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

@article{29894,
   author = {Vieira, Ronaldo Savioli Sumé and Kluźniak, Włodek and Abramowicz, Marek},
   article_number = {4},
   doi = {http://dx.doi.org/10.1103/PhysRevD.95.044008},
   keywords = {epicyclic frequencies; circular orbits; geodesic motion},
   language = {eng},
   issn = {2470-0010},
   journal = {Physical Review D},
   title = {Curvature dependence of relativistic epicyclic frequencies in static, axially symmetric spacetimes},
   url = {https://journals.aps.org/prd/abstract/10.1103/PhysRevD.95.044008},
   volume = {95},
   year = {2017}
}

TY  - JOUR
ID  - 29894
AU  - Vieira, Ronaldo Savioli Sumé - Kluźniak, Włodek - Abramowicz, Marek
PY  - 2017
TI  - Curvature dependence of relativistic epicyclic frequencies in static, axially symmetric spacetimes
JF  - Physical Review D
VL  - 95
IS  - 4
SP  - „044008-1“-„044008-6“
EP  - „044008-1“-„044008-6“
SN  - 24700010
KW  - epicyclic frequencies
KW  - circular orbits
KW  - geodesic motion
UR  - https://journals.aps.org/prd/abstract/10.1103/PhysRevD.95.044008
L2  - https://journals.aps.org/prd/abstract/10.1103/PhysRevD.95.044008
N2  - 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.
ER  - 

VIEIRA, Ronaldo Savioli Sumé, Włodek KLU$\backslash$'ZNIAK and Marek ABRAMOWICZ. Curvature dependence of relativistic epicyclic frequencies in static, axially symmetric spacetimes. \textit{Physical Review D}. 2017, vol.~95, No~4, p.~„044008-1“-„044008-6“, 6 pp. ISSN~2470-0010. Available from: https://dx.doi.org/10.1103/PhysRevD.95.044008.