Bertrand spacetimes as Kepler / oscillator potentials

Perlick’s classification of (3 + 1)-dimensional spherically symmetric and static spacetimes (M, η = − 1 V dt + g) for which the classical Bertrand theorem holds [Perlick V 1992 Class. Quantum Grav. 9 1009] is revisited. For any Bertrand spacetime (M, η) the term V (r) is proven to be either the intrinsic Kepler– Coulomb or the harmonic oscillator potential on its associated Riemannian 3manifold (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 Iway–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 curved spacetimes. PACS: 04.20.-q 02.40.Ky 02.30.Ik