On the nature of the force acting on a charged classical particle deviated from its geodesi

fromitsgeodesicpathinagravitationalfield

VesselinPetkov

ScienceCollege,ConcordiaUniversity1455DeMaisonneuveBoulevardWestMontreal,Quebec,CanadaH3G1M8E-mail:vpetkov@alcor.concordia.ca

23October2001

Abstract

Ingeneralrelativitythegravitationalfieldisamanifestationofspacetimecurvatureandunliketheelectromag-neticfieldisnotaforcefield.Aparticlefallinginagravitationalfieldisrepresentedbyageodesicworldlinewhichmeansthatnoforceisactingonit.Iftheparticleisatrestinagravitationalfield,however,itsworldlineisnolongergeodesicanditissubjectedtoaforce.Thenatureofthatforceisanopenquestioningeneralrelativity.TheaimofthispaperistooutlineanapproachtowardresolvingitinthecaseofclassicalchargedparticleswhichwasinitiatedbyFermiin1921.

arXiv:gr-qc/0005084v5 24 Oct 2001Generalrelativityprovidesaconsistentno-forceexplanationofgravitationalinteractionofbodiesfollowinggeodesicpaths.However,itissilentonthenatureoftheveryforceweregardasgravitational-theforceactinguponabodydeviatedfromitsgeodesicpathduetoitsbeingatrestinagravitationalfield.

Inbothspecialrelativity(inflatspacetime)andgeneralrelativity(incurvedspacetime)aparticleofferingnoresistancetoitsmotionisrepresentedbyageodesicworldline.Asthenon-resistantmotionofaparticleisregardedasinertialaparticlewhoseworldlineisgeodesicismovingbyinertia.Inbothspecialandgeneralrelativityaparticlewhoseworldlineisnotgeodesicispreventedfrommovingbyinertiaandthereforeissubjectedtoaninertialforce.Henceaparticlesupportedinagravitationalfieldisdeviatedfromitsgeodesicpath(i.e.preventedfrommovingbyinertia)whichmeansthattheforceactingonitisnotgravitationalbutinertialinorigin.

Themasscausingthespacetimecurvaturedeterminestheshapeoftheparticle’sgeodesicworldline,andingeneralwhichreferenceframesareinertial[7],buttheforcearisingwhentheparticleisdeviatedfromitsgeodesicpathoriginatesneitherfromthatmassnorfromthedistantmasses(asMachproposed).Thisforcehasthesameoriginastheforceactingonatestparticlepreventedfromfollowingageodesicpathinanemptyspacetime.Itshouldbestressedthatingeneralrelativitytheforceactingonaparticledeviatedfromitsgeodesicpathduetoitsbeingatrestinagravitationalfieldisnon-gravitationalinorigin.AsRindlerputit”ironically,insteadofexplaininginertialforcesasgravitational...inthespiritofMach,Einsteinexplainedgravitationalforcesasinertial”[8].Thisisthereasonwhy”thereisnosuchthingastheforceofgravity”ingeneralrelativity[9].Hereitwillbeshownthatacorollaryofgeneralrelativity-thatthepropagationoflightinagravitationalfieldisanisotropic-inconjunctionwiththeclassicalelectromagneticmasstheory[1]-[6]shedssomelightonthenatureoftheforceactingonaclassicalchargedparticledeviatedfromitsgeodesicpath.

Consideraclassicalelectron[10]atrestinthenon-inertialreferenceframeNgofanobserversupportedintheEarth’sgravitationalfield.FollowingLorentz[4]andAbraham[5]weassumethattheelectronchargeisuniformlydistributedonasphericalshell.Therepulsionofthechargeelementsofanelectroninuniformmotioninflatspacetimecancelsoutexactlyandthereisnonetforceactingontheelectron.Asweshallseebelow,however,theaverageanisotropicvelocityoflightinNg(i)givesrisetoaself-forceactingonanelectrondeviatedfromitsgeodesicpathbydisturbingthebalanceofthemutualrepulsionofitschargeelements,and(ii)makesafreeelectronfallinNgwithanaccelerationginordertobalancetherepulsionofitschargeelements.Noforceisactinguponafallingelectron(whoseworldlineisgeodesic)butifitispreventedfromfalling(i.e.deviatedfromitsgeodesicpath)theaveragevelocityoflightwithrespecttoitbecomesanisotropicanddisturbsthebalanceofthemutualrepulsionoftheelementsofitscharge

1

whichresultsinaself-forcetryingtoforcetheelectrontofall.ThisforceturnsouttobeequaltothegravitationalforceF=mgg,wheremg=U/c2representsthepassivegravitationalmassoftheclassicalelectronandUistheenergyofitsfield.AsthecoefficientmginfrontofgisexactlyequaltoU/c2(withoutthe4/3factor)itturnsoutthatthemassoftheclassicalelectronispurelyelectromagneticinoriginwhentheaverageanisotropicvelocityoflightinagravitationalfieldistakenintoaccount.

In1921Fermi[11]studiedthenatureoftheforceactingonachargeatrestinagravitationalfieldofstrengthgintheframeworkofgeneralrelativityandtheclassicalelectromagneticmasstheory.Thepotential

ϕ=

e

2gz

6

g?

rD

rA

?

2gt

2

=gr2/2c2

Cr-Figure1.ThreelightrayspropagateinanelevatoratrestintheEarth’sgravitationalfield.AfterhavingbeenemittedsimultaneouslyfrompointsA,C,andDtheraysmeetatB′(theraypropagatingfromDtowardB,butarrivingatB′,representstheoriginalthoughtexperimentconsideredbyEinstein).ThelightraysemittedfromAandCareintroducedinordertodeterminetheexpressionfortheaveragevelocityoflightinagravitationalfield.Ittakesthesamecoordinatetimet=r/cfortheraystotravelthedistancesDB′≈r,AB′=r+δ,andCB′=r−δ.ThereforetheaveragevelocityofthedownwardrayfromAtoB′iscAB′=(r+δ)/t≈c(1+gr/2c2);theaveragevelocityoftheupwardrayfromCtoB′iscCB′=(r−δ)/t≈c(1−gr/2c2).

Threelightraysareemittedsimultaneouslyintheelevator(representinganon-inertialreferenceframeNg)frompointsA,C,andDtowardpointB.TheemissionoftheraysisalsosimultaneousinareferenceframeI(alocalLorentzframe)whichismomentarilyatrestwithrespecttoNg.AtthemomentthelightraysareemittedIstartstofallinthegravitationalfield.AtthenextmomentanobserverinIseesthattheelevatormovesupwardwithanaccelerationg=|g|.ThereforeasseenfromIthethreelightraysarrivesimultaneouslynotatpointB,butatB′sinceforthetimet=r/ctheelevatormovesatadistanceδ=gt2/2=gr2/2c2.AsthesimultaneousarrivalofthethreeraysatthepointB′inIisanabsoluteevent(thesameinallreferenceframes)beingapointevent,itfollowsthattheraysarrivesimultaneouslyatB′asseenfromNgaswell.Sinceforthesamecoordinatetimet=r/cinNgthethreelightraystraveldifferentdistancesDB′≈r,AB′=r+δ,andCB′=r−δbeforearrivingsimultaneouslyatpointB′anobserverintheelevatorconcludesthattheaveragevelocityofthelightraypropagatingfromAtoB′isslightlygreaterthanc

2

cgAB′

=

r+δ

2c2

󰀁󰀁

.

′

TheaveragevelocitycgCB′ofthelightraypropagatingfromCtoBisslightlysmallerthanc

cgCB′

=

r−δ

2c2

.

Itiseasilyseenthattowithinterms∼c−2theaveragelightvelocitybetweenAandBisequaltothatbetweenAand

ggg

B′,i.e.cgAB=cAB′andalsocCB=cCB′:

󰀁rg

(2)cAB=

2c2and

cgCB

=

r

2c2

󰀁.

(3)

Astheaveragevelocities(2)and(3)arenotdeterminedwithrespecttoaspecificpointandsincethecoordinatetime

tisinvolvedintheircalculation,itisclearthattheexpressions(2)and(3)representtheaveragecoordinatevelocitiesbetweenthepointsAandBandCandB,respectively.

TheseexpressionsfortheaveragecoordinatevelocityoflightinNgcanbealsoobtainedfromthecoordinatevelocityoflightatapointinaparallelgravitationalfield.Ifthez-axisisantiparalleltotheelevator’saccelerationgthespacetimemetricinNghastheform[13]

󰀂

2gz2

ds=1+

Themetric(4)canbewritteninaformsimilarto(6)ifwechooser=r0+zwherer0isaconstant

󰀂

2GM

ds2=1−

.(5)c2

Noticethat(4)isthestandardspacetimeintervalinaparallelgravitationalfield[13],whichdoesnotcoincidewiththeexpressionforthespacetimeintervalinasphericallysymmetricgravitationalfield(i.e.theSchwarzschildmetric)[14,p.395]󰀂󰀄

󰀆2󰀇2GM22

ds2=1−dx+dy+dz.(6)

c2r

󰀁

c2r0

+

2gz

Usingthecoordinatevelocity(5)weobtainfortheaveragecoordinatevelocityoflightpropagatingbetweenAandB(Figure1)

󰀃󰀉󰀁󰀅1gzAg

cAB=c1+

2c2

andaszA=zB+r󰀉󰀁gzBg

cAB=c1+.(9)

2c2

FortheaveragecoordinatevelocityoflightpropagatingbetweenBandCweobtain

󰀁󰀉gzBg

(10)cBC=c1+

2c2sincezC=zB−r.WhenthecoordinateoriginisatpointB(zB=0)theexpressions(9)and(10)coincidewith(2)and(3).

ThereexistsathirdwaytoderivetheaveragecoordinatevelocityoflightinNg.Asthecoordinatevelocitycg(z)(5)iscontinuousontheinterval[zA,zB]inthecaseofweakparallelgravitationalfieldsonecancalculatetheaveragecoordinatevelocitybetweenAandB:

cgAB=

1

c2

+gr

dτA

wheredzB/dt=cg(zB)isthecoordinatevelocity(5)atB

=

dzB

dτA

󰀉gzB

c(zB)=c1+

g

c24

󰀁

dt.

SincezA=zB+rforthecoordinatetimedtwehave

󰀉gzA

dt=1−

󰀁󰀉

c2

−

gr

c2

orkeepingonlytheterms∼c−2

cgB

1−

gzB

c2

󰀁

󰀉gr=c1−

󰀁

2c2

.(11)

AsthelocalvelocityoflightatAiscitfollowsthatiflightpropagatesfromAtowardBitsaveragepropervelocitycgAB(asseenfromA)willbeequaltotheaveragepropervelocityoflightpropagatingfromBtowardAg

cBA(asseenfromA).Thus,asseenfromA,thebackandforthaveragepropervelocitiesoflighttravellingbetweenAandBarethesame.

NowletusdeterminetheaveragepropervelocityoflightbetweenBandAwithrespecttothesourcepointB.AlightsignalemittedatBasseenfromBwillhaveaninitial(local)velocitycthere.ThefinalvelocityofthesignalatAasseenfromBwillbe

dtdzA

cg=A

dt

󰀁

c2

anddτBisthepropertimeatB

󰀉gzB

dτB=1+

󰀁

c2

andtheaveragepropervelocityoflightpropagatingfromBtoAasseenfromBbecomes

󰀉grg

cBA(asseenfromB)=c1+

.2c2

AsseenfromapointPatadistancerfromBandlyingonalineforminganangleθwiththeaccelerationgtheaveragepropervelocityoflightfromBis

󰀂

grcosθg

cBP(asseenfromP)=c1+

󰀁

ThentheaveragepropervelocityoflightcomingfromBasseenfromapointdefinedbythepositionvectorroriginatingfromBhastheform󰀉g·rg

c¯=c1+

2c2

󰀄

oflightpropagatingfromBtoAasseenfromA(whichmeansthatthelocallightvelocityatAisc)canbewrittenas

󰀂󰀄

GMg

cBA(asseenfromA)=c1+.

2c2rBThevelocity(13)demonstratesthatthereexistsadirectionaldependenceinthepropagationoflightbetweentwopointsinanon-inertialframeofreferenceNgatrestinagravitationalfield.Thisanisotropyinthepropagationoflighthasbeenanoverlookedcorollaryofgeneralrelativity.Infact,uptonowneithertheaveragecoordinatevelocitynortheaveragepropervelocityoflighthavebeendefined.However,wehaveseenthattheaveragecoordinatevelocityisneededtoaccountforthepropagationoflightinagravitationalfield(toexplainthefactthattwolightsignalsemittedfrompointsA,andCinFigure1meetatB′,notatB).Wewillalsoseebelowthattheaveragepropervelocityoflightisnecessaryforthecorrectdescriptionofelectromagneticphenomenainagravitationalfield.

Theanisotropicvelocityoflight(13)leadstotwochangesinthescalarpotential

dϕg=

1

rg

(14)

ofachargeelementofanelectronatrestinNg;hereρisthechargedensity,dVgisavolumeelementofthechargeandrgisthedistancefromthechargetotheobservationpointdeterminedinNg.

First,analogouslytorepresentingrasr=ctinaninertialreferenceframe[17,p.416],rgisexpressedasrg=c¯gtinNg.Assumingthatg·r/2c2<<1(weakgravitationalfields)wecanwrite:

󰀉g·rg−1−1

1−(r)≈r

longerwithintheinformation-collectingsphere”[18,p.343]sweepingoverthechargeatthevelocityoflightcinI.BythesameargumenttheanisotropicvolumeelementdVgalsoappearsdifferentfromdVinNg:inadirectionoppositetogthevelocityoftheinformation-collectingsphere(whichpropagatesatthevelocityoflight(13)inNg)issmallerthancsinceforlightpropagatingagainstgwehaveg·r=−grin(13);thereforeanelementaryvolumedVgoftheelectronchargestayslongerwithinthesphereandcontributesmoretothepotentialinNg.

ConsiderachargeoflengthlatrestinNgplacedalongg.Thetimeforwhichtheinformation-collectingspheresweepsoverthechargeinNgis

∆tg

=

lc(1+g·r/2c2)

≈∆t󰀉1−g·r2c2

󰀁

.

Theanisotropicvolumeelementwhichcorrespondstosuchanapparentlengthlgisobviously

dVg

=dV󰀉1−

g·rρdVg

ρdV2

4πǫ0

4πǫ0

2c2

󰀁orifwekeeponlythetermsproportionaltoc−2weget

dϕg

=

ρ

c2

󰀁

dV.(17)

Asseenfrom(17)makinguseofdVginsteadofdVaccountsforthe1/2factorin(1).NowwecancalculatetheelectricfieldofachargeelementρdVgofanelectronatrestinNgbyusingonlythescalarpotential(17):

dEg=−∇dϕg

=

1r2−g·nc2r󰀁ρdV.(18)Thedistortionoftheelectricfield(18)iscausedbytheanisotropicvelocityoflight(13)inNg.Forthedistortedfield

ofthewholeelectronchargewefind

Eg

=1g·nr2−

c2r󰀁ρdV.(19)Theself-forcewithwhichthefieldofanelectroninteractswithanelementρdVg1ofitschargeis

dFgg1Eg=1g·nr2−c2r

󰀁ρ2

dVdVgself=ρdV1

.(20)

Duetothedistortedelectricfield(19)ofanelectronatrestinNgthemutualrepulsionofitschargeelementsdoesnot

cancelout.Asaresultanon-zeroself-forcewithwhichtheelectronactsuponitselfarises:

Fg

1g·nself=r2−c2r󰀁ρ2dVdVg1.(21)Aftertakingintoaccounttheexplicitform(16)ofdVg1(21)becomes

Fg

1g·nself=r2−

c2r

󰀁󰀉1−g·rAssumingasphericallysymmetricdistributionoftheelectroncharge[4]andfollowingthestandardprocedureofcalculatingtheself-force[21]weget:

U

Fg=self

󰀈󰀈

ρ2

8πǫo

1

4πǫor

A(r,t)=

g

c2

1−v·n/c

e

󰀉g·r

1−󰀁

󰀁

(25)

∂t

whichreducestotheCoulombfield[23]

=

e

r2

+

g·n

c2r

e

󰀉g·n+−

c2r

󰀁󰀅

,

.(27)r2

Thereforetheinstantaneouselectricfieldofafallingelectronisnotdistorted.Thisdemonstratesthattherepulsionofitschargeelementsisbalanced,thusproducingnoself-force.Thisresultshedslightonthequestionwhyingeneralrelativityanelectronisfallinginagravitationalfield”byitself”withnoforceactingonit.As(27)shows,theonlywayforanelectrontocompensatetheanisotropyinthepropagationoflightandtopreservetheCoulombshapeofitselectricfieldistofallwithanaccelerationg.Iftheelectronispreventedfromfallingitselectricfielddistorts(disturbingthebalanceoftherepulsionofitschargeelements),theself-force(24)appearsandtriestoforcetheelectrontofallinordertoeliminatethedistortionofitsfield;iftheelectronislefttofallitsCoulombfieldrestores,therepulsionofitschargeelementscancelsoutandtheself-forcedisappears[24].

Tosummarize,thestudyoftheopenquestioningeneralrelativity-whatisthenatureoftheforceactinguponaparticledeviatedfromitsgeodesicpath-bytakingintoaccounttheclassicalelectromagneticmasstheoryprovidesaninsightnotonlyintothatquestion(inthecaseoftheclassicalelectron)butalsointothequestionwhyafallingelectron(whoseworldlineisgeodesic)issubjectedtonoforce.

8

E=

References

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[4]H.A.Lorentz,TheoryofElectrons,2nded.(Dover,NewYork,1952).

[5]M.Abraham,TheClassicalTheoryofElectricityandMagnetism,2nded.(Blackie,London,1950).[6]F.Rohrlich,ClassicalChargedParticles,(Addison-Wesley,NewYork,1990).

[7]S.Weinberg,GravitationandCosmology:principlesandapplicationsofthegeneraltheoryofrelativity,(Wiley,

NewYork,1972),p.87.[8]W.Rindler,EssentialRelativity,2nded.(Springer-Verlag,NewYork,1977),p.244.[9]J.L.Synge,Relativity:thegeneraltheory,(Nord-Holand,Amsterdam,1960),p.109.

[10]Weconsideraclassicalelectronsincethereexistsnoquantum-mechanicaldescriptionoftheelectronstructure

atthismoment.However,ifaclassicaltreatmentsucceedsinexplainingthenatureoftheforceactinguponanelectronwhoseworldlineisnotgeodesic,thiswillimplythatacorrespondingquantum-mechanicaldescriptionofthatforceisalsopossible.[11]E.Fermi,NuovoCimento22,176(1921).

[12]V.Petkov,Ph.D.Thesis,(ConcordiaUniversity,Montreal,1997);seealsophysics/9909019.

[13]C.W.Misner,K.S.ThorneandJ.A.Wheeler,Gravitation,(Freeman,SanFrancisco,1973),p.1056.[14]H.OhanianandR.Ruffini,GravitationandSpacetime,2nded.,(NewYork,London:W.W.Norton,1994).[15]W.Rindler,Am.J.Phys.36,540(1968).

[16]Theequivalenceprinciplecanbeappliedonlytoregionsofdimensionsrinagravitationalfieldwhicharesmall

enough(suchthatgr/2c2≪1)inordertoensurethatthefieldisparallelthere.[17]D.J.Griffiths,IntroductiontoElectrodynamics,2nded.,(PrenticeHall,NewJersey,1989).

[18]W.K.H.PanofskyandM.Phillips,ClassicalElectricityandMagnetism,2nded.,(Addison-Wesley,Massachusetts,

London,1962).[19]R.P.Feynman,R.B.LeightonandM.Sands,TheFeynmanLecturesonPhysics,Vol.2,(Addison-Wesley,New

York,1964).[20]M.Schwartz,PrinciplesofElectrodynamics,(Dover,NewYork,1972).

[21]B.PodolskyandK.S.Kunz,FundamentalsofElectrodynamics,(MarcelDekker,NewYork,1969),p.288.[22]WeconsidertheinstantaneouselectronfieldinordertoseparateitsLorentzcontractionfromthedistortioncaused

bytheelectronaccelerationandtheanisotropicvelocityoflight.[23]Itisclearfromherethatafallingelectrondoesnotradiatesinceitselectricfielddoesnotcontaintheradiation

r−1terms[12].Ifthosetermswerepresentinthefieldofafallingelectronthiswouldconstituteacontradictionwiththeprincipleofequivalence.[24]Thebehaviouroftheclassicalelectroninagravitationalfieldasdescribedbygeneralrelativityandtheclassical

electromagneticmasstheorycanbesummarizedinthefollowingway:theworldlineofanelectronwhichpreservestheshapeofitsCoulombfieldisgeodesicandrepresentsafreenon-resistantlymovingelectron;ifthefieldofanelectronisdistorted,itsworldlineisnotgeodesicandtheelectronissubjectedtoaself-forceonaccountofitsowndistortedfield.

9