Bertrand spacetimes as Kepler/oscillator potentials
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.
Cite
@article{arxiv.0803.3430,
title = {Bertrand spacetimes as Kepler/oscillator potentials},
author = {Angel Ballesteros and Alberto Enciso and Francisco J. Herranz and Orlando Ragnisco},
journal= {arXiv preprint arXiv:0803.3430},
year = {2009}
}
Comments
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