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 that 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 [31

].)

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,

&tidle; τ = T (uB,ζ, ζ).

They determined the $L(uB,ζ, ¯ζ)$ that satisfies Eq. (4.1 ) 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, (4.1 ) and (4.2 ), 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 [12

, 39

]

By requiring that the new congruence be asymptotically shear-free, i.e., $σ0∗ = 0$ , we obtain the generalization of Eq. (4.1 ) 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,

Eq. (4.12

) 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 Eq. (4.2

In Section 4.2, we will discuss how to construct solutions of Eq. (4.15 ) 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

We now turn to these solutions and their properties.