Covariant perturbed wave equations in arbitrary type-D backgrounds

We present an approach to the fundamental tensorial quantities of general relativity which is inherently covariant and based on the irreducible representations of the Lorentz group, O(3,1). Using this technique, the wave equations appropriate to perturbing, massless, D(0,s), fields in an arbitrary curved background are studied and a relationship between the decoupling of (at least) one of the equations and the algebraic degeneracy of the spacetime is shown. It is then found that sufficient conditions for decoupling the equations determining both of the radiative components (the extremal helicities) are that the space be of type D. Using Plebański–Demiański coordinates to describe such an arbitrary vacuum spacetime (of type D), we separate the (decoupled) perturbation equations for the radiative components corresponding to spin s=0, 1/2, 1, and 2.