Black Holes and Solution Generating Symmetries in Gravity

curvature.png#mediaviewer/File:Spacetime_curvature.png, [1]. The colors are in-verted from original. A gravity theory in D dimensions with a spacetime metric that admits n com-muting Killing vectors can be dimensionally reduced to an effectively (D − n)-dimensional theory. The reduction is performed by using Kaluza-Klein techniques and, as it turns out, the lower dimensional theory reveals a hidden global symme-try, described by a group G, on the space of solutions. This symmetry is not seen in the original D-dimensional theory. When the solutions admit a sufficient num-ber of Killing vectors we can dimensionally reduce down to D = 3 where we get a non-linear sigma-model. This sigma-model contains scalar fields which originate from the D-dimensional metric and whatever other possible D-dimensional fields the theory may contain. These scalar fields can also be described as parameters in a coset space G/H which depends on the particular D-dimensional theory we started from. If one rewrites the sigma-model in terms of coset representative