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90 7 The Newman–Penrose and related formalismsMaxwell tensor is aligned with the multiple principal null direction of theWeyl tensor. Then it follows from the Bianchi identities (7.32a), (7.32b),and expressions (7.29), (5.12) that(2κ 0 Φ 1 Φ 1 +3Ψ 2 )σ =0, (−2κ 0 Φ 1 Φ 1 +3Ψ 2 )κ = 0 (7.54)(Kundt and Trümper 1962). If Ψ 2 = 0 (Petrov type III and more specialtypes), then κ = σ =0.IfΨ 2 ̸= 0 one obtainsκσ = 0, i.e. either κ or σmust vanish. Relation (7.52a) also leads to (7.54).Unfortunately, the Kundt–Thompson theorem does not directly specifythe most general matter distribution which would allow one to concludethat (A) ⇒ (B). For instance, the assumption that the Ricci tensor is ofpure radiation type,R ab = κ 0 Φ 2 k a k b = κ 0 T ab , (7.55)does not guarantee condition (C) for fields of Petrov type N ; in generalthe shear of k does not vanish, cp. §26.1. However, if T ab in (7.55)is the energy-momentum tensor of an electromagnetic (null) field, thecongruence k is necessarily shearfree because of the above corollary toTheorem 7.4.