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