guishablefromthatofanaccretingschwarzschildblackhole,accretionows onthedilatonicbackgroundexhibitsnoveleectsparticularlyastheextreme

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1 Chapter8 Adetailedanalysisofsphericalsteadystateadiabatichydrodynamicalaccretion ontoafourdimensionaldilatonicblackholeispresented.becausetheeventhorizonofadilatonicblackholepossessanareamuchsmallerthanaschwarzschild oneofthesamemass,andatthesametimeitsaccretionradiusisindistin- itisshownthatforanyequationofstateobeyingthecausalityconstraintand anysubsonicatinnityow,therealwaysexistatransonic,regularoverthe limitisapproached.byacombinationofnumericalandanalyticaltechniques, Accretionondilatonicblack guishablefromthatofanaccretingschwarzschildblackhole,accretionows onthedilatonicbackgroundexhibitsnoveleectsparticularlyastheextreme notanylongerjustied,theneedforincorporatingradiativetransporteects eventhorizonow.forbackgroundcorrespondingtoadilatonicblackhole ispointedout.itisarguedthatifsuchblackholesexistintheuniverse,they shouldhaveadistinctobservationalsignatureassociatedwiththem. approachingtheextremelimit,theasymptoticbehaviorofthetransonicow Furthermoreitisheatedenormously,sothattheprotoncomponentbecomes relativistic.asaconsequencewehavefoundthattheadiabaticassumptionis nearthehorizon,diersconsiderablyfromowsoccurringnearthehorizonofa Schwarzschildblackhole.Fortheformercasetheaccretingplasmaeventhough 8.1Introduction crossesthehorizonsupersonicaly,itisnotanylongerinthestateoffreefall. classconstitutesacounterexampletothepopularbelievethatisolatedblack Thediscoveryofanewclassofasymptoticallyatsphericalblackholesolutions oftheeinsteinnonvacuumeldequations[gib82,gm88,ghs91]caughtmany researchersbysurprise.thenewclassestablishesbeyondanydoubtsthatin generalnon-linearitiesineldcongurationsmayresistthepullofgravityand thuspeacefullycoexistwitharegulareventhorizon.furthermorethenew 209

2 210 holeshavealwaystheirexterioremptyapartfromanelectromagneticeld.the \hair"consistsofadilatoncoupledtoau(1)eld.inthedomainofouter communicationthegeometryofadilatonicblackhole(nicknameforthenew class)canbewritteninthefollowingmanner: ds2= (1 2Mr)dt2+(1 2Mr) 1dr2+g(r)d2; Accretionondilatonicblackhole recovered.introducinganewdimensionlessparameter=r blackholesectorcharacterizedbypositivemandvaluesofaintheinterval combinationoftheelectricormagneticchargeqandthevalueofthedilatonic whereg(r)=r2(1 QMr),M>0isthemassoftheholewhileQstandsfora [0,2].Morespecicallya=2characterizetheextremedilatonicsolution,a>2 coversthespaceofnakedsingularitieswhilefora=0theschwarzschildblackis eldatinnitynamelyq=q2e2o.weshalldenotehereafterbya=qm2.the (8.1) thescalarcurvaturerandproperareaaoofr=constantspheresonends R=a32(Mmpllpl) 24( 1) (2 a)2 3; 2Mandcomputing impliesfora>2thespacetimesingularity\liesoutsidetheeventhorizon". FurthermoreEqs.(8.1)-(8.3)showthatthedilatonicblackholepossessesafew wheremplandlplarerespectivelytheplanckmassandlength.expression(8.3) Ao() 4=12(Mmpllpl)2[(2 a)]; (8.2) distinctpropertiesworthmentioning.atrsttheextremecaseandinsharpcontrasttotheextremereissner-nordstromsolution,ischaracterizedbyapointlike singulareventhorizon.furthermorethermodynamicallyitpossessesnonzero Hawkingtemperatureandvanishingentropy,incontradistinctiontothethermodynamicalpropertiesofanextremeReissner-Nordstromsolution(zeroHawking temperature,nonvanishingentropy).awayfromtheextremelimit,arstlook correspondingareaforaschwarzschildblackhole.inadditioneq.(8.3)shows tremelyned-tunedvaluea 2=10 5thenforasolarmassblackholethe thatasoneapproachestheextremelimit,forinstanceevenassuminganex- twometrics"dieratonepoint".thedilatonicmetricendowstheproperarea oftheso(3)orbitswithlessareathanthecorrespondingschwarzschild.infact Eq.(8.3)showsthatadilatonicstateclosetotheextremelimitcouldpossess aneventhorizonwhoseproperarealiesmanyordersofmagnitudebellowthe holeshouldnotdierverymuchfromthewellknownschwarzschildone.the atthelineelementsineq.(8.1)suggests,thatthegeometryofadilatonicblack arearadius,liesmanyordersofmagnitudebelowtheplanckianvalues.thereforequantumgravitationaleectsontheblackholeexteriorcanbeignored.in factsuchnearlyextremestatesareratherbizzare.extremelycompactinsize, butatthesametimeexertingthesamegravitationalinuenceasasolarmass Schwarzschildblackholedoes.Besidesthispurelygeometricaldierence,asalreadystatedearlier,thedilatonicstatesareaccompaniedbyclassicalhair.From

3 tonicblackhole,residingsomewhereintheuniverse,likelytogetneutralizedvia preferentialaccretionofchargeoftheoppositesign.suchprocesspresumably astrophysicalprocesstakeplace.ontheotherhandthebackgroundmonopole electricormagneticeldmaybeofimportance.anelectricallychargeddila- 8.1Introduction theobservationalpointofview,thebackgrounddilatoneldisnotexpectedto playanysignicantrole,atleastfortherangeofenergyscaleswherevarious 211 willdriveatowardszero.itwouldbethenofsomeindependentinteresttond outwhatwillhappentothedilatoneld.forthemagneticallychargedhole, thebackgroundmonopoleeldcannotbetransformedaway.dependingonits strengthitisexpectedtoplaysignicantroleinastrophysicalprocessinvolving angularmomentumandchargedparticles.notehoweverforpurelyradialinfall theexertedlorentzforceisvanishingandthusitsroleisinsignicant. thepresentchapter,istoprobepossibleeectsuponhydrodynamicalows, thebackgroundmonopolemagneticeldisunimportant.sothemainfocusof spondingschwarzschildone.sinceaccretionofambientmatteristhedominant interactionmodeofblackholeswiththeexternalworld,itisnaturaltoexamine accretionphenomenaonamagneticallychargeddilatonicbackground.asapreliminarystep,inthepresentchapteradetailedexaminationofbondiadiabatic accretion[bon52],willbediscussed.recall,suchaccretionisradial,therefore withtherestoftheuniverseinanentirelydierentfashionthanthecorre- Inviewoftheabovedierencesonewonderswhetherthenewclassinteracts apossibility.inadditionpresenceofmultiplecriticalpointsisnotexcluded.in particularlythesecondpossibilityisenhancedinviewofthecloseconnection crosssectionalareaofthe\tube"wheretheowtakesplace[hol77,ht83]. betweenmultiplecriticalpointsandtherapidconvergence-divergenceinthe ForcomparisonpurposeswemayrecallthatBondiaccretiononaSchwarzschild backgroundisratherwellbehaved.ithasbeenestablishedthataslongasadiabaticityismaintainedandtheowissubsonicatinnity,therealwaysexist horizonisapproached.aprioriitisnotclearwhethersmoothaccretionows duetothetherapidreductionintheareaoftheso(3)spheresastheevent auniquetransonicowregularovertheeventhorizon.becauseofthewaya existforallvaluesofain[0,2].forinstance,developmentofstandingshocksis thingsbecomeunclearasadeviatesawayfromthezerovalueandinparticularly entersthemetric,itisexpectedthatbondiaccretiononadilatonicbackground tosharethesamepropertiesaswellprovidedaremainsclosetozero.however forthenearlyextremedilatonicbackgrounds. a,evenfortheextremalones.inthesamesectionanequivalentformulationof remindingthereaderoftherelevantequationsgoverningspherical,steadystate owsonadilatonicbackground.insection(8.4)wediscussthedelicateissue criticalpointofthesadletype,whichislocatedoutsidetheeventhorizon.the extremeone,admitsanadditionalcriticalpointresidingonthepointlikesingular eventhorizon.itisfoundthattheaccretionrateisinsensitivetoallvaluesof ofthecriticalpointsadmittedbytheowequations.itisshownthatforany dilatonicblackhole,therelevanthydrodynamicalequations,alwaysadmita Primarilymotivatedbytheabovequestions,webegintheChapterbyrst theowequationsarepresentedwhichshowsthatthecriticalpointisactually

4 behavioraswellasitsregularnaturenearthehorizonisdiscussedindetails.we foundthatthetransonicownearthehorizonmaydiersignicantlyfromthat occurringonaschwarzschildbackground.specicallywhileforthelatterthe uniquetransonicsolutionsubsonicatinnity,passingthroughthecriticalpoint andreachingsupersonicalytheeventhorizonisestablished.itsasymptotic 212 asonichorizon.withthehelpofnumericalintegrationtheexistenceofa Accretionondilatonicblackhole owisretardedsignicantly,resultingintoextremedenseplasma.consequences ofthisretardationisdiscussed.inparticularlytheadiabaticityassumptionis owisinafreefallstate,foranearlyextremedilatonicbackgroundthatisany longerso.becausetheevenhorizonscrosssectionisreducedconsiderablythe limit.thephysicalreasonsleadingtothisbehaviorispresentedindetails. putunderscrutiny.wepresentargumentsindicatingadiabaticityisnotany WenishtheChapterbycommentingonsomeopenproblemsanddiscussing longercompatiblewithowstakingplaceonholesapproachingtheextreme possibleobservationalsignaturesassociatedwithdilatonicblackholes. andaconservedbaryoncurrentj=nui.e.: WebeginbyconsideringaperfectuidmovingonthebackgroundofEq.(8.1). Theuidisconsideredtobeatestone,thuscausingnegligibledistortionon thebackgroundgeometry.itisdescribedbyaconservedstresstensor 8.2Bondiaccretiononadilatonicblackhole T=(+P)uu+Pg rt=0and r(nu)=0: (8.5) (8.4) andtemperaturerespectivelyasmeasuredinthelocalrestframeoftheuid Inabove;n;P;arethetotalmass-energydensity,baryonnumberdensityand pressurerespectivelyasmeasuredbyanobservercomovingwiththeuid.as anyheati.e.themotionisadiabatic.thusifs,taretheentropyperbaryon longasthereisnoexternalsupplyofenergyandirrespectivelyoftheequation ofstate,therstlawofthermodynamicscombinedwiththeconservationeqs. (8.5),(8.6)impliesthattheuidevolveswithoutitsconstituentsexchanging (8.6) then, Thecovariantconservationofthestresstensorisequivalentto: ur(n)+pur(1n)=turs=0: (+P)uru= rp uurp: ur+(+p)ru=0; (8.7) (8.8) (8.9)

5 8.2Bondiaccretiononadilatonicblackhole MakinguseofEq.(8.1),sphericalsymmetryandthesteadystatecharacterof theow,thecontinuityeq.(8.6)andeulereq.(8.9)yield: ddr(nug)=0; (8.10) 213 intoaccounttheadiabaticityassumptionthefollowingconservationlawscan obtainedfromeqs.(8.10)-(8.12): ustandsfortheradialcomponentoftheowi.e.uranduo=gttuo.taking Eq.(8.8)isautomaticallysatised,providedEqs.(8.9)-(8.12)hold.Inabove udu dr= 1 (+P)(duo (+P)dP dr)+uodp dr(1 2Mr+u2) Mr2: dr=0; (8.11) 4mnug=_M; (8.13) (8.12) energydowntothehorizonwhiletherighthandsideofeqs.(8.14),(8.15)are computedatinnity.conservationlawseqs.(8.14),(8.15)areactuallynot Theparametermstandsforamassscaleassociatedwiththebaryons._Mis aspacetimeindependentconstantrepresentingtheaccretionrateofrestmass (+P n)2(1 2Mr+u2)=(+P nuo= +P nj1: n)2j1; independentofeachother.theyaredierentwaysofexpressingthecovariant versionofbernoulliequationi.e.: (8.15) berecastas: introducingtheadiabaticspeedofsoundaviaa2=dp validforanygeometryadmittingakillingeldandanyowinvariantunder theactionoftheisometry.assuminganequationofstatep=p(n;s)and uara[+p n(gbcubc)]=0; u0=d1 djseqs.(8.10)-(8.12)can D=u2 a2(1 2Mr+u2) u0o= uoa2 n0= D2 nd2 D;where un ; (8.17) (8.18) (8.19) (8.16) D1= 1n[Mr2 Aa2(1 2Mr+u2)]; (8.20)

6 214 andaisdenedbyaddrlnjg(r)j=4 a D2=1u[ Mr2+Au2]; Accretionondilatonicblackhole hereaftereq.(8.18)asredundantandconcentrateoneqs.(8.16),(8.17).an Eqs.(8.16)-(8.21)arethebasicequationsdescribingtheow.Weshallignore 2M(2 a): (8.21) inspectionofthemshowsthattheyreducetothecorrespondingowequations ofstatedescribingtheaccretingplasma.theparameterkisaconstantand thesituationisreversedfortheclassofdilatonicblackholesapproachingthe extremelimit.infactattheextremelimitthesecondtermdiverges.following Bondiandothers,anytypeofradiationlosseswillbeinitialyignoredandthus adoptthepolytropicequationofstatep=p(n)=kn astherelevantequation rsttermwithinthesquarebracketineqs.(8.20),(8.21)dominatestherest, D1,D2showswhereasfortheSchwarzschildcaseandneartheeventhorizon,the nontrivialmannertheparameteraenterstheowequations.acloselookat forschwartzcildbackgroundprovidedg(r)=r2.itisworthhowevertonotethe thepolytropicindexwillsatisfy <5=3.Suchequationofstateactuallymakes thespeedofsoundandenergydensity: theowtobeisentropicandtherstlawimpliesthefollowingexpressionfor a2=dp dn d=n dnn +P= +Kn,=mn+P m+ Kn 1=( 1); 1: (8.22) disconnectedfromtheinterioroftheow.putitdierently:thesoundcone draggedinwards.thereforeanyobserverintheasymptoticregionissoundly manner,amannerwhichalsorevealsthatifcriticalpointexists,thennecessary aresonichorizons.namelyhorizonsdenedbythepropertythatoutgoingsound wavesemittedbythebackgroundowinteriortocriticalsphereareactually itisconvenienttoreformulatetheaboveowequationsinaslightlydierent Thusa2andarefunctionsofthebaryondensityalone.Fornumericalpurposes (8.23) istiltedinwardsforallpointslocatedinteriortothecriticalsphere.thisat asymptoticquantities.eliminatingthebaryondensityninfavorofthesound leastforthesteadyowsoccurs,wheneveralocalorthonormalobserverat owpassesviathecriticalpoint,allowsthedeterminationof_mintermsofthe beingidenticaltothatofsound.theimportanceofsonichorizonstoaccretion problemsiswellknown.asweshallpresentlyverifytherequirementthatthe restrelativetothecoordinatesystemeq.(8.1)ndthespeedoftheow orthonormalobserver,relatedtouvia speeda2andintroducingtheordinarythreevelocityv=dr u2=v21 (1 v2)(1 2Mr); dtmeasuredbyalocal (8.24)

7 8.3Thedeterminationofthecriticalpoints Eqs.(8.16),(8.17)transformedintothefollowingequivalentsystem: y(1 y)(v2 a2) v2(1 v2)2dv2 (1 v2)a2( 1 a2)da2 2y(1 y)(v2 a2) dy=y 2a2 1 v24 ya dy= y+24 ya 2 ya(1 y)+y(v2 a2) 2 yav2(1 y) 1 v2;(8.25) 215 whiledividingtherstonebya2,thesecondonebyv2andaddthemyield: wherey=1.addingthemtogetheroneobtains: [a2 1 a2]2 1v2 1 v2(1 y)(2 ay)2 y4 (1 v2);(8.26) wheretherighthandsidesofeqs.(8.27)-(8.28)areconstants.imposingv=0 (1 v2)( 1 a2)2 1 y =1 =2 (8.27) aty=0onegets1= 1 a212while2isingeneralafreeparameter. Itsrelationwiththeaccretionrateis 2=K m( 1)2 8mM2!2: _M (8.28) Inordertogetabetterfeelingabouttheglobalbehavioroftheow,weshould knowwhetherthedynamicalequations(8.16)-(8.18)admitcriticalpoints,andif TheaboveintegralsofmotionarejusttheconservationlawsshowninEqs. sotheircharacteri.e.whethertheyaresaddles,nodesetc.ongeneralgrounds, regulartransonicowwillbediscusseslateron. 8.3Thedeterminationofthecriticalpoints (8.13)-(8.14).Theirusefulnessinestablishingnumericallytheuniquenessofa smoothowsthataresubsonicatinnityandregularoverthehorizon(aconditionthatasweshallseeinthenextsectionrequiresu6=0)areexpectedtopass atradialinnityd<0whileforanyequationofstatesatisfyingthecausality constrainti.e.a2<1,itfollowsagainfromeq.(8.19)thatd>0atthehorizon.thereforetherewillbeatleastonepointattheblackholeexteriorwhere D(r)=0.Eqs.(8.16),(8.17)showsthatowsreachingsuchpointscharacterizedbyinnitygradientsinthevelocityandbaryondensity.Physicallytheow turnsoveranditscontinuationisconsideredasbeingunphysical.physically viacriticalpoints.thiscanbeinferredbynotingthateq.(8.19)indicatesthat importantowsreaching"turningoverpoints"mustsimultaneouslysatisfy: D=D1=D2=0: (8.29)

8 horizon,singlesauniquesolutiondeterminedsolelybytheboundaryconditions factdemandingtheowtopassvia,aswellasbeingregularovertheevent 216 Extendingtheowthroughthecriticalpointmaybeadelicateissue.Whether anunambiguousextensionispossibledependsuponthecharacterofthecritical points.however,asweshallseeinmoredetailsinthenextsectionthecritical pointsaresaddlesandforsuchcasestheextensionisfreeofambiguities.in Accretionondilatonicblackhole (8.17)fortheirdetermination.StartingfromEq.(8.29),oneinfersthatus;as atthepotentiallocationofthecriticalpointsrssatisfy: sonichorizons.inthesubsequentanalysisweshallemployequations(8.16), equaltothelocalspeedofsound.thusthecriticalhypersurfacesareactually ticationisofprimeimportance.bydenitionpotentialcriticalpointsofthe owequationssatisfyconstraints(8.29).notethatifinsteadof(8.16),(8.17) atinnity.inviewofthesignicanceofthecriticalpoints,theirproperiden- thealternativesetofequations(8.25),(8.26)isemployed,onendsthatatany criticalpoint,anorthonormalobservermeasuresthespeedoftheowbeing CombiningEqs.(8.30),(8.31)togetherwithBernoulliEq.(8.14)andinview of(8.22)thecoordinatesofthecriticalpointsrsandthecorrespondingbaryon density,nssatisfythefollowingalgebraicsystemofequations: a2s=u2s1 2Mrs+MAr2s 1: u2s=1amr2s; (8.31) (+P n)2(1 2Mr+M r2a)js=dp r2a)js=(+p dnn +Pjs; n)2j1: (8.32) theycanberewritteninthefollowingmanner; Intermsofthevariable=r 2(4 a)=(1 1+2 a 2 ah=+p 2Mdenedearlierand n=m( 1) 2(4 a))(dp 1 a2; dn)(n (8.33) Introducingthefunction a2jsh2j1 h2s=a2s[ 1 a2s 2(4 a)=h2j1 1 a21]2; h2js: +P)js; (8.36) (8.35) (8.34)

9 wheneq.(8.34)gives 8.3Thedeterminationofthecriticalpoints fromthelastequationwehave 2(4 a)=a2 1+1; 2 a 217 theeventhorizon.substitutingtheaboveexpressionsfor;ineq.(8.34) afterlongalgebraiccomputations,wegetthefollowingequationdetermining thesoundspeedatthecriticalpoints: Fromthislastequationweseethat>1,thusthecriticalpointliesoutside 8a4 2(1+a)a2a2 1 a2+aa2 1 a22=0:(8.38) = a2 (1 a2): (8.37) boundedaboveby 1seeEq.(8.22).Atthecriticalpoint,using Thisequationhasbeenstudiednumerically.Wehavefoundthatforvarious [0; 1).Noteforanypolytropicequationofstatethespeedofsoundis valuesofa21and =4=3therealwaysexistasolutionlyingintheinterval wegeth2=m2 1 1 a22=m2(1+2a2 a2s=2a211 1)+O(a4); =a2(1+2a O(a41;a4a); 1)+O(a4)and (8.40) (8.39) Furthermorethenumericalcomputationsconrmtheexpressions(8.41),(8.42). Insummarythereforetherealwaysexistonecriticalpointwhosecoordinate thatexpressions(8.41),(8.42)areidenticaltothoseoccurringforaccretion locationandspeedofsoundaregivenbyequations(8.41),(8.42).noticealso = a21+O(a41;a4a): (8.42) takingplaceonaschwarzschildbackground.utilizingtheinformationofthe owatthecriticalpointoneeasilycomputestheaccretionrateintermsof ofaappearsascorrectiontermoftheordera41a. i.e.thesamerateasifthebackgroundwouldhavebeenaschwarzschildblack holewiththesamemassm.theparameteriscombinationofthevariousnumericalfactors,isoforderunityandindependentupona,whileanycontribution asymptoticquantitiesatinnity: M=42 _ ( 1)M2n1a 3 1; (8.43)

10 218 Accretionondilatonicblackhole 8.4Thenatureofthecriticalpoints Havingestablishedtheexistenceofacriticalpoint,weshallnowexamineits topologicalnature.thisentailsanunderstandingofthewayvarioussolutions curvesofeqs.(8.16),(8.17)behaveintheneighborhoodofthecriticalpoint. Ultimatelyoneliketoshowthatforaowthatisstationaryatinnityaunique regularsolutionexistsreachingtheeventhorizon.forthatisnecessarythatthe criticalpointexhibitsaddlelikecharacter,whichbydenitionimpliesthatonly twoatleastc1solutionpassthroughthecriticalpoint.onedescribingaccretion andtheothera\stellarwind".howeveroutowsonablackholebackground areconsiderasunphysicalsinceasweshalldiscussfurtherlateron,theyare singularoveranonsingularhorizon.inorderunravelthenatureofthecritical point,itisconvenienttoturneqs.(8.16),(8.17)intoathreedimensional dynamicalsystem.introducingaparameterlalongthesolutioncurvesofthe system(8.16),(8.17)anddeningasonecolumnvector:~x=[r(l);u(l);n(l)] onegetstheequivalentthreedimensionalsystem (_~x)t=(d;d1;d2)t; (8.44) whereoverdotsigniesdierentiationwithrespecttol,whiletsigniestranspositionofthetherowvectors.inthisformulationthecriticalpointsappear asequilibriumpointsi.e.pointswheretherighthand-sideofeq.(8.44)isvanishing.accordingtohartmann-grobmanntheorem[per91],inthevicinityof acriticalpoint(equilibriumpoint)thesolutioncurvesof(8.44)arehomeomorphicallyequivalenttoitslinearizedversioni.e. _~x=f~x; (8.45) wherefstandsforthedierentialmatrixofthevectorvaluedfunctiondened bytherighthandsideof(8.44)andcomputedatthecriticalpoint.denoting byai;bi;ciwithi=1;2:3thepartialderivativesofd;d1;d2withrespectto r;u;nrespectivelyandtakingintoaccounteq.(8.29)onendsthefollowing expressionsvalidatthecriticalpoint: A1= 2Aa2u n;a2=2(1 a2) n;a3= u B1=u2 n;b3=au2 C1= 2A;C3=0:(8.46) Furthermoreintermsofthepartialderivatives,thecharacteristicequationde- nedbyfhasthefollowingform; [(A1 )(B2 )]+[B3C2+A2B1+A3C1] +[ A1B3C2+A2B3C1+A3B1C2 A3B2C1]=0 (8.47)

11 notingthata1= B1theremainingtwononzeroeigenvaluesaregivenby: 8.5Asymptoticanalysis Astraightforwardcomputationthenshowsthatthelastterminthecharacteristicequationisvanishing.Thereforeoneoftheeigenvaluesiszerowhileby =[ (A21+B3C2+A2B1+A3C1)]12: (8.48) 219 Expandingoutthequantitywithinthesquarerootonendsthatatthecritical pointtheeigenvaluesarerealandofoppositesign.diagonalizingthedierential matrixfandfollowingthestandardprocedure[per91],onemayshowexplicitly solutioncurvesexhibitasaddletypebehavior,implyingfurthertheexistence thatonlytwosolutionspassthroughthecriticalpoint.thereforelocallythe oftwodistinctsolutionspassingthroughthecriticalpoint.thusthecritical pointisofthesamenatureastheoneadmittedbythecorrespondingow equationsonaschwarzschildbackground.1wehaveestablishedtheexistence auniquetransonicow,subsonicatinnity,passingtroughthecriticalpoint foundmoreconvenienttoemployequations(8.27),(8.28).theyhavebeen plottedforvaluesof1appropriatetotypicalinterstellarmediumandvarious 1thereaexistacriticalvalueof2(andthusa\critical"accretionrate_ valuesoftheparameter2.numericallywehavefoundthatforanychoiceof andreachingthehorizon,byresortingtonumericaltechniques.forthatwe Ifforthemomentweignorethenumericalresultsdiscussedinlastparagraph, suchthataowsubsonicatinnity,passesthroughacriticalpointandreaches theeventhorizon.thevarioussolutioncurvesforv;aareshowinginfigures 8.1, Asymptoticanalysis M) theanalysisoftheaccretionowpresentedsofarisentirelyindependentupon placeastheowcrossestheeventhorizon.inparticularlyatthecrossing naturallyinferasetofconditionsobeyedbytheowatthehorizon.according tothestandardblackholephysics,onedemandsthatnothingpeculiartakes forallvaluesofaandirrespectivewhetheralieswithin[0,2].2however,its onablackholebackgroundoranakedsingularity.thecrucialelementthat dierentiatesbetweenthetwoisthesetofboundaryconditionsobeyedbythe owonthehorizonandsingularityrespectively.fortherstcase,onecan continuationfromthecriticalpointinwarddependswhetheritispropagating theparticularvalueofa.theowuptothecriticalpointisuniquelydetermined havebeenintroducedanddiscussedatsomelengthbythorne[tho81]and validfortheothercriticalpoint.theanalysisutilizeonlyrelations(8.29)andtheseareindependentofthelocationofthecriticalpoint. nophysicalscalarsareallowedtodiverge.thiskindofregularityconditions ofsingular\sphere". Thorneetal[TFZ81].Howeverforowsrunningonnakedsingularitiesthings areratherumbigious.itnotclearwhatconditionsaretobeimposedonthe 2Forthecaseofnakedsingularities,amustconstrainedsothatthecriticalpointliesoutside 1Althoughattentionhasbeenrestrictedtotheexteriorcriticalpoint,thesameconclusion

12 220 Accretionondilatonicblackhole obtainedfromplottingeqs.(8.27),(8.28).inorderexhibitclearlythesaddle Figure8.1:Figureshowsthetopologyofthevarioussolutioncurvesforthev2 a=1:8,a21=0:133andv1=0. characterofthecriticalpointandthetransonicsolutionwehavetaken =4=3, thepreviousgure. obtainedfromplottingeqs.(8.27),(8.28).theparametersarethesameasin Figure8.2:Figureshowsthetopologyofthevarioussolutioncurvesforthea2

13 8.5Asymptoticanalysis 221 a21=0:133andv1=0.notethatforrealisticboundaryconditionsa211, thecriticalpointliespracticallyonthehorizontalaxis. Figure8.3:Criticalsolutionsinthey v2plane,foraschwarzschildblackhole a=0andforanextremedilatoniconea=1:99.theotherparameters =4=3, Figure8.4:Thesameasinthepreviousgurebutinthey a2plane.

14 222 Accretionondilatonicblackhole singularity.noteinparticularlythatfora=2thesaddlecharacterofthe criticalpoint,atthespacetimesingularity,impliesthatoutowsarenotapriori excluded.thispointmayinterpretedasimplyingthatspacetimesingularities aretotallyunpredictable.hereafterweshallconcentrateonowsonblackhole backgrounds.forourproblem,itissucienttodemandthatthemagnitudeof thefour-accelerationoftheowaswellasthebaryondensitynareboundedat thehorizon.therstconditionimpliesthatthenongravitationalforcesacting upontheowarenitewhileaconsequenceofthesecondconditionsisthatall otherowparametersremainboundedaswell.astraightforwardcalculation showsthatthenonvanishingcomponentsofthefouraccelerationvectoraare @r+mr21 1 2Mr+2Mr2urur 1 2Mr#r: Intermsoftheordinaryvelocityvelocityvandtheparameteryintroduced earlier,onendsthefollowingexpressionforthemagnitudeaa: aa=1 4My2 (1 y)121 (1 v2)12 y2 2M(1 y)12v (1 v2)32dv dy: Acloselookattherighthandsideoftheaboveexpressionindicatesthatthe magnitudeofthefouraccelerationisunboundedonthehorizon,unlessv(y) exhibitsthefollowingbehavior; v(y)=1+dv dyjy=1(1 y)+o((y 1)2): Inthatcasewendthat aa(y=1)(dv dy)12y=1+o(y 1)2: HoweverfromEq.(8.25)onecaneasilyinferdv dyjy=1isnonzeroandboundedat thehorizon.inturnequation(8.24)showsthataregularowmusthaveradial velocityu=urnonvanishingonthehorizon.inthefollowingtheasymptotic behavioroftheownearthehorizoncompatiblewithu6=0andboundednwill bedeterminedforvariousvaluesoftheparameterabelongingto[0;2).from Eqs.(8.16),(8.17)onendsthefollowingequationsdescribingtheownear thehorizon1+:1udu d 1 22 u2a2d dlnjgj u2(1 a2); (8.49) 1ndn d1 22 u2d dlnjgj u2(1 a2): (8.50) Weshalllookforasymptoticsolutionsdescribingregularowsovertheevent horizon.itisclearthatthenatureoftheasymptoticsolutionsdependsuponthe

15 terminthenumeratorofeq.(8.49)maybeignored.inthatcasedemanding thatuisnonvanishingonthehorizon,onearrivesat 8.6Breakdownoftheadiabaticityassumption particularvalueofa.weshallexplicitlywritedowntheclosedformsolutionsfor thetwoextremecasesofthea,namelyforaclosetozeroandthediametrically oppositecasei.e.aextremelyclosetovalue2.fortheformercasethesecond 223 Howeveraswehavearguedearlierforanearlyextremedilatonicblackhole,the secondtermcannotanylongerbeignored.infactdominatestherstterm.in suchcaseonendsu()=a1g()a2 i.e.thewellknownfreefallasymptoticsolutiononshwarzchildbackground. u2()=1;n()=c32; Eq.(8.13)nearthehorizononendsthatA1;A2obey: wherea1;a2arearbitraryconstantsofintegration.demandingsatisfactionof whilebernoulliequationimplies: 1 a2;n()=a2g() 1 A1A2=_M; 1 a2; (8.51) Thesolutionareconsistentprovided+P= massesarenegligible.insuchcasewendthata2= 1=1=3wherewehave A2(+P)g()a2+1 1 a2=[+p 1Pi.e.mn0,ortherest n]1: (8.52) tionsshowsthatnearthehorizonthebaryondensityisgivenbythefollowing expression: takenthepolytropicindextoequalto4=3.manipulationoftheaboveequa- n()=[ Kn(1) ( 1)(+P)1_M g()]32: (8.54) (8.53) 8.6Breakdownoftheadiabaticityassumption relativisticandthus =53.ModelcalculationsfortheSchwarzschildholeshow Inmorerealisticmodelsofaccretinginterstellarmediumontoablackhole,the inowingplasmaconsistsofaprotonandanelectroncomponentofequaldensities.initiallyi.e.atradialinnity,bothcomponentareconsideredasnon thattheelectroncomponentbarelybecomesrelativisticwhiletheprotoncomponentremainsnonrelativisticallthewaydowntothehorizon.howeverfositiesimplyhightemperatures.assuminganidealplasmathenthetemperature measuredinthethelocalrestframeoftheuidisgivenbyt(r)=n(r) 1. theoneallowstheparameteratogetclosetotheextremevalues.buthighden- Theaboveasymptoticsolutionsindicatesthatn()becameextremelyhighas thedilatoniccase,thingsmaybedierent.theasymptoticsolutionsindicate

16 creation,electron-positronannihilationandproton-protoncollisionsleadingto pionproduction,totakeplace.becauseofthisimportantdierenceintheaccretionnearanextremedilatonicblackhole,itisnaturaltowonderwhetherthe 224 thatfordilatonicblackholesclosetotheextremelimitconditionsaresuchthat bothcomponentsoftheinowingplasmabecomehighlyrelativistic.physically thenoneexpectsthatmanycollisionssuchasthermalbremsstrahlung,pair Accretionondilatonicblackhole productionofphotonsduetovariousprocesses.thistaskhoweverentailsan duetovariousprocessesisnegligiblecomparativelytotheincreaseintheinternalenergydensityoftheplasmaduetotheworkdonebygravitationalforces. AsisknownBondiaccretiononaSchwarzschildbackgroundindeedfulllthis requirement.tocheckthevalidityoftheadiabaticityassumptionforaccretion onaanearlyextremedilatonicblackhole,onehastocomputeindetailsthe initiallyimposedadiabaticityassumptionisstilljustiedbytheobtainedsolution.thelargeproductionofradiationmaynotanylongerneglected.onthe elementofuncertaintymainlyduetoambiguitiesinthevariouscrosssections particularlyifoneallowsvaluesofathatleadintopionsproductionviaproton otherhandadiabaticitywouldbeconsistentprovidedtheoutgoingluminosity experimentatthesameregion.inviewofthefactthattheplasmaishighly colission.fortunatelywedonotneedtoenterintosuchnedetails.aswe relativistic,wemaysetwithoutlossofgenerality =4=3.Accordingtothe shallseetakingintoaccountonlythermalbremsstrahlungitwillbesucientto concludethatindeedadiabaticityitisnotconsistentwiththeasymptoticsolutionsandthustreatmentincorporatingradiativetransportisnecessary.since thesensitiveregionisneartheeventhorizonweshallperformourtheoretical calculationsofnovikovandthorne[nt73],agramoftheplasmainthelocal restframeoftheuidlossesradiationaccordingto: Comparingthetwooneconcludesthattheoutgoingluminosityoutweighsthe Ontheotherhandtherstlawimpliestherateofincreaseoftheinternalenergy bytheworkdonebygravityisgivenby d(n)=kn 2g()g() 16 efft12n() 1 2g() 14 internalenergygainingandthusbondiadiabaticaccretionstrictlyisnotany Insomesensethisisratherunderstandable.Theplasmaduetotheexcessive heatingemitsmuchmorefree-freeradiationinthecaseofextremedilatonic longerconsistentwithadilatonicblackholeaapproachingtheextremelimit. blackhole. 8.7Discussion Insummarytheresultsobtainedsofar,showthataccretiononadilatonicblack holemaybeofentirelydierentnaturethanaccretiononaschwarzschildone.

17 8.7Discussion Thecrucialelementisthevalueofa.ThebreakdownofBondiadiabaticaccretionindicatestheneedofinclusionofradiativetransporteects.Howevereven withtheinclusionofsucheectswebelievethatoutowingluminositywould bemuchlargerthantheonewouldhaveemergedifthebackgroundwasthat 225 ofaschwarzschildblackhole.sinceasisclearfromthesofardiscussionisthe geometryofthedilatonicblackholewhichcauseslargeluminositygeneration. Theincorporationofradiativetransporteectsaswellastheincorporationof thebackgroundmonopoleeld,arecurrentlyunderconsideration. ThisworkhasdoneincollaborationwithThomasZannias[VZ].

18 226 Accretionondilatonicblackhole

19 Bibliography [Hol77]T.E.Holzer,J.Geophys.Res.82(1977),23. [Gib82]G.W.Gibbons,Nucl.Phys.B207(1982),925. [GM88]G.W.GibbonsandK.Maeda,Nucl.Phys.B298(1988),741. [Bon52]H.Bondi,MNRAS112(1952),195. [GHS91]D.Garnkle,G.Horowitz,andA.Strominger,Phys.Rev.D43(1991), [HT83]S.R.HabbalandK.Tsinganos,J.Geophys.Res.88(1983), [TFZ81]K.S.Thorne,R.A.Flammang,andA.N.Zytkow,MNRAS194 [Per91]L.Perko,DierentialEquationsandDynamicalSystems,Spring- [NT73]I.D.NovikovandK.Thorne,inBlackHoles,GordonandBreach,N. Verlag,1991. Y.,1973. [VZ]N.VlahakisandT.Zannias,unpublished. [Tho81]K.S.Thorne,MNRAS194(1981),439. (1981),