1. Equilibrium configurations of perfect fluid in Reissner-Nordström-anti-de Sitter spacetimes Hana Kučáková, Zdeněk Stuchlík, Petr Slaný Institute of Physics, Silesian University at Opava RAGtime 10 15.-17. September 2008, Opava

2. Introduction investigating equilibrium configurations of perfect fluid in charged black-hole and naked-singularity spacetimes with an attractive cosmological constant (  < 0)investigating equilibrium configurations of perfect fluid in charged black-hole and naked-singularity spacetimes with an attractive 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-anti-de Sitter spacetimes four types with qualitatively different behavior of the effective potential of the geodetical motion and the circular orbitsfour 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 > r ph+ with unstable and then stable geodesics (for radius growing)AdS-BH-1 – one region of circular geodesics at r > r ph+ with unstable and then stable geodesics (for radius growing)

4. Types of the Reissner-Nordström-anti-de Sitter spacetimes Naked-singularity spacetimes AdS-NS-1 – two regions of circular geodesics, the inner one (r r ph+ ) contains both unstable and then stable circular geodesicsAdS-NS-1 – two regions of circular geodesics, the inner one (r r ph+ ) contains both unstable and then stable circular geodesics AdS-NS-2 – one region of circular orbits, subsequently with stable, then unstable and finally stable orbitsAdS-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 exclusivelyAdS-NS-3 – one region of circular orbits with stable orbits exclusively

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–(anti-)de Sitter spacetimesin Reissner–Nordström–(anti-)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 equipotential surfaces only, no disks are possible 2)an infinitesimally thin unstable ring exists 3)equilibrium configurations are possible, closed equipotential surfaces exist, one with the cusp that enables accretion from the toroidal disk into the black hole AdS-BH-1 : M = 1; e = 0.99; y = - 0.0001 l = 2.00 l = 3.10048313 l = 3.70

10. 4)the potential diverges, the cusp disappears, accretion into the black-hole is impossible 5)like in the previous case, equilibrium configurations are still possible, closed equipotential surfaces exist AdS-BH-1 : M = 1; e = 0.99; y = - 0.0001 l = 4.03557287 l = 5.00

11. 1)closed equipotential surfaces exist, equilibrium configurations are possible, one disk (1) only 2)the center of the second disk (2) appears, one equipotential surface with the cusp exists 3)the flow between the inner disk (1) and the outer one (2) is possible AdS-NS-1 : M = 1; e = 0.99; y = - 0.4 l = 1.30 l = 1.448272709327 l = 1.465

12. 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 5)like in the previous case, two separated disks exist 6)the disk 2 is infinitesimally thin AdS-NS-1 : M = 1; e = 0.99; y = - 0.4 l = 1.47233576 l = 1.50 l = 1.58113883

13. 7)the disk 1 exists only, equilibrium configurations are still possible, closed equipotential surfaces exist AdS-NS-1 : M = 1; e = 0.99; y = - 0.4 l = 1.60

14. 1)closed equipotential surfaces exist, equilibrium configurations are possible, one disk (1) only 2)the center of the second disk (2) appears, one equipotential surface with the cusp exists 3)the flow between the inner disk (1) and the outer one (2) is possible AdS-NS-2 : M = 1; e = 1.07; y = - 0.0001 l = 2.00 l = 2.94183736 l = 3.10

15. 4)the same values of the potential in the centers of both disks 5)the flow between the inner disk (1) and the outer one (2) is possible 6)the disk 1 is infinitesimally thin AdS-NS-2 : M = 1; e = 1.07; y = - 0.0001 l = 3.2181567 l = 3.41935796 l = 3.30

16. 7)the disk 2 exists only, equilibrium configurations are still possible, closed equipotential surfaces exist AdS-NS-2 : M = 1; e = 1.07; y = - 0.0001 l = 4.00

17. 1)there is only one center and one disk in this case, closed equipotential surfaces exist, equilibrium configurations are possible AdS-NS-3 : M = 1; e = 1.1; y = - 0.03 l = 3.00

18. Conclusions 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 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 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.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.

19. 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 ~