Constants of motion in stationary axisymmetric gravitational fields

TL;DR

This paper investigates the existence of constants of motion in stationary axisymmetric gravitational fields, establishing conditions for quadratic constants and proving the nonexistence of quartic constants, thus clarifying the integrability of such systems.

Contribution

It provides a systematic analysis of constants of motion in Newtonian analogues, proving uniqueness of quadratic constants and nonexistence of quartic ones in these fields.

Findings

Quadratic constants exist and are unique under certain conditions.

No nontrivial quartic constants exist in these gravitational fields.

Mass moments follow specific 'no-hair' recursion relations.

Abstract

The motion of test particles in stationary axisymmetric gravitational fields is generally nonintegrable unless a nontrivial constant of motion, in addition to energy and angular momentum along the symmetry axis, exists. The Carter constant in Kerr-de Sitter spacetime is the only example known to date. Proposed astrophysical tests of the black-hole no-hair theorem have often involved integrable gravitational fields more general than the Kerr family, but the existence of such fields has been a matter of debate. To elucidate this problem, we treat its Newtonian analogue by systematically searching for nontrivial constants of motion polynomial in the momenta and obtain two theorems. First, solving a set of quadratic integrability conditions, we establish the existence and uniqueness of the family of stationary axisymmetric potentials admitting a quadratic constant. As in Kerr-de Sitter spacetime, the mass moments of this class satisfy a "no-hair" recursion relation $M_{2l+2}=a^2 M_{2l}$, and the constant is Noether-related to a second-order Killing-St\"ackel tensor. Second, solving a new set of quartic integrability conditions, we establish nonexistence of quartic constants. Remarkably, a subset of these conditions is satisfied when the mass moments obey a generalized "no-hair" recursion relation $M_{2l+4}=(a^2+b^2)M_{2l+2}-a^2b^2 M_{2l}$. The full set of quartic integrability conditions, however, cannot be satisfied nontrivially by any stationary axisymmetric vacuum potential.