Coupled electromagnetic and gravitational linearized perturbations of the Kerr–Newman metric

A new gauge and tetrad invariant equation governing the coupled electromagnetic-gravitational perturbations for the Kerr–Newman metric and involving a closed 2-form is obtained. Operating on this equation with the complex vectorial formalism as developed by Crossman, we have found three equations for the gauge and tetrad invariant quantities χ1B(≡3ψ2Aφ0B−2φ1Aψ1B), χ−1B(≡3ψ2Aφ2B−2φ1Aψ3B), and χ0B(≡3ψ2Aφ1B−2φ1Aψ2B +2φ5/21Aφ̄−1/21Aφ̄1B).