Rotating dust solutions of Einstein's equations with 3-dimensional symmetry groups, Part 2: One Killing field spanned on u^{\alpha} and w^{\alpha}

TL;DR

This paper investigates rotating dust solutions of Einstein's equations with 3D symmetry groups, identifying new solutions and relating them to known classes, under specific symmetry assumptions involving Killing fields.

Contribution

It introduces a method to find rotating dust solutions with a single Killing field aligned with velocity and rotation, discovering three new solutions and classifying known ones.

Findings

Found three new solutions to Einstein's equations for rotating dust.

Solved Killing equations for metric components under specific symmetry assumptions.

Classified known solutions within the new framework.

Abstract

This is the second part of a series of 3 papers. Using the same method and the same coordinates as in part 1, rotating dust solutions of Einstein's equations are investigated that possess 3-dimensional symmetry groups, under the assumption that only one of the Killing fields is spanned on the fields of velocity $u^{\alpha}$ and rotation $w^{\alpha}$, while the other two define vectors that are linearly independent of $u^{\alpha}$ and $w^{\alpha}$ at every point of the spacetime region under consideration. The Killing fields are found and the Killing equations solved for the components of the metric tensor in every case that arises. The Einstein equations are simplified in a few cases, three (most probably) new solutions are found, and several classes of solutions known earlier are identified in the present scheme. They include those by Ozsv\'ath, Maitra, Ellis, King and Vishveshwara and Winicour. PACS numbers 04.20.-q, 04.20.Cv, 04.20.Jb, 04.40.+c