Gauge invariant perturbations of Petrov type D space-times

The Regge-Wheeler and Zerilli equations are satisfied by gauge invariant perturbations of the Schwarzschild black hole geometry. Both the perturbation of the imaginary part of $\Psi_2$ (a component of the Weyl curvature), and its time derivative, are gauge invariant and solve the Regge-Wheeler equation with different sources. The $\Psi_0$ and $\Psi_4$ perturbations of the Weyl curvature are not only gauge, but also tetrad, invariant. We explore the framework in which these results hold, and consider what generalizations may extend to the Kerr geometry, and presumably to Petrov type D space-times in general.