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

TL;DR

This paper derives the fourth post-Newtonian order Lagrangian and energy for a two-body gravitational system, including quadratic Newton constant terms, ensuring Lorentz invariance and proposing a center-of-mass framework.

Contribution

It provides the first derivation of the conservative two-body dynamics at fourth post-Newtonian order with quadratic Newton constant terms, maintaining Lorentz invariance.

Findings

Derived the 4PN Lagrangian and energy including quadratic Newton constant terms

Confirmed compatibility of these terms with Lorentz invariance

Proposed an ansatz for the center-of-mass position

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.