  We consider the axisymmetric, linear perturbations of Kerr-Newman black
holes, allowing for arbitrarily large (but subextremal) angular momentum and
electric charge. By exploiting the famous Carter-Robinson identities, developed
previously for the proofs of (stationary) black hole uniqueness results, we
construct a positive-definite energy functional for these perturbations and
establish its conservation for a class of (coupled, gravitational and
electromagnetic) solutions to the linearized field equations. Our analysis
utilizes the familiar (Hamiltonian) reduction of the field equations (for
axisymmetric geometries) to a system of wave map fields coupled to a
2+1-dimensional Lorentzian metric on the relevant quotient 3-manifold. The
propagating `dynamical degrees of freedom' of this system are entirely captured
by the wave map fields, which take their values in a four dimensional,
negatively curved (complex hyperbolic) Riemannian target space whereas the
base-space Lorentzian metric is entirely determined, in our setup, by elliptic
constraints and gauge conditions.
