Carter-like constants of motion in the Newtonian and relativistic two-center problems

TL;DR

This paper investigates the existence of Carter-like constants of motion in Newtonian and relativistic two-center problems, revealing that such constants exist in Newtonian gravity under specific conditions but not in the relativistic case, including first post-Newtonian order.

Contribution

It demonstrates the absence of Carter-like constants in the relativistic two-center problem, extending previous Newtonian results and analyzing the Bach-Weyl solution.

Findings

Carter-like constants exist in Newtonian two-center systems with specific multipole relations.

The relativistic Bach-Weyl solution does not admit a Carter-like constant.

No non-trivial Killing tensor exists for the relativistic two-center problem.

Abstract

In Newtonian gravity, a stationary axisymmetric system admits a third, Carter-like constant of motion if its mass multipole moments are related to each other in exactly the same manner as for the Kerr black-hole spacetime. The Newtonian source with this property consists of two point masses at rest a fixed distance apart. The integrability of motion about this source was first studied in the 1760s by Euler. We show that the general relativistic analogue of the Euler problem, the Bach-Weyl solution, does not admit a Carter-like constant of motion, first, by showing that it does not possess a non-trivial Killing tensor, and secondly, by showing that the existence of a Carter-like constant for the two-center problem fails at the first post-Newtonian order.