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The dynamics of the gravitational two-body problem at fourth post-Newtonian order and at quadratic order in the Newton constant

Stefano Foffa, Riccardo Sturani

TL;DR

The paper advances the conservative two-body dynamics of general relativity to fourth post-Newtonian order, deriving the effective action up to terms quadratic in Newton's constant using an EFT framework. It constructs the 4PN Lagrangian linear in accelerations via a double-zero reduction, computes the corresponding energy, and verifies Lorentz invariance and a consistent center-of-mass position, including GN^2 corrections. The authors provide explicit expressions for the GN and GN^2 contributions and demonstrate agreement with independent 4PN Hamiltonian results for circular orbits, validating their EFT approach. This work lays essential groundwork for the complete 4PN dynamics and has direct implications for high-precision gravitational-wave template modeling.

Abstract

We derive the conservative part of the Lagrangian and the energy of a gravitationally bound two-body system at fourth post-Newtonian order, up to terms quadratic in the Newton constant. We also show that such terms are compatible with Lorentz invariance and we write an ansatz for the center-of-mass position. The remaining terms carrying higher powers of the Newton constant are currently under investigation.

The dynamics of the gravitational two-body problem at fourth post-Newtonian order and at quadratic order in the Newton constant

TL;DR

The paper advances the conservative two-body dynamics of general relativity to fourth post-Newtonian order, deriving the effective action up to terms quadratic in Newton's constant using an EFT framework. It constructs the 4PN Lagrangian linear in accelerations via a double-zero reduction, computes the corresponding energy, and verifies Lorentz invariance and a consistent center-of-mass position, including GN^2 corrections. The authors provide explicit expressions for the GN and GN^2 contributions and demonstrate agreement with independent 4PN Hamiltonian results for circular orbits, validating their EFT approach. This work lays essential groundwork for the complete 4PN dynamics and has direct implications for high-precision gravitational-wave template modeling.

Abstract

We derive the conservative part of the Lagrangian and the energy of a gravitationally bound two-body system at fourth post-Newtonian order, up to terms quadratic in the Newton constant. We also show that such terms are compatible with Lorentz invariance and we write an ansatz for the center-of-mass position. The remaining terms carrying higher powers of the Newton constant are currently under investigation.

Paper Structure

This paper contains 15 sections, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Short-scale Lagrangian
  3. Integrating out gravity
  4. Order $G_N$
  5. Order $G_N^2$
  6. The 4PN Lagrangian linear in accelerations up to $G^2_N$
  7. 2PN double zeros evaluated on 2PN equations of motion
  8. 3PN double zeros evaluated on 1PN equations of motion
  9. 4PN double zeroes from $O(G)$ graphs
  10. Result of the double zero procedure
  11. Energy
  12. Lorentz invariance and center-of-mass position
  13. Boosts
  14. Center of mass position
  15. Conclusions

Figures (2)

  • Figure 1: The three diagrams contributing at order $G_N$. The $\phi$, $A$ and $\sigma$ propagators are represented respectively by blue dashed, red dotted and green solid lines. Propagators and vertices must be considered at appropriate $v$ order to pick the 4PN contribution of these graphs.
  • Figure 2: The diagrams contributing at order $G^2_N$ arranged in 4 lines. Diagrams in the $n^{\rm th}$ line enter the dynamics at $n$PN order. Propagators and vertices must be considered at appropriate $v$ order to pick the 4PN contribution of these diagrams.