Exact relativistic treatment of stationary counter-rotating dust disks II: Axis, Disk and Limiting Cases

TL;DR

This paper presents an exact relativistic solution for stationary counter-rotating dust disks, analyzing their properties and limits such as Newtonian, static, and ultra-relativistic cases, using advanced mathematical functions.

Contribution

It provides explicit solutions to Einstein's equations for counter-rotating dust disks with detailed analysis of their metric functions and limiting behaviors.

Findings

Solutions expressed via theta functions on genus 2 Riemann surfaces

Analysis of metric functions at the axis and disk

Identification of key physical limits like Newtonian and ultra-relativistic

Abstract

This is the second in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which can be interpreted as counter-rotating disks of dust. We discuss the class of solutions to the Einstein equations for disks with constant angular velocity and constant relative density which was constructed in the first part. The metric for these spacetimes is given in terms of theta functions on a Riemann surface of genus 2. We discuss the metric functions at the axis of symmetry and the disk. Interesting limiting cases are the Newtonian limit, the static limit, and the ultra-relativistic limit of the solution in which the central redshift diverges.