Collinear solution to the general relativistic three-body problem

TL;DR

This paper investigates the persistence of collinear solutions in the three-body problem within general relativity, revealing that such configurations are approximately maintained with relativistic corrections to spatial separations and Lagrange points.

Contribution

It demonstrates that the classical collinear solution persists under general relativity with specific corrections, extending the understanding of three-body dynamics in relativistic regimes.

Findings

Collinear solutions remain valid with relativistic corrections.

Relativistic adjustments to Sun-Jupiter Lagrange points L1, L2, L3 are quantified.

The solution's stability is affected by relativistic effects.

Abstract

The three-body problem is reexamined in the framework of general relativity. The Newtonian three-body problem admits Euler's collinear solution, where three bodies move around the common center of mass with the same orbital period and always line up. The solution is unstable. Hence it is unlikely that such a simple configuration would exist owing to general relativistic forces dependent not only on the masses but also on the velocity of each body. However, we show that the collinear solution remains true with a correction to the spatial separation between masses. Relativistic corrections to the Sun-Jupiter Lagrange points L1, L2 and L3 are also evaluated.