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Equilibrium configurations of perfect fluid in Reissner-Nordstr ö m-anti-de Sitter spacetimes

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

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Equilibrium configurations of perfect fluid in Reissner-Nordstr ö m-anti-de Sitter spacetimes

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  1. Equilibrium configurationsof perfect fluidin 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 fluidin charged black-hole and naked-singularity spacetimes withan attractive cosmological constant (< 0) • the line element of the spacetimes (the geometric units c = G =1) • dimensionless cosmological parameter and dimensionless charge parameter • dimensionless 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 orbits Black-hole spacetimes • AdS-BH-1 – one region of circular geodesics at r > rph+ 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 <rph-) consists of stable geodesics only, the outer one (r > rph+) contains both unstable and then stable circulargeodesics • AdS-NS-2 – one region of circular orbits, subsequently with stable, then unstableand finallystable orbits • AdS-NS-3 – one region of circular orbits with stable orbits exclusively

  5. 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( 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

  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 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

  7. 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 • relation for the equipotential surfaces • in Reissner–Nordström–(anti-)de Sitter spacetimes

  8. 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 ( = /2) • equipotential surfaces - meridional sections

  9. AdS-BH-1: M = 1; e = 0.99;y = - 0.0001 • open equipotential surfaces only, no disks are possible • an infinitesimally thin unstable ring exists • equilibrium configurations are possible, closed equipotential surfaces exist, one with the cusp that enables accretion from the toroidal disk into the black hole l = 2.00 l = 3.10048313 l = 3.70

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

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

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

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

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

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

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

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

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