Petrov D vacuum spaces revisited: Identities and Invariant Classification

TL;DR

This paper revisits Petrov D vacuum spaces, deriving new identities and providing a systematic classification using the GHP formalism, which simplifies the covariant differentiation order needed for their invariant classification.

Contribution

It introduces a complete set of identities and tables for GHP derivatives, reducing the differentiation order for Karlhede classification of Petrov D vacuum spaces.

Findings

Derived new identities for Petrov D vacuum spaces

Provided a complete involutive set of tables for GHP derivatives

Reduced the differentiation order needed for classification to two

Abstract

For Petrov D vacuum spaces, two simple identities are rederived and some new identities are obtained, in a manageable form, by a systematic and transparent analysis using the GHP formalism. This gives a complete involutive set of tables for the four GHP derivatives on each of the four GHP spin coefficients and the one Weyl tensor component. It follows directly from these results that the theoretical upper bound on the order of covariant differentiation of the Riemann tensor required for a Karlhede classification of these spaces is reduced to two.