Contents

7The Newman–Penrose and relatedformalisms7.1 The spin coefficients and their transformation lawsThe null tetrad formalism due to Newman and Penrose (1962) has provedvery useful in the construction of exact solutions, in particular for studyingalgebraically special gravitational fields (for some of the earliest examples,see Kinnersley (1969b), Talbot (1969), and Lind (1974)). Despitethe fact that we have to solve a considerably larger number of equationsthan arise when we use coordinates directly, this formalism has greatadvantages. All differential equations are of first order. Gauge transformationsof the tetrad can be used to simplify the field equations. Onecan extract invariant properties of the gravitational field without usinga coordinate basis. We give here an outline of this important approachto general relativity; see also Frolov (1977), Penrose and Rindler (1984,1986) and Stewart (1990).Using the complex null tetrad {e a } =(m, m, l, k), and recalling thedefinition (2.67),∇ b e a =Γ c abe c , (7.1)of the connection coefficients Γ c ab, we can define the so-called spin coefficients,12 independent complex linear combinations of the connection coefficients.Explicitly, the spin coefficients are defined in tensor and spinornotation as follows:−κ ≡ Γ 144 = k a;b m a k b = m a Dk a = o A ōḂo C ∇ A Ḃ o C ,−ρ ≡ Γ 142 = k a;b m a m b = m a¯δka = ι A ōḂo C ∇ A Ḃ o C ,−σ ≡ Γ 141 = k a;b m a m b = m a δk a = o A ῑḂo C ∇ A Ḃ o C ,−τ ≡ Γ 143 = k a;b m a l b = m a ∆k a = ι A ῑḂo C ∇ A Ḃ o C ,(7.2a)(7.2b)(7.2c)(7.2d)75