Astron. Astrophys. 363, 425–439 (2000) ASTRONOMYAND

ASTROPHYSICS

Equilibrium configurations of perfect fluid orbitingSchwarzschild–de Sitter black holes

Z. Stuchlık, P. Slany?, and S. Hledık

Silesian University, Department of Physics, Faculty of Philosophy and Science, Bezrucovo nam. 13, Opava, Czech Republic(zdenek.stuchlik@fpf.slu.cz; petr.slany@vsb.cz; stanislav.hledik@fpf.slu.cz)

Received 30 May 2000 / Accepted 15 September 2000

Abstract. The hydrodynamical structure of perfect fluid or-biting Schwarzschild–de Sitter black holes is investigated forconfigurations with uniform distribution of angular momentumdensity. It is shown that in the black-hole backgrounds admittingthe existence of stable circular geodesics, closed equipotentialsurfaces with a cusp, allowing the existence of toroidal accre-tion disks, can exist. Two surfaces with a cusp exist for theangular momentum density smaller than the one correspondingto marginally bound circular geodesics; the equipotential sur-face corresponding to the marginally bound circular orbit hasjust two cusps. The outer cusp is located nearby the static ra-dius where the gravitational attraction is compensated by thecosmological repulsion. Therefore, due to the presence of a re-pulsive cosmological constant, the outflow from thick accretiondisks can be driven by the same mechanism as the accretiononto the black hole. Moreover, properties of open equipotentialsurfaces in vicinity of the axis of rotation suggest a strong colli-mation effects of the repulsive cosmological constant acting onjets produced by the accretion disks.

Key words: accretion, accretion disks – black hole physics –gravitation – relativity – galaxies: quasars: general

1. Introduction

Recent observations give strong evidence that the energysources of quasars and active galactic nuclei are accretiondisks around central massive black holes (Abramowicz & Per-cival 1997; Blandford 1990). Similar, scaled down, accretiondisks appear in some extraordinary galactic binary systems con-taining a black hole (or a neutron star). In the accretion disks,the potential gravitational energy of matter orbiting the centralblack hole is liberated and transferred into heat, due to viscousstresses acting against shearing motion, and radiated (at leastpartly) away. During the process, angular momentum of the ac-creting matter has to be transported outwards.

It is well known that at low accretion rates the pressure isnegligible, and the accretion disk is geometrically thin. Its ba-

Send offprint requests to: Z. Stuchlık? Present address:Technical University at Ostrava, Department of

Physics, Faculty of Mining, Ostrava, Czech Republic

sic properties are determined by the circular geodesic motionin the black-hole background. The radiusrms of the marginallystable circular orbit represents the inner edge of the Kepleriandisks, since matter, following quasikeplerian orbits down torms,falls freely into the black hole from this radius (Novikov &Thorne 1973; Stoeger 1976). At high accretion rates, the pres-sure is relevant, and the accretion disk must be geometricallythick. Its basic properties are determined by equipotential sur-faces of test perfect fluid (i.e., perfect fluid that does not alterthe black-hole geometry) rotating in the black-hole background.The accretion is possible, if a toroidal equilibrium configurationof the test fluid containing a critical, self-crossing equipotentialsurface can exist in the background. The cusp of this equipo-tential surface corresponds to the inner edge of the disk, and theaccretion inflow of matter into the black hole is possible due toa mechanical non-equilibrium process, i.e., because of matterslightly overcoming the critical equipotential surface. The pres-sure gradients push the inner edge of the thick disks under theradiusrms (Kozlowski et al. 1978; Abramowicz et al. 1978).

The simplest, but quite illustrative case of the equipoten-tial surfaces of the test fluid can be constructed for the con-figurations with uniform distribution of the angular momen-tum density. This case is fully governed by the geometry of thespacetime, however, it contains all the characteristic features ofmore complex cases of the rotation of the fluid (Jaroszynski etal. 1980). Moreover, this case is also very important physicallysince it corresponds to marginally stable equilibrium configu-rations (Seguin 1975).

The equipotential surfaces were analyzed for bothSchwarzschild and Kerr black-hole spacetimes. The criticalclosed surfaces with a cusp can exist for angular momentum den-sity higher (lower) than the one corresponding to the marginallystable (bound) circular geodesic, and the location of the cuspshifts fromrms to the radius of the marginally bound geodesicorbit rmb. The cusp close to the horizon enables the inflow ofmatter into the black hole. However, the character of the equi-librium configurations does not allow outflow of matter andtransfer of the angular momentum for disks around isolatedblack holes. In binary systems the outflow is possible throughthe Lagrange point L2 – see, e.g., Novikov & Thorne (1973).

Very recently, a wide variety of cosmological observations(measurements of the present value of the Hubble parameter, de-

426 Z. Stuchlık et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes

tails of the anisotropy of the cosmic relic radiation, statistics ofgravitational lensing of quasars, and high-redshift supernovae)suggest a non-zero, repulsive cosmological constant (Krauss &Turner 1995; Ostriker & Steinhardt 1995; Krauss 1998). There-fore, it is interesting to clarify the influence of the repulsivecosmological constant on astrophysically relevant properties ofblack-hole spacetimes.

Here, we shall show that in the field of black-hole space-times with a repulsive cosmological constant the outflow ofmatter from the accretion disk is possible, because equipoten-tial surfaces with an outer cusp in vicinity of the so calledstatic radius can exist (beside the critical surfaces with the in-ner cusp nearby the horizon), if the mass of the black hole issmall enough to admit existence of the stable circular geodesics(Stuchlık & Hledık 1999). Moreover, if the uniform angular mo-mentum density of the equilibrium configuration correspondsto the marginally bound orbit of the background, the criticalequipotential surface has both the inner and outer cusps. In thissituation, any mechanical non-equilibrium in the thick disk leadsto both inflow into the black hole, and outflow from the disk nearthe static radius.

The plan of this paper is following. In Sect. 2, the ba-sic formulae for the equilibrium configurations of test per-fect fluid in a given stationary and axially symmetric back-ground are summarized, following the papers of Abramowiczand coworkers (Kozlowski et al. 1978; Abramowicz et al. 1978;Jaroszynski et al. 1980). In Sect. 3, the equipotential surfaces ofthe marginally stable configurations (having a uniform distribu-tion of angular momentum density) of the test perfect fluid aredetermined for the Schwarzschild–de Sitter black-hole space-times. For completeness, we include also discussion of the caseof the Schwarzschild–anti-de Sitter spacetimes with an attrac-tive cosmological constant. In Sect. 4, some concluding remarksare presented, and astrophysical consequences of the presentedresults are pointed out. We shall use the geometric system ofunits (c = G = 1), if not stated otherwise.

2. Boyer’s condition for equilibrium configurationsof test perfect fluid

We briefly summarize the well known results of a general the-ory of the equipotential surfaces inside any relativistic, differ-entially rotating, perfect fluid body (Boyer 1965; Abramow-icz 1974), applied to test configurations of perfect fluid rotatingin the stationary and axially symmetric spacetimes (Kozlowskiet al. 1978; Abramowicz et al. 1978; Jaroszynski et al. 1980).In the standard coordinate system the spacetimes are describedby the line element

ds2 = gtt dt2 + 2gtφ dtdφ+ gφφ dφ2 + grr dr2 + gθθ dθ2, (1)

where the metric coefficients depend neither on the time coor-dinate,t, nor the azimuthal coordinate,φ, i.e., the spacetimescontain timelike and azimuthal Killing vector fields∂/∂t and∂/∂φ.

We shall consider test perfect fluid rotating in theφdirection.Its four velocity vector fieldUµ has, therefore, only two non-zero components

Uµ = (U t, Uφ, 0, 0), (2)

which can be functions of the coordinatesr,θ. The stress-energytensor of the perfect fluid is

Tµν = (p + ε)UµUν + p δµ

ν , (3)

whereε andp denote the total energy density and the pressureof the fluid. The rotating fluid can be characterized by the vectorfields of the angular velocityΩ, and the angular momentum perunit mass (angular momentum density)`, defined by

Ω =Uφ

U t, ` = −Uφ

Ut. (4)

These vector fields are related by

Ω = − gtφ + `gtt

gφφ + `gtφ. (5)

In static spacetimes (gtφ = 0), the relation (5) reduces to asimple formula

Ω`

= − gtt

gφφ. (6)

The surfaces of constant` andΩ are called von Zeipel’scylinders. The family of von Zeipel’s cylinders does not de-pend on the assumed rotation law of the fluid,` = `(Ω), in thestatic spacetimes, but it will depend on the rotation law in thestationary spacetimes (withgtφ /= 0) (Kozlowski et al. 1978).

Projecting the energy conservation lawTµν;ν = 0 onto the

hypersurface orthogonal to the four velocityUµ by the projec-tion tensorhµν = gµν + UµUν , we obtain the relativistic Eulerequation in the form

∂µp

p + ε= −∂µ(lnUt) +

Ω ∂µ`

1 − Ω`, (7)

where

(Ut)2 =g2

tφ − gtt gφφ

gφφ + 2`gtφ + `2gtt. (8)

The solution of the relativistic Euler equation can be givenby Boyer’s condition determining the surfaces of constantpressure through the “equipotential surfaces” of the potentialW (r, θ) by the relations (Abramowicz et al. 1978)∫ p

0

dp

p + ε= Win − W, (9)

Win − W = ln(Ut)in − ln(Ut) +∫ `

`in

Ω d`

1 − Ω`; (10)

the subscript “in” refers to the inner edge of the disk. For analternative definition of Boyer’s condition see (Abramowiczet al. 1978; Fishbone & Moncrief 1976; Fishbone 1977). Theequipotential surfaces are determined by the condition

W (r, θ) = const, (11)

Z. Stuchlık et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes 427

and in a given spacetime can be found from Eq. (10), if a rota-tion law Ω = Ω(`) is given. The surfaces of constant pressurep(r, θ) = const are given by Eq. (9). The structure of thickaccretion disks can be obtained also in the framework of a verypractical and accurate Newtonian model for the gravitationalfield of a non-rotating black hole, known as the Paczynski–Wiitapotential (Paczynski & Wiita 1980; Abramowicz et al. 1980).

3. Equipotential surfacesof the marginally stable configurationsorbiting Schwarzschild–de Sitter black holes

Equilibrium configurations of test perfect fluid rotating aroundan axis of rotation in a given spacetime are determined bythe equipotential surfaces, where the gravitational and inertialforces are just compensated by the pressure gradient. (In an ax-ially symmetric spacetime, the axis of rotation coincides withthe axis of symmetry of the spacetime, while in a sphericallysymmetric spacetime the axis of rotation can be any radial line;usually, the coordinate system is chosen so that the rotation axiscorresponds toθ = 0.)

The equipotential surfaces can be closed or open. More-over, there is a special class of critical, self-crossing surfaces(with a cusp), which can be either closed or open. The closedequipotential surfaces determine stationary equilibrium config-urations. The fluid can fill any closed surface—at the surface ofthe equilibrium configuration pressure vanish, but its gradient isnon-zero (Kozlowski et al. 1978). On the other hand, the openequipotential surfaces are important in dynamical situations,e.g., in modeling jets (Lynden-Bell 1969; Blandford 1987). Thecritical, self-crossing closed equipotential surfacesWcusp areimportant in the theory of thick accretion disks, because accre-tion onto the black hole through the cusp of the equipotentialsurface located in the equatorial plane is possible due to thePaczynski mechanism.

According to Paczynski, the accretion into the black hole isdriven through the vicinity of the cusp due to a little overcomingof the critical equipotential surface,W = Wcusp, by the surfaceof the disk. The accretion is thus driven by a violation of thehydrostatic equilibrium, rather than by viscosity of the accretingmatter (Kozlowski et al. 1978).

It is well known that all characteristic properties of theequipotential surfaces for a general rotation law are reflectedby the equipotential surfaces of the simplest configurationswith uniform distribution of the angular momentum density` – see Jaroszynski et al. (1980). Moreover, these configura-tions are very important astrophysically, being marginally stable(Seguin 1975). Under the condition

`(r, θ) = const, (12)

holding in the rotating fluid, a simple relation for the equipo-tential surfaces follows from Eq. (10):

W (r, θ) = lnUt(r, θ), (13)

with Ut(r, θ) being determined by = const, and the metriccoefficients only.

The equipotential surfaces are described by the formulaθ =θ(r), which can be given by the differential equation

dθ

dr= −∂p/∂r

∂p/∂θ, (14)

which for the configurations with = const reduces to

dθ

dr= −∂Ut/∂r

∂Ut/∂θ. (15)

The influence of a non-zero cosmological constant on char-acter of the equipotential surfaces of the marginally stable con-figurations rotating around a black hole will be examined inthe simplest case of Schwarzschild–de Sitter spacetimes cor-responding to a repulsive cosmological constant,Λ > 0. (Forcompleteness, we briefly discuss the case of Schwarzschild––anti-de Sitter spacetimes corresponding to an attractive cos-mological constant,Λ < 0.)

In the standard Schwarzschild coordinates, the non-zerometric coefficients of the Schwarzschild–(anti)-de Sitter space-times are

− gtt = g−1rr = (1 − 2r−1 − yr2), (16)

gθθ = r2, (17)

gφφ = r2 sin2 θ. (18)

Here, the radial coordinater is expressed in units of the massparameterM , and the dimensionless cosmological constant pa-rameter

y = 13ΛM2 (19)

is introduced. It should be stressed that a static region exists inthe Schwarzschild–de Sitter spacetimes with subcritical valuesof

y < yc = 127 ; (20)

of course, the equilibrium configurations are possible only inthese spacetimes. Now, the equipotential surfaces are given bythe formulae

W (r, θ) = ln(1 − 2r−1 − yr2)1/2r sin θ[

r2 sin2 θ − (1 − 2r−1 − yr2)`2]1/2 (21)

and

dθ

dr= tan θ

[r(1−yr3) sin2 θ − (1−2r−1−yr2)2`2

]r

(r−2−yr3)2`2; (22)

for y = 0 these relations reduce to the well knownSchwarzschild formulae (Jaroszynski et al. 1980).

The best insight into the nature of the` = const configura-tions can be obtained by the examination of the behavior of thepotentialW (r, θ) in the equatorial plane (θ = π/2). There aretwo reality conditions ofW (r, θ = π/2):

1 − 2r−1 − yr2 ≥ 0, (23)

r2 − (1 − 2r−1 − yr2)`2 ≥ 0. (24)

428 Z. Stuchlık et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes

1. x 10-25 1. x 10-19 1. x 10-13 1. x 10-7 0.1y

1

1000

1. x 106

1. x 109

1. x 1012

1. x 1015r

Fig. 1. Characteristic radii of theSchwarzschild–de Sitter spacetimes asfunctions of the parametery. The blackhole (rh) and cosmological (rc) horizonsare given by bold solid lines, the staticradius (rs) by bold dotted line, the radii ofmarginally stable orbits (rms(i) andrms(o))by thin dashed lines, and marginally boundorbits (rmb(i) and rmb(o)) by thin solidlines.

The first condition is identical with the condition for the staticregions (located between the black-hole and cosmological hori-zons); the second condition can be expressed in the form

`2 ≤ `2ph(r; y) ≡ r3

r − 2 − yr3 . (25)

The function`2ph(r; y) is the effective potential of the photongeodesic motion; recall that ≡ Uφ/Ut corresponds to thedefinition of the impact parameter for photon’s geodesic motion– see Stuchlık & Hledık (1999). Further, the condition of thelocal extrema of the potentialW (r, θ = π/2) is identical withthe condition of vanishing of the pressure gradient (∂Ut/∂r =0,∂Ut/∂θ = 0). Since at the equatorial plane there is∂Ut/∂θ =0 independently of the = const, and

∂Ut(r, θ = π/2)∂r

=

[r(1−yr3) − (1−2r−1−yr2)2`2

](1−2r−1−yr2)1/2 [r2 − (1−2r−1−yr2)`2]3/2 , (26)

we arrive at the condition

`2 = `2K(r; y) ≡ r3(1 − yr3)(r − 2 − yr3)2

. (27)

The extrema ofW (r, θ = π/2) correspond to the points, wherethe fluid moves along a circular geodesic, since`2K(r; y) corre-sponds to the distribution of the angular momentum density ofthe circular geodesic orbits. Clearly,

Wextr(r, θ = π/2; y) = lnEc(r, y), (28)

where

Ec(r, y) =(

1 − 2r

− yr2) (

1 − 3r

)−1/2

(29)

is the specific energy of the circular geodesics. (Recall that thespecific energy of circular geodesics corresponds to the localextrema of the effective potentialVeff(r; `, y) of the geodesic

motion (Stuchlık & Hledık 1999).) The most important prop-erties of the potentialW (r, θ) are determined by its behaviorat the equatorial plane, and, especially, by the properties of thefunctions`2ph(r; y), and`2K(r, y). Discussion of these proper-ties enables us to give a classification of the Schwarzschild––(anti)-de Sitter spacetimes according to the properties of theequipotential surfaces of test perfect fluid. We shall separate thediscussion to the case of the Schwarzschild–de Sitter (y > 0),and Schwarzschild–anti-de Sitter spacetimes (y < 0). For thepure Schwarzschild spacetime (y = 0) the analysis can be foundin (Kozlowski et al. 1978).

3.1. Schwarzschild–de Sitter black holes

If y > 0, the function 2ph(r, y) diverges at the black-hole hori-

zon,rh, and the cosmological horizon,rc, determined by equal-ity in the condition (23). The horizons are given by the relations

rh =2√3y

cosπ + ξ

3, (30)

rc =2√3y

cosπ − ξ

3, (31)

where

ξ = cos−1(3√

3y)

. (32)

The radii of the horizons are illustrated in Fig. 1. The localminimum of`2ph(r, y) is located atrph = 3, independently ofy,and determines the unstable photon circular geodesic with theimpact parameter

`2ph(c) = `2ph(min)(y) ≡ 271 − 27y

. (33)

The function`2K(r; y), determining the Keplerian (geodesic)circular orbits, has a zero point at the so called static radiusrs(y) given by

rs = y−1/3, (34)

Z. Stuchlık et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes 429

1. x 10-25 1. x 10-19 1. x 10-13 1. x 10-7 0.1y

1

10

100

1000

10000

l

Fig. 2. The angular momentum density ofthe marginally stable (`ms(i) and `ms(o),solid line) and marginally bound (`mb,dashed line) orbits as functions of the pa-rametery of the Schwarzschild–de Sitterspacetimes. Note thatlim

y→0`mb = 4 /=

limy→0

`ms(i) = 3√

3/2, limy→0

`ms(o) = +∞.

1. x 10-25 1. x 10-19 1. x 10-13 1. x 10-7 0.1y

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

E

Fig. 3.The specific energy of the marginallystable (Ems(i) and Ems(o), solid line) andmarginally bound (Emb, bold dotted line)orbits as functions of the parameteryof the Schwarzschild–de Sitter spacetimes.Note that lim

y→0Ems(o) = lim

y→0Emb = 1,

limy→0

Ems(i) = 2√

2/3.

and it is not well defined atr > rs, being negative there. At thestatic radius (unstable) stationary equilibrium of test particles ispossible because the gravitational attraction of the black hole isjust compensated by the cosmological repulsion there.

The function`2K(r; y) diverges at the black-hole horizon:`2K(r → rh, y) → +∞; at the cosmological horizon, there is`2K(r → rc, y) → −∞. Since

∂`2K∂r

=r2[r − 6 + yr3(15 − 4r)]

(r − 2 − yr3)3, (35)

the local extrema of2K(r, y) are given by the condition

y = yms(r) ≡ r − 6r3(4r − 15)

, (36)

determining the marginally stable circular geodesics. The localmaximum ofyms(r) gives the critical value of the parameteryadmitting stable circular orbits

yms = 12/154 ∼ 0.000237. (37)

If y < yms, there exists an inner (outer) marginally stablecircular geodesic atrms(i) (rms(o)), see Fig. 1. The angularmomentum density of the marginally stable orbits`ms(i)(y),and`ms(o)(y), is simultaneously determined by Eqs. (27) and(36)—see Fig. 2. The specific energy of these orbitsEms(i)(y),andEms(o)(y), is simultaneously determined by Eqs. (29) and(36)—see Fig. 3. There is other special value ofy, correspond-ing to the situation, where the value of the minimum of`2ph(r; y)equals to the maximum of2K(r; y). We denote this valueye. Itcan be found that

ye = 1/118125 = 1/(33547) ∼ 0.00000846. (38)

In the Schwarzschild–de Sitter spacetimes, there is anotherimportant class of circular geodesics—namely the marginallybound orbits. These orbits exist in the Schwarzschild–de Sitterspacetimes admitting existence of the stable circular orbits, i.e.,spacetimes withy < yms. In these spacetimes, there exists aninner,rmb(i) (outer,rmb(o)), marginally bound orbit close to the

430 Z. Stuchlık et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes

a0.4 0.6 0.8 1 1.2 1.4 1.6

log r

10

20

30

40

50l2

lph2

lK2

lmb2

y = 5 x 10-6 < 1/118125

0.4 0.6 0.8 1 1.2 1.4 1.6log r

10

20

30

40

50l2

b0.4 0.6 0.8 1 1.2 1.4 1.6

log r

10

20

30

40

50

l2

lph2

lK2

lmb2

y = 1/118125

0.4 0.6 0.8 1 1.2 1.4 1.6log r

10

20

30

40

50

l2

c0.4 0.6 0.8 1 1.2 1.4 1.6

log r

10

20

30

40

50

l2

lph2

lK2

lmb2

y = 2 x 10-5 > 1/118125

0.4 0.6 0.8 1 1.2 1.4 1.6log r

10

20

30

40

50

l2

d0.4 0.6 0.8 1 1.2 1.4

log r

10

20

30

40

50

l2

lph2

lK2

y = 10-3

0.4 0.6 0.8 1 1.2 1.4log r

10

20

30

40

50

l2

Fig. 4a–d. Behavior of the functions 2ph(r; y) and `2K(r; y) in the four qualitatively different cases determining the four classes of the

Schwarzschild–de Sitter spacetimes with different properties of the equipotential surfaces (bothr and`2 are given in units ofM ). Figuresa–d reflect subsequently the cases0 < y < ye, y = ye, ye < y < yms, andyms < y < yc. In the shaded region, the equipotential surfacesare not defined in the equatorial plane of the spacetime, defined by the axis of rotation of the perfect fluid. The descending parts of the func-tion `2K(r; y) determine the cusps, while the growing parts determine central rings of the equilibrium configurations. The dotted line (`2ph(c))determines the impact parameter of the photon circular geodesic atr = 3.

black-hole horizon (static radius). These orbits are defined bythe condition

Emb(rmb(i), `mb) = Emb(rmb(o), `mb), (39)

and are determined by an appropriate numerical procedure(see Figs. 1–3). In the Schwarzschild spacetime (y = 0) themarginally bound orbit is located atrmb = 4, andEmb = 1—it is because the effective potential of the geodesic motionVeff → 1 at r → ∞ independently of the angular momentumdensity in the Schwarzschild spacetime. In the Schwarzschild––de Sitter spacetimes withyms < y < yc the marginally boundcircular orbits are not defined because only unstable circularorbits exist in these spacetimes; particles from them can alwaysescape to infinity.

We can distinguish four qualitatively different cases of thebehavior of the functions2

ph(r; y), `2K(r; y) which give fourclasses of the Schwarzschild–de Sitter black holes with differentcharacter of the equipotential surfaces of the rotating perfectfluid. These four classes are defined according to values of thecosmological parametery in the following way:

(A) 0 < y < ye,(B) y = ye,(C) ye < y < yms,(D) yms ≤ y < yc.

For these classes, the typical behavior of the functions`2ph(r; y),`2K(r; y), with y fixed, is given in Figs. 4(a)–(d). For complete-ness, the corresponding value of`mb(y) is exhibited in these

figures. Note that the descending parts of the curve`2K(r; y)(with y fixed) correspond to the unstable circular geodesics,while its growing part (if it exists) corresponds to the stable cir-cular geodesics. The extrema of`2K(r; y), if they exist, have animportant role: the minimumms(i), at rms(i), determines theinner marginally stable circular geodesic, while the maximum`ms(o), atrms(o), determines the outer marginally stable circulargeodesic.

Properties of the equipotential surfaces can be establishedeasily, using the behavior of the potentialW (r, θ) in the equa-torial plane. The properties of the potentialW (r, θ = π/2; y)are closely related to the properties of the effective poten-tial of the geodesic motion, and at their local extrema, lo-cated at the same radii, the condition (28) is satisfied. Further,W (r; θ = π/2, y) → −∞, if r → rh or r → rc. The topo-logical properties of the equipotential surfaces can be directlyinferred from the properties of the potentialW (r, θ = π/2; y).The local extrema of the potentialW (r, θ = π/2; y) are deter-mined by the condition

`2 = `2K(r; y); (40)

therefore, at the radii determined by the local extrema ofW (r, θ = π/2; y), perfect fluid follows free, geodesic circu-lar orbits. The maxima of the potential are determined by thedescending part of2K(r; y), they correspond to the cusps ofthe equipotential surfaces, and the matter moves along an un-stable geodesic orbit at the corresponding radii. The minimaof the potential are determined by the rising part of`2K(r; y),

Z. Stuchlık et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes 431

they correspond to the central rings of the equilibrium config-urations, and the matter moves along a stable geodesic orbit atthe corresponding radii.

Now, we give a complete survey of the behavior of theequipotential surfaces, and the related potentialW (r, θ =π/2; y). We start with the astrophysically most important case.

(A) 0 < y < ye. From Fig. 4a, we obtain nine qual-itatively different cases of the behavior of the potentialW (r, θ = π/2), and corresponding nine qualitatively dif-ferent families of the equipotential surfaces, according tothe values of = const. (In the following, we consider` > 0 only. This can be done due to the symmetry of thespacetimes under consideration.)(I) ` < `ms(i). Open surfaces only; no disks are possible.

Surface with the outer cusp exists. (Fig. 5a)(II) ` = `ms(i). An infinitesimally thin, unstable ring lo-

cated atrms(i) exists. An open surface with the outercusp exists. (Fig. 5b)

(III) `ms(i) < ` < `mb. Closed surfaces exist. Many equi-librium configurations without cusps are possible, andone with the inner cusp. An open surface with the outercusp exists. (Fig. 5c)

(IV) ` = `mb. Many equilibrium configurations withoutcusps are possible. There is an equipotential surface withboth the inner and outer cusps. Now, the mechanical non-equilibrium causes an inflow into the black hole, and anoutflow from the disk, with the same efficiency; it is themost interesting new feature of the accretion processescaused by the presence of a repulsive cosmological con-stant. (Fig. 5d)

(V) `mb < ` < `ph(c). Equilibrium configurations are pos-sible because closed equipotential surfaces exist. How-ever, accretion into the black hole is impossible becausethe equilibrium configurations (closed surfaces) have noinner cusp; the inner cusp has an open equipotential sur-face. The outer cusp belongs to a closed surface, and theoutflow from the disk is possible. (Fig. 5e)

(VI) ` = `(ph(c)). The potentialW (r, θ = π/2; y) divergesat the photon circular orbit located atr = 3, and the innercusp disappears. The closed equipotential surfaces stillexist, with the most extended one containing the outercusp that enables outflow from the disk. (Fig. 5f)

(VII) `ph(c) < ` < `ms(o). In the region defined by`2ph(r; y), the equipotential surfaces cannot reach theequatorial plane. The closed equipotential surfaces ex-ist, one with the outer cusp. (Fig. 5g)

(VIII) ` = `ms(o). An infinitesimally thin, unstable ringlocated atrms(o) exists (the center, and the outer cuspcoalesce). (Fig. 5h)

(IX) ` > `ms(o). Open equipotential surfaces exist only.There is no cusp in this case. (Fig. 5i).

(B) y = ye. For this special value ofy (Fig. 4b), we still obtainthe families of equipotential surfaces given by (A-I)–(A-V)and (A-IX). However, the case (A-VII) disappears, and thecases (A-VI) and (A-VIII) coalesce, giving the case

(X) ` = `ph(c) = `ms(o). The inner cusp just disappears,while the outer cusp coalesce with the center. (Fig. 5j)

(C) ye < y < yms. From Fig. 4c it follows that the intervalsof `, and the families of equipotential surfaces (A-I)–(A-IV)remain. The following new intervals of the angular momen-tum density must be introduced.(XI) `mb < ` < `ms(o). This case is equivalent to the case

(A-V).(XII) ` = `ms(o). There is the inner cusp of an open

equipotential surface, but the center and the outer cuspcoalesce—this corresponds to an infinitesimally thin un-stable ring, located atrms(o). (Fig. 5k)

(XIII) `ms(o) < ` < `ph(c). There are open surfaces only,one being with the inner cusp. (Fig. 5l)

(XIV) ` ≥ `ph(c). This case corresponds to the case (A-IX).(D) yms ≤ y < yc. For this interval ofy, the function

`2K(r; y) is descending everywhere (see Fig. 4d). Only max-ima of the potentialW (r, θ = π/2; y) are possible (if`2 < `2ph(c)), and open equipotential surfaces can existonly. Equilibrium configurations corresponding to toroidaldisks are not possible. This is quite natural result, since inthe spacetimes under consideration stable circular geodesicscannot exist. Now, there are only two different intervals ofthe parameter.(XV) ` < `ph(c). This family of equipotential surfaces cor-

responds to the case (A-I).(XVI) ` ≥ `ph(c). This family of equipotential surfaces

corresponds to the case (A-IX).

Values of the potential at the central ring and the cusps (pro-vided they exist) are given in Table 1. Note that the maximumdifference between the values of the potentialW on the bound-ary and at the center of the toroidal disk in the Schwarzschildspacetime is∆W = 0.0431 (Abramowicz et al. 1978). Com-paring this with the value∆Wi = ∆Wo = 0.0309 from Table 1characterizing the limiting accretion disk with` = `mb, we canconclude that the presence of a repulsive cosmological constantmakes the structure of the disk ‘smoother’.

The Schwarzschild casey = 0 was discussed in (Kozlowskiet al.1978) and will not be repeated here. We only mentionthat the critical self-crossing surface for the marginally boundconfigurationsWcusp(` = `mb; y = 0) = 0, while Wcusp(` =`mb, y > 0) < 0.

3.2. Schwarzschild–anti-de Sitter black holes

If y < 0, the function`2ph(r, y) diverges at infinity, and at theblack-hole horizon given by the relation

rh =

[−1

y+

(1y2 − 1

27y3

)1/2]1/3

+

[−1

y−

(1y2 − 1

27y3

)1/2]1/3

. (41)

The local minimum of 2ph(r; y) is again located atrph = 3,

and the impact parameter of the corresponding photon cir-

432 Z. Stuchlık et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes

(a) y = 10−6 ` = 2.34

0 0.5 1 1.5 2 2.5

log r

-0.25

-0.2

-0.15

-0.1

-0.05

0

W(r,

π/2)

0 0.5 1 1.5 2 2.5 3

( log r ) sin θ

- 3

- 2

- 1

0

1

2

3

(log

r)cos

θ

- 0.3

- 0.09

- 0.03

cusp

- 0.06

- 10.0

- 0.01

0.01

(b) y = 10−6 ` = 3.67403603

0 0.5 1 1.5 2 2.5

log r

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

W(r,

π/2)

0 0.5 1 1.5 2 2.5 3

( log r ) sin θ

- 3

- 2

- 1

0

1

2

3

(log

r)cos

θ

cent

- 0.046

- 0.025

cusp2

cusp1

- 0.06

- 10.0

- 0.010.01

(c) y = 10−6 ` = 3.84

0 0.5 1 1.5 2 2.5

log r

-0.08

-0.06

-0.04

-0.02

0

W(r,

π/2)

0 0.5 1 1.5 2 2.5 3

( log r ) sin θ

- 3

- 2

- 1

0

1

2

3

(log

r)cos

θ

cent

- 0.022

- 0.04

cusp2

cusp1

- 0.06

- 10.0

0.1

- 0.01

(d) y = 10−6 ` = 3.93920702

0 0.5 1 1.5 2 2.5

log r

-0.08

-0.06

-0.04

-0.02

0

W(r,

π/2)

0 0.5 1 1.5 2 2.5 3

( log r ) sin θ

- 3

- 2

- 1

0

1

2

3

(log

r)cos

θ

cent

- 0.035

- 0.025

cusps

- 0.06

- 10.0

- 0.010.01

(e) y = 10−6 ` = 4.18

0 0.5 1 1.5 2 2.5

log r

-0.075

-0.05

-0.025

0

0.025

0.05

W(r,

π/2)

0 0.5 1 1.5 2 2.5 3

( log r ) sin θ

- 3

- 2

- 1

0

1

2

3

(log

r)cos

θ

cent

0.0

- 0.025

cusp2

cusp1- 0.06

- 10.0

0.5

(f) y = 10−6 ` = 5.19622257

0 0.5 1 1.5 2 2.5

log r

-0.06

-0.04

-0.02

0

W(r,

π/2)

0 0.5 1 1.5 2 2.5 3

( log r ) sin θ

- 3

- 2

- 1

0

1

2

3

(log

r)cos

θ

cent

- 0.018

cusp

- 0.06

- 10.0

- 0.01

0.01

0.9

(g) y = 10−6 ` = 5.9

0 0.5 1 1.5 2 2.5

log r

- 0.04

- 0.03

- 0.02

- 0.01

0

0.01

0.02

W(r,

π/2)

0 0.5 1 1.5 2 2.5 3

( log r ) sin θ

- 3

- 2

- 1

0

1

2

3

(log

r)cos

θ

cent

- 0.015

cusp

- 0.06

- 10.0

- 0.01

0.01

0.8

(h) y = 10−6 ` = 7.12938367

1 1.5 2 2.5 3

log r

-0.15

-0.1

-0.05

0

0.05

0.1

W(r,

π/2)

0 0.5 1 1.5 2 2.5 3

( log r ) sin θ

- 3

- 2

- 1

0

1

2

3

(log

r)cos

θ

cent

cusp

- 0.06

- 10.0

0.0

0.4

Fig. 5a–h.Equipotential surfaces (meridional sections) for the marginally stable (` = const) configurations of test perfect fluid orbiting theSchwarzschild–de Sitter black-holes, and the related potentialW (r, θ = π/2; y). The radial coordinate is expressed in units ofM ; the logarithmicscale is used, in order to cover whole the range between the inner and outer cusps. The central black hole is shaded. The sequence of figuresa–lcovers all the possibilities of the behavior of the equipotential surfaces for black holes in spacetimes with a repulsive cosmological constant. Thesequencea–i gives successively all the possibilities for the behavior of the equipotential surfaces in the spacetimes of class A, with0 < y < ye,which is the astrophysically most plausible class. For the spacetimes of the classes B–D, the relevant sequences of the equipotential surfaces aredetermined in the text. The cusps of the toroidal disks correspond to the local maxima ofW (r, θ = π/2), the central rings correspond to theirlocal minima. The dashed lines give asymptotics ofW (r, θ = π/2), and determine the interval of radii where the equipotential surfaces cannotreach the equatorial plane.

Z. Stuchlık et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes 433

(i) y = 10−6 ` = 8.11

1 1.25 1.5 1.75 2 2.25 2.5

log r

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

W(r,

π/2)

0 0.5 1 1.5 2 2.5 3

( log r ) sin θ

- 3

- 2

- 1

0

1

2

3

(log

r)cos

θ

- 0.06

- 10.0

- 0.012

0.0

0.5

(j) y = 1/118125 ` = 5.19674637

0 0.5 1 1.5 2 2.5

log r

- 0.4

- 0.2

0

0.2

0.4

W(r,

π/2)

0 0.5 1 1.5 2 2.5

( log r ) sin θ

- 2

- 1

0

1

2

(log

r)cos

θ

cent

cusp

- 0.07

- 10.0

- 0.02

0.01

1.5

(k) y = 2× 10−5 ` = 4.62683526

0 0.5 1 1.5 2log r

-0.2

-0.1

0

0.1

0.2

W(r,

π/2)

0 0.5 1 1.5 2

( log r ) sin θ

- 2

- 1

0

1

2(log

r)cos

θ

cusp1

0.0

cent

cusp2

- 0.09

- 10.0

5.0

(l) y = 2× 10−5 ` = 4.98

0 0.5 1 1.5 2log r

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

W(r,

π/2)

0 0.5 1 1.5 2

( log r ) sin θ

- 2

- 1

0

1

2

(log

r)cos

θ

cusp

0.0

- 0.03

- 0.09

- 10.0

10.0

Fig. 5i–l.

0.4 0.6 0.8 1 1.2 1.4 1.6log r

10

20

30

40

50

l2

lph2

lK2

y = -10-6

0.4 0.6 0.8 1 1.2 1.4 1.6log r

10

20

30

40

50

l2

Fig. 6. Behavior of the functions 2ph(r; y) and `2K(r; y) for the

Schwarzschild–anti-de Sitter spacetimes, given fory = −10−6 (bothrand`2 are given in units ofM ). The dotted line determines`2ph(c), as inFig. 4. It is qualitatively similar to the pure Schwarzschild case (y = 0),and it has the same character for ally < 0. In the shaded region, theequipotential surfaces are not defined in the equatorial plane.

cular geodesic is given by Eq. (33). Ify < 0, there is nozero point of`2K(r, y) and `2K(r → rh, y) → +∞, `2K(r →∞, y) → +∞. Now, Eq. (36) determines only one marginallystable circular geodesic, close to the horizon. On the other hand,in the Schwarzschild–anti-de Sitter spacetimes the notion ofmarginally bound circular geodesic ceases any meaning becauseparticles from the unstable circular orbits never escape to infin-ity, since the effective potential diverges at infinity for each valueof the angular momentum density (Stuchlık & Hledık 1999).

If y < 0, the behavior of the functions2ph(r; y) and`2K(r; y)is qualitatively the same as in the Schwarzschild case. It is illus-trated in Fig. 6. The function2K(r; y) has a minimumms atrmscorresponding to the marginally stable circular geodesic. Theunstable geodesics are given by the descending part of`2K(r; y),while the stable are given by the rising part.

Now, it is immediately clear that for all of theSchwarzschild–anti-de Sitter spacetimes we always obtain fourpossible cases of the behavior of the potentialW (r, θ = π/2; y)and four corresponding families of the equipotential surfaces;notice thatW (r, θ = π/2, y) → ∞ asr → ∞. These cases aregiven by the following intervals of:

(I) ` < `ms. There are open equipotential surfaces only.(Fig. 7a)

(II) ` = `ms. An infinitesimally thin unstable ring is located atrms. (Fig. 7b)

(III) `ms < ` ≤ `ph(c). Closed equipotential surfaces exist, onewith the cusp that enables accretion from the toroidal diskinto the black hole. (Fig. 7c)

(IV) ` > `ph(c). Closed equipotential surfaces exist, but nowith a cusp at the equatorial plane. In vicinity of the hori-zon (in region limited by radii determined by the equation`2ph(r; y) = `2) the equipotential surfaces cannot cross theequatorial plane. (Fig. 7d)

Values of the potential at the cusp and the central ring (pro-vided they exist) are given in Table 2.

434 Z. Stuchlık et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes

(a) y = −10−6 ` = 2.18

0 0.5 1 1.5 2 2.5

log r

-0.1

-0.05

0

0.05

0.1

W(r,

π/2)

0 0.5 1 1.5 2 2.5 3

( log r ) sin θ

- 3

- 2

- 1

0

1

2

3

(log

r)cos

θ

0.4

0.05

0.0

- 0.025

- 0.08

- 0.3

(b) y = −10−6 ` = 3.67443285

0 0.5 1 1.5 2 2.5

log r

-0.15

-0.1

-0.05

0

0.05

0.1

W(r,

π/2)

0 0.5 1 1.5 2 2.5 3

( log r ) sin θ

- 3

- 2

- 1

0

1

2

3

(log

r)cos

θ

cent

cusp

0.4

0.07

0.0

- 0.022

- 0.045- 0.4

(c) y = −10−6 ` = 4.12

0 0.5 1 1.5 2 2.5

log r

-0.1

-0.05

0

0.05

0.1

W(r,

π/2)

0 0.5 1 1.5 2 2.5 3

( log r ) sin θ

- 3

- 2

- 1

0

1

2

3

(log

r)cos

θ

cent

cusp

0.4

- 0.01

- 0.03

(d) y = −10−6 ` = 5.8

0 0.5 1 1.5 2 2.5 3

log r

- 0.05

0

0.05

0.1

0.15

0.2

W(r,

π/2)

0 0.5 1 1.5 2 2.5 3

( log r ) sin θ

- 3

- 2

- 1

0

1

2

3

(log

r)cos

θ

cent

- 0.013

0.0

0.07

0.4

Fig. 7a–d.Equipotential surfaces (meridional sections) for the marginally stable (` = const) configurations of test perfect fluid orbiting theSchwarzschild–anti-de Sitter black holes, and the related potentialW (r, θ = π/2, y), given fory = −10−6. The behavior of the equipotentialsurfaces has the same character for ally < 0. There are four possibilities described in the text. We express the radial coordinate in units ofM ,and use the logarithmic scale. The central black hole is shaded. Notice the special shape of the equipotential surfaces with a cusp, resembling afalling wave. The dashed lines give asymptotics ofW (r, θ = π/2), and determine the interval of radii where the equipotential surfaces cannotreach the equatorial plane.

Table 1. Radii of the inner cusp (rcu(i)), outer cusp (rcu(o)), and the central ring (rcent), the corresponding values of the potential (Wcu(i),Wcu(o), Wcent), and the differences (∆Wi = Wcu(i) − Wcent, ∆Wo = Wcu(o) − Wcent) for the equilibrium configurations with= const inthe Schwarzschild–de Sitter spacetimes. (Radii and` are in units of mass parameterM , while W and∆W are in units ofc2.)

y ` rcu(i) rcent rcu(o) Wcu(i) Wcent Wcu(o) ∆Wi ∆Wo

10−6 2.3400 none none 98.2201 none none −0.01496 none none10−6 3.6740 6.00195 6.00195 95.3473−0.05895 −0.05895 −0.01454 0 0.0441010−6 3.8400 4.41949 8.82178 94.8669−0.03464 −0.04940 −0.01448 0.01476 0.0349210−6 3.9392 4.13295 9.87591 94.5644−0.01443 −0.04537 −0.01443 0.03094 0.0309410−6 3.9900 4.02028 10.3897 94.4048−0.00251 −0.04358 −0.01441 0.04107 0.0291710−6 4.1800 3.70860 12.2516 93.7790 0.05255−0.03797 −0.01433 0.09052 0.0236410−6 5.1962 3 22.6904 89.4730 ∞ −0.02193 −0.01378 ∞ 0.0081510−6 5.9000 none 31.4330 85.0665 none −0.01664 −0.01328 none 0.0033710−6 7.1294 none 62.1768 62.1768 none −0.01197 −0.01197 none 010−6 8.1100 none none none none none none none none2 × 10−5 4.6268 3.29004 22.2247 22.2247 0.27754−0.03271 −0.03271 0.31025 02 × 10−5 4.9800 3.09207 none none 0.71569 none none none none10−3 2.1800 none none 9.06615 none none −0.15977 none none1/118125 5.1967 3 30 30 ∞ −0.02451 −0.02451 ∞ 0

4. Conclusions

The new phenomena in the structure of equilibrium configura-tions of test perfect fluid, caused by the presence of a repulsivecosmological constant, can be summarized in the following way.

1. There is always an equipotential surface with a cusp for` = 0. It is always an open surface.

2. The position of the outer cusp of the equipotential surfacewith ` = 0 is just atrcusp(` = 0) = rs = 3

√1/y. The value

of the potential at the cusp is given by

Wmax(rs, ` = 0, y) = lnEc(rs, ` = 0, y). (42)

Because

Ec(r = rs, ` = 0, y) =(1 − 3y1/3

)1/2, (43)

Z. Stuchlık et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes 435

Table 2. Radii of the cusp and central ring, the corresponding values of the potential and their difference for equilibrium configurations with` = const in the Schwarzschild–anti-de Sitter spacetimes.

y ` rcu(i) rcent rcu(o) Wcu(i) Wcent Wcu(o) ∆Wi ∆Wo

−10−6 2.1800 none none none none none none none none−10−6 3.6744 5.99806 5.99806 none−0.05884 −0.05884 none 0 none−10−6 4.1200 3.79293 11.6194 none 0.03332−0.03940 none 0.07272 none−10−6 5.8000 none 28.4736 none none −0.01629 none none none

we find

Wcusp(` = 0) =12

ln(1 − 3y1/3

). (44)

3. The accretion disks around black holes can exist when aninner cusp will appear near the black-hole horizon, in ad-dition to the outer cusp, located nearby the static radiusrs = 3

√1/y.

4. Closed equipotential surfaces, necessary for the existence oftoroidal accretion disks, can exist for` ∈ (`ms(i), `ms(o)).Here ms(i) (`ms(o)) corresponds to the local minimum (max-imum) of the function 2

K(r; y), giving the minimum (max-imum) value` of stable circular geodesic (Keplerian) or-bits. The closed surfaces can exist in the spacetimes withy < yms ∼ 0.000237.

5. Accretion onto the central black hole by the Paczynskimechanism is possible, if∈ (`ms(i), `mb); The value mbcorresponds to the marginally bound circular geodesics.Now, they are determined nontrivially: by the conditionthat for ` = `mb both W (r, θ = π/2) and the effectivepotential of geodesic motionVeff(r; `mb, y) have two lo-cal maxima with the same value (recall thatWmax(r, θ =π/2) = lnVeff(max) there). In this case, outflow fromthe accretion disk is possible through both cusps, if themechanical equilibrium is destroyed for both the cusps,i.e., if both equipotential surfaces with cusp are filled up:W > Wcusp(o) > Wcusp(i). If Wcusp(i) < W < Wcusp(o),the accretion flow is directed down the black hole only.

6. We stress that for = `mb, the equipotential surface withW = Wmax(rmb(i), y) = Wmax(rmb(o), y) has two cusps.The mass outflow due to mechanical non-equilibrium, i.e.,overfilling of the (both-sided) marginally closed equipoten-tial surface, is equally efficient for the inflow down the blackhole and the outflow near the static radius. Of course, wecould expect significant differences in details of the accre-tion inflow near the black-hole horizon, and the outflow nearthe static radius.

7. The outer cusp of the configuration with` = `mb, andWcusp = lnEmb, i.e., the limiting equilibrium configura-tion which enables accretion into the Schwarzschild–de Sit-ter black holes, is located atr = rmb(o). It is quite interest-ing that such configurations will approach the static radius,however, they cannot exceed the static radius (rmb(o) → rsif y → 0). Notice that `mb ∼ `ms(i) (Fig. 2), whileEmb ∼ Ems(o) (Fig. 3).

8. For` ∈ (`mb, `ms(o)), the accretion flow down the hole is“switched-off”, because an open self–crossing equipotential

surface withW = Wcusp(i) appears under the inner edge ofthe toroidal configuration in the equatorial plane. However,the outflow through the cusp near the static radius can stilloccur due to a possible mechanical non-equilibrium.

9. Toroidal structures of equipotential surfaces, leading toequilibrium configurations of perfect fluid, cannot exist justif ` > `ms(o). Then, an inner cusp, nearby the black-hole horizon, still exists for equipotential surfaces withW = Wcusp(i) > 0. However, these equipotential sur-faces are always open, and can exist in spacetimes withye < y < yc.

10. The behavior of the open equipotential surfaces along theaxis of rotation gives an important effect—the surfaces be-come significantly narrower while approaching the static ra-dius and the cosmological horizon. This behavior suggestsa strong collimation effect on jets, caused by the influenceof a repulsive cosmological constant.

In the case of Schwarzschild–anti-de Sitter spacetimes thesituation is different. The presence of an attractive cosmologi-cal constant brings no qualitatively new phenomena in compar-ison with the Schwarzschild case, concerning the character ofthe equilibrium configurations related to accretion disks. No-tice, however, the special shape (resembling a falling wave) ofthe closed equipotential surfaces which manifests in an illus-trative way the interplay of the gravitational, cosmological, andcentrifugal forces. Moreover, there exist no open equipotentialsurfaces around the rotation axis in these spacetimes.

From the astrophysical point of view, the most importantphenomena were discovered in spacetimes with a repulsive cos-mological constant, if they admit stable circular geodesic orbits.The first is the presence of an outer cusp of toroidal disks nearbythe static radius which enables outflow of mass and angular mo-mentum from the accretion disks by the Paczynski mechanism,i.e., due to a violation of the hydrostatic equilibrium. This is thesame mechanism that drives the accretion into the black holethrough the inner cusp. (Recall that outflow from toroidal disksaround a Schwarzschild or Kerr black hole by the Paczynskimechanism is impossible because no outer cusp of toroidaldisks exists in the asymptotically flat black-hole spacetimes(Kozlowski et al. 1978; Abramowicz et al. 1978; Jaroszynskiet al. 1980).) The second is the possibility of strong collimationeffects on jets escaping along the rotation axis of toroidal disksfollowing the open equipotential surfaces that are narrowingstrongly when approaching the static radius (and the cosmo-logical horizon). We give an explicit illustration of these twoprincipally new phenomena caused by the repulsive cosmolog-

436 Z. Stuchlık et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes

Thick accretion disk around

a Schwarzschild black hole,

y = 0,

`(r, ϑ) = 3.96 < `mb

Thick accretion disk around

a Schwarzschild–de Sitter

black hole, y = 10−6,`(r, ϑ) = const = `mb

jet

collimated jet

@@@@

accretion disk

-outer cusp

*

static radius

Fig. 8. The structure of an accretion diskwith a jet is compared in meridian sections.The radial coordinate is expressed in unitsof M , but the logarithmic scale is not usedhere, since we are interested in the regionsnear the static radius where both the outercusp and the collimation effect are evident.

ical constant in Fig. 8. Of course, both of those very interestingphenomena deserve further, more detailed studies. Further, therunaway instability of the toroidal disks with respect to the out-flow through the outer cusp, and the influence of self–gravitationon their structure, have to be examined. We plan these studiesin near future.

It is interesting to find astrophysically plausible situationsin which these two phenomena could be relevant. We shouldconsider their role in

(a) quasars and active galactic nuclei during the present periodof expansion of the Universe,

(b) accretion processes onto primordial black holes during thevery early stages of expansion of the Universe, when phasetransitions connected to symmetry breaking of physical in-teractions due to Higgs mechanism (e.g., the breaking ofelectroweak interactions) could take place, and the effec-tive cosmological constant can have values in many ordersexceeding its present value (Kolb & Turner 1990).

Recent cosmological observations give strong indicationsthat the present value of the vacuum energy density is(Krauss 1998)

ρvac(0) ≈ 0.65ρcrit(0) (45)

Z. Stuchlık et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes 437

Table 3. Characteristic radii of the Schwarzschild–de Sitter black-hole spacetimes (in units of mass parameterM ). The parameterS =rmb(o)/rmb(i) determines the relative extension of the toroidal accretion disks with` = const = `mb. The table ends aty = yms ∼ 0.000237,corresponding to the marginal spacetime allowing stable circular geodesics. In spacetimes withy > yms, stable circular geodesics are notallowed, and both thick and thin accretion disks cannot exist.

y rh rc rs rms(i) rms(o) rmb(i) rmb(o) S

1 × 10−30 2 1 × 1015 1 × 1010 6 6.30 × 109 4 1 × 1010 1.05 × 109

1 × 10−29 2 3.16 × 1014 4.64 × 109 6 2.92 × 109 4 4.64 × 109 4.87 × 108

1 × 10−28 2 1 × 1014 2.15 × 109 6 1.36 × 109 4 2.15 × 109 2.26 × 108

1 × 10−27 2 3.16 × 1013 1 × 109 6 6.30 × 108 4 1 × 109 1.05 × 108

1 × 10−26 2 1 × 1013 4.64 × 108 6 2.92 × 108 4 4.64 × 108 4.87 × 107

1 × 10−25 2 3.16 × 1012 2.15 × 108 6 1.36 × 108 4 2.15 × 108 2.26 × 107

1 × 10−24 2 1 × 1012 1 × 108 6 6.30 × 107 4 1 × 108 1.05 × 107

1 × 10−23 2 3.16 × 1011 4.64 × 107 6 2.92 × 107 4 4.64 × 107 4.87 × 106

1 × 10−22 2 1 × 1011 2.15 × 107 6 1.36 × 107 4 2.15 × 107 2.26 × 106

1 × 10−21 2 3.16 × 1010 1 × 107 6 6.30 × 106 4 1 × 107 1.05 × 106

1 × 10−20 2 1 × 1010 4.64 × 106 6 2.92 × 106 4 4.64 × 106 4.87 × 105

1 × 10−19 2 3.16 × 109 2.15 × 106 6 1.36 × 106 4 2.15 × 106 2.26 × 105

1 × 10−18 2 1 × 109 1 × 106 6 6.30 × 105 4 1 × 106 1.05 × 105

1 × 10−17 2 3.16 × 108 4.64 × 105 6 2.92 × 105 4 4.64 × 105 4.87 × 104

1 × 10−16 2 1 × 108 2.15 × 105 6 1.36 × 105 4 2.15 × 105 2.26 × 104

1 × 10−15 2 3.16 × 107 1 × 105 6 6.30 × 104 4 1 × 105 1.05 × 104

1 × 10−14 2 1 × 107 4.64 × 104 6 2.92 × 104 4 4.64 × 104 4.87 × 103

1 × 10−13 2 3.16 × 106 2.15 × 104 6 1.36 × 104 4 2.15 × 104 2.26 × 103

1 × 10−12 2 1 × 106 1 × 104 6 6.30 × 103 4.001 1 × 104 1.05 × 103

1 × 10−11 2 3.16 × 105 4.64 × 103 6 2.92 × 103 4.003 4.64 × 103 4.87 × 102

1 × 10−10 2 1 × 105 2.15 × 103 6 1.36 × 103 4.006 2.15 × 103 2.26 × 102

1 × 10−9 2 3.16 × 104 1 × 103 6 6.30 × 102 4.012 9.95 × 102 1.05 × 102

1 × 10−8 2 1 × 104 4.64 × 102 6 2.92 × 102 4.026 4.59 × 102 48.61 × 10−7 2 3.16 × 103 2.15 × 102 6 1.36 × 102 4.058 2.10 × 102 22.51 × 10−6 2 1 × 103 1 × 102 6.002 62.2 4.132 9.46 10.45 × 10−6 2 4.46 × 102 58.5 6.01 36.0 4.247 53.0 5.988.466 × 10−6 2 3.43 × 102 49.1 6.02 30.0 4.306 43.5 4.991 × 10−5 2.0001 3.16 × 102 46.4 6.02 28.3 4.328 40.8 4.702 × 10−5 2.0002 2.23 × 102 36.8 6.04 22.2 4.446 31.1 3.681 × 10−4 2.0008 1 × 102 21.5 6.24 12.2 5.082 15.3 1.962 × 10−4 2.0016 70 17.1 6.72 8.89 6.097 9.84 1.322.370 × 10−4 2.0019 64 16.2 7.50 7.50 7.568 7.568 1

with present values of the critical energy densityρcrit(0), andHubble parameterH0 given by

ρcrit(0) =3H2

0

8π, H0 = 100h km s−1 Mpc−1. (46)

Taking value of the dimensionless parameterh ∼ 0.7, we ar-rive at the present value of the “relict” repulsive cosmologicalconstant

Λ0 = 8πρvac(0) ≈ 1.1 × 10−56 cm−2. (47)

Having this value ofΛ0, we can determine the mass parameterof the spacetime corresponding to any given value ofy, andall the relevant parameters of the equilibrium configurations.The results concerning the important radii characterizing theSchwarzschild–de Sitter spacetimes withΛ = Λ0 are summa-rized in Table 3 and Table 4.

We can clearly see that the relict cosmological constantΛ0 ∼ 1.1×10−56 cm−2 puts a natural limit on the size of equi-librium configurations rotating around black holes. In fact, the

outer edge of the accretion disks, where the outflow goes throughthe outer cusp of the toroidal structure, is located nearby thestatic radius. It is quite interesting that for black holes of masses∼108M–109M, corresponding to black holes located in thecentral parts of quasars and active galactic nuclei, the outer edgeof the largest accretion disks is located atrmb(o) ∼ 50–100 kpc,and is comparable with maximum extension of large galaxies.Note that extension of quasikeplerian, thin accretion disks islimited by the outer marginally stable circular orbit; ify is smallenough (y ≤ 10−8), it can be shown that

rms(o) ∼ 0.63rs, (48)

and dimensions of these disks are comparable to the static ra-dius, too. Therefore, the relict repulsive cosmological constantcan radically influence the behavior of accretion disks in largegalaxies with active nuclei, and can even be connected to thelimit of extension of these large galaxies.

Moreover, it is clear that the collimation effect of the repul-sive cosmological constant could be relevant in these situations,

438 Z. Stuchlık et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes

Table 4. Mass parameter and the radiusrmb(o) determining the outer edge of toroidal disks with` = const = `mb in the Schwarzschild––de Sitter black-hole spacetimes, given for (a) the relict repulsive cosmological constant indicated by recent cosmological observationsΛ0 ∼0.65Λcrit(0) ∼ 1.1×10−56cm−2, (b) the primordial effective cosmological constantΛew ∼ 0.028 cm−2, and (c) the other possible primordialeffective cosmological constantΛqc ∼ 2.8 × 10−10 cm−2.

y Λ0 Λew Λqc

M rmb(o) M rmb(o) M rmb(o)

[M] [kpc] [g] [cm] [g] [cm]

1 × 10−30 1.1 × 108 56 1.4 × 1014 0.00011 1.4 × 1018 1.11 × 10−29 3.5 × 108 82 4.4 × 1014 0.00016 4.4 × 1018 1.61 × 10−28 1.1 × 109 130 1.4 × 1015 0.00024 1.4 × 1019 2.41 × 10−27 3.5 × 109 170 4.4 × 1015 0.00034 4.4 × 1019 3.41 × 10−26 1.1 × 1010 250 1.4 × 1016 0.00048 1.4 × 1020 4.81 × 10−25 3.5 × 1010 360 4.4 × 1016 0.0007 4.4 × 1020 7.01 × 10−24 1.1 × 1011 530 1.4 × 1017 0.001 1.4 × 1021 101 × 10−23 3.5 × 1011 780 4.4 × 1017 0.0015 4.4 × 1021 151 × 10−22 1.1 × 1012 1100 1.4 × 1018 0.0022 1.4 × 1022 221 × 10−21 3.5 × 1012 1700 4.4 × 1018 0.0032 4.4 × 1022 321 × 10−20 1.1 × 1013 2500 1.4 × 1019 0.0048 1.4 × 1023 481 × 10−19 3.5 × 1013 3600 4.4 × 1019 0.007 4.4 × 1023 701 × 10−18 1.1 × 1014 5300 1.4 × 1020 0.01 1.4 × 1024 1001 × 10−17 3.5 × 1014 7800 4.4 × 1020 0.015 4.4 × 1024 1501 × 10−16 1.1 × 1015 11000 1.4 × 1021 0.022 1.4 × 1025 2201 × 10−15 3.5 × 1015 17000 4.4 × 1021 0.032 4.4 × 1025 3201 × 10−14 1.1 × 1016 25000 1.4 × 1022 0.048 1.4 × 1026 4801 × 10−13 3.5 × 1016 36000 4.4 × 1022 0.07 4.4 × 1026 7001 × 10−12 1.1 × 1017 53000 1.4 × 1023 0.1 1.4 × 1027 10001 × 10−11 3.5 × 1017 78000 4.4 × 1023 0.15 4.4 × 1027 15001 × 10−10 1.1 × 1018 110000 1.4 × 1024 0.22 1.4 × 1028 22001 × 10−9 3.5 × 1018 170000 4.4 × 1024 0.32 4.4 × 1028 32001 × 10−8 1.1 × 1019 240000 1.4 × 1025 0.47 1.4 × 1029 47001 × 10−7 3.5 × 1019 350000 4.4 × 1025 0.68 4.4 × 1029 68001 × 10−6 1.1 × 1020 500000 1.4 × 1026 0.97 1.4 × 1030 97005 × 10−6 2.5 × 1020 630000 3.1 × 1026 1.2 3.1 × 1030 120008.5 × 10−6 3.2 × 1020 670000 4 × 1026 1.3 4 × 1030 130000.00001 3.5 × 1020 690000 4.4 × 1026 1.3 4.4 × 1030 130000.00002 5 × 1020 740000 6.2 × 1026 1.4 6.2 × 1030 140000.0001 1.1 × 1021 810000 1.4 × 1027 1.6 1.4 × 1031 160000.0002 1.6 × 1021 740000 2 × 1027 1.4 2 × 1031 140000.00024 1.7 × 1021 620000 2.1 × 1027 1.2 2.1 × 1031 12000

because the largest observed jets extend to distances∼200 kpc(Blandford 1990), exceeding dimensions of the “seed” galaxy(comparable to the static radius).

It is well known (Carroll & Ostlie 1996) that dimensions oflarge galaxies, of both spiral and elliptical type, are in the interval50–100 kpc, while the extremely large elliptical galaxies of cDtype extend up to 1000 kpc. Thus, we can conclude that toroidaldisks around a central hole of massM ∼ 109 M have sizescomparable with the large galaxies and can be related to size-limits on these galaxies. On the other hand, such disks are wellinside the cD elliptical galaxies; in order to obtain an accretiondisk of dimension∼ 1000 kpc, mass parameter of the centralblack hole have to be∼1012 M.

Of course, if the mass of a protogalactic disk related to aquasar is higher than the mass of the central black hole, theself-gravitational effects of the disk itself have to be taken intoconsideration. Nevertheless, we can expect that even in the sit-

uation like this the repulsive cosmological constant keeps thepresence of the outer cusp enabling outflows of matter from thedisk. On the other hand, the collimation effect on jets could beefficient even for small toroidal disks, with outer edge locateddeeply under the static radius. In such disks the self-gravitationaleffects could usually be neglected.

In the case of accretion onto primordial black holes in thevery early universe, with assumed high values of repulsive cos-mological constant, we can expect even stronger effects. Con-sidering the electroweak phase transition atTew ∼ 100 GeV,we obtain an estimate of the primordial effective cosmologicalconstant

Λew ∼ 0.028 cm−2. (49)

Considering the quark confinement atTqc ∼ 1 GeV, we obtainan estimate of the primordial cosmological constant

Λqc ∼ 2.8 × 10−10 cm−2. (50)

Z. Stuchlık et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes 439

It follows from the Table 4 that the accretion onto primor-dial black holes of massM > Mew ∼ 2 × 1027 g, andM > Mqc ∼ 2 × 1031 g, respectively, is then forbidden in thedisk regime because no equilibrium configurations of perfectfluid are allowed in the corresponding Schwarzschild–de Sitterbackgrounds. Of course, the accretion can be realized in qua-sispherical regime in these spacetimes, however, its characterrepresents an open problem.

Acknowledgements.This work has been supported by the GACR GrantNo.202/99/0261, by the Committee for Collaboration of Czech Repub-lic with CERN and by the Bergen Computational Physics Laboratoryproject, an EU Research Infrastructure at the University of Bergen,Norway, supported by the European Community – Access to ResearchInfrastructure Action of the Improving Human Potential Programme.Two of authors (Z. S. and S. H.) would like to acknowledge the perfecthospitality and excellent working conditions at the CERN’s TheoryDivision and the Institute of Physics of the University of Bergen.

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