Bertrand spacetimes as Kepler/oscillator potentials

TL;DR

This paper revisits Perlick's classification of spherically symmetric static spacetimes where classical Bertrand theorem applies, showing that the potential must be Kepler-Coulomb or harmonic oscillator type on associated Riemannian manifolds, extending Euclidean dynamics to curved spaces.

Contribution

It proves that Bertrand spacetimes' potentials are either Kepler-Coulomb or harmonic oscillator types on their Riemannian manifolds, identifying specific spaces where these potentials apply.

Findings

Potential V(r) is either Kepler-Coulomb or harmonic oscillator type.

Explicit identification of classical spaces of constant curvature, Darboux, and Iwai-Katayama spaces.

Extension of Euclidean Kepler and oscillator dynamics to curved 3D spaces.

Abstract

Perlick's classification of (3+1)-dimensional spherically symmetric and static spacetimes (\cal M,\eta=-1/V dt^2+g) for which the classical Bertrand theorem holds [Perlick V Class. Quantum Grav. 9 (1992) 1009] is revisited. For any Bertrand spacetime (\cal M,\eta) the term V(r) is proven to be either the intrinsic Kepler-Coulomb or the harmonic oscillator potential on its associated Riemannian 3-manifold (M,g). Among the latter 3-spaces (M,g) we explicitly identify the three classical Riemannian spaces of constant curvature, a generalization of a Darboux space and the Iwai-Katayama spaces generalizing the MIC-Kepler and Taub-NUT problems. The key dynamical role played by the Kepler and oscillator potentials in Euclidean space is thus extended to a wide class of 3-dimensional curved spaces.