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

TL;DR

This paper reviews explicit solutions to the stationary axisymmetric Einstein-Maxwell equations that describe charged dust disks, highlighting solution construction methods and physical properties such as charge and gyromagnetic ratio.

Contribution

It introduces a systematic approach to generate solutions for charged dust disks using SU(2,1) invariance and Harrison transformations, extending pure vacuum solutions.

Findings

Solutions always have non-zero total charge

Solutions exhibit a gyromagnetic ratio of 2

Hyperelliptic theta functions characterize the most complex solutions

Abstract

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. 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. The Harrison transformed hyperelliptic solutions are discussed.