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8/3/2019 Zdenìk Stuchlík, Hana Kuèáková and Petr Slaný- Equilibrium con gurations of perfect fluid in Reissner-Nordström-de Sitter spacetimes
Recall that Reissner–Nordström–(anti-)de Sitter (RN(a)dS) black-hole spacetimes and
some RNdS black-hole spacetimes a region containing stable circular geodesics exists,
which allows accretion processes in the disk regime. On the other hand, around some
naked singularities even two separated regions with stable circular geodesics exist. The
inner regionis limited from belowby particleswith zero angular momentum that are located
in stable equilibrium positions (Stuchlík and Hledík, 2002).
The hydrodynamical structure of perfect fluid orbiting RNdS black holes (and naked-
singularities) is investigated for configurations with uniform distribution of angular mo-
mentum density. In the black-hole and the naked-singularity backgrounds admitting the
existence of stable circular geodesics, closed equipotential surfaces with a cusp, allowing
the existence of toroidal accretion disks, can exist (Stuchlík et al., 2000).
It is well known that at low accretion rates the pressure is negligible, and the accretion
disk is geometrically thin. Its basic properties are determined by the circular geodesic mo-
tion in the black-hole (naked-singularity) background (Novikov and Thorne, 1973). Athigh accretion rates, the pressure is relevant, and the accretion disk must be geometric-
ally thick (Abramowicz et al., 1988). Its basic properties are determined by equipotential
surfaces of test perfect fluid (i.e., perfect fluid that does not alter the black-hole geometry)
rotating in the black-hole (naked-singularity) background.
The accretion is possible, if a toroidal equilibrium configuration of the test fluid contain-
ing a critical, self-crossing equipotential surface can exist in the background. The cusp
of this equipotential surface corresponds to the inner edge of the disk, and the accretion
inflow of matter into the black hole is possible due to a mechanical non-equilibrium process,
i.e., because of matter slightly overcoming the critical equipotential surface. The pressure
gradients push the inner edge of the thick disks under the radius r ms, which corresponds tomarginally stable circular geodesic (Kozłowski et al., 1978; Abramowicz et al., 1978).
The simplest, but quite illustrative case of the equipotential surfaces of the test fluid can
be constructed for the configurations with uniform distribution of the angular momentum
density. This case is fully governed by the geometry of the spacetime, however, it contains
all the characteristic features of more complex cases of the rotation of the fluid (Jaroszyński
et al., 1980). Moreover, this case is also very important physically since it corresponds to
and the turning points of the radial motion are determined by the condition
E 2 = V 2eff (r ; L, y, e) .
The radial motion of photons (m = 0) is determined by a “generalized effective potential”
2ph(r ; y, e) related to the impact parameter . The motion is allowed, if
2 ≤ 2ph(r ; y, e) ≡r 4
r 2 − 2r + e2 − yr 4,
the condition 2 = 2ph(r ; y, e) gives the turning points of the radial motion (Stuchlík and
Hledík, 2002).
The special case of e = 0 has been extensively discussed in Stuchlík and Hledík (1999).
Therefore, we concentrate our discussion on the case e2 > 0. The effective potentials
V 2eff (r ; L, y, e) and 2ph(r ; y, e) define turning points of the radial motion at the static
regions of the RN(a)dS spacetimes. (At the dynamic regions, where the inequalities
V 2eff (r ; L, y, e) < 0 and 2ph(r ; y, e) < 0 hold, there are no turning points of the radialmotion.) Effective potential V 2eff is zero at the horizons, while 2 diverges there. At r = 0,
V 2eff → +∞, while 2ph = 0. Circular orbits of uncharged test particles correspond to
local extrema of the effective potential (∂V eff /∂r = 0). Maxima (∂2V eff /∂r 2 < 0) determ-
ine circular orbits unstable with respect to radial perturbations, minima (∂ 2V eff /∂r 2 > 0)
determine stable circular orbits. The specific energy and specific angular momentum of
particles on a circular orbit, at a given r , are determined by the relations (Stuchlík and
Hledík, 2002)
E c(r ; y, e) =1 − 2/r + e2/r 2 − yr 2
1 − 3/r + 2e2/r 2
1/2 , Lc(r ; y, e) = r − e2 − yr 4
1 − 3/r + 2e2
/r 2
1/2
.
(The minus sign for Lc is equivalent to the plus sign in spherically symmetric spacetimes.)
4 BOYER’SCONDITION FOR EQUILIBRIUM CONFIGURATIONS OF TESTPERFECT FLUID
We consider test perfect fluid rotating in the φ direction. Its four velocity vector field U µ
has, therefore, only two non-zero components
U µ = (U t , 0, 0,U φ) ,
8/3/2019 Zdenìk Stuchlík, Hana Kuèáková and Petr Slaný- Equilibrium con gurations of perfect fluid in Reissner-Nordström-de Sitter spacetimes
The equipotential surfaces can be closed or open. Moreover, there is a special class of
critical, self-crossing surfaces (with a cusp), which can be either closed or open. The closed
equipotential surfaces determine stationary equilibrium configurations. The fluid can fill
any closed surface – at the surface of the equilibrium configuration pressure vanish, but its
gradient is non-zero(Kozłowski et al., 1978). Thecritical, self-crossingclosed equipotential
surfaces W cusp are important in the theory of thick accretion disks, because accretion onto
the black hole through the cusp of the equipotential surface located in the equatorial plane
is possible due to the Paczyński mechanism.
According to Paczyński, the accretion into the black hole is driven through the vicinity of
the cusp due to a little overcoming of the critical equipotential surface, W = W cusp, by the
surface of the disk. Theaccretion is thus driven by a violation of the hydrostatic equilibrium,
rather than by viscosity of the accreting matter (Kozłowski et al., 1978).
All characteristic properties of the equipotential surfaces for a general rotation law are
reflected by the equipotential surfaces of the simplest configurations with uniform distri-bution of the angular momentum density (Jaroszyński et al., 1980). Moreover, these
configurations are very important astrophysically, being marginally stable (Seguin, 1975).
Under the condition
(r , θ ) = const ,
holding in the rotating fluid, a simple relation for the equipotential surfaces follows from
Eq. (3):
W (r , θ ) = lnU t (r , θ ) ,
withU t (r , θ ) being determined by = const, and the metric coefficients only.
Theequipotentialsurfaces are described by theformula θ = θ (r ), given by the differential
equation (Stuchlík et al., 2000)
dθ
dr = −
∂ p/∂r
∂ p/∂θ,
which for the configurations with = const reduces to
dθ
dr = −
∂U t /∂r
∂U t /∂θ.
The equipotential surfaces are given by the formula
W (r ; θ, y, e) = ln(1 − 2/r + e2/r 2 − yr 2)1/2r sin θ
r 2 sin2 θ − (1 − 2/r + e2/r 2 − yr 2)21/2 .
The best insight into the nature of the = const configurations can be obtained by the
examination of the behaviour of the potential W (r , θ ) in the equatorial plane (θ = π/2).
The condition of the local extrema of the potential W (r , θ = π/2, y, e) is identical with the
condition of vanishing of the pressure gradient (∂U t /∂r = 0, ∂U t /∂θ = 0). The extrema
of W (r , θ = π/2, y, e) correspond to the points, where the fluid moves along a circular
geodesic (Stuchlík et al., 2000).
8/3/2019 Zdenìk Stuchlík, Hana Kuèáková and Petr Slaný- Equilibrium con gurations of perfect fluid in Reissner-Nordström-de Sitter spacetimes
Equilibriumconfigurations of perfectfluid inRNdS spacetimes 439
6 CLASSIFICATION OF THE REISSNER–NORDSTRÖM–DE SITTERSPACETIMES
Seven types of the RNdS spacetimes with qualitatively different behaviour of the effectivepotential of the geodetical motion (and the circular orbits) exist. The description of the
types of the Reissner–Nordström (RN) spacetimes with a positive cosmological constant
( y > 0) according to the properties of the circular geodesics can be given in the following
way (Stuchlík and Hledík, 2002):
dS-BH-1 One region of circular geodesics at r > r ph+ with unstable then stable and
finally unstable geodesics (for radius growing).1
dS-BH-2 One region of circular geodesics at r > r ph+ with unstable geodesics only.
dS-NS-1 Two regions of circular geodesics, the inner region consists of stable geodesics
only, the outer one contains subsequently unstable, then stable and finally unstable circu-
lar geodesics.dS-NS-2 Two regions of circular orbits, the inner one consist of stable orbits, the outer
one of unstable orbits.
dS-NS-3 One region of circular orbits, subsequently with stable, unstable, then stable
and finally unstable orbits.
dS-NS-4 One region of circular orbits with stable and then unstable orbits.
dS-NS-5 No circular orbits allowed.
7 PROPERTIES OF EQUILIBRIUM CONFIGURATIONS OF PERFECT FLUID
We shall discuss the perfect fluid configurations in the framework of the RNdS spacetime
classification due to circular geodesic properties. Of course, only the spacetimes admit-
ting existence of stable circular geodesics are taken into account, since the equilibrium
configurations are allowed only in these spacetimes (Stuchlík and Hledík, 2002).
The behaviour of the potential W (r , θ = π/2), and corresponding equipotential surfaces
(meridional sections) are given, according to the values of = const, and illustrated by
representative sequences of figures. The radial coordinate is expressed in units of M . The
cusps of the toroidal disks correspond to the local maxima of W (r , θ = π/2), the central
rings correspond to their local minima.
7.1 dS-BH-1 ( M = 1, e = 0.5, y = 10−6 )
(1) Open surfaces only, no disks are possible, surface with the outer cusp exists
( = 3.00);
(2) an infinitesimally thin, unstable ring exists ( = 3.55378053);
(3) closed surfaces exist, many equilibrium configurations without cusps are possible,
one with the inner cusp( = 3.75);
1 Type dS-BH-1 means asymptotically de Sitter black-hole spacetime of type 1; in the following, the notation has
to be read in an analogous way.
8/3/2019 Zdenìk Stuchlík, Hana Kuèáková and Petr Slaný- Equilibrium con gurations of perfect fluid in Reissner-Nordström-de Sitter spacetimes
Equilibriumconfigurations of perfectfluid inRNdS spacetimes 443
7.3 dS-NS-2 ( M = 1, e = 1.02, y = 0.01 )
(1) There are only one centre and one disk in this case, closed equipotential surfaces
exist, one with the cusp, the outflow from the disk is possible ( = 4.00);(2) the potential diverges, the cusp disappears, equilibrium configurations are possible
(closed surfaces exist), but the outflow from the disk is impossible ( = 4.25403109);
(3) the situation is similar to the previous case ( = 5.00);
(4) the disk is infinitesimally thin ( = 6.40740525);
(5) no disk is possible, open equipotential surfaces only ( = 7.00).
1 1.5 2 3 5 7
r
-1.5
-1
-0.5
0
0.5
1
1.5
2
W
( r ,
θ
=
π / 2 )
0 0.1 0.2 0.3 0.4 0.5 0.6
(log r) sin θ
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
( l o g
r )
c o s
θ
cent
-0.3
3.0
cusp
0.0
(1)
1 1.5 2 3 5 7
r
-1.5
-1
-0.5
0
0.5
1
1.5
2
W
( r ,
θ
=
π / 2 )
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
(log r) sin θ
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
( l o g
r )
c o s
θ
cent
-0.3
3.0
3.0
0.0
(2)
1 1.5 2 3 5 7
r
-1.5
-1
-0.5
0
0.5
1
1.5
2
W
( r ,
θ
=
π / 2 )
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
(log r) sin θ
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
( l o g
r )
c o s
θ
cent
-0.3
3.0
3.0
0.0
(3)
r
-1.5
-1
-0.5
0
0.5
1
1.5
2
W
( r ,
θ
=
π / 2 )
0 0.2 0.4 0.6
(log r) sin θ
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
( l o g
r )
c o s
θ
cent
-0.3
3.0
0.0
(4)
r
-1.5
-1
-0.5
0
0.5
1
1.5
2
W
( r ,
θ
=
π / 2 )
0 0.2 0.4 0.6 0.8
(log r) sin θ
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
( l o g
r )
c o s
θ
-0.3
3.0
0.0
(5)
7.4 dS-NS-3 ( M = 1, e = 1.07, y = 0.0001 )
(1) Closed surfaces exist, one with the outer cusp, equilibrium configurations are pos-
sible ( = 2.50);
(2) the second closed surface with the cusp, and the centre of the second disk appear,
the inner disk (1) is inside the outer one (2) ( = 2.93723342);
(3) two closed surfaces with a cusp exist, the inner disk is still inside the outer one
( = 3.00);
8/3/2019 Zdenìk Stuchlík, Hana Kuèáková and Petr Slaný- Equilibrium con gurations of perfect fluid in Reissner-Nordström-de Sitter spacetimes
The motion above the outer horizon of black-hole backgrounds has the same character
as in the SdS spacetimes for asymptotically de Sitter spacetimes. There is only one static
radius in these spacetimes. No static radius is possible under the inner black-hole horizon,
no circular geodesics are possible there.
The motion in the naked-singularity backgrounds has similar character as the motion in
the field of RN naked singularities. However, in the case of RNdS, two static radii canexist,
while the RN naked singularities contain one static radius only. The outer static radius
appears due to the effect of the repulsive cosmological constant. Stable circular orbits exist
in all of the naked-singularity spacetimes. There are even two separated regions of stable
circular geodesics in some cases. The inner one is limited by the inner static radius from
bellow, where particles with zero angular momentum (in stable equilibrium positions) are
located. In the asymptotically de Sitter naked-singularity spacetimes, two regions of stable
circular orbits can exist, if e2 < 275/216, and y < 0.00174 (Stuchlík and Hledík, 2002).
Then two separated tori are possible in these spacetimes.
ACKNOWLEDGEMENTS
This work was supported by the Czech grant MSM 4781305903 and by the Czech Ministry
of Education under the project LC06014.
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8/3/2019 Zdenìk Stuchlík, Hana Kuèáková and Petr Slaný- Equilibrium con gurations of perfect fluid in Reissner-Nordström-de Sitter spacetimes