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Petrov D vacuum spaces revisited: Identities and invariant classification. (English) Zbl 1166.83015

Summary: 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.

MSC:

83E05 Geometrodynamics and the holographic principle
83C15 Exact solutions to problems in general relativity and gravitational theory

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