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Chapter8 GravitationalAttractionandUnificationofForces Inchapter6,thereaderwasaskedtotemporarilyconsiderallforcestoberepulsive.Thiswasa simplification which allowed the calculations in chapter 6 to proceed without addressing the morecomplicatedsubjectofattraction.Inchapter7,vacuumenergy/pressurewasintroduced asanessentialconsiderationinthegenerationofallforces,butespeciallyforcesthatproduce attraction. In this chapter we are going to attempt to give a conceptually understandable explanationfortheforceofattractionexertedbygravitywhenabodyisheldstationaryrelative toanotherbody. Physical Interpretation of General Relativity: Einstein’s general relativity has passed numerousexperimentalandmathematicaltests.Thismathematicalsuccesshasconvincedmost physicists to accept the physical interpretation usually associated with these equations. However, the most obvious problem with the physical interpretation is examined in the following quotes. The first is from B. Haisch of the California Institute of Physics and Astrophysics. “The mathematical formulation of general relativity represents spacetime as curvedduetothepresenceofmatter….Geometrodynamicsmerelytellsyouwhat geodesicafreelymovingobjectwillfollow.Butifyouconstrainanobjecttofollow somedifferent path or not to move atall , geometrodynamicsdoes not tell you howorwhyaforcearises….Logicallyyouwinduphavingtoassumethataforce arisesbecausewhenyoudeviatefromageodesicyouareaccelerating,butthatis exactlywhatyouaretryingtoexplaininthefirstplace:Whydoesaforcearisewhen youaccelerate?…Thismerelytakesyouinalogicalfullcircle.” Talkingaboutcurvedspacetime,thebookPushingGravity M.R.Edwards states: “Logically,asmallparticleatrestonacurvedmanifoldwouldhavenoreasontoend itsrestunlessaforceactedonit.Howeversuccessfulthisgeometricinterpretation maybeasamathematicalmodel,itlacksphysicsandacausalmechanism.” Generalrelativitydoesnotexplainwhymass/energycurvesspacetimeorwhythereisaforce whenanobjectispreventedfromfallingfreelyinagravitationalfield.Ifrestraininganobject from following a geodesic is the equivalent of acceleration, then apparently the gravitational forceisintimatelytiedtothepseudoforcegeneratedwhenamassisaccelerated.Inthestandard model,particlespossessnointrinsicinertia.TheygaininertiafromaninteractionwiththeHiggs field.IstheHiggsfieldalsonecessarytogenerateagravitationalforcewhenaparticlewithout The Universe Is Only Spacetime ©2012 [email protected] 8-1 intrinsicinertiaispreventedfromfollowingthegeodesic?IstheHiggsfieldalwaysaccelerating towards a mass in an endless flow that attempts to sweep along a stationary particle? The standardmodeldoesnotincludegravity.Isgravityaforce? Thepointofthesequestionsistoshowthattherearelogicalproblemswiththephysicalmodel normally associated with general relativity. The equations of general relativity accurately describegravityonamacroscopicscale.However,theseequationsaresilentastothephysical interpretation,especiallyatascale spatialortemporal wherequantummechanicstakesover. Gravitational Nonlinearity Examined: Previously we reasoned that spacetime must be a nonlinear medium for waves in spacetime and gravity is the result of this nonlinearity. At distance fromarotarwecalculatedthegravitationalforceusingoneofthe5wave‐amplitude equationsF A2ω2Z /c.InthiscalculationwesubstitutedA Aβ2 Lp/ 2 Tpωc 2.Also theangularfrequencyωisequaltotherotar’sComptonfrequencyω ωc.Atdistance oftwo ofthesamerotarsweobtained F Gm2/ .Thisisthecorrectmagnitudeofthegravitational force between two rotars of mass m, but there are two problems. First: the equation F A2ω2Z /c is for a traveling wave striking a surface. This traveling wave implies the radiationofpowerthatisnothappening.Second:awaveinspacetimetravelingatthespeedof lightgeneratesarepulsiveforceifitinteractsinawaythatthewaveisdeflectedorabsorbed. Gravity is obviously an attractive force. We have the magnitude of the force correct, but the modelmustberefinedsothatthereisnolossofpowerandsothattheforceisanattraction. Thereareseveralstepsinvolved,anditisprobablydesirabletobeginwithabriefreview.Recall thatwearedealingwithdipolewavesinspacetimewhichmodulateboththerateoftimeand volume. There are two ways that we can express the amplitude of the dipole in spacetime: displacement amplitude and strain amplitude. The maximum displacement of spacetime allowed by quantum mechanics is a spatial displacement of Planck length or a temporal displacementofPlancktime.Sincetheseareoscillationamplitudes,wesometimesusetheterm “dynamic Planck length Lp” or “dynamic Planck time Tp”. As previously explained, these distortionsofspacetimeproduceastraininspacetime.Thestrainisadimensionlessnumber equivalenttoΔl/lorΔt/tInthiscaseΔl/l Lp/ andΔt/t Tpωc. Inchapter5weimaginedahypotheticalperfectclockplacedatapointonthe“Comptoncircle” ofarotarasillustratedinfigure5‐1.Thisistheimaginarycirclewithradiusequaltotherotar radius .Thisclock hereaftercalledthe“dipoleclock”withtimeτd wascomparedtothetime onanotherclockthatwecalledthe“coordinateclock” withtime tc .Thiscoordinateclockis measuringtherateoftimeiftherewasnospacetimedipolepresent.Itisalsopossibletothink ofthecoordinateclockaslocatedfarenoughfromtherotatingdipolethatitdoesnotfeelany significanttimefluctuations.Figure5‐3showsthedifferenceintheindicatedtime Δt τd‐tc. Thedipoleclockspeedsupandslowsdownrelativetothecoordinateclockandthemaximum difference is dynamic Planck time Tp. Therefore Tp is the temporal displacement amplitude. The Universe Is Only Spacetime ©2012 [email protected] 8-2 ThereisalsospatialdisplacementamplitudethatisequaltodynamicPlancklengthLp.Thestrain ofspacetimeproducedbythesedisplacementsofspacetimeisdepictedinfigure5‐4.Withinthe rotarvolume,thestrainofspacetimeisdesignatedbythestrainamplitude Aβ Tpωc Lp/ . Thisisequivalenttothemaximumslopeofthesinewavewhichoccursatthezerocrossingpoints infigure5‐3. Nonlinear Effects: The above review now has brought us to the point where we can ask interestingquestions:Doesthedipoleclockalwaysreturntoperfectsynchronizationwiththe coordinate clock at the completion of each cycle? Does the volume oscillation of spacetime produce a net change in the average volume near a rotar? Since we are going to initially concentrateonexplainingthegravitationalforcebetweentwofundamentalparticles,wewill initiallyconcentrateontheeffectontime.Therefore,doestherateoftimeoscillationcausethe dipoleclocktoshowanetlossoftimecomparedtothecoordinateclock?Ifspacetimehasno nonlinearity then the clocks would remain substantially synchronized. However, if there is nonlinearity,thedipoleclockwouldslowlylosetime. Aspreviouslyexplained,thestrainofspacetime instantaneousslopeinfigure5‐3 hasalinear componentandanonlinearcomponent.Theproposedspacetimestrainequationforapointon theedgeoftherotatingdipoleis: Strain Aβsinωt Aβsinωt 2… higherordertermsignored Thelinearcomponentis“Aβsinωt”andthefirstterminthenonlinearcomponentis Aβsinωt 2. TherewouldalsobehigherordertermswhereAβisraisedtohigherpowers,butthesewouldbe sosmallthattheywouldbeundetectableandwillbeignored.Thisnonlinearcomponentcanbe expanded: Aβsinωt 2 Aβ2sin2ωt ½Aβ2–½Aβ2cos2ωt The Universe Is Only Spacetime ©2012 [email protected] 8-3 Figure 8‐1 plots the linear component Aβ sinωt and the nonlinear component Aβ sinωt 2 separately.Itcanbeseenthatthenonlinearcomponentisasmalleramplitudebecause Aβ 1 and squaring this produces a smaller number. Also the nonlinear component is at twice the frequency of the linear component. Most importantly, the nonlinear component is always positive.Makinganelectricalanalogy,thiscanbethoughtofasifthenonlinearwavehasanAC componentandaDCcomponent.Itisobviousthatwhenthelinearandnonlinearwavesare added together, the sum will produce an unsymmetrical wave that is biased in the positive direction. Itwasnecessarytousesomeartisticlicenseinordertoillustratetheseconceptsinfigure8‐1. ForfundamentalrotarsthevalueofAβisroughlyintherangeof10‐20.ThismeansthatAβ2 10‐40 andthereforeAβisapproximately1020timeslargerthanAβ2.Itwouldbeimpossibletoseethe plotofAβ2sin2ωtwithoutartificiallyincreasingthisrelativeamplitude.Therefore,theassumed valueinthisfigureisAβ 0.2.InthiscasethedifferencebetweenAβandAβ2isonlyafactorof5 ratherthanafactorofroughly1020.Therefore,forfundamentalrotarsitisnecessarytomentally decreasetheamplitudeofthenonlinearwavebyroughlyafactorofroughly1020. The Universe Is Only Spacetime ©2012 [email protected] 8-4 Whenweaddthetwowavestogetherweobtaintheplotinfigure8‐2.Becauseoftheartistic license,itisvisuallyobviousthatthisisanunsymmetricalwave.Thereisalargerareaunderthe positiveportionofthewavethantheareaunderthenegativeportionofthewave.Thepeak amplitude for the positive portion is Aβ Aβ2 while the negative portion has peak negative The Universe Is Only Spacetime ©2012 [email protected] 8-5 amplitude of: – Aβ Aβ2. If this was a plot of electrical current, we would say that this unsymmetricalwavehadaDCbiasonanACcurrent.Tousetheanalogyfurther,itisasifthe nonlinearitycausesspacetimetohavetheequivalentofasmallDCbiasinitsstress. Thedipoleclockdoesnotreturntosynchronizationwiththecoordinateclockeachcycle.Figure 8‐3 attempts to illustrate this with a greatly exaggerated plot of the difference between the coordinate clock and the dipole clock. The “X” axis of this figure is time as indicated on the coordinateclockwhilethe“Y”axisisthedifferencebetweenthecoordinateclockandthedipole clock t–τ .Ifthetwoclocksranatexactlythesamerateoftime,theplotwouldbeastraight linealongthe“X”axis.Normallythisplotforafewcyclesshouldlooklikeasinewavesimilarto figure5‐3.However,thepurposeoffigure8‐3istoillustratethatovertimethecoordinateclock pulls ahead of the dipole clock or the dipole clock loses time . Therefore, for purposes of illustration,thiseffectoftheaccumulatedtimedifferencehasbeenexaggeratedbyafactorof roughly1022. The unsymmetrical strain plot in figure 8‐2 produces a net loss of time on the dipole clock relativetothecoordinateclock.Inthefirstquartercycleoffigure8‐3,thecoordinateclockfalls behindthedipoleclockbyanamountapproximatelyequaltoPlancktime.Thisoccurswhenthe fast lobe of the rotating dipole passes the dipole clock first. However, with each cycle, the coordinateclockgainsasmallamountoftimeonthedipoleclock.Theamountoftimegained percycleisillustratedbythegaplabeled“Singlecycletimeloss”.ThisisequaltoTp2ωcwhichis about2.2 10‐66sforanelectron. Thepointoffigure8‐3istoillustratethecontributionofthenonlineareffect.Thenonlinear wavewithstrainof Aβ2sin2ωtatdistance producesthecontributionthatcausesthenetloss oftimeforthedipoleclockrelativetothecoordinateclock.Thisnettimedifferencebetweenthe twoclocks aftersubtractingAβsinωt isshownasthewavylinelabeled“nonlinearcomponent”. Theaverageslopeofthislineisequaltothegravitationalmagnitudefortherotarvolumewhich hasbeendesignatedasAβ2 βq.Foranelectronthisslopeisabout1.75x10‐45whichmeansthat ittakesabout30secondsforthecoordinateclocktohaveanettimegainofPlancktimeoverthe dipoleclock.Thistakesabout4x1021cyclesratherthan4cyclesasillustratedinfigure8‐3.If wesubtractedthenonlinearwavecomponentfromfigure8‐3,wewouldbeleftwithasinewave withamplitudeofTp. Theslopeonthisnonlinearwavecomponentis Aβ2atdistance whichisobtainedfromthe strainequation–theimportantpartishighlightedbold Aβsinωt Aβsinωt 2 Aβsinωt–½Aβ2cos2ωt ½ Aβ2 Derivation of Curved Spacetime: TheDCequivalentterm non‐oscillatingterm isAβ2.Thisis the nonlinear strain in spacetime produced by the rotar at distance . This is an important The Universe Is Only Spacetime ©2012 [email protected] 8-6 conceptsinceitrelatestocurvedspacetime.Beforeperusingthisthoughtfurtheritisnecessary tointroduceanewsymbol: .Thenaturalunitoflengthforarotaris λc.Thereforewewill designatetheradialdistancefromarotarnotinunitsoflengthsuchasmeters,butasthenumber ofreducedComptonwavelengths numberofrotarradiusunits . ≡ ħ For example, the non‐oscillating strain in spacetime produced by a rotar should decrease proportionalto1/ .Wecantestthisideasincetheproposalisthatthenon‐oscillatingstrainin spacetime at distance λc where 1 is equal toAβ2 and this decreases as 1/ . We will evaluateAβ2/ 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. Thissimpleevaluationisanothersuccessofthismodelbecausetheterm Gm/c2ristheweak gravitycurvatureofspacetimeproducedbymassmatdistancedr.Forexample,thepreviously defined gravitational magnitude is β ≡ 1 – dτ/dt . The weak gravity temporal distortion of spacetimeis:dt/dτ 1 Gm/c2r .Inflatspacetimedt/dτ 1,sotheweakgravitycurvature termisβ Gm/c2r .Forfundamentalparticles rotars atdistance thistermisintherange of10‐40,sothisisvirtuallyexact. The question of how matter “causes” curved spacetime has been a major topic in general relativityandquantumgravity.Nowweseethemechanismofhowdipolewavesinspacetime producebothmatterandcurvedspacetime.Thisusesequationsfromquantummechanicsto deriveanequationfromgeneralrelativity.Thisisnotonlyasuccessfultestofthespacetime based model, but it is also a prediction of this model of the mechanism that achieves curved spacetime. Oscillating Component of Gravity: There is proposed to be another residual gravitational effectthathasnotbeenobservedbecauseitisaveryweakoscillationatafrequencyinexcessof 1020Hz.Infigure8‐3thenon‐linearwavecomponentisshownasawavylinelabeled“nonlinear component”.Wecaninteractwiththenon‐oscillatingpartofthislineresponsibleforgravity, but there is also a residual nonlinear oscillating component. At distance this oscillating componenthasamplitude Aβ2andfrequency 2ωc.Whathappenstothisoscillatingcomponent beyond intheexternalvolume?Weknowthatthefewfrequenciesthatformstableandsemi The Universe Is Only Spacetime ©2012 [email protected] 8-7 stable rotars exist at resonances with the vacuum fluctuations of spacetime which eliminate energy loss. If the amplitude of the oscillating component was Aβ2/ , then there would be continuous radiation of energy. Energetic composite particles such as protons or neutrons wouldradiateawayalltheirenergyinafewmillionyears.Inthechapter10ananalogywillbe made to the energy density of a rotar’s electric field. The amplitude term for the oscillating componentofgravitywillthenbeproposedtoscaleasAβ2/ 2.Thiswouldbeanextremelysmall amplitude and furthermore it is a standing wave that does not radiate energy. However, this oscillatingcomponentwouldtheoreticallygiveenergydensitytoagravitationalfield.Theenergy density of a gravitational field and its contribution to producing curved spacetime will be discussedattheendofchapter10.Theoscillatingcomponentofagravitationalfieldmayalsobe importantintheevolutionoftheuniverse.Thiswillbediscussedinchapters13and14. Summary: Since we need to bring together several different components to achieve gravitational attraction, we will do another review. This time we will emphasize the role of vacuumenergy,circulatingpower,thecancelingwaveandnon‐oscillatingstraininspacetime.A rotarisarotatingspacetimedipoleimmersedinaseaofvacuumenergywhichisequivalentto a vacuum pressure. This vacuum energy/pressure is made up of very high energy density 1046J/m3 dipolewavesinspacetimethatlackangularmomentum.Therotaralsohasahigh energydensitythatisattemptingtoradiateawayenergyattherateoftherotar’scirculating power.Therotarsurvivesbecauseitexistsatoneofthefewfrequenciesthatachievearesonance withthevacuumenergy/pressure. Thisresonancecreatesanewwavethathasacomponentthatpropagatesradiallyawayfromthe rotating dipole and a component that propagates radially towards the rotating dipole. Tangential wave components are also created, but these add incoherently and effectively disappear. The resonant wave that is propagating away from the dipole cancels out the fundamental radiation from the dipole. Besides having the correct frequency and phase to produce destructive interference, the canceling wave also must match the rotar’s circulating power.Thismeansthatthecorrectpressureisgeneratedfromthevacuumenergy/pressure thatisrequiredtocontaintheenergydensityoftherotar.Onlyafewfrequenciesthatformstable rotarscompletelysatisfytheseconditions. Forexample,anelectronhasacirculatingpowerofabout64millionwatts.Inordertocancel thismuchpowerfrombeingradiatedfromtherotarvolume,thecancelationwavegeneratedin the vacuum energy must have an outward propagating component of 64 million watts attemptingtoleavetherotar’svolumeandaninwardpropagatingcomponentofthesamepower. Therecoilfromtheoutwardpropagatingcomponentprovidesthepressurerequiredtostabilize the rotating dipole that is the rotar the electron . This pressure can be thought of as being carriedbytheinwardpropagatingcomponentthatreplenishestherotatingdipole. The Universe Is Only Spacetime ©2012 [email protected] 8-8 Ifitwaspossibletoseethisprocess,wewouldnotseeoutwardorinwardpropagatingwaves. Wewouldonlyseethesumofthesetwowaveswhichisastandingwavewhichdecreasesin amplitudewithdistancefromthecentralrotar.Astandingwaveintherotar’sexternalvolume means that no power is being radiated. These standing waves cause the rotar’s electric field discussedlater .Wewouldalsoseethattherewasaslightnon‐oscillatingresidualstrainin spacetimewithstrainamplitudeofAβ2/ gm/c2r. Newtonian Gravitational Force Equation:Therearestilltwomorestepsbeforewearriveat theexplanationthatgivesthecorrectattractingforceatarbitrarydistancebetweentworotars. We will start by assuming two of the same rotars mass m separated by distance r. It was previously explained that deflecting all of a rotar’s circulating power generates the rotar’s maximumforceFm.Arotaralwaysdependsonthepressureofthespacetimefieldtocontainits circulatingpower.Whentherotarisisolated,theforcerequiredtodeflectthecirculatingpower isbalanced.However,agravitationalfieldproducesagradientinthegravitationalmagnitude dβ/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.Imaginemassm1beingarotar rotatingdipole attemptingtodispersebutbeingcontainedbypressuregeneratedwithinthe vacuumenergy/pressurepreviouslydiscussed.Thispressureexactlyequalsthedispersiveforce of the dipole wave rotating at the speed of light. However, if there is a gradient in the gravitationalmagnitude dβ/drthenthereisagradientacrosstherotarwhichwewillcall Δβ. Thisaffectsthenormalizedspeedoflightandthenormalizedunitofforceonoppositesidesof therotar.Recallfromchapter3wehad: Co ГCgnormalizedspeedoflighttransformation Fo ГFgnormalizedforcetransformation Г 1 βapproximationconsideredexactforrotars Therefore,becauseofthestraininspacetime,thetwosidesoftherotar separatedby are livingunderwhatmightbeconsideredtobedifferentstandardsforthenormalizedspeedoflight andnormalizedforce.Onanabsolutescale,ittakesadifferentamountofpressuretostabilize theoppositesidesoftherotarbecauseofthegradientΔβacrosstherotar.Thenetdifferencein thisforceistheforceofgravityexertedontherotar. Wewillfirstcalculatethechangeingravitationalmagnitude Δβacrosstherotarradius ofa rotar when it is in the gravitational field of another similar rotar another rotar of the same The Universe Is Only Spacetime ©2012 [email protected] 8-9 mass .Inotherwords,wewillcalculatethedifferenceinβatdistanceranddistancer from arotarofmassm. Δβ approximationvalidif r Theforceexertedbyvacuumenergy/pressureonoppositehemispheresoftherotarisequalto the maximum force Fm m2c3/ħ. The difference in the force absolute value exerted on oppositesidesoftherotaristhemaximumforcetimesΔβ.Therefore,theforcegeneratedbytwo rotarsofmassmseparatedbydistanceris: F ΔβFm ħ ħ ħ Ifwehavetwodifferentmassrotars mass m1andmass m2 ,thenwecanconsidermass m1in thegravitationalfieldofmassm2.Inthiscase, andFmareformassm1andΔβischangeinthe gravitationalmagnitudefrommassm2acrosstherotarradius frommassm1. Fg ΔβFm Fg ħ ħ ħ Newtoniangravitationalforceequationderivedfromadipolewavemodel Gravitational Attraction:WehavederivedtheNewtoniangravitationalequationfromstarting assumptions,butwestillhavenotshownthatthisisaforceofattraction.However,fromthe previousconsiderations,thislaststepiseasy.Thereisaslightlydifferentpressurerequiredto stabilizetherotardependingonthelocalvalueofβorГ inweakgravityГ 1 β .Thiscanbe consideredasadifferenceinnetforceexertedbyvacuumenergyonthehemisphereoftherotar thatisfurthestfromtheotherrotarcomparedtothehemispherethatisnearesttheotherrotar. The furthest hemisphere has a smaller average value of Г than the nearest hemisphere. The normalizedspeedoflightisgreaterandthenormalizedforceexertedonthefarthesthemisphere mustbegreatertostabilizetherotar.ThisproducesanetforceinthedirectionofincreasingГ. Themagnitudeofthisforceis F Gm1m2/r2andthevectorofthisforceisinthedirectionof increasingГ towardstheothermass . Weconsiderthistobeaforceofattractionbecausethetworotarswanttomigratetowardseach other increasingГ .However,theforceisreallycomingfromthevacuumenergyexertinga repulsive pressure. There is greater normalized pressurebeing exerted on the side with the lower Г. The two rotars are really being pushed together by a force of repulsion that is unbalanced. The Universe Is Only Spacetime ©2012 [email protected] 8-10 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:Wewilldoaplausibilitycalculationtoseeifweobtain roughlythecorrectgravitationalforceforanelectronintheearth’sgravitationalfieldbasedon the above explanation. We will be using values for the electron’s energy density and the electron’smaximumforcethatwerepreviouslycalculatedbyignoringdimensionlessconstants. Therefore, we will continue with this plausibility calculation that ignores dimensionless constants. An electron has internal energy of Ei 8.19x10‐14 J and a rotar radius of 3.86x10‐13 m. Ignoring dimensionless constants, this gives an energy density of about Ei/ 1.4x1024J/m3.Thisrotarmodelofanelectronisexertingapressureofroughly1.4x1024 N/m2.Thispressureoverareaλc2producestherotar’smaximumforcewhichforanelectronis Fm 0.212N obtainedfromℙq 1.4x1024N/m2 3.86x10‐13m 2 0.212N . Theweakgravitygravitationalmagnitudeis: β Gm/c2r.Fortheearth m 5.96x1024kgand theequatorialradiusis:r 6.37x106m.Therefore,atthesurfaceoftheearththegravitational magnitudeis:β 6.95x10‐10.Toobtainthegradientinthismagnitudewedividebytheearth’s equatorial radius 6.37x106 m to obtain a gradient of dβ/dr 1.091x10‐16/m. The change in gravitationalmagnitudeΔβacrosstherotarradius 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 the spacetimefield.However,thehomogeneousspacetimefieldinzerogravityismodifiedbythe earth’sgravitationalfield.Aspreviouslycalculated,gravityaffectsnotonlytherateoftimeand propervolume,butalsotheunitofforce,energy,etc.Thepreviouslycalculatednormalizedforce transformationis:Fo ГFg.Thegradientintheearth’sgravitationalfieldmeansthataslightly differentvalueofΓexistsonoppositesidesoftheelectron.Thisismoreconvenientlyexpressed asadifferenceinthegravitationalmagnitudeΔβthatexistsacrosstheelectron’sradiusλc.There will be a slight difference in the force exerted by the spacetime field exerted on opposite hemispheresoftheelectron rotar .Calculatingthisdifferenceshouldequalthemagnitudeof thegravitationalforceontheelectron. F ΔβFm 4.213x10‐29x0.212N 8.89x10‐30N Wewillnowcheckthisbycalculationtheforceexertedonanelectronbytheearth’sgravityusing F mgwheretheearth’sgravitationalaccelerationis:g 9.78m/s2 F mg 9.1x10‐31kgx9.78m/s2 8.89x10‐30N The Universe Is Only Spacetime ©2012 [email protected] 8-11 Success!TheanswerobtainedfromthecalculationusingF ΔβFmisexactlycorrect.Apparently theignoreddimensionlessconstantscancel.Thisisanothersuccessfulplausibilitytest. At the beginning of this chapter two quotes were presented that pointed out that general relativitydoesnotidentifythesourceoftheforcethatoccurswhenaparticleisrestrainedfrom followingthegeodesic.M.R.Edwardsstates:“Howeversuccessfulthisgeometricinterpretation maybeasamathematicalmodel,itlacksphysicsandacausalmechanism.”Theideasproposed in this book give a conceptually understandable explanation for both the magnitude and the vectordirectionofthegravitationalforce.Thegravitationalforcewasobtainedfromthestarting assumptionswithoutusinganalogyofacceleration. Electrostatic Force at Arbitrary Distance:Thestrainamplitudeofthespacetimewaveinside therotarvolumehasbeendesignatedwiththesymbol Aβ Lp/λc Tpωc.Wehavejustshown thatthegravitationaleffectexternaltotherotarvolumescaleswithAg Aβ2/ gm/c2r.This is the gravitational curvature of spacetime produced by a rotar with radius λc and angular frequencyωc.Nowwewillexaminetheelectromagneticeffectonspacetimeproducedbythe effectofthefundamentalwavewithamplitudeAβ notsquared .Fromchapter6weknowthat thisamplitudeisassociatedwiththeelectrostaticforce.Nowwewillextendthistoarbitrary distance.Asbefore,weneedtomatchtheknownamplitudeatdistance λc.Thisisachievedby scalingdistanceusing ≡ r/λcbecause 1atdistance λc.Wewillagainassumethatthe electrostatic amplitude AE decreases as 1/ for the electrostatic force we assume AE Aβ/ Lp/λc λc/r .WewillusetheequationF = kA2ω2Z /c and also insert the following: F = FE, ω = ωc, Z = Zs = c3/G, r/λc, = kλc2 and ħc = ⁄4 = /c = ħ Therefore, we have generated the Coulomb law equation where the charge is q qp Planck charge . It should not be surprising that the charge obtained is Planck charge rather than ħ about11.7timeschargee andisbased elementarycharge e.Planckchargeis qp 4 on the permittivity of free space εo. Planck charge is known to have a coupling constant to photonsof1whileelementarychargeehasacouplingconstanttophotonsofα,thefinestructure constant. This calculation is actually the maximum possible electrostatic force which would require a coupling constant of 1. The symbol FE implies the electrostatic force between two PlanckchargeswhileFeimpliestheelectrostaticforcebetweentwoelementarychargese.The conversionisFE Feα‐1. Wewillcontinuetousetheequation: F = kA2ωc2Zs /c even though it implies the emission of power which is striking area andexertingarepulsiveforce.Thisisnothappeningbutthe useofF = kA2ωc2Zs /c gives the correct magnitude of forces. This simplified equation allows a lot of quick calculations to be made which give correct magnitude. The Universe Is Only Spacetime ©2012 [email protected] 8-12 Calculation with Two Different Mass Particles:Untilnowwehaveassumedtwoofthesame mass/energyparticleswhenwecalculated Fgand FE.Nowwewillassumetwodifferentmass particles m1andm2 ,butwewillkeeptheassumptionthatbothparticleshavePlanckcharge. When we have two different mass particles, this means that we have two different reduced . The single radial separation r now Compton wavelengths λc1 ħ/ c and λc1 ħ/ becomes two differentvalues 1 r/λc1 and 2 r/λc2. Also, there would be twodifferent strainamplitudesAβ1 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:Inchapter6wefoundthatthereisalogicalconnectionbetweenthe gravitationalforceandtheelectromagneticforce.However,thosecalculationsweredoneonly for separation distance equal to λc. Now we will generate some more general equations for arbitrary separation distance expressed as , the number of reduced Compton wavelengths. Thefollowingequationscouldbemadeassumingtwodifferentmassparticles,butitiseasierto returntotheassumptionofbothparticleshavingthesamemassbecausethenwecandesignate asinglevalueof separatingtheparticles.SomeofthefollowingequationsassumePlanck charge qp with force designation FE rather than charge e designated with force Fe. The conversionisFE Feα–1. WewillstartbyconvertingtheNewtongravitationalequationandtheCoulomblawequationso thattheyarebothexpressedinnaturalunits.Thismeansthatboththeforcesandtheparticle’s energywillbeinPlanckunits Fg Fg/Fp, FE FE/Fp, Ei Ei/Ep, .Alsoseparationdistance willbeexpressedintheparticlesnaturalunitoflength,thenumber r/λcofreducedCompton wavelengths. Also, weassume two particles each have the same mass/energy and they both havePlanckcharge. Convertbothequations:FE qp2/4πεor2andFg Gm2/r2intoequationsusingFg;FEand . Substitutions:r ħc/Ei;m Ei/c2;Ei EiEp Ei ħ ⁄ ħ FE Fg = ħ ħ The Universe Is Only Spacetime ©2012 ħ ħ [email protected] Ei2/ 2 Ei4/ 2 8-13 (Fg ) FE 2 2 2 Ei4 Theequation (Fg 2) FE 2 2clearlyshowsthatevenwitharbitraryseparationdistancethe squarerelationshipbetweenFgandFEstillexists.Itisinformativetostatethesesameequations intermsofpowerbecauseafundamentalassumptionofthisbookisthatthereisonlyonetruly fundamentalforceFr Pr/c.Ifthisiscorrect,thenwewouldexpectthattheforcerelationship between rotars would also be a simple function of the rotar’s circulating power: ⁄ħ.Toconvert PctodimensionlessPlanckunits Pc Pc/PpwedividebyPlanckpower Pp c5/G.Notethesimplicityoftheresult. FE Pc/ 2 Fg Pc2/ 2 So far we have used dimensionless Planck units because they show the square relationship between forces mostclearly. However, we will nowswitch and use equations with standard units. Thenextequationwillfirstbeexplained withanexample.Wewillassumeeithertwo electrons or two protons both charge e and we hold them apart at an arbitrary separation distancer.Asbefore,thisseparationdistancewillbedesignatedusingthenumber ofreduced Comptonwavelengths,therefore r Nλc.Protonsarecompositeparticles,butwecanstilluse theminthisexampleifweusetheproton’stotalmasswhencalculating . Nowweimaginealogscaleofforce.Atoneendofthisforcescaleweplacethelargestpossible forcewhichisPlanckforce Fp c4/G.Attheotherendofthislogscaleofforceweplacethe gravitational force Fg which is weakest possible force between the two particles either 2 electronsor2protons .Nowforthemagicalpart!Exactlyhalfwaybetweenthesetwoextremes onthelogscaleofforceisthecompositeforceFe α–1.Inwords,thisistheelectrostaticforceFe between the two particles times the number of reduced Compton wavelengths times the inverseofthefinestructureconstant α‐1 137 .Particlephysicistsliketotalkaboutvarious symmetries.Iamclaimingthatthereisaforcesymmetrybetweenthegravitationalforce,Planck forceandthecompositeforceFe α–1.Theequationforthisis: Itisinformativetogiveanumericalexamplewhichillustratesthisequation.Supposethattwo electronsareseparatedby68nanometers thisdistancesimplifiesexplanations .Theelectrons experience both a gravitational force Fg and an electrostatic force Fe. The electrons have λc 3.86x10‐13 m, therefore this separation is equivalent to 1.76x105 reduced Compton wavelengths. The gravitational force between the two electrons at this distance would be Fg 1.2x10‐56NandtheelectrostaticforcewouldbeFe 5x10‐14N.Alsoα –1 137socombining The Universe Is Only Spacetime ©2012 [email protected] 8-14 Fe, andα –1wehave: Fe α–1 1.2x10‐6N.AlsoPlanckforceis Fp c4/G 1.2x1044N.To summarizeandseethesymmetrybetweentheseforces,wewillwritetheforcesasfollows: Fg 1.2 10 NFgfortwoelectronsat6.8x10‐8mis1050timessmallerthanFe /α Fe α –1 1.2 10 NFe /αfortwoelectronsat6.8x10‐8m Fp 1.2 10 NFp Planckforce is1050timeslargerthanpreviousFe /α AnotherwayofstatingthisrelationshipwouldsetPlanckforceequalto1.ThereforewhenFp 1 thenFe α –1 10‐50andFg 10‐100.Thenumericalvaluesarenotimportantbecausedifferent mass particles or different separation distance could be used. The important point is the symmetrybetweenFp,FeandFgwhenweinclude andα‐1inthecompositeforceFe α –1. Force Ratios Fg/FE and Fg/Fe α-1: Nextwewillshowhowthewavestructureofparticlesand forcesdirectlyleadstoequationswhichconnecttheelectrostaticforceandgravity.Previously westartedwiththewave‐amplitudeequation F = kA2ωc2Zs /cwhichisapplicabletowavesin spacetime. In this equation A isstrain amplitude, ωc is Comptonangular frequency, Zs is the impedanceofspacetimeZs c3/Gand isparticlearea.Wehaveshownthattheinsertingthe strainamplitudetermA Aβ/ intoF = kA2ωc2Zs /cgivestheelectrostaticforcebetweentwo PlanckchargesFE,.Wehavealsoshownthatgravityisanonlineareffectwhichscaleswithstrain amplitudesquared Aβ2 .InsertingA Aβ2/ intothisequationgivesthegravitationalforceFg betweentwoequalmassparticles.SincewehaveequationswhichgenerateFEandFg,,weshould be able to generate new equations which give the ratio of forces Fg/FE. In the following Aβ Lp/λc Tpωc.Forgravity,A Ag Aβ2/ andforelectrostaticforceA AE Aβ/ . Fg = k(Aβ2/ )2ωc2Zs /c Fg = gravitationalforcebetweentwoofthesamemassparticles FE = k(Aβ/ )2ωc2Zs /c FE theelectrostaticforcebetweentwoparticleswithPlanckcharge Setcommontermsequaltoeachother: (kωc2Zs /c) = (kωc2Zs /c) 2 2 where: Aβ = Lp/λc = Tpωc = rotar strain amplitude 2 2 Theequation Fg/FE Aβ2showsmostclearlythevalidityofthespacetimebasedmodelofthe universeproposedhere.Recallthatallfermionsandbosonsarequantizedwaveswhichproduce thesamedisplacementofspacetime.ThespatialdisplacementisequaltoPlancklength Lpand the temporal displacement is Planck time Tp. Even though all waves produce the same displacementofspacetime,differentparticleshavedifferentwavestrainamplitudesbecausethe strainamplitudeisthemaximumslope maximumstrain producedbythewave.Thereforea rotar’s strain amplitude is Aβ 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 = Aβ2. The Universe Is Only Spacetime ©2012 [email protected] 8-15 All the previous equations relating Fg and FE either specified either a specific separation or specifiedseparationdistanceusing .However,Fg/FE = Aβ2 doesnotspecifyseparation.This ispossiblebecausetheratioofthegravitationalforcetotheelectrostaticforceisindependentof distance.Forexample,theratioforanelectronisFg/Fe 2.4x10‐43.However,whenweadjust forthecouplingconstantassociatedwithchargee,theratiobecomes:Fg/Feα‐1 1.75x10‐45.The rotarstrainamplitudeforanelectronis Aβ Lp/λc 4.185x10‐23.Therefore Aβ2 1.75x10‐45 Clearly,thederivationofthisequationandthephysicsbehinditgivestrongproofofthewave‐ based structure of particles and forces. Next we will extend the relationship between these forcesonemoresteptobringanewperspective. ħ ħ ħ or: Theequation Fg/FE = Rs/λc isveryinterestingbecause Fg/Fe isaratioofforcesand Rs/λc isaratio ofradii.Recallthat Rs≡ Gm/c2 istheSchwarzschildradiusofarotarbecauseitwouldforma blackholerotatingatthespeedoflight.SuchablackholehashalftheSchwarzschildradiusofa non‐rotatingblackholetherefore Rs Gm/c2.Also,λc = ħ/mc istheradiusoftherotarmodelof afundamentalparticle.Forexample,foranelectronRs 1.24x10‐54mandanelectron’srotar radiusis:λc 3.85x10‐13m.Thesetwonumbersseemcompletelyunrelated,yettogetherthey equaltheelectron’sforceratioFg/Feα‐1 1.75x10‐45 Rs/λc. However,asthefollowingequationsshow,therearetwoamazingconnectionsbetweenRsand λc.First, Rsλc Lp2.Thesecond isRs λc‐1 Inwords,thissaysthattherotar’sSchwarzschild radius Rs≡ Gm/c2)istheinverseofthereducedComptonradiuswhenbothareexpressed in thenaturalunitsofspacetimewhicharedimensionlessPlanckunits Rsandλcunderlined . Rsλc ħ ħ Rs λc‐1equivalentto: ⁄ ⁄ The Schwarzschild radius comes from general relativity and is considered to be completely unconnected to quantum mechanics. A particle’s reduced Compton wavelength comes from quantum mechanics and is considered to be completely unconnected to general relativity. However,whentheyareexpressedinnaturalunits dimensionlessPlanckunits thetworadii . TherelationshipsbetweenRsand arejusttheinverseofeachother Rs 1/λc.Also The Universe Is Only Spacetime ©2012 [email protected] 8-16 λc are compatible with the wave‐based rotar model of fundamental particles but they are incompatiblewiththemessengerparticlehypothesisofforcetransmission. TosummarizealltheequationsequaltoFg/FE,plusafewmorewehave: ⁄ ⁄ ⁄ Aβ2 Rs2 λc–2 ωc2= Ei 2 Pc All the previous force equations also work with composite particles such as protons if the proton’stotalmassisusedtocalculatethevarioustermssuchas λc ħ/mc.Forexample,two protonshaveFg/Feα‐1 5.9x10‐39atanyseparationdistance.AlsoforprotonsRs/λc 5.9x10‐39 and Lp/λc 2 5.9x10‐39. Iwanttopauseforamomentandreflectontheimplicationsofallthepreviousequationswhich showtherelationshipbetweentheelectrostaticforceandthegravitationalforce.Tome,they clearlyimplyseveralthings.Theseare: 1 Gravitycanbeexpressedasthesquareoftheelectrostaticforcewhenseparationdistance isexpressedas multiplesofthereducedComptonwavelengthλc. 2 The equations relating the gravitational and electrostatic forces imply that they both scaleasafundamentalfunctionofaparticle’squantummechanicalpropertiessuchas ComptonwavelengthorComptonfrequency. 3 All the connections between the gravitational force and the electrostatic force are proposedtobetraceabletoarotargeneratingaComptonfrequencystandingwaveinthe surrounding volume. Spacetime is a nonlinear medium, so a single standing wave has bothafundamentalcomponent electrostatic andanonlinearcomponent gravity . 4 Theelectromagneticforceisuniversallyrecognizedasbeingarealforce.Theequations showthatgravityiscloselyrelated.Therefore,gravityisalsoarealforce. 5 These equations are incompatible with virtual photons transferring the electrostatic forceorgravitonstransferringthegravitationalforce.Theequationsalsoappeartobe incompatiblewithstringtheory. 6 There is a quantum mechanical connection between a particle’s rotar radius and its Schwarzschildradius. DerivationoftheEquations:IwanttorelateabriefstoryaboutthefirsttimethatIproveda relationshipbetweenthegravitationalandelectrostaticforces.Aspreviouslystated,theinitial ideathatledtothisbookwasthatlightconfinedinahypothetical100%reflectingboxwould exhibitthesameinertiaasaparticlewiththesameenergy.Theotherideasinchapter1followed quickly and I was struck with the idea that these connections between confined light and particles were probably not a coincidence. I will now skip ahead several years when I was methodically inventing a model of the universe using only the properties of 4 dimension spacetime. I had the idea of dipole waves in spacetime and quantized angular momentum forming a “rotar” that was one Compton wavelength in circumference and rotating at the The Universe Is Only Spacetime ©2012 [email protected] 8-17 particle’sComptonfrequency.Thisimpliedasphericalvolumewithradiusofλc 3.86x10‐13.I hadindependentlyderivedZs c3/Gwhichwaskeytoalltheotherequations.Ihaddeveloped thedimensionlesswaveamplitudeforanelectronwhichwas Aβ 4.185x10‐23.Animportant successwastosubstitutethesevaluesintoE A2ω2ZV/candIobtainedEi 8.19x10‐14Jforan electron. Thenextlogicalstepwastofindtheforcethatwouldexistbetweentwoelectronsatadistance equaltoλc.ThisdistancewasimpliedbecauseIwasusingtheonlyamplitudethatIknewwhich correspondedtoadistanceequaltoλc.WhenImadenumericalsubstitutionsforanelectroninto theequationF A2ω2Z /cIobtainedF 0.212N.Thiswasabout137timesgreaterthanthe electrostatic force between two electrons at the separation distance λc. I was quite happy becauseIrealizedthatthiswouldbethecorrectforceifthechargewasPlanckchargeqprather thanelementarycharge e.Thiswasactuallyapreferableresultbecauseitcorrespondedtoa couplingconstantof1,whichwasreasonableforthiscalculation.Thiswasunderstandableand itwasquiteexciting. NextIthoughtaboutgravity.Iknewthatgravitywasvastlyweakerthantheotherforcesand only had asingle polarity. I was reminded about the optical Kerr effectdiscussed in the last chapter.Thisnonlineareffectscaleswithamplitudesquared electricfieldsquared andalways produces an increase in the index of refraction of the transparent material a single polarity effect .Gravityhasasinglepolarityandisvastlyweakerthantheelectrostaticforce.Therefore, IdecidedtocalculatetheforceusingF A2ω2Z /c.andset:A Aβ2 1.75x10‐45,ωc 7.76x1020 λc2 1.49x10‐25m2.Usingapocketcalculator,therewasaEureka s‐1,Zs 4.04x1035kg/sand moment when I got the answer which exactly equaled the gravitational force between two electronsatthisseparation F 3.71x10‐46N . ThisstoryistoldbecauseIwanttosupporttheclaimthatthepredictioncamefirst.Itisoften saidthattheproofoftheaccuracyofanewidealiesinwhetheritcanmakeapredictionrevealing somepreviouslyunknownfact.Usuallytheproofrequiresanexperiment,butinthiscaseitwas asimplecalculation.Tomyknowledgethisisthefirsttimethatthegravitationalforcehasbeen calculatedfromfirstprincipleswithoutreferencingacceleration. Gravitational Rate of Time Gradient:Intheweakfieldlimit,itisquiteeasytoextrapolate fromthegravitationalmagnitude β producedbyasinglerotarataparticularpointinspaceto thetotalgravitationalmagnitudeproducedbymanyrotars.Naturemerelysumsthemagnitudes of all the rotars at a point in space without regard to the direction of individual rotars. The gravitationalaccelerationgwaspreviouslydeterminedtobe: g c2dβ/dr –c2d dτ/dt /dr. The Universe Is Only Spacetime ©2012 [email protected] 8-18 Agravitationalaccelerationof1m/s2requiresarateoftimegradientof1.11x10‐17secondsper secondpermeter.Theearth’sgravitationalaccelerationof9.8m/s2impliesaverticalrateoftime gradientof1.09x10‐16meter‐1.Thismeansthattwoclocks,withaverticalseparationofone meterattheearth’ssurface,willdifferintimeby1.09x10‐16seconds/second.Similarly,there is also a spatial gradient. A gravitational acceleration also implies that there is a difference betweencircumferentialradiusRandradiallengthL.Intheearth’sgravitythisspatialdifference is1.09x10‐16meters/meter β 1–dR/dL . Outofcuriosity,wecancalculatehowmuchrelativevelocitywouldberequiredtoproducea timedilationequivalenttoaonemeterelevationchangeintheearth’sgravity.Ifelevation2is1 meterhigherthanelevation1,then:dt1/dt2 1–1.09x10‐16.Usingspecialrelativity: v 1 setdt1/dt2 1–1.09x10‐16 v 4.4m/s This4.4m/svelocityisexactlythesamevelocityasafallingobjectachievesafterfallingthrough adistanceof1meterintheearth’sgravity.Carryingthisonestepfurther,anobserveringravity perceivesthataclockinaspaceshipinzerogravityhasthesamerateoftimeasaclockingravity, ifthespaceshipismovingatarelativevelocityofve,thegravity’sescapevelocity.Forexample, an observer on earth would perceive that a spaceship in zero gravity moving tangentially at about40,000km/hrhasthesamerateoftimeasaclockontheearth.Ontheotherhand,an observerinthespaceshipperceivesthataclockontheearthisslowedtwiceasmuchasifthere wasonlygravityoronlyrelativemotion. Equivalence of Acceleration and Gravity Examined:AlbertEinsteinassumedthatgravity couldbeconsideredequivalenttoacceleration.Thisassumptionobviouslyleadstothecorrect mathematicalequations.However,onaquantummechanicallevel,isthisassumptioncorrect? Todayitiscommonlybelievedthatanacceleratingframeofreferenceisthesameasgravity. This is associated with the geometric interpretation of gravity. A corollary to this is that an inertialframeofreferenceeliminatesgravity.Theimplicationisthatgravityisnotasrealasthe electromagneticforcewhichcannotbemadetodisappearmerelybychoosingaparticularframe ofreference.Itiscommonforexpertsingeneralrelativitytoconsidergravitytobeageometric effectratherthanatrueforce. Theconceptspresentedinthisbookfundamentallydisagreewiththisphysicalinterpretationof general relativity. There is no disagreement with the equations of general relativity. The previous pages have shown that it is possible to derive the gravitational force using wave properties 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 the gravitationalforceandtheelectrostaticforce.Inparticular,Iwillreferencethetwoequations The Universe Is Only Spacetime ©2012 [email protected] 8-19 whichdealtwithtwodifferentmassparticles m1and m2 .Theconcludingstatementinthis analysiswas: “Note that the only difference between the intermediate portions of these two equations is and the that the gravitational force Fg has the strain amplitude terms squared ( 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 and thereforegravityisalsoarealforce.Theimplicationisthatwhenwegetdowntoanalyzing wavesinspacetimethereisadifferencebetweengravityandacceleration. Anargumentnotinvolvingdipolewavesinspacetimegoesasfollows:Aparticleinfreefallina gravitational field has not eliminated the effect of gravity. The particle is just experiencing offsettingforces.Thegravitationalforceisstillpresentinfreefallbutitisbeingoffsetbythe inertialpseudo‐forcecausedbytheacceleratingframeofreference.Thesetwoopposing“forces” just offset each other and give the impression that there is no force. The acceleration also producesanoffsettingrateoftimegradientandanoffsettingspatialeffect.Thepseudo‐forceof inertiahasnotbeeneliminatedandthegravitationalforcehasnotbeeneliminated.Einstein obtainedthecorrectequationsofgeneralrelativitybyassumingthatgravitywasthesameof acceleration.Sincetheyexactlyoffseteachotherinfreefall,thisassumptiongavethecorrect equationsbutthephysicalinterpretationiswrong.Onthequantummechanicalscaleinvolving waves,gravityisdifferentthanacceleration.Onewaytoprovethatgravityisatrueforceisto showthatagravitationalfieldpossessenergydensity.Thiswillbediscussedinchapter10. “Grav” Field in the Rotar volume:Theabovediscussionofgravitationalaccelerationfroma rate of time gradient prepares us to return to the subject of the “grav field” inside the rotar volumeofarotar.Recallthattherotatingdipolethatformstherotarvolumeofanisolatedrotar wasshowninfigure5‐1.Thisrotatingdipolewavehastwolobesthathavedifferentratesof timeanddifferenteffectsonpropervolume.Thedifferenceintherateoftimebetweenthetwo lobesproducesarotatingrateoftimegradientthatwasdepictedinfigure5‐2.Arotarisvery sensitivetoarateoftimegradient.Arateoftimegradientof1.11x10‐17seconds/second/meter causes a rotar to accelerate at 1 m/s2 and the acceleration scales linearly with rate of time gradient. Therefore, the rotating rate of time gradient in the center of a rotar model can be consideredtobearotatingaccelerationfieldthathassimilaritiestoarotatinggravitationalfield. Wenormallyencountertherateoftimegradientinagravitationalfield.Thisistheresultofa nonlinearity that produces a static stress in spacetime. A static rate of time gradient has frequencyofω 0andnoenergydensity.However,inanactualgravitationalfieldthereisalso theoscillatingcomponentofagravitationalfieldandthisdoeshaveenergydensitythatwillbe discussedlater.Ifarotatinggravitationalfieldissomehowgenerated,thensucharotatingfield The Universe Is Only Spacetime ©2012 [email protected] 8-20 wouldalsohaveenergydensity.Therotatingrateoftimegradient rotatinggravfield thatis presentnearthecenterofarotardoeshaveenergydensitythatwillbecalculatednext. In a time period of 1/ωc, the fast time lobe of the dipole gains Planck time displacement amplitude Tp andtheslowtimelobelosesPlancktime Tp.Theselobesareseparatedby 2 . Therefore,inatimeof1/ωcthereisatotaltimedifferenceof2Tpacrossadistanceof2 .The rateoftimegradientpermeteris: – Theaccelerationproducedbythisrateoftimegradient gravacceleration g istherateoftime gradienttimesc2.Thefollowingareseveralequalitiesforgravacceleration g: – g g gravaccelerationand c2 Lpωc2 Aβ2 p ħ p c/tp /ħ Planckacceleration Comparison of Grav Acceleration and Gravitational Acceleration: Howdoestherotating grav acceleration at the center of a rotar’s rotar volume g compare with the non‐rotating, gravitationalacceleration gq attheedgeofthesamerotar’srotarvolume?. gq Aβ2ωccrotar’snon‐rotatinggravitationalaccelerationatdistance fromthecenter g Aβωccrotar’srotating ωc gravaccelerationatthecenterofarotar Aβratioofgq staticgravitationalaccelerationat torotatinggravacceleration g ForanelectronAβ 4.18x10‐23,sotherotatinggravfieldatthecenteroftheelectronisabout 2 1022timesstrongerthanthenon‐rotatinggravitationalfieldatdistance .Thisresultsin an electron having a rotating grav acceleration of: g 9.73 x 106 m/s2. The gravitational acceleration notrotating ofanelectronatdistance is:gq 4.07x10‐16m/s2.Thereforethe gravaccelerationintherotarvolumeofanelectronisaboutamilliontimesgreaterthanthe gravitationalaccelerationatthesurfaceoftheearth.Theearth’sgravityisnotrotatingandisa nonlineareffect.Theelectron’sgravfieldisrotatingandisafirstordereffectresultingfromthe rateoftimegradientestablishedintheelectron’srotatingdipolewave. Recall the incredibly small difference in the rate of time that exists between the lobes of an electron. It would take 50,000 times the age of the universe for the hypothetical lobe clock runningattherateoftimeinsidetheslowlobetoloseonesecondcomparedtothecoordinate The Universe Is Only Spacetime ©2012 [email protected] 8-21 clock.Thedifferencebetweentherateoftimeontheslowlobeclockandthecoordinateclockis comparabletothedifferenceintherateoftimeexhibitedbyanelevationchangeofabout4x10‐7 m in the earth’s gravity. The reason that the rotating grav field has a million times larger acceleration than the earth is because this difference in the rate of time occurs over approximatelyamilliontimesshorterdistance ~4x10‐13m .Therotatingrateoftimegradient inside a rotar is a first order effect related to Aβ while the non‐rotating gravitational field producedbytherotarisasecondordereffectrelatedtoAβ2. Conservation of Momentum in the Grav Field:Itwouldappearthattheconceptofagravfield mustviolatetheconservationofmomentum.Anexamplewillillustratethispoint.Supposethat asmallneutralparticle suchasaneutralmeson wandersintothecenterofanelectron’srotar volume.Evenifthemassofthemesonis1000timeslargerthantheelectron,therotatinggrav fieldoftheelectronshouldproducethesameaccelerationoftheneutralparticle.Thiswouldbe aviolationoftheconservationofmomentumunlessthedisplacementproducedbytherotating gravfieldisequaltoorlessthanPlancklength theuncertaintyprincipledetectablelimit .We willcalculatethemaximumdisplacement thattakesplaceinatimeperiodof:t 1/ωc.We choosethistimeperiodbecausetherotatingvectorofthegravfieldischangingbyoneradianin atimeperiodof1/ωc.Hypotheticallytheneutralparticlewouldnutateinacirclewitharadius relatedto ignoringdimensionlessconstants . ½at2 Lp Lp maximumradialdisplacementproducedbyarotar’srotatinggravfield Thereforeanymass/energyrotaralwaysproducesthesamedisplacementequaltoPlancklength ignoringdimensionlessconstants inthetimerequiredforthegravfieldtorotateoneradian. Thisdisplacementispermittedbyquantummechanicsandisnotaviolationoftheconservation ofmomentum.Thisisanothersuccessfulplausibilitytest. Energy Density in the Rotating Grav Field:Anacceleratingfieldthatisrotatingpossesses energydensity.Itwouldhypotheticallybepossibletoextractenergyfromsuchafieldifthefield producedanutationthatwaslargerthanthequantummechanicallimitofPlancklength.No energycanbeextractedfromarotar’srotatinggravfieldbecausethenutationisatthequantum mechanicallimitofdetection.However,thisfieldstillpossessesenergydensity. Previouslywedesignatedthestrainamplitudeofarotaras Aβ Lp/ Tpωc ωc/ωp.These wereoriginallydefinedintermsofthestrainamplitudeofadipolewavethatisonewavelength in circumference. This definition tended to imply that the energy density of a rotar was distributedaroundthecircumference.However,itisproposedthattherotatinggradientthatis present at the center of the rotar model can also be characterized as having a dimensionless The Universe Is Only Spacetime ©2012 [email protected] 8-22 amplitudeof Aβ Lp/ Tpωc.Thisamplitude Aβisjustintheformofarotatingrateoftime gradientandarotatingspatialgradient.Thespatialgradientfromthelobetothecenterisstill Lp/ .TherateoftimegradientisstillrelatedtoTpωc,althoughthisishardertosee.Previously wesubstitutedAβ,ωcandZsintoU A2ω2Z/candobtainedarotar’senergydensityintherotar volume Uq kmc2/. If we ignorethedimensionlessconstant k, this is the rotar’s internal energyinthevolumeofacubethatis onaside.HerearesomeotherequalitiesforUq. Uq Uq ħ Aβ4Upset ħ g 2 Thereforetherotating“gravfield”hasthesameenergydensityastheenergydensityoftheentire rotar Ei/λc3 . If we broaden the definition of Aβ so that it also defines rotating rate of time gradients and rotating spatial gradients, then the energy density of the rotar model becomes homogeneous.Theenergydensitynearthecenterofarotaristhesameastheenergydensity neartheedge.Thisenergydensityisjustintwodifferentforms.Inchapter6weattemptedto calculatetheangularmomentumofarotar.Ifweassumedthatalltheenergywasconcentrated neartheedgeofahoopwithradius ,thenweobtainedananswerofangularmomentumofħ. However,ifweassumedthattheenergywasdistributedmoreuniformly likeadisk andalso rotationinasingleplane,thentherotarmodelwouldhaveangularmomentumof½ħ.Thefact that energy is contained in the grav field does smooth out the energy distribution, thereby tendingtowardstheanswerof ½ħ.However,aspreviouslyexplained,thereisalsoachaotic rotationwherethereisanexpectationrotationalaxisbutotherrotationaldirectionsareallowed withlessprobability.Also.theenergydensitydistributionwithinarotardoesnotendabruptly ataradialdistanceofλc.Thedetailsthatresultinnetangularmomentumof½ħwillhavetobe workedoutbyothers. If a rotating grav field has energy density, does a static gravitational field also have energy density? This question will be examined in chapter 10 after some additional concepts are introducedabouttheoscillatingcomponentofgravity Energy Density in Dipole Waves:Theaboveinsightsintothegravfieldalsohaveimplications foranyPlanckamplitudedipolewaveinspacetime,notjusttherotatingdipolesthatformrotars. Iamgoingtotalkaboutdipolewavesinspacetimebutstartoffbymakingananalogytosound waves in a gas. Sound waves can be depicted with a sinusoidal graph of pressure. The compression regions have pressure above the local norm and the rarefaction regions have pressure below the local norm. These can be represented as a sine wave maximum and minimum.However,ifagraphwastobedrawnshowingthekineticenergyofthemoleculesin thegas,themaximumkineticenergyoccursinthenoderegionsbetweenthepressuremaximum andminimum.Akineticenergygraphdepictingmotion velocity leftandrightwouldhavea 90°phaseshifttothepressuregraph.Theenergyinthesoundwaveisbeingconvertedfrom The Universe Is Only Spacetime ©2012 [email protected] 8-23 kineticenergy particlemotion toenergyintheformofhighorlowpressuregas.Whenthese twoformsofenergyareaddedtogether,thenasoundwavewithaplanewavefronthasauniform totalenergydensity sin2θ cos2θ 1 .Theenergyisjustbeingexchangedbetweentwoforms. This concept of energy being exchanged between two different forms also applies to dipole waves in spacetime. In one form, energy exists because the vacuum energy of spacetime is distortedsothatthereareregionswheretherateoftimeisfasterorslowerthanthelocalnorm. Perhapsthisisanalogoustothecompressionandrarefactionrepresentationofasoundwave. Theregionsbetweenthemaximumandminimumratesoftimehavethegreatestgradientinthe rateoftime.Thesearethegravfieldregionsandtheyareanalogoustoregionsinthesound wavewherethegasmoleculeshavethegreatestkineticenergy.Addingtogetherthetwoforms ofenergydensitypresentineithersoundwavesordipolewavesinspacetimeproducesatotal energydensitywithoutthecharacteristicwaveundulations. The waves in spacetime have sometimes been discussed emphasizing either the temporal characteristics rate of time gradients, etc. or emphasizing the spatial characteristics for exampleLp/r .Actuallybothcharacteristicsarealwayspresent;itissometimeseasiertoexplain usingjustonecharacteristic.Therefore,thegravfieldcouldhavebeenexplainedemphasizing thepropervolumegradientratherthantherateoftimegradient. Gravitational Potential Energy Storage:Whenwelookatthegravitationaleffectthatarotar hasonspacetime,weconcludethattheslowingoftherateoftimealsoproducesaslowingofthe normalizedspeedoflight Co ГCgfromchapter3 .Toreachthisconclusionwemustassume thatproperlengthisconstant,evenwhenthereisachangeinГ.Thisisanunspokenassumption forphysicsthatdoesnotinvolvegeneralrelativity. The effect on the rate of time andon the normalized speed of light ultimately effects energy, force,mass,etc.aspreviouslydiscussed.ThereasonforbringingthisupnowisthatIwantto addressgravitationalpotentialenergy.Gravitationalpotentialenergyisconsideredanegative energythathasitsmaximumvalueinzerogravityanddecreaseswhenamassisloweredinto gravity.Whatphysicallychangeswhenarotariselevatedorloweredingravity? Inchapter3itwasfoundthatsubstitutingthenormalizedspeedoflightCgandthenormalized massMgintotheequationE mc2givesenergythatscalesinverselywithgravitationalgamma Γ rest frame of reference . We illustrated this concept by calculating the difference in the internalenergyofa1kgmassforanelevationofsealevelandonemeterabovesealevel.The calculateddifferenceinthenormalizedinternalenergywas9.8Jouleswhichisexactlythesame asthegravitationalpotentialenergy. ThischangeinenergyisduetothechangeinthenormalizedspeedoflightaffectingtheCompton frequencyoftherotarasseenfromzerogravity.Forexample,afreeelectroninzerogravityhas The Universe Is Only Spacetime ©2012 [email protected] 8-24 aComptonangularfrequencyof7.76x1020s‐1.Earth’sgravityhasβ 7x10‐10.Afreeelectron inearth’sgravityhasanormalizedComptonangularfrequencythatisslowerthanazerogravity electronbyabout5.4x1011radianspernormalizedsecond 7x10‐10x7.76x1020s‐1 .This lower Compton frequency decreases the normalized internal energy of an electron and decreases the gravity non‐oscillating strain generated by an electron. The non‐oscillating strainisresponsiblefortherotar’sgravity,soarotaratrestingravitycontributeslessgravityto thetotalgravitythanthesamerotaratrest sametemperature inzerogravity. Foranotherexample,wewillcalculatethechangeintheinternalenergyofanelectronwhenit iselevated1meterintheearth’sgravitationalfield.Inchapter3thereisasectiontitled “Energy Transformation and Calculation”wherethedifferenceinthegravitationalgammawascalculated for a 1 meter elevation change near the earth’s surface. This was expressed as Γ2 – Γ1 1.091x10‐16 dt2/dt1.SincethereducedComptonfrequencyofanelectronisabout7.76x1020 s‐1,thismeansa1meterelevationchangewillproduceafrequencychange: Δωc 7.7634x1020s‐1x1.0915x10‐16 84,737s‐1 ΔE Δωcħ 84,690s‐1xħJs 8.936x10‐30J Wewillcomparethistothegravitationalpotentialenergystoredwhenanelectroniselevated1 meterintheearth’sgravitationalfieldwithaccelerationofg 9.81m/s2. ΔE meΔhg 9.109x10‐31kgx1mx9.81m/s2 8.936 10‐30J Theseconceptsalsoleadtoaphysicalexplanationforpotentialenergy.Inchapter3theconcept of potential energy was related to a reduction in the normalized speed of light reducing the E mc2internalenergy.Nowwegoonestepfurtherandtracethegravitationalpotentialenergy toachangeintherotationalfrequencyofarotarinagravitationalfield.Thisnotonlyaffectsthe internalenergyoftherotar,butitalsoaffectstheamountofgravitygeneratedbytherotar. Whenwehavelookedatmass,energy,inertiaandgravityfromthepointofviewofazerogravity observer,wehaveseenadifferencenotobviouslocally.SinceachangeinГaffectsmassand energydifferently zerogravityobserverperspective andsincegravityscaleswithenergy not restmass ,tobetechnicallycorrectthegravitationalequationsshouldbewrittenintermsof energy,notmass.Thetransformationsandinsightsprovidedherehaveforcedustorecognize thattheterm“mass”isaquantificationofinertia.Massisnotsynonymouswithmatterandmass scalesdifferentlythanenergywhenviewedbyanobserverusingthezerogravitycoordinaterate oftime. The Universe Is Only Spacetime ©2012 [email protected] 8-25 Personal Note: Iwanttotelltwoshortpersonalstoriesthatrelatetothischapter.Thefirststoryhastodowith thegravitationaleffectonrotarfrequency.Inormallyrunalmosteveryday.Someofthebest ideasinthisbookcametomeduringthesedailyruns.ManyyearsagoIusedtorunonflatground buttopreservemykneesInowrunupasteepsectionofahillandwalkdowntothestarting point.Irepeatthisfor½hourwhichtypicallyisabout15roundtrips.AsIrunupthehillIoften amawarethattheworkthatIamdoingisultimatelyresultinginanincreaseintheCompton frequencyofalltheelectronsandquarksinmybody.Locallythereisnomeasurablechangein the Compton frequencies of these rotars, but using the absolute time scale of a zero gravity observer, I am increasing the frequency of these particles. It somehow is comforting to understandthephysicsofrunningupahill.Theconceptof“gravitationalpotentialenergy”has beendemystified.Inowunderstandwhyitisdifficulttorunupahill. The second story is about the experience of resolving a mystery about gravity. When I was initiallywritingthisbook,IthoughtthatIhadunlockedthekeytounderstandinggravitywhenI haddevelopedtheconceptspresentedinchapter6 theconceptsthatarenowregardedasbeing oversimplified .ThemagnitudeofthegravitationalforcewascorrectandIthoughtIcouldeasily extrapolate to larger distances and larger mass. Then it occurred to me that the vector was wronganditwasobviousthatIwasmissingothermajorconcepts.Iwasfarfromfinishingmy questtoexplainimportantaspectsofgravity. My initial reaction was to try to rationalize changes that would make the simplified model explain the correct vector attraction rather than repulsion . This thought process was somethingliketryingtoreverseengineergravity.Iwasattemptingtoworkbackwardsfromthe desiredresult attraction tofindthechangestothemodelthatwouldgivethedesiredresult.I spent a long time working backwards from result to cause, but it was getting nowhere. Therefore,infrustrationIreturnedtotheapproachthatIhadpreviouslyusedtodevelopthe modeltothatpoint.Thatapproachmerelymovedforwardfromthestartingassumption the universeisonlyspacetime andacceptedthelogicalextensionsofthisassumption.OnceIgot backonthistrack,Irealizedthattheenergydensityoftherotarmodelimpliedpressure.When Itookthelogicalstepstocontainthispressure,Ieventuallyobtainednotonlythegravitational force with the correct vector but also obtained improved insights into the strong force, the electromagneticforceandthestabilityoffundamentalparticles. While other people are attempting to adjust models to explain specific physical effects, I am finding that logically extending the starting assumption gives unexpected explanations. The expandedmodelexplainsdiverseeffectsnotinitiallyunderconsideration.Theseexperiences havegivenmeagreatdealofconfidenceinthismodelandapproach. The Universe Is Only Spacetime ©2012 [email protected] 8-26