The intrinsic derivative and centrifugal forces in

arXiv:gr-qc/0106014v15Jun2001InternationalJournalofModernPhysicsD,Vol.6,No.1(1997)143–198[revised2001]fcWorldScientificPublishingCompanyTHEINTRINSICDERIVATIVEANDCENTRIFUGALFORCESINGENERALRELATIVITY:II.APPLICATIONSTOCIRCULARORBITSINSOMEFAMILIARSTATIONARYAXISYMMETRICSPACETIMESDONATOBINIIstitutoperApplicazionidellaMatematicaC.N.R.,I–80131Napoli,ItalyandInternationalCenterforRelativisticAstrophysics,UniversityofRome,I–00185Roma,ItalyPAOLOCARINI∗GP-B,HansenLabs,StanfordUniversity,Stanford,CA94305,USAandInternationalCenterforRelativisticAstrophysics,UniversityofRome,I–00185Roma,ItalyROBERTT.JANTZENDepartmentofMathematicalSciences,VillanovaUniversity,Villanova,PA19085,USAandInternationalCenterforRelativisticAstrophysics,UniversityofRome,I–00185Roma,ItalyReceived17December1996Thetoolsdevelopedinaprecedingarticleforinterpretingspacetimegeometryintermsofallpossiblespace-plus-timesplittingapproachesareappliedtocircularorbitsinsomefamiliarstationaryaxisymmetricspacetimes.Thishelpsgiveamoreintuitivepictureoftheirrotationalfeaturesincludingspinprecessioneffects,andputsrelatedworkofAbramowicz,deFelice,andothersoncircularorbitsinblackholespacetimesintoamoregeneralcontext.Keywords:gravitoelectromagnetism–inertialforces1.Introduction“Rotatingspacetimes”havecapturedpeople’simaginationseversince“rigid”rotationsinMinkowskispacetimewereconsideredwithinthetheoryofgeneralrelativity.Eventhissimpleexamplewhichisthefoundationofthe“fictitious”centrifugalandCoriolisforcesinclassicalphysicshasledtoitsshareofconfusionaboutrotationinrelativity.G¨odel’sdiscovery1ofthespacetimewhichbearshisnamecertainlyaddedfueltothefire,whichwasagainstokedbythediscoveryoftherotatingblackholesolutionofKerr2anditsgeneralizations.3,4Thelanguageofgravitoelectromagnetism,5specializedintheprecedingcom-panionarticle6(tobereferredtohereas[BCJ1])forstationaryaxiallysymmetricspacetimes,helpsustounderstandtheeffectsofrotationaswellasthoseofaccel-erationandspatialcurvatureinthesethreeclassicspacetimeexamples.Indeedthelinesofforceofthevariousgravitoelectromagneticvectorfields,especiallyinthe∗PresentAddress:PhysicsDepartment,AmherstCollege,Amherst,MA01002,USA.12D.Bini,P.CariniandR.T.JantzenKerrspacetimes,helpgiveamoretangiblewayofinterpretingthebehavioroftestparticlemotionsinthegravitationalfieldofthesespacetimes.HerewefocusonthesimplercaseofcircularorbitsfollowingKillingtrajecto-riesinthesespacetimes,confiningourattentiontotheequatorialplaneintheKerrspacetimes.7,9Thistestparticlemotionisanexampleof“purelytransverse”rela-tiveacceleration.10Byexploringtherolesplayedbytheradialspatialgravitationalforces,oneobtainsaclearerpictureoftheactionandinterrelationshipsofthevar-iousgravitoelectric(GE),gravitomagnetic(GM),andspacecurvature(SC)forcesthatonemaydefinewithineachspacetimeaswellasofthecorrespondencesonemayestablishbetweenthesedifferentspacetimes.Presentingthedetailsoftheseapplicationsalsohelpsmakemoreconcretethesomewhatabstractbutpowerfullanguageofgravitoelectromagnetismitself,whichcanbevaluableininterpretingthegeometryofotherspacetimes.Inparticular,previousdiscussionsofcircularorbitsinblackholespacetimesbyAbramowicz,deFelice,andothers11–34arefitintoamoregeneralpicturewhichhelpstoclarifytheirparticularanalysesofthebehaviorofcertainpropertiesoftheseorbits.Eachofthesethreeclassesofspacetimeshavenaturalstationaryaxisymmet-ricnonlinearreferenceframes,i.e.,threadedslicings(hypersurfacefoliationswithtransversalcongruencesofcurves)ofthespacetimewhichareadaptedtotwoKillingvectorfieldsassociatedwitha2-dimensionalstationaryaxisymmetrygroup.ThenonlinearreferenceframesfortherotatingMinkowskiandG¨odelspacetimeshaveanadditionaltranslationalsymmetrymakingthemcylindricallysymmetricaswell.Ineachcasethethreadingandslicingfamiliesoftestobserversassociatedwiththesenonlinearreferenceframesaretiedtothegeometryofthespacetimeandhelpelucidateitsproperties.InthecaseofKerr,thenonlinearreferenceframeassociatedwithBoyer-Lind-quistcoordinates{t,r,θ,φ}38hasasitsthreadingobservers(followingthetimecoordinatelineKillingtrajectories)thedistantlynonrotatingobserversorstaticobservers,whiletheslicingobservers(movingnormaltothetimecoordinatehy-persurfaces)arethelocallynonrotatingobserversorzero-angular-momentumob-servers.Bothobserverfamiliesareaccelerated,thethreadingobserversopposingthedraggingalongactionoftherotatingblackhole,whiletheslicingobserversaredraggedalongbytheholewithrespecttospatialinfinity.IntheG¨odelspacetime,thenonlinearreferenceframeofcylindricalcoordinates{t,ρ,φ,z}hasthethread-ingobserversmovingalongthegeodesicflowlinesoftherotatingdustsource,whiletheacceleratedslicingobserversopposetheglobalrotationofthespacetime.IntherotatingMinkowskispacetime,theacceleratedthreadingobserversareuniformlyro-tating,whilethegeodesicslicingobserversareaglobalfamilyofinertialobserversassociatedwiththeusualtimelinesofanorthonormalCartesiancoordinatesystem.Eachofthesethreespacetimeexamplesexhibitsdifferentconfigurationsofthevari-ousgravitoelectromagneticfieldswhosecomparisonoffersinsightsaboutthenatureofthespacetimesthemselves.TheG