Lukash plane waves, revisited

TL;DR

This paper revisits the Lukash plane wave spacetime, analyzing its geometry, coordinate transformations, and global structure, revealing its relation to cosmological models and horizon structures.

Contribution

It provides a detailed analysis of the Lukash metric, including coordinate transformations, isometry groups, and global structure, expanding understanding of its cosmological and geometric properties.

Findings

Derived the isometry group of the Lukash spacetime.

Identified the spacetime's relation to Milne cosmology and Rindler wedge.

Explained the coordinate transformations and global structure.

Abstract

The Lukash metric is a homogeneous gravitational wave which at late times approximates the behaviour of a generic class of spatially homogenous cosmological models with monotonically decreasing energy density. The transcription from Brinkmann to Baldwin-Jeffery-Rosen (BJR) to Bianchi coordinates is presented and the relation to a Sturm-Liouville equation is explained. The 6-parameter isometry group is derived. In the Bianchi VII range of parameters we have two BJR transciptions. However using either of them induces a mere relabeling of the geodesics and isometries. Following pioneering work of Siklos, we provide a self-contained account of the geometry and global structure of the spacetime. The latter contains a Killing horizon to the future of which the spacetime resembles an anisotropic version of the Milne cosmology and to the past of which it resemble the Rindler wedge.