Curvature invariants in type-N spacetimes

Abstract

Scalar curvature invariants are studied in type-N solutions of the vacuum Einstein equations with, in general, a non-vanishing cosmological constant . Zeroth-order invariants, which include only the metric and Weyl (Riemann) tensor, either vanish or are constants depending on . All higher-order invariants containing covariant derivatives of the Weyl (Riemann) tensor are also shown to be trivial if a type-N spacetime admits a non-expanding and non-twisting null geodesic congruence. However, in the case of expanding type-N spacetimes we discover a non-vanishing scalar invariant, which is quartic in the second derivatives of the Riemann tensor. We use this invariant to demonstrate that both the linearized and third-order type-N twisting solutions recently discussed in literature contain singularities at large distances and thus cannot describe radiation fields outside bounded sources.