# Harmonic oscillations of neutral particles in the $\gamma$-metric

## TL;DR

This paper analyzes the harmonic oscillations of test particles in the $b3$-metric, a static axially symmetric solution of Einstein's equations, comparing their frequencies with those in Schwarzschild and Kerr geometries.

## Contribution

It determines the fundamental frequencies of small harmonic oscillations around stable circular orbits in the $b3$-metric and compares them with known solutions, revealing indistinguishability in certain frequency regimes.

## Key findings

- Identifies radial, latitudinal, and azimuthal oscillation frequencies in the $b3$-metric.
- Shows that some frequencies are indistinguishable from Kerr or Schwarzschild solutions.
- Highlights limitations in differentiating central objects based on oscillation frequencies.

## Abstract

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.