Mimicking Kerr's multipole moments

Multipole moments carry a lot of information about the gravitational field. Nonetheless, knowing all the multipole moments of an object does not determine conclusively the nature of the object itself. In particular, the multipole moments of the Kerr spacetime are not unique. Here we construct several physically motivated Newtonian objects with multipole moments identical to those of Kerr. Moreover, we also provide a description of how to include post-Newtonian corrections to these objects without changing their multipole moments.