Chaos in an Exact Relativistic 3-body Self-Gravitating System

TL;DR

This paper derives an exact Hamiltonian for a relativistic 3-body self-gravitating system in one dimension, analyzes its phase space, and explores the effects of relativity on its chaotic and regular motions.

Contribution

It provides the first exact relativistic Hamiltonian for a 3-body self-gravitating system and investigates the impact of relativity on its dynamical behavior and phase space structure.

Findings

Identifies periodic, quasi-periodic, and chaotic motions in the relativistic system.

Shows the relativistic phase space structure remains similar to the non-relativistic case.

Finds a post-Newtonian KAM breakdown at b7 a 0.26, leading to chaos.

Abstract

We consider the problem of three body motion for a relativistic one-dimensional self-gravitating system. After describing the canonical decomposition of the action, we find an exact expression for the 3-body Hamiltonian, implicitly determined in terms of the four coordinate and momentum degrees of freedom in the system. Non-relativistically these degrees of freedom can be rewritten in terms of a single particle moving in a two-dimensional hexagonal well. We find the exact relativistic generalization of this potential, along with its post-Newtonian approximation. We then specialize to the equal mass case and numerically solve the equations of motion that follow from the Hamiltonian. Working in hexagonal-well coordinates, we obtaining orbits in both the hexagonal and 3-body representations of the system, and plot the Poincare sections as a function of the relativistic energy parameter $\eta $. We find two broad categories of periodic and quasi-periodic motions that we refer to as the annulus and pretzel patterns, as well as a set of chaotic motions that appear in the region of phase-space between these two types. Despite the high degree of non-linearity in the relativistic system, we find that the the global structure of its phase space remains qualitatively the same as its non-relativisitic counterpart for all values of $\eta $ that we could study. However the relativistic system has a weaker symmetry and so its Poincare section develops an asymmetric distortion that increases with increasing $\eta $. For the post-Newtonian system we find that it experiences a KAM breakdown for $\eta \simeq 0.26$: above which the near integrable regions degenerate into chaos.