Equations of motion of self-gravitating $N$-body systems in the first post-Minkowskian approximation

TL;DR

The paper develops a $1$PM treatment of $N$-body self-gravitating systems in GR, recasting the dynamics in a PN-like framework to enable direct comparison with high-order PN results, and provides both Lagrangian and Hamiltonian formulations in harmonic coordinates. It derives the linearized gravitational field and the corresponding conservative equations of motion, constructs conserved energy and momentum to $O(G)$, and specializes to the equal-mass two-body problem to obtain explicit expressions and connections to previous formalisms via canonical transformations. A generalized Lagrangian linear in accelerations is developed, with a PF-based method to relate to the $4$PN harmonic-coordinate Lagrangian and a 5PN extension; in the Hamiltonian picture, the equal-mass case is mapped to the LSB08 Hamiltonian, and the center-of-mass description is formulated to $O(G)$ for the relative motion. Overall, the work provides a coherent PM-to-PN bridge, validates known scattering results at $1$PM, and offers a structured route to cross-check high-order PN terms with PM-based expressions.

Abstract

We revisit the problem of the equations of motion of a system of $N$ self-interacting massive particles (without spins) in the first post-Minkowskian (1PM) approximation of general relativity. We write the equations of motion, gravitational field and associated conserved integrals of the motion in a form suitable for comparison with recently published post-Newtonian (PN) results at the 4PN order. We show that the Lagrangian associated with the equations of motion in harmonic coordinates is a generalized one, and compute all the terms linear in $G$ up to 5PN order. We discuss the Hamiltonian in the frame of the center of mass and exhibit a canonical transformation connecting it to previous results directly obtained with the Hamiltonian formalism of general relativity. Finally we recover the known result for the gravitational scattering angle of two particles at the 1PM order.

Figures (1)

• Figure 1: Left panel: The circular orbits in the two branches corresponding to $\epsilon=1$ (PN regime) and $\epsilon=-1$ (UR regime) in Eq. \ref{['gamma0']}. The point with minimal circular radius corresponds to $r_\text{min} = (2+\sqrt{2})\,\frac{G m}{c^2}$ and $\gamma_\text{min}=\frac{1}{2}(4+\sqrt{2})^{1/2}$. Right panel: Stability criterium of circular orbits. The unstable region ($\Omega_0^2<0$) is for $1.02\lesssim\gamma_0\lesssim 1.21$ and surrounds the minimal radius $r_\text{min}$ in the figure on the left (indeed $\gamma_\text{min}\simeq 1.16$).