3-Body Dynamics in a (1+1) Dimensional Relativistic Self-Gravitating System

TL;DR

This study explores the complex dynamics of a three-particle relativistic self-gravitating system in one dimension, revealing diverse phase space behaviors including chaos, and compares relativistic effects to non-relativistic models.

Contribution

It derives a Hamiltonian for the three-particle system, analyzes its dynamics across mass ratios, and identifies new chaotic regions absent in equal-mass cases.

Findings

Identification of three distinct phase space regions: annulus, pretzel, and chaos.

Relativistic corrections modify the size and structure of phase space regions.

Unequal masses introduce additional chaotic behaviors not seen in equal-mass systems.

Abstract

The results of our study of the motion of a three particle, self-gravitating system in general relativistic lineal gravity is presented for an arbitrary ratio of the particle masses. We derive a canonical expression for the Hamiltonian of the system and discuss the numerical solution of the resulting equations of motion. This solution is compared to the corresponding non-relativistic and post-Newtonian approximation solutions so that the dynamics of the fully relativistic system can be interpretted as a correction to the one-dimensional Newtonian self-gravitating system. We find that the structure of the phase space of each of these systems yields a large variety of interesting dynamics that can be divided into three distinct regions: annulus, pretzel, and chaotic; the first two being regions of quasi-periodicity while the latter is a region of chaos. By changing the relative masses of the three particles we find that the relative sizes of these three phase space regions changes and that this deformation can be interpreted physically in terms of the gravitational interactions of the particles. Furthermore, we find that many of the interesting characteristics found in the case where all of the particles share the same mass also appears in our more general study. We find that there are additional regions of chaos in the unequal mass system which are not present in the equal mass case. We compare these results to those found in similar systems.