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Vertical stability of circular orbits in relativistic razor-thin disks

Ronaldo S. S. Vieira1,*, Javier Ramos-Caro2,†, and Alberto Saa3,‡

  • 1Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Universidade de São Paulo, 05508-090 São Paulo, SP, Brazil
  • 2Departamento de Física, Universidade Federal de São Carlos, São Carlos, 13565-905 SP, Brazil
  • 3Departamento de Matemática Aplicada, Universidade Estadual de Campinas, Campinas, 13083-859 SP, Brazil

  • *rss.vieira@usp.br
  • †javier@ufscar.br
  • ‡asaa@ime.unicamp.br
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Phys. Rev. D 94, 104016 – Published 7 November, 2016

DOI: https://doi.org/10.1103/PhysRevD.94.104016

Abstract

During the last few decades, there has been a growing interest in exact solutions of Einstein equations describing razor-thin disks. Despite the progress in this area, the analytical study of geodesic motion crossing the disk plane in these systems has not yet been well developed. In the present work, we propose a definite vertical stability criterion for circular equatorial timelike geodesics in static, axially symmetric thin disks, possibly surrounded by other structures preserving axial symmetry. It turns out that the strong energy condition for the disk stress-energy content is sufficient for the vertical stability of these orbits. Moreover, adiabatic invariance of the vertical action variable gives us an approximate third integral of motion for oblique orbits that deviate slightly from the equatorial plane. This new approximate third integral certainly points to a better understanding of the analytical properties of these orbits. The results presented here, derived for static spacetimes, may be a starting point to study the motion around rotating, stationary razor-thin disks. Our results also allow us to conjecture that the strong energy condition should be sufficient to assure transversal stability of periodic orbits for any singular timelike hypersurface, provided it is invariant under the geodesic flow.

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Phys. Rev. D 94, 104016– Published 7 November, 2016
  • Received 11 September 2016

© 2016 American Physical Society

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