Spacetime Encodings II - Pictures of Integrability

TL;DR

This paper visually explores geodesic orbits in stationary axisymmetric vacuum spacetimes, highlighting features of integrability and chaos, and discusses the mathematical structures underlying these orbits with examples from general relativity.

Contribution

It provides a visual and conceptual analysis of geodesic integrability in SAV spacetimes, emphasizing the potential existence of a fourth constant of motion.

Findings

Evidence of orbit-crossing structures suggesting a quartic constant of motion.

Illustrations of the connection between Hamilton-Jacobi equations and Killing tensors.

Insights into the geometric features distinguishing integrable and chaotic orbits.

Abstract

I visually explore the features of geodesic orbits in arbitrary stationary axisymmetric vacuum (SAV) spacetimes that are constructed from a complex Ernst potential. Some of the geometric features of integrable and chaotic orbits are highlighted. The geodesic problem for these SAV spacetimes is rewritten as a two degree of freedom problem and the connection between current ideas in dynamical systems and the study of two manifolds sought. The relationship between the Hamilton-Jacobi equations, canonical transformations, constants of motion and Killing tensors are commented on. Wherever possible I illustrate the concepts by means of examples from general relativity. This investigation is designed to build the readers' intuition about how integrability arises, and to summarize some of the known facts about two degree of freedom systems. Evidence is given, in the form of orbit-crossing structure, that geodesics in SAV spacetimes might admit, a fourth constant of motion that is quartic in momentum (by contrast with Kerr spacetime, where Carter's fourth constant is quadratic).