1202.5228 (Charalampos Markakis)

Constants of motion in stationary axisymmetric gravitational fields    [PDF]

Charalampos Markakis

The motion of a test particle in a stationary axisymmetric gravitational field is generally nonintegrable unless, in addition to the energy and angular momentum about the symmetry axis, an extra nontrivial constant of motion exists. We use a direct approach to systematically search for a nontrivial constant of motion polynomial in the momenta. By solving a set of quadratic integrability conditions, we establish the existence and uniqueness of the family of stationary axisymmetric Newtonian potentials admitting a nontrivial constant quadratic in the momenta. Although such constants do not arise from a group of diffeomorphisms, they are Noether-related to symmetries of the action and associated with irreducible rank-2 Killing-St\"ackel tensors. The multipole moments of this class of potentials satisfy a no-hair recursion relation $M_{2l+2}=a^2 M_{2l}$ and the associated quadratic constant is the Newtonian analogue of the Carter constant in a Kerr-de Sitter spacetime. We further explore the possibility of invariants quartic in the momenta associated with rank-4 Killing-St\"ackel tensors and derive a new set of quartic integrability conditions. We show that a subset of the quartic integrability conditions are satisfied by potentials whose even multipole moments satisfy a generalized no-hair recursion relation $M_{2l+4}=(a^2+b^2)M_{2l+2}-a^2b^2 M_{2l}$. However, the full set of quartic integrability conditions cannot be satisfied nontrivially by any Newtonian stationary axisymmetric vacuum potential. We thus establish the nonexistence of irreducible invariants quartic in the momenta for motion in such potentials.

View original: http://arxiv.org/abs/1202.5228