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Chapter8

GravitationalAttractionandUnificationofForcesInchapter6,thereaderwasaskedtotemporarilyconsiderallforcestoberepulsive.Thiswasasimplificationwhichallowedthecalculations inchapter6 toproceedwithoutaddressing themorecomplicatedsubjectofattraction.Inchapter7,vacuumenergy/pressurewasintroducedasanessentialconsiderationinthegenerationofallforces,butespeciallyforcesthatproduceattraction. In this chapter we are going to attempt to give a conceptually understandableexplanationfortheforceofattractionexertedbygravitywhenabodyisheldstationaryrelativetoanotherbody.Physical Interpretation of General Relativity: Einstein’s general relativity has passednumerousexperimentalandmathematicaltests.Thismathematicalsuccesshasconvincedmostphysicists to accept the physical interpretation usually associated with these equations.However, the most obvious problem with the physical interpretation is examined in thefollowing quotes. The first is from B. Haisch of the California Institute of Physics andAstrophysics.“The mathematical formulation of general relativity represents spacetime as

curvedduetothepresenceofmatter….Geometrodynamicsmerelytellsyouwhatgeodesicafreelymovingobjectwillfollow.Butifyouconstrainanobjecttofollowsomedifferentpath ornot tomoveatall ,geometrodynamicsdoesnot tellyouhoworwhyaforcearises….Logicallyyouwinduphavingtoassumethataforcearisesbecausewhenyoudeviatefromageodesicyouareaccelerating,butthatisexactlywhatyouaretryingtoexplaininthefirstplace:Whydoesaforcearisewhenyouaccelerate?…Thismerelytakesyouinalogicalfullcircle.”

Talkingaboutcurvedspacetime,thebookPushingGravity M.R.Edwards states:“Logically,asmallparticleatrestonacurvedmanifoldwouldhavenoreasontoend

itsrestunlessaforceactedonit.Howeversuccessfulthisgeometricinterpretationmaybeasamathematicalmodel,itlacksphysicsandacausalmechanism.”

Generalrelativitydoesnotexplainwhymass/energycurvesspacetimeorwhythereisaforcewhenanobjectispreventedfromfallingfreelyinagravitationalfield.Ifrestraininganobjectfrom followingageodesic is theequivalentof acceleration, thenapparently thegravitationalforceisintimatelytiedtothepseudoforcegeneratedwhenamassisaccelerated.Inthestandardmodel,particlespossessnointrinsicinertia.TheygaininertiafromaninteractionwiththeHiggsfield.IstheHiggsfieldalsonecessarytogenerateagravitationalforcewhenaparticlewithout

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intrinsicinertiaispreventedfromfollowingthegeodesic?IstheHiggsfieldalwaysacceleratingtowards amass in an endless flow that attempts to sweep along a stationary particle? Thestandardmodeldoesnotincludegravity.Isgravityaforce?Thepointofthesequestionsistoshowthattherearelogicalproblemswiththephysicalmodelnormally associated with general relativity. The equations of general relativity accuratelydescribegravityonamacroscopicscale.However,theseequationsaresilentastothephysicalinterpretation,especiallyatascale spatialortemporal wherequantummechanicstakesover.Gravitational Nonlinearity Examined: Previouslywe reasoned that spacetimemust be anonlinearmedium for waves in spacetime and gravity is the result of this nonlinearity. Atdistance fromarotarwecalculatedthegravitationalforceusingoneofthe5wave‐amplitudeequationsF A2ω2Z /c.InthiscalculationwesubstitutedA Aβ2 Lp/ 2 Tpωc 2.Alsotheangularfrequencyωisequaltotherotar’sComptonfrequencyω ωc.Atdistance oftwoofthesamerotarsweobtainedF Gm2/ .Thisisthecorrectmagnitudeofthegravitationalforce between two rotars of mass m, but there are two problems. First: the equationF A2ω2Z /c is for a traveling wave striking a surface. This traveling wave implies theradiationofpowerthatisnothappening.Second:awaveinspacetimetravelingatthespeedoflightgeneratesarepulsiveforceifitinteractsinawaythatthewaveisdeflectedorabsorbed.Gravity isobviouslyanattractive force. Wehave themagnitudeof the forcecorrect,but themodelmustberefinedsothatthereisnolossofpowerandsothattheforceisanattraction.Thereareseveralstepsinvolved,anditisprobablydesirabletobeginwithabriefreview.Recallthatwearedealingwithdipolewavesinspacetimewhichmodulateboththerateoftimeandvolume. There are twoways thatwe can express the amplitude of thedipole in spacetime:displacement amplitude and strain amplitude. The maximum displacement of spacetimeallowed by quantum mechanics is a spatial displacement of Planck length or a temporaldisplacementofPlancktime.Sincetheseareoscillationamplitudes,wesometimesusetheterm“dynamic Planck length Lp” or “dynamic Planck time Tp”. As previously explained, thesedistortionsofspacetimeproduceastraininspacetime.ThestrainisadimensionlessnumberequivalenttoΔl/lorΔt/tInthiscaseΔl/l Lp/ andΔt/t Tpωc.Inchapter5weimaginedahypotheticalperfectclockplacedatapointonthe“Comptoncircle”ofarotarasillustratedinfigure5‐1.Thisistheimaginarycirclewithradiusequaltotherotarradius .Thisclock hereaftercalledthe“dipoleclock”withtimeτd wascomparedtothetimeonanotherclockthatwecalledthe“coordinateclock” withtimetc . Thiscoordinateclockismeasuringtherateoftimeiftherewasnospacetimedipolepresent.Itisalsopossibletothinkofthecoordinateclockaslocatedfarenoughfromtherotatingdipolethatitdoesnotfeelany

significanttimefluctuations.Figure5‐3showsthedifferenceintheindicatedtimeΔt τd‐tc.Thedipoleclockspeedsupandslowsdownrelativetothecoordinateclockandthemaximumdifference is dynamicPlanck timeTp. ThereforeTp is the temporaldisplacement amplitude.

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ThereisalsospatialdisplacementamplitudethatisequaltodynamicPlancklengthLp.Thestrainofspacetimeproducedbythesedisplacementsofspacetimeisdepictedinfigure5‐4.Withintherotarvolume,thestrainofspacetimeisdesignatedbythestrainamplitudeAβ Tpωc Lp/ .Thisisequivalenttothemaximumslopeofthesinewavewhichoccursatthezerocrossingpointsinfigure5‐3.Nonlinear Effects: The above review now has brought us to the point where we can askinterestingquestions:Doesthedipoleclockalwaysreturntoperfectsynchronizationwiththecoordinate clock at the completion of each cycle? Does the volume oscillation of spacetimeproduce a net change in the average volume near a rotar? Since we are going to initiallyconcentrateonexplainingthegravitational forcebetweentwofundamentalparticles,wewillinitiallyconcentrateontheeffectontime.Therefore,doestherateoftimeoscillationcausethedipoleclocktoshowanetlossoftimecomparedtothecoordinateclock?Ifspacetimehasnononlinearity then the clockswould remain substantially synchronized. However, if there isnonlinearity,thedipoleclockwouldslowlylosetime.Aspreviouslyexplained,thestrainofspacetime instantaneousslopeinfigure5‐3 hasalinearcomponentandanonlinearcomponent.Theproposedspacetimestrainequationforapointontheedgeoftherotatingdipoleis:Strain Aβsinωt Aβsinωt 2… higherordertermsignored Thelinearcomponentis“Aβsinωt”andthefirstterminthenonlinearcomponentis Aβsinωt 2.TherewouldalsobehigherordertermswhereAβisraisedtohigherpowers,butthesewouldbesosmallthattheywouldbeundetectableandwillbeignored.Thisnonlinearcomponentcanbeexpanded:Aβsinωt 2 Aβ2sin2ωt ½Aβ2–½Aβ2cos2ωt

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Figure 8‐1 plots the linear component Aβ sinωt and the nonlinear component Aβ sinωt 2separately.ItcanbeseenthatthenonlinearcomponentisasmalleramplitudebecauseAβ 1andsquaring thisproducesa smallernumber. Also thenonlinearcomponent isat twice thefrequency of the linear component. Most importantly, the nonlinear component is alwayspositive.Makinganelectricalanalogy,thiscanbethoughtofasifthenonlinearwavehasanACcomponentandaDCcomponent. It isobviousthatwhenthe linearandnonlinearwavesareadded together, the sumwill produce an unsymmetricalwave that is biased in the positivedirection.Itwasnecessarytousesomeartisticlicenseinordertoillustratetheseconceptsinfigure8‐1.ForfundamentalrotarsthevalueofAβisroughlyintherangeof10‐20.ThismeansthatAβ2 10‐40

andthereforeAβisapproximately1020timeslargerthanAβ2.ItwouldbeimpossibletoseetheplotofAβ2sin2ωtwithoutartificiallyincreasingthisrelativeamplitude.Therefore,theassumedvalueinthisfigureisAβ 0.2.InthiscasethedifferencebetweenAβandAβ2isonlyafactorof5ratherthanafactorofroughly1020.Therefore,forfundamentalrotarsitisnecessarytomentallydecreasetheamplitudeofthenonlinearwavebyroughlyafactorofroughly1020.

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Whenweaddthetwowavestogetherweobtaintheplotinfigure8‐2.Becauseoftheartisticlicense,itisvisuallyobviousthatthisisanunsymmetricalwave.Thereisalargerareaunderthepositiveportionofthewavethantheareaunderthenegativeportionofthewave. Thepeakamplitude for the positive portion is Aβ Aβ2while the negative portion has peak negative

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amplitude of: – Aβ Aβ2. If this was a plot of electrical current, we would say that thisunsymmetricalwavehadaDCbiasonanACcurrent.Tousetheanalogyfurther,itisasifthenonlinearitycausesspacetimetohavetheequivalentofasmallDCbiasinitsstress.Thedipoleclockdoesnotreturntosynchronizationwiththecoordinateclockeachcycle.Figure8‐3 attempts to illustrate thiswith a greatly exaggerated plot of the difference between thecoordinate clockand thedipole clock. The “X” axisof this figure is timeas indicatedon thecoordinateclockwhilethe“Y”axisisthedifferencebetweenthecoordinateclockandthedipoleclock t–τ .Ifthetwoclocksranatexactlythesamerateoftime,theplotwouldbeastraightlinealongthe“X”axis.Normallythisplotforafewcyclesshouldlooklikeasinewavesimilartofigure5‐3.However,thepurposeoffigure8‐3istoillustratethatovertimethecoordinateclockpulls ahead of the dipole clock or the dipole clock loses time . Therefore, for purposes ofillustration,thiseffectoftheaccumulatedtimedifferencehasbeenexaggeratedbyafactorofroughly1022.The unsymmetrical strain plot in figure 8‐2 produces a net loss of time on the dipole clockrelativetothecoordinateclock.Inthefirstquartercycleoffigure8‐3,thecoordinateclockfallsbehindthedipoleclockbyanamountapproximatelyequaltoPlancktime.Thisoccurswhenthefast lobe of the rotating dipole passes the dipole clock first. However, with each cycle, thecoordinateclockgainsasmallamountoftimeonthedipoleclock.Theamountoftimegainedpercycleisillustratedbythegaplabeled“Singlecycletimeloss”.ThisisequaltoTp2ωcwhichisabout2.2 10‐66sforanelectron.Thepointof figure8‐3 is to illustratethecontributionof thenonlineareffect. ThenonlinearwavewithstrainofAβ2sin2ωtatdistance producesthecontributionthatcausesthenetlossoftimeforthedipoleclockrelativetothecoordinateclock.Thisnettimedifferencebetweenthetwoclocks aftersubtractingAβsinωt isshownasthewavylinelabeled“nonlinearcomponent”.TheaverageslopeofthislineisequaltothegravitationalmagnitudefortherotarvolumewhichhasbeendesignatedasAβ2 βq.Foranelectronthisslopeisabout1.75x10‐45whichmeansthatittakesabout30secondsforthecoordinateclocktohaveanettimegainofPlancktimeoverthedipoleclock.Thistakesabout4x1021cyclesratherthan4cyclesasillustratedinfigure8‐3.Ifwesubtractedthenonlinearwavecomponentfromfigure8‐3,wewouldbeleftwithasinewavewithamplitudeofTp.TheslopeonthisnonlinearwavecomponentisAβ2atdistance whichisobtainedfromthestrainequation–theimportantpartishighlightedbold

Aβsinωt Aβsinωt 2 Aβsinωt–½Aβ2cos2ωt ½ Aβ2

Derivation of Curved Spacetime: TheDCequivalentterm non‐oscillatingterm isAβ2.Thisisthenonlinearstrain inspacetimeproducedby therotaratdistance . This isan important

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conceptsinceitrelatestocurvedspacetime.Beforeperusingthisthoughtfurtheritisnecessarytointroduceanewsymbol: . Thenaturalunitoflengthforarotarisλc. Thereforewewilldesignatetheradialdistancefromarotarnotinunitsoflengthsuchasmeters,butasthenumberofreducedComptonwavelengths numberofrotarradiusunits .

≡ ħ

For example, the non‐oscillating strain in spacetime produced by a rotar should decreaseproportionalto1/ .Wecantestthisideasincetheproposalisthatthenon‐oscillatingstraininspacetimeatdistance λc where 1 isequal toAβ2andthisdecreasesas1/ . WewillevaluateAβ2/ andcallthisthegravitationalamplitudeAg.

Ag

Therefore we have succeeded in producing the previously discussed weak gravity gravitational magnitude β ≈ Gm/c2r. This is the curvature of spacetime that we associate with gravity.ThissimpleevaluationisanothersuccessofthismodelbecausethetermGm/c2r istheweakgravitycurvatureofspacetimeproducedbymassmatdistancedr.Forexample,thepreviouslydefinedgravitationalmagnitude isβ≡1– dτ/dt . Theweakgravity temporaldistortionofspacetimeis:dt/dτ 1 Gm/c2r .Inflatspacetimedt/dτ 1,sotheweakgravitycurvaturetermisβ Gm/c2r .Forfundamentalparticles rotars atdistance thistermisintherangeof10‐40,sothisisvirtuallyexact.The question of how matter “causes” curved spacetime has been a major topic in generalrelativityandquantumgravity.Nowweseethemechanismofhowdipolewavesinspacetimeproducebothmatterandcurvedspacetime.Thisusesequationsfromquantummechanicstoderiveanequationfromgeneralrelativity. Thisisnotonlyasuccessfultestofthespacetimebasedmodel,but it isalsoapredictionof thismodelof themechanismthatachievescurvedspacetime.Oscillating Component of Gravity: There is proposed to be another residual gravitationaleffectthathasnotbeenobservedbecauseitisaveryweakoscillationatafrequencyinexcessof1020Hz.Infigure8‐3thenon‐linearwavecomponentisshownasawavylinelabeled“nonlinearcomponent”.Wecaninteractwiththenon‐oscillatingpartofthislineresponsibleforgravity,but there is also a residual nonlinear oscillating component. At distance this oscillatingcomponenthasamplitudeAβ2andfrequency2ωc.Whathappenstothisoscillatingcomponentbeyond intheexternalvolume?Weknowthatthefewfrequenciesthatformstableandsemi

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stable rotars exist at resonanceswith the vacuum fluctuationsof spacetimewhich eliminateenergy loss. If the amplitude of the oscillating componentwasAβ2/ , then therewould becontinuous radiation of energy. Energetic composite particles such as protons or neutronswouldradiateawayalltheirenergyinafewmillionyears.Inthechapter10ananalogywillbemade to theenergydensityof a rotar’s electric field. Theamplitude term for theoscillatingcomponentofgravitywillthenbeproposedtoscaleasAβ2/ 2.Thiswouldbeanextremelysmallamplitudeand furthermore it isastandingwave thatdoesnotradiateenergy.However, thisoscillatingcomponentwouldtheoreticallygiveenergydensitytoagravitationalfield.Theenergydensity of a gravitational field and its contribution to producing curved spacetime will bediscussedattheendofchapter10.Theoscillatingcomponentofagravitationalfieldmayalsobeimportantintheevolutionoftheuniverse.Thiswillbediscussedinchapters13and14.Summary: Since we need to bring together several different components to achievegravitational attraction,wewill doanother review. This timewewill emphasize the roleofvacuumenergy,circulatingpower,thecancelingwaveandnon‐oscillatingstraininspacetime.Arotarisarotatingspacetimedipoleimmersedinaseaofvacuumenergywhichisequivalenttoa vacuum pressure. This vacuum energy/pressure is made up of very high energy density1046J/m3 dipolewavesinspacetimethatlackangularmomentum.Therotaralsohasahigh

energydensitythat isattemptingtoradiateawayenergyat therateof therotar’scirculatingpower.Therotarsurvivesbecauseitexistsatoneofthefewfrequenciesthatachievearesonancewiththevacuumenergy/pressure.Thisresonancecreatesanewwavethathasacomponentthatpropagatesradiallyawayfromtherotating dipole and a component that propagates radially towards the rotating dipole.Tangential wave components are also created, but these add incoherently and effectivelydisappear. The resonant wave that is propagating away from the dipole cancels out thefundamental radiation from the dipole. Besides having the correct frequency and phase toproducedestructive interference, the cancelingwave alsomustmatch the rotar’s circulatingpower. Thismeansthatthecorrectpressureisgeneratedfromthevacuumenergy/pressurethatisrequiredtocontaintheenergydensityoftherotar.Onlyafewfrequenciesthatformstablerotarscompletelysatisfytheseconditions.Forexample,anelectronhasacirculatingpowerofabout64millionwatts.Inordertocancelthismuchpowerfrombeingradiatedfromtherotarvolume,thecancelationwavegeneratedinthe vacuum energy must have an outward propagating component of 64 million wattsattemptingtoleavetherotar’svolumeandaninwardpropagatingcomponentofthesamepower.Therecoilfromtheoutwardpropagatingcomponentprovidesthepressurerequiredtostabilizetherotatingdipole that is therotar theelectron . Thispressurecanbe thoughtofasbeingcarriedbytheinwardpropagatingcomponentthatreplenishestherotatingdipole.

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Ifitwaspossibletoseethisprocess,wewouldnotseeoutwardorinwardpropagatingwaves.Wewouldonlyseethesumofthesetwowaveswhichisastandingwavewhichdecreasesinamplitudewithdistancefromthecentralrotar.Astandingwaveintherotar’sexternalvolumemeans thatnopower isbeing radiated.These standingwaves cause the rotar’s electric fielddiscussedlater . Wewouldalsoseethattherewasaslightnon‐oscillatingresidualstraininspacetimewithstrainamplitudeofAβ2/ gm/c2r. Newtonian Gravitational Force Equation:Therearestilltwomorestepsbeforewearriveattheexplanationthatgivesthecorrectattractingforceatarbitrarydistancebetweentworotars.Wewill startbyassuming twoof the same rotars massm separatedbydistance r. Itwaspreviously explained that deflecting all of a rotar’s circulating power generates the rotar’smaximumforceFm.Arotaralwaysdependsonthepressureofthespacetimefieldtocontainitscirculatingpower.Whentherotarisisolated,theforcerequiredtodeflectthecirculatingpowerisbalanced.However,agravitationalfieldproducesagradientinthegravitationalmagnitudedβ/dr.When a first rotar is in the gravitational field of a second rotar, there is a gradient dβ/dr that exists across the rotar radius of the first rotar. This means that there is a slight difference in the force exerted by vacuum energy/pressure on opposite sides of the first rotar. This difference in force produces a net force that we know as the force of gravity.Thiswillberestatedinadifferentwaybecauseofitsimportance.Imaginemassm1beingarotarrotatingdipole attemptingtodispersebutbeingcontainedbypressuregeneratedwithinthevacuumenergy/pressurepreviouslydiscussed.Thispressureexactlyequalsthedispersiveforceof the dipole wave rotating at the speed of light. However, if there is a gradient in thegravitationalmagnitudedβ/drthenthereisagradientacrosstherotarwhichwewillcallΔβ.Thisaffectsthenormalizedspeedoflightandthenormalizedunitofforceonoppositesidesoftherotar.Recallfromchapter3wehad:Co ГCgnormalizedspeedoflighttransformationFo ГFgnormalizedforcetransformationГ 1 βapproximationconsideredexactforrotarsTherefore,becauseofthestraininspacetime,thetwosidesoftherotar separatedby arelivingunderwhatmightbeconsideredtobedifferentstandardsforthenormalizedspeedoflightandnormalizedforce.Onanabsolutescale,ittakesadifferentamountofpressuretostabilizetheoppositesidesoftherotarbecauseofthegradientΔβacrosstherotar.Thenetdifferenceinthisforceistheforceofgravityexertedontherotar.WewillfirstcalculatethechangeingravitationalmagnitudeΔβacrosstherotarradius ofarotarwhen it is in thegravitational fieldof another similar rotar another rotarof the same

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mass .Inotherwords,wewillcalculatethedifferenceinβatdistanceranddistancer fromarotarofmassm.

Δβ approximationvalidif r

Theforceexertedbyvacuumenergy/pressureonoppositehemispheresoftherotarisequaltothe maximum force Fm m2c3/ħ. The difference in the force absolute value exerted onoppositesidesoftherotaristhemaximumforcetimesΔβ.Therefore,theforcegeneratedbytworotarsofmassmseparatedbydistanceris:

F ΔβFm ħ

ħ

ħ

Ifwehavetwodifferentmassrotars massm1andmassm2 ,thenwecanconsidermassm1inthegravitationalfieldofmassm2.Inthiscase, andFmareformassm1andΔβischangeinthegravitationalmagnitudefrommassm2acrosstherotarradius frommassm1.

Fg ΔβFm

ħ

ħ

ħ

Fg Newtoniangravitationalforceequationderivedfromadipolewavemodel

Gravitational Attraction:WehavederivedtheNewtoniangravitationalequationfromstartingassumptions,butwestillhavenotshownthatthisisaforceofattraction.However,fromthepreviousconsiderations,thislaststepiseasy.ThereisaslightlydifferentpressurerequiredtostabilizetherotardependingonthelocalvalueofβorГ inweakgravityГ 1 β .Thiscanbeconsideredasadifferenceinnetforceexertedbyvacuumenergyonthehemisphereoftherotarthatisfurthestfromtheotherrotarcomparedtothehemispherethatisnearesttheotherrotar.The furthesthemispherehasasmalleraveragevalueofГ than thenearesthemisphere. Thenormalizedspeedoflightisgreaterandthenormalizedforceexertedonthefarthesthemispheremustbegreatertostabilizetherotar.ThisproducesanetforceinthedirectionofincreasingГ.ThemagnitudeofthisforceisF Gm1m2/r2andthevectorofthisforceisinthedirectionofincreasingГ towardstheothermass .Weconsiderthistobeaforceofattractionbecausethetworotarswanttomigratetowardseachother increasingГ . However,theforceisreallycomingfromthevacuumenergyexertingarepulsivepressure. There isgreaternormalizedpressurebeingexertedonthesidewith thelower Г. The two rotars are really being pushed together by a force of repulsion that isunbalanced.

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CorollaryAssumption:The force of gravity is the result of unsymmetrical pressure exerted on a rotar by vacuum energy. This is unbalanced repulsive force that appears to be an attractive force.Example: Electron in Earth’s Gravity:Wewilldoaplausibilitycalculationtoseeifweobtainroughlythecorrectgravitationalforceforanelectronintheearth’sgravitationalfieldbasedonthe above explanation. We will be using values for the electron’s energy density and theelectron’smaximumforcethatwerepreviouslycalculatedbyignoringdimensionlessconstants.Therefore, we will continue with this plausibility calculation that ignores dimensionlessconstants. An electron has internal energy of Ei 8.19x10‐14 J and a rotar radius of 3.86x10‐13m. Ignoring dimensionless constants, this gives an energy density of about

Ei/ 1.4x1024J/m3.Thisrotarmodelofanelectronisexertingapressureofroughly1.4x1024N/m2.Thispressureoverareaλc2producestherotar’smaximumforcewhichforanelectronisFm 0.212N obtainedfromℙq 1.4x1024N/m2 3.86x10‐13m 2 0.212N .Theweakgravitygravitationalmagnitudeis:β Gm/c2r.Fortheearthm 5.96x1024kgandtheequatorialradiusis:r 6.37x106m.Therefore,atthesurfaceoftheearththegravitationalmagnitudeis:β 6.95x10‐10.Toobtainthegradientinthismagnitudewedividebytheearth’sequatorial radius6.37x106m toobtainagradientofdβ/dr 1.091x10‐16/m. Thechange ingravitationalmagnitudeΔβacrosstherotarradius 3.862x10‐13m ofanelectronis:Δβ 1.091x10‐16/m 3.862x10‐13m 4.213x10‐29Δβacrossanelectron’s The electron’s internal pressure is being stabilized by the pressure being exerted by thespacetimefield.However,thehomogeneousspacetimefieldinzerogravityismodifiedbytheearth’sgravitationalfield.Aspreviouslycalculated,gravityaffectsnotonlytherateoftimeandpropervolume,butalsotheunitofforce,energy,etc.Thepreviouslycalculatednormalizedforcetransformationis:Fo ГFg.Thegradientintheearth’sgravitationalfieldmeansthataslightlydifferentvalueofΓexistsonoppositesidesoftheelectron.ThisismoreconvenientlyexpressedasadifferenceinthegravitationalmagnitudeΔβthatexistsacrosstheelectron’sradiusλc.Therewill be a slight difference in the force exerted by the spacetime field exerted on oppositehemispheresoftheelectron rotar .Calculatingthisdifferenceshouldequalthemagnitudeofthegravitationalforceontheelectron.F ΔβFm 4.213x10‐29x0.212N 8.89x10‐30NWewillnowcheckthisbycalculationtheforceexertedonanelectronbytheearth’sgravityusingF mgwheretheearth’sgravitationalaccelerationis:g 9.78m/s2F mg 9.1x10‐31kgx9.78m/s2 8.89x10‐30N

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Success!TheanswerobtainedfromthecalculationusingF ΔβFmisexactlycorrect.Apparentlytheignoreddimensionlessconstantscancel.Thisisanothersuccessfulplausibilitytest.At the beginning of this chapter two quotes were presented that pointed out that generalrelativitydoesnotidentifythesourceoftheforcethatoccurswhenaparticleisrestrainedfromfollowingthegeodesic.M.R.Edwardsstates:“Howeversuccessfulthisgeometricinterpretationmaybeasamathematicalmodel,itlacksphysicsandacausalmechanism.”Theideasproposedin thisbookgivea conceptuallyunderstandable explanation forboth themagnitudeand thevectordirectionofthegravitationalforce.Thegravitationalforcewasobtainedfromthestartingassumptionswithoutusinganalogyofacceleration.Electrostatic Force at Arbitrary Distance:ThestrainamplitudeofthespacetimewaveinsidetherotarvolumehasbeendesignatedwiththesymbolAβ Lp/λc Tpωc.WehavejustshownthatthegravitationaleffectexternaltotherotarvolumescaleswithAg Aβ2/ gm/c2r.Thisis the gravitational curvature of spacetime produced by a rotar with radius λc and angularfrequencyωc.NowwewillexaminetheelectromagneticeffectonspacetimeproducedbytheeffectofthefundamentalwavewithamplitudeAβ notsquared .Fromchapter6weknowthatthisamplitude isassociatedwiththeelectrostatic force.Nowwewillextendthistoarbitrarydistance.Asbefore,weneedtomatchtheknownamplitudeatdistanceλc.Thisisachievedbyscalingdistanceusing ≡r/λcbecause 1atdistanceλc.Wewillagainassumethattheelectrostatic amplitude AE decreases as 1/ for the electrostatic force we assumeAE Aβ/ Lp/λc λc/r .WewillusetheequationF = kA2ω2Z /c and also insert the following: F = FE, ω = ωc, Z = Zs = c3/G, r/λc, = kλc

2 and ħc = 4⁄

= /c = ħ

Therefore,wehavegenerated theCoulomb lawequationwhere thecharge isq qp Planckcharge . It should not be surprising that the charge obtained is Planck charge rather thanelementarychargee.Planckchargeisqp 4 ħ about11.7timeschargee andisbasedon the permittivity of free space εo. Planck charge is known to have a coupling constant tophotonsof1whileelementarychargeehasacouplingconstanttophotonsofα,thefinestructureconstant. This calculation is actually themaximumpossible electrostatic forcewhichwouldrequirea coupling constantof1. The symbolFE implies theelectrostatic forcebetween twoPlanckchargeswhileFeimpliestheelectrostaticforcebetweentwoelementarychargese.TheconversionisFE Feα‐1.Wewillcontinuetousetheequation:F = kA2ωc

2Zs /c even though it implies the emission of power which is striking area andexertingarepulsiveforce.ThisisnothappeningbuttheuseofF = kA2ωc

2Zs /c gives the correct magnitude of forces. This simplified equation allows a lot of quick calculations to be made which give correct magnitude.

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Calculation with Two Different Mass Particles:Untilnowwehaveassumedtwoofthesamemass/energyparticleswhenwecalculatedFgandFE.Nowwewillassumetwodifferentmassparticles m1andm2 ,butwewillkeeptheassumptionthatbothparticleshavePlanckcharge.Whenwe have two differentmass particles, thismeans thatwe have two different reducedComptonwavelengths λc1 ħ/ c andλc1 ħ/ .The single radial separation r nowbecomestwodifferentvalues 1 r/λc1and 2 r/λc2. Also, therewouldbe twodifferentstrainamplitudesAβ1 Lp/λc1andAβ2 Lp/λc2aswellasacompositearea kλc1λc2.

k

Note that the only difference between the intermediate portions of these two equations is that the gravitational force Fg has the strain amplitude terms squared ( and the electrostatic force FE has the strain amplitude terms not squared ( . The tremendous difference between the gravitational force and the electrostatic force is due to a simple difference in exponents.Unification of Forces: Inchapter6wefoundthatthereisa logicalconnectionbetweenthegravitationalforceandtheelectromagneticforce.However,thosecalculationsweredoneonlyfor separationdistance equal to λc. Nowwewill generate somemore general equations forarbitraryseparationdistanceexpressedas , thenumberof reducedComptonwavelengths.Thefollowingequationscouldbemadeassumingtwodifferentmassparticles,butitiseasiertoreturntotheassumptionofbothparticleshavingthesamemassbecausethenwecandesignateasinglevalueof separatingtheparticles.SomeofthefollowingequationsassumePlanckcharge qpwith force designation FE rather than charge e designatedwith force Fe. TheconversionisFE Feα–1.WewillstartbyconvertingtheNewtongravitationalequationandtheCoulomblawequationsothattheyarebothexpressedinnaturalunits.Thismeansthatboththeforcesandtheparticle’senergywillbeinPlanckunits Fg Fg/Fp,FE FE/Fp,Ei Ei/Ep, .Alsoseparationdistancewillbeexpressedintheparticlesnaturalunitoflength,thenumber r/λcofreducedComptonwavelengths. Also,weassumetwoparticleseachhavethesamemass/energyandtheybothhavePlanckcharge.Convertbothequations:FE qp2/4πεor2andFg Gm2/r2intoequationsusingFg;FEand .

Substitutions:r ħc/Ei;m Ei/c2;Ei EiEp Ei ħ ⁄

FE ħ

ħ ħ Ei

2/ 2

Fg = ħ

ħ

Ei4/ 2

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(Fg2) FE 2 2 Ei4

Theequation(Fg

2) FE 2 2clearlyshowsthatevenwitharbitraryseparationdistancethesquarerelationshipbetweenFgandFEstillexists.ItisinformativetostatethesesameequationsintermsofpowerbecauseafundamentalassumptionofthisbookisthatthereisonlyonetrulyfundamentalforceFr Pr/c.Ifthisiscorrect,thenwewouldexpectthattheforcerelationshipbetweenrotarswouldalsobeasimple functionof therotar’scirculatingpower:

ħ⁄ .ToconvertPctodimensionlessPlanckunits Pc Pc/PpwedividebyPlanckpowerPp c5/G.Notethesimplicityoftheresult.FE Pc/ 2Fg Pc

2/ 2So farwe have used dimensionless Planck units because they show the square relationshipbetween forcesmostclearly. However,wewillnowswitchanduseequationswithstandardunits.Thenextequationwillfirstbeexplainedwithanexample.Wewillassumeeithertwoelectronsor twoprotons bothchargee andwehold themapartatanarbitraryseparationdistancer.Asbefore,thisseparationdistancewillbedesignatedusingthenumber ofreducedComptonwavelengths,thereforer Nλc.Protonsarecompositeparticles,butwecanstillusetheminthisexampleifweusetheproton’stotalmasswhencalculating . Nowweimaginealogscaleofforce.AtoneendofthisforcescaleweplacethelargestpossibleforcewhichisPlanckforceFp c4/G. Attheotherendofthislogscaleofforceweplacethegravitational force Fg which is weakest possible force between the two particles either 2electronsor2protons .Nowforthemagicalpart!ExactlyhalfwaybetweenthesetwoextremesonthelogscaleofforceisthecompositeforceFe α–1.Inwords,thisistheelectrostaticforceFebetween the two particles times the number of reduced Comptonwavelengths times theinverseofthefinestructureconstant α‐1 137 .Particlephysicistsliketotalkaboutvarioussymmetries.Iamclaimingthatthereisaforcesymmetrybetweenthegravitationalforce,PlanckforceandthecompositeforceFe α–1.Theequationforthisis:

Itisinformativetogiveanumericalexamplewhichillustratesthisequation.Supposethattwoelectronsareseparatedby68nanometers thisdistancesimplifiesexplanations .Theelectronsexperience both a gravitational force Fg and an electrostatic force Fe. The electrons haveλc 3.86x10‐13m, therefore this separation is equivalent to 1.76x105 reducedComptonwavelengths. The gravitational force between the two electrons at this distance would beFg 1.2x10‐56NandtheelectrostaticforcewouldbeFe 5x10‐14N.Alsoα –1 137socombining

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Fe, andα –1wehave:Fe α–1 1.2x10‐6N.AlsoPlanckforceisFp c4/G 1.2x1044N.Tosummarizeandseethesymmetrybetweentheseforces,wewillwritetheforcesasfollows:Fg 1.2 10 NFgfortwoelectronsat6.8x10‐8mis1050timessmallerthanFe /αFe α –1 1.2 10 NFe /αfortwoelectronsat6.8x10‐8mFp 1.2 10 NFp Planckforce is1050timeslargerthanpreviousFe /αAnotherwayofstatingthisrelationshipwouldsetPlanckforceequalto1.ThereforewhenFp 1thenFe α –1 10‐50andFg 10‐100.Thenumericalvaluesarenotimportantbecausedifferentmass particles or different separation distance could be used. The important point is thesymmetrybetweenFp,FeandFgwhenweinclude andα‐1inthecompositeforceFe α –1.Force Ratios Fg/FE and Fg/Fe α-1:Nextwewillshowhowthewavestructureofparticlesandforcesdirectlyleadstoequationswhichconnecttheelectrostaticforceandgravity.Previouslywestartedwiththewave‐amplitudeequation F = kA2ωc

2Zs /cwhichisapplicabletowavesinspacetime. In thisequationA isstrainamplitude,ωc isComptonangular frequency,Zs is theimpedanceofspacetimeZs c3/Gand isparticlearea.WehaveshownthattheinsertingthestrainamplitudetermA Aβ/ intoF = kA2ωc

2Zs /cgivestheelectrostaticforcebetweentwoPlanckchargesFE,.Wehavealsoshownthatgravityisanonlineareffectwhichscaleswithstrainamplitudesquared Aβ2 .InsertingA Aβ2/ intothisequationgivesthegravitationalforceFgbetweentwoequalmassparticles.SincewehaveequationswhichgenerateFEandFg,,weshouldbe able to generate new equations which give the ratio of forces Fg/FE. In the followingAβ Lp/λc Tpωc.Forgravity,A Ag Aβ2/ andforelectrostaticforceA AE Aβ/ .Fg = k(Aβ2/ )2ωc2Zs /c Fg = gravitationalforcebetweentwoofthesamemassparticles FE = k(Aβ/ )2ωc

2Zs /c FE theelectrostaticforcebetweentwoparticleswithPlanckcharge Setcommontermsequaltoeachother: (kωc

2Zs /c) = (kωc2Zs /c)

2

where: Aβ = Lp/λc = Tpωc = rotar strain amplitude

22 2

TheequationFg/FE Aβ2showsmostclearlythevalidityofthespacetimebasedmodeloftheuniverseproposedhere.Recallthatallfermionsandbosonsarequantizedwaveswhichproducethesamedisplacementofspacetime.ThespatialdisplacementisequaltoPlancklengthLpandthe temporal displacement is Planck time Tp. Even though all waves produce the samedisplacementofspacetime,differentparticleshavedifferentwavestrainamplitudesbecausethestrainamplitudeisthemaximumslope maximumstrain producedbythewave.Thereforearotar’sstrainamplitude isAβ Lp/λc Tpωc. Now we discover that the force produced by particles with strain amplitude Aβ reveal their connection to the underlying physics because Fg/FE = Aβ

2.

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All theprevious equations relating Fg and FE either specified either a specific separationorspecifiedseparationdistanceusing .However,Fg/FE = Aβ2 doesnotspecifyseparation.Thisispossiblebecausetheratioofthegravitationalforcetotheelectrostaticforceisindependentofdistance.Forexample,theratioforanelectronisFg/Fe 2.4x10‐43.However,whenweadjustforthecouplingconstantassociatedwithchargee,theratiobecomes:Fg/Feα‐1 1.75x10‐45.TherotarstrainamplitudeforanelectronisAβ Lp/λc 4.185x10‐23.ThereforeAβ2 1.75x10‐45Clearly,thederivationofthisequationandthephysicsbehinditgivestrongproofofthewave‐based structure of particles and forces. Nextwewill extend the relationshipbetween theseforcesonemoresteptobringanewperspective.

ħ

ħ

ħ

or:

Theequation Fg/FE = Rs/λc isveryinterestingbecause Fg/Fe isaratioofforcesand Rs/λc isaratioofradii.Recallthat Rs≡ Gm/c2 istheSchwarzschildradiusofarotarbecauseitwouldformablackholerotatingatthespeedoflight.SuchablackholehashalftheSchwarzschildradiusofanon‐rotatingblackholetherefore Rs Gm/c2.Also,λc = ħ/mc istheradiusoftherotarmodelofafundamentalparticle.Forexample,foranelectronRs 1.24x10‐54mandanelectron’srotarradiusis:λc 3.85x10‐13m.Thesetwonumbersseemcompletelyunrelated,yettogethertheyequaltheelectron’sforceratioFg/Feα‐1 1.75x10‐45 Rs/λc.However,asthefollowingequationsshow,therearetwoamazingconnectionsbetweenRsandλc.First,Rsλc Lp2.ThesecondisRs λc‐1Inwords,thissaysthattherotar’sSchwarzschildradius Rs≡Gm/c2)istheinverseofthereducedComptonradiuswhenbothareexpressedinthenaturalunitsofspacetimewhicharedimensionlessPlanckunits Rsandλcunderlined .

Rsλc ħ

ħ

Rs λc‐1equivalentto: ⁄ ⁄ The Schwarzschild radius comes from general relativity and is considered to be completelyunconnected to quantummechanics. A particle’s reduced Comptonwavelength comes fromquantum mechanics and is considered to be completely unconnected to general relativity.However,whentheyareexpressedinnaturalunits dimensionlessPlanckunits thetworadiiarejusttheinverseofeachother Rs 1/λc.Also .TherelationshipsbetweenRsand

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λc are compatible with the wave‐based rotar model of fundamental particles but they areincompatiblewiththemessengerparticlehypothesisofforcetransmission.TosummarizealltheequationsequaltoFg/FE,plusafewmorewehave:

⁄ ⁄ ⁄ Aβ2 Rs

2 λc–2 ωc2= Ei

2 Pc All the previous force equations also work with composite particles such as protons if theproton’stotalmassisusedtocalculatethevarioustermssuchasλc ħ/mc.Forexample,twoprotonshaveFg/Feα‐1 5.9x10‐39atanyseparationdistance.AlsoforprotonsRs/λc 5.9x10‐39and Lp/λc 2 5.9x10‐39. Iwanttopauseforamomentandreflectontheimplicationsofallthepreviousequationswhichshowtherelationshipbetweentheelectrostaticforceandthegravitationalforce.Tome,theyclearlyimplyseveralthings.Theseare:

1 Gravitycanbeexpressedasthesquareoftheelectrostaticforcewhenseparationdistanceisexpressedas multiplesofthereducedComptonwavelengthλc.

2 The equations relating the gravitational and electrostatic forces imply that they bothscaleasafundamentalfunctionofaparticle’squantummechanicalpropertiessuchasComptonwavelengthorComptonfrequency.

3 All the connections between the gravitational force and the electrostatic force areproposedtobetraceabletoarotargeneratingaComptonfrequencystandingwaveinthesurroundingvolume.Spacetime isanonlinearmedium,soasinglestandingwavehasbothafundamentalcomponent electrostatic andanonlinearcomponent gravity .

4 Theelectromagneticforceisuniversallyrecognizedasbeingarealforce.Theequationsshowthatgravityiscloselyrelated.Therefore,gravityisalsoarealforce.

5 These equations are incompatible with virtual photons transferring the electrostaticforceorgravitonstransferringthegravitationalforce.Theequationsalsoappeartobeincompatiblewithstringtheory.

6 There is a quantummechanical connection between a particle’s rotar radius and itsSchwarzschildradius.

DerivationoftheEquations:IwanttorelateabriefstoryaboutthefirsttimethatIprovedarelationshipbetweenthegravitationalandelectrostaticforces.Aspreviouslystated,theinitialideathatledtothisbookwasthatlightconfinedinahypothetical100%reflectingboxwouldexhibitthesameinertiaasaparticlewiththesameenergy.Theotherideasinchapter1followedquickly and I was struck with the idea that these connections between confined light andparticleswere probably not a coincidence. Iwill now skip ahead several yearswhen Iwasmethodically inventing a model of the universe using only the properties of 4 dimensionspacetime. I had the idea of dipole waves in spacetime and quantized angular momentumforming a “rotar” that was one Compton wavelength in circumference and rotating at the

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particle’sComptonfrequency.Thisimpliedasphericalvolumewithradiusofλc 3.86x10‐13.IhadindependentlyderivedZs c3/Gwhichwaskeytoalltheotherequations.IhaddevelopedthedimensionlesswaveamplitudeforanelectronwhichwasAβ 4.185x10‐23. AnimportantsuccesswastosubstitutethesevaluesintoE A2ω2ZV/candIobtainedEi 8.19x10‐14Jforanelectron.Thenextlogicalstepwastofindtheforcethatwouldexistbetweentwoelectronsatadistanceequaltoλc.ThisdistancewasimpliedbecauseIwasusingtheonlyamplitudethatIknewwhichcorrespondedtoadistanceequaltoλc.WhenImadenumericalsubstitutionsforanelectronintotheequationF A2ω2Z /cIobtainedF 0.212N.Thiswasabout137timesgreaterthantheelectrostatic force between two electrons at the separation distance λc. I was quite happybecauseIrealizedthatthiswouldbethecorrectforceifthechargewasPlanckchargeqpratherthanelementarychargee. Thiswasactuallyapreferableresultbecauseitcorrespondedtoacouplingconstantof1,whichwasreasonableforthiscalculation.Thiswasunderstandableanditwasquiteexciting.NextIthoughtaboutgravity.Iknewthatgravitywasvastlyweakerthantheotherforcesandonlyhadasinglepolarity. Iwasremindedabout theopticalKerreffectdiscussed in the lastchapter.Thisnonlineareffectscaleswithamplitudesquared electricfieldsquared andalwaysproducesan increase in the indexof refractionof the transparentmaterial a singlepolarityeffect .Gravityhasasinglepolarityandisvastlyweakerthantheelectrostaticforce.Therefore,IdecidedtocalculatetheforceusingF A2ω2Z /c.andset:A Aβ2 1.75x10‐45,ωc 7.76x1020s‐1,Zs 4.04x1035kg/sand λc2 1.49x10‐25m2.Usingapocketcalculator,therewasaEurekamoment when I got the answer which exactly equaled the gravitational force between twoelectronsatthisseparation F 3.71x10‐46N .ThisstoryistoldbecauseIwanttosupporttheclaimthatthepredictioncamefirst.Itisoftensaidthattheproofoftheaccuracyofanewidealiesinwhetheritcanmakeapredictionrevealingsomepreviouslyunknownfact.Usuallytheproofrequiresanexperiment,butinthiscaseitwasasimplecalculation.Tomyknowledgethisisthefirsttimethatthegravitationalforcehasbeencalculatedfromfirstprincipleswithoutreferencingacceleration.Gravitational Rate of Time Gradient: Intheweakfieldlimit,itisquiteeasytoextrapolatefromthegravitationalmagnitudeβproducedbyasinglerotarataparticularpointinspacetothetotalgravitationalmagnitudeproducedbymanyrotars.Naturemerelysumsthemagnitudesof all the rotarsat apoint in spacewithout regard to thedirectionof individual rotars. Thegravitationalaccelerationgwaspreviouslydeterminedtobe:g c2dβ/dr –c2d dτ/dt /dr.

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Agravitationalaccelerationof1m/s2requiresarateoftimegradientof1.11x10‐17secondspersecondpermeter.Theearth’sgravitationalaccelerationof9.8m/s2impliesaverticalrateoftimegradientof1.09x10‐16meter‐1. Thismeansthattwoclocks,withaverticalseparationofonemeterattheearth’ssurface,willdifferintimeby1.09x10‐16seconds/second.Similarly,thereis also a spatial gradient. A gravitational acceleration also implies that there is a differencebetweencircumferentialradiusRandradiallengthL.Intheearth’sgravitythisspatialdifferenceis1.09x10‐16meters/meter β 1–dR/dL .Outofcuriosity,wecancalculatehowmuchrelativevelocitywouldberequiredtoproduceatimedilationequivalenttoaonemeterelevationchangeintheearth’sgravity.Ifelevation2is1meterhigherthanelevation1,then:dt1/dt2 1–1.09x10‐16.Usingspecialrelativity:

v 1 setdt1/dt2 1–1.09x10‐16

v 4.4m/sThis4.4m/svelocityisexactlythesamevelocityasafallingobjectachievesafterfallingthroughadistanceof1meterintheearth’sgravity.Carryingthisonestepfurther,anobserveringravityperceivesthataclockinaspaceshipinzerogravityhasthesamerateoftimeasaclockingravity,ifthespaceshipismovingatarelativevelocityofve,thegravity’sescapevelocity.Forexample,anobserveronearthwouldperceive that a spaceship in zerogravitymoving tangentially atabout40,000km/hrhasthesamerateoftimeasaclockontheearth. Ontheotherhand,anobserverinthespaceshipperceivesthataclockontheearthisslowedtwiceasmuchasiftherewasonlygravityoronlyrelativemotion.Equivalence of Acceleration and Gravity Examined: AlbertEinsteinassumedthatgravitycouldbeconsideredequivalenttoacceleration.Thisassumptionobviouslyleadstothecorrectmathematicalequations.However,onaquantummechanicallevel,isthisassumptioncorrect?Todayit iscommonlybelievedthatanacceleratingframeofreferenceisthesameasgravity.This is associatedwith thegeometric interpretationof gravity. A corollary to this is thataninertialframeofreferenceeliminatesgravity.Theimplicationisthatgravityisnotasrealastheelectromagneticforcewhichcannotbemadetodisappearmerelybychoosingaparticularframeofreference.Itiscommonforexpertsingeneralrelativitytoconsidergravitytobeageometriceffectratherthanatrueforce.Theconceptspresentedinthisbookfundamentallydisagreewiththisphysicalinterpretationofgeneral relativity. There is no disagreement with the equations of general relativity. Theprevious pages have shown that it is possible to derive the gravitational force using waveproperties and the impedance of spacetime. This is completely different than acceleration.Numerous equations in this book have shown that there is a close connection between thegravitationalforceandtheelectrostaticforce.Inparticular,Iwillreferencethetwoequations

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whichdealtwithtwodifferentmassparticles m1andm2 . Theconcludingstatement inthisanalysiswas:“Note that the only difference between the intermediate portions of these two equations is that the gravitational force Fg has the strain amplitude terms squared ( and the electrostatic force FE has the strain amplitude terms not squared ( . The tremendous difference between the gravitational force and the electrostatic force is due to a difference in exponents.”

Surely this qualifies as proof that gravity is closely related to the electrostatic force andthereforegravityisalsoarealforce.Theimplicationisthatwhenwegetdowntoanalyzingwavesinspacetimethereisadifferencebetweengravityandacceleration.Anargumentnotinvolvingdipolewavesinspacetimegoesasfollows:Aparticleinfreefallinagravitational field has not eliminated the effect of gravity. The particle is just experiencingoffsettingforces.Thegravitationalforceisstillpresentinfreefallbutitisbeingoffsetbytheinertialpseudo‐forcecausedbytheacceleratingframeofreference.Thesetwoopposing“forces”just offset each other and give the impression that there is no force. The acceleration alsoproducesanoffsettingrateoftimegradientandanoffsettingspatialeffect.Thepseudo‐forceofinertiahasnotbeeneliminatedandthegravitational forcehasnotbeeneliminated. Einsteinobtainedthecorrectequationsofgeneralrelativitybyassumingthatgravitywasthesameofacceleration.Sincetheyexactlyoffseteachotherinfreefall,thisassumptiongavethecorrectequationsbutthephysicalinterpretationiswrong.Onthequantummechanicalscaleinvolvingwaves,gravityisdifferentthanacceleration.Onewaytoprovethatgravityisatrueforceistoshowthatagravitationalfieldpossessenergydensity.Thiswillbediscussedinchapter10. “Grav” Field in the Rotar volume:Theabovediscussionofgravitationalaccelerationfromarateof timegradientpreparesus to return to the subjectof the “grav field” inside the rotarvolumeofarotar.Recallthattherotatingdipolethatformstherotarvolumeofanisolatedrotarwasshowninfigure5‐1. Thisrotatingdipolewavehastwolobesthathavedifferentratesoftimeanddifferenteffectsonpropervolume.Thedifferenceintherateoftimebetweenthetwolobesproducesarotatingrateoftimegradientthatwasdepictedinfigure5‐2.Arotarisverysensitivetoarateoftimegradient.Arateoftimegradientof1.11x10‐17seconds/second/metercauses a rotar to accelerate at 1m/s2 and the acceleration scales linearlywith rate of timegradient. Therefore, the rotating rateof timegradient in the centerof a rotarmodel canbeconsideredtobearotatingaccelerationfieldthathassimilaritiestoarotatinggravitationalfield.Wenormallyencountertherateoftimegradientinagravitationalfield.Thisistheresultofanonlinearity that produces a static stress in spacetime. A static rate of time gradient hasfrequencyofω 0andnoenergydensity.However,inanactualgravitationalfieldthereisalsotheoscillatingcomponentofagravitationalfieldandthisdoeshaveenergydensitythatwillbediscussedlater.Ifarotatinggravitationalfieldissomehowgenerated,thensucharotatingfield

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wouldalsohaveenergydensity.Therotatingrateoftimegradient rotatinggravfield thatispresentnearthecenterofarotardoeshaveenergydensitythatwillbecalculatednext.In a time period of 1/ωc, the fast time lobe of the dipole gains Planck time displacementamplitudeTp andtheslowtimelobelosesPlancktimeTp.Theselobesareseparatedby2 .Therefore,inatimeof1/ωcthereisatotaltimedifferenceof2Tpacrossadistanceof2 .Therateoftimegradientpermeteris:

–

Theaccelerationproducedbythisrateoftimegradient gravacceleration g istherateoftimegradienttimesc2.Thefollowingareseveralequalitiesforgravacceleration g:

g –

c2 Lpωc2 Aβ2 pħ

g gravaccelerationand p c/tp /ħ PlanckaccelerationComparison of Grav Acceleration and Gravitational Acceleration: Howdoestherotatinggravaccelerationat thecenterofa rotar’s rotarvolume g comparewith thenon‐rotating,gravitationalacceleration gq attheedgeofthesamerotar’srotarvolume?.gq Aβ2ωccrotar’snon‐rotatinggravitationalaccelerationatdistance fromthecenterg Aβωccrotar’srotating ωc gravaccelerationatthecenterofarotar

Aβratioofgq staticgravitationalaccelerationat torotatinggravacceleration g

ForanelectronAβ 4.18x10‐23,sotherotatinggravfieldatthecenteroftheelectronisabout2 1022timesstrongerthanthenon‐rotatinggravitationalfieldatdistance .Thisresultsinan electron having a rotating grav acceleration of: g 9.73 x 106m/s2. The gravitationalacceleration notrotating ofanelectronatdistance is:gq 4.07x10‐16m/s2.Thereforethegravacceleration intherotarvolumeofanelectron isaboutamilliontimesgreaterthanthegravitationalaccelerationatthesurfaceoftheearth.Theearth’sgravityisnotrotatingandisanonlineareffect.Theelectron’sgravfieldisrotatingandisafirstordereffectresultingfromtherateoftimegradientestablishedintheelectron’srotatingdipolewave.Recall the incredibly small difference in the rate of time that exists between the lobes of anelectron. Itwould take50,000 times the ageof the universe for thehypothetical lobe clockrunningattherateoftimeinsidetheslowlobetoloseonesecondcomparedtothecoordinate

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clock.Thedifferencebetweentherateoftimeontheslowlobeclockandthecoordinateclockiscomparabletothedifferenceintherateoftimeexhibitedbyanelevationchangeofabout4x10‐7m in the earth’s gravity. The reason that the rotating grav field has amillion times largeracceleration than the earth is because this difference in the rate of time occurs overapproximatelyamilliontimesshorterdistance ~4x10‐13m .Therotatingrateoftimegradientinside a rotar is a first order effect related to Aβ while the non‐rotating gravitational fieldproducedbytherotarisasecondordereffectrelatedtoAβ2.Conservation of Momentum in the Grav Field:Itwouldappearthattheconceptofagravfieldmustviolatetheconservationofmomentum.Anexamplewillillustratethispoint.Supposethatasmallneutralparticle suchasaneutralmeson wandersintothecenterofanelectron’srotarvolume.Evenifthemassofthemesonis1000timeslargerthantheelectron,therotatinggravfieldoftheelectronshouldproducethesameaccelerationoftheneutralparticle.ThiswouldbeaviolationoftheconservationofmomentumunlessthedisplacementproducedbytherotatinggravfieldisequaltoorlessthanPlancklength theuncertaintyprincipledetectablelimit .Wewillcalculatethemaximumdisplacement thattakesplaceinatimeperiodof:t 1/ωc.Wechoosethistimeperiodbecausetherotatingvectorofthegravfieldischangingbyoneradianinatimeperiodof1/ωc.Hypotheticallytheneutralparticlewouldnutateinacirclewitharadiusrelatedto ignoringdimensionlessconstants .

½at2

Lp

Lp maximumradialdisplacementproducedbyarotar’srotatinggravfieldThereforeanymass/energyrotaralwaysproducesthesamedisplacementequaltoPlancklengthignoringdimensionlessconstants inthetimerequiredforthegravfieldtorotateoneradian.Thisdisplacementispermittedbyquantummechanicsandisnotaviolationoftheconservationofmomentum.Thisisanothersuccessfulplausibilitytest.Energy Density in the Rotating Grav Field: Anacceleratingfieldthat isrotatingpossessesenergydensity.Itwouldhypotheticallybepossibletoextractenergyfromsuchafieldifthefieldproducedanutationthatwaslargerthanthequantummechanical limitofPlancklength. Noenergycanbeextractedfromarotar’srotatinggravfieldbecausethenutationisatthequantummechanicallimitofdetection.However,thisfieldstillpossessesenergydensity.PreviouslywedesignatedthestrainamplitudeofarotarasAβ Lp/ Tpωc ωc/ωp.Thesewereoriginallydefinedintermsofthestrainamplitudeofadipolewavethatisonewavelengthin circumference. This definition tended to imply that the energy density of a rotar wasdistributedaroundthecircumference.However,itisproposedthattherotatinggradientthatispresentat thecenterof therotarmodelcanalsobecharacterizedashavingadimensionless

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amplitudeofAβ Lp/ Tpωc.ThisamplitudeAβisjustintheformofarotatingrateoftimegradientandarotatingspatialgradient.ThespatialgradientfromthelobetothecenterisstillLp/ .TherateoftimegradientisstillrelatedtoTpωc,althoughthisishardertosee.PreviouslywesubstitutedAβ,ωcandZsintoU A2ω2Z/candobtainedarotar’senergydensityintherotarvolumeUq kmc2/. Ifwe ignorethedimensionlessconstantk, this is therotar’s internalenergyinthevolumeofacubethatis onaside.HerearesomeotherequalitiesforUq.

Uq ħ

Aβ4Upsetħ

g2

Uq

Thereforetherotating“gravfield”hasthesameenergydensityastheenergydensityoftheentirerotar Ei/λc3 . Ifwe broaden the definition of Aβ so that it also defines rotating rate of timegradientsandrotatingspatialgradients, then theenergydensityof therotarmodelbecomeshomogeneous.Theenergydensitynearthecenterofarotaristhesameastheenergydensityneartheedge.Thisenergydensityisjustintwodifferentforms.Inchapter6weattemptedtocalculatetheangularmomentumofarotar.Ifweassumedthatalltheenergywasconcentratedneartheedgeofahoopwithradius ,thenweobtainedananswerofangularmomentumofħ.However,ifweassumedthattheenergywasdistributedmoreuniformly likeadisk andalsorotationinasingleplane,thentherotarmodelwouldhaveangularmomentumof½ħ.Thefactthat energy is contained in the grav field does smooth out the energy distribution, therebytendingtowardstheanswerof½ħ.However,aspreviouslyexplained,thereisalsoachaoticrotationwherethereisanexpectationrotationalaxisbutotherrotationaldirectionsareallowedwithlessprobability.Also.theenergydensitydistributionwithinarotardoesnotendabruptlyataradialdistanceofλc.Thedetailsthatresultinnetangularmomentumof½ħwillhavetobeworkedoutbyothers.If a rotating grav field has energy density, does a static gravitational field also have energydensity? This question will be examined in chapter 10 after some additional concepts areintroducedabouttheoscillatingcomponentofgravityEnergy Density in Dipole Waves:TheaboveinsightsintothegravfieldalsohaveimplicationsforanyPlanckamplitudedipolewaveinspacetime,notjusttherotatingdipolesthatformrotars.Iamgoingtotalkaboutdipolewavesinspacetimebutstartoffbymakingananalogytosoundwaves in a gas. Sound waves can be depicted with a sinusoidal graph of pressure. Thecompression regions have pressure above the local norm and the rarefaction regions havepressure below the local norm. These can be represented as a sine wave maximum andminimum.However,ifagraphwastobedrawnshowingthekineticenergyofthemoleculesinthegas,themaximumkineticenergyoccursinthenoderegionsbetweenthepressuremaximumandminimum.Akineticenergygraphdepictingmotion velocity leftandrightwouldhavea90°phaseshifttothepressuregraph.Theenergyinthesoundwaveisbeingconvertedfrom

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kineticenergy particlemotion toenergyintheformofhighorlowpressuregas.Whenthesetwoformsofenergyareaddedtogether,thenasoundwavewithaplanewavefronthasauniformtotalenergydensity sin2θ cos2θ 1 .Theenergyisjustbeingexchangedbetweentwoforms.This concept of energy being exchanged between two different forms also applies to dipolewaves in spacetime. In one form, energy exists because the vacuumenergy of spacetime isdistortedsothatthereareregionswheretherateoftimeisfasterorslowerthanthelocalnorm.Perhapsthisisanalogoustothecompressionandrarefactionrepresentationofasoundwave.Theregionsbetweenthemaximumandminimumratesoftimehavethegreatestgradientintherateoftime. Thesearethegravfieldregionsandtheyareanalogoustoregionsinthesoundwavewherethegasmoleculeshavethegreatestkineticenergy.Addingtogetherthetwoformsofenergydensitypresentineithersoundwavesordipolewavesinspacetimeproducesatotalenergydensitywithoutthecharacteristicwaveundulations.The waves in spacetime have sometimes been discussed emphasizing either the temporalcharacteristics rate of time gradients, etc. or emphasizing the spatial characteristics forexampleLp/r .Actuallybothcharacteristicsarealwayspresent;itissometimeseasiertoexplainusingjustonecharacteristic.Therefore,thegravfieldcouldhavebeenexplainedemphasizingthepropervolumegradientratherthantherateoftimegradient.Gravitational Potential Energy Storage:Whenwelookatthegravitationaleffectthatarotarhasonspacetime,weconcludethattheslowingoftherateoftimealsoproducesaslowingofthenormalizedspeedoflight Co ГCgfromchapter3 .Toreachthisconclusionwemustassumethatproperlengthisconstant,evenwhenthereisachangeinГ.Thisisanunspokenassumptionforphysicsthatdoesnotinvolvegeneralrelativity.Theeffectontherateof timeandonthenormalizedspeedof lightultimatelyeffectsenergy,force,mass,etc.aspreviouslydiscussed.ThereasonforbringingthisupnowisthatIwanttoaddressgravitationalpotentialenergy.Gravitationalpotentialenergyisconsideredanegativeenergythathasitsmaximumvalueinzerogravityanddecreaseswhenamassisloweredintogravity.Whatphysicallychangeswhenarotariselevatedorloweredingravity?Inchapter3itwasfoundthatsubstitutingthenormalizedspeedoflightCgandthenormalizedmassMgintotheequationE mc2givesenergythatscalesinverselywithgravitationalgammaΓ rest frame of reference . We illustrated this concept by calculating the difference in theinternalenergyofa1kgmassforanelevationofsealevelandonemeterabovesealevel.Thecalculateddifferenceinthenormalizedinternalenergywas9.8Jouleswhichisexactlythesameasthegravitationalpotentialenergy.ThischangeinenergyisduetothechangeinthenormalizedspeedoflightaffectingtheComptonfrequencyoftherotarasseenfromzerogravity.Forexample,afreeelectroninzerogravityhas

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aComptonangularfrequencyof7.76x1020s‐1.Earth’sgravityhasβ 7x10‐10.Afreeelectroninearth’sgravityhasanormalizedComptonangularfrequencythatisslowerthanazerogravityelectronbyabout5.4x1011radianspernormalizedsecond 7x10‐10x7.76x1020s‐1 . Thislower Compton frequency decreases the normalized internal energy of an electron anddecreases the gravity non‐oscillating strain generated by an electron. The non‐oscillatingstrainisresponsiblefortherotar’sgravity,soarotaratrestingravitycontributeslessgravitytothetotalgravitythanthesamerotaratrest sametemperature inzerogravity.Foranotherexample,wewillcalculatethechangeintheinternalenergyofanelectronwhenitiselevated1meterintheearth’sgravitationalfield.Inchapter3thereisasectiontitled “Energy Transformation and Calculation”wherethedifferenceinthegravitationalgammawascalculatedfor a 1 meter elevation change near the earth’s surface. This was expressed as Γ2 – Γ1 1.091x10‐16 dt2/dt1.SincethereducedComptonfrequencyofanelectronisabout7.76x1020s‐1,thismeansa1meterelevationchangewillproduceafrequencychange:Δωc 7.7634x1020s‐1x1.0915x10‐16 84,737s‐1ΔE Δωcħ 84,690s‐1xħJs 8.936x10‐30JWewillcomparethistothegravitationalpotentialenergystoredwhenanelectroniselevated1meterintheearth’sgravitationalfieldwithaccelerationofg 9.81m/s2.ΔE meΔhg 9.109x10‐31kgx1mx9.81m/s2 8.936 10‐30JTheseconceptsalsoleadtoaphysicalexplanationforpotentialenergy.Inchapter3theconceptof potential energywas related to a reduction in thenormalized speedof light reducing theE mc2internalenergy.Nowwegoonestepfurtherandtracethegravitationalpotentialenergytoachangeintherotationalfrequencyofarotarinagravitationalfield.Thisnotonlyaffectstheinternalenergyoftherotar,butitalsoaffectstheamountofgravitygeneratedbytherotar.Whenwehavelookedatmass,energy,inertiaandgravityfromthepointofviewofazerogravityobserver,wehaveseenadifferencenotobviouslocally. SinceachangeinГaffectsmassandenergydifferently zerogravityobserverperspective andsincegravityscaleswithenergy notrestmass ,tobetechnicallycorrectthegravitationalequationsshouldbewrittenintermsofenergy,notmass.Thetransformationsandinsightsprovidedherehaveforcedustorecognizethattheterm“mass”isaquantificationofinertia.Massisnotsynonymouswithmatterandmassscalesdifferentlythanenergywhenviewedbyanobserverusingthezerogravitycoordinaterateoftime.

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Personal Note: Iwanttotelltwoshortpersonalstoriesthatrelatetothischapter.Thefirststoryhastodowiththegravitationaleffectonrotarfrequency.Inormallyrunalmosteveryday.Someofthebestideasinthisbookcametomeduringthesedailyruns.ManyyearsagoIusedtorunonflatgroundbuttopreservemykneesInowrunupasteepsectionofahillandwalkdowntothestartingpoint.Irepeatthisfor½hourwhichtypicallyisabout15roundtrips.AsIrunupthehillIoftenamawarethattheworkthatIamdoingisultimatelyresultinginanincreaseintheComptonfrequencyofalltheelectronsandquarksinmybody.Locallythereisnomeasurablechangeinthe Compton frequencies of these rotars, but using the absolute time scale of a zero gravityobserver, I am increasing the frequency of these particles. It somehow is comforting tounderstandthephysicsofrunningupahill.Theconceptof“gravitationalpotentialenergy”hasbeendemystified.Inowunderstandwhyitisdifficulttorunupahill.The second story is about the experienceof resolvingamystery about gravity. When Iwasinitiallywritingthisbook,IthoughtthatIhadunlockedthekeytounderstandinggravitywhenIhaddevelopedtheconceptspresentedinchapter6 theconceptsthatarenowregardedasbeingoversimplified .ThemagnitudeofthegravitationalforcewascorrectandIthoughtIcouldeasilyextrapolate to largerdistancesand largermass. Then it occurred tome that thevectorwaswronganditwasobviousthatIwasmissingothermajorconcepts.Iwasfarfromfinishingmyquesttoexplainimportantaspectsofgravity.My initial reaction was to try to rationalize changes that wouldmake the simplifiedmodelexplain the correct vector attraction rather than repulsion . This thought process wassomethingliketryingtoreverseengineergravity.Iwasattemptingtoworkbackwardsfromthedesiredresult attraction tofindthechangestothemodelthatwouldgivethedesiredresult.Ispent a long time working backwards from result to cause, but it was getting nowhere.Therefore,infrustrationIreturnedtotheapproachthatIhadpreviouslyusedtodevelopthemodeltothatpoint.Thatapproachmerelymovedforwardfromthestartingassumption theuniverseisonlyspacetime andacceptedthelogicalextensionsofthisassumption.OnceIgotbackonthistrack,Irealizedthattheenergydensityoftherotarmodelimpliedpressure.WhenItookthelogicalstepstocontainthispressure,Ieventuallyobtainednotonlythegravitationalforcewith the correct vector but also obtained improved insights into the strong force, theelectromagneticforceandthestabilityoffundamentalparticles.While otherpeople are attempting to adjustmodels to explain specific physical effects, I amfinding that logically extending the starting assumption gives unexpected explanations. Theexpandedmodelexplainsdiverseeffectsnotinitiallyunderconsideration. Theseexperienceshavegivenmeagreatdealofconfidenceinthismodelandapproach.