Carter-like constants of the motion in Newtonian gravity and electrodynamics

Abstract

For a test body orbiting an axisymmetric body in Newtonian gravitational theory with multipole moments Q_L, (and for a charge in a non-relativistic orbit about a charge distribution with the same multipole moments) we show that there exists, in addition to the energy and angular momentum component along the symmetry axis, a conserved quantity analogous to the Carter constant of Kerr spacetimes in general relativity, if the odd-L moments vanish, and the even-L moments satisfy Q_2L = m (Q_2/m)^L. Strangely, this is precisely the relation among mass moments enforced by the no-hair theorems of rotating black holes. By contrast, if Newtonian gravity is supplemented by a multipolar gravitomagnetic field, whose leading term represents frame-dragging (or if the electrostatic field is supplemented by a multipolar magnetic field), we are unable to find an analogous Carter-like constant. This further highlights the very special nature of the Kerr geometry of general relativity.

4 pages

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• Anomalous Dimers in Quantum Mixtures near Broad Resonances: Pauli Blocking, Fermi Surface Dynamics and Implications
• A Newtonian analogue of Kerr black hole
• Mimicking Kerr's multipole moments
• Carter-like constants of motion in the Newtonian and relativistic two-center problems
• Tidally-induced nonlinear resonances in EMRIs with an analogue model
• Chaotic particle motion around a homogeneous circular ring