The Fermat Principle in General Relativity and App

arXiv:math-ph/9906023v125Jun1999THEFERMATPRINCIPLEINGENERALRELATIVITYANDAPPLICATIONS∗F.GiannoniDipartimentodiMatematicaeFisicaUniversit´adiCamerinoe-mail:[e-mail]tematicaPolitecnicodiBari,Italye-mail:[e-mail]´aticaeEstat´ısticaUniversidadedeSaoPaulo,Brazile-mail:[e-mail]seageneralversionofFermat’sprin-cipleforlightraysinGeneralRelativityandacurveshorten-ingmethodtowritetheMorserelationsforlightraysjoininganeventwithasmoothtimelikecurveinaLorentzianman-ifoldwithboundary.Asaphysicalmeaning,onecanapplytheMorserelationstohaveamathematicaldescriptionofthegravitationallenseffectinaverygeneralcontext.∗ThefirsttwoauthorswerepartiallysponsoredbyresearchfundsM.U.R.S.T.(Italy),andthethirdauthorbyCNPq(Brazil),Processon.301410/95-0.1INTRODUCTIONTheFermat’sPrincipleinClassicalOpticsstatesthatthetrajectoryofalightrayfromasourceAtoatargetBissuchthatitisaminimizer,orbetter,astationarycurveforthetraveltimeamongallthepathsjoiningthepointsAandB.ThisvariationalprinciplecanbeextendedinthecontextofGeneralRelativitywherethetrajectoryofalightrayundertheactionofthegravitationalfieldinvacuumisgivenbyanullgeodesicinaLorentzianmanifoldmodellingthespace–timegeneratedbyagravitationalmassdistribution.AformulationoftheFermat’sprincipleisgivenoncethefollowingdataaredetermined:1.asetoftrialcurvesjoiningthelightsourceandtheobserver;2.afunctionalthatassociatestoeachtrialcurvearealnumber,whichhastoberelatedtoameasurementofthetimepassedfromtheinstantatwhichthephotondepartedfromthelightsourcetotheinstantatwhichthephotonarrivestotheobserver.AmathematicalproofoftheFermat’sprincipleconsistsinprovingthatthetrajectoryofalightraysischaracterizedasastationarypointofthetimefunctionalinthesetoftrialcurves.Thegeodesicsinasemi-Riemannianmanifoldarecharacterizedassolutionsofdifferentialequations,andthelocaltheoryofthelightrayscanbedevelopedintermsofsystemsofdifferentialequationsinIRn.However,thevariationalapproachhastheadvantageofprovidingtechniquesforprovingglobalexistenceresults,andalsoforproducingseveralkindsofestimatesonthenumberofsolutions,givenintermsofthetopologyofthespaceoftrialcurves.TothisaiminthispaperweprovetheMorseRelationsforlightrays,thatwillbepresentedindetailsinSection1.Wenowproceedtoageneraldiscussionofthemathematicalproblem,itsphysicalapplicationsandapresentationofthenewresultsthatwillbeproveninthispaper.WefixaLorentzianmanifold(M,g)thatisthemathematicalmodelofourrelativisticspacetime,andweassumethatMisendowedwithatimeorientationgivenbythechoiceofacontinuoustimelikevectorfieldWonM.Suchassumptionisindeedverymild;namely,givenanyLorentzianmanifold,thereexistsalwaysatwo-foldcoveringfMofMthatadmitsatimeorientation(cf.[19]),andclearlythereisatwo-to-onecorrespondencebetweenthegeodesicsinfMandthoseonM.Ifwewanttostudythelightraysemittedbysomesourceatagiventimeinthepast,representedbyaneventpofM,andreachinganobserversometimesduringitslife,whoseworldlineisgivenbyatimelikecurveγinM,thenweneedtodetermineallthelightlikefuturepointinggeodesicsjoiningpandγinM.Weareassumingherethatboththesourceandthereceiversarepointlike,i.e.theyhavedimensionswhichareneglectiblewithrespecttotheirdistance;avariationalprincipleforlightraysbetweenaspatiallyextendedsourceandaspatiallyextendedreceivermaybefoundin[23].Inanalogywiththeprincipleinclassicaloptics,thesetoftrialcurvesischosentobethesetofallpossiblefuturepointingtrajectoriesjoiningthesourcewiththeobserver,andthatarerunatthespeedoflight.Thisamountstosayingthatatrialcurveisacurvewhosetangentvectoriseverywhereinthelightcone,anditbelongstothesamehalflightconeasthevectorfieldW.Thechoiceoftheregularitytoimposeonthetrialcurvesand,mostofall,thechoiceofthefunctionaltobeextremized2areratherdelicatequestions,whichhavedeepconsequencesforthemathematicaltheorytobedeveloped.Thefirstrelativisticformulationoftheprinciple,validinthecaseofastaticspacetime,isduetoWeyl(see[33]);thevalidityofthegeneralrelativisticFermat’sprinciplewassuccessivelyextendedtothecaseofstationaryspacetimesbyLevi–Civita(see[15]).Forconformallystationaryspacetimes,analternativeformulationoftheprincipleisgivenin[6].ThefirstattempttoextendtheFermat’sprinciplebeyondthe(conformally)stationarycaseisduetoUhlenbeck(see[31]),whoconsideredaLorentzianmanifolddiffeomorphictoaspace-timesplittingM0×IRandatimedependentmetricwhichisdiagonalwithrespecttothisproduct.Thevariationalprincipleprovenin[31]employsthetimefunctionalgivenbythetheprojectionontothesecondfactorcalculatedatthefinalpointofeachtrialcurve.Suchfunctionaldoesnotdependontheparameterizationofthetrialcurveas,forinstance,thelengthfunctionalforcurvesinaRiemannianmanifold,andthislackofrigiditymakesitadifficulttasktoobtainresultsofexistenceandmultiplicityofcriticalpoints.Forthisreason,inordertoprovetheclassicalMorserelationstheauthoremploysanactionfunctionalwhoseLagrangianfunctiondependsquadraticallybythevelocities.ThiskindoffunctionalhasastrictrelationshipwiththeenergyfunctionalforRiemanniangeodesics,obtainedbyremovi