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Energy and periastron advance of compact binaries on circular orbits at the fourth post-Newtonian order

Laura Bernard, Luc Blanchet, Alejandro Bohé, Guillaume Faye, Sylvain Marsat

TL;DR

The paper advances the 4PN description of non-spinning compact binaries in harmonic coordinates by treating the non-local tail term and infrared regularization ambiguities. It introduces two ambiguity parameters to capture IR prescription effects, derives conserved energy and angular momentum for non-local dynamics, and fixes these parameters by matching with gravitational self-force results, achieving agreement with ADM-like Hamiltonians. Through both reduced canonical and Delaunay–Poincaré analyses, it demonstrates the physical equivalence of the harmonic-coordinate Lagrangian and the ADM Hamiltonian, including a crucial DC tail contribution to the energy for circular orbits. The work solidifies the circular-orbit 4PN energy and periastron precession predictions and lays groundwork for extending the framework to 4.5PN and beyond, with future attention to IR regularization and radiation-field aspects.

Abstract

In this paper, we revisit and complete our preceding work on the Fokker Lagrangian describing the dynamics of compact binary systems at the fourth post-Newtonian (4PN) order in harmonic coordinates. We clarify the impact of the non-local character of the Fokker Lagrangian or the associated Hamiltonian on both the conserved energy and the relativistic periastron precession for circular orbits. We show that the non-locality of the action, due to the presence of the tail effect at the 4PN order, gives rise to an extra contribution to the conserved integral of energy with respect to the Hamiltonian computed on shell, which was not taken into account in our previous work. We also provide a direct derivation of the periastron advance by taking carefully into account this non-locality. We then argue that the infra-red (IR) divergences in the calculation of the gravitational part of the action are problematic, which motivates us to introduce a second ambiguity parameter, in addition to the one already assumed previously. After fixing these two ambiguity parameters by requiring that the conserved energy and the relativistic periastron precession for circular orbits are in agreement with numerical and analytical gravitational self-force calculations, valid in the limiting case of small mass ratio, we find that our resulting Lagrangian is physically equivalent to the one obtained in the ADM Hamiltonian approach.

Energy and periastron advance of compact binaries on circular orbits at the fourth post-Newtonian order

TL;DR

The paper advances the 4PN description of non-spinning compact binaries in harmonic coordinates by treating the non-local tail term and infrared regularization ambiguities. It introduces two ambiguity parameters to capture IR prescription effects, derives conserved energy and angular momentum for non-local dynamics, and fixes these parameters by matching with gravitational self-force results, achieving agreement with ADM-like Hamiltonians. Through both reduced canonical and Delaunay–Poincaré analyses, it demonstrates the physical equivalence of the harmonic-coordinate Lagrangian and the ADM Hamiltonian, including a crucial DC tail contribution to the energy for circular orbits. The work solidifies the circular-orbit 4PN energy and periastron precession predictions and lays groundwork for extending the framework to 4.5PN and beyond, with future attention to IR regularization and radiation-field aspects.

Abstract

In this paper, we revisit and complete our preceding work on the Fokker Lagrangian describing the dynamics of compact binary systems at the fourth post-Newtonian (4PN) order in harmonic coordinates. We clarify the impact of the non-local character of the Fokker Lagrangian or the associated Hamiltonian on both the conserved energy and the relativistic periastron precession for circular orbits. We show that the non-locality of the action, due to the presence of the tail effect at the 4PN order, gives rise to an extra contribution to the conserved integral of energy with respect to the Hamiltonian computed on shell, which was not taken into account in our previous work. We also provide a direct derivation of the periastron advance by taking carefully into account this non-locality. We then argue that the infra-red (IR) divergences in the calculation of the gravitational part of the action are problematic, which motivates us to introduce a second ambiguity parameter, in addition to the one already assumed previously. After fixing these two ambiguity parameters by requiring that the conserved energy and the relativistic periastron precession for circular orbits are in agreement with numerical and analytical gravitational self-force calculations, valid in the limiting case of small mass ratio, we find that our resulting Lagrangian is physically equivalent to the one obtained in the ADM Hamiltonian approach.

Paper Structure

This paper contains 13 sections, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Ambiguity parameters in the Fokker Lagrangian
  3. Conserved integrals for a non-local Hamiltonian
  4. Integral of energy
  5. Integral of angular momentum
  6. Conserved energy for circular orbits
  7. Derivation using reduced canonical variables
  8. Derivation using Delaunay-Poincaré variables
  9. Periastron advance for circular orbits
  10. Conclusions
  11. The 4PN Lagrangian in harmonic and ADM coordinates
  12. Fourier coefficients of the quadrupole moment
  13. On Sommerfeld's method of contour integrals

Figures (1)

  • Figure 1: Left panel: Path of integration of the integrals $I^{n,m}$ corresponding to the standard, logarithmic-free case. Right panel: Path of integration of the integrals $K^{n,m}$ for the logarithmic case. The branch cut $\mathbb{R}^{-}$ of the logarithm function is avoided.