Post-Newtonian effects in N-body dynamics: Relativistic precession and conserved quantities in hierarchical triple systems

TL;DR

This paper demonstrates that including post-Newtonian cross terms in the equations of motion is essential for accurately conserving energy and angular momentum in hierarchical triple systems over relativistic precession timescales.

Contribution

It clarifies the importance of PN cross terms for consistent long-term relativistic dynamics in hierarchical triples.

Findings

Total energy is conserved with PN cross terms included.

Angular momentum component remains conserved over precession timescale.

Neglecting cross terms leads to inaccuracies in long-term evolution.

Abstract

Conventional approaches to incorporating general relativistic effects into the dynamics of N-body systems containing central black holes, or of hierarchical triple systems with a relativistic inner binary, may not be adequate when the goal is to study the evolution of the system over a timescale related to relativistic secular effects, such as the precession of the pericenter. For such problems, it may necessary to include post-Newtonian "cross terms" in the equations of motion in order to capture relativistic effects consistently over the long timescales. Cross terms are post-Newtonian (PN) terms that explicitly couple the two-body relativistic perturbations with the Newtonian perturbations due to other bodies in the system. In this paper, we show that the total energy and the normal component of total angular momentum of a hierarchical triple system is manifestly conserved to Newtonian order over the relativistic pericenter precession timescale of the inner binary if and only if PN cross-term effects in the equations of motion are taken carefully into account.