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Harmonic oscillations of neutral particles in the $\gamma$-metric
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We consider a well-known static, axially symmetric, vacuum solution of Einstein equations belonging to Weyl's class and determine the fundamental frequencies of small harmonic oscillations of test particles around stable circular orbits in the equatorial plane. We discuss the radial profiles of frequencies of the radial, latitudinal (vertical), and azimuthal (Keplerian) harmonic oscillations relative to the comoving and distant observers and compare with the corresponding ones in the Schwarzschild and Kerr geometries. We show that there exist latitudinal and radial frequencies of harmonic oscillations of particles moving along the circular orbits for which it is impossible to determine whether the central gravitating object is described by the slowly rotating Kerr solution or by a slightly deformed static space-time.
Forward citations
Cited by 2 Pith papers
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Newtonian Shirokov Effect: Epicyclic Frequency Splitting from Mass Multipoles
The Newtonian quadrupole and octupole split epicyclic frequencies (Ωθ²−Ωr²=6GMJ2R²/r0⁵, ω+²−ω−²≈6G|O|/r0⁶) but the dipole does not, establishing a selection rule for multipoles.
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Constraining quadrupole deformations with relativistic effects
In the Zipoy-Voorhees spacetime, the Shapiro time delay and the Shirokov oscillation frequencies acquire corrections from the quadrupole deformation parameter q, with the delay correction appearing at first order in q.
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