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

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]

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

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

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

Schwarzschild-de Sitter geometry Line element

Schwarzschild-de Sitter geometry Equatorial plane

Schwarzschild-de Sitter geometry Embedding diagrams Schwarzschild Schwarzschild-de Sitter

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

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

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

Pseudo-Newtonian approach Geodesic motion RelativisticPseudo-Newtonian

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

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

Pseudo-Newtonian approach Toroidal structures Euler equation Potential Integration

Shape of structure Comparison

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

Adiabatic index y=10 -6 y= y=  =5/3 9.5x x x x x x  =3/2 1.8x x x x x  =7/5 2.8x x x x x x10 -7 Central density of structure Comparison

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

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

NewtonianRelativistic Footnote Pseudo-Newtonian definition

Relativistic potential Newtonian potential Shape of structure Newtonian potential

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