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Post-Newtonian corrections to the motion of spinning bodies in NRGR

Rafael A. Porto

TL;DR

This work extends NRGR, an effective field theory approach to gravity in the post-Newtonian regime, to include spin and permanent multipole moments of extended bodies. By formulating a spinful worldline action and deriving spin-graviton interactions, the authors obtain leading spin-orbit and spin-spin potentials, a quadrupole-spin coupling, and the corresponding quadrupole radiation formula, while also clarifying the equivalence of different spin supplementary conditions. They analyze divergences and classify finite-size effects as tidal (logarithmic, RG-flow) or self-induced (power-law, scale-independent), deriving their scaling and showing how they renormalize within NRGR. The results provide a systematic framework for incorporating spin and internal structure into PN gravitational-wave templates, with explicit predictions for higher-order effects and clear guidance for future extensions.

Abstract

In this paper we include spin and multipole moment effects in the formalism used to describe the motion of extended objects recently introduced in hep-th/0409156. A suitable description for spinning bodies is developed and spin-orbit, spin-spin and quadrupole-spin Hamiltonians are found at leading order. The existence of tidal, as well as self induced finite size effects is shown, and the contribution to the Hamiltonian is calculated in the latter. It is shown that tidal deformations start formally at O(v^6) and O(v^10) for maximally rotating general and compact objects respectively, whereas self induced effects can show up at leading order. Agreement is found for the cases where the results are known.

Post-Newtonian corrections to the motion of spinning bodies in NRGR

TL;DR

This work extends NRGR, an effective field theory approach to gravity in the post-Newtonian regime, to include spin and permanent multipole moments of extended bodies. By formulating a spinful worldline action and deriving spin-graviton interactions, the authors obtain leading spin-orbit and spin-spin potentials, a quadrupole-spin coupling, and the corresponding quadrupole radiation formula, while also clarifying the equivalence of different spin supplementary conditions. They analyze divergences and classify finite-size effects as tidal (logarithmic, RG-flow) or self-induced (power-law, scale-independent), deriving their scaling and showing how they renormalize within NRGR. The results provide a systematic framework for incorporating spin and internal structure into PN gravitational-wave templates, with explicit predictions for higher-order effects and clear guidance for future extensions.

Abstract

In this paper we include spin and multipole moment effects in the formalism used to describe the motion of extended objects recently introduced in hep-th/0409156. A suitable description for spinning bodies is developed and spin-orbit, spin-spin and quadrupole-spin Hamiltonians are found at leading order. The existence of tidal, as well as self induced finite size effects is shown, and the contribution to the Hamiltonian is calculated in the latter. It is shown that tidal deformations start formally at O(v^6) and O(v^10) for maximally rotating general and compact objects respectively, whereas self induced effects can show up at leading order. Agreement is found for the cases where the results are known.

Paper Structure

This paper contains 29 sections, 83 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. NRGR
  3. Basic philosophy
  4. Effective theory of extended objects
  5. The Post-Newtonian expansion
  6. Internal degrees of freedom I: Spinning particle
  7. Basics
  8. Action principle and EOM
  9. Constraints and angular-velocity/spin relationship
  10. NRGR for spinning bodies
  11. Power counting
  12. Feynman rules: spin-graviton vertex
  13. Leading order graviton exchange
  14. EOM
  15. Internal degrees of freedom II: Permanent multipole moments
  16. ...and 14 more sections

Figures (9)

  • Figure 1: Leading order mass vertex. the dashed line represents a potential graviton, whereas the wavy line stands for the full graviton propagator.
  • Figure 2: Leading order spin-graviton vertex interaction. The blow represents an spin insertion.
  • Figure 3: Leading order spin-orbit interaction. Diagram a) takes into account a $v^1$ mass insertion. Diagram b) correspond to the $v^3$ spin-graviton vertex.
  • Figure 4: Leading order spin-spin interaction.
  • Figure 5: Leading order quadrupole-spin one graviton exchange. The black square represents a quadrupole insertion.
  • ...and 4 more figures