Homogeneous Plane Waves

TL;DR

This paper classifies all homogeneous plane wave metrics in any dimension, identifying geodesically complete and singular cases, and explores their embedding in string theory along with their symmetries and conserved quantities.

Contribution

It provides a complete classification of homogeneous plane wave metrics, including new time-dependent solutions and their isometry algebras, with applications to string theory.

Findings

Identified all homogeneous plane wave solutions in arbitrary dimensions.

Found a family of geodesically complete and singular HPWs.

Connected the geometry of HPWs to invariants of time-dependent oscillators.

Abstract

Motivated by the search for potentially exactly solvable time-dependent string backgrounds, we determine all homogeneous plane wave (HPW) metrics in any dimension and find one family of HPWs with geodesically complete metrics and another with metrics containing null singularities. The former generalises both the Cahen-Wallach (constant $A_{ij}$) metrics to time-dependent HPWs, $A_{ij}(t)$, and the Ozsvath-Sch\"ucking anti-Mach metric to arbitrary dimensions. The latter is a generalisation of the known homogeneous metrics with $A_{ij}\sim 1/t^2$ to a more complicated time-dependence. We display these metrics in various coordinate systems, show how to embed them into string theory, and determine the isometry algebra of a general HPW and the associated conserved charges. We review the Lewis-Riesenfeld theory of invariants of time-dependent harmonic oscillators and show how it can be deduced from the geometry of plane waves. We advocate the use of the invariant associated with the extra (timelike) isometry of HPWs for lightcone quantisation, and illustrate the procedure in some examples.