1. PSEUDO-NEWTONIAN TOROIDAL STRUCTURES IN SCHWARZSCHILD-DE SITTER SPACETIMES Jiří Kovář Zdeněk Stuchlík & Petr Slaný Institute of Physics Silesian University in Opava Czech Republic Hradec nad Moravicí, September, 2007 This work was supported by the Czech grant MSM 4781305903

2. Introduction Discription of gravity Newtonian > Newtonian gravitational potential (force) General relativistic > curvature of spacetime (geodesic equation) Pseudo-Newtonian > pseudo - Newtonian gravitational potential (force) Schwarzschild-de Sitter spacetime Geodesic motion [Stu-Kov, Inter. Jour. of Mod. Phys. D, in print] Toroidal perfect fluid structures [Stu-Kov-Sla, in preparation for CQG]

3. Introduction Newtonian central GF Poisson equationGravitational potential r-equation of motionEffective potential

4. Einstein’s equations Line element r-equation of motion Effective potential Introduction Relativistic central GF

5. Gravitational potential Paczynski-Wiita r-equation of motion Effective potential Introduction Pseudo-Newtonian central GF

6. Schwarzschild-de Sitter geometry Line element

7. Schwarzschild-de Sitter geometry Equatorial plane

8. Schwarzschild-de Sitter geometry Embedding diagrams Schwarzschild Schwarzschild-de Sitter

9. Schwarzschild-de Sitter geometry Geodesic motion horizons marginally bound (mb) marginally stable (ms)

10. Pseudo-Newtonian approach Potential definition Potential and intensity Intensity and gravitational force

11. Pseudo-Newtonian approach Gravitational potential NewtonianRelativistic Pseudo-Newtonian y=0, P-W potential

12. Pseudo-Newtonian approach Geodesic motion RelativisticPseudo-Newtonian

13. Pseudo-Newtonian approach Geodesic motion exact determination of - horizons - static radius - marginally stable circular orbits - marginally bound circular orbits small differences when determining - effective potential (energy) barriers - positions of circular orbits

14. Relativistic approach Toroidal structures Perfect fluidEuler equation Potential Integration (Boyer’s condition)

15. Pseudo-Newtonian approach Toroidal structures Euler equation Potential Integration

16. Shape of structure Comparison

17. Mass of structure Comparison Pseudo-Newtonian mass Relativistic mass Polytrop – non-relativistic limit

18. Adiabatic index y=10 -6 y=10 -10 y=10 -28  =5/3 9.5x10 -25 9.9x10 -25 4.0x10 -23 3.9x10 -23 3.6x10 -14 3.5x10 -14  =3/2 1.8x10 -24 2.3x10 -22 2.2x10 -22 1.9x10 -10 1.8x10 -10  =7/5 2.8x10 -24 3.0x10 -24 9.4x10 -22 9.1x10 -22 5.3x10 -7 5.0x10 -7 Central density of structure Comparison

19. exact determination of - cusps of tori - equipressure surfaces small differences when determining - potential (energy) barriers - mass and central densities of structures Pseudo-Newtonian approach Toroidal structures

20. GRPN Fundamental Easy and intuitive Precise Approximative for some problems Approximative for other problems Conclusion

21. NewtonianRelativistic Footnote Pseudo-Newtonian definition

22. Relativistic potential Newtonian potential Shape of structure Newtonian potential

23. Thank you Acknowledgement To all the authors of the papers which our study was based on To you