On the symmetry orbits of black holes in non-linear sigma models

Breitenlohner, Maison and Gibbons claimed some time ago that all bona-fide four dimensional asymptotically flat non-degenerate black holes are in a symmetry orbit of theSchwarzschild/Kerrblackholeinalargesetoftheoriesofgravityandmatter. Their argument involved reducing the theory on a time-like Killing vector field and analysing the resulting three dimensional sigma model of maps to a symmetric space G/H. In the construction of their proof, they conjectured the existence of a suitable H-transformation that always remove the electromagnetic charges of the four dimensional black hole solution. We show in this short note that such a transformation does not exist in general, and discuss a set of boundary conditions on the horizon for the scalar fields in the sigma model that yield black holes for which the result by Breitenlohner, Maison and Gibbonscan beapplied.