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All-Order Structure of Static Gravitational Interactions and the Seventh Post-Newtonian Potential

Giacomo Brunello1,2,*, Manoj K. Mandal3,4,†, Pierpaolo Mastrolia3,4,‡, Raj Patil5,6,§, Matteo Pegorin3,4,5,∥, Sid Smith3,4,7,¶, and Jan Steinhoff5,**

  • *Contact author: giacomo.brunello@sns.it
  • †Contact author: manojkumar.mandal@unipd.it
  • ‡Contact author: pierpaolo.mastrolia@unipd.it
  • §Contact author: raj.patil@aei.mpg.de
  • ∥Contact author: matteo.pegorin.1@phd.unipd.it
  • Contact author: sid.smith@ed.ac.uk
  • **Contact author: jan.steinhoff@aei.mpg.de
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Phys. Rev. Lett. 137, 201401 – Published 12 November, 2026

DOI: https://doi.org/10.1103/kcxq-7zsk

Abstract

We present a closed formula for the computation of static post-Newtonian corrections to the two-body gravitational dynamics at any odd order, assuming the lower-order results are known. The formula is derived within a correlation function framework and exploits the Z2 symmetry of the static sector, leading to a novel theoretical interpretation of the factorization theorem. As an application, we compute the gravitational interaction of two compact coalescing objects at the seventh post-Newtonian order in the static limit, which receives contributions from seven-loop graphs at order O(GN8v0), and find complete agreement with the results obtained using the diagrammatic approach of the factorization theorem.

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Phys. Rev. Lett. 137, 201401– Published 12 November, 2026
  • Received 15 May 2026
  • Accepted 25 September 2026
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