Interpretation of the Siklos solutions as exact gravitational waves in the anti-de Sitter universe

TL;DR

This work interprets the Siklos class of exact type N vacuum solutions with $\Lambda<0$ as genuine gravitational waves propagating in anti-de Sitter space. By analyzing geodesic deviation in carefully chosen frames, the author shows the wave is transverse with two polarizations and that the propagation direction rotates in the transverse plane at angular speed $\omega=\sqrt{-\Lambda/3}$; in the limit $\Lambda\to0$ this rotation vanishes. The Kaigorodov space-time is singled out as the simplest homogeneous representative, providing explicit geodesics and deviation solutions that illuminate the cosmological analogue of flat-space homogeneous pp-waves. The results offer physical intuition for waves in AdS and suggest numerical-relativity test beds for cosmological gravitational radiation. All mathematical notation is retained and emphasized with explicit expressions for the wave amplitudes and their polarization structure.

Abstract

The Siklos class of solutions of Einstein's field equations is investigated by analytical methods. By studying the behaviour of free particles we reach the conclusion that the space-times represent exact gravitational waves propagating in the anti-de Sitter universe. The presence of a negative cosmological constant implies that the 'background' space is not asymptotically flat and requires a 'rotating' reference frames in order to fully simplify and view the behaviour of nearby test particles. The Kaigorodov space-time, which is the simplest representative of the Siklos class, is analyzed in more detail. It is argued that it may serve as a 'cosmological' analogue of the well-known homogeneous pp-waves in the flat universe.