Triangular solution to general relativistic three-body problem for general masses

TL;DR

This paper revisits the relativistic three-body problem, deriving conditions for stable triangular configurations under general relativity and showing they differ from classical solutions, especially in size and shape.

Contribution

It provides a new post-Newtonian solution for three finite masses in a triangular configuration, extending previous restricted cases and highlighting relativistic corrections.

Findings

Post-Newtonian triangular solutions are not always equilateral.

Relativistic configurations are smaller than Newtonian ones.

The solution recovers previous restricted three-body results.

Abstract

Continuing work initiated in an earlier publication [Ichita, Yamada and Asada, Phys. Rev. D 83, 084026 (2011)], we reexamine the post-Newtonian effects on Lagrange's equilateral triangular solution for the three-body problem. For three finite masses, it is found that a triangular configuration satisfies the post-Newtonian equation of motion in general relativity, if and only if it has the relativistic corrections to each side length. This post-Newtonian configuration for three finite masses is not always equilateral and it recovers previous results for the restricted three-body problem when one mass goes to zero. For the same masses and angular velocity, the post-Newtonian triangular configuration is always smaller than the Newtonian one.