Lukash plane waves, revisited

TL;DR

The paper analyzes Lukash plane waves, a Ricci-flat pp-wave of BVII_h symmetry, providing a self-contained treatment of its geometry, isometries, and global structure. It develops a Siklos-based route from Brinkmann through BJR to BVII_h coordinates, uncovering a Sturm-Liouville framework that governs the coordinate transformations via a P-matrix and revealing a 6th Killing vector that enforces homogeneous BVII_h structure. It demonstrates multiple BJR transcriptions within the BVII range, their interrelations under u → 1/u duality, and explicit examples including Minkowski/Milne and BVII cases, illustrating how geodesics, Carroll symmetries, and Killing horizons behave. The global picture shows the wave emanating from a Killing-horizon-split spacetime into Milne- and Rindler-type regions, highlighting deep connections between gravitational waves and spatially homogeneous cosmologies. Overall, the work links exact gravitational wave solutions to BVII_h cosmologies, yielding a rich structure of symmetries and horizons with potential implications for late-time cosmological behavior and horizon physics.

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.

Figures (5)

• Figure 1: The projections to the transverse plane of the geodesics in BJR coordinates, unfolded to "time" $u$. The trajectories obtained by choosing $s_+$ or $s_-$ look substantially different.
• Figure 2: (i) Consistently with \ref{['+-BJRx']} and \ref{['+-BJRy']}, changing the parameter $u_+\equiv u$ into $u_-\equiv u^{-1}$ and multiplying by appropriate $\pm$ signs as in \ref{['+-uxyv']} carries the transverse components of a geodesic associated with $s_+$ into those associated with $s_-$. (ii) For null geodesics the vertical components $v(u_{\pm})$ match also, see \ref{['+-BJRv']} .
• Figure 3: Pulling back to Brinkmann coordinates the transverse components of the BJR geodesics associated with $s_+$ or $s_-$ [or equivalently, the coefficients $\boldsymbol{\beta}$ of the isometries] are different.
• Figure 4: The transverse projections of the geodesics in Brinkmann coordinates are taken into each other under \ref{['Bgeo+-']}.
• Figure 5: Transverse components of 4-parameter families of "translation-boost type" isometries/geodesics for Bianchi-VII-type Lukash waves. When $C=\kappa\to \infty$ they manifestly "flatten out.