A "Helium Atom" of Space: Dynamical Instability of the Isochoric Pentahedron

TL;DR

This paper investigates the chaotic dynamics of a specific polyhedron called the equifacial pentahedron within a volume-preserving Hamiltonian framework, revealing dominant local instability and strong chaos characteristics.

Contribution

It provides the first detailed analysis of the classical dynamics and chaos properties of the equifacial pentahedron, a key shape in loop quantum gravity.

Findings

Borders between configurations act as separatrices.

Locally unstable regions dominate the phase space.

The system exhibits strong chaos with positive Lyapunov exponents.

Abstract

We present an analysis of the dynamics of the equifacial pentahedron on the Kapovich-Millson phase space under a volume preserving Hamiltonian. The classical dynamics of polyhedra under such a Hamiltonian may arise from the classical limit of the node volume operators in loop quantum gravity. The pentahedron is the simplest nontrivial polyhedron for which the dynamics may be chaotic. We consider the distribution of polyhedral configurations throughout the space and find indications that the borders between certain configurations act as separatrices. We examine the local stability of trajectories within this phase space and find that locally unstable regions dominate although extended stable regions are present. Canonical and microcanonical estimates of the Kolmogorov-Sinai entropy suggest that the pentahedron is a strongly chaotic system. The presence of chaos is further suggested by calculations of intermediate time Lyapunov exponents which saturate to non zero values.