Curvature invariants in type-III spacetimes

TL;DR

This paper generalizes previous results on curvature invariants in vacuum type-III spacetimes, showing conditions under which these invariants vanish or do not vanish based on spacetime properties like expansion or twist.

Contribution

It extends the analysis of curvature invariants to include non-zero cosmological constant and identifies conditions affecting their vanishing in type-III spacetimes.

Findings

All derivatives of the Weyl tensor invariants vanish if the spacetime admits a non-expanding, non-twisting null congruence.

A non-vanishing invariant with first derivatives of the Weyl tensor exists when the spacetime has expansion or twist.

Results apply to vacuum solutions with a cosmological constant Lambda.

Abstract

The results of paper [1] are generalized for vacuum type-III solutions with, in general, a non-vanishing cosmological constant Lambda. It is shown that all curvature invariants containing derivatives of the Weyl tensor vanish if a type-III spacetime admits a non-expanding and non-twisting null geodesic congruence. A non-vanishing curvature invariant containing first derivatives of the Weyl tensor is found in the case of type-III spacetime with expansion or twist.