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

TL;DR

This paper demonstrates the existence of Carter-like conserved quantities in Newtonian gravity and electrostatics for axisymmetric bodies with specific multipole moment relations, highlighting the unique nature of Kerr spacetime in general relativity.

Contribution

It identifies conditions under which Carter-like constants exist in Newtonian and electrostatic systems, revealing a surprising link to black hole no-hair theorems.

Findings

Carter-like constants exist when odd multipole moments vanish and even moments follow a specific relation.

No Carter-like constants are found in Newtonian systems with multipolar gravitomagnetic or magnetic fields.

The results emphasize the special properties of Kerr geometry in general relativity.

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.