Zdenìk Stuchlík, Hana Kuèáková and Petr Slaný- Equilibrium congurations of perfect fluid in Reissner-Nordström-de Sitter spacetimes

8/3/2019 Zdenìk Stuchlík, Hana Kuèáková and Petr Slaný- Equilibrium con gurations of perfect fluid in Reissner-Nordström-de Sitter spacetimes

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Proceedings of RAGtime 8/9,15–19/19–21September, 2006/2007,Hradec nad Moravicí, Opava, Czech Republic 433S.HledíkandZ. Stuchlík, editors, SilesianUniversity inOpava,2007,pp. 433–447

Equilibriumconfigurations of perfect fluidinReissner–Nordström–de Sitter spacetimes

Zdeněk Stuchlík, Hana Kučáková and Petr Slaný

Institute of Physics, Faculty of Philosophy & Science, Silesian University in Opava,

Bezručovo nám. 13, CZ-746 01 Opava, Czech Republic

ABSTRACT

Marginally stable perfect fluid tori with uniform distribution of specific angular mo-

mentum are determined in the Reissner–Nordström–de Sitter black-hole and naked-

singularity spacetimes. Perfect fluid toroidal configurations are allowed only in the

spacetimesadmitting existence of stable circular geodesics. The configurations with

equipotential surfaces crossing itself in a cusp allow accretion (inner cusp) and/or

excretion (outer cusp) of matter from the toroidal configuration. The classification

of the Reissner–Nordström–de Sitter spacetimes according to the properties of the

marginally stable tori is given.

1 INTRODUCTION

Many observations suggest that the energy sources in quasars and active galactic nuclei

are accretion discs orbiting massive black-holes. However, despite the cosmic censorship

hypothesis (Penrose, 1969) that is not probed yet, existence of naked singularities related

to the black hole solutions of the Einstein equations is not excluded (see, e.g., de Felice and

Yunqiang, 2001) and is still worth of consideration.

The accretion discs could be geometrically thin with low accretion rates and negligible

pressure, characterized by quasicircular geodetical motion, or geometrically thick with high

accretion rates and relevant pressure gradients that could be, in the basic approximation,

determined by equipotential surfaces of test perfect fluid orbiting the central object (Ab-

ramowicz, 1998).

The presence of a repulsive cosmological constant (dark, or vacuum, energy) Λ0 ∼

10−56cm−2 indicated by wide range of cosmological tests (Spergel et al., 2003) could

influence significantly the properties of the accretion discs (Stuchlík, 2005), as shown both

for the Schwarzschild–de Sitter (SdS) spacetimes (Stuchlík and Hledík, 1999; Stuchlík

et al., 2000) and Kerr–de Sitter (KdS) spacetimes (Stuchlík and Slaný, 2004; Slaný and

Stuchlík, 2005).

Geodetical motion and related thick accretion disc properties in the Reissner–Nord-

ström–de Sitter (RNdS) spacetimes were studied in Stuchlík and Hledík (2002). Since

some characteristics of the geodetical motion in RNdS spacetimes differ from those in SdS

and KdS spacetimes, we shall study here properties of equilibrium tori in the RNdS

spacetimes.

978-80-7248-419-5 © 2007 – SU in Opava. All rights reserved.



434 Z.Stuchlík, H.Kučákováand P. Slaný

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

marginally stable equilibrium configurations (Seguin, 1975).

2 PROPERTIES OF THE REISSNER–NORDSTRÖM–(ANTI-)DE SITTERSPACETIMES

In the standard Schwarzschild coordinates (t , r , θ , φ ), and the geometrical units (c =

G = 1), the RNdS (Λ > 0), and RN(a)dS (Λ < 0) spacetimes are given by the line

element (Stuchlík and Hledík, 2002)

ds2 = −

1 −

2 M

r +

Q2

r 2−

Λ

3r 2

dt 2

+ 1 −2 M

r

+Q2

r 2

−Λ

3

r 2−1

dr 2 + r 2(dθ2 + sin2 θ dφ2) , (1)



Equilibriumconfigurations of perfectfluid inRNdS spacetimes 435

and the related electromagnetic field is given by the four-potential

Aµ =Q

r δt

µ

.

Here, M denotes mass and Q denotes electric charge of the spacetimes. However, it is

convenient to introduce a dimensionless cosmological parameter

y ≡ 13

Λ M 2 ,

a dimensionless charge parameter

e ≡Q

M ,

and dimensionless coordinates t → t / M , r → r / M . It is equivalent to putting M = 1.

3 GEODETICAL MOTION

Motion of uncharged test particles and photons is governed by the geodetical structure of

the spacetime. The geodesic equation reads

D pµ

dλ= 0 ,

where pµ ≡ d xµ/dλ is the four-momentum of a test particle (photon) and λ is the affine

parameter related to the proper time τ of a test particle by τ = λ/m.

It follows from the central symmetry of the geometry Eq. (1) that the geodetical motion

is allowed in the central planes only. Due to existence of the time Killing vector field

ξ(t ) = ∂/∂ t and the axial Killing vector field ξ(φ) = ∂/∂φ, two constants of the motion must

exist, being the projections of the four-momentum onto the Killing vectors (Stuchlík and

Hledík, 2002):

pt = gt µ pµ = − ˜ E ,

pφ = gφµ pµ = Φ .

In the spacetimes with Λ = 0, the constants of motion ˜ E and Φ cannot be interpreted as

energy and axial component of the angular momentum at infinity since the geometry is not

asymptotically flat.It is convenient to introduce specific energy E , specific axial angular momentum L and

impact parameter by the relations

E =˜ E

m, L =

Φ

m, =

Φ

˜ E .

Choosing the plane of the motion to be the equatorial plane (θ = π/2 being constant along

the geodesic), we find that the motion of test particles (m = 0) can be determined by an

“effective potential” of the radial motion

V 2

eff (r ; L, y, e) ≡ 1 −

2

r +

e2

r 2− yr 21 +

L2

r 2 .




Since

ur 2 = dr

dτ 2

= E 2 − V 2eff (r ; L, y, e) ,

the motion is allowed where

E 2 ≥ V 2eff (r ; L, y, e) ,

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) ,




which can be functions of the coordinates r , θ . The stress-energy tensor of the perfect fluid

is

T µν = ( p + )U µU ν + p δµν ,

where and p denote the total energy density and the pressure of the fluid. The rotating

fluid can be characterized by the vector fields of the angular velocity Ω , and the angular

momentum per unit mass (angular momentum density) , defined by

Ω =U φ

U t , = −

U φ

U t .

Projecting the energy conservation law T µν

;ν= 0 onto the hypersurface orthogonal to

the four velocityU µ by the projection tensor hµν = gµν + U µU ν , we obtain the relativistic

Euler equation in the form (Stuchlík et al., 2000)

∂µ p

p + = −∂µ(lnU t ) +

Ω ∂µ

1 − Ω,

where

(U t )2 =

g2t φ − gt t gφφ

gφφ + 2gt φ + 2gt t

.

The solution of the relativistic Euler equation can be given by Boyer’s condition determ-

ining the surfaces of constant pressure through the “equipotential surfaces” of the potentialW (r , θ ) by the relations (Abramowicz et al., 1978)

p

0

d p

p + = W in − W , (2)

W in − W = ln(U t )in − ln(U t ) +

in

Ω d

1 − Ω; (3)

the subscript “in” refers to the inner edge of the disk. The equipotential surfaces are

determined by the condition

W (r , θ ) = const ,

and in a given spacetime can be found from Eq. (3), if a rotation law Ω = Ω() is given.

The surfaces of constant pressure p(r , θ ) = const are given by Eq. (2).

5 EQUIPOTENTIAL SURFACES OF THE MARGINALLY STABLECONFIGURATIONS

Equilibrium configurations of test perfect fluid rotating around an axis of rotation (θ = 0)

in a given spacetime are determined by the equipotential surfaces, where the gravitational

and inertial forces are just compensated by the pressure gradient (Stuchlík et al., 2000).




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).




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.






1 5 10 50 100 500

r

-0.075

-0.05

-0.025

0

0.025

0.05

0.075

0.1

W

( r ,

θ

=

π / 2 )

0 0.5 1 1.5 2 2.5 3

(log r) sin θ

-3

-2

-1

0

1

2

3

( l o g

r

)

c o s

θ1.0

-1.0

-0.01

-0.05

0.02

(9)

7.2 dS-NS-1 ( M = 1, e = 1.02, y = 0.00001 )

(1) Closed surfaces exist, one with the outer cusp, equilibrium configurations are pos-sible ( = 2.00);

(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) ( = 3.04327472);

(3) two closed surfaces with a cusp exist, the inner disk is still inside the outer one

( = 3.15);

(4) closed surface with two cusps exists, two disks meet in one cusp, the flow between

disk 1 and disk 2, and the outflow from disk 2 are possible ( = 3.2226824);

(5) the disks are separated, the outflow from disk 1 into disk 2 only, and the outflow

from disk 2 are possible ( = 3.55);

(6) the cusp 1 disappears, the potential diverges, two separated disks still exist( = 3.91484803);

(7) like in the previous case, the flow between disk1 and disk 2 is impossible, the outflow

from disk 2 is possible ( = 4.40);

(8) disk 1 exists, so does an infinitesimally thin, unstable ring exists (region 2)

( = 4.9486708);

(9) disk 1 exists only, there are no surfaces with a cusp ( = 5.15);

(10) disk 1 is infinitesimally thin ( = 5.39574484);

(11) no disks, open equipotential surfaces only ( = 6.00).

1 5 10 50 100

r

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

W

( r ,

θ

=

π / 2 )

0 0.5 1 1.5 2

(log r) sin θ

-2

-1

0

1

2

( l o g

r )

c o s

θ

cent1

cusp

-0.05

-0.05

-0.3

-0.3

10.0

-0.025

0.0

(1)

1 5 10 50 100

r

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

W

( r ,

θ

=

π / 2 )

0 0.5 1 1.5 2

(log r) sin θ

-2

-1

0

1

2

( l o g

r )

c o s

θ

cent1cent2

cusp2

cusp1

cusp1

-0.04

-0.04

-0.3

10.0

-0.025

0.0

(2)

(plots continued on the next page)






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);






1 2 5 10 20 50 100

r

-0.5

-0.25

0

0.25

0.5

0.75

1

W

( r ,

θ

=

π / 2 )

0 0.25 0.5 0.75 1 1.25 1.5 1.75

(log r) sin θ

-2

-1

0

1

2

( l o g

r )

c o s

θ

-0.1

-0.3

1.0 -0.0550.0

(9)

7.5 dS-NS-4 ( M = 1, e = 1.07, 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 ( = 3.00);

(2) an infinitesimally thin, unstable ring exists ( = 3.63788074);

(3) no disk is possible, no cusp, open equipotential surfaces exist only ( = 3.80).

1 1.5 2 3 5 7

r

-0.5

0

0.5

1

1.5

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

-0.5

-0.3

-0.3

3.0

cusp

0.0

(1)

1 1.5 2 3 5 7

r

-0.5

0

0.5

1

1.5

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

cent

-0.5

-0.33.0

cusp

0.0

(2)

1 1.5 2 3 5 7

r

-0.5

0

0.5

1

1.5

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

-0.5

-0.33.0 0.0

(3)

8 CONCLUSIONS

The RNdS spacetimes can be separated into seven types of spacetimes with qualitatively

different character of the geodetical motion. In five of them toroidal disks can exist, because

in these spacetimes stable circular orbits exist.

The presence of an outer cusp of toroidal disks nearby the static radius which enables

outflow of mass and angular momentum from the accretion disks by the Paczyński mechan-

ism, i.e., due to a violation of the hydrostatic equilibrium. This is the same mechanism that

drives the accretion into the black hole through the inner cusp (Stuchlík et al., 2000).




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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Documents

Zdenìk Stuchlík, Hana Kuèáková and Petr Slaný- Equilibrium congurations of perfect fluid in Reissner-Nordström-de Sitter spacetimes