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[doc] Spin-Spin Interactions in Gauge Theory of Gravity, Violation of Weak Equivalence Principle and New Classical Test of General Relativity

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[doc] Spin-Spin Interactions in Gauge Theory of Gravity, Violation of Weak Equivalence Principle and New Classical Test of General Relativity

    Spin-Spin Interactions in Gauge Theory of

    Gravity, Violation of Weak Equivalence Principle and New Classical Test of General

    Relativity

Commun.Theor.Phys.(Beijing,China)49(2008)PP.15331540

    ;?ChinesePhysicalSocietyVo1.49,No.6,June15,2008

    ;Spin.SpinInteractionsinGaugeTheoryofGravity,ViolationofWeakEquivalence

    ;PrincipleandNewClassicalTestofGeneralRelativity

    ;WUNing

    ;InstituteofHighEnergyPhysics,P.O.Box9181,theChineseAcademyofS

    ciences,Beijing100049,China

    ;(ReceivedJuly6,2007)

    ;AbstractForalongtime.ithasbeengenerallybefievedthatspinspininteract

    ionscanonexistinatheorywhere

    ;Lorentzsymmetryisgauged.andatheorywithspinspininteractionsj8notp

    erturbativelyrenormafizable.Butthis

    ;isnottrue.Bystudyingthemotionofaspinningparticleingravitationalfield,itisfoundthatthereexistspinspin

    ;interactionsingaugetheoryofgravity.Itsmechanismisthataspinningparticle

willgenerategravitomagneticfield

    ;inspace-time,andthisgravitomagneticfieldwillinteractwiththespinofanoth

    spin erparticle,whichwillcausespin

    ;interactions.So,spinspininteractionsaretransmittedgravitationalfield.eformofspinspininteractionsinpost

    ;Newtonianapproximationsisdeduced.ThisresultcallalsobededucedfromthePapapetrouequation.Tj1j8kindof

    ;interactionwillnotaffecttherenormalizabilityofthetheory.Thespin.spininteractionswillviolatetheweakequivalence

    ;principle.andtheviolationeffectsaredetectable.Anexpedmentisproposedtodetecttheeffectsoftheviolationofthe

    ;weakequivalenceprinciple.

    ;PACSnumbers:04.25.Nx,04.20.Cv,04.80.04.25.g,04.60.m

    ;Keywords:spinspininteractions,weakequivalenceprinciple,equationofmotionofaspinningparticle

    ;experimentaltestofgravitytheory,post-Newtonianapproximation ;1Introduction

    ;Itisknownthatthesourceofgravitationalinterac-

    ;tionsisenergy-momentam.InNewton’sclassicaltheory

    ;ofgravity,ljthemotionofatestparticleingravitational

    ;fieldisdrivenbythegravitationalforceontheparticle.

    ;andthespinoftheparticlewillnotaffectitsmotionin

;gravity.Ingeneralrelativity.,themotionofatestpax-

    ;ticleingravitationalfeldisdeterminedbygeodesicequa- ;tion.Butthemotionofaspinningparticleisnotgiven ;bygeodesicequation.Ingeneralrelativity,themotionof ;aspinningparticleisgivenbythePapapetrouequation, ;whichisdeducedfromtheBianchiidentitiesandunder ;theassumptionofpole-dipoleapproximation.4’J

    ;QuantumGaugeTheoryofGravity(QGTG)wasfirst ;proposedin2001.8llJItisaquantamtheoryofgravity

    ;proposedintheframeworkofquantumgaugefieldtheory. ;In2003,QuantumGaugeGeneralRelativity(QGGR) ;wasproposedintheframeworkofQGTG.[12--14]Unlike ;Einstein’sgeneraltheoryofrelativity,thecornerstoneof

    ;QGGRisthegaugeprinciple,nottheprincipleofequiva- ;lence,whichwillcausefar-reachinginfluencetothetheory ;ofgravity.InQGGR,thefieldequationofgravitational ;gaugefieldisjusttheEinstein’sfieldequation,soinclas.

    ;sicallevel,wecansetupitsgeometricalformulation. ;[151

    ;andQGGRreturnstoEinstein’sgeneralrelativityinclas-

    ;sicalleve1.Thefieldequationofgravitationalgaugefield ;inQGGRisthesameasEinstein’sfieldequationingen-

    ;eralrelativisotwoequationshavethesamesolutions, ;thoughmathematicalexpressionsofthetwoequationsare ;completelydifferent.Forclassicaltestsofgravity,QGGR ;givesthesametheoreticalpredictionsasthoseofGR,[16j ;andfornon.relativisticproblems,QGGRcanreturnto ;Newton’sclassicaltheoryofgravity.[17]BasedontheCOU-

    ;plingbetweenthespinofaparticleandgravitoelectro- ;magneticfield,theequationofmotionofspincanbeob. ;tainedinQGGR.InpostNewtonianapproximations,this ;E-mail:wuning@mail.ihep.ac.cn

    ;equationofmotionofspinthoseofGR.[QGGRisgivesoutthesameresultsas ;aperturbativelyrenormaliz

    ;ablequantamtheory,andbasedonit,quantumeffects ;ofgravity[192andgravitationalinteractionsofsome ;basicquantumfields[23,24canbeexplored.Unification

    ;offundamentalinteractionsincludinggravitycanbeful- ;filledinasemidirectproductgaugegroup.2528JIfwe

    ;usethemassgenerationmechanismwhichisproposedin ;literature,29,30jwecanproposeanewtheo1.

    ;ongravity,

    ;whichcontaiasmassivegravitonandtheintroductionof ;massivegravitondoesnotaffectthestrictlocalgravita-

    ;tionalgaugesymmetryoftheactionanddoesnotaffect ;thetraditionallong-rangegravitationalforce.1JTheex ;istenceofmassivegravitonwillhelpustounderstandthe ;possibleoriginofdarkmatter.

    ;Theequivalenceprincipleisgenerallybelievedtobe ;oneofthemilestoneofEinstein’sgeneralrelativity.It

    ;hasbeentestedbymanyexperiments.[32361AUthese

    ;experimentsgivenullresultfortheviolationofequiva- ;lenceprinciple.Ontheotherhand,itisgenerallybelieved ;thatspin-spininteractionbetweenrotatingbodieswillvi- ;plateequivalenceprinciple.Zhangeta1.proposedaphe. ;nomenologicalmodelforspin-spininteractions.37]Later.

    ;Zhangandhiscollaboratordesignedadoublefreefall ;experimenttotestequivalenceprinciplewithfreefalling ;gyroscopes.,JTherehasalsobeenaninterestinspin- ;gravitationalcouplingforalongtime.[21,4044]

    ;Inapreviouspaper,theequationofmotionofaspin- ;ningtestparticleingravitationalfieldisdeducedbased ;onthecouplingbetweenthespinoftheparticleandthe ;gravitomagneticfield.[44JInthispaper.wewillstudythe ;postNewtonianapproximationofthatequation.Thepost ;NewtonianapproximationofthePapapetrouequationis

    ;alsostudied.Then,weapplytheequationtotheprob. ;1emofgyroscopewhichismovingaroundtheearth.Fi. ;nally,qualitativeresultsonthemotionofthegyroscope ;

    ;1534WUNingVol,49

    ;aregiven,andbasedonthiscalculation,allexperimentis ;proposedtodetecttheeffectsoftheviolationoftheweak ;equivalenceprinciple.

    ;2EquationofMotionofaSpinningParticle

    ;Forthesakeofintegrity,wegiveasimpleintroduction ;toQGGRandintroducesomebasicnotations,whichis ;usedinthispaper.DetailsonQGGRcallbefoundin ;literatures[8】一[14and[161.Ingaugetheoryofgravity, ;themostfundamentalquantityisgravitationalgaugefield ;(),whichisthegaugepotentialcorrespondingtograv

    ;itationalgaugesymmetry.Gaugefieldx)isavector ;inthecorrespondingLiealgebra.whichiscalledgravita- ;tionalLiealgebra.soc()=c()l0(,ot=0,1,2,3), ;whereCfx)isthecomponentfieldand=-iO/Ox

    ;isthegeneratorofglobalgravitationalgaugegroup.The ;gravitationalgaugecovariantderivativeisgivenby ;Du=igx)=G竺以,(1)

    ;wheregisthegravitationalcouplingconstantandma- ;trixG=(G)=(gc2).itsinversematrixis

    ;G:11(IgC)=(G).UsingmatrixGandG,

    ;wecandefinetwoimportantcompositeoperators ;g=GuG~,,(2)

    ;=?7GG,(3)

    ;whicharewidelyusedinQGGR.InQGGR,spacetime

    ;isalwaysfiatandspa~e-timemetricisalwaysMinkowski ;metric,sogandgaaxenolongerspace~timemetric. ;Theyaxeonlytwocompositeoperatorswhichconsistof ;gravitationalgaugefield.Thefieldstrengthofgravita- ;tionalgaugefieldisdefinedby

    ;where

    ;()全二1

    ;iD.=%()?,(4)

    ;%=GG.(5)

    ;InQGGR,gravitationalgaugefieldisaspin-2ten- ;sorfield.Thefieldequationofgravitationalgaugefield ;givenbytheleastactionprincipleisequivalent ;.

    ;tothe

    ;Einstein’sfieldequationingeneralrelativity.[0]

    ;Accordingtoliterature[441,theequationofmotionof ;aspinningtestparticleingravitationalfieldis ;[()

    ;

    ;(g.po-5)],

    ;wherejoisthespintensoroftheparticle,and ;T

    ;=dT+rd

    ;T,D’

    ;orequivalently

    ;(p+g%)=()

    ;Equation(6)canbewrittenintothefollowingform ;:

    ;,D-r

    ;(6)

    ;(7)

    ;(8)

    ;(9)

    ;wheref2istheinteractionforceoriginatedfromtheCOU- ;plingofspinandgravitomagneticfield, ;=

    ;()

;

    ;()](10)

    ;3PostNewtonianApproximation

    ;Now,letusdiscussthepostNewtonianapproximation. ;ThestandardprocedureofthepostNewtonianapproxi

    ;mationcanbefoundintextbooks.[45,461Ingaugetheory ;ofgravity,postNewtonianapproximationisdoneinasim- ;ilarway.Letusconsiderthecaseofagyroscopemoving ;aroundtheearth.ThenitspostNewtonianapproxima- ;tionsgive

    ;g00:?7002+0(),gij=?Tij25ijdP+0(),

    ;0:g0’=+0(),U.:1+0(.),

    ;:+.(.),0

    ;1

    ;,

    ;0

    ;

    ;,

    ;whereandaregivenbythefollowingrelations, ;西:

    ;GM

    ;

;e

    ;r

    ;=

    ;(×)

    ;(12)

    ;(13)

    ;Intheaboverelations,Meandaremassandspin

    ;angularmomentumofthesource.Theleadingtermsof

    ;gravitationalgaugefieldsare ;=:一曲+o(),

    ;9C0~=+0(),

    ;=一夕=

    ;+.().

    ;Thegravitationalgaugefieldstrengths

    ;pansion

    ;)+.()

    ;:

    ;10+

    ;.

    ;()

    ;(14)

    ;havetheeX.

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