Generating rotating fields in general relativity

I present a new method to generate rotating solutions of the Einstein–Maxwell equations from static solutions, give several examples of its application, and discuss its general properties. When dealing with exact stationary solutions of the Einstein equations, one sometimes stumbles on the questions, quite easy to ask, but rather difficult to answer: Given some static solution, what is the family of non-static (rotating) solutions which are near —in some sense— this static solution? On how many parameters do these solutions depend? And (last but not least) how can one practically generate these rotating solutions from the static solution? In principle, these questions can be answered in the context of the Geroch group [1]. Let us recall that the 4–dimensional stationary Einstein (resp. Einstein–Maxwell) equations are invariant under the group O(2,1) (resp. SU(2,1)) of generalized Ehlers–Harrison transformations [2]. In the case of stationary axisymmetric solutions, with two commuting Killing vectors ∂t and ∂φ, the combination of the invariance transformations associated with a given direction in the Killing 2–plane with rotations in this plane leads to the infinite–dimensional Geroch group. These transformations allow in principle the generation of all solutions of the stationary axisymmetric Einstein (or Einstein–Maxwell) problem, which is thus completely integrable. This generation of stationary axisymmetric solutions can be achieved in a Talk presented at the 10 Russian Gravitational Conference, Vladimir (Russia) 20-27 June 1999 Present address: Laboratoire de Physique Théorique LAPTH, B.P.110, F-74941 Annecy-le Vieux cedex, France. E-mail: [email protected]