Advertisement

Neutral particle trajectory in the Kerr field

  • Enamul HaqueEmail author
  • M. D. I. Bhuyan
Article
  • 3 Downloads

Abstract

In this paper, the neutral particle trajectory around a rotating black hole in the four-dimensional Kerr space–time is studied. The equation of motion is derived by using the Kerr metric within the slowly rotating and weak-field limit. The stability conditions of the orbital motion of the neutral particle are deduced. For small amplitude of motion, the approximate solutions of the resulting differential equations are obtained by the perturbation method of Lindstedt and Poincaré. These are then expressed as power series for some small perturbation parameters. The angular frequencies are also found in the corresponding order of the perturbation. By using angular frequencies, an expression of the perihelion advance is derived. To find the orbital equation similar to Newtonian orbit, the coordinate system is transformed into isotropic coordinates. The Newtonian orbital equation is derived and hence the eccentricity of the orbit is obtained. The resulting expression of circular motion shows the relationship between energy and angular momentum of a spinning particle in the Kerr field.

Keywords

Kerr metric Lindstedt and Poincaré method perihelion advance Boyer Lindiquist coordinates isotropic coordinates 

References

  1. Bardeen J. M., Press W. H., Teukolsky S. A. 1972, Astrophys. J., 178, 347ADSCrossRefGoogle Scholar
  2. Carter B. 1966, Phys. Rev., 141, 1242ADSMathSciNetCrossRefGoogle Scholar
  3. Carter B. 1968, Phys. Rev., 174, 1559ADSCrossRefGoogle Scholar
  4. Carter B. 1971, Phys. Rev. Lett., 26, 331ADSCrossRefGoogle Scholar
  5. Chandrasekhar S. 1998, The Mathematical Theory of Black Holes, vol. 69Google Scholar
  6. Clifford M. W. 2018, Theory and Experiment in Gravitational Physics (Cambridge University Press)zbMATHGoogle Scholar
  7. Damour T., Nagar A., Pollney D., Reisswig C. 2012, Phys. Rev. Lett., 108, 131101ADSCrossRefGoogle Scholar
  8. Darwin C. 1959, Proc. R. Soc. London, 249, 180ADSMathSciNetCrossRefGoogle Scholar
  9. Darwin C. 1961, Proc. R. Soc. London, 263, 39ADSMathSciNetCrossRefGoogle Scholar
  10. Do-Nhat T. 1995, Canadian J. Phys., 73, 608ADSCrossRefGoogle Scholar
  11. Do-Nhat T. 1998, Phys. Lett. A, 238, 328ADSMathSciNetCrossRefGoogle Scholar
  12. Duncombe R. L. 1956, Astron. J., 61, 174ADSCrossRefGoogle Scholar
  13. Einstein A. 1915, Albert Einstein: Akademie-Vorträge: Sitzungsberichte der Preußischen Akademie der Wissenschaften 1914–1932, vol. 831Google Scholar
  14. Graves J. C., Brill D. R. 1960, Phys. Rev., 120, 1507ADSMathSciNetCrossRefGoogle Scholar
  15. Hawking S. W. 1966, Proc. R. Soc. London, Series A: Math. Phys. Sci., 294, 511ADSCrossRefGoogle Scholar
  16. Hawking S. W. 1967, Proc. R. Soc. London, Series A: Math. Phys. Sci., 300, 187ADSCrossRefGoogle Scholar
  17. Hobson M. P., Efstathiou G. P., Lasenby A. N. 2006, General Relativity: An Introduction for Physicists (Cambridge University Press)CrossRefGoogle Scholar
  18. Hoenselaers C. 1976, Prog. Theor. Phys., 56, 324ADSCrossRefGoogle Scholar
  19. Hulse R. A., Taylor J. H. 1975, Neutron stars, black holes, and binary X-ray sources, vol. 48, p. 433Google Scholar
  20. Kerr R. P. 1963, Phys. Rev. Lett., 11, 237ADSMathSciNetCrossRefGoogle Scholar
  21. Marinca V., Herisanu N. 2012, Nonlinear Dynamical Systems in Engineering: Some Approximate Approaches (Springer Science & Business Media)zbMATHGoogle Scholar
  22. Misner C. W., Thorne K. S., Wheeler J. A., Kaiser D. I. 1973, Gravitation (Macmillan)Google Scholar
  23. Morin D. 2008, Introduction to Classical Mechanics: With Problems and Solutions (Cambridge University Press)CrossRefGoogle Scholar
  24. Nayfeh A. H., Mook D. T. 2008, Nonlinear Oscillations (John Wiley & Sons)zbMATHGoogle Scholar
  25. Newman E. T., Janis A. I. 1965, J. Math. Phys., 6, 915ADSCrossRefGoogle Scholar
  26. Penrose R. 1965, Phys. Rev. Lett., 14, 57ADSMathSciNetCrossRefGoogle Scholar
  27. Poincaré H. 1992, New Methods of Celestial Mechanics (Springer Science & Business Media)Google Scholar
  28. Shapiro I. I. et al. 1972, Phys. Rev. Lett., 28, 1594ADSCrossRefGoogle Scholar
  29. Teukolsky S. A. 2015, Class. Quantum Gravit., 32, 124006ADSCrossRefGoogle Scholar
  30. Weisfeld M. 1964, J. Math. Anal. Appl., 8, 282MathSciNetCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of PhysicsMawlana Bhashani Science and Technology UniversitySantoshBangladesh

Personalised recommendations