Table Of ContentUniversality andbackreaction inageneral-relativisticaccretion ofsteady fluids
JanuszKarkowski,1 BoguszKinasiewicz,1 PatrykMach,1 EdwardMalec,1,2,3 andZdobysławS´wierczyn´ski4
1M.Smoluchowski InstituteofPhysics, Jagiellonian University, Reymonta4, 30-059Krako´w, Poland
2AEI MPI, Golm, Germany
3TPI FSU, Jena, Germany
4Pedagogical University, Podchora¸z˙ych 1, Krako´w, Poland
Thesphericallysymmetricsteadyaccretionofpolytropicperfectfluidsontoablackholeisthesimplestflow
modelthatcandemonstratetheeffectsofbackreaction. Theanalyticandnumericalinvestigationrevealsthat
backreaction keeps intact most of the characteristics of the sonic point. For any such system, with the free
parameterbeingtherelativeabundanceofthefluid,themassaccretionrateachievesmaximalvaluewhenthe
massofthefluidis1/3ofthetotalmass.Fixingthetotalmassofthesystem,oneobservestheexistenceoftwo
weaklyaccretingregimes,oneover-abundantandtheotherpoorinfluidcontent.
6 The spherical steady accretion of gas onto a gravitational ThearealvelocityU ofacomovingparticledesignatedby
0 center has been investigated in the Newtonian context by coordinates(r,t)isgivenbyU(r,t)= ∂tR = R(trK Kr).From
0 N 2 − r
Bondi[1]andintheSchwarzschildspace-timebyMichel[2], theEinsteinconstraintequationsonehas
2
Shapiro and Teukolsky [3], and others [4]. The Newtonian
n
criticalaccretionflow was shownto be stable by Balazs and 2m(R)
a pR=2 1 +U2 (2)
J others [5], while the stability of the steady flow in the fixed r − R
6 Schwarzschild background has been analyzed by Moncrief
[6]. The general-relativistic description including backreac- and∂R(R2U) R2trK = 0. We usehereandinwhatfollows
−
2 tion has been formulated in [8]. There are two reasons for therelation∂r = √α(pR/2)∂Rinordertoeliminatethediffer-
v inspecting steady flows in the full generality, when infalling entiationwith respectthecomovingradiusr. Thequasilocal
79 gasmodifiesthespace-timegeometry,thatisincludingback- mass m(R) obeys the equation ∂Rm(R) = 4πR2̺. The mass
0 reaction[7]. First, onecan see the effectsofbackreactionin accretionrateism˙ =(∂t−(∂tR)∂R)m(R)=−4πNR2U(̺+ p˜).
9 asimplebutnontrivialaccretionmodel.Second,gravitational A more familiar form of that is m˙ = 4πR2nU, where n is
−
0 collapse of the fluid which starts as a flow dominated by a thebaryonicdensity(seebelow).Astandardconditionforthe
5 steadyaccretion,canbebettercontrolled,andthatwouldal- steady collapsing fluid is that all its characteristics are con-
/0 lowonetohaveaninsightintoformationofgravitationalsin- stant at a fixed R (see [9]). In analytical terms ∂tX|R=const =
c gularities. This paper focuses on the issue of backreaction. (∂t (∂tR)∂R)X = 0,whereX = ̺,U,a...[10]. TheEinstein
q We show that the backreaction strongly influences the mass evo−lutionequation∂tU = p2R2∂RN/4 m(R)N/R2 4πNRp˜
- − −
r accretionrate. andtheenergyconservationequation∂t̺= NtrK(̺+p˜)be-
g We will consider a spherically symmetric compactball of comeordinarydifferentialequationswithres−pectR,assuming
:
v afluidfallingontoanon-rotatingblackhole. Theblackhole steady flow. Selfgravitating steady fluids still allow for the
Xi providesthe simplest choice for the central object since one change in time of some geometric quantities, like the mean
can neglect the occurrence of shock waves. The fluid is re- curvature p or the area of the black hole. It is easy to show
r
a garded to be steady in a sense defined later and by a black thatforasteadyflow∂Rm˙ =0[8].
holeweunderstandtheexistenceofanapparenthorizon. For Byablackholeismeantaregionwithinanapparenthori-
thedetailedderivationofequationsthereadershouldconsult zon to the future, i.e., a region enclosed by an outermost
[8]. Herewegiveonlyabriefdescription,focusingattention sphereSA onwhichtheopticalscalarθ+ ≡ p2R +U vanishes
on the physicalassumptions. We will use comovingcoordi- [12]. (The other condition, that θ pR U > 0 for all
nates spheres outside S , is satisfied triv−ia≡lly2for−a steady accre-
A
tion.) That means that on S the ratio 2m /R becomes
ds2 = N2dt2+αdr2+R2 dθ2+sin2θdφ2 , (1) A BH AH
1, where R is the areal radiusof the apparenthorizonand
− AH
(cid:16) (cid:17)
m m(R ) is the mass of the black hole. We shall spe-
where the lapse N, α and the areal radius R are functions ciaBlHize≡to poAlyHtropic perfect fluids p˜ = K̺Γ, with Γ being a
of a coordinate radius r and an asymptotic time variable t.
constant,(1 Γ 5/3).
ThenonzerocomponentsoftheextrinsiccurvatureKij ofthe ≤ ≤
Furthermore,assumetheradiusR oftheballoffluidand
t = const slices read Krr = 2N1α∂tα, Kφφ = Kθθ = ∂NtRR = boundary data are such that U ∞ m(R ) a . These
21(trK−Krr). HeretrK = N1∂tln √αR2 . Themeancurvature boundary conditions are need|ed∞i|n≪ordeRr∞∞to g≪lue t∞he steady
of two-spheres of constant radi(cid:16)us r, em(cid:17) bedded in a Cauchy fluidwiththeexternalSchwarzschildgeometry(seeadiscus-
hypersurface is p = 2∂rR. The energy-momentumtensor of sionlater). Themomentumconservationequation Tµ = 0
R√α µ r
theperfectfluidreadsTµν = (p˜+̺)uµuν+ p˜gµν,(uµuµ = 1) —incomovingcoordinates ∇
−
whereuµ denotesthefour-velocityofthefluid, p˜ isthepres-
sureand̺theenergydensityinthecomovingframe. N∂ p˜+(p˜+̺)∂ N =0 (3)
R R
2
—canbeintegrated,yielding(withN(R )=1) the standard junction conditions — continuity of the metric
∞
and of the transversal extrinsic curvature, and the condition
Γ+a2
a2 = Γ+ ∞. (4) N = 1 —requirenowtheappropriatechoiceofthe slicing
− Nκ of∞the exteriorregion. We verifiedin a numberofnumerical
Here κ = Γ 1. (4) canbe regardedas the general-relativistic calculationsthat for short evolution times (t R /a ) and
−Γ undertheformerboundaryconditionsthebul≪kof∞thefl∞uidis
versionoftheBernoulliequation.Anotherusefulrelationthat
followsfrom(3)isN = Cn . Hereappearsthebaryonicnum- steady. (Notice, that this observationindicatesthat such ini-
berdensityn=Cexp ̺p˜+d̺s 1 .Thebaryonicnumberden- tialdatacanbeusefulinordertomodelthegravitationalcol-
̺ s+p˜(s) lapse.)Theasymptoticmassmofthespacetimewouldbewell
sity works, forperfectRb∞arotropes,asanintegrationfactor; if
approximatedbym ,thesumofthemassesoftheblackhole
T is conserved, then (nuµ) = 0. This representation of ∞
µν µ and the quasi-stationary gas, for a suitably chosen transition
∇
thelapsefunctionshowsthatthelapseN isconstantalongan
zone.
orbitoftheconstantarealradiusR.
Innumericalcalculationsitwillbeconvenienttorepresent
In the general-relativistic case the sonic point is defined
themassm(R)intheform
as a location where the length of the spatial velocity vec-
tor U~ = U/(pR/2) equals a. Therefore at a sonic point R
|U| |=|12pR|a.|In the Newtonian limit this coincides with the m(R)=m−4πZR ∞drr2̺. (7)
standard requirement U = a. In the following we will de-
note by the asterisk a|ll v|alues referring to the sonic points, Noticethatm( ) = m m. Assumingsteadyflow,onecan
∞ ∞ ≈
e.g. a , U , etc. The fourcharacteristics, a , U , m /R and obtainfollowingexpressionforthelapseN [8]
c are∗rela∗ted [8], a2 1 3m +c = U2 =∗ m ∗+c∗, w∗here
c∗ =2πR2p˜ . Thein∗fa(cid:16)ll−ve2loR∗c∗ityU∗(cid:17)reads∗ 2R∗∗ ∗ N = pR β(R),
p(R )R
∗ ∗ ∗ ∞ ∞
R2 1+ Γ 1/(Γ−1) β(R) = exp 16π R∞(p˜+̺) ds . (8)
U =U∗R∗2 1+ aaΓ22∗ . (5) The full description of t h−e steaZdRy accretionpt2hsa!t will be dis-
Here U is the negative square root. From the relation be-
cussedlaterisgivenbyequations(2),(7),(4),(5)and(8);this
∗
tweenthepressureandtheenergydensity,oneobtains,using
isequivalenttotheformersystem.
equation(4)
Oneoftwomainresultsofthispaperisrealizationthatsig-
nificant information about the full system with backreaction
where̺th=e c̺o∞n(sata/nat∞̺)2/(Γis−1e)q=ua̺l∞to−thaΓe2∞m+asasΓ2∞Nd+eκn1sitΓy−11o,f a c(o6l-) ctahaenNabbanaemdcokelbrleyetta,ailcUnetteiodbnethbgreioivmnue(gRgnhi)Γgtnh∈oer(iae1ndv,.5.e/sAt3isg]s,auatmisoyenmtohpfatsotttetiahcdedysaofltnaoiw̺c∞spwoainintdht
lapsingfluidatthebo∞undaryR .Thesteadyfluidisdescribed is∞located |ou∞ts|i≪de thR∞e∞a≪ppar∞ent horizon, i.e., U2 < 0.25.
∞
byequations(4–6).Theyconstituteanintegro-algebraicsys- Then the sonic point parametersa2, U2 andm /∗R in the
tem of equations, with a bifurcationpointat the sonic point, abovemodelwithbackreactionare∗the∗sameas∗in∗thetest
wheretwobranches(identifiedasaccretionorwind)docross;
fluidaccretionwiththesameasymptoticdata[11].
that is a well known feature of that problem, present also in
Sketchofthereasoning.Wewillapplythebootstrap-type
modelswithatestfluid[1]–[5],[8].Thatrequiressomecau- argument. Let us assume temporarily that U2 < 0.1, that is
tionindoingnumericsandacarefulselectionofthesolution R > 5m . This implies pR/2 √0.7, a2 ∗ a2 1/7 and
binratenrcmh.sNofoqtiucaentthitaitetshtehsaetaerqeusatteioandsyainretheexpsreenssseedofetxhcelufosirvmeelyr 16∗π RR∞(p∗˜+̺)p12sds≤80m−4m9R(R∗≥) forR≥R∗.≤ ∗ ≤
IfRm m(R ) (49/80)m(R ∗),thentheintegrandpresentin
definition. theexp−onent∗of≤thefunctionβ∗(r)issmallerthanm /R . That
This collapsing compact cloud can be connected to the ∗ ∗
Swcohuwldarbzestchhaitldenegxitneereiorer.dbTyhae”didaeeamlomn”odwehloojfusatfistxeeasdbyouflnodw- inturnmeansthatN ≥ q1− 4Rm∗∗ +U∗2and
ary data conditions; a similar picture might well be valid in Γ+a2
somebinaries. Alternatively,onecanimagineaninitialcon- a2∗ ≤−Γ+ 1 4m ∞+U2κ. (9)
figuration with a steady fluid annulus around the black hole q − R∗∗ ∗
and an externaltransition zone thatis not steady (since both
Solutions of this inequality are bounded from above by the
pressureandthemassdensitymustbemadetovanishthere)
solutionoftheequation
and that can expand outward and inward with the speed of
sound, perturbing the external layers of the steady interior. Γ+a2
Tlehssetbraonusnidtiaornyzcoanneb,etoctohneneexctteerdnaaltSRc∞h,wvairazsacnhialldmgoesotmmeatsrys-; a˜2 =−Γ+ 1 2m˜ +∞U˜2κ, (10)
− R
q ∗
3
wherem˜ =2m ,atthepointwherea˜2/ 1+3a˜2 =m˜/(2R )= ingfluidcanbeexpressedasbelow(seeEq.(6.1)in[8])
∗ ∗
U˜2. NotethatR > 2m˜; thisallowsus(cid:16)touse t(cid:17)heestimateof
Theorem2of[8]∗. Thusonehas R2 a2 (25(−Γ3Γ1)) a2 1+3a2
m˙ =πm2̺ ∗ ∗ − 1+ ∗ ∗. (14)
∗ ∞m2 a2 ! Γ! a3
2a2 ∗ ∞ ∞
a2∗ <a˜2 ≤ 5−3Γ+ 3a˜2∞4(ΓΓ−(11+)23(a˜92Γ)−7). (11) TThheiswachhoileevdeesp(efinxdienngcme)ona̺m∞axisimcounmtaaintemdin=th2emf/a3ctaonrdmte2∗̺n∞ds.
to zero at both ends: i) m m (when∗the density ̺ tends
Insertingthisestimateinto(6),onecanboundthematterden- ∗ →
to zero) and ii) m /m 0 (when the mass of black hole is
sity, ̺(R) ̺ 1+C(Γ,a )/ 5 3Γ+a2 . Here the con- ∗ →
stantCiso≤fthe∞o(cid:16)rderof10.∗Th(cid:16)isi−nturnall∗o(cid:17)w(cid:17) sonetoreplace negligibleincomparisontothemassofthefluid).Inthisplace
weinvokedtheassumptionsconcerningboundaryconditions.
theboundontheexponentofβbyamuchbetteroneoftheor-
Thus, one arrives at our second main result: Amongst
der(m /R )(R /R ); thisisclearlynegligibleincomparison
steadilyaccretingsystemsofthesameR ,a andmthose
tom /R∗ (∗save∗for∞asmallregionaroundΓ=5/3)andonecan willbemostefficient forwhichm = 2m∞/3.∞Thefactor2/3
repe∗atth∗e previousargument, butthis time assuming β = 1. isuniversal—independentofthepa∗rametersR ,Γanda .
Thiswouldlead to the improvedestimate ofa and ̺outside Sincethemassofthefluidm isclosetom ∞m andm∞=
Rue=so5fRU∗.2Faunrdthoefrrmefi/nmem, aenndt—toathlleΓe’xsteinnstihoeniinntteorvbailgg(1e,r5v/a3l-] mf + mBH, the above means thaft the maximum− of∗the mass
accretion takes place when the mass of the fluid is a half of
—would∗bedoneb∗ysplittingtheintegralofβintotwointe-
the mass of the black hole. We would like to point that in
R 5R R
grals, R∞... = R ∗...+ 5R∞... and separately estimating (14)themassm isonlyapproximatelyconstant,andhencem˙
eachteRrm∗ . R ∗ R ∗ is not strictly co∗nstant in time. The fulfilmentof m˙τ m ,
Usingtheprecedingestimates,onegets whereτistheevolutiontime,canbeunderstoodasyeta≪nothe∗r
criterionforthevalidityofthesteadyflowapproximation.
a2 m R2 m
c = ∗2πR2̺ Ca2 ∗ ∗ ∗. (12) In the remaining part of the paper the asymptotic mass m
∗ Γ ∗ ∗ ≤ ∗R R2 ≪ R has been normalized to 1. In order to test the above results,
∞ ∞ ∗
weperformedaseriesofnumericalcalculations,accordingto
Thatimpliesthatbothc andtheexponentofβcanbeputto
thefollowingscenario. i)FixavalueoftheadiabaticindexΓ
∗
zeroinallrelationsbetweencharacteristicsofthesonicpoint. andtheasymptoticspeedofsounda2;changeinasystematic
InsertingthisinformationintotheBernoulliequation(4)one way the asymptotic mass density. ii∞) For the same Γ choose
obtainsthatatasonicpoint[8] newa2 andchangetheasymptoticmassdensity. iii)Choose
new Γ∞and repeat the preceding procedure. Below we shall
1+y(3Γ 1)=3 a2 +Γ yΓ2+Γ1, (13) describeonlyasmallsubsetofthenumericalevidence.
− (cid:16) ∞ (cid:17)
wherey=1 3m /(2R ). Coefficientsofthisalgebraicequa-
tiondonotd−epend∗onth∗easymptoticmassdensityandthere- TABLEI:Characteristicsofthesonicpointa2, U ,R fortheΓ =
fore y is ̺ -independent. The sonic mass m and the sonic 4/3polytropeanda2 =0.1. Thelastcolumns∗ho|w∗s|the∗arealradius
radiusR c∞learlydependon̺ buttheirratio∗isconstant. In of the apparent hori∞zon. The mass accretion rate (second column)
fact,m /∗R mustbeexactlyth∞esameasinthecasewhenthe achievesmaximumatmBH ≈0.663.
backrea∗ctio∗n can be neglected, that is when the mass of the mf/m m˙ R U a2 RAH
∗ ∗ ∗
fluidoutsidetheblackholeissmallincomparisontothetotal 4.186 10−32 4.156 10−48 4.290 0.34137 0.179184 1.993
× ×
mass. Thesameconclusionholdstruealsoforotherparame- 4.186 10−17 4.156 10−33 4.290 0.34137 0.179184 1.993
× ×
tersofthesonicpoint,thefluidvelocityU andthespeedof 0.041 3.815 10−18 4.110 0.341369 0.179182 1.906
×
sound a . In conclusion: a2, U2 and m /R∗ can be inferred 0.083 6.978 10−18 3.931 0.341371 0.179183 1.821
×
fromas∗uitablesteadyfloww∗ith∗atestflu∗id.∗ 0.167 1.152 10 17 3.571 0.341371 0.179182 1.656
−
×
It follows from the above discussion that the mass den- 0.251 1.397 10−17 3.212 0.341371 0.179183 1.492
×
sity changes moderatelyoutside the sonic point. In fact, for 0.333 1.469 10 17 2.859 0.341371 0.179183 1.326
−
×
R [R ,10R ] ̺(R) C(R)̺ with C(R ) being of the or- 0.418 1.403 10 17 2.493 0.341399 0.179184 1.156
−
der∈of 1∗0 and∗C(10R≤) being s∞lightly large∗r than 1. Assum- 0.628 8.612×10 18 1.594 0.341369 0.179182 0.740
−
∗ ×
ing that R 10R one can show (taking into account the 0.837 2.187 10 18 0.696 0.341394 0.179185 0.322
∞ ≫ ∗ × −
fact,thattheregion(10R ,R )hasamuchlargervolumethan 0.879 1.264 10 18 0.516 0.341399 0.179184 0.239
−
(R ,10R ))thatunderthe∗ pr∞ecedingassumptionsthemassof 0.962 1.277×10 19 0.156 0.34137 0.179187 0.072
−
ou∗terlay∗ersofthefluidm−m∗ =4π RR∞drr2̺=γ̺∞ witha 0.983 2.382××10−20 0.067 0.341399 0.179185 0.031
constantγ determinedinpracticeonlRy∗byR . Alternatively,
m /m = 1 ̺ (γ/m);theratiom /misalin∞earlydecreasing Itappearsthatthecharacteristicsa2,U2,m /R ofthesonic
fu∗nctionof−̺ ∞. ∗ point practically do not depend, fo∗r a ∗give∗n Γ∗and a2, on
Therateof∞themassaccretionm˙ withinthesteadilyaccret- theenergydensity̺ (exemplaryresultsareshownin∞Table
∞
4
1), suggesting a precision of 10 5. The more reliable mea- tionreadsN Γ 1 a2 =Γ 1 a2.Theanalyticarguments
−
sureofthenumericalerroristhedifferencebetweenthemass ofthetypepr(cid:16)ese−nte−dab∞o(cid:17)vecan−be−appliedandconclusionsare
m =1(assumedintheequations)andthemassfoundnumer- thesame. Similar resultsshouldbetrue forNewtonianmas-
ically. Our results(see Table 1) suggestthatnumericalerror sivefluidsforΓsignificantlysmallerthan5/3.
is smaller than 0.5 %. The quantity c is negligible in com-
∗ Inconclusion,inthesimplemodelofaccretionwithback-
parisontoothersonicpointparameters.Themassm behaves
like m = 1 γ̺ (Fig. 1). Andfinally, theextrem∗umofm˙ reactionconsideredhere,onecangetallparametersdescrib-
is achi∗evedw−hen∞m 2/3(Fig. 2). Notably, numericalre- ingthe sonic point, with theexceptionof its locationR and
∗ ≈ massm simplyfromarelatedpolytropicmodelwithth∗etest
sults extendthe regimein whichthe universalityis observed
fluidha∗vingthesameindexΓandthesameasymptoticspeed
intofluidscharacterizedbythespeedofsounda exceeding
1. This is remarkable, since that implies R < R∞ and the of sound a . The main resultis that the mass accretion rate
AH ∞
∗ m˙ achievesmaximumatm /m 1/2;thereforethereexist
theoreticalanalysispresentedabovewouldnotbevalid. f BH
≈
twodifferentregimes,m /m 1andm /m 1, with
f BH f BH
≪ ≫
1 lowaccretion. Incontrasttothat,inthetestfluidapproxima-
tionthequantitym˙ growswith̺ (thatiswithm /m ). The
0.9 f BH
∞
mass accretion rate m˙ sets the upperlimit for the luminosity
0.8
of the system. Hence this simple analysis of accretion sug-
0.7
geststheexistenceoftwoweaklyluminousregimes:onerich
0.6 influidwithm /m 1andtheotherwithasmallamount
f BH
m ≫
/ 0.5 of fluid, m /m 1. We noticed also that the normalized
mf f BH ≪
0.4 efficiencyofaccretion(definedasm˙/mBH)goestozero.
0.3
Results of this paper show the importance of the backre-
0.2 action. We believethatthequalitativefeaturesdemonstrated
0.1 bythe sphericallysymmetricmodel– the dependenceof the
0 mass accretion on mf/mBH and the existence of two weakly
0.0·100 5.0·10−20 1.0·10−19 1.5·10−19 2.0·10−19 2.5·10−19 accretingregimes–willappearinthe descriptionsofaccret-
̺
∞ ingsystemswithangularmomentum.
FIG. 1: The dependence of mf 1 m on the asymptotic mass
density. m ≈ − ∗ Acknowledgements.EMthanksBerndSchmidtfordiscus-
sions. This paper is partially supported by the MNII grant
1PO3B01229.
1.6·10−17
1.4·10−17
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0.0 100
· 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [6] V.Moncrief,Astrophys.J.235,1038(1980).
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FIG.2:Thedependenceofm˙ ontheratio(m m )/m. [8] E.Malec, Phys.Rev.D60,104043(1999).
− ∗
[9] R. Courant andK.O.Friedrichs, Supersonic FlowandShock
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[10] Noticethat(∂ (∂R)∂ ) ∂ , whereT istheMisner-Sharp
with backreaction(when m < m) can be only smaller from t − t R ∝ T
themassaccretionratewith∗outbackreaction(whenm = m), time, see: C. W. Misner and D. H. Sharp, Phys. Rev. 136B,
571(1964).
∗
with the same asymptotic parametersρ , a , m. Numerical
[11] Inthetestfluidapproximationtheparametersofthesonicpoint
resultsconfirmtheprecedingargument∞that∞γdependsessen- do not depend on the central mass. See Theorem 2 and Eqs.
tiallyonlyonthesizeofsupport. (5.10–5.11)in[8].
Inthecaseofpolytropeswith p˜ = KnΓtheBernoulliequa- [12] E.MalecandN.O’Murchadha,Phys.Rev.D50,6033(1994).
