Bertrand spacetimes as Kepler/oscillator potentials

Abstract

Perlick's classification of (3+1)-dimensional spherically symmetric and static spacetimes (\cal M,η=-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,η) 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.

17 pages. Comments on the differences between our results/approach and those given by Perlick have been added together with three new references. To appear in Class. Quantum Grav

References in corpus (8)

• Superintegrable Systems in Darboux spaces
• A maximally superintegrable system on an n-dimensional space of nonconstant curvature
• Hidden symmetry of hyperbolic monopole motion
• Multi-center MICZ-Kepler system, supersymmetry and integrability
• N-dimensional sl(2)-coalgebra spaces with non-constant curvature
• Anisotropic inharmonic Higgs oscillator and related (MICZ-)Kepler-like systems
• Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: I. Darboux Spaces DI and DII
• Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: II. Darboux Spaces DIII and DIV

Cited by in corpus (2)

• Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles
• Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds