Download Chapter 8 Gravitational Attraction and Unification of Forces

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
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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.
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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
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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.
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Whenweaddthetwowavestogetherweobtaintheplotinfigure8‐2.Becauseoftheartistic
license,itisvisuallyobviousthatthisisanunsymmetricalwave.Thereisalargerareaunderthe
positiveportionofthewavethantheareaunderthenegativeportionofthewave.Thepeak
amplitude for the positive portion is Aβ Aβ2 while 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 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
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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
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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.
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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
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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.
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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: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
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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.
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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
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ħ
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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
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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.
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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.
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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.
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