Travelling waves in the expanding spatially homogeneous space-times

TL;DR

The paper presents new classes of traveling wave solutions in Einstein and Einstein-Maxwell equations within expanding, homogeneous spacetimes, highlighting their unique properties and potential for studying nonlinear wave interactions in curved backgrounds.

Contribution

It introduces traveling wave solutions that do not admit null Killing vectors and can exist in curved, expanding backgrounds, expanding the understanding of wave propagation in General Relativity.

Findings

Solutions depend on arbitrary functions of null coordinates.

Waves can propagate without scattering in curved backgrounds.

Solutions enable analysis of nonlinear wave interactions with curvature.

Abstract

Some classes of the so called "travelling wave" solutions of Einstein and Einstein - Maxwell equations in General Relativity and of dynamical equations for massless bosonic fields in string gravity in four and higher dimensions are presented. Similarly to the well known pp-waves, these travelling wave solutions may depend on arbitrary functions of a null coordinate which determine the arbitrary profiles and polarizations of the waves. However, in contrast with pp-waves, these waves do not admit the null Killing vector fields and can exist in some curved (expanding and spatially homogeneous) background space-times, where these waves propagate in certain directions without any scattering. Mathematically, some of these classes of solutions arise as the fixed points of Kramer-Neugebauer transformations for hyperbolic integrable reductions of the mentioned above field equations, or, in the other cases, -- after imposing of the ansatz that these waves do not change the part of spatial metric transversal to the direction of wave propagation. It is worth to note that strikingly simple forms of all presented solutions make possible a consideration of nonlinear interaction of these waves with the background curvature and singularities as well as a collision of sandwiches of such waves with solitons or with each others in the backgrounds where such travelling waves may exist.