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Spin(1)Spin(2) Effects in the Motion of Inspiralling Compact Binaries at Third Order in the Post-Newtonian Expansion

Rafael A. Porto, Ira Z. Rothstein

TL;DR

The paper develops an effective field theory approach (NRGR) to spinning compact binaries and computes the spin-spin and spin-orbit potentials to third post-Newtonian order using a covariant spin supplementary condition within a Routhian formalism. By postponing the SSC to the end of the calculation, the authors maintain a canonical phase-space structure and reproduce the known MP equations, deriving a full 3PN spin-spin potential and spin EOM. They then demonstrate precise equivalence between the covariant-SSC Routhian results and earlier Newton-Wigner SSC results obtained at the action level, including a detailed treatment of frame transformations and precession. They also outline how S^2 (finite-size) corrections enter at higher order and discuss renormalization of divergences into finite-size parameters, with plans for future work on complete S^2 contributions.

Abstract

We use effective field theory techniques to compute the potentials due to spin-spin and spin-orbit effects, from which the spin(1)spin(2) contribution to the motion of spinning compact binaries to third Post-Newtonian (PN) order follow. We use a formalism which allows us to impose the spin supplementarity condition (SSC) in a canonical framework to all orders in the PN expansion. We explicitly show the equivalence with our previous results, obtained using the Newton-Wigner SSC at the level of the action for spin-spin and spin-orbit potentials reported in arXiv:gr-qc/0604099 and arXiv:0712.2032[gr-qc] respectively.

Spin(1)Spin(2) Effects in the Motion of Inspiralling Compact Binaries at Third Order in the Post-Newtonian Expansion

TL;DR

The paper develops an effective field theory approach (NRGR) to spinning compact binaries and computes the spin-spin and spin-orbit potentials to third post-Newtonian order using a covariant spin supplementary condition within a Routhian formalism. By postponing the SSC to the end of the calculation, the authors maintain a canonical phase-space structure and reproduce the known MP equations, deriving a full 3PN spin-spin potential and spin EOM. They then demonstrate precise equivalence between the covariant-SSC Routhian results and earlier Newton-Wigner SSC results obtained at the action level, including a detailed treatment of frame transformations and precession. They also outline how S^2 (finite-size) corrections enter at higher order and discuss renormalization of divergences into finite-size parameters, with plans for future work on complete S^2 contributions.

Abstract

We use effective field theory techniques to compute the potentials due to spin-spin and spin-orbit effects, from which the spin(1)spin(2) contribution to the motion of spinning compact binaries to third Post-Newtonian (PN) order follow. We use a formalism which allows us to impose the spin supplementarity condition (SSC) in a canonical framework to all orders in the PN expansion. We explicitly show the equivalence with our previous results, obtained using the Newton-Wigner SSC at the level of the action for spin-spin and spin-orbit potentials reported in arXiv:gr-qc/0604099 and arXiv:0712.2032[gr-qc] respectively.

Paper Structure

This paper contains 16 sections, 91 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review of Spin in GR
  3. The Routhian approach for spinning bodies in NRGR
  4. The effective action
  5. An Example: The spin--orbit effects at leading order
  6. The Equivalence of Methodologies
  7. The PN frame versus the local frame in the covariant SSC
  8. Imposing the SSC before or after calculating the EOM
  9. The 3PN spin--spin Potential
  10. The spin Hamiltonian in the NW SSC to 3PN
  11. The spin equation of motion to 3PN order
  12. The Precession equation to 3PN and the equivalence with our previous results using the NW SSC
  13. Adding ${\bf S}^2$ terms
  14. Conclusions
  15. On the Routhian formalism
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Leading order spin--spin interaction.
  • Figure 2: Diagrams contributing to 3PN order which do not involve non--linear interactions. The blob represents a spin insertion and the cross corresponds to a propagator correction.
  • Figure 3: Non--linear contributions to the 3PN spin--spin potential.