Gravitational-wave lensing beyond the diffraction integral: proper-time methods and the bound lens
Abstract
Gravitational waves (GWs) can have wavelengths comparable to the Schwarzschild radius of their lenses, and detectors record their phase, so lensing requires wave optics. Its standard tool, the diffraction integral, follows from the particle path integral of a scalar wave on a static weak field, under the eikonal (one-way) and paraxial approximations, plus the thin-lens limit. We give a self-contained account of how the proper-time (worldline) representation overcomes these steps. An auxiliary proper time makes the wave equation exactly first order, with frequency as 1/ℏ, so geometric optics and, for paraxial thin-lens paths, the diffraction integral follow from stationarity of an exact integral. For a point mass, the Duru-Kleinert (DK) gauge and Kustaanheimo-Stiefel map give the exact propagator; for a distant observer the DK time and fibre angle become the radius and azimuth of a lens plane, the result is the point-lens amplification factor with the offset set by the excess path, and geometric optics requires w≫1 and wy≫1. This agreement reflects the Coulomb potential, not the approximations: the exact eikonal condition, εE≃λ̅/d+GM/c2d+𝒪(sin2θ/2), fails near the lens and at large angles, and the one-way step is a non-relativistic limit. We then treat the bound lens, a GW source orbiting its lens, which emits a backscattered echo and moves during the lensing delays. Its exact time-domain signal gives each worldline its own retarded time and Doppler factor, so no single amplification factor describes it. In the diffraction regime the Shapiro phase alone misses all lensing beyond it, geometric optics misses 46-100% of it near conjunction at w=0.3, the paraxial geometry fails on the far side, and the static amplification factor along the orbit errs by ≈(1−2) wM/R near conjunction and by the echo amplitude elsewhere.
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BibTeX
@misc{braga2026gravitationalwave,
title = {{Gravitational-wave lensing beyond the diffraction integral: proper-time methods and the bound lens}},
author = {Ginevra Braga and Raul Jimenez},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05152},
eprint = {2611.05152},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05152},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gravitational-wave lensing beyond the diffraction integral: proper-time methods and the bound lens
AU - Ginevra Braga
AU - Raul Jimenez
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05152
UR - https://arxiv.org/abs/2611.05152
AB - Gravitational waves (GWs) can have wavelengths comparable to the Schwarzschild radius of their lenses, and detectors record their phase, so lensing requires wave optics. Its standard tool, the diffraction integral, follows from the particle path integral of a scalar wave on a static weak field, under the eikonal (one-way) and paraxial approximations, plus the thin-lens limit. We give a self-contained account of how the proper-time (worldline) representation overcomes these steps. An auxiliary proper time makes the wave equation exactly first order, with frequency as 1/ℏ, so geometric optics and, for paraxial thin-lens paths, the diffraction integral follow from stationarity of an exact integral. For a point mass, the Duru-Kleinert (DK) gauge and Kustaanheimo-Stiefel map give the exact propagator; for a distant observer the DK time and fibre angle become the radius and azimuth of a lens plane, the result is the point-lens amplification factor with the offset set by the excess path, and geometric optics requires w≫1 and wy≫1. This agreement reflects the Coulomb potential, not the approximations: the exact eikonal condition, ε_E≃λ̅/d+GM/c^2d+𝒪(sin^2θ/2), fails near the lens and at large angles, and the one-way step is a non-relativistic limit. We then treat the bound lens, a GW source orbiting its lens, which emits a backscattered echo and moves during the lensing delays. Its exact time-domain signal gives each worldline its own retarded time and Doppler factor, so no single amplification factor describes it. In the diffraction regime the Shapiro phase alone misses all lensing beyond it, geometric optics misses 46-100% of it near conjunction at w=0.3, the paraxial geometry fails on the far side, and the static amplification factor along the orbit errs by ≈(1−2) wM/R near conjunction and by the echo amplitude elsewhere.
ER -