On the non-linearity of the subsidiary systems

HelmutFriedrichAlbert-Einstein-Institut

Max-Planck-Institutf¨urGravitationsphysik

AmM¨uhlenberg114476GolmGermany

February7,2008

Abstract

InhyperbolicreductionsoftheEinsteinequationstheevolutionofgaugeconditionsorconstraintquantitiesiscontrolledbysubsidiarysystems.Wepointoutaclassofnon-linearitiesinthesesystemswhichmayhavethepotentialofgeneratingcatastrophicgrowthofgaugeresp.constraintviolationsinnumericalcalculations.

1Introduction

MostnumericalcalculationsofsolutionstoEinstein’sfieldequationsarebeingplaguedbyanundesirablyfastgrowthofconstraintviolations.AhugevarietyofreducedequationshasbeenderivedfromEinstein’sequationswiththehopeoffindingversionswithstablepropagationpropertiesandtherehavebeensug-gestedmodifcationsoftheequationswhichwerehopedtoforcebackthesolutiontotheconstraintmanifold(wereferto[6]forfurtherdicussionandreferences).Morerecently,therehavebeenperformedstabilityanalysesofthesubsidiarysystemswhichcontroltheevolutionofthegaugeconditionsortheconstraintquantities(cf.[1],[8]andthereferencesgiventhere).Theseledtorequirementsonthecoefficientsoftheequationswhichinthecaseof[1]resultedingeometricconditionsonthefoliationunderlyingtheevolutionbythemainsystem.Nev-ertheless,thefieldappearstobewideopen,mostofthesuggestedremediesare

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experimental,andthecauseoftheproblemsisnotunderstood.In[1],[8]thesubsidiarysystemshavebeenconsideredaslinearsystemsongivenspace-times.Incontrast,wewishtoemphasizeinthisnotethenon-linearityofthesubsidiarysystem,whichappearstohaveapotentialofgeneratingcatastrophicconstraintviolations.Tounderstandtheextenttowhichthesenon-linearitiesmayaffectthenumericalconstructionofspace-timesandtodevelop,ifnecessary,waystoavoidtheireffects,furtherinvestigationsareneeded.

2Thehyperbolicreduction

WeshalluseafewfactsaboutthehyperbolicreductionprocedurebywhichthegeometricinitialvalueproblemforEinstein’sfieldequationsisreducedtoaCauchyproblemforahyperbolicsystem.Thereexistnowmanyversionsofit,whosegeneralunderlyingstructureare,however,moreorlessthesame.Becauseitleadstoconciseexpressions,wewillusethemetriccoeffficientsasbasicunknownsandconsidertherepresentationofthefieldequationsinwhichtheirevolutionisgovernedbyasystemofwaveequations.Toemphasizetheindependenceofourdiscussionofanyparticularcoordinatesystemweshallemploythenotionofagaugesourcefunctionintroducedin[2].Wereferto[3],[4]formoredetailsonthereductionprocedure.

Let(M,g)denoteasmooth4-dimensionalLorentzspacewithsmoothspace-likeCauchyhypersurfaceSandU∋xλof→R4.Fµ(xλ)∈VasmoothmapofanopensubsetUintoanothersubsetVWithfunctionsxµandtheir(4-dimensional)differentialsdxνprescribedonsomeopensubsetWofSonecansolvenearWtheCauchyproblemforthesemi-linearsystemofwaveequations

∇ν∇νxµ=−Fµ(xλ).

IfthedxµhavebeenchosenlinearlyindependentonWthesolutionwillprovideasmoothcoordinatesystemxλonsomeneighbourhoodofWinM.Intermsofthesecoordinatestheequationsabovetaketheform

−Γµ(xλ)=−Fµ(xλ),

wheretheΓµdenotethecontractedChristoffelsymbolsΓµ=gνηΓνµηofgµν.Thisshows(ignoringsubtletiesarisinginsituationsoflowdifferentiability)thatthecontractedChristoffelsymbolscanlocallybemadetoagreewithanypre-scribedsetoffunctionsFµandthatthesefunctionandtheinitialdatadeterminethecoordinatesuniquely.Werefertothesefunctionsasgaugesourcefunctions.Assumenowthat(M,g)istobeobtainedbysolvingaCauchyproblemforEinstein’svacuumfieldequations.WeshallderivethepropertieswhichwillhelpusformulatethisproblemasaCauchyproblemforhyperbolicequations.ThecontractedChristoffelsymbolsareofparticularinteresttousbecausetheRiccitensorofgcanbewrittenintheformRµν=−

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HerethecontractedChristoffelsymbols(andthefunctionsFν=gνµFµconsid-eredinthefollowing)arebeingformallytreatedasiftheydefinedavectorfield(which,ofcourse,theydonot).ThusΓν=gνµΓµand∇µΓν=∂µΓν−ΓµλνΓλ.Thediscussionabovesuggestsreplacingin(2.1)thefunctionsΓνbyfreelycho-sengaugesourcefunctionsFν(sothattheresultingexpressionwilldependingeneralonthecoordinatesxλnotanylongeronlythroughthegµν).Withthisreplacementthevacuumfieldequationstaketheform

F

0=Rµν≡−

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Rgµν)=∇µ∇µQν+RµνQµ,

(2.4)

andthusasystemofwaveequationsforthequantitiesQν.Werefertothissystemassubsidiarysystem.

ThedetailedanalysisshowsthatiftheCauchyproblemforthemainevolutionsystemisarrangedsothattheinitialdatasatisfytheconstraintsandthegaugeconditionQν=0onS,itfollowsforthesolutionofthemainevolutionsystemthatalsodQνandthusany‘timederivative’∂tQνtransversetoSvanishesonS.TheuniquenesspropertyofthesubsidiarysystemthereforeimpliesthatthesolutiontothemainevolutionsystemdoesindeedsatisfyQν=0onthedomainofdependenceofSwithrespecttog.ThisreducesthelocalCauchyproblemforEinstein’sfieldequationstotheproblemofsolvingequation(2.2),whichthustakesthecentralroleintheanalyticdiscussion.Ofthesubsidiarysystemonlythehomogeneityandtheresultinguniquenesspropertyareneeded.

3Thenon-linearityofthesubsidiaryequation

Themainevolutionsystemisalsocentralinnumericaldiscussions.BecausetheinitialdataQµand∂tQνontheinitialhypersurfaceScomewithanerror,theevolutionpropertiesofthesubsidiarysystemwill,however,alsobecomeimportant.

Thereisnowaytorelateanumericalsolutiontothesolutionofthecontinuumproblemonewantstoapproximate.Togetsomeideahowerrorsinfiltrateinto

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thevarioussystems,itisusefultoconsiderananalogyaccessibletoanalyticmethods.WeassumethemainevolutionsystemtobesatisfiedbyfieldsgµνofclassC3withanerrortermEµνofclassC1sothat

Rµν=∇(µQν)+Eµν.

(3.1)

NothingwillbeassumedabouttheoriginandstructureofthiserrorandtheerrorsintheinitialdataQµand∂tQνonS.

UsingtheBianchiidentitywith(3.1)givestheanalogue

∇µ∇µQν+RµνQµ=−2∇µ(E1

µν−

2

gµνEρρ󰀅

,

itfollowsfromthesplittingabovethat

∇µ󰀄

f1

µν−

errorsundercontrol.However,byitselfthiswillbeoflimiteduse.Inserting(3.1)into(3.2)and,observingthesplitting,into(3.4),gives

∇µ∇µQν+Qλ(∇(λQν)+Eλν)=−2∇µ(Eµν−

1

(c−

Q02)withc=2b+√a2.TheintegrationgivesQ′0=a=const.ifb=0,

√ctanh22a√Q′if0=2b>−a2,Q′0=0=c+atanh2

󰀃󰀁√

a

if2b<−a2.Thesolutionsthusre-|c|tan−2+arctan

′

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mainboundedfort≥0ifb≥0orifa≥0and−a2<2b<0,whiletheydeveloppolesatsomet∗>0ifb<0anda≤0orif2b<−a2anda>0.(Ifthecorrespondinginitialdatawouldbemodifiedoutsidetheintersectionofthehypersurface{t=0}withthebackwardlightconeofthepoint(t∗,xa),thesolutionwouldstillbecomesingularat(t∗,xa).)

WenotethatingeneralanyKillingvectorfieldKofavacuumsolutiongµνsatisfiesequation(2.4)withQµ=Kµinthecoordinatesxµ.Inthepresentcasethisgivessolutionswhichgrowlinearly.

Forusthefollowingobservationisimportant:

WithourassumptionongµνitfollowsthateachneighbourhoodoftheinitialdataQa=0and∂tQa=0containsinitialdatafor(2.4)forwhichthesolutionQa(xµ)becomeunboundedatsomefinitex0=t∗>0.

|c|

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4Concludingremarks

Theonlypurposeofthediscussionabovewastoindicatethebehaviourofsolu-tionstothesubsidiarysystem.Weexpecttofindasimilarsingularbehaviourofthesolutionstoequation(2.4)ifthelatterisgivenwithageneralsmoothmetricgµν(extendingsufficientlyfarintothefuture).Itwouldbeinterestingtoknowwhetherthesingulardataset,i.e.thesubsetofdatawhichdeterminesingularsolutionsofequation(2.4),isopenoroflowerdimensioninthesetofalldataifthemetricgisgiven.Whilethesingulardatasetitselfwilldependonthemetricg,suchacharacterizationmaybeindependentofthechosenmetricandmayhelpavoidenteringthesingularsector.

InaconsistentdiscussionofthegrowthofQµonewouldhavetoconsiderthemainsystemandthesubsidiarysystem

Rµν=∇(µQν),

∇µ∇µQν+Qλ∇(λQν)=0,

asacoupledsystem.Themainequationcanbestudiedindependently.Thesubsidiarysystem,implicitinthemainsystem,dependsonthemetricdefinedbythelatter.IfQµtendstogrow,themainsystemwillreacttoit,whetherforthebetterortheworthisnotclear.Whetherwithchanginggµνthenon-linearityofthesubsidiarysystemcanstillimplyablowupofQµatafinitetimeneedstobeanalysed.

Forthispurposeitmightbeinterestingtostudyundersimplifyingassumptionssuchassphericalsymmetrywhetherthemodelsystem

Rµν=∇(µqν),

∇µ∇µqν+qλ∇(λqν)=0,

consideredasEinsteinequationscoupledtoasourcefieldgivenbyavectorfieldqµ,willdevelopablowupforsuitabledata.Thereductionoftheseequationsisobtainedbyaslightmodificationoftheonedescribedabove.Thesystemcanbesimplifiedfurtherbyassumingqµtobeadifferentialqµ=∇µfofsomefunctionf.Thesecondequationwillthenbeimpliediffsatisfies

∇µ∇µf+∇µf∇µf=const.,

whichisawavemapequationinthecasewheretheconstantontherighthandsidevanishes.Inanycasetheresultsmightleadtoanidentifcationofamechanismresponsibleforthegrowthofconstraintviolationsandtothedevelopmentofmethodstoavoidthem.

SomeauthorsaddtermsbuiltfromQνtothemainsystem,whichappeartoreducethegrowthoftheconstraintviolationsincertaincalculation[5],[7].Thismayberelatedtothedifferenteffectsofthenon-linearitieswhichareobtainedintheappropriatelymodifiedsubsidiarysystems.

Thereareavailablenowmanydifferenttypesofreductions.Dependingonseveralchoices,thesubsidiarysystemmaycontrolthepreservationofthegaugeorthepreservationofconstraintsoramixturethereof.Inspiteofthedifferentappearanceoftheresultingmainandsubsidiarysystemsweexpectthatsimilar

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non-linearitiesastheonediscussedabovewilloccurinanysubsidiarysystem,though,dependingonthesystem,theymayhavedifferenteffects.

Inanumericalschemeforthesecondorderwaveequationsthesubsidiarysys-tem,whichisofthirdorderinthemetric,can,ofcourse,hardlybeidentifiedanylongerasakindofidentityandtherelationsbetweenthetwosystemsisobscured.Butifthenon-linearityofthesubsidiarysystemcanhavefornon-vanishinginitialdataQµand∂tQµdrasticeffectsinthecontinuummodel,theyarelikelytobereflectedinnumericalcalculations.

References

[1]J.Frauendiener,T.Vogel.Onthestabilityofconstraintpropagation.

http://de.arXiv.org/abs/gr-qc/0410100[2]H.Friedrich.OnthehyperbolicityofEinstein’sandothergaugefieldequa-tions.Comm.Math.Phys.100(1985)525–543.[3]H.Friedrich.HyperbolicReductionsforEinstein’sEquations.Class.Quan-tumGrav.13(1996)1451–1469.[4]H.Friedrich,A.Rendall.TheCauchyProblemfortheEinsteinEquations.

In:B.Schmidt(ed.):Einstein’sFieldEquationsandTheirPhysicalImpli-cations.Springer,LectureNotesinPhysics,Berlin2000.[5]C.Gundlach.Privatecommunication.

[6]L.Lehner,O.Reula.Statusquoandopenproblemsinthenumericalcon-structionofspace-times.In:P.T.Chru´sciel,H.Friedrich(eds.):TheEinsteinequationsandthelargescalebehaviourofgravitationalfields.Birkh¨auser,Basel,2004.[7]F.Pretorius.Privatecommunication.

[8]G.Yoneda,H.Shinkai.Diagonalizabilityofconstraintpropagationmatri-ces.Class.QuantumGrav.20(2003)L31-L36.

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