New variables for the Lema^{i}tre-Tolman-Bondi dust solutions

a

r

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i

v

:

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-qc/0105081v1 23 May 2001NewvariablesfortheLemaˆıtre-Tolman-Bondidustsolutions.

RobertoA.Sussman†andLuisGarc´ıaTrujillo‡

†InstitutodeCienciasNucleares,ApartadoPostal70543,UNAM,M´exicoDF,04510,M´exico.

‡InstitutodeF´ısica,UniversidaddeGuanajuato,Leon,Guanajuato,M´exico.

Were-examinetheLemˆaitre-Tolman-Bondi(LTB)solutionswithadustsourceadmittingsymme-

trycenters.Weconsiderasfreeparametersofthesolutionstheinitialvaluefunctions:Yi,ρi,(3)Ri,

obtainedbyrestrictingthecurvatureradius,Y≡√

g

θθ,containingtwoarbitraryfunctionsoftheradialcoordinate:M(r)andE(r).Theintegrationofthe

dynamicalequationyieldsathirdarbitraryfunction,t

bb(r).Thefirsttwoofthesefunctions(MandE)areusualy

identifiedastherelativisticanaloguesofthenewtonianmassandthelocalenergyassociatedwiththecomovingdust

shells,thethirdfunction(t

bb)canbeinterpretedasthe“bangtime”,markingthepropertimefortheinitial(or

final)singularityforeachcomovingoberver.ThefunctionsM,E,t

bbarethenthefreeparametersofthesolutions,

thoughanyoneofthemcanalwaysbeusedtofixtheradialcoordinate,thustherearerealyonlytwofreefunctions.

Regularityconditions,relatedtowellbehavedsymmetrycenters,abscenceofshellcrossingsingularitiesandsurface

layers,canbegivenasrestrictionsonthesefunctionsandtheirradialgradients.Thesefunctionsareusualyselected

bymeansofconvenientansatzesbasedontheinterpretationsdescribedaboveand/orthecompatibilitywithregularity

conditionsorobservationalcriteria.Oncethefreeparametershavebeenselectedwehaveacompletelydetermined

LTBmodel.

SincethestandardfreefunctionsleadtoaconsistentdescriptionofLTBmodels,thesefreefunctionsareusedinmost

literaturedealingwiththesesolutions(see[22]foracomprehensiveandautoritativereviewofthisliterature).There

arepapersconsideringspecificmodificationsofthestandardfreefunctionsinordertoformulateinitialconditions

forstudyingthecensorshipofsingularities[20],oraimingatan“intialvalue”formulationofLTBmodels[9].We

believethatnewvariables,definedalonganarbitraryandregular“initial”hypersurfacet=t

i,leadtoamoreintuitive

understandingofLTBsolutions.Therefore,weproposeinthispaperusinganewsetofbasicfreefunctionsalongthe

followingsteps

•Considertherestmassdensity,ρ,the3-dimensionalRicciscalar,(3)R,andthecurvatureradius,Y,allofthem

evaluatedalonganarbitraryCauchyhypersurface,T

i,markedbyconstantcosmictimet=t

i.Thisleadsto:

ρ

i(r),(3)R

i(r)andY

i(r),wherethesubindex

iindicatesevaluationalongatt=t

i(weshallusethisconvention

henceforth).

1•RescaleYwithY

i,leadingtoanadimensionalscalefactory=Y/Y

i,sothatwecandistinguishbetweenthe

regularvanishingofY(atasymmetrycenter,nowgivenasY

i=0)andacentralsingularity(nowcharacterized

asy=0).Likewise,wecandistinguishbetweenthetwodifferentsituationswhenY′=0:regularvanishing

orsurfacelayersifY′

i=0forasinglevalueofr(thusrestrictinggradientsofρ

iand(3)R

i)andshellcrossings

whenΓ=(Y′/Y)/(Y′

i/Y

i)=0.

•Amongthethreefreefunctions,ρ

i(r),(3)R

i(r)andY

i(r),wefixtheradialcoordinate(aswellasthetopology

ofT

i)byaconvenientselectionofY

i,usingtheremainingtwofunctionsasinitialvaluefunctions.

•Fromthelatterfunctionswecanconstructsuitablevolumeaveragesandcontrastfunctions,allofwhichex-

pressibleintermsofinvariantscalarsdefinedinT

i.Theseauxiliaryfunctionsprovideaclearcutandprecise

characterizationofthenatureoftheinitialinhomogeneityaroundasymmetrycenter,asoverdensities(“lumps”)

orunderdensities(“voids”)ofρ

iand(3)R

i.

TheoldfunctionsM,E,t

bbandtheirgradientscanalwaysberecoveredandexpressedintermsoftheabovementioned

volumeaveragesandcontrastfunctions.Hence,allknownregularityconditionsgivenintheoldvariablescaneasilyberecastintermsofthenewones.Inparticular,theknownconditionsforavoidingshellcrossingsingularities[15](seealso[16]and[17])haveamoreappealingandelegantformwiththenewvariables,sincewiththeoldvariablesthe

signsofthegradientsM′,E′,t′

bbdonotdistinguishmanifestlybetweenlumpsorvoids.Intermsofthenewvariables,

itisevidentthat,ingeneral,shellcrossingsareavoidedonlyforlumpsofρ

iand(3)R

i.However,someparticular

casesofinitialconditions,suchasa“simmultaneousbangtime”(t′

bb=0),doallowforsomeinitialconditionsgiven

asvoidstoevolvewithoutshellcrossings.Ifweconsideranevolutioninthetimeranget>t

i,then(underspecific

restrictions)initialvoidscanalsoevolvewithoutshellcrossings.Allthesefeaturescouldalsoemergewiththeoldfree

functions,butthenewvariablesprovideamorestraightforwardandintuitivecharacterizationoftheeffectofinitial

conditionsintheregularevolutionofLTBmodels.

ThenewapproachtoLTBdustsolutionspresentedinthispaperaccounts,notonlyforaninitialvaluetreatment

onaregularCauchyhypersurfaceT

i,buttoanewformalismbasedonrescalingtheevolutionofthemodelsto

variablesdefinedinthishypersurface.Someaspectsofthisformalism(averageandcontrastfunctions)havebeen

developedelsewhere(see[26]and[28]forapplicationsofLTBmetricswithviscousfluidsources).Variablessimilarto

theonesintroducedherehavebeendefinedin[2],[20],[9],[23]and[10](thelatterinconnectionwithSzekeresdust

solutions,seeequations2.4.8to2.4.14of[22]).MenaandTavakol[11]haveclaimedthatsomeofthesevariablesare

notcovariant,thussuggestingcovariantexpressionsfordensitycontrasts(seealso[24]).Althoughweshowthatthe

newvariablescanbeexpressedintermsofinvariantscalars,andsoarecoordinateindependentandcanbecomputed

foranycoordinatesystem,wedonotclaimthatthedensityandcurvatureaveragesandcontrastfunctionsintroduced

herehaveaclear“covariantinterpretation”,orthattheyareunique.However,whatwecancertainlyclaimisthat

thesenewvariablesemergenaturalyfromthefieldequations,provideanadequateinitialvaluetreatmentandyielda

clearandconcisedescriptionofregularityconditions.

Thepaperisorganizedasfollows:sectionIIreviewsthebasicexpressions(metric,evolutionequationandits

solutions)characteristicofLTBmodelsintheusualvariables.ThenewvariablesareintroducedinsectionIII,

leadingtonewformsforthemetric,therestmassdensityandthesolutionsoftheevolutionequation.Volume

averagesandcontrastfunctionsalongT

iaredefinedinsectionIV.SectionVexaminesindetailgenericproperties

ofthenewvariables:regularity,symmetrycentersandgeometricfeaturesofthehypersurfaceT

i(subsectionVA),

characterizationofsingularities(subsectionVB),choiceofradialcoordinateandhomeomorphicclass(topology)ofT

i

(subsectionVC),expressingthenewvariablesintermsofinvariantscalars(subsectionVD),FLRWandSchwarzschild

(subsectionlimitsVE).InsectionVIwelookatthecharacterizationoftheinhomogneityofT

iintermsdensityand

3-curvature“lumps”and“voids”anditsrelationwiththecontrastandaveragefunctions.InsectionVIIweusethe

newvariablesinordertolookattheconditionstoavoidshellcrossings,bigbangtimes,regularityoftheselected

hypersurfaceT

iandsimmultaneousbigbangtimes.Asummaryoftheno-shell-crossingconditionsisgiveninTable

1.SectionVIIIprovidessimplefunctionalansatzesforthenewvariablesandtheirauxiliaryquantities(volume

averages,contrastfunctions,aswellastheoldvariablesandthebigbangtimes).Asimplestep-by-stepguideline

isprovidedfortheconstructionofLTBmodelsandforplottingallrelevantquantitiesderivedfromtheseansatzes.

Thecharacterizationofintialconditionsas“lumps”and“voids”,aswellasthefulfilmentoftheno-shell-crossing

conditionsisillustratedandtestedgraphicalyin12figuresthatcomplementtheinformationprovidedinthetext.

Finaly,theAppendixprovidesausefulsummary(intermsoftheusualvariables)ofthegenericgeometricfeaturesof

thesolutions(regularityconditions,singularities,symmetrycenters,surfacelayers).

Someimportanttopicshavebeenleftoutinthispaper:timeevolutionoflumpsandvoidsbygeneralizingvolume

averagesandcontrastfunctionstoarbitrarytimes(t=t

i),modelswithoutcenters,modelswithamixeddynamics

(ellipticandhyperbolic,etc),matchingsofvariousLTBmodels,etc.Theexaminationofthesetopicswiththe

formalismpresentedhereiscurrentlyinprogress[29].

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