Harrison transformation of hyperelliptic solutions and charged dust disks

TL;DR

This paper applies the Harrison transformation to hyperelliptic solutions of the Ernst equation to generate new Einstein-Maxwell solutions, specifically modeling charged dust disks with various physical properties and limits.

Contribution

It provides explicit analytic expressions for transformed solutions and explores their physical interpretations and limiting behaviors.

Findings

Derived new solutions representing charged dust disks with currents

Analyzed extreme and ultrarelativistic limits of these solutions

Connected solutions to physical scenarios like charged matter and streams

Abstract

We use a Harrison transformation on solutions to the stationary axisymmetric Einstein equations to generate solutions of the Einstein-Maxwell equations. The case of hyperelliptic solutions to the Ernst equation is studied in detail. Analytic expressions for the metric and the multipole moments are obtained. As an example we consider the transformation of a family of counter-rotating dust disks. The resulting solutions can be interpreted as disks with currents and matter with a purely azimuthal pressure or as two streams of freely moving charged particles. We discuss interesting limiting cases as the extreme limit where the charge becomes identical to the mass, and the ultrarelativistic limit where the central redshift diverges.