Extreme Reissner-Nordström black holes can support a "superhair"

The spin-$\frac{3}{2}$ perturbations of the extreme Reissner-Nordström black holes are analyzed by linearizing the field equations of O(2) extended supergravity theory with respect to the Rarita-Schwinger fields. It is proven that, in contrast to all other black holes, the extreme Reissner-Nordström geometry does admit a class of static, regular, nontrivial spin-$\frac{3}{2}$ perturbations. The spin-$\frac{3}{2}$ fields which belong to this class are shown to carry the conserved spinor supercharge as a new parameter. The linearized form of a possible exact black-hole solution of the O(2) supergravity theory which possesses a mass, central charges, as well as a supercharge is thereby obtained.