Differentially rotating disks of dust

TL;DR

This paper introduces a new family of solutions to Einstein's equations describing differentially rotating dust disks, extending previous models to include variable angular velocities and complex mathematical representations.

Contribution

It generalizes the Neugebauer-Meinel solution to include differential rotation, using Riemann theta functions and Baecklund transformations for a broader class of disk models.

Findings

Solutions exhibit both increasing and decreasing angular velocity profiles.

Mathematically related to Jacobi's inversion problem and expressed via Riemann theta functions.

Includes a special subfamily connected to Baecklund transformations.

Abstract

We present a three-parameter family of solutions to the stationary axisymmetric Einstein equations that describe differentially rotating disks of dust. They have been constructed by generalizing the Neugebauer-Meinel solution of the problem of a rigidly rotating disk of dust. The solutions correspond to disks with angular velocities depending monotonically on the radial coordinate; both decreasing and increasing behaviour is exhibited. In general, the solutions are related mathematically to Jacobi's inversion problem and can be expressed in terms of Riemann theta functions. A particularly interesting two-parameter subfamily represents Baecklund transformations to appropriate seed solutions of the Weyl class.