Equilibrium configurations
of perfect fluid
in Reissner-Nordström de Sitter
spacetimes
Hana Kučáková, Zdeněk Stuchlík, Petr Slaný
Institute of Physics, Silesian University at Opava
RAGtime 9
19.-21. September 2007, Hradec nad Moravicí
Introduction
• investigating equilibrium configurations of perfect fluid
in charged black-hole and naked-singularity spacetimes with
a repulsive cosmological constant ( > 0)
• the line element of the spacetimes (the geometric units c = G = 1)
1
 2M Q 2  2  2  2M Q 2  2 
2
ds  1 
 2  r  dt  1 
 2  r  dr 2  r 2 d   sin 2  d 
r
r
 
r
r
 



• dimensionless cosmological parameter and dimensionless charge
parameter
1
Q
2
y
3
M
e
M
• dimensionless coordinates
t t M
r r M

Types of the Reissner-Nordström de Sitter
spacetimes
• 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
2 1/ 2 


3
8e 

 

rph (e)  1  1 
2 
9  


Types of the Reissner-Nordström de Sitter
spacetimes
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
Test perfect fluid
• does not alter the geometry
• rotating in the  direction – its four velocity vector field U  has,
therefore, only two nonzero components U  = (U t, 0, 0 , U )
• the stress-energy tensor of the perfect fluid is
T     p   U U   p
( 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 , and the angular momentum density l
U
 t
U

U
Ut
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, )
• 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
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–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
2 1/ 2
2
2
2
2
2 1/ 2
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 ( = /2)
• equipotential surfaces - meridional sections
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
2) an infinitesimally thin, unstable ring exists
3) closed surfaces exist, many equilibrium configurations without cusps are possible,
one with the inner cusp
l = 3.00
l = 3.55378053
l = 3.75
dS-BH-1: M = 1; e = 0.5; y = 10-6
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
5) accretion into the black-hole is impossible, the outflow from the disk is possible
6) the potential diverges, the inner cusp disappears
l = 3.8136425
l = 4.00
l = 4.96797564
dS-BH-1: M = 1; e = 0.5; y = 10-6
7) the closed equipotential surfaces still exist, one with the outer cusp
8) an infinitesimally thin, unstable ring exists (the center, and the outer cusp coalesce)
9) open equipotential surfaces exist only, there is no cusp in this case
l = 6.00
l = 7.11001349
l = 10.00
dS-NS-1: M = 1; e = 1.02; y = 0.00001
1) closed surfaces exist, one with the outer cusp, equilibrium configurations are
possible
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)
3) two closed surfaces with a cusp exist, the inner disk is still inside the outer one
l = 2.00
l = 3.04327472
l = 3.15
dS-NS-1: M = 1; e = 1.02; y = 0.00001
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
5) the disks are separated, the outflow from disk 1 into disk 2 only, and the outflow
from disk 2 is possible
6) the cusp 1 disappears, the potential diverges, two separated disks still exist
l = 3.2226824
l = 3.55
l = 3.91484803
dS-NS-1: M = 1; e = 1.02; y = 0.00001
7) like in the previous case, the flow between disk 1 and disk 2 is impossible, the
outflow from disk 2 is possible
8) disk 1 exists, also an infinitesimally thin, unstable ring exists (region 2)
9) disk 1 exists only, there are no surfaces with a cusp
l = 4.40
l = 4.9486708
l = 5.15
dS-NS-1: M = 1; e = 1.02; y = 0.00001
10) disk 1 is infinitesimally thin
11) no disks, open equipotential surfaces only
l = 5.39574484
l = 6.00
dS-NS-2: M = 1; e = 1.02; y = 0.01
1) there is only one center and one disk in this case, closed equipotential surfaces
exist, one with the cusp, the outflow from the disk is possible
2) the potential diverges, the cusp disappears, equilibrium configurations are possible
(closed surfaces exist), but the outflow from the disk is impossible
3) the situation is similar to the previous case
l = 4.00
l = 4.25403109
l = 5.00
dS-NS-2: M = 1; e = 1.02; y = 0.01
4) the disk is infinitesimally thin
5) no disk is possible, open equipotential surfaces only
l = 6.40740525
l = 7.00
dS-NS-3: M = 1; e = 1.07; y = 0.0001
1) closed surfaces exist, one with the outer cusp, equilibrium configurations are
possible
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)
3) two closed surfaces with a cusp exist, the inner disk is still inside the outer one
l = 2.50
l = 2.93723342
l = 3.00
dS-NS-3: M = 1; e = 1.07; y = 0.0001
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
5) the disks are separated, the outflow from disk 1 into disk 2 only, and the outflow
from disk 2 is possible
6) an infinitesimally thin, unstable ring exists (region 1), also disk 2
l = 3.0411677
l = 3.20
l = 3.42331737
dS-NS-3: M = 1; e = 1.07; y = 0.0001
7) one cusp, and disk 2 exists only, the outflow from disk 2 is possible
8) an infinitesimally thin, unstable ring exists (region 2)
9) no disk, no cusp, open equipotential surfaces only
l = 3.50
l = 3.59008126
l = 3.80
dS-NS-4: M = 1; e = 1.07; y = 0.01
1) there is only one center and one disk in this case, closed equipotential surfaces
exist, one with the cusp, the outflow from the disk is possible
2) an infinitesimally thin, unstable ring exists
3) no disk is possible, no cusp, open equipotential surfaces exist only
l = 3.00
l = 3.63788074
l = 3.80
Conclusions
• 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 blackhole 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.
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
~ The End ~