Periodic orbits and their gravitational wave radiations in $\gamma$-metric

TL;DR

This paper investigates how the $oldsymbol{\gamma}$-metric affects periodic orbits and their gravitational wave signatures, revealing that deviations from spherical symmetry can significantly alter waveforms and potentially constrain spacetime parameters.

Contribution

It analyzes the impact of the $oldsymbol{\gamma}$-metric on orbit classifications and gravitational waveforms, highlighting observable effects of deviations from the Schwarzschild solution.

Findings

Deviations from $oldsymbol{\gamma=1}$ alter orbit radii and angular momentum.

Waveforms exhibit phase shifts and amplitude modulations due to $oldsymbol{\gamma eq 1}$.

Zoom-whirl structures in waveforms become more complex with larger zoom numbers.

Abstract

The $\gamma$-metric, also known as Zipoy-Voorhees spacetime, is a static, axially symmetric vacuum solution to Einstein's field equations characterized by two parameters: mass and the deformation parameter $\gamma$. It reduces to the Schwarzschild metric when $\gamma = 1$. In this paper, we explore potential signatures of the $\gamma$-metric on periodic orbits and their gravitational-wave radiation. Periodic orbits are classified by a rotational number specified by three topological numbers $(z, w, v)$, each triple corresponding to characteristic zoom-whirl behavior. We show that deviations from $\gamma=1$ alter the radii and angular momentum of bound orbits and thereby shift the $(z, w, v)$ taxonomy. We also compute representative gravitational waveforms for certain periodic orbits and demonstrate that $\gamma \neq 1$ can induce phase shifts and amplitude modulations correlated with changes in the zoom-whirl structure. In particular, larger zoom numbers lead to increasingly complex substructures in the waveforms, and finite deviations from $\gamma=1$ can significantly modify these features. Our results indicate that precise measurements of waveform morphology from extreme-mass-ratio inspirals may constrain deviations from spherical symmetry encoded in $\gamma$.