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76 7 The Newman–Penrose and related formalismsν ≡ Γ 233 = l a;b m a l b = m a ∆l a = −ι A ῑḂι C ∇ A Ḃ ι C ,(7.2e)µ ≡ Γ 231 = l a;b m a m b = m a δl a = −o A ῑḂι C ∇ A Ḃ ι C , (7.2f)λ ≡ Γ 232 = l a;b m a m b = m a¯δla = −ι A ōḂι C ∇ A Ḃ ι C ,π ≡ Γ 234 = l a;b m a k b = m a Dl a = −o A ōḂι C ∇ A Ḃ ι C ,(7.2g)(7.2h)−ε ≡ 1 2 (Γ 344 − Γ 214 )= 1 2 (k a;bl a k b − m a;b m a k b )= 1 2 (la Dk a − m a Dm a )=o A ōḂι C ∇ A Ḃ o C , (7.2i)−β ≡ 1 2 (Γ 341 − Γ 211 )= 1 2 (k a;bl a m b − m a;b m a m b )= 1 2 (la δk a − m a δm a )=o A ῑḂι C ∇ A Ḃ o C , (7.2j)γ ≡ 1 2 (Γ 433 − Γ 123 )= 1 2 (l a;bk a l b − m a;b m a l b )= 1 2 (ka ∆l a − m a ∆m a )=−ι A ῑḂo C ∇ A Ḃ ι C , (7.2k)α ≡ 1 2 (Γ 432 − Γ 122 )= 1 2 (l a;bk a m b − m a;b m a m b )= 1 2 (ka¯δla − m a¯δma )=−ι A ōḂo C ∇ A Ḃ ι C , (7.2l)where we have used the notation (3.82), i.e.D ≡ k a ∇ a = −o A ōḂ∇ A Ḃ , ∆ ≡ la ∇ a = −ι A ῑḂ∇ A Ḃ , (7.3)δ ≡ m a ∇ a = −o A ῑḂ∇ A Ḃ , ¯δ ≡ m a ∇ a = −ι A ōḂ∇ A Ḃ ,for the directional derivatives D, ∆,δ, δ . Some of these spin coefficientshave already been introduced in (6.20). From the spinor expressions forthe spin coefficients and from the relationo Aι A =1 =⇒ ι C ∇ A Ḃ o C = oC ∇ A Ḃ ι C(7.4)it follows that all connection coefficients can be expressed in terms of the12 complex spin coefficients (7.2). One can also give the spin coefficientsas partial derivatives, e.g. κ = k [a,b] m a k b . These formulae can be obtainedfrom the commutator relations (see (7.6) below) and are given e.g. inCocke (1989).