Numerical integration of a relativistic two-body problem via a multiple scales method

TL;DR

This paper develops an analytical approach using multiple scales to study the relativistic two-body problem, revealing small energy variations and planar motion, with a periodic solution more general than previous methods, applicable to relativistic binary systems.

Contribution

It introduces a novel multiple scales method to analytically solve the relativistic two-body problem, extending previous approaches and providing insights into long-term binary evolution.

Findings

Angular momentum is not conserved but motion remains planar.

Energy experiences small relativistic-induced variations.

A more general periodic solution is derived.

Abstract

We offer an analytical study on the dynamics of a two-body problem perturbed by small post-Newtonian relativistic term. We prove that, while the angular momentum is not conserved, the motion is planar. We also show that the energy is subject to small changes due to the relativistic effect. We also offer a periodic solution to this problem, obtained by a method based of separation of timescales. We demonstrate that our solution is more general than the method developed in the book by Brumberg (1991). The practical applicability of this model may be studies of the long-term evolution of relativistic binaries (neutron stars or black holes).