4.1 Asymptotically shear-free NGCs and the good-cut equation

We saw in Section 3 that shear-free NGCs in Minkowski space could be constructed by looking at their properties near ℑ+, in one of two equivalent ways. The first was via the stereographic angle field, L (uB,ζ,ζ¯), which gives the directions the null rays make at their intersection with ℑ+. The condition for the congruence to be shear-free was that L must satisfy
˙ ∂(uB)L + LL = 0. (163 )
We required solutions that were all nonsingular (regular) on the (ζ,ζ¯) sphere. (This equation has in the past most often been solved via twistor methods [19Jump To The Next Citation Point].)

The second was via the complex cut function, uB = G (τ, ζ, &tidle;ζ), that satisfied

2 ∂(τ)G = 0. (164 )
The regular solutions were easily given by
&tidle; a ˆ &tidle; uB = G (τ,ζ,ζ) = ξ (τ)la(ζ, ζ) (165 )
with inverse function,
τ = T (uB,ζ, &tidle;ζ).

They determined the ¯ L(uB, ζ,ζ) that satisfies Equation (163View Equation) by the parametric relations

L (uB,ζ,ζ¯) = ∂ (τ)G (τ,ζ, &tidle;ζ), (166 ) a ˆ &tidle; uB = ξ (τ )la(ζ,ζ),
or by
¯ ¯ L (uB,ζ, ζ) = ∂(τ)G (τ,ζ,ζ)|τ=T(uB,ζ,&tidle;ζ),

where ξa(τ) was an arbitrary complex world line in complex Minkowski space.

It is this pair of equations, (163View Equation) and (164View Equation), that will now be generalized to asymptotically-flat spacetimes.

In Section 2, we saw that the asymptotic shear of the (null geodesic) tangent vector fields, la, of the out-going Bondi null surfaces was given by the free data (the Bondi shear) σ0(u ,ζ, ¯ζ) B. If, near ℑ+, a second NGC, with tangent vector ∗a l, is chosen and then described by the null rotation from a l to ∗a l around na by

∗a a --a - a - a l = l + bm + bm + bbn , (167 ) m ∗a = ma + bna, ∗a a n = n , b = − L ∕r + O (r −2),
with L (uB,ζ,ζ¯) an arbitrary stereographic angle field, then the asymptotic Weyl components transform as
ψ∗00 = ψ00 − 4L ψ01 + 6L2 ψ02 − 4L3 ψ∗30+ L4ψ04, (168 ) ψ∗0 = ψ0 − 3L ψ0 + 3L2 ψ0 − L3ψ0 , (169 ) 1∗0 10 20 2 03 4 ψ2 = ψ 2 − 2L ψ3 + L ψ4, (170 ) ψ∗0 = ψ0 − L ψ0, (171 ) 3∗0 30 4 ψ4 = ψ 4, (172 )
UpdateJump To The Next Update Information and the (new) asymptotic shear of the null vector field ∗a l is given by [6Jump To The Next Citation Point26Jump To The Next Citation Point]
σ0∗ = σ0 − ∂ L − L ˙L. (173 ) (uB)

By requiring that the new congruence be asymptotically shear-free, i.e., σ0∗ = 0, we obtain the generalization of Equation (163View Equation) for the determination of ¯ L (uB,ζ,ζ ), namely,

∂(uB)L + LL˙= σ0(uB,ζ, ¯ζ). (174 )
To solve this equation we again complexify + ℑ to + ℑℂ by freeing ¯ζ to &tidle;ζ and allowing uB to take complex values close to the real.

Again we introduce the complex potential τ = T(uB, ζ, &tidle;ζ) that is related to L by

∂ (uB)T + L T˙= 0, (175 )
with its inversion,
&tidle; uB = G (τ,ζ,ζ). (176 )
Equation (174View Equation) becomes, after the change in the independent variable, uB ⇒ τ = T (uB,ζ, ¯ζ), and implicit differentiation (see Section 3.1 for the identical details),
∂2 G = σ0(G, ζ, &tidle;ζ). (177 ) (τ)
This, the inhomogeneous good-cut equation, is the generalization of Equation (164View Equation).

In Section 4.2, we will discuss how to construct solutions of Equation (177View Equation) of the form, &tidle; uB = G (τ, ζ,ζ); however, assuming we have such a solution, it determines the angle field ¯ L (uB,ζ,ζ ) by the parametric relations

L (uB,ζ,ζ&tidle;) = ∂ (τ)G, (178 ) u = G (τ,ζ,ζ&tidle;). B
We return to the properties of these solutions in Section 4.2.
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