Equilibrium configurations of perfect fluid
in Reissner-Nordström-(anti-)de Sitter
spacetimes
Hana Kučáková, Zdeněk Stuchlík & Petr Slaný
Institute of Physics, Faculty of Philosophy and Science, Silesian University at Opava,
Bezručovo nám. 13, CZ-746 01 Opava,
Czech Republic
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Introduction
• investigating equilibrium configurations of perfect fluid in charged
black-hole and naked-singularity spacetimes with a nonzero cosmological
constant (Λ ≠ 0)
• the line element of the spacetimes (the geometric units c = G = 1)
1
 2M Q 2 Λ 2  2  2M Q 2 Λ 2 
2
ds  1 
 2  r  dt  1 
 2  r  dr 2  r 2 d   sin 2 Θ d 
r
r
 
r
r
 




• dimensionless cosmological parameter and dimensionless charge parameter
y
1
ΛM 2
3
e
Q
M
• dimensionless coordinates
r r M
t t M
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Test perfect fluid
• does not alter the geometry
• rotating in the ϕ direction – its four velocity vector field U μ has, therefore,
only two nonzero components U μ = (U t, 0 , 0 , U ϕ)
• the stress-energy tensor of the perfect fluid is
T     p   U U   p
(ɛ 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 density ℓ
U
 t
U

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U
Ut
3
Equipotential surfaces
• the solution of the relativistic Euler equation can be given by Boyer’s
condition determining the surfaces of constant pressure through the
“equipotential surfaces” of the potential W (r, θ)
• the equipotential surfaces are determined by the condition
W r,   const
• equilibrium configuration of test perfect fluid rotating around an axis of
rotation in a given spacetime are determined by the equipotential surfaces,
where the gravitational and inertial forces are just compensated by the
pressure gradient
• 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
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Equilibrium configurations
• 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
• configurations with uniform distribution of angular momentum density
 r,   const
• relation for the equipotential surfaces
W r ,   ln U t r , 
• in Reissner–Nordström–(anti-)de Sitter spacetimes
1  2 / r  e / r  yr  r sin 
W r ;  y, e   ln
r sin   1  2 / r  e / r  yr  
2
2
2 1/ 2
2
2
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2
2
2
2 1/ 2
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Behaviour of the equipotential surfaces,
and the related potential
• according to the values of
 r,   const
• region containing stable circular geodesics → accretion processes in the
disk regime are possible
• behaviour of potential in the equatorial plane (θ = π/2)
• equipotential surfaces - meridional sections
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Types of the Reissner-Nordström-de Sitter
spacetimes (RNdS)
• seven types with qualitatively different behavior of the effective potential
of the geodetical motion and the circular orbits
Black-hole spacetimes
• dS-BH-1 – one region of circular geodesics at r > rph+ with unstable then
stable and finally unstable geodesics (for radius growing)
• dS-BH-2 – one region of circular geodesics at r > rph+ with unstable
geodesics only
3   8e 2 

rph (e)  1  1 
2 
9 

1/ 2
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


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Types of the Reissner-Nordström-de Sitter
spacetimes (RNdS)
Naked-singularity spacetimes
• 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 circular 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
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Types of the Reissner-Nordström-anti-de
Sitter spacetimes (RNadS)
• four types with qualitatively different behavior of the effective potential of
the geodetical motion and the circular orbits
Black-hole spacetimes
• AdS-BH-1 – one region of circular geodesics at r > rph+ with unstable and
then stable geodesics (for radius growing)
Naked-singularity spacetimes
• AdS-NS-1 – two regions of circular geodesics, the inner one (r < rph-)
consists of stable geodesics only, the outer one (r > rph+) contains both
unstable and then stable circular geodesics
• AdS-NS-2 – one region of circular orbits, subsequently with stable, then
unstable and finally stable orbits
• AdS-NS-3 – one region of circular orbits with stable orbits exclusively
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RNdS black-hole spacetimes
1) open surfaces only, no disks are possible, surface with the outer cusp exists (M = 1; e = 0.5; y = 10-6;
ℓ = 3.00)
2) an infinitesimally thin, unstable ring exists (M = 1; e = 0.5; y = 10-6; ℓ = 3.55378053)
3) closed surfaces exist, many equilibrium configurations without cusps are possible, one with the
inner cusp (M = 1; e = 0.5; y = 10-6; ℓ = 3.75)
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RNdS black-hole spacetimes
4) there is an equipotential surface with both the inner and outer cusps, the mechanical nonequilibrium
causes an inflow into the black hole, and an outflow from the disk, with the same efficiency (M = 1;
e = 0.5; y = 10-6; ℓ = 3.8136425)
5) accretion into the black-hole is impossible, the outflow from the disk is possible (M = 1; e = 0.5; y =
10-6; ℓ = 4.00)
6) the potential diverges, the inner cusp disappears, the closed equipotential surfaces still exist, one
with the outer cusp (M = 1; e = 0.5; y = 10-6; ℓ = 6.00)
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RNdS black-hole spacetimes
7) an infinitesimally thin, unstable ring exists (the center, and the outer cusp coalesce) (M = 1; e = 0.5;
y = 10-6; ℓ = 7.11001349)
8) open equipotential surfaces exist only, there is no cusp in this case (M = 1; e = 0.5; y = 10-6; ℓ =
10.00)
9) an infinitesimally thin, unstable ring exists (the center, and the outer cusp coalesce), surface with
the inner cusp exists as well, accretion into the black-hole is impossible (M = 1; e = 1.02; y = 10-4;
ℓ =3.7920002388)
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RNdS naked-singularity spacetimes
1) closed surfaces exist, one with the outer cusp, equilibrium configurations are possible (M = 1; e =
1.02; y = 10-5; ℓ = 2.00)
2) the second closed surface with the cusp, and the center of the second disk appears, the inner disk (1)
is inside the outer one (2) (M = 1; e = 1.02; y = 10-5; ℓ = 3.04327472)
3) two closed surfaces with a cusp exist, the inner disk is still inside the outer one (M = 1; e = 1.02; y =
10-5; ℓ = 3.15)
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RNdS naked-singularity spacetimes
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 is possible (M = 1; e = 1.02; y = 10-5; ℓ = 3.2226824)
5) the disks are separated, the outflow from disk 1 into disk 2, and the outflow from disk 2 is possible
(M = 1; e = 1.02; y = 10-5; ℓ = 3.55)
6) the cusp 1 disappears, the potential diverges, two separated disks still exist, the flow between disk 1
and disk 2 is impossible, the outflow from disk 2 is possible (M = 1; e = 1.02; y = 10-5; ℓ = 4.40)
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RNdS naked-singularity spacetimes
7) disk 1 exists, also an infinitesimally thin, unstable ring exists (region 2) (M = 1; e = 1.02; y = 10-5;
ℓ = 4.9486708)
8) the potential diverges, the cusp disappears, equilibrium configurations are possible (closed surfaces
exist), but the outflow from the disk is impossible (M = 1; e = 1.02; y = 10-2; ℓ = 5.00)
9) an infinitesimally thin, unstable ring exists (region 1), also disk 2 (M = 1; e = 1.07; y = 10-4; ℓ =
3.42331737)
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RNdS naked-singularity spacetimes
10) one cusp, and disk 2 exists only, the outflow from disk 2 is possible (M = 1; e = 1.07; y = 10-4; ℓ =
3.50)
11) an infinitesimally thin, unstable ring exists (region 2) (M = 1; e = 1.07; y = 10-4; ℓ = 3.59008126)
12) no disk, no cusp, open equipotential surfaces only (M = 1; e = 1.07; y = 10-4; ℓ = 3.80)
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RNdS naked-singularity spacetimes
13) the disks are separated, the outflow from disk 1 into disk 2 (an infinitesimally thin, unstable ring),
and the outflow from disk 2 is possible (M = 1; e = 0.5; y = 10-4; ℓ = 3.6138361382)
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RNadS black-hole spacetimes
1) open equipotential surfaces only, no disks are possible (M = 1; e = 0.99; y = - 10-4; ℓ = 2.00)
2) an infinitesimally thin unstable ring exists (M = 1; e = 0.99; y = - 10-4; ℓ = 3.10048313)
3) equilibrium configurations are possible, closed equipotential surfaces exist, one with the cusp that
enables accretion from the toroidal disk into the black hole (M = 1; e = 0.99; y = - 10-4; ℓ = 3.70)
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RNadS black-hole spacetimes
4) the potential diverges, the cusp disappears, accretion into the black-hole is impossible, equilibrium
configurations are still possible, closed equipotential surfaces exist (M = 1; e = 0.99; y = - 10-4; ℓ =
5.00)
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RNadS naked-singularity spacetimes
1) closed equipotential surfaces exist, equilibrium configurations are possible, one disk (1) only (M =
1; e = 0.99; y = - 0.4; ℓ = 1.30)
2) the center of the second disk (2) appears, one equipotential surface with the cusp exists (M = 1; e =
0.99; y = - 0.4; ℓ = 1.448272709327)
3) the flow between the inner disk (1) and the outer one (2) is possible (M = 1; e = 0.99; y = - 0.4; ℓ =
1.465)
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RNadS naked-singularity spacetimes
4) the potential diverges, no equipotential surface with the cusp exists, the disks are separated, the flow
between the disk 1 and the disk 2 is impossible (M = 1; e = 0.99; y = - 0.4; ℓ = 1.50)
5) the disk 1 is infinitesimally thin (M = 1; e = 1.07; y = - 10-4; ℓ = 3.41935796)
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Conclusions (RNdS)
•
•
•
•
The Reissner–Nordström–de Sitter 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 mechanism, i.e., due to a violation of the hydrostatic equilibrium.
The motion above the outer horizon of black-hole backgrounds has the same
character as in the Schwarzschild–de Sitter 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 Reissner–Nordström naked singularities. However, in the
case of Reissner–Nordström–de Sitter, two static radii can exist, while the
Reissner–Nordström 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.
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Conclusions (RNadS)
•
•
•
The Reissner–Nordström–anti-de Sitter spacetimes can be separated into four
types of spacetimes with qualitatively different character of the geodetical
motion. In all of them toroidal disks can exist, because
in these spacetimes stable circular orbits exist.
The motion above the outer horizon of black-hole backgrounds has the same
character as in the Schwarzschild–anti-de Sitter spacetimes.
The motion in the naked-singularity backgrounds has similar character as the
motion in the field of Reissner–Nordström naked singularities. Stable circular
orbits exist in all of the naked-singularity spacetimes.
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References
• Z. Stuchlík, S. Hledík. Properties of the Reissner-Nordström spacetimes
with a nonzero cosmological constant. Acta Phys. Slovaca, 52(5):363-407,
2002
• Z. Stuchlík, P. Slaný, S. Hledík. Equilibrium configurations of perfect fluid
orbiting Schwarzschild-de Sitter black holes. Astronomy and Astrophysics,
363(2):425-439, 2000
Thank you for your attention!
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