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Gravitational spin-orbit and aligned spin$_1$-spin$_2$ couplings through third-subleading post-Newtonian orders

Andrea Antonelli, Chris Kavanagh, Mohammed Khalil, Jan Steinhoff, Justin Vines

TL;DR

This work advances the spin-dependent two-body problem in general relativity by deriving the third-subleading PN spin-orbit (4.5PN) and aligned spin$_1$-spin$_2$ (5PN) dynamics from PM scattering combined with gravitational self-force results. By exploiting a simple mass dependence of the scattering angle in the aligned-spin case and employing a canonical mass-shell framework, the authors relate unbound scattering data to bound radial actions and invariant quantities (redshift and spin precession) via the first law of BBH mechanics. They fix the remaining ambiguities using first-order self-force results and construct an effective-one-body Hamiltonian, implementing the new PN corrections and demonstrating improved agreement with numerical relativity in binding energies. The work showcases a powerful PM–GSF–PN synthesis that refines waveform-model inputs and opens avenues for extending to higher-spin sectors and alternate cross-validations with EFT approaches. Overall, the results meaningfully enhance analytical models used in gravitational-wave data analysis and offer a pathway toward higher-accuracy waveform predictions for spinning binaries.

Abstract

The study of scattering encounters continues to provide new insights into the general relativistic two-body problem. The local-in-time conservative dynamics of an aligned-spin binary, for both unbound and bound orbits, is fully encoded in the gauge-invariant scattering-angle function, which is most naturally expressed in a post-Minkowskian (PM) expansion, and which exhibits a remarkably simple dependence on the masses of the two bodies (in terms of appropriate geometric variables). This dependence links the PM and small-mass-ratio approximations, allowing gravitational self-force results to determine new post-Newtonian (PN) information to all orders in the mass ratio. In this paper, we exploit this interplay between relativistic scattering and self-force theory to obtain the third-subleading (4.5PN) spin-orbit dynamics for generic spins, and the third-subleading (5PN) spin$_1$-spin$_2$ dynamics for aligned spins. We further implement these novel PN results in an effective-one-body framework, and demonstrate the improvement in accuracy by comparing against numerical-relativity simulations.

Gravitational spin-orbit and aligned spin$_1$-spin$_2$ couplings through third-subleading post-Newtonian orders

TL;DR

This work advances the spin-dependent two-body problem in general relativity by deriving the third-subleading PN spin-orbit (4.5PN) and aligned spin-spin (5PN) dynamics from PM scattering combined with gravitational self-force results. By exploiting a simple mass dependence of the scattering angle in the aligned-spin case and employing a canonical mass-shell framework, the authors relate unbound scattering data to bound radial actions and invariant quantities (redshift and spin precession) via the first law of BBH mechanics. They fix the remaining ambiguities using first-order self-force results and construct an effective-one-body Hamiltonian, implementing the new PN corrections and demonstrating improved agreement with numerical relativity in binding energies. The work showcases a powerful PM–GSF–PN synthesis that refines waveform-model inputs and opens avenues for extending to higher-spin sectors and alternate cross-validations with EFT approaches. Overall, the results meaningfully enhance analytical models used in gravitational-wave data analysis and offer a pathway toward higher-accuracy waveform predictions for spinning binaries.

Abstract

The study of scattering encounters continues to provide new insights into the general relativistic two-body problem. The local-in-time conservative dynamics of an aligned-spin binary, for both unbound and bound orbits, is fully encoded in the gauge-invariant scattering-angle function, which is most naturally expressed in a post-Minkowskian (PM) expansion, and which exhibits a remarkably simple dependence on the masses of the two bodies (in terms of appropriate geometric variables). This dependence links the PM and small-mass-ratio approximations, allowing gravitational self-force results to determine new post-Newtonian (PN) information to all orders in the mass ratio. In this paper, we exploit this interplay between relativistic scattering and self-force theory to obtain the third-subleading (4.5PN) spin-orbit dynamics for generic spins, and the third-subleading (5PN) spin-spin dynamics for aligned spins. We further implement these novel PN results in an effective-one-body framework, and demonstrate the improvement in accuracy by comparing against numerical-relativity simulations.

Paper Structure

This paper contains 13 sections, 160 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The mass dependence of the scattering angle
  3. From the unbound scattering angle to the bound radial action via canonical Hamiltonian dynamics
  4. The canonical Hamiltonian and/or the mass-shell constraint
  5. The scattering angle
  6. The radial action
  7. Third-subleading post-Newtonian spin-orbit and spin$_1$-spin$_2$ couplings
  8. The first law of BBH mechanics
  9. Comparison with self-force results
  10. Effective-one-body Hamiltonian and comparison with numerical relativity
  11. Conclusions
  12. The nonspinning 4PN terms in the bound radial action through sixth order in eccentricity
  13. Kerr-geodesic variables

Figures (2)

  • Figure 1: Binding energy versus the velocity parameter $v_\omega$ for the SO contribution to the EOB (left panels) and PN-expanded (right panels) binding energies for mass ratios $q=1$ (top panels) and $q=1/3$ (bottom panels).
  • Figure 2: Binding energy versus the velocity parameter $v_\omega$ for the S$_1$S$_2$ contribution to the EOB (left panels) and PN-expanded (right panels) binding energies for mass ratios $q=1$ (top panels) and $q=1/3$ (bottom panels). The NR error is indicated by the shaded regions.