Chaos in a 3-body Self-Gravitating Cosmological Spacetime

Abstract

We investigate the equal-mass 3-body system in general relativistic lineal gravity in the presence of a cosmological constant $Λ$. The cosmological vacuum energy introduces features that do not have a non-relativistic counterpart, inducing a competing expansion/contraction of spacetime that competes with the gravitational self-attraction of the bodies. We derive a canonical expression for the Hamiltonian of the system and discuss the numerical solution of the resulting equations of motion. As for the system with $Λ=0$, we find that the structure of the phase space yields a rich 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. However unlike the $Λ=0$ case, we find that a negative cosmological constant considerably diminishes the amount of chaos in the system, even beyond that of the $Λ=0$ non-relativistic system. By contrast, a positive cosmological constant considerably enhances the amount of chaos, typically leading to KAM breakdown.