Original article
Timelike and null equatorial geodesics in the Bonnor-Sackfield relativistic disk
Published 2011-01-31
Keywords
- General relativity,
- exact solutions,
- equations of motion
How to Cite
González, G. A., & López-Suspes, F. (2011). Timelike and null equatorial geodesics in the Bonnor-Sackfield relativistic disk. Revista Integración, Temas De matemáticas, 29(1), 59–72. Retrieved from https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/2430
Abstract
A study of timelike and null equatorial geodesics in the Bonnor-Sackfield relativistic thin disk is presented. The motion of test particles in the equatorial plane is analyzed, both for the newtonian thin disk modelas for the corresponding relativistic disk. The nature of the possible orbits is studied by means of a qualitative analysis of the effective potential and by numerically solving the motion equation for radial and non-radial equatorial trajectories. The existence of stable, unstable and marginally stable circular orbits is analyzed, both for the newtonian and relativistic case. Examples of the numerical results, obtained with some simple values of the parameters, are presented.
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References
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[2] Bi˘cák J. and Ledvinka T., “Relativistic disks as sources of the Kerr metric”, Phys. Rev. Lett. 71 (1993), no. 11, 1669–1672.
[3] Bi˘cák J., Lynden-Bell D. and Katz J., “Relativistic disks as sources of static vacuum spacetimes”, Phys. Rev. D 47 (1993), 4334–4343.
[4] Bi˘cák J., Lynden-Bell D. and Pichon C., “Relativistic discs and flat galaxy models”, Monthly Notice Roy Astronom. Soc. 265 (1993), no. 1, 126–144.
[5] Bonnor W.A. and Sackfield A., “The interpretation of some spheroidal metrics”, Comm. Math. Phys. 8 (1968), no. 4, 338–344.
[6] D’Afonseca L.A., Letelier P.S., and Oliveira S.R., “Geodesics around Weyl-Bach’s ring solution”, Class. Quantum Grav. 22 (2005), no. 17, 3803–3814.
[7] González G. and Espitia O.A., “Relativistic static thin disks: The counterrotating model”, Phys. Rev. D 68 (2003), no. 10, 104028-1 – 104028-8.
[8] González G. and Letelier P.S., “Rotating relativistic thin disks”, Phys. Rev. D 62 (2000), 064025-1 – 064025-8.
[9] González G. and Letelier P.S., “Relativistic static thin discs with radial stress support”, Class. Quantum Grav. 16 (1999), no. 2, 479–494.
[10] Kramer D., Stephani H., Herlt E. and McCallum M., Exact Solutions of Einsteins’s Field Equations, Cambridge University Press, Cambridge-England, 2000
[11] Lemos J.P.S., “Self-similar relativistic discs with pressure”, Class. Quantum Grav. 6 (1989), no. 9, 1219–1230.
[12] Lemos J.P.S. and Letelier P.S., “Superposition of Morgan and Morgan discs with a Schwarzschild black hole”, Class. Quantum Grav. 10 (1993), no. 6, L75–L78.
[13] Lemos J.P.S. and Letelier P.S., “Exact general relativistic thin disks around black holes”, Phys. Rev. D 49 (1994), 5135–5143.
[14] Lemos J.P.S. and Letelier P.S., “Two families of exact disks with a central black hole”, Int. J. Modern Phys. D 5 (1996), no. 1, 53–63.
[15] Letelier P.S. and Oliveira S.R., “Exact selfgravitating disks and rings: a solitonic approach”, J. Math. Phys. 28 (1987), no. 1, 165–170.
[16] Lynden-Bell D. and Pineault S., “Relativistic disks - 1. Counter rotating disks”, Monthly Notice Roy Astronom. Soc. 185 (1978), 679–694.
[17] Lynden-Bell D. and Pineault S., “Relativistic disks - 11. Self-similar disks in rotation”, Monthly Notice Roy Astronom. Soc. 185 (1978), 695–712.
[18] Morgan T. and Morgan L., “The Gravitational Field of a Disk”, Phys. Rev. 183 (1969), 1097–1101.
[19] Morgan L. and Morgan T., “Gravitational Field of Shells and Disks in General Relativity”, Phys. Rev. D 2 (1970), 2756–2761.
[20] Morse P.M. and Feshbach H., Methods of Theoretical Physics, McGraw-Hill, New York, 1953.
[21] Semerák O., “Gravitating discs around a Schwarzschild black hole. III”, Class. Quantum Grav. 20 (2003), no. 9, 1613–1634.
[22] Semerák O. and Zá˘cek M., “Oscillations of static discs around Schwarzsch ˘ ild black holes: Effect of self-gravitation”, Publ. Astron. Soc. Japan 52 (2000), 1067–1074.
[23] Semerák O., Zá˘cek M. and Zellerin T., “The structure of superposed Weyl ˘ fields”, Monthly Notice Roy Astronom. Soc. 308 (1999), 691–704.
[24] Voorhees B.H., “Static Axially Symmetric Gravitational Fields”, Phys. Rev. D 2 (1970), 2119–2122.
[25] Weyl H., “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter”, Ann. Physik 54 (1917), 117–145.
[26] Weyl H., “Bemerkung über die axialsymmetrischen Lösungen der Einsteinschen Gravitationsgleichungen”, Ann. Physik 59 (1919), 185–189.
[27] Zipoy D.M., “Topology of Some Spheroidal Metrics”, J. Math. Phys. 7 (1966), 1137–1143