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

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• A new exactly solvable quantum model in N dimensions
• Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability
• Higher-order superintegrability of separable potentials with a new approach to the Post-Winternitz system
• Curved manifolds with conserved Runge-Lenz vectors
• Second Hopf map and Yang-Coulomb system on 5d (pseudo)sphere
• Galactic Dark Matter and Bertrand Space-times
• A new approach to the higher-order superintegrability of the Tremblay-Turbiner-Winternitz system
• A Family of Exactly Solvable Radial Quantum Systems on Space of Non-Constant Curvature with Accidental Degeneracy in the Spectrum
• Classical and quantum higher order superintegrable systems from coalgebra symmetry
• On two superintegrable nonlinear oscillators in N dimensions
• Geometrical Formulation of Relativistic Mechanics
• The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories
• MICZ Kepler Systems in Noncommutative Space and Duality of Force Laws
• Possible potentials responsible for stable circular relativistic orbits
• Geodesic Congruences and Their Deformations in Bertrand Space-times
• New superintegrable models with position-dependent mass from Bertrand's Theorem on curved spaces
• The classical Darboux III oscillator: factorization, Spectrum Generating Algebra and solution to the equations of motion
• Superintegrable systems in non-Euclidean plane: hidden symmetries leading to linearity
• Symmetries and integrals of motion of a superintegrable deformed oscillator
• An infinite family of Dunkl type superintegrable curved Hamiltonians through coalgebra symmetry: Oscillator and Kepler-Coulomb models
• Curvature as an integrable deformation
• A maximally superintegrable deformation of the N-dimensional quantum Kepler-Coulomb system
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• Superintegrable potentials on 3D Riemannian and Lorentzian spaces with non-constant curvature
• Bertrand Spaces and Projected Closed Orbits in General Relativity
• Exactly solvable deformations of the oscillator and Coulomb systems and their generalization
• Isochrone spacetimes