On the Integrability of the Geodesic Flow on a Friedmann-Robertson-Walker Spacetime

TL;DR

This paper demonstrates that the geodesic flow on Friedmann-Robertson-Walker spacetimes is completely integrable, utilizing the separability of the Hamilton-Jacobi equation and the Killing fields of the metric.

Contribution

It establishes the complete integrability of geodesic flow on FRW spacetimes and constructs invariant submanifolds for closed universes, advancing understanding of spacetime symmetries.

Findings

Geodesic flow is completely integrable in the Liouville-Arnold sense.

The Hamilton-Jacobi equation is fully separable on the cotangent bundle.

Constructed invariant submanifolds for spatially closed universes.

Abstract

We study the geodesic flow on the cotangent bundle of a Friedman-Robertson-Walker spacetime (M, g). On this bundle, the HamiltonJacobi equation is completely separable and this separability leads us to construct four linearly independent integrals in involution i.e. Poisson commuting amongst themselves and pointwise linearly independent. These integrals involve the six linearly independent Killing fields of the background metric g. As a consequence, the geodesic flow on an FRW background is completely integrable in the Liouville-Arnold sense. For the case of a spatially closed universe we construct families of invariant by the flow sub manifolds.