1. 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

2. Introduction investigating equilibrium configurations of perfect fluid in charged black-hole and naked-singularity spacetimes with a repulsive cosmological constant (  > 0)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)the line element of the spacetimes (the geometric units c = G = 1) dimensionless cosmological parameter and dimensionless charge parameterdimensionless cosmological parameter and dimensionless charge parameter dimensionless coordinatesdimensionless coordinates

3. 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 orbitsseven 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 > r ph+ with unstable then stable and finally unstable geodesics (for radius growing)dS-BH-1 – one region of circular geodesics at r > r ph+ with unstable then stable and finally unstable geodesics (for radius growing) dS-BH-2 – one region of circular geodesics at r > r ph+ with unstable geodesics onlydS-BH-2 – one region of circular geodesics at r > r ph+ with unstable geodesics only

4. 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 geodesicsdS-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 orbitsdS-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 orbitsdS-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 orbitsdS-NS-4 – one region of circular orbits with stable and then unstable orbits dS-NS-5 – no circular orbits alloweddS-NS-5 – no circular orbits allowed

5. Test perfect fluid does not alter the geometrydoes 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  )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 (  and p denote the total energy density and the pressure of the fluid)the stress-energy tensor of the perfect fluid is (  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 lthe rotating fluid can be characterized by the vector fields of the angular velocity , and the angular momentum density l

6. 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 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 conditionthe equipotential surfaces are determined by the condition 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 gradientequilibrium 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 openthe 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

7. Equilibrium configurations the closed equipotential surfaces determine stationary equilibrium configurationsthe 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-zerothe 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 densityconfigurations with uniform distribution of angular momentum density relation for the equipotential surfacesrelation for the equipotential surfaces in Reissner–Nordström–de Sitter spacetimesin Reissner–Nordström–de Sitter spacetimes

8. Behaviour of the equipotential surfaces, and the related potential according to the values ofaccording to the values of region containing stable circular geodesics -> accretion processes in the disk regime are possibleregion containing stable circular geodesics -> accretion processes in the disk regime are possible behaviour of potential in the equatorial plane (  =  /2)behaviour of potential in the equatorial plane (  =  /2) equipotential surfaces - meridional sectionsequipotential surfaces - meridional sections

9. 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 dS-BH-1 : M = 1; e = 0.5; y = 10 -6 l = 3.00 l = 3.55378053 l = 3.75

10. 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 dS-BH-1 : M = 1; e = 0.5; y = 10 -6 l = 3.8136425 l = 4.00 l = 4.96797564

11. 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 dS-BH-1 : M = 1; e = 0.5; y = 10 -6 l = 6.00 l = 7.11001349 l = 10.00

12. 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 dS-NS-1 : M = 1; e = 1.02; y = 0.00001 l = 2.00 l = 3.04327472 l = 3.15

13. 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 dS-NS-1 : M = 1; e = 1.02; y = 0.00001 l = 3.2226824 l = 3.55 l = 3.91484803

14. 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 dS-NS-1 : M = 1; e = 1.02; y = 0.00001 l = 4.40 l = 4.9486708 l = 5.15

15. 10)disk 1 is infinitesimally thin 11)no disks, open equipotential surfaces only dS-NS-1 : M = 1; e = 1.02; y = 0.00001 l = 5.39574484 l = 6.00

16. 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 dS-NS-2 : M = 1; e = 1.02; y = 0.01 l = 4.00 l = 4.25403109 l = 5.00

17. 4)the disk is infinitesimally thin 5)no disk is possible, open equipotential surfaces only dS-NS-2 : M = 1; e = 1.02; y = 0.01 l = 6.40740525 l = 7.00

18. 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 dS-NS-3 : M = 1; e = 1.07; y = 0.0001 l = 2.50 l = 2.93723342 l = 3.00

19. 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 dS-NS-3 : M = 1; e = 1.07; y = 0.0001 l = 3.0411677 l = 3.20 l = 3.42331737

20. 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 dS-NS-3 : M = 1; e = 1.07; y = 0.0001 l = 3.59008126 l = 3.80 l = 3.50

21. 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 dS-NS-4 : M = 1; e = 1.07; y = 0.01 l = 3.00 l = 3.63788074 l = 3.80

22. 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 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 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 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.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.

23. References Z. Stuchlík, S. Hledík. Properties of the Reissner-Nordström spacetimes with a nonzero cosmological constant. Acta Phys. Slovaca, 52(5):363-407, 2002Z. 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, 2000Z. 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 ~