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First Look at Quartic-in-Spin Binary Dynamics at Third Post-Minkowskian Order

Dogan Akpinar1,*, Fernando Febres Cordero2,†, Manfred Kraus3,‡, Alexander Smirnov4,5,§, and Mao Zeng1,∥

  • *Contact author: dogan.akpinar@ed.ac.uk
  • †Contact author: ffebres@hep.fsu.edu
  • ‡Contact author: mkraus@fisica.unam.mx
  • §Contact author: asmirnov@srcc.msu.ru
  • ∥Contact author: mao.zeng@ed.ac.uk
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Phys. Rev. Lett. 135, 041602 – Published 23 July, 2025

DOI: https://doi.org/10.1103/c2dh-tj4v

Abstract

We compute the conservative and radiation-reaction contributions to classical observables in the gravitational scattering between a spinning and a spinless black hole to the fourth order in spin and third order in the gravitational constant. The conservative results are obtained from two-loop amplitudes for the scattering process of a massive scalar with a massive spin-s field (s=0, 1, 2) minimally coupled to gravity, employing the recently introduced spin interpolation method to resolve all spin-Casimir terms. The two-loop amplitude exhibits a spin-shift symmetry in both probe limits, which we conjecture to be a sign of yet unknown integrability of Kerr orbits through the quartic order in spin and to all orders in the gravitational constant. We obtain the radial action from the finite part of the amplitude and use it to compute classical observables, including the impulse and spin kick. This is done using the recently introduced covariant Dirac brackets, which allow for the computation of classical scattering observables for general (nonaligned) spin configurations. Finally, employing the radiation-reaction amplitude proposed by Alessio and Di Vecchia, together with the Dirac brackets, we obtain radiation-reaction contributions to observables at all orders in spin and beyond the aligned-spin limit. We find agreement with known results up to the quadratic order in spin for both conservative and radiation-reaction contributions. Our results advance the state of the art in the understanding of spinning binary dynamics in general relativity and demonstrate the power and simplicity of the Dirac bracket formalism for relating scattering amplitudes to classical observables.

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Outline

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Phys. Rev. Lett. 135, 041602– Published 23 July, 2025
  • Received 27 February 2025
  • Accepted 17 June 2025
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