Periodic orbits and their gravitational wave radiations in $γ$-metric

TL;DR

This work investigates equatorial geodesics and gravitational-wave emission in the γ-metric (Zipoy-Voorhees) by treating the central object as a BH mimicker with mass parameter $m$ and deformation parameter $\gamma$, where $M=\gamma m$ and $\gamma=1$ recovers Schwarzschild. Equatorial bound orbits are analyzed through the effective potential $V_{\text{eff}} = F\left[1+ \frac{L^2 F}{r(r-2m)}\right]$, yielding numerically the MBO and ISCO radii and their angular momenta, both increasing with $\gamma$. Periodic orbits are classified by the triplet $(z,w,v)$ via the rational ratio $q = \omega_\phi/\omega_r - 1 = w + v/z$, with $L$ interpolated between $L_{\text{isco}}$ and $L_{\text{mbo}}$, and morphology shown to depend sensitively on $\gamma$ and on the zoom number $z$. Gravitational waves are computed under the adiabatic approximation using the quadrupole formula, yielding $h_{ij} = \frac{2 m_\star}{D_L}(a_i x_j + a_j x_i + 2 v_i v_j)$ and the detector-frame polarizations $h_+$ and $h_\times$, which exhibit phase shifts and amplitude modulations as $\gamma$ deviates from 1 and as $z$ increases. These results suggest that precise EMRI waveform morphologies could constrain deviations from spherical symmetry encoded in $\gamma$ with future space-based detectors.

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 != 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.

Figures (11)

• Figure 1: The behaviors of dimensionless quantities $F$, $G$ and $m^{-2}H$ as functions of $r/m$. For the upper panel we choose $\gamma=0.5$ while for the lower panel we choose $\gamma=1.5$. The function $F$ for the Schwarzschild case (denoted as $F_{\text{Sch}}$) is also exhibited in here as a comparison.
• Figure 2: The behaviors of $V_{\text{eff}}$ as a function of the dimensionless variable $r/m$ by choosing different values for $L$. For the upper panel, we set $\gamma=0.5$ while for the lower panel we set $\gamma=1.5$.
• Figure 3: The behaviors of dimensionless quantities $L_{\text{mbo}}/m$ and $r_{\text{mbo}}/m$ for a changing $\gamma$.
• Figure 4: The behaviors of dimensionless quantities $E_{\text{isco}}$, $L_{\text{isco}}/m$ and $r_{\text{isco}}/m$ for a changing $\gamma$.
• Figure 5: The relation between the rational number $q$ and the energy $E$ for different choices of $\epsilon$ [cf. \ref{['Lexpression']}] and $\gamma$. Notice that, for those curves in each panel we have $\gamma=0.5, 0.6, 0.8, 1.2$ respectively from left to right (red, orange, blue and purple curve, respectively).