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arXiv · 0803.3430

Bertrand spacetimes as Kepler/oscillator potentials

Angel Ballesteros·Alberto Enciso·Francisco J. Herranz·Orlando Ragnisco
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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.

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Angel Ballesteros, Alberto Enciso, Francisco J. Herranz, Orlando Ragnisco. 2008-06-12. Bertrand spacetimes as Kepler/oscillator potentials. https://doi.org/10.1088/0264-9381%2F25%2F16%2F165005

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