Curvature invariants in type N spacetimes

TL;DR

This paper investigates scalar curvature invariants in type N vacuum solutions of Einstein's equations, revealing conditions under which these invariants vanish or are non-trivial, and identifying a new invariant related to spacetime singularities.

Contribution

It introduces a new scalar invariant for expanding type N spacetimes and analyzes the singularity structure of certain solutions.

Findings

Zero-order invariants vanish or are constant depending on Lambda.

Non-vanishing invariant exists in expanding type N spacetimes.

Certain solutions contain singularities at large distances.

Abstract

Scalar curvature invariants are studied in type N solutions of vacuum Einstein's equations with in general non-vanishing cosmological constant Lambda. Zero-order invariants which include only the metric and Weyl (Riemann) tensor either vanish, or are constants depending on Lambda. Even all higher-order invariants containing covariant derivatives of the Weyl (Riemann) tensor are 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 linearized and the third order type N twisting solutions recently discussed in literature contain singularities at large distances and thus cannot describe radiation fields outside bounded sources.