesnotcontributetothegravitationalacceleration.Enlargingtheradiusoftheimaginarysphericalvolumeincreasestheenclosedmass.Theneteffectisthatthecollapseofthedustparticleshappenseverywhere.However, gravitational acceleration does not happen in the abstract. To have gravitationalacceleration,thecloudofdustparticlesmusthaveadefinablerateoftimegradient.Ittakesarateof timegradient of about1.11 10‐17 seconds/second/meter toproduce accelerationof1m/s2.Itwouldbepossibletodrawathreedimensionalmapshowingtheequivalentofisobarlineswithinthecloudexceptdepictingregionsofconstantrateoftime.Adustcloudoragalaxydoeshaverateoftimecontoursthatcanbemapped. However,theuniversehas andalwayshad thesamerateoftimeeverywhereonthelargescaleaddressedbythecomovingcoordinatesystem.Isitpossibletoproposeamodeloftheuniversethatdoesnothavearateoftimegradientonthescaleofthecomovingcoordinatesystem?Furthermore,doessuchamodelfollowlogicallyfromstartingtheuniverseasPlanckspacetime?Thesequestionswillbeexaminedbystartingwithathoughtexperiment.Dust Particle Cloud Thought Experiment: Supposethatthepreviouslymentionedcloudofdustparticlesisuniformlydistributedinspaceoveravolumeaboutthesizeoftheearth.Also,supposethatitispossibletoturnoffthegravityofallthedustparticlesinthecloud.Thisisanunrealisticassumptionforacloudofdustbutitwillbeshownthatitisnotunrealisticforthebeginningoftheuniverse. Thereforeattempttofollowthishypotheticalthoughtexperiment.Afterturningoffthegravityofeachdustparticle,thegravitationalmagnitudewithinthecloudwouldquicklydroptoβ 0whichisequivalenttoГ 1.Theproperrateoftimeeverywherewithinthecloudwouldbeequaltothecoordinaterateoftime,thereforewewouldstartwithdτ dtorГ dt/dτ 1.Thereobviouslywouldbenorateoftimegradient.


Next,thegravityofalltheparticlesissimultaneouslyturnedon.Atspeedoflightpropagation,therewouldbeaperiodoftimewheretheproperrateoftimedτwouldbeslowingdownwhilethegravitationalinfluenceofsuccessivelymoredistantparticlesisbecomingestablished.Forexample,ittakesabout20millisecondsforthegravityofaparticleatthecenterofthisearthsizecloudtoaffecttherateoftimeofparticlesneartheouter“surface”ofthesphericalcloudandviceversa.Itwouldtakeabout40msforthemostdistantparticlesonoppositesidesofthecloudtomake gravitational contact. Therefore, it takes about 40 ms for the mature gravitationalacceleration maturerateoftimegradient withinthecloudtobecomefullyestablished.SincewearedefiningthegravitationalgammaasГ dt/dτ,wecouldcharacterizetheestablishmentofgravityataparticularlocationinthecloudbyreferringtothevalueofГasafunctionoftimeafterthegravitywasturnedon.Supposethatwelookjustatthefirstfewmillisecondsafterturningonthegravity.Alsowewillexamineseveralpoints testpoints withinthecloudthatarecloseenoughtothecenterofthecloudthatthereisnotenoughtimetoestablishgravitationalcommunicationwiththeboundarycondition that occurs at the outer surface of the cloud. In the first few milliseconds thegravitationalГisincreasingexactlythesamewayateachofthesetestpoints.Eventhoughtherateofpropertimeisslowing Гisincreasing ,thereisnogradientintherateofpropertime.This isbecauseall thetestpoints andallotherpoints far fromthesurface haveundergoneexactlythesameamountofgravitationalinteractionwiththeirsurroundingparticles.Thelackofarateoftimegradientmeansthattherewouldbenogravitationalaccelerationattemptingtocollapse thecloudduring the first fewmillisecondsaftergravity is turnedon in this thoughtexperiment.Thereisnotendencytowardsgravitationalcollapseonlyduringthenonequilibriumcondition of an increasing value of the background Г. Within the dust cloud there is flatspacetimewhileГisincreasing.Thisflatspacetimedoesnotrequirea“criticaldensity”.Thelowdensityof thedustcloud isvastly less thanmeeting theconditionsofa “criticaldensity”,yetduringthenonequilibriumphaseofincreasingГthedustclouddoesachieveflatspacetime nogravitational acceleration . In fact, any homogeneous density can achieve flat spacetimeprovidedthatthenonequilibriumconditionofanincreasingbackgroundГissomehowachieved.Immature Gravity: This thought experiment describes a condition that will be called“immaturegravity”orthenonequilibriumconditionwherethegravitationalinfluenceofdistantmass/energy is in theprocessofbeingestablished. This is tobe contrasted to the “maturegravity”conditionwheretherehasbeensufficienttimetoestablishthegravitationalgradientsresultingfromtheexistenceofdistantboundaryconditions distantchangesindensity .Inthisexample,afterabout40mstherearedifferencesintherateoftimethroughoutthecloudbecauseofdifferentdistancestothecloudboundary.Therateoftimegradientproducesgravitationalacceleration thatwas previouslymissing. This is themature gravity conditionwe normallyassumewhenwetalkabouttheuniformgravitationalcollapsethatweexpectfromacloudofparticles.Theseconceptscanbeillustratedusingthefollowingfigures.


Figure13‐2showsavolumedeepwithinthedustcloud.Supposethatthisisasnap‐shotabout1msafterthegravitywas“turnedon”. Thecirclelabeled“particlehorizonforpointO”wouldhave a radius of about 300,000mwhich represents the speed of light gravitational contactestablishedfrompoint“O”after1ms.Thishorizonisactuallyanexpandingsphereandthefigureisacrosssectionalrepresentationofamomentintime.Thereisalsoanotherpointdesignated“X”thatisrelativelyclosetopoint comparedtotheparticlehorizon.Forexample,supposepoint“X”isseparatedby30,000mfrompoint“O”.Thisseparationdistanceisabout10%ofthedistancetotheparticlehorizonafter1ms. However, itshouldbenotedthatpoint“X”hasadifferentexpandingparticlehorizondesignatedbythedoublelinecircle.After1msthisparticlehorizonisalsoasphereabout300,000minradius,butcenteredonpoint“X”.

Figure13‐2alsoshowsa“testvolume”.Thisisaanimaginaryspherethatiscenteredonpoint“O”andhasaradiusequaltothedistancebetweenpoint“O”andpoint“X” about30,000m .Therefore,after1mstherehasbeenenoughtimeforthecenterofthis30,000mradiussphericalvolumetoestablishgravitationalcontactwithalltheotherdustparticleswithinthetestvolume0.2msisrequiredforoppositeedgestoestablishgravitationalcontact .Nowwearegoingtoexaminethegravitationalforcesondustparticleswithinthetestvolumeifthetestvolumeisisolatedasshowninthelowerrighthandcorneroffigure13‐2.Thisisolatedillustrationshowsbothpoints “O” and “X”. If this small test volume is isolatedas shown, thenafter0.2msallparticles in the test volume have sufficient time to establish gravitational contact with theboundaryofthetestvolume.Ifthetestvolumeisisolatedasshown,the“mature”rateoftime


gradientwouldbeestablishedwithinthetestvolumeafter0.2ms.Inthiscase,particleswithinthistestvolumewouldstarttoundergoagravitationalcollapseandadustparticleatpoint“X”experiencesagravitationalforcetowardspoint“O”asshownbythearrow.

Figure13‐3showsadifferentperspectivecenteredonpoint“X”.Thereforetheparticlehorizoninfigure13‐3isaspherecenteredonpoint“X”.Nowwearegoingtoexaminetheforcesonpoint“X”.Inthepreviousfigure,weconcludedthatadustparticleatpoint“X”hadaforcetowardspoint“O”after0.2mswhenthetestvolumewasisolated.Fromthisnewperspectiveshowninfigure13‐3,supposethatweweretophysicallyremovethetestvolumeasshowntotheright.Now1msaftergravityhasbeen“turnedon”thedustparticleatpoint“X”hasunbalancedforcefromthecomparablevolumetoitsleftthatisdesignatedas“thesymmetricalvolumethatoffsetsthetestvolume”.Thelengthanddirectionofthisarrowrepresentsthegravitationalforceonadustparticle.Thisforcearrowispointingintothe“symmetricalvolume”andexactlyoffsetsthelengthanddirectionoftheforcearrowpointingtowardspoint“O”inthetestvolume.Withthetestvolumeremoved,point“X”wouldexperienceagravitational force in thedirectionof thearrowpointingtothecenterofthe“symmetricalvolume”.However,ifthetestvolumeispresentandpoint“X”issurroundedbyahomogeneousdistributionofparticles,thenthereisnonetforceon a dust particle located at point “X”. There is also no rate of time gradient and no netgravitationalacceleration.


The point of figures 13‐2 and 13‐3 is to illustrate that the immature gravity increasingbackground Г creates a unique condition where there is no rate of time gradient and notendencyforgravitationalcollapse.Bothpoint“O”andpoint“X”hadtheirownparticlehorizonsthatwerepartlyoverlapping,butalsoeachpointhadahorizonthatwasuniquelycenteredonthem. This is the conditionwhere all the gravitational forces are balanced and there is notendency towardsa gravitational collapse. Anotherwayof saying this is that the conditionsshowncreatearateoftimethatisthesameeverywhereasdefinedbyamidpointobserveratanygiveninstant.Itisironicthattheconditionwherethereisnorateoftimegradient midpointobserverperspective onlyhappensinimmaturegravitywheretherateoftimeiscontinuouslydecreasingrelativetoanoutsideconstantrateoftime.Observable Effects of an Increasing Г:TheslowingoftherateoftimewhenГisincreasingwouldnotbeobviouswithinthecloudwhileitishappeningbecauseitwouldnotbepossibletocomparetheproperrateoftimewithcoordinaterateoftimeusingspeedoflightcommunication.However, itwill be shown that the condition of a homogeneous increase in the backgroundgravitationalgammawouldtheoreticallyproduce1 anincreaseinthepropervolumeofthedustcloudand2 wouldproduceaslightredshiftinradiationcomingfromotherpartsofthedustcloud.Theproofforthisstatementwillbegiveninthenextchapteronceaneasiermodelforanalysishasbeenintroduced.Implications of an Increasing Гu in the Universe:Atthestartofthischapter,thestatementwasmadethatbeforewecanattempttounderstandtheapparentaccelerationintheexpansionoftheuniverse,itisfirstnecessarytounderstandwhythereisanyexpansionintheuniverse.Itisproposedthatthereasonfortheapparentexpansionoftheuniverseisthatthebackgroundgravitational gamma Гu of the universe is presently increasing. Furthermore, Гuwas alwaysincreasingoverthelifetimeoftheuniversebuttherateof increasehasbeendifferentduringdifferentepochs.Thepurposeofthepreviousthoughtexperimentistointroducetheconceptofimmaturegravityusingamodelthatiseasiertounderstandthantheentireuniverse.However,acloudofduststartingwith thegravity turnedoff isnotaperfectanalogy for theuniverse. Thedustcloudillustrates how it is possible for there to be no rate of time gradient midpoint observerperspective aslongasthebackgroundГisincreasing,buttheexampledescribesaconditionwherethereisonlyaminutechangeinГ.Scaledoesmatter.ThechangeinГuexperiencedbytheuniverseinitsfirstmomentsaftergravityis“turnedon” thestartoftheBigBang comparedtothedustcloudoverasimilartimeperiodisadifferenceofmorethanafactorof1030.Thisenormousdifferenceintroducesmanyotherdifferencesbetweentheuniverseandthecloudthatwillbedescribed.Tomyknowledge,noonehasmathematicallyanalyzedtheimplicationsoflivinginauniversewherethebackgroundГuisincreasingasafunctionoftheageoftheuniverse.TheSchwarzschild


solutiontoEinstein’s fieldequationassumedasinglesourceofstaticgravity inanotherwiseempty universe. The solution describes the effect on spacetime that surrounded this staticsource of gravity. The Friedman equation assumes a uniform distribution of matter in theuniverse. However,even theFriedmansolutionassumesamaturegravitationaldistribution.ThestandardBigBangmodelalsoimpliesamaturegravitationaldistribution.Itisproposedthatthisisanerroneousassumptionthatultimatelyleadstotheneedtoinventdarkenergy.NotetotheReader: Wewilldivertfromthedustcloudexampleandimmaturegravityforashort time to introduce additional properties of spacetime. Wewill then combine all theseconceptstodevelopamodeloftheexpandinguniverse.Energy Density of Planck Spacetime:Untilnow,thisbookhasignorednumericalfactorsnear1.Fromobservationsandcalculations,theenergydensityoftheuniverseasafunctionoftimeiswellknown.Inorderforatheoreticalmodeltobecredible,itisnecessarytobeabletomatchobservations. Therefore, it is necessary to introduce a missing numerical factor into thedefinitionofPlanckenergydensitytodeterminetheenergydensityofPlanckspacetime.ItisnecessarytocarefullydefinetheenergydensityofPlanckspacetimebecausethisisproposedtobethestartingenergydensityoftheuniverseandthisvalueaffectstheevolutionoftheuniverse.PlanckenergydensityisdefinedasUp c7/ħG2 4.6 10113J/m3.ThisdefinitiondescribestheenergydensitythatwouldoccurifPlanckenergyEpwascontainedinacubewithdimensionsofPlanck lengthon a side Up Ep/Lp3 . Thisdefinition lacksdimensionlessnumerical factors.EventhoughtheenergydensityofPlanckspacetimeiscloselyrelatedtoPlanckenergydensity,theydifferbyadimensionlessconstant.Forcosmology,amorereasonabledefinitionofenergydensitywouldbebasedonthevolumeofaspherethatisPlancklengthinradiusratherthanacubethatisPlancklengthonaside.Thedifferencebetweenthesetwovolumesis 4/3 π.Zero Point Energy: There is one other dimensionless numerical factor that must also beincluded.Itisproposedthatvacuumenergyisequivalenttozeropointenergy.Zeropointenergyisthelowestenergythataquantummechanicalsystemmayhave. Theenergyofthegroundstateofaquantumharmonicoscillatoris:E ½ħωzeropointenergyItisproposedthatPlanckspacetimewasthehighestzeropointenergydensitypossible. Thedipolewaves in spacetime can be visualized as quantum harmonic oscillators. The highestpossiblezeropointenergydensitycanbevisualizedasaquantumoscillatorthatisoscillatingatthehighestpossiblefrequencyinthesmallestpossiblevolume.Thehighestpossiblefrequency

isPlanckangularfrequency:ωp c /ħG.Thezeropointenergyofthisoscillatoris:

E ½ħωp ½ ħc /G ½Ep 9.78 108J.


Therefore,thehighestenergydensityofthespacetimefieldistohavethisenergy ½Ep inthevolumeofaspherethatisPlancklengthinradius.ThisenergydensityofPlanckspacetimewillbeidentifiedbythesymbolUps.

Ups≡½ ħ

k’Upwhere:k’≡

Ups 5.53 10112J/m3Ups≡energydensityofPlanckspacetimeThe correction factor3/8πwill alsobeused frequentlyandwill be identifiedby the symbolk’ 3/8π.AspherePlancklengthinradiuswhichhassphericalPlanckenergydensitycontainsenergyof½Ep 9.78 108J.However,wewilloftenroundthisofftoaboutabillionJoules 109J .Theconstantk’ 3/8πistheconversionfactorbetweenPlanckspacetimeenergydensityUpsandPlanckenergydensity Up Ups/k’ .Theseconceptsareexpressedusingenergydensity U ratherthanmassdensity ρ becausemassisameasurementofinertia.Wavesinspacetimearenaturallyexpressedinunitsofenergydensityratherthanunitsofmassdensitywhichimpliesinertia.Difference between Planck Spacetime and Vacuum Energy:PlanckspacetimeatthestartoftheBigBangpossessedthesameproperenergydensityasvacuumenergytoday. Therefore,whatisthedifference?Oneofthekeydifferencesishiddeninthequalificationof“properenergydensity”.PlanckspacetimehadabackgroundgammaoftheuniverseofГu 1whiletodaythevalueofГuisanextremelylargenumberthatwillbecalculatedlater.ThislargevalueofГuaffectseverythingincludingtherateoftimeandourstandardofaunitofenergy.EventhoughoneJouletodayappearstobethesameasoneJouleabillionyearsagooroneJouleattheBigBang,thesearecomparisonsofpropervaluesthatdonottakeintoaccounttheeffectofadifferentvalueofГu.OnanabsolutescaleofenergythatcompensatesforthechangeinГu,oneJouletodayisfarlessenergythanoneJoulewhenГu 1atthestartoftheBigBang explainedbelow .Anotherdifferenceisthatvacuumenergyhasnoquantizedangularmomentum nospin whileatthestartoftheBigBang100%oftheenergyinPlanckspacetimepossessedquantizedangularmomentum.Thelackofquantizedangularmomentuminvacuumenergymeansthatthisisaperfect superfluid thatdoesnot interactwithourobservableuniverseexcept throughsubtlequantummechanicalinteractions.Therearenofermions halfintegerspin orbosons integerspin in“pure”vacuumenergy.Wewouldsaythatvacuumenergyhasatemperatureofabsolutezero.Eventhoughthereisenergydensity,vacuumenergyisincapableofgivingkineticenergytemperature toparticles.Ontheotherhand,inPlanckspacetimeatthestartoftheBigBangeachofthe½Epunitsofenergy zeropointenergy possessedħofangularmomentum.Thisresultsinthehighestpossibleinteractionwithotherquantizedunitsofenergy.ThetemperatureofPlanckspacetimehadthehighestpossibletemperatureforzeropointenergywhichisequalto½Plancktemperature ½ p 7 1031°K .


Proposed Alternative Model of the Beginning of the Universe: The alternative modelproposedherestartstheuniversenotasasingularitybutasPlanckspacetimewithvolumeandfiniteenergydensity.RecallthatPlanckspacetimehasasphericalcorrectionfactorandazeropoint energy correction factor that total 3/8π 0.12 compared to Planck energy density.However, Planck energy density eliminates numerical factors by assuming a cubic volume.Therefore,Planckspacetimeisthehighestzeropointenergydensityachievable. What istheimpliedrateoftime ortheimpliedcoordinatespeedoflight requiredtoachievethehighestpossibleenergydensityonanabsolutescale?Thisquestioncanalsobestatedasfollows:WhatvalueofthegravitationalgammaisrequiredtoachievePlanckspacetime?Thehighestpossibleenergydensity on an absolute scale that takes into consideration the valueof Г can only beachievedifГ 1.IfГ 1thentherateoftimeisslowerthancoordinaterateoftimeandtheunitofenergyislessthanthecoordinateunitofenergy.Thetheoreticallargestenergydensityonanabsolutescalecanonlybeachievedatthetheoreticalfastestrateoftimewhichisalsothefastestcoordinatespeedoflight.Therefore,Planckspacetimehaswhatmight seem likeacontradiction. Thehighestpossibleenergydensitystartswitharateoftimethatwewouldexpecttofindinahypotheticalemptyuniverse noslowingoftimecausedbygravitationalfields .Ifwearetrulyatthebeginningoftime,thentherewasnopriortimeforgravitytobecomeestablished.ThereisnocontradictionforГ 1ifthiswasthestartingcondition.The source of Planck spacetime is an open question. Perhaps spacetimemerely came intoexistenceasPlanckspacetimeorperhapsthisisaphasechangeinarepeatingcycle.Ineithercase,weassumethatPlanckspacetimestartswitharateoftimecommensuratewithГu 1.ThisassumptionmeansthatatthestartoftheBigBangtherateofpropertimeequaledtherateofcoordinatetime dτu dtatthebeginning .Extending the Dust Cloud Example:Previouslywehadthethoughtexperimentinvolvingacloud of dust distributed in space. In this thought experiment we imagined eliminating allgravitationaleffectsfromthiscloudofparticles gravityturnedoff .ThismeansthatwestartedwithabackgroundgravitationalgammaofГ 1.Thenwe“turnedon”gravityeverywhere.Thestarting condition for the entire universe is proposed to be similar in some respects to thisthoughtexperiment.Planckspacetimehadanenergydensityroughly10100timeslargerthantheenergydensityofthedustcloudbutPlanckspacetimewasnotasingularity.Planckspacetimehadanextremelyuniformenergydensitybecausetheenergydensitywasdeterminedbythelimitationsofspacetimeitself.Therefore,itisproposedthattheuniversestartedwiththefiniteenergydensityofPlanckspacetime.Thisquantifiableconditionwasthestartingconditionfortheevolutionoftheuniverse.


Therewasnopreexisting gravitypresent in this volume, sowhen time started thewaves inspacetimebegantodistortbecauseofthenonlinearityofspacetime.ThebackgroundГuoftheuniversebegantorapidlyincreasefromthestartingconditionofГu 1ofPlanckspacetime.Thisissimilartogravitybeing“turnedon”inthethoughtexperiment,exceptthechangesbroughtaboutbyanincreasingГuwerevastlylargerthanoccurredinthethoughtexperiment.Ratherthananalmostimperceptibleeffectonpropervolumeinthethoughtexperiment,therewasatremendousincreaseinthepropervolumeoftheuniverseasГurapidlyincreased.Therateoftimeintheuniverseslowedandthecoordinatespeedoflightslowed.Allofthisproducedtheimmaturegravitycondition.Thisimmaturegravitylacksgravitationalaccelerationonthelargescaleofhomogeneityaspreviouslydescribedinthedustcloudexample.Comparison to the Big Bang Model:Wearenowbeginningtoresolvethedifferencebetweentwomodelsoftheuniverse.TheBigBangmodeldoesnotattempttoexplaintheprocesswhichresults incosmicexpansion.Matter isnotexpandingintoapreexistingvoid. Instead,theBigBangmodelhasavastexpansioninthevolumeoftheuniversebutnoexplanationisgivenastohow new volume is created. The proposed model of the universe that starts with Planckspacetime will be called the “spacetime transformation model”. This model attributes theexpansioninpropervolumetoachangeinthepropertiesofspacetimethatcanbequantifiedasacontinuousincreaseinГu.Mostimportant,thereisauniformincreaseinГueverywherewhichimpliesthatthereisnolargescalegravitationalgradientattemptingtocollapsetheuniverse.ThiscontrastswiththeBigBangmodeloftheuniversewhichhasgravityattemptingtoslowdowntheexpansionoftheearlyuniverse. ThefollowingquotebyP.J.E.Peeblesinthebook“PrinciplesofPhysicalCosmology”describesthecurrentreasoning.

“Newton’s ironspheretheoremsays that theNewtoniangravitationalaccelerationinside a hollow spherical mass vanishes. The relativistic generalization is thatspacetime is flat in a hole centered inside a spherically symmetric distribution ofmatter.Nowweareinapositiontofindtherelationbetweenthemassdensity,ρ,andthelocalrateofexpansionorcontractionofthematerial.Weareconsideringaspatiallyhomogeneousandisotropicmassdistribution.Supposethematterwithinthespacewithinasphereofradiusr isremovedandset tooneside. Thenspacetimeis flatwithinthesphere. Nowreplacethematter. Ifr issmallenough,wehaveplacedasmall amount of material into flat spacetime. Therefore, we can use Newtonianmechanicstodescribethegravitationalaccelerationofthematerial.”

Thisgoeson to show that the sphericalvolumewithhomogeneousdensityhasgravitationalaccelerationthatattemptstocontractthedistributedmass.Whenthisreasoningisappliedtotheentireuniverseitnaturallyresultsingravityopposingtheexpansionoftheuniverse.Eveniftheuniverseisundergoingacceleratedexpansionbecauseofdarkenergy,theBigBangmodel


hasgravityattemptingtocontracttheuniverse.Thissoundsreasonableandithasbeenacceptedbygenerationsofphysicists.However,itisproposedtobewrong.IfweassumethattheuniversehasimmaturegravityandanincreasingГu,thenthefallacyintheabovereasoningisthatremovingasphericalvolumeofhomogeneousdensitymaterialfromtheuniversewouldnot leaveavoidwithflatspacetime. Instead,theremainingvoidwouldhavenegativecurvaturewhichhasgravitationalaccelerationawayfromthecenterofthevoid.Figure13‐3showedthatpoint“X”ontheedgeofthevoidhasgravitationalaccelerationawayfrompoint“O”atthecenterofthevoid.Point“X”infigure13‐3isnotunique.Allotherpointswithinthevoidalsohavegravitationalaccelerationawayfrompoint“O”atthecenterofthehypotheticalvoid.Thisnegativecurvatureeffectcanbeunderstoodsinceeachpointinthevoidcanbethoughtofasexisting inasphericalparticlehorizoncenteredonthem. Therefore theseotherpointswithinthevoidaresimilartopoint“X”andalsohaveagravitationalaccelerationvectorpointingaway from the central point “O”. These vectors exactly offset the gravitational accelerationtowardspoint“O”causedbythetestvolume.Replacingthetestvolumewouldresultinoffsettingvectorsthateliminateanygravitationalacceleration.Theimmaturegravitydescribedherehasnotendencytowardsgravitationalcollapseoftheuniverseonthecurrentscaleofhomogeneity~300millionlightyears .Thepreviousparagraphmightbemisunderstoodifquicklyread,sohereisanotherattempt.Onthelargescaleof300millionlightyears,thereisflatspacetime.However,onthesmallerscaleof 1million light years there canbepositively curved spacetimenear a galaxyornegativelycurved spacetime in a volume of spacewithout anymatter. Supposematterwas uniformlydistributed throughout the universe so that every cubic meter of space possessed uniformenergydensityofabout10‐9J/m3.Also,theuniverseextendsperhapsinfinitely,butdefinitelyfarbeyondoutpresentparticlehorizon.Thentheuniversewouldhaveflatspacetimeevenonthesmallscale.Thematterineachsphericalvolume1meterinradiuswouldbeproducingagravitational fieldwhichwouldtendtocollapsethemattertothecenterof thesphere if thatspherical volumewas isolated like the “test volume” in figure 13‐3. However, in the actualuniversetheimmaturegravityconditionisproducinganoffsettingnegativecurvature,sothecombinationresultsinflatspacetime.Low Density Regions of the Universe:Onthescalesmallerthan300millionlightyearstheuniverseiscurrentlynothomogeneous.Therearevolumescontaininggalacticclusterswhichhave above average density and volumes of space between the galactic clusterswith belowaveragedensity.Thesevolumeswithbelowaveragedensityshouldexhibitnegativecurvatureanalogoustothevoidpreviouslydiscussed.Alowdensityregionintheuniversewouldhaveasimilarity to theconceptofanelectronic “hole” ina semiconductor electronsandholes . Avolumeofasemiconductorthatismissinganelectronistheequivalentofapositiveelectricalcharge.Similarly,alowdensityregionintheuniverse missingmasscomparedtotheaverage exhibits the repulsive properties as if themissingmass had anti‐gravity properties had an


inverserateoftimegradient .Thiseffectonlyhappensiftheuniversehastheimmaturegravitycharacteristic.Iflowdensityregionsoftheuniverseexhibitsomethinglikeanti‐gravity,thiswouldsubstantiallyspeedtherateofformationofstarsandgalaxiesintheearlyuniverse.Anymatterwithinthelowdensityregionsoftheearlyuniversewouldtendtobeexpelled.Furthermore,theanti‐gravityaccelerationwouldextendintotheedgesofsurroundinghigherdensityregions.Thiswouldtendtodrive thesehighdensityregions toevengreaterdensity. Theaveragedensityof theearlyuniversewasmuchgreaterthantoday.Thereforetherepulsivegravitationalaccelerationoflowdensity regions of the early universewould bemuch greater than today. Modeling galaxyformationintheearlyuniverseshouldtakethisproposedeffectintoconsideration.Scaling Factor Based on Planck Spacetime au:Thecomovingcoordinatesystemuses“ao”torepresentthepresentscalingfactorofthecoordinatesystemandthescalingisusuallybasedontheconventionofconsideringthecurrentscaleoftheuniverseasao 1.Thisisconvenientsinceamodeloftheuniversethatstartswithasingularitydoesnothaveanyabsolutescalingfactorfrom the start of the Big Bang. However, the concept that the universe started as PlanckspacetimemeansthatthespacetimetransformationmodeldoeshaveadefinableinitialsizeandrateoftimeatthestartoftheBigBang.Itisnaturalthatwewouldchoosethisabsolutestartingconditionasthebasisofourscalingfactor.Thereforewewillusethesymbol“au”torepresentthescalingfactorrelativetothescaleoftheuniversewhenitwasPlanckspacetimeatthestartoftheBigBang.Therefore the spacetime transformation model sets au ≡ 1 for Planck spacetime when theuniversewas1unitofPlancktimeoldandГu 1.Accordingtothisproposal,atthestartoftheBigBangPlanckspacetimehadenergydensityofaboutUps 5.53 10112J/m3and100%ofthisenergywas“observable”.AnotherwayofsayingthisisthatalltheenergyinPlanckspacetimehadquantizedangularmomentum.Todayalmostalltheenergydensityintheuniverseisintheformofvacuumenergywhichlacksanyquantizedangularmomentum.Thereforeonlyroughly1 part in 10122 today has quantized angular momentum and is “observable” to us and outinstruments.Weonlyinteractwiththevastenergydensityinvacuumenergythroughquantummechanicaleffects.Thesenumbersthenbecomethebasisforgaugingthescaleoftheuniverseatanyothertimeinthehistoryoftheuniverse.Cosmic Expansion from Гu:WhathappenstothecomovingcoordinatesystemwhenthereisanincreaseinthebackgroundgravitationalgammaoftheuniverseГu?Inchapter2wedidathought experiment involving two concentric shells. The changes that occur whenmass isintroduced inside the inner shell are: 1 the rate of proper time dτu slowed compared tocoordinatetimedtand2 thepropervolumebetweenthetwoshells increased. We live inauniversewhichisproposedtohavestartedwithabackgroundgravitationalgammaГu 1andthisvaluethenstartedtoincreaseatthebeginningoftime attheBigBang .Wealsoknowthat


the proper volume of the universe rapidly increased initially and the proper volume of theuniversecontinuestoincreasetoday theso‐calledcosmicexpansion .Assumingthattheuniverseisonlyspacetime,thereappearstobeonlyonewaythattheuniversecan rearrange itself to increase its proper volume. I am proposing that the reason for theobservedcosmicexpansion is that thebackgroundgravitationalgammaГuof theuniverse iscontinuously increasing towards Гu ∞. This is a transformation in the characteristics ofspacetime. When the background gravitational gamma of the universe Гu increases, propervolume increases, the rate of time decreases and the coordinate speed of light decreasesproportional to1/Гu2. Itwill be shown later that themodelbeingdeveloped results innewmass/energycontinuouslycomingintoviewatourparticlehorizon. Theproposal isthattherate of time continues to get slower, butwe are unaware of this change becausewe cannotdirectlycompareourrateoftimetoeithertherateoftimeyesterdayortherateoftimeinahypothetical static universe that is not undergoing any change. This obviously is a differentmodelthantheΛ‐CDMmodelcurrentlyconsideredtobethecosmologicalstandard.The proposal is that the expansion of the proper volume of the universe is caused by the background gravitational gamma of the universe Гu increasing.We can see if this proposal is compatiblewith theRobertson‐Walkermetric hereafterR‐Wmetric . The highly successful R‐W metric has been called the standard model of moderncosmology. It assumes an isotropic and homogeneous universe and a spherical coordinatesystemthatexpandswiththe“Hubbleflow”sothatthecoordinatesystemremainsintheCMBrestframeeverywhere.ThestandardwayofwritingtheR‐Wmetricis:

dS2 c2dt2–a2 t –

r2 dθ2 sin2θdϕ2

wherea t isthescalefactoroftheuniverseandrisphysicaldistance.Theconventionistousethecurrentscalefactoroftheuniverseasao 1.Thereforea t isusuallyscaledintothefutureorpastfromthecurrentscalingfactorand“r”issetequaltothecurrentdistancebetweentwopointswhichcanbeconsideredaunitoflength.However,itisalsopossibletochoosesomeotherstandardfor“a”and“r”ifthereisacompellingreason.Inourcase,wehaveaverycompellingreasontochooseadifferentstandardfor“a”and“r”becauseunliketheΛ‐CDMmodel,wehaveanexactscalefortheuniverseatthebeginningofthebeginningoftime atthestartoftheBigBang .Thereforethisscalefactorwhichchangeswithtimebutwasau 1whentheuniversewasPlanckspacetimewillbedesignatedau t .AlsothephysicallengthbetweentwopointswilluseasitsstandardthelengthwhentheuniversewasPlanckspacetime Гu 1 . Thislengthstandard coordinatelength willbedesignated . Therefore,rewritingtheR‐Wmetricwiththesechangeswehave:


dS2 c2dt2–au2 t –

2dΩ2 R‐WmetricwheredΩ2 dθ2 sin2θdϕ2

The spatial curvature term is k and in the general case “k” can take values of 1, 0 or – 1.However, analysis of the CMB using experimental data from WMAP has established thatspacetimeonthelargescaleisflattobetterthan0.4%experimentalaccuracy.Therefore,wearegoingtoassumeaflatuniverseandsetk 0. Wecandeterminethespatialmetric fora flatuniversebysettingdt 0andfollowaradialraybysettingΩ 0.

dS2 c2dt2–au2 t –

2dΩ2 setk 0,dt 0,Ω 0anddS cdτ dL

dL2 au2 t d 2

au t

Theproposedphysicalinterpretationofthisisthatthereasonthatthescalefactorincreasesisbecauseproperlengthcontractsrelativetocoordinatelength whentheuniversehadГu 1orcomparedtoaunitof lengthinahypotheticalemptyuniverse. IfaunitofproperlengthLissmallerthanaunitofcoordinatelength ,thenprogressingintheradialdirectiontherateofchangeinproperlengthdLwillbegreaterthantherateofchangeind .Thisratioisequaltothescalefactorau t .Sincethespacetimefieldishomogeneousandisotropiconthelargescale,this applies to all spatial directions since the choice of a “radial” direction is arbitrary. TherelationshiptothegravitationalbackgroundoftheuniverseГuisproposedtobe:

SeventhStartingAssumption:Гu t au t

Itisnotpossibletoconclusivelyprovethisequation.Therefore,thisequationisanassumption.However,itispossibletosupportitsvaliditybyshowingthatitiscompatiblewithobservationsandgives a compellingmodelof theuniverse. Thepropervolumeof theuniverse is indeedexpanding. This equation is a key step in explaining why the universe is undergoing whatappearstobeacosmicexpansion.TheBigBangmodeldoesnotgofarenoughtoexplaintheunderlyingphysicsbehindthepropervolumeexpansionoftheuniverse.ThisequationallowsustoestimatethevalueofГuoftheuniversetodaybecauseweknowtheinitialscaleofPlanckspacetimeandwecanestimatetheexpansionthathasoccurredsincetheuniversehadtheenergydensityofPlanckspacetime.WewillproceedbyassumingГu autoobtainotherinsights.Iftheanswersobtainedbytheuseofthisequationarereasonable,thiscanalsoserveasacheckontheequation.TheappealoftheconceptthattheuniverseisundergoinganincreaseinГuisthat:1 itexplainstheapparentcosmicexpansionoftheuniverse2 itgivesthecorrectredshifttolightfromdistant


galaxies provenlater ,3 itgivesauniversewithauniformrateoftimeonthecomovinggridand4 iteliminatesthemysteryofexpandingspacesupposedlycarryinggalaxiesawayfromusatfasterthanthespeedoflight5 itisoneofseveralstepsthateliminatetheneedfordarkenergy6 itsolvesthemysteryofhowtheenergydensityofvacuumfluctuations zeropointenergy canremainconstantwhenthepropervolumeoftheuniverseexpands.Coordinate Rate of Time dt:Asexplainedpreviously,thedefinitionofthecoordinaterateoftime designateddt usedbythespacetimetransformationmodelistherateoftimewhenГu 1.ThisonlyoccurredinouruniverseduringthefirstPlanckunitoftimeintheageoftheuniverse.Therefore,thecoordinateclockcanbeimaginedasaclockthatcontinuestorunattherateoftimethatwaspresentatthebeginningoftime.Thisisalsotherateoftimethatwouldoccurinahypothetical empty universe where always Гu 1. After the first unit of Planck time, thebackgroundГuof theuniversebegan to increase Гu 1 and therateofproper time in theuniversedτudecreasedrelativetothecoordinaterateoftime.Thegravitationalgammaoftheuniversecanalsobedefinedbythefollowingtemporalrelationship:Гu dt/dτuCoordinate and Hybrid Speed of Light:Thesymbol“dt”intheR‐Wmetricrepresentstherateoftimeinthecomovingcoordinatesystem.ThisisactuallytheproperrateoftimeintheCMBrestframeoftheuniverse.Weneedtosubstituteadifferentsymbolbecausewewanttouse“dt”asthecoordinaterateoftimewhenГu 1.TherateoftimeusedintheR‐Wmetricisactuallytheproperrateoftimeintheuniverse therateoftimeonthecosmicclock .ThereforeintheR‐Wmetricwewillmakethefollowingsubstitutiondt→dτu.Alsomakingthesubstitutionofauand previouslydiscussed,wehave:

dS2 c2dτu2–au2 t –

2dΩ2modifiedR‐Wmetric

Combiningthesesubstitutionswithsubstitutionspreviouslyexplainedwehave:Set:dS 0,dΩ 0,k 0,au t Гuc2dτu2 au2 t d 2

c au t setau t Гu t and ≡C

C Г

hybridspeedoflight coordinatelengthandproperrateoftime

Next we will determine the coordinate speed of light d /dt which references thecoordinaterateoftimedtandtherateofchangeofcoordinatelengthd bothinahypotheticalemptyuniversewhereГu 1.ThedefinitionofdtusedfortheremainderofthisbookistherateoftimeinahypotheticalГu 1universe.Therefore:dt≡Гu t dτu.


c Гu t setdτu Г

c Гu2 t set

Г

coordinatespeedoflight

Therearetimeswhenitisnecessarytousethecoordinatespeedoflight butusuallythehybridspeedoflightCisadequateandsimpler.Whenpossible,itismoreconvenientandintuitivetousetheproper comoving rateoftime.Ifwearedeterminingthetimerequiredforlighttotraveladistancebetween two stationarypoints,weusuallywant theanswerexpressed inunitsofpropertimethatwouldberecordedonthecosmicclock. ThedifferencebetweenthehybridspeedoflightC d /dτuandtheproperspeedoflightc dL/dτuisthedifferencebetweenaunitofcoordinatelength andaunitofproperlengthL.Thereasonthatittakesmoretimeforlighttotravelbetweentwodistantgalaxiestodaythanitdidabillionyearsagoisthatthehybridspeed of light is less today than itwas a billion years ago. The distance between these twogalaxiesisconstantwhenmeasuredinunitsofcoordinatelength .Aslowinghybridspeedoflightimpliesthataunitofproperlength, suchas1metertoday iscontractingrelativetoaunitofcoordinatelength suchas1meterwhenГu 1 .Reconciliation With Proper Length Being Constant:Clearlythephysicalinterpretationthatis emerging is that aunit of proper length contractswhenГu increases. In chapter3 itwasproposed that itwasanacceptablebasisofacoordinatesystemtoconsideraunitofproperlength,suchasameter,tobethesameeverywhereintheuniverse.AmeterinalocationwithalargegravitationalГwasconsideredtobethesameasameterinalocationfarfromanysourceofgravitywhichwewerepreviouslycalling“zerogravity”.Howisitpossibletoreconciletheviewthatproper length iscontractingasГu increaseswith theviewinchapter3 thatproperlengthisconstanteverywhereinthecurrentuniverse?Theansweristhatthe“normalized”coordinatesystemofchapter3usedtheequationLo Lg.ThisdoesnotimplythatLoandLgareconstantovertime.Itmerelysaysthatthiscoordinatesystemassumesthatinastationaryframeofreference,aunitoflengthinazerogravitylocationLo is the same as a unit of length in a locationwith gravity Lg at a given instant in timemidpointobserverperspective .BothoftheselengthscanbesimultaneouslycontractingastheuniverseagesandyetmaintainLo Lg.Morewillbesaidaboutthisinchapter14.Planck Sphere:Wewillnowdefinetermsnecessarytoquantifyandanalyzetheevolutionoftheuniverse.PlanckspacetimehasenergydensityequivalenttoaspherethatisPlancklengthinradius containing themaximumzeropointenergypermittedby thepropertiesof spacetime.Thismaximumenergyisequalto½Planckenergy ½Ep 9.78 108J .Mostimportant,Planck


spacetime has Гu 1 and all the energy possesses quantized angular momentum. Theimplicationsoftheseconditionswillbeexplainedlater.WewillcallthebasicquantizedunitofPlanckspacetimea“Plancksphere”. BesideshavingaradiusofPlancklengthitalsohasquantizedangularmomentumofħ perhaps½ħ .Thechoicebetweenthesetwowillhavetobeworkedoutbyothers,butħismorereasonableandwewilluseħinthefollowingdiscussion.TheconceptofaPlancksphereisintroducedprimarilybecausethisisaconvenientreferencetohelpusvisualizetheapparentexpansionoftheuniverseinunitsofproperlength.Wecandescribewhathappenstotheradius,energyandangularmomentumthatareinitiallyinthePlancksphere.InthiswaytheevolutionofthePlanckspherebecomesaneasyreferenceforwhathappenstoallthespacetimeintheuniverse.Angular Momentum in Planck Spacetime:Besideshavingenergydensity,Planckspacetimealsomusthaveadistributeddensityofquantizedangularmomentum. OurcurrentuniversepossessesquantizedangularmomentumprimarilyintheformofCMBphotons.CurrentlytheCMBhasthephotondensityofa2.725°Kblackbodycavitywhichisabout4 108photons/m3.Theangularmomentum ħ possessedbythisvastnumberofphotonsexceedsbyafactorofabout 108 the internal angular momentum possessed by baryonic matter in the universeignoringdarkmatter . Ifwe simply imagine reversing time, all theCMBphotonswouldbecompressedandblueshifteduntileachphotonwasreducedtoaspherethatisPlancklengthinradius.Thisideacanbecheckedwithaplausibilitycalculationthatwillbemadelater.Starting the Universe from Planck Spacetime:Theremainderofthischapterwilllookattheevolutionoftheuniverse.ThiscombinestheconceptsofanincreasingГuandPlanckspacetimewiththeBigBangmodelofexpandingspacetime.ThiscombinationallowsustointroducenewideasinthefamiliarcontextoftheBigBangmodel.InthenextchapterwewilldroptheBigBangmodel and switch completely to the proposed spacetime transformationmodel that ismorecompatiblewiththepremisethattheuniverseis andalwayswas onlyspacetime.WearesettingthestageforthestartoftheBigBang.InthisthoughtexperimentwearegoingtowitnesstheBigBangfromthevantagepointofaninfinitelysmallpointthatisatthecenterofaPlancksphereatthestartoftheBigBang.FromthisvantagepointwearegoingtopayparticularattentiontowhathappenstotheapproximatelyonebillionJoules ½Ep thatistheenergyofthePlanckspherethatsurroundsourinfinitelysmallvantagepoint.AllofthisenergyisinitiallywithinadistanceofPlancklengthofourlocation.Wewillimaginethatwecanwatchtheevolutiontheuniverse watchtheBigBang fromourinfinitelysmallvantagepointwithinPlanckspacetime. WearealsogoingtoequipourselveswithtwoclocksthatwillstartrunningthemomentthattimestartstoprogressforwardatthestartoftheBigBang.BothoftheseclocksaredigitalclocksthatruninquantizedincrementsofPlancktime ~5.4 10‐44secondsteps .Oneoftheseclockswillbecalledthe“cosmicclock”


thatmeasurestheproperageoftheuniverseinthecomovingframeofreference.Timeonthecosmicclock,expressedindimensionlessPlanckunitsoftime,willbedesignatedwiththesymbolτu boldandunderlineddesignatesPlanckunits anddefinedas:τu≡τu/tp.Theotherclockwillbecalledthe“Г 1clock”.ThisisahypotheticalclockthatrunsattherateoftimeofanemptyuniversewherethebackgroundgravitationalГuisalwaysequalto1.TimeontheГ 1clockalsoisinPlanckunitsoftimeanddesignatedastu.SincetheuniversestartedwithГu 1,theГ 1clockcanalsobethoughtofasaclockthatcontinuedrunningattherateoftimethatwaspresentatthebeginningoftheBigBang.Thisclockiskeepingcoordinatetimewhichisthe“dt”termintheequation:Гu dt/dτu.The Big Bang Starts:Nowforthebigevent:TimestartsandtheBigBanghasbegun.Actually,theobservable,properenergydensitydefiningPlanckspacetimeonlyexistsinthisformduringthefirstunitofPlancktime.Afterthisbrieftimetheobservableenergydensity energywithspin rapidly drops to less than Ups 5.53 10112 J/m3 and Гu rapidly increases Гu 1 .However,thetotalenergydensityoftheuniverseremainsconstantwhenweaddtogetherthevacuumenergydensity nospin plustheobservableenergydensity withspin asexplainedlater.Itisnotpossibletotalkaboutwhathappens“before”timestartstoprogress whentu 0 because this implies that the rate of time has stopped and there is no motion of waves inspacetime.Wewilladoptthepositionthattherewasnotime“before”tu 1.Inorderforthespacetimefieldtobeenergetic,theremustbemovingdipolewavesinspacetime theremustbedynamicspacetime .Thereisnoenergyinstaticspacetime.IfthespinineachPlancksphereisħ,thenthisistheequivalentofonephotonpersphere.InitiallytheenergyofthissinglephotonperPlanckspherewouldbeequaltothemaximumzeropointenergywhichis½Ep.AnotherwayofsayingthisisthatalltheenergyofPlanckspacetimehadquantizedangularmomentumandwas“observable”energy.Todayonlyabout1partin10122ofthetotalenergyintheuniversehasquantizedangularmomentumandis“observable”.WearegoingtoassumethattimestartssimultaneouslyeverywherewithinthePlanckspacetime.Thisobviouslyrequireseitherfasterthanspeedoflightcommunicationorsomeotherwayofsynchronizingthestartoftime.Theonlyjustificationforthisassumptionisthatobservationstoday indicateasimultaneous initiationof theBigBang. Perhaps thiscanberationalizedbyassuming that in this highest possible energy density, the entire universe was like a singleparticlethatexhibitedtheinstantaneouscommunicationofentanglementorunity.Wewillnotdwell on this point. This assumption seems more understandable than starting with asingularity,butultimatelythesimultaneousinitiationoftheBigBang simultaneousstartingoftime isjustastartingassumption.


AttheinstantthattheBigBangstarts,ourimaginedvantagepointiscompletelyisolated.TherehasbeennotimeforanycommunicationwithanyofthesurroundingenergyinPlanckspacetimeand therehasbeennononlineardistortionof thewaves in spacetime that constitutePlanckspacetime. Thismeans thatat thestartof theBigBang there isnopreexistinggravitationaleffects–nopreexistinggravitationalaccelerationandforthe firstunitofPlancktime,Гu 1everywhereintheuniverse.Therefore,forthefirstunitofPlancktime,thecosmicclockandtheГ 1clockbothrunatthesamerateoftime dτu dt .Thisstartingconditionissimilartothedustcloudthoughtexperimentwherewestartedwithgravity“turnedoff”.Increasing Value of Гu:ThefirsttickofPlancktimewaseasybecauseithappenswithГu 1.Our“particlehorizon”wasequaltoPlancklengthwhichistheboundaryofthePlancksphere.Thequestionis:WhathappenswithsubsequentticksasourparticlehorizonexpandsbeyondtheboundaryofourPlancksphere?SpeedoflightcontactisbeingmadewithadjacentwavesinspacetimejustbeyondtheboundaryofourPlancksphere.ThenonlinearityofspacetimebeginstoaffectthesewavescausinganincreaseinthebackgroundgravitationalgammafromГu 1toГu 1.Therateoftimeslowsandpropervolumeincreases,butthereisnotimegradient.Allpointsareexperiencingthesamerateoftimesothereisnotendencyforgravitationalattractionorgravitationalcollapse.Thisisthestartoftheimmaturegravityconditionthatcontinuestoday.AuniformlyincreasingГueliminatesthetimegradientnecessaryforgravitationalattraction.For the first unit of Planck time the cosmic clock and the Г 1 clock were synchronized.However,all subsequent ticksexhibit the fact that thecosmicclockhasarateof time that isslowingrelativetotheГ 1clock.AlsotheproperdistancetotheboundaryofthePlancksphereincreases.BothoftheseeffectsaretheresultofГuincreasingwhichinturnistheresultofthenonlineareffectsofspacetimeaccumulating.ThewavesinspacetimeresponsiblefortheenergydensityofPlanckspacetimehavebeguntodistort.Unlikethegravityproducedbyanisolatedmass,insidetheearlyuniversethestressedspacetimeisahomogeneousdistributionofenergythatextendsbeyondourparticlehorizon.ThiswavedistortionwouldproduceanincreaseinthebackgroundgravitationalgammaГusimilartotheincreaseinthebackgroundГinthedustcloudthoughtexperiment.However,becausetheuniversestartedwiththemaximumpossiblequantizedenergydensity,theeffectsonvolumeandtherateoftimearevastlylargerthaninthedustcloudthoughtexperiment.Spacetime Evolves into Different Types of Energy: Thedistortion freeoscillating spacetimePlanckspacetime thatwaspresentduringthefirsttickofthecosmicclockbeginstoexhibitdistortionforallsubsequentticks.Thenonlinearityofspacetimecausesthewavesinspacetimetodistort. Thedistortedwavescanbe throughofasbeingsplit into threecomponentparts.Thesethreecomponentsare:


1 Thedistortionfreecomponentoftheoriginalwavesinspacetimethatretaintheoriginalquantized angular momentum spin . This is proposed to become the source ofeverythingintheuniversethatwecansensetoday allfermionsandbosons .

2 Thenon‐oscillatingcomponent theAβ2componentofchapter6 thatresultswhenthewavesinspacetimearedistortedbythenonlinearityofspacetime.ThisisresponsibleforgravitationaleffectsandalsoresponsibleforthebackgroundГuoftheuniverse.

3 Thehighfrequencyoscillatingcomponentofgravity theAβcos2ωtcomponentdescribedin chapter 6 proposed to be responsible for vacuum energy. This component lacksangularmomentumandbecomes thevacuum fluctuations componentof theuniversetoday.

Therefore startingwith Planck spacetimemeans that any nonlinear distortion caused by aninteractionofwavesinspacetime gravitationalcontactwithsurroundingenergy mustincreaseГu.However,asГuincreasesthisalsoincreasesproperdistancebetweenstationarypointsanddecreasestherateoftime.Newenergyisalwaysaddingitsgravitationalinfluenceastheparticlehorizonexpandsandnewenergycomesintoview.AsГuincreases,theproperdistancetotheparticlehorizonincreasesevenfasterthanthespeedoflightbecauseourunitoflengthisalsoshrinkingrelativetocoordinatelength .ThevalueofГuincreasestowardsinfinity.Avoiding Gravitational Collapse: What prevents Planck spacetime from collapsing into asingularity? In fact,whenτu 1eachPlancksphereappears tocontain theexactamountofenergyrequiredtobecomeablackholewithaSchwarzschildradiusequaltoPlancklength.Eventodaytheproperenergydensityofspacetimeisapproximatelythesameastheenergydensityof Planck spacetime. According to general relativity, energy in any form produces gravity.Therefore, general relativity considers the energydensity of the spacetime field requiredbyquantummechanics tobea ridiculously largenumber ~10120 times larger than the criticalenergydensityoftheuniverse .Whatpreventstheenergydensityofthespacetimefieldfromcollapsingintoablackhole?Apartialanswerwillbegivenhereandadditionaldetailswillbegiveninthenextchapter.Thinkoftheenergydensityofthespacetimefieldasbeingthenecessaryingredienttogivethespacetimefieldpropertiessuchasaspeedoflight,impedance,agravitationalconstant,elasticity,etc.Thereforeitisanoversimplificationtosaythatenergyinanyformcreatesgravity.Themorecorrect statement would be that any form of energy that possesses quantized angularmomentum creates gravity. In otherwords, fermions and bosons create gravity. The dipolewavesinspacetimethatformvacuumenergylackangularmomentumandarepartofthefabricofspacetime.Theyhavesuperfluidpropertiesandareashomogeneousasquantummechanicsallows.Theyarealsothenecessaryingredientrequiredtosupporttheexistenceoffermionsandbosons.


When rotars with their quantized angular momentum are introduced into an otherwisehomogeneousvolumeofthespacetimefield,thisisaddinganadditionalenergydensitytothepreviouslyhomogeneousenergydensityofthespacetimefield.Forexample,iftherewasarotarwithPlanckenergy,thenthisrotarwouldhaveaquantumradiusequaltoPlancklength.Theenergydensityofsucharotarwouldbeequaltotheenergydensityofanequivalentvolumeofthespacetimefield.Therefore,introducingsucharotarintoapreviouslyhomogeneousvolumeofthespacetimefieldwoulddoubletheenergydensityofthatvolumeofthespacetimefield.Thisdoublingreachesthetheoreticallimitofthepropertiesofthespacetimefieldandresultsinthecreationofablackhole.Therearenofundamentalparticles rotars withPlanckmass/energy. Theconditionswhichcreateastellarsizeblackholeorasupermassiveblackholedonotmatchthetotalenergydensityof thespacetime field ~10113 J/m3 . However, theydomatchthe interactiveenergydensityUipreviouslydiscussedinchapter4.Recallfromchapter4thefollowing:

TheinteractiveenergydensityUi Fp/λ2isaverylargeenergydensity. Howdoesthisenergydensitycomparetotheenergydensityofablackhole symbolUbh ? We will designate the black hole’s energy as Ebh and its definedSchwarzschildradiusasRs≡Gm/c2.Ignoringnumericalfactorsnear1wehave:

Ubh

Therefore,sinceUi Fp/λ2andUbh Fp/Rs2itcanbeseenthatifλ RsthenUi Ubh.Addingadditionalenergytoaspecificvolumebyintroducingmatterincreasesthetotalenergydensityandproducesthegravitationaleffectsdescribedbygeneralrelativity.Thelargestforcethatthevacuumenergy/pressurecanexertisPlanckforce Fp c4/G .Withtherotarmodelitwasstatedthattheenergydensityofarotarimpliedaspecificpressure.Thispressureneededtobecontainedbyanoffsettingpressureexertedbyvacuumenergy/pressure. Likewise, thematterofalargerbodysuchasastarhasacollectiveenergydensitythatmustbestabilizedbyoffsettingpressurefromvacuumenergy/pressure.Thepressureonastarcanbeconsideredastwoopposingforcesexertedonoppositehemispheresofthestar.ThemaximumforcethatthespacetimefieldcanexertisPlanckforce.Thereforetheconditionswhichformablackholeoccurwhenthespacetimefieldisaskedtoexertthismaximumforce.Theenergydensityofablackhole ignoringnumericalfactorsnear1 isUbh Fp/Rs2.Thepointisthattheenergydensityofspacetimedoesnotformablackhole.Instead,ablackholeformswhenadditionalenergydensitymatter inintroducedtoavolumeandequalstheinteractiveenergydensitywhereλ RsintheequationUi Fp/λ2.


WhentheuniversestartedatthebeginningoftheBigBang τu 1 ,100%oftheenergyofPlanckspacetimepossessedquantizedangularmomentum.TherewasnovacuumenergyduringthefirstunitofPlancktime.Therefore,thetotalenergydensityofaPlanckspheredidnotformablackholeandtheparticlehorizonwasequaltoPlancklength.ThepossibilityofformingblackholesonlyoccurredlaterwhensomeoftheenergyofPlanckspacetimewasconvertedtovacuumenergy.Thenitbecamepossibletooverloadaparticularvolumeofthespacetimefieldwiththecombinationofvacuumenergyandenergywithquantizedangularmomentum multiplerotarsand/orphotons .Inchapter14wewillseehowitispossiblefortheproperenergydensityofthespacetimefieldtoremainconstantwhilethepropervolumeoftheuniverseexpands.Three Epochs in the Evolution of the Universe:Therateofexpansionoftheuniverse rateofchange of Гu is currently believed to have gone through 3 different phases. The followingcalculationsusethecurrentestimatesfortheratesofexpansionoftheuniverseandthecurrentestimates of age of the universe when transitions occur. As improved estimates becomeavailable,theyshouldbesubstitutedforthefollowingpreliminaryestimates.ThefirstphaseistheradiationdominatedepochandiscurrentlybelievedtohavestartedwiththeBigBangandendedroughly70,000yearsaftertheBigBang.Duringthisepoch,thescalingfactoroftheuniversegrewas:au t τu

1/2.Nextcamethematterdominatedepochfromroughly70,000yearsaftertheBigBangtoabout5billionyearsaftertheBigBang.Thematterdominatedepochisbelievedtohaveascalingfactorthatgrewwithaslopecharacteristicofτu

2/3.Howeveritisnotaccuratetosaythatau t τu

2/3becauseduringtheradiationdominatedepochtheslopewasdifferent. Therefore,wewill say thatduring thematterdominatedphase the change inscalingfactorisΔau t Δτu

2/3.Fromabout5billionyearsuntilthepresentweareinwhatisreferredtoasthe“darkenergy”dominatedepoch.Duringthisphaseafirstapproximationofthechangescalingfactorcanbedescribedas:Δau t Δτu. However,thechangeinscalingfactorappears to be more complex than this because the change in the scale factor is probablyaccelerating.TheΛ‐CDMmodelofcosmologyhasparametersthatcanbeadjustedtopermitittomatchobservations.Fortunately,thekeypointsoftheproposedspacetimetransformationmodelcanbestatedwithoutrequiringapreciseequationforthescalefactorinthe“darkenergy”dominatedepoch.Radiation Dominated Epoch: ThestandardBigBangtheory withoutinflation hasthescalefactor of the universe increase proportional to the square root of time during the radiationdominatedepochau t τu

1/2.Eventhemodeloftheuniversethatincludesinflationstartsoffwiththescalefactorincreasingproportionaltoτu

1/2betweenthestartoftheBigBangandabout10‐37secondafterthestart.Theinflationmodelthenhasabriefperiodwhereavastexponentialexpansiontakesplace.ThisisthenfollowedbyareturntoexpansionproportionaltoΔτu

1/2.The spacetime wave model proposed here does not need the hypothetical inflationaryexponentialexpansiontomaketheuniversehomogeneous,isotropicandspatiallyflat.Allofthis


automatically follows from starting the universe at the highest possible energy density thatspacetimecansupport‐Planckspacetime.Thisenergydensityisashomogeneousasquantummechanics allows. These quantum mechanical fluctuations are traceable to the Plancklength/time limitation of wave amplitude in the model of the universe based on waves inspacetime.Thesequantumfluctuationsprovidethesmallamountofinhomogeneityrequiredtoseedtheeventualgravitationalformationofstarsandgalaxies.Thesymbolsandequationsthatwewillbeusingtocharacterizetheradiationdominatedepochare:

au t Гu τu1/2 τ /t radiationdominatedepochrelationships

The Universe at τu = 9:Wewillillustratesomeimportantconceptsabouttheearlystagesoftheexpansionoftheuniversewithanexample.PreviouslyanimaginaryvantagepointatthecenterofaPlanckspherewasdescribed.Supposethatwereturntothatinfinitelysmallvantagepointandlookatthechangesthatoccur9unitsofPlancktime τu 9 afterthestartoftheBigBang.Fromτu

1/2 Гuweobtainthatatτu 9thebackgroundgravitationalgammaoftheuniverseis:Гu 3. Thishasaprofoundeffectonspacetime. Thescale factorof theuniverserelative toPlanck spacetime au has tripled.Therefore, thePlanck sphere that startedwitha radiusofPlancklengthlpnowhasaproperradius3timesaslarge r 3lp . However,thecoordinateradiusalwaysassumesГu 1,thereforethecoordinateradiusofthePlancksphere measuredinunitsof remainsconstantat1unitofPlancklength 1 TheHubblespherewithradiusrharoundtheoriginpoint,hasaproperradiusofrh cτusowhenτu 9thenrh 9lp.TheHubblespheredefinesthedistancewhereobjectsseemtoberecedingattheproperspeedoflight.Theproperdistancetotheparticlehorizon designatedrph ataninstantintimeislargerthantheHubblesphereatthatsameinstantintime.Thisisbecausetheinstantaneousproperdistancetotheparticlehorizonincludesdistancethathasexpandedafteraspeedof lightsignalhaspassedanypoint. DuringtheradiationdominatedepochitcanbeshownthattheproperdistancetotheparticlehorizonisalwaystwicetheradiusoftheHubblesphere.Atageoftheuniverseτu,theradiusoftheparticlehorizonisrph 2cτu 18lpwhenτu 9orτu 4.85 10‐43 s. Today thevalueofГu ismuch larger than3 calculated later .Therefore the proper dimensions quotedwhen τu 9 and Гu 3would be different todaybecauseГuismuchlarger.However,thecoordinatedimensions measuredinunitsof alwaysareconstantbecausetheydonotscalewithГu.Any signalobtained from theparticlehorizonhas infinite redshift but a signal emitted frominsidetheparticlehorizonhaslessofaredshift.Thegravityofanapparentlyrecedingmassislessthanthegravityofastationarymassatthesamedistance.Therefore,theredshiftatvariousdistancesalsoaffectsgravity. It isthecombinationofthegravityofalltheenergyinsidetheparticlehorizon that together is increasing thebackgroundgammaatourobservationpoint.Sinceallotherobservationpointsareexperiencingasimilargravitationaleffect,thebackground


gravitational gammaof the universe is increasing homogeneously because Planck spacetimestartedoffatthemaximumhomogeneitypermittedbyquantummechanics.Wewillnowreturntoourexampleoftheuniversewhenitwas9unitsofPlancktimeold τu 9 .Atthisinstantwehad:Гu τu

1/2 dt/dτu 3.TherateofpropertimeshownonthecosmicclockhasslowedtoathirdtherateoftimeshownontheГ 1clock.AlsothehybridspeedoflightC d /dτuhasslowedtoathirdtheproperspeedoflightexpressedbytheuniversalconstantc dL/dτubecauseofthedifferencebetweend anddL.Therefore:

C Г c/τu

1/2 assumesradiationdominatedepoch

ThePlancksphereinitiallycontainedabout109JoulesduringthefirstunitofPlancktimewhentheuniversewasPlanckspacetime. Allthisenergyinitiallypossessedħofquantizedangularmomentum.Thereforetheseinitialwaveswerephotonsinmaximumconfinement.However,energeticphotonscanbeconvertedtoparticles.Forexample,ithasbeenexperimentallyproventhattwo511,000eVphotonscanbeconvertedtoanelectron/positronpair.Whenτu 9andГu 3thephoton‐likewaveshavebeenredshiftedbyafactorof3.Therefore,thepropervolumeof thePlanck spherehas increasedbya factorof27and the “observableenergy” possessingquantizedangularmomentum inthePlanckspherehasdecreasedbecauseofredshiftbyafactorof3toabout0.33billionJoules.Wheredidthedifferenceinenergygo?Lost Energy Becomes Vacuum Energy:Whenphysicsstudentshearaboutthecosmicredshiftdecreasingtheenergyofphotons,theyoftenaskwheretheenergywent.Answersofteninvolveadiscussionofstretchedwavelengths,andeventuallyimplythatconservationofenergydoesnot apply to the cosmic expansion of the universe. The proposed spacetime transformationmodelgivesadifferentanswer.Whenthepropervolumeoftheuniverseexpands,aphotonlosesenergybutretains100%ofitsproperangularmomentum.Thereforethelostenergypossessesnoangularmomentum.Itisproposedthatalltheobservableenergylostbyphotonsisbeingconvertedtovacuumenergyzeropointenergy thatisinasuperfluidstatethatcannotpossessangularmomentum.Infact,itisnotonlyphotonsthatarelosingenergytocosmicexpansion.Neutrinosandotherrelativisticparticles movingrelativetothelocalCMBrestframe arealsolosingsubstantialamountsofobservablekineticenergy.Thislostenergyisalsobeingconvertedtovacuumenergy.Inthenextchapterwewillattempttocalculatetheratioofvacuumenergydensitytoobservableenergydensityintheuniversetoday.Radiation-Matter Equality Transition: The radiation dominated epoch ended when theenergydensityinradiationfelltoeventuallyequaltheenergydensityofmatterintheuniverse.The proper energy ofmatter such as an electron does not decreasewhen Гu increases so


eventuallyradiationenergydensity falls toequal theaverageenergydensityofmatter. Thistransitioniscalledtheradiation/matterequalityandoccurredabout70,000yearsaftertheBigBang.BeforetheWMAPprobeaccuratelymeasuredtheCMB,thistransitionwasthoughttohaveoccurredearlier,perhaps10,000yearsaftertheBigBang.However,analysisoftheWMAPdatahasdeterminedthattheenergycontentoftheuniverseat380,000yearsaftertheBigBangwas15%radiation,10%neutrinos,12%baryonicmatterand63%darkmatter.Sincebothradiationandneutrinoenergydecreaseinenergyatvirtuallythesamerate assumingnonewsources ,the breakdown at 380,000 years can be generalized as 25% radiation‐like energy and 75%energyinmatter. Thesenumbersimplytheradiation/matterequalityoccurredabout70,000yearsaftertheBigBang τu 4.09 1055Planckunitsoftime .Itispossibletocalculatethevalueofthebackgroundgravitationalgammaattheradiation/matterequalitytransition.Thisbackgroundgammaisimportantforlatercalculationsandwillbedesignatedas:Гeq.Гu τu

1/2setτu 4.09 1055 70,000years Гeq 6.4 1027Гeq Гuattheradiation/matterequalitytransitionTherefore,about70,000yearsaftertheBigBangtheabsolutescalefactoroftheuniversewas:au Гeq 6.4 1027.ThismeansthatthePlancksphereincreasedinradiusfromPlancklength~10‐35m toabout10‐7m obtainedfromlp 6.4 1027 10‐7m .Furthermore,theredshiftthatoccurredduringtheradiationdominatedepochdecreasedtheproperenergyofradiationbyafactorequaltoГeq.Thismeansthatthe“observable”energyinthePlanckspheredecreasedby a factor equal to Гeq from about a billion Joules to about 1.53 10‐19 J. All themissingobservableenergy i.e.energypossessingangularmomentum wasconvertedtovacuumenergy.ThereforethetotalenergyinthePlancksphere observableenergyplusvacuumenergy,bothmeasuredinunitsofcoordinateenergy remainedconstantatabout9.78 108Joulesattheendoftheradiationdominatedepoch discussedlater .Thiscanbeconfusingsincetherearetwodifferentstandards forenergy thesameway that thereare twodifferentstandardsof lengthproper and coordinate . The conclusion is that the energy density of the universe has notchangedfromtheBigBangtotodayifvacuumenergyisincludedintheenergydensity.Whathaschangedisthatthe“observable”energydensityoftheuniverse excludesvacuumenergy hasdecreasedbyafactorofabout10122.After theradiationdominatedepochended, thenextphasewas thematterdominatedepochwhichlastedfromabout70,000yearstoabout5billionyearsaftertheBigBang.ThescalefactorandГu duringthisepochscalesproportionaltoΔτu

2/3.Thecurrentepochisusuallyreferredtoasthedarkenergydominatedepochandextendsfromabout5billionyearstothepresent.ThescalefactorandГuduringthisepochmightbelinearlyincreasingproportionaltoΔτuoritmightbeincreasingfasterthanlinearlyinwhichcaseitwouldbeanexponentialincrease.Thisisthefamousacceleratingexpansionoftheuniverseandtheexactrepresentationofthisphasewillbelefttotheastrophysicists.


Гuo Calculation from Expansion:OurobjectivehereistodeterminethecurrentvalueofthebackgroundgravitationalgammafortheuniversedesignatedГuo.WehavecalculatedthevalueГeq 6.4 1027 for the radiation dominated epoch. The remaining part of the calculationrequiredtodetermineГuoisgreatlyhelpedbythefactthattheredshiftfromtheradiation/matterequalitytothepresenthasbeenexperimentallymeasuredbyWMAP. Thefollowingquoteisfromthe5YearWMAPreport4;“TheequalityredshiftzeqisoneofthefundamentalobservablesthatonecanextractfromtheCMBpowerspectrum5”.Theterm“equalityredshiftzeq”isthevalueoftheredshiftsincetheradiation/mattertransition equality tothepresent.ThisredshifthasbeenmeasuredbyWMAPandfoundtobe:zeq 3253 88.Thereforethisnumberincludesallof the redshift that occurred during both thematter dominated epoch and the dark energydominatedepochuntilnow.Thissinglenumberincludesevenacceleratedexpansion.Thereforesincetheredshiftzeqequalsthechangeinscalingfactorsinceequality,wecansimplymultiplythis timesthepreviouslycalculatedvalueofГupriortoradiation/matterequalitytoobtainthetotalvalueofthebackgroundgravitationalГuoftheuniversetoday.WemakeuseofthecontentionthatГu authereforethecurrentvaluesareГuo auo.Гuo Гeqzeq 6.4 1027 3253Гuo 2.1 1031ГuocalculatedfromcosmologicalexpansionThis calculationofГuoandauo simplyextended thestartingassumption theuniverse isonlyspacetime .ThismeansthatwebegintheuniversewithPlanckspacetimeatoneunitofPlancktimeold τu 1 rather thanstartingwithasingularityat τu 0. Thecalculatedexpansionincorporated the radiation dominated epoch, thematter dominated epoch and the so called“darkmatter”epochwhentherewasacceleratedexpansion.Thecalculationdidnotincorporatean “inflation” phase. Inflation is an ad hoc assumption necessitated when the universe ispresumedtostartfromasingularity.Tokeep trackof theexpansion since theuniversewasτu 1,wewill be referencingasourstandardofcomparisonthepropertiesofaPlanckspherewhenτu 1.Forexample,theproperradiusofthePlanckspherewasPlancklengthlpatthattimeandΓu 1.ThecalculatedvalueofГuo 2.1 1031saysthattheproperradiusofthePlanckspheretodayhasincreasedbyafactorofabout2.1 1031times.Therefore,thepropervolumeofthePlanckspherehasincreasedbyafactorofabout1094times 2.1 1031cubed .Гuo Calculation from Planck Temperature: One advantage of the proposed spacetimetransformationmodeloftheuniverseisthatitdescribesthestartingconditionoftheuniversein

4 Five-Year Wilkinson Microwave Anisotropy Probe ∗ Observations: Cosmological Interpretation, E. Komatsu1 et.al. The Astrophysical Journal Supplement Series, 180:330–376, 2009 February


awaythatwecanmakebothpredictionsandplausibilitycalculationsthatcheckthemodel.Thecurrent characteristics of the universe have been experimentally measured. It is only thephysical interpretationof thesemeasurements that isbeingcalled intoquestion. Wecandosimplecalculationsextrapolatingfromtheproposedstartingconditionstoseeifthemodel isreasonable.ThefirstoftheseplausibilitycalculationsdeterminesthevalueofГuobycomparingthetemperatureofPlanckspacetimetothecurrenttemperatureoftheCMB.TheratioofthesetwotemperaturesshouldbeequaltoГuo.Thereasoningisthatthetemperatureoftheuniversechanged inverselywith the scaling factorwhichmeans that the change in temperature alsoscaledinverselywiththechangeinГu.Toexpressthisrelationshipwewillusethefollowingnewsymbolsps temperatureofPlanckspacetime: ps ½Ep/kB 7.08 1031°Kuo currenttemperatureoftheuniverse CMBtemperature uo 2.725°K

Гuo currentvalueofthebackgroundgravitationalgammaoftheuniverse

Гuo 7.08 1031°K/2.725°K 2.6 1031Гuofromratiooftemperatures

Thisisafantasticresult.FromtheratiooftemperaturesweobtainГuo 2.6 1031comparedtoГuo 2.1 1031fromtheredshiftcalculation.Thissupportstheassumptionthatthestartingcondition for the Big Bangwas Planck spacetime and not a singularity or any other energydensityinexcessofPlanckenergydensity.Гuo Calculation from Total Energy Density:Thereisanotherwaythatwecancalculatethevalue of Гuo using the energy density of Planck spacetime and comparing this to the totalobservableenergydensityoftheuniversetoday. Theenergydensityoftheuniversetodayiscommonlythoughttobeabout8.46 10‐10J/m3 equivalentto9.4 10‐27kg/m3 .However,thisisacalculatednumberbasedontheconceptthattheuniversemustmaintainacriticaldensity.Only about 27.9% of this energy density or 2.36 10‐10 J/m3 is “observable” mass/energyconsisting of fermions and bosons including dark matter . Dark matter is considered“observable”becausethegravitationaleffectsofdarkmatterareclearlyobservable. Onlytheobservable 2.36 10‐10 J/m3 is traceable toPlanck spacetime. The remaining approximately72.1%ofthe“critical”energydensity 8.46 10‐10J/m3 iscurrentlyattributedtodarkenergy.Almostallthisdarkenergyhassupposedlybeenaddedtotheuniversesincetheuniversewasabout 5 billion years old. It did not originate with Planck spacetime and therefore will beexcluded from this calculation. Therefore, the current energy density of the universe thatexcludesdarkenergyisabout2.36 10‐10J/m3. Thiswillbecalledthe“currentlyobservableenergydensityoftheuniverse”anddesignatedwiththesymbolUobs.ThereisanotherwayofcalculatingthevalueofГuousingonlyUps,Uobsandzeq.ThereasoningisthattheuniversestartedwithPlanckspacetimewithenergydensityofUps 5.53 10112J/m3andtodaytheobservableenergydensityisUobs 2.36 10‐10J/m3.Therearetworeasonsfor


thischangeinenergydensity.First,thevolumeofeachPlancksphereincreasedbyafactorofГuo3becausetheradiusofthePlancksphereincreasedbyafactorofГuo.Secondly,thecosmicredshiftreducedtheenergyinthePlancksphereduringtheradiationdominatedepoch.ThisepochstartedwithГu 1andendedroughlyattheradiation/matterequalitywithabackgroundgamma designated as Гeq. We do not experimentally know the value of Гeq, but we doexperimentally know the redshift that has occurred since the radiation/matter equality. Aspreviously stated, this redshift value has been measured by WMAP and found to bezeq 3253 88.ThevalueofГuoisГuo ГeqzeqthereforeГeq Гuo/zeq.TheinitialenergydensityofPlanckspacetimewasreducedbothbecauseofexpansion afactorofГuo3 andbecauseofcosmicredshift afactorofГeq .WecannowcalculatethevalueofГuousingthisknowledge.Ups/Uobs Гuo3 ГeqsetГeq Гuo/zeqUps/Uobs Гuo4/zeqГuo Upszeq/Uobs 1/4setUps 5.53 10112J/m3Uobs 2.36 10‐10J/m3zeq 3253Гuo 2.95 1031ГuocalculatedfromUps,UobsandzeqThis is another successful plausibility calculation that supports themodel that the universestartedasPlanckspacetimebecause thisvalueofГuogenerallyagreeswith theprevious twovaluesofГuo.Calculation from Energy Density of the CMB:TheCMBcurrentlyhasenergydensityequalto theenergydensityof2.725°Kblackbodyradiation:UCMB 4.2 10‐14 J/m3. Throughtheentirelifetimeoftheuniverse,theenergydensityinradiationhasscaledproportionalto1/a4.Thereforeifweonlytracktheenergydensityofradiationwehaveanextremelysimplifiedmodeloftheuniverse.Thisassumptionignoresallthefermionsandotherbosonswhichhaveabout5,000timesgreaterenergydensitythantheenergydensityofphotonsintheCMB.Therefore,thisgreatlysimplifiedassumptionrepresentsaroughupperlimitestimateofГuo. WeassumethattheuniversestartedasPlanckspacetimewithenergydensityofUps 5.53 10112J/m3andthrough expansion achieved the current energy density of just the UCMB. The cosmologicalredshiftplusincreasedvolumeresultedina1/a4scalingofenergydensity.Гuo Ups/UCMB 1/4 5.53 10112/4.2 10‐14 1/4 3.4 1031IfinditsurprisingthatthissimplifiedestimatethatincludesonlyradiationachievesarelativelyclosevalueforГuo.Thereasonisthatthegreatestexpansionfactoroccurredwhentheuniversewasradiationdominatedthereforemostoftheenergylossinfactscaledas1/a4.ThisГuovaluewouldmorecloselyapproachthepreviousthreevaluesifweincludedtherelativisticenergyofparticles in the early universe which are similar to radiation because they also exhibit arelativistic“redshift”.


Calculation from Spin: There isone lastcalculationthat I findamazing. It isbasedontheassumptionthatquantizedspinshouldbeapproximatelyconserved.Theterm“quantizedspin”isbestdefinedwithanexample.Twogammarayphotons,eachwithħspin,caninteracttoformanelectronandapositron.Thesetwofermionseachhavespinof½ħ.Theconceptofquantizedspinignoresspindirectionandmakesnodistinctionbetweenħspinand½ħspin.Todeterminethe totalnumberofquantizedspinunitswemerelyadd together thenumberofbosonsandfermions ħand½ħunitsofspinaretreatedthesame .WeknowthedensityofquantizedspinunitsinPlanckspacetimeandweknowthedensityofphotons and fermions in theuniverse today except for darkmatter . The spacetimebasedmodeloftheuniversedoesnotrequiretheexchangeofvirtualparticlestotransmitforcessonoquantizedspinisallottedtovirtualparticles.However,theassumptionthatthereislargescalepreservationofquantizedspinisquestionable.Forexample,twophotonscanbeabsorbedandreemitted as a single photon. However, with black body radiation there is large scalepreservationofthetotalnumberofphotonsbecausethenumerousabsorptionsandreemissionsaverageout.Forthiscalculationtobeaccuratewewillmakethequestionableassumptionthatquantizedspinhasasimilaraveraging.Iftheanswerisreasonable,thenthiswillsupporttheaccuracyofthisassumption.PlanckspacetimehadħspinineachPlanckspherewithinitialvolumeof 4π/3 lp3 1.77 10‐104m3 or a quantized spin density of 5.65 10103 spin units/m3. Todaymatter dominates theuniversebutthequantizedspinunitscontainedinobservablematterissmallcomparedtothequantizedspin in theCMB. Ordinarymatter is4.6%of the “criticaldensity”of theuniverse.Therefore the density of ordinary matter has a density of about 4.3 10‐28 kg/m3. This isequivalenttoabout¼hydrogenatompercubicmeter.Sinceahydrogenatomhas3quarksandoneelectron,¼hydrogenatompercubicmeterisequivalenttoabout1quantizedspinunitpercubicmeter.Recallthatthespacetimebasedmodelhasnovirtualphotons,gravitonsorgluons.Iftherereallyaregravitons, etc. then this calculation shouldbewrongbymanyordersofmagnitude. Thequantized spin in the photons of the CMB dominate baryonic matter because the 2.725°KblackbodyCMBphotondensityis4.2 108photons/m3comparedtoabout1fermion/m3forordinarymatter.Thephotondensityofstarlightisalsoconsideredinsignificantbecauseithasbeenestimated6thatthedensityofstarlightphotonsislessthan1%ofthephotondensityoftheCMB.Wewillalsotemporarilyignorethespininneutrinosanddarkmatter.Therefore,withtheseexceptionsthequantizedspinoftheCMBdominatestheuniverse.IfeachPlanckspherecontainedoneunitofquantizedspininPlanckspacetimeandiftherewaspreservationofquantizedspin,thenwewouldexpectthattheexpandedPlanckspherewouldstillonaveragecontainoneunitofquantizedspin.Thebestestimateofthecurrentproperradius

6 Paul Davies, 1982 The Accidental Universe, Cambridge University Press ISBN 0-521-24212-6


ofthisPlancksphereis:lpГuo lp 2.6 1031 4.2 10‐4moravolumeofabout3.1 10‐10m3.ThisestimateusesthevalueofГuoobtainedfromtheratiooftemperatures Гuo 2.6 1031 .Thisnumber represents themiddleof the rangeofvaluesand isobtained from the simplestcalculation.Therefore,assumingspinpreservationwewouldexpectthatthecurrentdensityofquantizedspinunitswouldbe3.2 109/m3.ItappearsasiftheCMBphotondensityof4.2 108photons/m3 is low by a factor of about 7.6. However this is actually very good agreementconsideringthatthevolumehasexpandedbyafactorofabout1094from:Гuo3 2.6 1031 3 1094.Infact,comparedto1094,thissmallfactorcouldbeexplainedbynotincludinganumericalfactornear1suchas2π.Dark Matter Calculation:Eventhoughweshouldbesatisfiedwithanagreementthatisquiteaccurateconsideringthatitcoversarangeof1094,westillhavenotaccountedforneutrinosanddarkmatter.Supposethatweassumethatstatisticallythereisvirtuallyprefectpreservationofquantizedspin.Thisassumptiongivesustheopportunitytoestimatetheenergyofadarkmatterparticle rotar . Using 4.2 108 photons/m3 and an expanded Planck sphere volume of3.1 10‐10m3/spherewehave:4.2 108photons/m3 3.1 10‐10m3/sphere 0.13photons/sphereAssumingspinpreservationandГuo 2.6 1031,thereshouldbe1spinunitperspherethereforewearemissing0.87spinunits/sphere. Theaveragedensityofdarkmatterintheuniverseisabout2.2 10‐27kg/m3oranenergydensityof1.9 10‐10J/m3.ThecurrentvolumeofaPlancksphereis3.1 10‐10m3,thereforecombiningthesewehave:1.9 10‐10J/m3 3.1 10‐10m3/spinunit 1/0.87 6.8 10‐20J/spinunitTheimpliedenergyperdarkmatterfundamentalparticleis6.8 10‐20J 0.4eV.Perhapsthisisacoincidence,butoutofthemanyordersofmagnitudeinvolvedinthiscalculation,theanswerofabout0.4eVisintheenergyrangecurrentlyattributedtoneutrinosanditisfarremovedfromthe 1010eVenergyrangeassumedforthehypotheticalWIMPdarkmatterparticlemodel.Forexample,aphotometricredshiftsurvey7hassetanupperlimitof0.28eV/c2onthesumofthemassesofthethreetypesofneutrinosknowntoexist.AnanalysisofthePlanckspaceprobedatahassetanupperlimit 0.23eV/c2 onthismasssum8.Allowingforthespeculativenatureofthisresult,thissurprisingresultwarrantsanewexaminationofdarkmattercandidates.Can Neutrinos Be Dark Matter?Neutrinoshavebeendiscountedaspossibleexplanationsofdarkmatter because they are considered to be “hot darkmatter” propagating at velocity in

7 Thomas, S.A.; (2010). "Upper Bound of 0.28 eV on Neutrino Masses from the Largest Photometric Redshift Survey". Physical Review Letters 105 (3): 031301.arXiv:0911.5291 8 Planck Collaboration, "Planck 2013 results. XVI. Cosmological parameters". Astronomy & Astrophysics 1303: 5076.arXiv:1303.5076


excessof0.1c.Theyhavesuchlowrestmassthatevenatatemperatureofabout1°K,theywouldhaveavelocitymuchgreaterthantheescapevelocityofagalaxy.Forexample,theMilkyWaygalaxyhasanescapevelocityofroughly600,000km/s.Also,intheearlyuniverseahighdensityofultra‐relativisticneutrinoswouldexcessivelyhomogenizetheCMBdistribution. However,the leading candidate for dark matter a WIMP also has problems. For example, thehypothetical fundamental particle of a WIMP ~1010 to 1012 eV must not exhibit anyelectromagnetic or strong interaction. Therefore these hypothetical particles would notexperience collisions with baryonic matter except through the unlikely event of a weakinteraction.Mostimportant,theWIMPscannotlosekineticenergybyEMradiationtoformastablegravitationallyboundorbit.ThinkabouttherequiredpropertiesofaWIMPparticle.Itmusthavesuchahighprobabilityofbeingformedthatintheearlyuniverse,thatover80%oftheenergythatwasdestinedtobecomeafermionwouldhavehadtogointotheformationofthesehighlyenergeticparticles.Inallotherrespects, the lowest energyparticles are themost stable andhave thehighestprobabilityofsurvivingtotoday.IfaWIMPisdarkmatter,thenthistrendshouldbereversed.Ahighenergyislandofstabilitywouldhavetoexistandover80%oftherestmassintheuniversewouldhavetocurrentlyexistasWIMPs.Nowlookat theeasewithwhichneutrinosareproduced. Theyare formedwiththebirthofalmosteverynewparticle.Theyarethelowestenergyfermionknownandtheyareknowntohave long term stability. Also, the lowmass/energy of neutrinosmore easily forms amassdistributionaroundagalaxythatischaracteristicofthedarkmatterspherical“halo”providedthattheycansomehowbeslowedtofitthedefinitionofcolddarkmatter.However,aplausiblemethodforthemtolosekineticenergyandcooltolessthanabout0.01°Kmustbesuggested.Neutrinosat2.7°Kwouldgreatlyexceedtheescapevelocityfromgalaxies.However,neutrinosatatemperatureintherangeof0.01°Kwouldhavetherequiredproperties.Neutrinoshavethefollowingadvantages:1 Theyareknowntoexist,2 Theyareknowntobeproducedinlargenumbersintheearlyuniverseand3 Theyareknowntoonlyinteractwithordinarymatterby theweak force. Using theknownpropertiesofneutrinos, thedensityofneutrinoshasbeenestimated9atabout3/11oftheCMBphotondensityperspecies.Thereforethe 3 species of neutrinos would have a total density of about 82% of the photon density~ 3.4 108/m3 for 3 species . This is not quite the density required to satisfy themissingquantizedspin requires~2.8 109/m3 ,butitiswithinanorderofmagnitude.Eventhoughthisisnotenough,theknownformationmechanismsmakethemthemostabundantfermionintheuniverse.A rotarmodel of a neutrinowith internal energy of 5 10‐20 J about 0.3 eV would have aquantumradiusof ħc/Ei 6.3 10‐7moraquantumvolumeofabout10‐18m3.Theenergy

9 J Rich; Fundamentals of Cosmology (second edition 2009); Springer, Heidelberg, New York


densityofa0.3eVrotarwouldbeabout1024timessmallerthantheenergydensityofanelectron.Theenormousdifference inenergydensityofneutrinoscompared toanyother fundamentalparticlesuggests thepossibility thatperhapsneutrinosalsopossessothercharacteristicsnotobservedwithotherfundamentalparticles.Neutrinoflavoroscillationhasbeenwelldocumentedexperimentally.However,itisdifficulttoexplainhowneutrinoscanchangetheirrestmassiftheyarevisualizedaspropagatingthroughanemptyvacuum.Theusualexplanationinvokestheneutrinosexistingsimultaneouslyinthreestatesthatdifferslightlyinmass/energy.ThesethreestateswouldhavethreeslightlydifferentdeBrogliefrequencieswhentheyaremovingrelativetoanobserver.Thethreefrequenciescanthenconstructivelyanddestructivelyinterferewitheachotherproducingwhatisachangeinneutrino flavor mass . However, this explanation has problems. It is difficult to devise anexplanationwhichallowsforachange inrestmasswhilemaintainingbothaconservationofenergyandaconservationofmomentum.I would like to propose a new consideration which is best explained by an example. The so called “MSW effect” occurs when neutrinos pass through matter such as a star and achieve a resonance with electrons. This is analogous to introducing a different index of refraction for each of the 3 neutrino flavors and results in an increase in flavor oscillation frequency. The mystery of neutrinos changing their mass ceases if we postulate an interaction with other particles because conservation laws hold if other particles are introduced.

Suppose that we assume that there is some mechanism by which neutrinos can be cooled to a temperature less than 0.01 degrees K. Then neutrinos would exhibit the properties of dark matter. At the earth’s distance from the galactic center, the energy density of dark matter is about 3x1014 eV/m3. If we assume neutrinos to have energy of about 0.3 eV, then this works out to about 1015 neutrinos/m3 at the earth. With this density, the experiments conducted on earth are not measuring the flavor oscillation rate in a total vacuum. Our best vacuum would still contain about 1015 neutrinos/m3. The interesting point is that an interaction between neutrinos would produce acceleration and deceleration of fermions with rest mass. This would generate gravitational waves and cooling of neutrinos. Perhaps this is an over simplification, but a flavor oscillation which involves an interaction between neutrinos with different energy resulting in a slight mass change would involve acceleration and deceleration of matter. This should produce a small gravitational wave. If this happens, it would represent a loss of kinetic energy relative to the cosmic microwave background (CMB) rest frame of reference. Neutrinos which originated at the Big Bang could possibly be cooled by gravitational wave emission to a temperature far below the temperature of the CMB photons. If this additional cooling occurs to an extent that neutrino temperature is lowered to less than about 0.01 degree K, then neutrinos would qualify as being cold dark matter. Summary of the Гuo Calculations: We took a deviation in the discussion of darkmatter.However, nowwewill return to the earlier discussion of the Гuo calculation. We have now


calculatedthevalueofГuoseveraldifferentwaysandtheyallgiveaboutthesameanswer.Theclosenessoftheresultsobtainedwithverydifferentapproachessupportsthemodelbeingused.WewillusethevalueГuo 2.6 1031asrepresentativeoftherangeofvaluescalculatedhereforfuturecalculations.Usingthisvalue,todayeachPlanckspherehasexpandedtoaproperradiusoflp Гuo 0.42mm.However,measuredincoordinateunitswhichpresumeГu 1,thePlanckspherestillhasaradiusof 1.Italsoappearstostillhaveonaverage1unitofquantizedspin.TheanalysisofthespinperPlanckspherewillbemadeinthenextchapter.

Graph Showing the Evolution of Гu and Observable Energy: Figure 13‐4 shows twosuperimposedgraphsthatusethesametimelinebutdifferentYaxisscales.NotefirstthattheXaxisisalogscaleoftheageoftheuniverseinseconds.Thetimescaleextendsfrom5 10‐44swhich is1unitofPlanck time to1020secondswhich isanageofabout3 trillionyears. Thepresentageoftheuniverseisabout4.3 1017seconds ~13.8 109years andisdesignatedwithanarrowandaverticaldashedline.ThegraphlinedesignatedГuisthevalueofthebackgroundgravitationalgammaoftheuniverseГu .ThisgraphusestherightscalewhichisalogscaleextendingfromГu 1toaboutГu 1032.It canbeseen from thisgraph thatГu 1at a timeof τu 5 10‐44 secondsandendswithГu 2.6 1031atthepresent τu 4.3 1017s .Notethatthereisaslightchangeintheslopeof


thislineasitcrossestheverticallinedesignatedR‐Mtransition.Thisisthetransitionbetweenaradiationdominateduniverseandamatterdominateduniversethatoccurredabout70,000yearsafterthebeginningoftheuniverse theBigBang . Thereshouldalsobeanotherslopechangeatthetransitionbetweenthematterdominatedepochandthedarkenergydominatedepoch.However,thisslopechangeissosmallthatitdoesnotshowuponthislog‐loggraph.Theothergraph designated “Joules” uses the left scaleand is theobservableenergy in thePlancksphere alsoalogscale .AspherePlancklength lp inradiusstartedwithabout1billionJouleswhentheuniversewasPlanckspacetimeforthefirstunitofPlancktime.OvertheageoftheuniverseasГuincreased,theproperradiusofthissphereincreasedandequaledГulp.TheproperenergyofthisimaginaryPlanckspheredecreasedduringtheradiationdominatedepochby a factor of 6.4 1027 from about 109 J to about 1.5 10‐19 J and then the proper energygenerallyhasremainedconstantsincetheendoftheradiationdominatedepoch.Therefore,notetheflatlinefromanageofabout70,000yearstothepresent.However,theenergyonanabsolutescalewhereГu 1continuestodecreaseasГuincreases.Forexample,thedecreasingrateoftimemakesitappearthattheComptonfrequencyoffundamentalparticlesisconstant.However,ifwetimedtheComptonfrequencyofanelectronusingtheГ 1clock,wewouldfindthatonthisabsolutetimescale,eventheComptonfrequencyofanelectronwouldbeslowingdowneachsecond.Itisnotnoticeablebecauseourcosmicclockisslowingdownatthesamerate.

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