3.1 The flat-space good-cut equation and good-cut functions

In Section 2, we saw that in the NP formalism, two of the complex spin coefficients, the optical parameters ρ and σ of Eqs. (2.23View Equation) and (2.24View Equation), play a particularly important role in their description of an NGC; namely, they carry the information of the divergence, twist and shear of the congruence.

From Eqs. (2.23View Equation) and (2.24View Equation), the radial behavior of the optical parameters for general shear-free NGCs, in Minkowski space, is given by

iΣ − r ρ = --------, σ = 0, (3.1 ) r2 + Σ2
where Σ is the twist of the congruence. A more detailed and much deeper understanding of the shear-free congruences can be obtained by first looking at the explicit coordinate expression, Eq. (2.19View Equation), for all flat-space NGCs:
--a xa = uB (ˆla + ˆna) − Lmˆ − ¯L ˆma + (r∗ − r0)ˆla, (3.2 )
where ¯ L(uB, ζ,ζ) is an arbitrary complex function of the parameters w ¯ y = (uB, ζ,ζ); r0, also an arbitrary function of ¯ (uB,ζ, ζ), determines the origin of the affine parameter; and ∗ r can be chosen freely. Most frequently, to simplify the form of ρ, r0 is chosen as
( ) 1- ¯ ¯ ˙¯ ¯ ˙ r0 ≡ − 2 ∂L + ∂L + L L + LL . (3.3 )
At this point, Eq. (3.2View Equation) describes an arbitrary NGC with (u ,ζ, ¯ζ) B labeling the geodesics and ∗ r the affine distance along the individual geodesics; later ¯ L (uB,ζ,ζ) will be chosen so that the congruence is shear-free. The tetrad (--a ˆla,ˆna,mˆa, ˆm) is given by Eqs. (1.1View Equation1.2View Equation), see [40Jump To The Next Citation Point].

There are several important comments to be made about Eq. (3.2View Equation). The first is that there is a simple geometric meaning to the parameters ¯ (uB, ζ,ζ): they are the values of the Bondi coordinates of + ℑ, where each geodesic of the congruence intersects ℑ+. The second concerns the geometric meaning of L. At each point of ℑ+, consider the past light cone and its sphere of null directions. Coordinatize that sphere (of null directions) with stereographic coordinates. The function ¯ L (uB,ζ,ζ ) is the stereographic angle field on ℑ+ that describes the null direction of each geodesic intersecting ℑ+ at the point (uB,ζ, ¯ζ). The values L = 0 and L = ∞ represent, respectively, the direction along the Bondi la and na vectors. This stereographic angle field completely determines the NGC.

The twist, Σ, of the congruence can be calculated in terms of L (uB,ζ, ¯ζ) directly from Eq. (3.2View Equation) and the definition of the complex divergence, Eq. (2.20View Equation), leading to

1 { -- -- -- -- } iΣ = -- ∂L + LL˙− ∂L − L ˙L . (3.4 ) 2
We now demand that L be a regular function of its arguments (i.e., have no infinities), or, equivalently, that all members of the NGC come from the interior of the spacetime and not lie on ℑ+ itself.

It has been shown [12Jump To The Next Citation Point] that the condition on the stereographic angle field L for the NGC to be shear-free is that

∂L + LL˙ = 0. (3.5 )
Our task is now to find the regular solutions of Eq. (3.5View Equation). The key to doing this is via the introduction of a new complex variable τ and complex function [39Jump To The Next Citation Point, 40Jump To The Next Citation Point],
¯ τ = T(uB, ζ,ζ). (3.6 )
T is related to L by the CR equation (related to the existence of a CR structure on ℑ+; see Appendix B):
˙ ∂(uB)T + LT = 0. (3.7 )

Remark 5. The following ‘gauge’ freedom becomes useful later. ∗ τ → τ = F (τ), with F analytic, leaving Eq. (3.7View Equation) unchanged. In other words,

( ) ( ) τ∗ = T ∗ uB, ζ, ¯ζ ≡ F T (uB,ζ, ¯ζ) , (3.8 )
leads to
∂ (uB)T ∗ = F ′∂(uB)T, ˙∗ ′ ˙ T = F T, ∂(uB)T ∗ + L T˙∗ = 0.

We assume, in the neighborhood of real ℑ+, i.e., near the real u B and &tidle;ζ = ζ¯, that T (u ,ζ, &tidle;ζ) B is analytic in the three arguments &tidle; (uB,ζ,ζ). The inversion of Eq. (3.6View Equation) yields the complex analytic cut function

&tidle; uB = G (τ,ζ,ζ). (3.9 )
Though we are interested in real values for uB, from Eq. (3.9View Equation) we see that for arbitrary τ it may take complex values. Shortly, we will also address the important issue of what values of τ are needed for real uB.

Returning to the issue of integrating the shear-free condition, Eq. (3.5View Equation), using Eq. (3.6View Equation), we note that the derivatives of T, ∂(uB)T and ˙T can be expressed in terms of the derivatives of G (τ,ζ, ¯ζ) by implicit differentiation. The uB derivative of T is obtained by taking the uB derivative of Eq. (3.9View Equation):

′ 1 1 = G (τ,ζ, ¯ζ)T˙⇒ T˙ = ---′, (3.10 ) (G )
while the ∂(uB)T derivative is found by applying ∂(uB) to Eq. (3.9View Equation),
0 = G ′(τ,ζ,ζ¯)∂ T + ∂ G, (3.11 ) (uB) (τ) ∂ T = − --∂(τ)G---. (uB ) G ′(τ,ζ,ζ¯)
When Eqs. (3.10View Equation) and (3.11View Equation) are substituted into Eq. (3.7View Equation), one finds that L is given implicitly in terms of the cut function by
L (uB,ζ,ζ¯) = ∂ (τ)G (τ,ζ, ¯ζ), (3.12 ) ¯ ¯ uB = G (τ,ζ,ζ) ⇔ τ = T (uB,ζ,ζ). (3.13 )

Thus, we see that all information about the NGC can be obtained from the cut function ¯ G (τ,ζ,ζ).

By further implicit differentiation of Eq. (3.12View Equation), i.e.,

¯ 2 ¯ ′ ¯ ∂ (uB)L(uB, ζ,ζ) = ∂(τ)G(τ,ζ,ζ ) + ∂(τ)G (τ,ζ,ζ) ⋅ ∂(uB)T, ˙L(uB, ζ, ¯ζ) = ∂ G′(τ,ζ, ¯ζ) ⋅ ˙T, (τ)
using Eq. (3.7View Equation), the shear-free condition (3.5View Equation) becomes
∂2 G (τ,ζ, ¯ζ) = 0. (3.14 ) (τ)
This equation will be referred to as the homogeneous Good-Cut Equation and its solutions as flat-space Good-Cut Functions (GCFs). In the next Section 4, an inhomogeneous version, the Good-Cut Equation, will be found for asymptotically shear-free NGCs. Its solutions will also be referred to as GCFs.

From the properties of the ∂2 operator, the general regular solution to Eq. (3.14View Equation) is easily found: G must contain only l = 0 and l = 1 spherical harmonic contributions; thus, any regular solution will be dependent on four arbitrary complex parameters, za. If these parameters are functions of τ, i.e., a a z = ξ (τ), then we can express any regular solution G in terms of the complex world line a ξ (τ) [39Jump To The Next Citation Point, 40Jump To The Next Citation Point]:

√2-ξ0(τ) 1 uB = G (τ,ζ, ¯ζ) = ξa(τ)ˆla(ζ, ¯ζ) ≡-------- − -ξi(τ)Y10i. (3.15 ) 2 2
The angle field L(uB,ζ, ¯ζ) then has the form
a L (uB,ζ,ζ¯) = ∂ (τ)G (τ,ζ, ¯ζ) = ξ (τ)ˆma (ζ, ¯ζ), (3.16 ) a ˆ ¯ uB = ξ (τ )la(ζ,ζ). (3.17 )

Thus, we have our first major result: every regular shear-free NGC in Minkowski space is generated by the arbitrary choice of a complex world line in what turns out to be complex Minkowski space. See Eq. (2.66View Equation) for the connection between the l = (0,1) harmonics in Eq. (3.15View Equation) and the Poincaré translations. We see in the next Section 4 how this result generalizes to regular asymptotically shear-free NGCs.

Remark 6. We point out that this construction of regular shear-free NGCs in Minkowski space is a special example of the Kerr theorem (cf. [67Jump To The Next Citation Point]). Writing Eqs. (3.16View Equation) and (3.17View Equation) as

-- - -- a + bζ + bζ + cζζ uB = -----------------, - 1 + ζζ- -- (b + cζ) − ζ(a + bζ) L = --------------------, 1 + ζζ
where the (a(τ ),b(τ),c(τ),d(τ )) are simple combinations of the ξa (τ ), we then find that
-- - -- L + uB ζ = b + cζ, -- uB − L ζ = a + bζ.
Noting that the right-hand side of both equations are functions only of τ and -- ζ, we can eliminate the τ from the two equations, thereby constructing a function of three variables of the form
-- -- F (L + uBζ, uB − L ζ,ζ) = 0.

This is a special case of the general solution to Eq. (3.5View Equation), which is the Kerr theorem.

In addition to the construction of the angle field, L (uB, ζ, ¯ζ), from the GCF, another quantity of great value in applications, obtained from the GCF, is the local change in u B as τ changes, i.e.,

&tidle; ′ V (τ,ζ,ζ) ≡ ∂τG = G . (3.18 )

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