Initial Value Problems and Signature Change

arXiv:gr-qc/9501026v22Jul1995gr-qc/9501026DAMTPR95/1January1995,RevisedJune1995InitialValueProblemsandSignatureChangeL.J.Altya⋆andC.J.Fewstera,b∗a)DepartmentofAppliedMathematicsandTheoreticalPhysics,UniversityofCambridge,CambridgeCB39EW,UnitedKingdom.b)Institutf¨urTheoretischePhysik,Universit¨atBern,Sidlerstrasse5,CH-3012Bern,SwitzerlandAbstractWemakearigorousstudyofclassicalfieldequationsona2-dimensionalsigna-turechangingspacetimeusingthetechniquesofoperatortheory.Boundaryconditionsatthesurfaceofsignaturechangearedeterminedbyformingself-adjointextensionsoftheSchr¨odingerHamiltonian.WeshowthattheinitialvalueproblemfortheKlein–Gordonequationonthisspacetimeisill-posedinthesensethatitssolutionsareunstable.Furthermore,iftheinitialdataissmoothandcompactlysupportedawayfromthesurfaceofsignaturechange,thesolutionhasdivergentL2-normafterfinitetime.⋆e-mailaddress:[e-mail]∗e-mailaddress:[e-mail]1IntroductionAnaturalgeneralisationofgeneralrelativityisobtainedbyrelaxingthere-strictionthatthemetrictensoriseverywhereLorentzian.Thissubjecthasproducedaspiriteddebateonthenatureofthejunctionconditionswhichmustbesatisfiedbythemetrictensorandmatterfieldsatthesurfaceofsignaturechange[1,2,3,4,5,6,7,8,9,10].Partofthedifficultyinfind-ingaconsensusinthisdebateisthatmuchdependsonwhatoneregardsaspathological.Oneapproach[4,5,7,8,9]hasbeentoassumecertaindifferentiabilitycon-ditionsonthespacetimemetricandmatterfieldsandthenchoosejunctionconditionswhichprovideaboundedenergymomentumtensor.Forexam-ple,consideradiscontinuouslysignaturechangingmetricthatsatisfiestheEinsteinequations,andassumethatthemetricisC2everywhere(exceptatthesurfaceofsignaturechangewhereonlytheinducedmetricisrequiredtobeC2).Thentheenergymomentumtensorisboundedatthesurfaceofsignaturechangeifandonlyifthesecondfundamentalformofthatsurfacevanishes†[1,4,8].Asasecondexample,considerascalarfieldpropagat-ingonaspacetimethatcontinuouslychangessignature.AssumingthatthescalarfieldiseverywhereC3thenthecomponentofitsmomentumnormaltothesurfaceofsignaturechangemustvanishatthatsurface(seelemma3in[9]).Thisalsoimpliesthattheenergymomentumtensorofthescalarfieldisboundedatthesurfaceofsignaturechange.Asecondapproach[2,6,10](andtheonewewilladoptinthispaper)istorelaxsomeoftheassumptionsunderwhichtheabovejunctionconditionsarederived.Wewillstudysignaturechange,withnocouplingbetweengravityandmatter,undertheassumptionthatthematterfieldsarecontinuous,butnotnecessarilydifferentiable,atthesurfaceofsignaturechange‡.Inthissituation,therearenopre-ordainedjunctionconditionsonthematterfields†Foracontinuouslysignaturechangingmetric,theconditionsareslightlystronger,andwerecommendthatthereadernotestherigorousworkin[8](inparticularseetheorems3and4).‡Thisassumptionisnotunreasonableandtherearemanyparallelsinphysics;forexample,asoundwavethatpassesbetweentwodifferentmediaisonlyC0attheinterfacebetweenthemedia.1atthesurfaceofsignaturechange.Thisphilosophywasusedin[10],whereavarietyofdifferentjunctionconditionswereusedtostudythepropagationofclassicalfieldsonafixedspacetimewhosesignaturechangedfromLorentziantoKleinian.Itwasshownthatawiderangeoffieldbehaviourcouldbegeneratedinthisway.Ultimately,ofcourse,onewantstochoosethejunctionconditionswhichbestmodelthephysicalscenario.Unfortunately,thescatteringpropertiesofasurfaceofsignaturechangeareunknown,andinthissituation,itmakessensetochooseboundaryconditionswhichprovidethemostwell-behavedsystem.If,withsuchachoice,thesystemisstillpoorlydefinedthenonehasgroundsforconcludingthatthesituationwhichisbeingmodelledisunphysical.However,ifthesystemisfairlywell-defined,oneshouldobtainsomereasonablephysicalpredictions,whichcanthenbeusedtoreassessthetheory.Onemethodofobtainingawell-behavedsystemistoimposesmoothnessconditionsasmentionedabove.Inthispaper,weinvestigateanotherpossibilitywhichutilisesHilbertspacetechniques.Forsimplicity,westudya2-dimensionalspacetimewithsignaturechangefrom(+−)to(++),andexamineclassicalfieldsthatpropagate,withnointeraction,onthebackgroundspacetime§.Weidentify+withtime,andthereforethistypeofsignaturechangehasmoresimilaritywithaLorentziantoKleiniansignaturetransition[10]thanaLorentziantoRiemanniansigna-turetransition.TheexistenceofaglobaltimeparameterallowsustowritethematterfieldequationsinHamiltonianform.WethendefinethematterfieldHamiltoniansasHilbertspaceoperatorssothatanyjunctionconditionswhichareappliedtoamatterfieldareequivalenttoboundaryconditionssatisfiedbyfunctionsinthedomainofitsHamiltonian.Inordertoobtainawellbehavedsystem,werequiretheSchr¨odingerHamilto-nian(ie.minustheLaplacianonthebackgroundspacetime)tobeself-adjoint,andwealsorequiretheboundaryconditionstotakeageneralformwhichenforcescontinuityofthefunctionsinitsdomain.Thischoiceprovestobehighlysuccessful,asitsinglesoutauniquesetofboundaryconditionsforthe§ThebackgroundspacetimethatwestudysatisfiesthevacuumEinsteinequationsexactly(ie.therearenodistributionaltermsatthesurfaceofsignaturechange).Thiscanbecheckedeitherbydirectcalculationorbyusingtheorem4of[8].2matterfields.Theseboundaryconditionswereprevious