PSEUDO-NEWTONIAN TOROIDAL STRUCTURESIN SCHWARZSCHILD-DE SITTER SPACETIMES

Jiří KovářZdeněk Stuchlík & Petr Slaný

Institute of PhysicsSilesian University in Opava

Czech Republic

Hradec nad Moravicí, September, 2007

This work was supported by the Czech grant MSM 4781305903

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 equation

Gravitational potential

r-equation of motion Effective potential

ikikikik kTgRgR 21

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

Newtonian Relativistic

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 fluid Euler 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=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 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

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

GR PN

Fundamental

Easy and intuitive

Precise

Approximative for some problems

Approximative for other problems

Conclusion

Newtonian Relativistic

Footnote Pseudo-Newtonian definition

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