Asymptotically shear-free NGCs and the good-cut equation

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

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 [19

].)

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

The regular solutions were easily given by

with inverse function,

τ = T (uB,ζ, &tidle;ζ).

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

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, (163 ) and (164 ), 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 )

Update

and the (new) asymptotic shear of the null vector field $∗a l$ is given by [6

, 26

]

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

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

with its inversion,

Equation (174

) 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),

This, the inhomogeneous good-cut equation, is the generalization of Equation (164

In Section 4.2, we will discuss how to construct solutions of Equation (177 ) 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

We return to the properties of these solutions in Section 4.2.