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82 7 The Newman–Penrose and related formalismsDΨ 3 − ¯δΨ 2 − DΦ 21 + δΦ 20 − 112 ¯δR =− κΨ 4 +2(ρ − ε)Ψ 3 +3πΨ 2 − 2λΨ 1+(2ᾱ − 2β − ¯π)Φ 20 − 2(¯ρ − ε)Φ 21 − 2πΦ 11 +2µΦ 10 +¯κΦ 22 , (7.32g)∆Ψ 1 − δΨ 2 − ∆Φ 01 + ¯δΦ 02 − 112 δR =νΨ 0 +2(γ − µ)Ψ 1 − 3τΨ 2 +2σΨ 3+(¯τ − 2 ¯β +2α)Φ 02 +2(¯µ − γ)Φ 01 +2τΦ 11 − 2ρΦ 12 − ¯νΦ 00 , (7.32h)DΦ 11 − δΦ 10 − ¯δΦ 01 +∆Φ 00 + 1 8 DR =(2γ − µ +2¯γ − ¯µ)Φ 00 +(π − 2α − 2¯τ)Φ 01 +(¯π − 2ᾱ − 2τ)Φ 10+2(ρ +¯ρ)Φ 11 +¯σΦ 02 + σΦ 20 − ¯κΦ 12 − κΦ 21 ,(7.32i)DΦ 12 − δΦ 11 − ¯δΦ 02 +∆Φ 01 + 1 8 δR =(−2α +2¯β + π − ¯τ)Φ 02 +(¯ρ +2ρ − 2¯ε)Φ 12 +2(¯π − τ)Φ 11+(2γ − 2¯µ − µ)Φ 01 +¯νΦ 00 − ¯λΦ 10 + σΦ 21 − κΦ 22 ,(7.32j)DΦ 22 − δΦ 21 − ¯δΦ 12 +∆Φ 11 + 1 8 ∆R =(ρ +¯ρ − 2ε − 2¯ε)Φ 22 +(2¯β +2π − ¯τ)Φ 12 +(2β +2¯π − τ)Φ 21− 2(µ +¯µ)Φ 11 + νΦ 01 +¯νΦ 10 − ¯λΦ 20 − λΦ 02 .(7.32k)The consistency, completeness and integrability of the Newman–Penrose formalism has been considered in a number of papers, e.g.Papapetrou (1971a, 1971b) and Edgar (1980, 1992). As given here, theequations are a set of differential equations for the tetrad components e aiwith respect to a coordinate basis {∂/∂x i }, the spin coefficients (7.2) andthe Riemann tensor components (7.10)–(7.20), the corresponding equationsbeing respectively either the commutator relations (7.6) togetherwith the rigid frame condition dg ab = 0 or the definitions (7.2), the Ricci(Newman–Penrose) equations (7.21) and the Bianchi equations (7.32). Itis (implicitly) assumed that the connection coefficients and Riemann tensorcomponents not mentioned explicitly in (7.2) and (7.10)–(7.20) canbe found from the symmetry relations Γ (ab)c = 0 and (3.26).There is redundancy between these equations in the sense that someof them (or combinations of some of them) are integrability conditionsfor others. The underlying reason is that (7.6) and (7.21) are versionsof the Cartan structure equations (2.76) and (2.85), which have as integrabilityconditions the first and second Bianchi identities, i.e. d 2 ω a =0