Spin-3/2 perturbations of algebraically special solutions of the Einstein–Maxwell equations

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Abstract
The equations for the spin- (3)/(2) perturbations of the solutions of the Einstein–Maxwell equations given by the linearized O(2) extended supergravity are considered. It is shown that for each geodetic and shear-free principal null direction of the background electromagnetic field there exists a gauge-invariant quantity made out of the spin- (3)/(2) field that satisfies a decoupled equation. In the case of type-D solutions with a nonsingular aligned electromagnetic field it is explicitly shown that the decoupled equations associated with the two principal null directions admit separable solutions and the separated functions obey certain differential relations.