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

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 $\gamma$.