Gaussian coordinate systems for the Kerr metric

M. Novello and E. Bittencourt1

Abstract

We present a class of Gaussian coordinate systems for the Kerr metric obtained from the relativistic Hamilton-Jacobi equation. We discuss the Cauchy problem of such a coordinate system. In the appendix, we present the JEK (Jordan-Ehlers-Kundt) formulation of General Relativity - the so-called quasi-Maxwell equations - which acquires a simpler form in the Gaussian coordinate system. We show how this set of equations can be used to regain the internal metric of the Schwarzschild solution and, with this in mind, we suggest a possible way to find out a physically significant internal solution for the Kerr metric.

References

  1. A. Burinskii, E. Elizalde, S. R. Hildebrandt and G. Magli, Regular Sources of the Kerr-Schild Class for Rotating and Nonrotating Black Hole Solutions, Phys. Rev. D 65, 064039 (2002).
  2. I. Dymnikova, Spinning Superconducting Electrovacuum Soliton, Phys. Lett. B 639, 369 (2006).
  3. A. Burinskii, Regularized Kerr-Newman Solution as a Gravitating Soliton, J. Phys. A: Math. Theor. 43 (2010).
  4. J. M. Salim, Ph. D thesis, CBPF/Rio de Janeiro (1982); M. Novello et al., Minimal Closed Set of Observables in the Theory of Cosmological Perturbations, Phys. Rev. D 51, 450 (1995).
  5. A. Lichnerowicz, Ondes et Radiations Electromagnetiques et Gravitationelles en Relativite Generale, Ann. Mat. Pure et Appl. 50, 1 (1960).
  6. S. W. Hawking, The Chronology Protection Conjecture, Phys. Rev. D 46, 603 (1992).
  7. W. Rindler, Relativity: Special, General, and Cosmological (Oxford University Press, 2001); C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, Sec. 6.6 (W. H. Freeman, San Francisco, 1973).
  8. S. Hawking and G. F. R. Ellis, The Large-Scale Structure of Space-Time (Cambridge University Press, Great Britain, 1973).
  9. T. Padmanabhan, Gravity and the Thermodynamics of Horizons, gr-qc/0311036; V. Belinski, A. Helfer, and N. F. Svaiter, Round Table at the XIV Brazilian School of Cosmology and Gravitation (2010), in preparation.
  10. E. Newman et al., Metric of a Rotating, Charged Mass, J. Math. Phys. 6, 918 (1965).
  11. S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York, 1983).
  12. M. Visser, The Kerr Spacetime: A Brief Introduction, arXiv: 0706.0622.
  13. B. Carter, Global Structure of the Kerr Family of Gravitational Fields, Phys. Rev. 174, 1559 (1968).
  14. R. Kerr, Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics, Phys. Rev. Lett. 11, 237 (1963).
  15. R. Adler, M. Bazin, and M. Schiffer, Introduction to General Relativity, Chap. 6 (McGraw-Hill, New York, 1975).
  16. S. Dai and C. Guan, Maximally Symmetric Subspace Decomposition of the Schwarzschild Black Hole, gr-qc/0406109.
  17. M. Novello, N. F. Svaiter, and M. E. X. Guimar aes, Synchronized Frames for Godel's Universe, Gen. Rel. Grav. 25, 137 (1993).
  18. E. Bittencourt, Gaussian coordinate systems for the Schwarzschild and Kerr Metrics, Master's Dissertation, CBPF/Rio de Janeiro (2009), in Portuguese; M. Novello, Gaussian Coordinate Systems, unpublished (1982).
  19. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, USA (1989)
  20. H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics, 3rd ed. (Addison Wesley, San Francisco, 2001)
  21. V. A. Belinski, I. M. Khalatnikov, and E. M. Lifshitz, A General Solution of the Einstein Equations with a Time Singularity, Adv. Phys. 31, 639 (1982).
  22. E. M. Lifshitz and I. M. Khalatnikov, Investigations in Relativistic Cosmology, Adv. Phys. 12, 185 (1963).
  23. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, p. 826, (Freeman, San Francisco (1973); in details: I. D. Novikov, Doctoral Dissertation, (Shternberg Astronomical Institute, Moscow, 1963).
  24. A. Friedmann, Uber die Krummung des Raumes, Zeitschrift fur Physik 10, 377 (1922); A. Friedmann, On the Curvature of Space, Gen. Rel. Grav. 31, 1991 (1999).
  25. E. Kasner, Geometrical Theorems on Einstein's Cosmological Equations, Am. J. Math. 43, 217 (1921).
  26. L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields, (Butterworth-Heinemann, London, 1975).
For more information about this paper please visit Springer's Home Page of this paper.



Back to The Contents Page