An Inertial Frame is a frame that is not accelerating (in the sense of - PDF document

An Inertial Frame is a frame that is not accelerating (in the sense of proper acceleration that would be detected by In1916,usingthegeneraltheoryofrelativity,deSitter20alsocalculatedtheadditionalframe-draggingeffectconsistinginthemuchsmallerMercuryÕsperihelionshiftduetotheSunÕsspin(Box1).In1918,LenseandThirring21formulatedtheweak-fieldandslow-motiondescriptionofframedraggingontheorbitofatestparticlearoundaspinningbody,nowknownastheLenseÐThirringeffect(Box1andFig.3).Butframe-draggingisextremelysmallforSolarSystemobjects,sotomeasureitseffectontheorbitofasatelliteweneedtomeasurethepositionofthesatellitetoextremelyhighTheaccuratemeasurementofdistancesisafundamentaltaskinscienceandtechnology.Laser-rangingisthemostaccuratetechniqueformeasuringdistancestotheMoon22,23andtoartificialsatellitessuchasLAGEOS(lasergeodynamicssatellite)24.Short-durationlaserpulsesareemittedfromlasersonEarthandthenreflectedbacktotheemit-tinglaser-rangingstationsbyretro-reflectorsontheMoonoronarti-ficialsatellites.Bymeasuringthetotalround-triptraveltimewearetodayabletodeterminetheinstantaneousdistanceofaretro-reflectorontheLAGEOSsatelliteswithaprecisionofafewmillimetres25.LAGEOS24waslaunchedbyNASAin1976andLAGEOS2waslaunchedbytheItalianSpaceAgencyandNASAin1992,ataltitudesofapproximately5,900kmand5,800kmrespectively.TheLAGEOSsatellitesÕorbitscanbepredicted,overa15-dayperiod,withanuncertaintyofjustafewcentimetres16,25.TheLenseÐThirringdragBTime flowT2BCAT1BCAT3CAFigure1|Frame-draggingeffectsonclocksbyarotatingmass.Iftwoclocks,ortwinsAandC,flyallaroundaspinningbody,evenveryslowly,andathirdoneBawaitsthematthestartingpoint,fixedrelativetotheÔdistantstarsÕ(aÔfixedstarÕisshowninblue,andT1,T2andT3arethreeconsecutiveinstantsoftime),thenwhentheymeetagain,thetwinAthatwastravellinginthedirectionoppositetotherotationofthecentralbody,wouldbeyoungerrelativetothetwinBawaitingatthestartingpoint.OntheotherhandtwinC,travellinginthesamedirectionofrotationofthebody,wouldbeolderwithrespecttothestandingtwinBandtothetwinArotatingintheoppositedirection10Ð12.Forthistimedilation,duetothespinofthecentralbody,tooccur,theywouldnotneedtomovenearthespeedoflight(asinthecaseofthewell-knownÔtwin-paradoxÕofspecialrelativity).Forexample,iftwosuchtwinsmeetagain,havingflownarbitrarilyslowlyaroundthewholeEarthinoppositedirectionsontheequatorialplaneandexactlyatthesamealtitude,thedifferenceintheiragesduetotheEarthÕsspinwouldbeapproximately10216s(foranaltitudeofabout6,000km),whichwouldbeinprincipledetectableifnotfortheother,muchlarger,relativisticclockeffects.However,frame-draggingdoesproducerelevanteffectsonlightandmatter ,thatis,amagneticneedle,whichtendstobealignedalongB.Inpanelbisthe ofacentralbodyandframe-draggingVofatestgyroscopeS. altitude, the difference in their ages due to the EarthÕs spin would be approximately 10-16 s. TheLAGEOSsatellitesAspinninggyroscopedefinesveryaccuratelybyitsrotationanaxisfixedrelativetothelocalinertialframes.Similarly,theorbitalplaneofaplanet,moonorsatelliteisahugegyroscopethatÔfeelsÕgeneralrelativisticeffects.Oneoftheearlytriumphs3,7,8ofthegeneraltheoryofrelativitywasthepredictionoftheprecessionofMercuryÕsperi-helionbythemassoftheSun,whosedeviationfromthenewtonianresultwasbecomingsomethingofanembarrassmenttoastronomers.In1916,usingthegeneraltheoryofrelativity,deSitter20alsocalculatedtheadditionalframe-draggingeffectconsistinginthemuchsmallerMercuryÕsperihelionshiftduetotheSunÕsspin(Box1).In1918,LenseandThirring21formulatedtheweak-fieldandslow-motiondescriptionofframedraggingontheorbitofatestparticlearoundaspinningbody,nowknownastheLenseÐThirringeffect(Box1andFig.3).Butframe-draggingisextremelysmallforSolarSystemobjects,sotomeasureitseffectontheorbitofasatelliteweneedtomeasurethepositionofthesatellitetoextremelyhighTheaccuratemeasurementofdistancesisafundamentaltaskinscienceandtechnology.Laser-rangingisthemostaccuratetechniqueformeasuringdistancestotheMoon22,23andtoartificialsatellitessuchasLAGEOS(lasergeodynamicssatellite)24.Short-durationlaserpulsesareemittedfromlasersonEarthandthenreflectedbacktotheemit-tinglaser-rangingstationsbyretro-reflectorsontheMoonoronarti-ficialsatellites.Bymeasuringthetotalround-triptraveltimewearetodayabletodeterminetheinstantaneousdistanceofaretro-reflectorontheLAGEOSsatelliteswithaprecisionofafewmillimetres25.LAGEOS24waslaunchedbyNASAin1976andLAGEOS2waslaunchedbytheItalianSpaceAgencyandNASAin1992,ataltitudesofapproximately5,900kmand5,800kmrespectively.TheLAGEOSsatellitesÕorbitscanbepredicted,overa15-dayperiod,withanuncertaintyofjustafewcentimetres16,25.TheLenseÐThirringdragBTime flowT2BCAT1BCAT3CAFigure1|Frame-draggingeffectsonclocksbyarotatingmass.Iftwoclocks,ortwinsAandC,flyallaroundaspinningbody,evenveryslowly,andathirdoneBawaitsthematthestartingpoint,fixedrelativetotheÔdistantstarsÕ(aÔfixedstarÕisshowninblue,andT1,T2andT3arethreeconsecutiveinstantsoftime),thenwhentheymeetagain,thetwinAthatwastravellinginthedirectionoppositetotherotationofthecentralbody,wouldbeyoungerrelativetothetwinBawaitingatthestartingpoint.OntheotherhandtwinC,travellinginthesamedirectionofrotationofthebody,wouldbeolderwithrespecttothestandingtwinBandtothetwinArotatingintheoppositedirection10Ð12.Forthistimedilation,duetothespinofthecentralbody,tooccur,theywouldnotneedtomovenearthespeedoflight(asinthecaseofthewell-knownÔtwin-paradoxÕofspecialrelativity).Forexample,iftwosuchtwinsmeetagain,havingflownarbitrarilyslowlyaroundthewholeEarthinoppositedirectionsontheequatorialplaneandexactlyatthesamealtitude,thedifferenceintheiragesduetotheEarthÕsspinwouldbeapproximately10216s(foranaltitudeofabout6,000km),whichwouldbeinprincipledetectableifnotfortheother,muchlarger,relativisticclockeffects.However,frame-draggingdoesproducerelevanteffectsonlightandmatter ,thatis,amagneticneedle,whichtendstobealignedalongB.Inpanelbisthe ofacentralbodyandframe-draggingVofatestgyroscopeS. ÕÕ effects. For weak gravitational fields there is an analogy between the equations that govern the forces on a spinning electric charge (magnetic moment µ) moving through a magnetic field and the forces on a spinning mass moving through the field of a rotating mass. The magnetic dipole will feel a force due to the magnetic field and rotate. In a similar manner the spinning mass will experience a force and precess. TheLAGEOSsatellitesAspinninggyroscopedefinesveryaccuratelybyitsrotationanaxisfixedrelativetothelocalinertialframes.Similarly,theorbitalplaneofaplanet,moonorsatelliteisahugegyroscopethatÔfeelsÕgeneralrelativisticeffects.Oneoftheearlytriumphs3,7,8ofthegeneraltheoryofrelativitywasthepredictionoftheprecessionofMercuryÕsperi-helionbythemassoftheSun,whosedeviationfromthenewtonianresultwasbecomingsomethingofanembarrassmenttoastronomers.In1916,usingthegeneraltheoryofrelativity,deSitter20alsocalculatedtheadditionalframe-draggingeffectconsistinginthemuchsmallerMercuryÕsperihelionshiftduetotheSunÕsspin(Box1).In1918,LenseandThirring21formulatedtheweak-fieldandslow-motiondescriptionofframedraggingontheorbitofatestparticlearoundaspinningbody,nowknownastheLenseÐThirringeffect(Box1andFig.3).Butframe-draggingisextremelysmallforSolarSystemobjects,sotomeasureitseffectontheorbitofasatelliteweneedtomeasurethepositionofthesatellitetoextremelyhighTheaccuratemeasurementofdistancesisafundamentaltaskinscienceandtechnology.Laser-rangingisthemostaccuratetechniqueformeasuringdistancestotheMoon22,23andtoartificialsatellitessuchasLAGEOS(lasergeodynamicssatellite)24.Short-durationlaserpulsesareemittedfromlasersonEarthandthenreflectedbacktotheemit-tinglaser-rangingstationsbyretro-reflectorsontheMoonoronarti-ficialsatellites.Bymeasuringthetotalround-triptraveltimewearetodayabletodeterminetheinstantaneousdistanceofaretro-reflectorontheLAGEOSsatelliteswithaprecisionofafewmillimetres25.LAGEOS24waslaunchedbyNASAin1976andLAGEOS2waslaunchedbytheItalianSpaceAgencyandNASAin1992,ataltitudesofapproximately5,900kmand5,800kmrespectively.TheLAGEOSsatellitesÕorbitscanbepredicted,overa15-dayperiod,withanuncertaintyofjustafewcentimetres16,25.TheLenseÐThirringdragBTime flowT2BCAT1BCAT3CAFigure1|Frame-draggingeffectsonclocksbyarotatingmass.Iftwoclocks,ortwinsAandC,flyallaroundaspinningbody,evenveryslowly,andathirdoneBawaitsthematthestartingpoint,fixedrelativetotheÔdistantstarsÕ(aÔfixedstarÕisshowninblue,andT1,T2andT3arethreeconsecutiveinstantsoftime),thenwhentheymeetagain,thetwinAthatwastravellingin JHb!SFigure2|Frame-draggingandthegravitomagneticanalogyofthegeneraltheoryofrelativitywithelectrodynamics.Inthegeneraltheoryofrelativity,freelyfallingtest-gyroscopes,thatis,sufficientlysmallandaccuratespinningtops,determinetheaxesofthelocal,non-rotating,inertialframes1Ð3,wheretheequivalenceprincipleholdsÑthatis,wherethegravitationalfieldislocallyÔunobservableÕandallthelawsofphysicsarethelawsofthespecialtheoryofrelativity2.Therefore,ifwerotatewithrespecttothesegyroscopes,wefeelcentrifugalforces,eventhoughwemaynotrotateatallwithrespecttotheÔdistantstarsÕ,contrarytooureverydayintuition.Indeed,agyroscopeisdraggedbyspinningmasses,thatis,itsorientationchangeswithrespecttothe approximately 5,900 km and 5,800 km respectively. Thirring GRACE satellites have reduced the orbital uncertainties due to the modeling errors in the non-spherical EarthÕs gravitational field to only a few per cent of the LenseÐThirring effect. Experimental Tests of Frame-Dragging oftheorbitalplanesofLAGEOSandLAGEOS2(Box1)is26,27approximately31milliarcsecondsperyear,correspondingatthe ByfarthemainperturbationoftheirorbitalplanesisduetotheEarthÕsdeviationsfromsphericalsymmetry33.Inparticular,theflat-teningoftheEarthÕsgravitationalpotentialproducesalargeper-turbationoftheLAGEOSnode28Ð30,82.Butthankstotheobservationsofthegeodeticsatellites,theEarthÕsshapeanditsgrav-itationalfieldareextremelywellknown.Forexample,theflatteningoftheEarthÕsgravitationalpotentialistodaymeasured34withanuncertaintyofonlyaboutonepartin107.ToeliminatetheorbitaluncertaintiesduetotheerrorsintheEarthÕsgravitymodels,theuseofbothLAGEOSandLAGEOS2wasproposed27.However,itwasnoteasytoconfidentlyassesstheaccuracyoftheearliermeasure-ments35,36oftheLenseÐThirringeffectwiththeLAGEOSsatellites,giventhelimitingfactoroftheuncertaintyofthegravitymodelsavailablein1998.Theproblem37,38oftheuncertaintiesintheEarthÕsgravityfieldwasovercomeinMarch2002whenthetwinGRACE(gravityrecoveryandclimateexperiment)39,40spacecraftofNASAwerelaunchedinapolarorbitatanaltitudeofapproximately400kmandabout200Ð250kmapart.Thespacecraftrangetoeachotherusingradarandtheyaretrackedbytheglobalpositioningsatellites.TheGRACEsatelliteshavegreatlyimprovedourknowledgeoftheEarthÕsgravitationalfield.Indeed,byusingthetwoLAGEOSsatellitesandtheGRACEEarthgravitymodels34,theorbitaluncertaintiesduetothemodellingerrorsinthenon-sphericalEarthÕsgravitationalfieldareonlyafewpercentoftheLenseÐThirringeffect Ja3(1{e2)3=2whereJistheangularmomentumofthecentralbody,athesemi-majoraxisoftheorbitingtest-particleandeitsorbitaleccentricity.TherateofchangevLÐT ofthecentralbodyis:VS~3(J:^rr)^rr{Jr3where^rristhepositionunit-vectorofthetest-gyroscopeandrisitsradialdistancefromthecentralbody.ThegeodeticprecessionVgeodeticofatest-gyroscopeduetoitsvelocityv,orbitingataradialdistancerfromamassM,is: J4corresponds,accordingtothe2004GRACE(GFZ)Earthgravitymodel34,toonly3%oftheLenseÐThirringeffect.NATUREjVol449j6September2007REVIEWS43 geodetic. Its theoretical value is ~ 6,6 arcsec per year about an axis orthogonal to the Gravity Probe B orbital plane. Unfortunately due to unexpected large drifts of the gyroscopesÕ spin axes the geodetic precession was only measured to a precision of 1.5% (10-5 expected) and the error on the measurement of frame dragging guidestar(ref.45andseehttp://einstein.stanford.edu/).Ifwespinanelectricallychargedsphereweproduceamagneticfield.Anychangeoforientationofthesphererotationaxiswould .ForaspinningspheretheLondonmomentisdirectedalongthespinaxisofthesphere.EachofthefourGravityProbeBgyroscopesconsisted45,46(seehttp://einstein.stanford.edu/)ofaquartzsphere(rotor)ofradius1.9cm,designedtobesphericalandtohaverelativeinhomogeneitiesofitsdensitytoafewpartsinamillion.Eachrotorwascoveredwithaverythinfilmofniobium,thatis,asuperconductoratthetemperatureoftheexperiment(about2K),andwasspinningatapproximately4,000r.p.m.,sothatthespinningsuperconductorlayergeneratedaLondonmagneticmomentalignedalongthespinaxisofthegyro.Therotorswereencircledbyasuperconductingloop.Thevariationsofthemagneticfluxthroughtheloopweremeasuredbythechangesofcurrentinasuperconductingquantuminterferencedevice(SQUID).On14April2007,afterabout18monthsofdataanalysis(seehttp://einstein.stanford.edu/),thefirstGravityProbeBresultswerepresented.TheGravityProbeBexperimentmeasuredthegeodeticprecessionwithanaccuracyoftheorderof1.5%.Indeed,theGravityProbeBteamdiscoveredunexpectedlargedriftsofthegyroscopesÕspinaxesandestimatedtheunmodelledsystematicerrorstobeoftheorderof100milliarcsecondsperyear,correspondingtoanuncer-taintyoftheorderoftwoandhalftimestheframe-draggingbytheEarthspin.However,byadditionalmodellingofthesystematicerrors,theGravityProbeBteamaimstoachieve(ref.50andseehttp://einstein.stanford.edu/)anuncertaintyofabout5milliarcse-condsperyearthatwouldcorrespondtoameasurementofthegeo-deticprecessionwithabout0.1%accuracyandofframe-draggingbytheEarthspinwithabout13%accuracy. ,Thegreensolidlineisthetheoretical,LenseÐThirring,predictionofgeneralrelativity.c,Theredsolidlineisthebest-fitlinethroughtheobservedresiduals(inblack)andthebluesolidlinerepresentstheuncertaintyinthecombinednodallongitudesoftheLAGEOSsatellitesfromthelargesterrorsourceduetotheuncertaintyintheEarthÕsevenzonalharmonicofdegreefour,J4(correspondingtoapproximately3%oftheLenseÐThirringeffectaccordingtothe2004GRACE(GFZ)Earthgravitymodel34;seeFig.3).Theobservedslopeoftheredlineis0.99