Differentially rotating disks of dust: Arbitrary rotation law

TL;DR

This paper develops a method to generate solutions for the gravitational field of differentially rotating dust disks in general relativity, allowing for arbitrary rotation laws and including realistic and exotic disk configurations.

Contribution

It introduces a new class of solutions to the Ernst equation based on limiting Backlund transformations, enabling modeling of arbitrary differential rotation in dust disks.

Findings

Solutions depend on two real analytic functions on [0,1]

Method allows approximation of gravitational fields for given boundary conditions

Includes examples with realistic angular velocity profiles and ergoregions

Abstract

In this paper, solutions to the Ernst equation are investigated that depend on two real analytic functions defined on the interval [0,1]. These solutions are introduced by a suitable limiting process of Backlund transformations applied to seed solutions of the Weyl class. It turns out that this class of solutions contains the general relativistic gravitational field of an arbitrary differentially rotating disk of dust, for which a continuous transition to some Newtonian disk exists. It will be shown how for given boundary conditions (i. e. proper surface mass density or angular velocity of the disk) the gravitational field can be approximated in terms of the above solutions. Furthermore, particular examples will be discussed, including disks with a realistic profile for the angular velocity and more exotic disks possessing two spatially separated ergoregions.