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Sixth post-Newtonian local-in-time dynamics of binary systems

Donato Bini, Thibault Damour, Andrea Geralico

TL;DR

This work extends a hybrid analytic framework that splits conservative binary dynamics into nonlocal and local pieces to 6PN order, synthesizing PN, PM, MPM, EFT, SF, EOB, and Delaunay averaging. It derives the full local 6PN Hamiltonian (151 coefficients) with only four nonlinear-in-ν coefficients remaining undetermined, and cross-checks the 1SF redshift and the mass-ratio dependence of scattering angles against known PM results, validating the approach. The authors translate nonlocal h-route results into a canonically equivalent EOB description, connect SF data to PN/PM structures via q_n(γ,ν) and χ_n(γ,ν), and reveal a hidden, ν-dependent simplicity in the radial action that underpins the 6PN dynamics. They also provide gauge-invariant representations and detailed tables of EOB potentials, scattering angles, and radial-action coefficients, offering a robust, cross-validated picture of local binary dynamics with broad implications for gravitational-wave modeling.

Abstract

Using a recently introduced method [Phys.\ Rev.\ Lett.\ {\bf 123}, 231104 (2019)], which splits the conservative dynamics of gravitationally interacting binary systems into a non-local-in-time part and a local-in-time one, we compute the local part of the dynamics at the sixth post-Newtonian (6PN) accuracy. Our strategy combines several theoretical formalisms: post-Newtonian, post-Minkowskian, multipolar-post-Minkowskian, effective-field-theory, gravitational self-force, effective one-body, and Delaunay averaging. The full functional structure of the local 6PN Hamiltonian (which involves 151 numerical coefficients) is derived, but contains four undetermined numerical coefficients. Our 6PN-accurate results are complete at orders $G^3$ and $G^4$, and the derived $O(G^3)$ scattering angle agrees, within our 6PN accuracy, with the computation of [Phys.\ Rev.\ Lett.\ {\bf 122}, no. 20, 201603 (2019)]. All our results are expressed in several different gauge-invariant ways. We highlight, and make a crucial use of, several aspects of the hidden simplicity of the mass-ratio dependence of the two-body dynamics.

Sixth post-Newtonian local-in-time dynamics of binary systems

TL;DR

This work extends a hybrid analytic framework that splits conservative binary dynamics into nonlocal and local pieces to 6PN order, synthesizing PN, PM, MPM, EFT, SF, EOB, and Delaunay averaging. It derives the full local 6PN Hamiltonian (151 coefficients) with only four nonlinear-in-ν coefficients remaining undetermined, and cross-checks the 1SF redshift and the mass-ratio dependence of scattering angles against known PM results, validating the approach. The authors translate nonlocal h-route results into a canonically equivalent EOB description, connect SF data to PN/PM structures via q_n(γ,ν) and χ_n(γ,ν), and reveal a hidden, ν-dependent simplicity in the radial action that underpins the 6PN dynamics. They also provide gauge-invariant representations and detailed tables of EOB potentials, scattering angles, and radial-action coefficients, offering a robust, cross-validated picture of local binary dynamics with broad implications for gravitational-wave modeling.

Abstract

Using a recently introduced method [Phys.\ Rev.\ Lett.\ {\bf 123}, 231104 (2019)], which splits the conservative dynamics of gravitationally interacting binary systems into a non-local-in-time part and a local-in-time one, we compute the local part of the dynamics at the sixth post-Newtonian (6PN) accuracy. Our strategy combines several theoretical formalisms: post-Newtonian, post-Minkowskian, multipolar-post-Minkowskian, effective-field-theory, gravitational self-force, effective one-body, and Delaunay averaging. The full functional structure of the local 6PN Hamiltonian (which involves 151 numerical coefficients) is derived, but contains four undetermined numerical coefficients. Our 6PN-accurate results are complete at orders and , and the derived scattering angle agrees, within our 6PN accuracy, with the computation of [Phys.\ Rev.\ Lett.\ {\bf 122}, no. 20, 201603 (2019)]. All our results are expressed in several different gauge-invariant ways. We highlight, and make a crucial use of, several aspects of the hidden simplicity of the mass-ratio dependence of the two-body dynamics.

Paper Structure

This paper contains 21 sections, 181 equations, 1 figure, 14 tables.

Table of Contents

  1. Introduction
  2. Nonlocal action at the 6PN order
  3. Computing the Delaunay average of the nonlocal-in-time h-route Hamiltonian
  4. The 2PN-accurate multipole moments in harmonic coordinates
  5. Generic 2PN quasi-Keplerian parametrization of elliptic motion (valid in all coordinates)
  6. 2PN expressions of the orbital parameters in harmonic coordinates
  7. 2PN quasi-Keplerian orbital parameters in EOB coordinates
  8. Evaluating the multipole moments along the orbit
  9. Computing the Delaunay-average of the nonlocal Hamiltonian in harmonic coordinates
  10. Deriving the h-route nonlocal EOB Hamiltonian
  11. Computing the 1SF time-averaged redshift to eighth order in eccentricity and deriving its EOB counterpart
  12. Determining the local part of the EOB potentials at order $\nu^1$
  13. Using the mass-ratio dependence of the scattering angle to determine most of the $\nu^{n\geq 2}$ structure of the 6PN $f$-route local Hamiltonian
  14. Going from the $p_r$-gauge to the energy-gauge
  15. Computing the f-route local scattering angle
  16. ...and 6 more sections

Figures (1)

  • Figure 1: Schematic representation of the irreducible information contained, at each post-Minkowskian level (keyed by a power of $u=GM/r$), in the local dynamics. Each vertical column of dots describes the post-Newtonian expansion (keyed by powers of $p^2$) of an energy-dependent function parametrizing the scattering angle. The various columns at a given post-Minkowskian level correspond to increasing powers of the symmetric mass-ratio $\nu$. See text for details.