Bertrand spacetimes as Kepler/oscillator potentials

Details Bertrand spacetimes as Kepler/oscillator potentials Bertrand spacetimes as Kepler/oscillator potentials

2008-03-24

arxiv.org/abs/0803.3430v2

Class. Quantum Grav. 25 (2008) 165005(13)

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

17 pages. Comments on the differences between our results/approach and those given by Perlick have been added together with th

Scientific paper

10.1088/0264-9381/25/16/165005

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.

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