  We derive the equations governing the linear stability of Kerr-Newman
spacetime to coupled electromagnetic-gravitational perturbations. The equations
generalize the celebrated Teukolsky equation for curvature perturbations of
Kerr, and the Regge-Wheeler equation for metric perturbations of
Reissner-Nordstr\"om. Because of the "apparent indissolubility of the coupling
between the spin-1 and spin-2 fields", as put by Chandrasekhar, the stability
of Kerr-Newman spacetime can not be obtained through standard decomposition in
modes. Due to the impossibility to decouple the modes of the gravitational and
electromagnetic fields, the equations governing the linear stability of
Kerr-Newman have not been previously derived. Using a tensorial approach that
was applied to Kerr, we produce a set of generalized Regge-Wheeler equations
for perturbations of Kerr-Newman, which are suitable for the study of
linearized stability by physical space methods. The physical space analysis
overcomes the issue of coupling of spin-1 and spin-2 fields and represents the
first step towards an analytical proof of the stability of the Kerr-Newman
black hole.
