Chaos in a Relativistic 3-body Self-Gravitating System

TL;DR

This paper derives the Hamiltonian for a relativistic 3-body system in lineal gravity, showing that despite non-linearity, the system remains non-chaotic with fractal orbit structures, contrasting with expectations of chaos at high energies.

Contribution

It provides an exact Hamiltonian and equations of motion for a relativistic 3-body problem, revealing non-chaotic behavior and fractal orbit structures in a highly non-linear setting.

Findings

Relativistic effects lead to more tightly-bound, higher-frequency orbits.

No evidence of chaos or breakdown at high energies in equal-mass case.

Existence of a fractal structure in the orbit space.

Abstract

We consider the 3-body problem in relativistic lineal gravity and obtain an exact expression for its Hamiltonian and equations of motion. While general-relativistic effects yield more tightly-bound orbits of higher frequency compared to their non-relativistic counterparts, as energy increases we find in the equal-mass case no evidence for either global chaos or a breakdown from regular to chaotic motion, despite the high degree of non-linearity in the system. We find numerical evidence for a countably infinite class of non-chaotic orbits, yielding a fractal structure in the outer regions of the Poincare plot.