https://doi.org/10.1140/epjc/s10052-026-15606-2
Regular Article - Theoretical Physics
Periodic orbits and their gravitational wave radiations in 
-metric
1
Basic Research Center for Energy Interdisciplinary, College of Science, China University of Petroleum-Beijing, 102249, Beijing, China
2
Beijing Key Laboratory of Optical Detection Technology for Oil and Gas, China University of Petroleum-Beijing, 102249, Beijing, China
3
Institute for Theoretical Physics and Cosmology, Zhejiang University of Technology, 310032, Hangzhou, China
4
United Center for Gravitational Wave Physics (UCGWP), Zhejiang University of Technology, 310032, Hangzhou, China
a
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Received:
19
November
2025
Accepted:
22
March
2026
Published online:
7
April
2026
Abstract
The -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 . It reduces to the Schwarzschild metric when 
. In this paper, we explore potential signatures of the -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 
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 
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 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 .
© The Author(s) 2026
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