Spin and Rotation in General Relativity

arXiv:gr-qc/0102101v122Feb2001SpinandRotationinGeneralRelativityLewisH.RyderPhysicsLaboratory,UniversityofKentatCanterburyCanterbury,KentCT27NR,UKE-mail:[e-mail]ntofPhysicsandAstronomy,UniversityofMissouri-ColumbiaColumbia,Missouri65211,USAE-mail:[e-mail]7,20081GeneralremarksBroadlyspeaking,the“roleofspinandrotationinGeneralRelativity”coverstwotopics;thebehaviorofspinningparticlesinGR—this“spin”beingeitherclassicalorquantummechanical,andthephysicsassociatedwith(noninertial)rotations.Thepaperspresentedtothissessioncoverboththeseaspectsofthesubject.Tothenon-specialist,themostfamiliarheadinginthisgeneralareaistheLense-Thirringeffect,aprecessionaleffectwhichispredicted(thoughsofarnotobserved)totakeplaceclosetoarotatingbody.ThepapersofCiufoliniandTeyssandierarebothdevotedtothiseffect.Alsowell-knownisthe(gravitomagnetic)clockeffect;asitsnameimplies,thisisconcernedwithtimenonintegrability,ratherthanwithprecession.TartagliaandMalekibothaddressthistopic;andwelive,moreover,atatimewhenboththeLense-Thirringandgravitomagneticclockeffectshavetheenticingpossibilityofexperimentalconfirmationinthenearfuture.Connectionswithgravitationalwaves(alsosoontobedetected?)aremade,thoughinverydifferentwaysbyCooperstock,Slagter,andSuzukiandMaeda.ThebehaviorofspinningparticlesinvarioustypesofgravitationalfieldsisanalyzedbyMohseni,TuckerandWang,byMukhopadhyayandbySinghandPapini,thelattertwocontributionsbeingconcernedspecificallywithDiracpar-ticles,whileWhiteandRaine—aloneinthissession—consideratheorywithtorsion.Bini,GemelliandRuffinigiveamoregeneralaccountofthedescrip-tionofspinningparticlesinGR.Inanonquantumcontext,WuandXuinclude1rotationalmotionsintheiranalysisofhydrodynamicequationsinGeneralRel-ativity.Thesecondsubjectofthissession,concernedwithrotationeffects,hasspawned,inrecentyears,theinterestingphenomenonofspin-rotationcoupling.Thiseffectisdescribedinthefollowingsectionsofthisintroduction,andpro-videsthemotivationforRyder’spaper,whichshowsthatitisconsistentwithspecialrelativity.2Spin-rotationcouplingInclassicalmechanics,theinertialpropertiesofmattercanbeillustratedthroughnumerousphenomenainvolvingframesofreferenceundergoingtranslationalandrotationalaccelerations.Innonrelativisticquantummechanics,thestateofaparticleisprimarilycharacterizedbyitsinertialmass.Ontheotherhand,theirreducibleunitaryrepresentationsoftheinhomogeneousLorentzgrouparecharacterizedbybothmassandspin.Thusinrelativisticquantummechanicstwoindependenttypesofinertiaareexpected:theinertiathatisprimarilyduetomass-energyandthepurelyquantuminertiaduetointrinsicspin.Thephenomenonofspin-rotationcouplingillustratestheinertiaofintrinsicspin.LetusfirstconsiderarealmasslessscalarfieldΦ(x)inMinkowskispacetimesuchthat2Φ=0.ImagineanobserverrotatingwithfrequencyΩaboutthez-axisinthe(x,y)-plane.Thecoordinatetransformationtotherestframeoftherotatingobserverisgivenbyt′=t,z′=z,andx′=xcosΩt+ysinΩt,y′=−xsinΩt+ycosΩt.(1)WenotethatthiscoordinatetransformationinfactcoincideswiththegeodesicframesetupalongtheworldlineofthenoninertialobserverthatremainsatrestattheoriginofspatialcoordinatesbutrefersobservationstothespatialaxesrotatingwithfrequencyΩ.Intermsofthesphericalpolarcoordinates(r,θ,ϕ),thistransformationcanbeexpressedasr′=r,θ′=θandϕ′=ϕ−Ωt.WesupposethatthescalarfieldcanbeexpressedintermsofasuperpositionofFouriermodesasΦ(x)=ReXkα(k)ei(k·r−ωt),(2)whereω=ck.Writingaplanewaveintermsofsphericalwaveseik·r=4πXlmiljl(kr)Y∗lm(ˆk)Ylm(ˆr),(3)wefindthatupontransformationtotherotatingsystemoftheobserver,Φ(x)=Φ′(x′)andthatΦ′(x′)=4πReXklmα(k)iljl(kr)Y∗lm(ˆk)Ylm(ˆr′)e−i(ω−mΩ)t.(4)2Herejl,l=0,1,2,...,aresphericalBesselfunctionsandwehaveusedthefactthatforsphericalharmonicsYlm(θ,ϕ)=eimΩtYlm(θ′,ϕ′).(5)Thusthefrequenciesofthemodesasmeasuredbytherotatingobserveraregivenbyω′=γ(ω−mΩ),wherem=0,±1,±2,...,andtheLorentzfactorγaccountsfortimedilationsincedt=γdτ.ComparingthisresultwiththeDopplereffectω′D=γ(ω−v·k),wefindfromv=Ω×rthatω′D=γ(ω−l·Ω),wherel=r×kand¯hl=Listhe(orbital)angularmomentum.Thatis,ourresultforω′goesovertoω′DintheJWKBlimit.Letusnextconsideranelectromagneticfieldintheinertialframecharacter-izedbythevectorpotentialinagaugesuchthatAμ=(0,A)withA(x)=ReXka(k)ei(k·r−ωt),(6)whereω=ckasbeforeandk·a(k)=0bytransversality.ThetransformationofthevectorpotentialtotherotatingframeimpliesthatA′0(x′)=0andA′(x′)=A(x).WritingA(x)intermsofvectorsphericalharmonics,wefindA(x)=4πReXkJlMilhY∗JlM(ˆk)·a(k)ijl(kr)YJlM(ˆr)e−iωt,(7)whereJandMarethetotalangularmomentumparametersofthefieldwithJ=1,2,3,...,andM=0,±1,±2,....ItfollowsthatintherotatingsystemwehaveA′(x′)=4πReXkJlMilhY∗JlM(ˆk)·a(k)ijl(kr)YJlM(ˆr′)e−i(ω−MΩ)t,(8)sothatasbeforethenewfrequenciesareω′=γ(ω−MΩ)withM=0,±1,±2,...,foreachmodewithfrequencyωintheinertialframe.Todistinguishtheroleoftheintrinsicspinofthephotonfromthatoftheorbitalpartintheformulaforω′,letusconsiderthecaseofnormalincidence,wheretheorbitalcontributionvanishes.Tothisend,weconsideracircularly-polarizedplanemonochromaticelectromagneticwavepropagatingalongthez-axisgivenbyA=Reha