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 that 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. (4.1 )
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 [31Jump To The Next Citation Point].)

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

∂2 G = 0. (4.2 ) (τ)
The regular solutions were easily given by
a uB = G (τ,ζ, &tidle;ζ) = ξ (τ)ˆla(ζ, &tidle;ζ) (4.3 )
with inverse function,
&tidle; τ = T (uB,ζ, ζ).

They determined the L(uB,ζ, ¯ζ) that satisfies Eq. (4.1View Equation) by the parametric relations

L (uB,ζ,ζ¯) = ∂ (τ)G (τ,ζ, &tidle;ζ), (4.4 ) uB = ξa (τ )ˆla(ζ, &tidle;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, (4.1View Equation) and (4.2View 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(uB,ζ, ¯ζ). 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

- - l∗a = la + bma + bma + bbna, (4.5 ) ∗a a a m = m + bn , n ∗a = na, −2 b = − L ∕r + O (r ),
with L (uB,ζ,ζ¯) an arbitrary stereographic angle field, then the asymptotic Weyl components transform as
∗0 0 0 2 0 3 0 4 0 ψ 0 = ψ0 − 4Lψ 1 + 6L ψ 2 − 4L ψ3 + L ψ4, (4.6 ) ψ ∗10= ψ01 − 3Lψ02 + 3L2ψ03 − L3 ψ04, (4.7 ) ψ ∗0= ψ0− 2Lψ0 + L2ψ0, (4.8 ) 2∗0 20 03 4 ψ 3 = ψ3 − Lψ 4, (4.9 ) ψ ∗0= ψ0, (4.10 ) 4 4
and the (new) asymptotic shear of the null vector field l∗a is given by [12Jump To The Next Citation Point, 39Jump To The Next Citation Point]
σ0∗ = σ0 − ∂ L − L ˙L. (4.11 ) (uB)

By requiring that the new congruence be asymptotically shear-free, i.e., σ0∗ = 0, we obtain the generalization of Eq. (4.1View Equation) for the determination of ¯ L (uB,ζ,ζ), namely,

∂(uB)L + LL˙= σ0(uB,ζ, ¯ζ). (4.12 )
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, (4.13 )
with its inversion,
u = G (τ,ζ, &tidle;ζ). (4.14 ) B
Eq. (4.12View 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). (4.15 ) (τ)
This, the inhomogeneous good-cut equation, is the generalization of Eq. (4.2View Equation).

In Section 4.2, we will discuss how to construct solutions of Eq. (4.15View Equation) of the form, uB = G (τ,ζ,ζ&tidle;); however, assuming we have such a solution, it determines the angle field L (u ,ζ, ¯ζ) B by the parametric relations

&tidle; L (uB,ζ,ζ ) = ∂ (τ)G, (4.16 ) uB = G (τ,ζ,ζ&tidle;).
We now turn to these solutions and their properties.
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