1. Equilibrium configurations of perfect fluid in Reissner-Nordström-(anti-)de Sitter spacetimes Hana Kučáková, Zdeněk Stuchlík & Petr Slaný MG12 Paris, 12-18 July 2009 Institute of Physics, Faculty of Philosophy and Science, Silesian University at Opava, Bezručovo nám. 13, CZ-746 01 Opava, Czech Republic

2. Introduction investigating equilibrium configurations of perfect fluid in charged black-hole and naked-singularity spacetimes with a nonzero cosmological constant (Λ ≠ 0) 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) the line element of the spacetimes (the geometric units c = G = 1) dimensionless cosmological parameter and dimensionless charge parameter dimensionless cosmological parameter and dimensionless charge parameter dimensionless coordinates dimensionless coordinates MG12 Paris, 12-18 July 2009 2

3. Test perfect fluid does not alter the geometry 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 ) 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 the stress-energy tensor of the perfect fluid is ( ɛ and p denote the total energy density and the pressure of the fluid) ( 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 the rotating fluid can be characterized by the vector fields of the angular velocity Ω, and the angular momentum density ℓ MG12 Paris, 12-18 July 2009 3

4. 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 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 the 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 gradient 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 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 MG12 Paris, 12-18 July 2009 4

5. Equilibrium configurations the closed equipotential surfaces determine stationary 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 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 configurations with uniform distribution of angular momentum density relation for the equipotential surfaces relation for the equipotential surfaces in Reissner–Nordström–(anti-)de Sitter spacetimes in Reissner–Nordström–(anti-)de Sitter spacetimes MG12 Paris, 12-18 July 2009 5

6. Behaviour of the equipotential surfaces, and the related potential according to the values of according to the values of region containing stable circular geodesics → accretion processes in the disk regime are possible region containing stable circular geodesics → accretion processes in the disk regime are possible behaviour of potential in the equatorial plane (θ = π/2) behaviour of potential in the equatorial plane (θ = π/2) equipotential surfaces - meridional sections equipotential surfaces - meridional sections MG12 Paris, 12-18 July 2009 6

7. 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 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 > 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 only dS-BH-2 – one region of circular geodesics at r > r ph+ with unstable geodesics only MG12 Paris, 12-18 July 2009 7

8. 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-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-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-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-4 – one region of circular orbits with stable and then unstable orbits dS-NS-5 – no circular orbits allowed dS-NS-5 – no circular orbits allowed MG12 Paris, 12-18 July 2009 8

9. 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 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 > 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) Naked-singularity spacetimes AdS-NS-1 – two regions of circular geodesics, the inner one (r r ph+ ) contains both unstable and then stable circular geodesics AdS-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 orbits 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 AdS-NS-3 – one region of circular orbits with stable orbits exclusively MG12 Paris, 12-18 July 2009 9

10. RNdS black-hole spacetimes MG12 Paris, 12-18 July 2009 10 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)

11. RNdS black-hole spacetimes MG12 Paris, 12-18 July 2009 11 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)

12. RNdS black-hole spacetimes MG12 Paris, 12-18 July 2009 12 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)

13. RNdS naked-singularity spacetimes MG12 Paris, 12-18 July 2009 13 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)

14. RNdS naked-singularity spacetimes MG12 Paris, 12-18 July 2009 14 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)

15. RNdS naked-singularity spacetimes MG12 Paris, 12-18 July 2009 15 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)

16. RNdS naked-singularity spacetimes MG12 Paris, 12-18 July 2009 16 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)

17. RNdS naked-singularity spacetimes MG12 Paris, 12-18 July 2009 17 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)

18. RNadS black-hole spacetimes MG12 Paris, 12-18 July 2009 18 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)

19. RNadS black-hole spacetimes MG12 Paris, 12-18 July 2009 19 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)

20. RNadS naked-singularity spacetimes MG12 Paris, 12-18 July 2009 20 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)

21. RNadS naked-singularity spacetimes MG12 Paris, 12-18 July 2009 21 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)

22. 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 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. MG12 Paris, 12-18 July 2009 22

23. 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 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. MG12 Paris, 12-18 July 2009 23

24. 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, 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 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 MG12 Paris, 12-18 July 2009 24 Thank you for your attention!