  A new version of the Teukolksy Master Equation, describing any massless field
of different spin $s=1/2,1,3/2,2$ in the Kerr black hole, is presented here in
the form of a wave equation containing additional curvature terms. These
results suggest a relation between curvature perturbation theory in general
relativity and the exact wave equations satisfied by the Weyl and the Maxwell
tensors, known in the literature as the de Rham-Lichnerowicz Laplacian
equations. We discuss these Laplacians both in the Newman-Penrose formalism and
in the Geroch-Held-Penrose variant for an arbitrary vacuum spacetime.
Perturbative expansion of these wave equations results in a recursive scheme
valid for higher orders. This approach, apart from the obvious implications for
the gravitational and electromagnetic wave propagation on a curved spacetime,
explains and extends the results in the literature for perturbative analysis by
clarifying their true origins in the exact theory.
