Rotating dust solutions of Einstein's equations with 3-dimensional symmetry groups; Part 1: Two Killing fields spanned on u^{\alpha} and w^{\alpha }

TL;DR

This paper classifies rotating dust solutions of Einstein's equations with a 3D symmetry group, focusing on cases with two Killing fields spanned on velocity and rotation, identifying known solutions and introducing new invariant definitions.

Contribution

It provides a comprehensive classification of such solutions using Plebański's formalism, including new invariant characterizations of known solutions.

Findings

Lanczos and G"odel solutions identified as special cases

New invariant definitions for solutions are introduced

Classification considers all orientations of symmetry group orbits

Abstract

For a rotating dust with a 3-dimensional symmetry group all possible metric forms can be classified and, within each class, explicitly written out. This is made possible by the formalism of Pleba\'nski based on the Darboux theorem. In the resulting coordinates, the Killing vector fields (if any exist) assume a special form. Each Killing vector field may be either spanned on the fields of velocity and rotation or linearly independent of them. By considering all such cases one arrives at the classification. With respect to the structures of the groups, this is just the Bianchi classification, but with all possible orientations of the orbits taken into account. In this paper, which is part 1 of a 3-part series, all solutions are considered for which two Killing fields are spanned on velocity and rotation. The solutions of Lanczos and G\"{o}del are identified as special cases, and their new invariant definitions are provided. In addition, a new invariant definition is given of the Ozsvath class III solution.