Number of revolutions of a particle around a black hole: Is it infinite or finite?

  • Yuri V.  Pavlov
  • Oleg B. ZaslavskiiEmail author
Research Article


We consider a particle falling into a rotating black hole. Such a particle makes an infinite number of revolutions n from the viewpoint of a remote observer who uses the Boyer–Lindquist type of coordinates. We examine the behavior of n when it is measured with respect to a local reference frame that also rotates due to dragging effect of spacetime. The crucial point consists here in the observation that for a nonextremal black hole, the leading contributions to n from a particle itself and the reference frame have the same form being in fact universal, so that divergences mutually cancel. As a result, the relative number of revolutions turns out to be finite. For the extremal black hole this is not so, n can be infinite. Different choices of the local reference frame are considered, the results turn out to be the same qualitatively. For illustration, we discuss two explicit examples—rotation in the flat spacetime and in the Kerr metric.


Black holes Kerr metric Rotating Frames 



This work was supported by the Russian Government Program of Competitive Growth of Kazan Federal University. The work of Yu. P. was supported also by the Russian Foundation for Basic Research, Grant No. 15-02-06818-a. The work of O. Z. was also supported by SFFR, Ukraine, Project No. 32367.


  1. 1.
    Kerr, R.P.: Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11, 237–238 (1963)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Boyer, R.H., Lindquist, R.W.: Maximal analytic extension of the Kerr metric. J. Math. Phys. 8, 265–281 (1967)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chandrasekhar, S.: The Mathematical Theory of Black Holes. Oxford University Press, Oxford (1983)zbMATHGoogle Scholar
  4. 4.
    Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields. Pergamon Press, Oxford (1983)zbMATHGoogle Scholar
  5. 5.
    Bañados, M., Silk, J., West, S.M.: Kerr black holes as particle accelerators to arbitrarily high energy. Phys. Rev. Lett. 103, 111102 (2009)ADSCrossRefGoogle Scholar
  6. 6.
    Zaslavskii, O.B.: Acceleration of particles as universal property of rotating black holes. Phys. Rev. D 82, 083004 (2010)ADSCrossRefGoogle Scholar
  7. 7.
    Grib, A.A., Pavlov, Y.V.: On particle collisions in the gravitational field of the Kerr black hole. Astropart. Phys. 34, 581–586 (2011)ADSCrossRefGoogle Scholar
  8. 8.
    Zaslavskii, O.B.: Acceleration of particles by black holes: kinematic explanation. Phys. Rev. D 84, 024007 (2011)ADSCrossRefGoogle Scholar
  9. 9.
    Grib, A.A., Pavlov, Y.V.: Comparison of particle properties in Kerr metric and in rotating coordinates. Gen. Relativ. Gravit. 49, 78 (2017)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Dolan, C.: New form of the Kerr solution. Phys. Rev. D 61, 067503 (2000)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Natário, J.: Painlevé–Gullstrand coordinates for the Kerr solution. Gen. Relativ. Gravit. 41, 2579–2586 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Zaslavskii, O.B.: Near-horizon circular orbits and extremal limit for dirty rotating black holes. Phys. Rev. D 92, 044017 (2015)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Tanatarov, I.V., Zaslavskii, O.B.: Dirty rotating black holes: regularity conditions on stationary horizons. Phys. Rev. D 86, 044019 (2012)ADSCrossRefGoogle Scholar
  14. 14.
    Frolov, V.P., Novikov, I.D.: Black Hole Physics: Basic Concepts and New Developments. Kluwer Academic Publishers, Dordrecht (1998)CrossRefzbMATHGoogle Scholar
  15. 15.
    Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. Freeman, San Francisco (1973)Google Scholar
  16. 16.
    Bardeen, J.M., Press, W.H., Teukolsky, S.A.: Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radiation. Astrophys. J. 178, 347–369 (1972)ADSCrossRefGoogle Scholar
  17. 17.
    Thorne, K.S.: Disk-accretion onto a black hole. II. Evolution of the hole. Astrophys. J. 191, 507–519 (1974)ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Institute of Problems in Mechanical EngineeringRussian Academy of SciencesSt. PetersburgRussia
  2. 2.N.I. Lobachevsky Institute of Mathematics and MechanicsKazan Federal UniversityKazanRussia
  3. 3.Department of Physics and TechnologyKharkov V.N. Karazin National UniversityKharkivUkraine

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