On explicit solutions to the stationary axisymmetric Einstein-Maxwell equations describing dust disks

We review explicit solutions to the stationary axisymmetric Einstein-Maxwell equations which can be interpreted as disks of charged dust. The disks of finite or infinite extension are infinitesimally thin and constitute a surface layer at the boundary of an electro-vacuum. The Einstein-Maxwell equations in the presence of one Killing vector are obtained by using a projection formalism. This leads to equations for three-dimensional gravity where the matter is given by a $SU(2,1)/S[U(1,1)\times U(1)]$ nonlinear sigma model. The $SU(2,1)$ invariance of the stationary Einstein-Maxwell equations can be used to construct solutions for the electro-vacuum from solutions to the pure vacuum case via a so-called Harrison transformation. It is shown that the corresponding solutions will always have a non-vanishing total charge and a gyromagnetic ratio of 2. Since the vacuum and the electro-vacuum equations in the stationary axisymmetric case are completely integrable, large classes of solutions can be constructed with techniques from the theory of solitons. The richest class of physically interesting solutions to the pure vacuum case due to Korotkin is given in terms of hyperelliptic theta functions. Harrison transformed hyperelliptic solutions are discussed. As a concrete example we study the transformation of a family of counter-rotating dust disks. To obtain algebro-geometric solutions with vanishing total charge which are of astrophysical relevance, three-sheeted surfaces have to be considered. The matter in the disk is discussed following Bi\v{c}\'ak et al. We review the `cut and glue' technique where a strip is removed from an explicitly known spacetime and where the remainder is glued together after displacement. The discontinuities of the normal derivatives of the metric at the glueing hypersurface lead to infinite disks. If the energy conditions are satisfied and if the pressure is positive, the disks can be interpreted in the vacuum case as made up of two components of counter-rotating dust moving on geodesics. In electro-vacuum the condition of geodesic movement is replaced by electro-geodesic movement. As an example we discuss a class of Harrison-transformed hyperelliptic solutions. The range of parameters is identified where an interpretation of the matter in the disk in terms of electro-dust can be given