Rotating Bianchi type V dust models generalizing the k = -1 Friedmann model

TL;DR

This paper investigates rotating Bianchi type V dust cosmological models, deriving their field equations, symmetries, and limits, including the relation to the k = -1 Friedmann model, and explores integrability and constraints.

Contribution

It provides a detailed analysis of rotating Bianchi type V dust models, including their differential equations, symmetries, and the relation to known cosmological solutions, with new insights into their integrability.

Findings

The k = -1 Friedmann model is a limit of the rotating Bianchi V models.

The field equations reduce to three second-order ODEs with a first integral.

No polynomial first integrals of degree 1 or 2 in derivatives exist for the system.

Abstract

The Einstein equations for one of the hypersurface-homogeneous rotating dust models are investigated. It is a Bianchi type V model in which one of the Killing fields is spanned on velocity and rotation (case 1.2.2.2 in the classification scheme of the earlier papers). A first integral of the field equations is found, and with a special value of this integral coordinate transformations are used to eliminate two components of the metric. The k = -1 Friedmann model is shown to be contained among the solutions in the limit of zero rotation. The field equations for the simplified metric are reduced to 3 second-order ordinary differential equations that determine 3 metric components plus a first integral that algebraically determines the fourth component. First derivatives of the metric components are subject to a constraint (a second-degree polynomial with coefficients depending on the functions). It is shown that the set does not follow from a Lagrangian of the Hilbert type. The group of Lie point-symmetries of the set is found, it is two-dimensional noncommutative. Finally, a method of searching for first integrals (for sets of differential equations) that are polynomials of degree 1 or 2 in the first derivatives is applied. No such first integrals exist. The method is used to find a constraint (of degree 1 in first derivatives) that could be imposed on the metric, but it leads to a vacuum solution, and so is of no interest for cosmology.