Separation of variables for massive spin-2 fields on the Kerr black hole

Abstract

Massive spin-2 fields on the Kerr black hole generally obey coupled partial differential equations. We show that these equations separate through two commuting angular operators acting on five components of the field, its curl and its linearised Weyl tensor. A joint angular eigenfunction fixes the relative profiles of the five polarisations and leaves one second-order radial equation. The separation constants satisfy an algebraic relation that follows from the angular operators themselves. The construction removes the obstruction to a Teukolsky-type equation by a radial transport identity, then eliminates the additional radial solution through a first integral. It provides ordinary differential equations for massive spin-2 fields on Kerr, and, equivalently, for Kaluza-Klein perturbations of the Kerr black string.