Hana Kučáková , Zdeněk Stuchlík & Petr Slaný

1. Equilibriumconfigurationsofperfect fluid in Reissner-Nordström-(anti-)de Sitterspacetimes Institute of Physics,Faculty of Philosophy and Science, Silesian University atOpava,Bezručovonám. 13, CZ-74601 Opava, Czech Republic Hana Kučáková,Zdeněk Stuchlík& Petr Slaný MG12 Paris, 12-18 July 2009

2. Introduction • investigating equilibrium configurations of perfect fluidin chargedblack-hole and naked-singularity spacetimes witha nonzero cosmological constant (Λ≠0) • the line element of the spacetimes (the geometric units c = G = 1) • dimensionless cosmological parameter and dimensionless charge parameter • dimensionlesscoordinates MG12 Paris, 12-18 July 2009

3. Test perfect fluid • does not alter the geometry • rotating in the ϕ direction – its four velocity vector field Uμ has, therefore, only two nonzero components Uμ = (Ut,0,0 ,Uϕ) • the stress-energy tensor of the perfect fluid is (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 MG12 Paris, 12-18 July 2009

4. Equipotentialsurfaces • 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 • 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 isa special class of critical, self-crossing surfaces (with a cusp), which can be either closed or open MG12 Paris, 12-18 July 2009

5. Equilibriumconfigurations • 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 • relation for the equipotential surfaces • in Reissner–Nordström–(anti-)de Sitterspacetimes MG12 Paris, 12-18 July 2009

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

7. TypesoftheReissner-Nordström-de Sitterspacetimes (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 MG12 Paris, 12-18 July 2009

8. TypesoftheReissner-Nordström-de Sitterspacetimes (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 MG12 Paris, 12-18 July 2009

9. Types of theReissner-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 MG12 Paris, 12-18 July 2009

10. RNdSblack-hole spacetimes open surfacesonly, no disks are possible, surface with the outer cusp exists(M = 1; e = 0.5;y = 10-6; ℓ= 3.00) an infinitesimally thin, unstable ring exists(M = 1; e = 0.5;y = 10-6; ℓ= 3.55378053) 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) MG12 Paris, 12-18 July 2009

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

12. RNdSblack-hole spacetimes an infinitesimally thin, unstable ring exists (the center, and the outer cusp coalesce) (M= 1; e = 0.5;y = 10-6; ℓ =7.11001349) open equipotential surfaces exist only, there is no cusp in this case (M= 1; e = 0.5;y = 10-6; ℓ =10.00) 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) MG12 Paris, 12-18 July 2009

13. RNdSnaked-singularity spacetimes closed surfaces exist, one with the outer cusp, equilibrium configurations are possible (M= 1; e = 1.02;y = 10-5; ℓ =2.00) 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) 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) MG12 Paris, 12-18 July 2009

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

15. RNdSnaked-singularity spacetimes disk 1 exists, also an infinitesimally thin, unstable ring exists (region 2) (M= 1; e = 1.02;y = 10-5; ℓ = 4.9486708) 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) an infinitesimally thin, unstable ring exists (region 1), also disk 2 (M= 1; e = 1.07;y = 10-4;ℓ= 3.42331737) MG12 Paris, 12-18 July 2009

16. RNdSnaked-singularity spacetimes 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) an infinitesimally thin, unstable ring exists (region 2) (M= 1; e = 1.07;y = 10-4; ℓ =3.59008126) no disk, no cusp, open equipotential surfaces only (M= 1; e = 1.07;y = 10-4;ℓ= 3.80) MG12 Paris, 12-18 July 2009

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

18. RNadSblack-hole spacetimes openequipotential surfacesonly, no disks are possible (M= 1; e = 0.99;y= -10-4; ℓ =2.00) an infinitesimally thin unstable ring exists (M= 1; e = 0.99;y = - 10-4; ℓ= 3.10048313) equilibrium configurations are possible, closed equipotentialsurfaces exist, one with the cuspthatenablesaccretionfromthetoroidal disk intotheblackhole (M= 1; e = 0.99;y = - 10-4; ℓ= 3.70) MG12 Paris, 12-18 July 2009

19. RNadSblack-hole spacetimes the potential diverges, the cusp disappears, accretion into the black-hole is impossible, equilibrium configurations are stillpossible, closed equipotentialsurfaces exist (M= 1; e = 0.99;y= -10-4; ℓ =5.00) MG12 Paris, 12-18 July 2009

20. RNadSnaked-singularity spacetimes closed equipotentialsurfaces exist, equilibrium configurations are possible, one disk (1) only (M= 1; e = 0.99;y= -0.4; ℓ =1.30) the center of the second disk (2) appears, one equipotentialsurface with the cuspexists (M= 1; e = 0.99;y = - 0.4; ℓ =1.448272709327) 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) MG12 Paris, 12-18 July 2009

21. RNadSnaked-singularity spacetimes the potential diverges,no equipotentialsurfacewiththe 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) the disk 1 is infinitesimally thin (M= 1; e = 1.07;y = - 10-4; ℓ=3.41935796) MG12 Paris, 12-18 July 2009

22. Conclusions (RNdS) • TheReissner–Nordström–de Sitterspacetimescanbeseparatedintoseventypesofspacetimeswithqualitativelydifferentcharacterofthegeodeticalmotion. In fiveofthemtoroidaldiskscanexist, becausein these spacetimesstablecircularorbitsexist. • The presence ofanoutercuspoftoroidaldisksnearbythe static radiuswhichenablesoutflowofmassandangularmomentumfromtheaccretiondisks by thePaczyńskimechanism, i.e., due to a violationofthehydrostaticequilibrium. • Themotionabovetheouterhorizonofblack-hole backgrounds has thesamecharacter as in theSchwarzschild–de Sitterspacetimesforasymptotically de Sitterspacetimes. Thereisonlyone static radius in these spacetimes. No static radiusispossibleundertheinnerblack-hole horizon, no circulargeodesics are possiblethere. • Themotion in thenaked-singularity backgrounds has similarcharacter as themotion in thefieldofReissner–Nordströmnakedsingularities. However, in the case ofReissner–Nordström–de Sitter, two static radii canexist, whiletheReissner–Nordströmnakedsingularitiescontainone static radiusonly. Theouter static radiusappearsdue to theeffectoftherepulsivecosmologicalconstant. Stablecircularorbitsexist in allofthenaked-singularity spacetimes. There are eventwoseparatedregionsofstablecirculargeodesics in somecases. MG12 Paris, 12-18 July 2009

23. Conclusions (RNadS) • TheReissner–Nordström–anti-de Sitterspacetimescanbeseparatedintofourtypesofspacetimeswithqualitativelydifferentcharacterofthegeodeticalmotion. In allofthemtoroidaldiskscanexist, becausein these spacetimesstablecircularorbitsexist. • Themotionabovetheouterhorizonofblack-hole backgrounds has thesamecharacter as in theSchwarzschild–anti-de Sitterspacetimes. • Themotion in thenaked-singularity backgrounds has similarcharacter as themotion in thefieldofReissner–Nordströmnakedsingularities. Stablecircularorbitsexist in allofthenaked-singularity spacetimes. MG12 Paris, 12-18 July 2009

24. References • Z. Stuchlík, S. Hledík. PropertiesoftheReissner-Nordströmspacetimeswith 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! MG12 Paris, 12-18 July 2009