Gravitational multipoles, antipodal matching relations, and the logarithmic soft graviton theorem
Abstract
We derive the classical logarithmic soft graviton theorem for the scattering of massive particles in four-dimensional asymptotically flat spacetime using a position-space analysis of the Weyl tensor. Starting from the Iyer-Damour multipole solution in harmonic gauge, we obtain an infinite tower of antipodal matching relations across spatial infinity for all five Newman-Penrose Weyl scalars, at linear order in G and for the leading matter-induced logarithms at order G2. We connect the relation relevant to the logarithmic soft theorem to the radiative-gauge formulation of Boschetti-Campiglia and Compère-Robert. We then compute the required asymptotic fields directly from scattering data, including matter contributions and nonlinear effects responsible for graviton drag. Combining these results with the Newman-Penrose evolution equations yields position-space proofs of the classical leading and logarithmic soft graviton theorems, independently confirming the frequency-space derivation of the latter originally performed by Laddha and Sen.
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BibTeX
@misc{compere2026gravitational,
title = {{Gravitational multipoles, antipodal matching relations, and the logarithmic soft graviton theorem}},
author = {Geoffrey Compère and Dima Fontaine and Wen-Bin Liu and Kevin Nguyen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39253},
eprint = {2609.39253},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39253},
}
RIS
TY - UNPB
TI - Gravitational multipoles, antipodal matching relations, and the logarithmic soft graviton theorem
AU - Geoffrey Compère
AU - Dima Fontaine
AU - Wen-Bin Liu
AU - Kevin Nguyen
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39253
UR - https://arxiv.org/abs/2609.39253
AB - We derive the classical logarithmic soft graviton theorem for the scattering of massive particles in four-dimensional asymptotically flat spacetime using a position-space analysis of the Weyl tensor. Starting from the Iyer-Damour multipole solution in harmonic gauge, we obtain an infinite tower of antipodal matching relations across spatial infinity for all five Newman-Penrose Weyl scalars, at linear order in G and for the leading matter-induced logarithms at order G^2. We connect the relation relevant to the logarithmic soft theorem to the radiative-gauge formulation of Boschetti-Campiglia and Compère-Robert. We then compute the required asymptotic fields directly from scattering data, including matter contributions and nonlinear effects responsible for graviton drag. Combining these results with the Newman-Penrose evolution equations yields position-space proofs of the classical leading and logarithmic soft graviton theorems, independently confirming the frequency-space derivation of the latter originally performed by Laddha and Sen.
ER -