Cosmology in three dimensions: steps towards the general solution

TL;DR

This paper explores the properties and solutions of three-dimensional relativistic cosmology using covariant and first-order formalisms, providing new solutions, analyzing perturbations, and studying specific spacetime structures.

Contribution

It introduces new solutions for 3D cosmologies, extends known models, and applies covariant and first-order formalisms to analyze general relativistic spacetimes in three dimensions.

Findings

Derived the general solution for dust-filled 3D cosmologies with cosmological constant.

Extended known solutions to include non-comoving dust and scalar fields.

Analyzed asymptotic behavior and singular structures of 3D cosmological models.

Abstract

We use covariant and first-order formalism techniques to study the properties of general relativistic cosmology in three dimensions. The covariant approach provides an irreducible decomposition of the relativistic equations, which allows for a mathematically compact and physically transparent description of the 3-dimensional spacetimes. Using this information we review the features of homogeneous and isotropic 3-d cosmologies, provide a number of new solutions and study gauge invariant perturbations around them. The first-order formalism is then used to provide a detailed study of the most general 3-d spacetimes containing perfect-fluid matter. Assuming the material content to be dust with comoving spatial 2-velocities, we find the general solution of the Einstein equations with non-zero (and zero) cosmological constant and generalise known solutions of Kriele and the 3-d counterparts of the Szekeres solutions. In the case of a non-comoving dust fluid we find the general solution in the case of one non-zero fluid velocity component. We consider the asymptotic behaviour of the families of 3-d cosmologies with rotation and shear and analyse their singular structure. We also provide the general solution for cosmologies with one spacelike Killing vector, find solutions for cosmologies containing scalar fields and identify all the PP-wave 2+1 spacetimes.