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
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