New Perspectives On the Relevance of Gravitation f

arXiv:math-ph/0610082v29Nov2006NewPerspectivesOntheRelevanceofGravitationfortheCovariantDescriptionofElectromagneticallyPolarizableMediaT.Dereli∗DepartmentofPhysics,Ko¸cUniversity34450Sarıyer,˙Istanbul,TurkeyJ.Gratus†DepartmentofPhysics,LancasterUniversityandtheCockcroftInstituteLancasterLA14YB,UK,R.W.Tucker‡DepartmentofPhysics,LancasterUniversityandtheCockcroftInstituteLancasterLA14YB,UKFebruary7,2008AbstractByrecognisingthatstress-energy-momentumtensorsarefunda-mentallyrelatedtogravitationinspacetimeitisarguedthattheclassi-calelectromagneticpropertiesofasimplepolarisablemediummaybeparameterisedintermsofaconstitutivetensorwhosepropertiescaninprinciplebedeterminedbyexperimentsinnon-inertial(accelerating)∗E.mail:[e-mail]†E.mail:[e-mail]‡E.mail:[e-mail]1framesandinthepresenceofweakbutvariablegravitationalfields.Afterestablishingsomegeometricnotation,discussionisgiventoba-sicconceptsofstress,energyandmomentuminthevacuumwheretheusefulnotionofadriveformisintroducedinordertoassociatetheconservationofcurrentsinvolvingthefluxofenergy,momentumandangularmomentumwithspacetimeisometries.Thedefinitionofthestress-energy-momentumtensorisdiscussedwithparticularreferencetoitssymmetrybasedonitsroleasasourceofrelativis-ticgravitation.Generalconstitutivepropertiesofmaterialcontinuaareformulatedintermsofspacetimetensorsincludingthosethatde-scribemagneto-electricphenomenainmovingmedia.Thisleadstoaformulationofaself-adjointconstitutivetensordescribing,ingeneral,inhomogeneous,anisotropic,magneto-electricbulkmatterinarbitrarymotion.Thequestionofaninvariantcharacterisationofintrinsicallymagneto-electricmediaisexplored.AnactionprincipleisestablishedtogeneratethephenomenologicalMaxwellsystemandtheuseofvari-ationalderivativestocalculatestress-energy-momentumtensorsisdiscussedinsomedetail.Therelationofthisresulttotensorspro-posedbyAbrahamandothersisdiscussedintheconcludingsectionwheretherelevanceofthewholeapproachtoexperimentsonmatterinnon-inertialenvironmentswithvariablegravitationalandelectro-magneticfieldsisstressed.MSCcodes:83D05,83C40,83C35Keywords:Stress-energy-momentumtensor,constitutiverelations,vari-ational,electromagnetic,polarisable,magneto-electric,gravitation,Maxwell’sequations.21IntroductionThelawsofquantum-electrodynamicshavebeendevisedtodescribetheelec-tromagneticinteractionswithmatteraccordingtothetenetsofrelativisticquantumfieldtheory.HoweverMaxwell’sclassicalequationsremainmanda-toryforthedescriptionofavastamountofnaturalphenomena.Thisver-satilityisinpartduetothesupplementaryconstitutiverelationsthatarenecessarytoaccommodatethewiderangeofmaterialsthatrespondtoelec-tromagneticfields.Althoughinprinciplesuchrelationscanbederivedfromtheunderlyingquantumdescriptionofmatter,inmanypracticalsituationsonemustrelyonexperimentalguidancetoascertaintheclassicalresponseofmaterialstosuchfields.Oncetheunifyingpowerofaspacetimeformulationofphysicalphenom-enabecameapparentwithEinstein’srelativisticworldview,thenaturalmathematicaltoolfordescribingconstitutiveresponsesbecamethetotalstress-energy-momentumtensorforallmatterandfields.EarlysuggestionsbyMinkowski[1]andAbraham[2]forthestructureofitselectromagneticcomponentinsimplemediainitiatedalongdebateinvolvingboththeoreticalandexperimentalcontributionsthatcontinuestothecurrenttime(seee.g.[3]),[4],[5],[6],[7],[8],[10],[22],[23]).Althoughitiswidelyrecognisedthatthiscontroversyisanargumentaboutdefinitions[11]andthattherelativemeritsofalternativedefinitionsareundecidablewithoutacomplete(experi-mentallyverifiable)covariantdescriptionofrelativisticcontinuummechanicsformatterandfields,itremainsimportanttoclarifythemanyconflictingargumentsthathaveappearedovertheyearsandtooffernewinsightsthatmayhelpinmodellingtheelectromagneticpropertiesofmovingmediaintheabsenceofaviableorcompletedescriptionoffield-particleinteractionsatamorefundamentallevel.Somewaytowardsthisgoalisofferedby(covariant)averagingmeth-ods[18],[17].Thesehoweveryieldnon-symmetricstress-energy-momentumtensorsforelectromagneticfieldsinsimplemedia.Ifthetotalstress-energy-momentumistoremainsymmetricthisimpliesthatotherasymmetriccon-tributionsmustcompensateandnoguidanceisofferedtoaccountforsuchmaterialinducedasymmetries.Theneedforasymmetrictotalstress-energy-momentumtensorisoftenattributedtoconservationoftotalangularmo-mentumdespitethefactthatsuchglobalconservationlawsmaynotexistinarbitrarygravitationalfields.Althoughthemagnitudeofgravitationalin-teractionsmaybetotallyinsignificantcomparedwiththescaleofthosedue1toelectromagnetism,gravitydoeshaverelevanceinestablishingthegeneralframework(viathegeometryofspacetime)forclassicalfieldtheoryandinparticularthisframework[14],[12]offersthemostcogentmeanstodefinethetotalstress-energy-momentumtensorasthesourceofrelativisticgrav-itation.Thisinturnmayberelatedtoavariationalformulation[16]ofthefullycoupledfieldsystemofequationsthatunderpintheclassicaldescriptionofinteractingmatterintermsoftensor(andspinor)fieldsonspacetime.Inthisarticlestress-energy-momentumtensorsaredefinedasvariationalder