Rotating dust solutions of Einstein's equations with 3-dimensional symmetry groups, Part 3: All Killing fields linearly independent of u^{\alpha} and w^{\alpha}

TL;DR

This paper investigates rotating dust solutions of Einstein's equations with 3D symmetry groups, focusing on cases where Killing vectors are linearly independent of velocity and rotation, but finds no new solutions or progress with Einstein's equations.

Contribution

It systematically analyzes symmetry conditions for rotating dust solutions, identifying Killing fields but does not find new solutions or advance the Einstein equations.

Findings

Killing fields are identified under the specified conditions.

No new solutions to Einstein's equations are found.

The analysis does not progress in solving Einstein's equations for these cases.

Abstract

This is the third and last part of a series of 3 papers. Using the same method and the same coordinates as in parts 1 and 2, rotating dust solutions of Einstein's equations are investigated that possess 3-dimensional symmetry groups, under the assumption that each of the Killing vectors is linearly independent of velocity $u^{\alpha}$ and rotation $w^{\alpha}$ at every point of the spacetime region under consideration. The Killing fields are found and the Killing equations are solved for the components of the metric tensor in every case that arises. No progress was made with the Einstein equations in any of the cases, and no previously known solutions were identified. A brief overview of literature on solutions with rotating sources is given.