Gravitational waves in vacuum spacetimes with cosmological constant. II. Deviation of geodesics and interpretation of non-twisting type N solutions

TL;DR

This paper analyzes the behavior of gravitational waves in vacuum spacetimes with a cosmological constant, focusing on non-twisting type N solutions, and interprets their effects as exact transverse waves propagating on various backgrounds.

Contribution

It provides a detailed analysis of non-twisting type N vacuum solutions with cosmological constant, interpreting them as exact gravitational waves with specific polarization modes.

Findings

Non-twisting type N solutions behave as exact transverse gravitational waves.

Solutions with positive Lambda illustrate the cosmic 'no-hair' conjecture.

Explicit coordinate forms and properties of these solutions are derived.

Abstract

In a suitably chosen essentially unique frame tied to a given observer in a general spacetime, the equation of geodesic deviation can be decomposed into a sum of terms describing specific effects: isotropic (background) motions associated with the cosmological constant, transverse motions corresponding to the effects of gravitational waves, longitudinal motions, and Coulomb-type effects. Conditions under which the frame is parallelly transported along a geodesic are discussed. Suitable coordinates are introduced and an explicit coordinate form of the frame is determined for spacetimes admitting a non-twisting null congruence. Specific properties of all non-twisting type N vacuum solutions with cosmological constant Lambda (non-expanding Kundt class and expanding Robinson-Trautman class) are then analyzed. It is demonstrated that these spacetimes can be understood as exact transverse gravitational waves of two polarization modes "+" and "x", shifted by pi/4, which propagate "on" Minkowski, de Sitter, or anti-de Sitter backgrounds. It is also shown that the solutions with Lambda>0 may serve as exact demonstrations of the cosmic "no-hair" conjecture in radiative spacetimes with no symmetry.