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Curvature dependence of relativistic epicyclic frequencies in static, axially symmetric spacetimes

Journal article
Authors R. S. S. Vieira
W. Kluzniak
Marek A Abramowicz
Published in Physical Review D
Volume 95
Issue 4
ISSN 1550-7998
Publication year 2017
Published at Department of Physics (GU)
Language en
Links https://doi.org/10.1103/PhysRevD.95...
Keywords maclaurin spheroids, Astronomy & Astrophysics, Physics
Subject categories Physical Sciences, Astronomy, Astrophysics and Cosmology

Abstract

The sum of squared epicyclic frequencies of nearly circular motion (omega(2)(r) + omega(2)(theta).) in axially symmetric configurations of Newtonian gravity is known to depend both on the matter density and on the angular velocity profile of circular orbits. It was recently found that this sum goes to zero at the photon orbits of Schwarzschild and Kerr spacetimes. However, these are the only relativistic configurations for which such a result exists in the literature. Here, we extend the above formalism in order to describe the analogous relation for geodesic motion in arbitrary static, axially symmetric, asymptotically flat solutions of general relativity. The sum of squared epicyclic frequencies is found to vanish at photon radii of vacuum solutions. In the presence of matter, we obtain that omega(2)(r) +omega(2)(theta) > 0 for perturbed timelike circular geodesics on the equatorial plane if the strong energy condition holds for the matter-energy fluid of spacetime; in vacuum, the allowed region for timelike circular geodesic motion is characterized by the inequality above. The results presented here may be of use to shed light on general issues concerning the stability of circular orbits once they approach photon radii, mainly the ones corresponding to stable photon motion. RAMOWICZ MA, 1990, MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, V245, P733

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