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56 4 The Petrov classificationI✠II✠ ❄❄III ✲ N✲❄❄D✲ ❄OFig. 4.1. Penrose diagramAny two of the following conditions for a null direction k imply the third(Hall 1973)C abc[d k f] k b k c =0, R abc[d k f] k b k c =0, R a[b k c] k a =0. (4.26)The last condition in (4.26) means that k is a Ricci eigendirection (§5.1).Type D is characterized by the existence of two double principal nulldirections, k and l,C abc[d k f] k b k c =0⇐⇒ Ψ 0 =Ψ 1 =0, Ψ 2 ̸= 0,C abc[d l f] l b l c =0⇐⇒ Ψ 4 =Ψ 3 =0, Ψ 2 ̸= 0.(4.27)Type O (zero Weyl tensor) does not single out any null directions.In the Penrose diagram (Fig. 4.1) the arrows point in the direction ofincreasing multiplicity of the principal null directions; every arrow indicatesone additional degeneration.The classification in terms of principal null directions can be formulatedin terms of spinors (Penrose 1960). The completely symmetric spinorΨ ABCD can be written as a symmetrized product of one-index spinors,which are uniquely determined apart from factors. The proof is an applicationof the fundamental theorem of algebra: any polynomial may befactorized over C into linear forms,Ψ ABCD ζ A ζ B ζ C ζ D =(o A ζ A )(β B ζ B )(γ C ζ C )(ι D ζ D ). (4.28)The Petrov types are then characterized by the criteria:Type I :Ψ ABCD ∼ o (A β B γ C ι D) ,D :Ψ ABCD ∼ o (A o B ι C ι D) ,II :Ψ ABCD ∼ o (A o B γ C ι D) ,(4.29)III :Ψ ABCD ∼ o (A o B o C ι D) ,N :Ψ ABCD ∼ o (A o B o C o D)(k a ←→ o A ōḂ, l a ←→ ι A ῑḂ).