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Next to leading order spin-orbit effects in the motion of inspiralling compact binaries

Rafael A. Porto

TL;DR

This work delivers a complete NRGR-based calculation of the next-to-leading order spin-orbit contribution to the gravitational potential for inspiralling compact binaries, using a covariant spin supplementary condition and confirming consistency with previous findings. The analysis proceeds via a systematic Feynman-diagram approach, including one-graviton exchange and nonlinear gravitational interactions (seagulls and three-graviton couplings), with divergences handled by standard regularization. The spin dynamics are recast into a precession form, yielding a corrected precession frequency and demonstrating equivalence with prior results up to a total time-derivative through a spin redefinition, while also validating the Newton-Wigner SSC in this self-gravitating system. These results complete the spin dynamics within the EFT framework and lay the groundwork for the spin contributions to the energy flux and phase evolution at 3PN in upcoming work. The methods and findings have direct implications for improving gravitational-wave templates and parameter estimation for spinning binaries.

Abstract

Using effective field theory (EFT) techniques we calculate the next-to-leading order (NLO) spin-orbit contributions to the gravitational potential of inspiralling compact binaries. We use the covariant spin supplementarity condition (SSC), and explicitly prove the equivalence with previous results by Faye et al. in arXiv:gr-qc/0605139. We also show that the direct application of the Newton-Wigner SSC at the level of the action leads to the correct dynamics using a canonical (Dirac) algebra. This paper then completes the calculation of the necessary spin dynamics within the EFT formalism that will be used in a separate paper to compute the spin contributions to the energy flux and phase evolution to NLO.

Next to leading order spin-orbit effects in the motion of inspiralling compact binaries

TL;DR

This work delivers a complete NRGR-based calculation of the next-to-leading order spin-orbit contribution to the gravitational potential for inspiralling compact binaries, using a covariant spin supplementary condition and confirming consistency with previous findings. The analysis proceeds via a systematic Feynman-diagram approach, including one-graviton exchange and nonlinear gravitational interactions (seagulls and three-graviton couplings), with divergences handled by standard regularization. The spin dynamics are recast into a precession form, yielding a corrected precession frequency and demonstrating equivalence with prior results up to a total time-derivative through a spin redefinition, while also validating the Newton-Wigner SSC in this self-gravitating system. These results complete the spin dynamics within the EFT framework and lay the groundwork for the spin contributions to the energy flux and phase evolution at 3PN in upcoming work. The methods and findings have direct implications for improving gravitational-wave templates and parameter estimation for spinning binaries.

Abstract

Using effective field theory (EFT) techniques we calculate the next-to-leading order (NLO) spin-orbit contributions to the gravitational potential of inspiralling compact binaries. We use the covariant spin supplementarity condition (SSC), and explicitly prove the equivalence with previous results by Faye et al. in arXiv:gr-qc/0605139. We also show that the direct application of the Newton-Wigner SSC at the level of the action leads to the correct dynamics using a canonical (Dirac) algebra. This paper then completes the calculation of the necessary spin dynamics within the EFT formalism that will be used in a separate paper to compute the spin contributions to the energy flux and phase evolution to NLO.

Paper Structure

This paper contains 16 sections, 71 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Spin effects in the Effective Field Theory approach
  3. The (covariant) spin supplementarity condition
  4. The next-to-leading order spin-orbit potential
  5. One-graviton exchange
  6. Non-linear gravitational interactions
  7. Seagulls
  8. Three-graviton couplings
  9. Divergences
  10. Assembling the pieces
  11. Spin-orbit dynamics
  12. Precession equation
  13. Comparison with other methods
  14. The Newton-Wigner SSC
  15. Conclusions
  16. ...and 1 more sections

Figures (4)

  • Figure 1: One-graviton contributions linear in spin at ${\cal O}(v^5)$. A dashed line represents a potential graviton, a blob account for spin insertions at each PN order from Eqs. (\ref{['sgnr1']}--\ref{['sgnr2']}), and a black dot stands for 'propagator' corrections as explained in the text.
  • Figure 2: Non-linear gravitational effects due the couplings in Eqs. (\ref{['sgh2v4']}, \ref{['sgh2v5']}) and (\ref{['h2m']}, \ref{['h2mv']}).
  • Figure 3: Non-linear gravitational effects due to the three-graviton couplings. A black dot represents an extra power of $v$ from a time derivative in the three graviton vertex (see text.)
  • Figure 4: Type of divergences in the three-graviton diagrams