  The Carter tensor is a Killing tensor of the Kerr-Newman spacetime, and its
existence implies the separability of the wave equation. Nevertheless, the
Carter operator is known to commute with the D'Alembertian only in the case of
a Ricci-flat metric. We show that, even though the Kerr-Newman spacetime
satisfies the non-vacuum Einstein-Maxwell equations, its curvature and
electromagnetic tensors satisfy peculiar properties which imply that the Carter
operator still commutes with the wave equation. This feature allows to adapt to
Kerr-Newman the physical-space analysis of the wave equation in Kerr by
Andersson-Blue, which avoids frequency decomposition of the solution by
precisely making use of the commutation with the Carter operator.
  We also extend the mathematical framework of physical-space analysis to the
case of the Einstein-Maxwell equations on Kerr-Newman spacetime, representing
coupled electromagnetic-gravitational perturbations of the rotating charged
black hole. The physical-space analysis is crucial in this setting as the
coupling of spin-1 and spin-2 fields in the axially symmetric background
prevents the separation in modes as observed by Chandrasekhar, and therefore
represents an important step towards an analytical proof of the stability of
the Kerr-Newman black hole.
