Constants of motion in stationary axisymmetric gravitational fields

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-Stäckel 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 $M_{2l+4}=(a^2+b^2)M_{2l+2}-a^2{b}^2{M}_{2l}$. The full set of quartic integrability conditions, however, cannot be satisfied nontrivially by any stationary axisymmetric vacuum potential.

13 pages, matches published version

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• Non-linear effects in EMRI dynamics and their imprints on gravitational waves
• A Newtonian analogue of Kerr black hole
• Non-Linear Effects in Non-Kerr spacetimes
• Integrable and Superintegrable Potentials of 2d Autonomous Conservative Dynamical Systems
• Reducibility of Valence-3 Killing Tensors in Weyl's Class of Stationary and Axially Symmetric Space-Times