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 Equations (27View Equation) and (28View 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 Equations (27View Equation) and (28View Equation), the radial behavior of the optical parameters for general shear-free NGCs, in Minkowski space, is given by

iΣ − r ρ = --2----2, σ = 0, (71 ) r + Σ
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, Equation (23View Equation), for all flat-space NGCs:
a ˆa a --a ¯ a ∗ ˆa x = uB (l + ˆn ) − Lmˆ − L ˆm + (r − r0)l, (72 )
where L(uB, ζ, ¯ζ) is an arbitrary complex function of the parameters yw = (uB, ζ, ¯ζ); r0, also an arbitrary function of (uB,ζ, ¯ζ), determines the origin of the affine parameter; r∗ can be chosen freely. Most frequently, to simplify the form of ρ, r0 is chosen as
1 ( ˙ ) r0 ≡ − -- ∂¯L + ∂¯L + L ¯L + ¯LL˙ . (73 ) 2
At this point, Equation (72View Equation) describes an arbitrary NGC with (uB,ζ,ζ¯) 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 (ˆa a a --a l,nˆ , ˆm ,ˆm) is given by [27Jump To The Next Citation Point]

√ -- ˆa -----2----( ¯ ¯ ¯ ¯) l = 2(1 + ζ¯ζ) 1 + ζ ζ,ζ + ζ,iζ − iζ,− 1 + ζζ , (74 ) √ -- a -----2----( ¯ ¯ ¯ ¯) ˆn = 2(1 + ζ¯ζ) 1 + ζ ζ,− (ζ + ζ),iζ − iζ,1 − ζζ , √ -- ˆma = -----2----(0,1 − ¯ζ2,− i(1 + ¯ζ2),2¯ζ) . 2(1 + ζ¯ζ)
There are several important comments to be made about Equation (72View 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 Equation (72View Equation) and the definition of the complex divergence, Equation (24View Equation), leading to

1 { -- -- -- -- } iΣ = -- ∂L + LL˙− ∂L − L ˙L . (75 ) 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 [6Jump 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. (76 )
Our task is now to find the regular solutions of Equation (76View Equation). The key to doing this is via the introduction of a new complex variable τ and complex function [26Jump To The Next Citation Point27Jump To The Next Citation Point],
¯ τ = T(uB, ζ,ζ). (77 )
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. (78 )

Remark: The following ‘gauge’ freedom becomes useful later. τ ⇒ τ∗ = F (τ), with F analytic, leaving Equation (78View Equation) unchanged. In other words,

∗ ∗( ¯) ( ¯) τ = T uB, ζ,ζ ≡ F T (uB,ζ,ζ ) , (79 )
leads to
∂ T ∗ = F ′∂ T, (uB) (uB) T˙∗ = F ′ ˙T, ∂ T ∗ + L T˙∗ = 0. (uB)

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 Equation (77View Equation) yields the complex analytic cut function

u = G (τ,ζ, &tidle;ζ). (80 ) B
Though we are interested in real values for uB, from Equation (80View Equation) we see that for arbitrary τ in general it would take complex values. Shortly, we will also address the important issue of what values of τ are needed for real u B.

Returning to the issue of integrating the shear-free condition, Equation (76View Equation), using Equation (77View 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 Equation (80View Equation):

1 1 = G′(τ,ζ, ¯ζ)T˙⇒ T˙ = ----, (81 ) (G ′)
while the ∂ T (uB) derivative is found by applying ∂ (uB) to Equation (80View Equation),
0 = G ′(τ,ζ,ζ¯)∂ (uB)T + ∂ (τ)G, (82 ) ∂ G ∂(uB )T = − ---(τ)----. G ′(τ,ζ,ζ¯)
When Equations (81View Equation) and (82View Equation) are substituted into Equation (78View Equation), one finds that L is given implicitly in terms of the cut function by
L (uB,ζ,ζ¯) = ∂ (τ)G (τ,ζ, ¯ζ), (83 ) uB = G (τ,ζ,ζ¯) ⇔ τ = T (uB,ζ,ζ¯). (84 )

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 Equation (83View Equation), i.e.,

∂ (uB)L(uB, ζ, ¯ζ) = ∂2(τ)G(τ,ζ,ζ¯) + ∂(τ)G ′(τ,ζ, ¯ζ) ⋅ ∂(uB)T, ˙ ¯ ′ ¯ ˙ L(uB, ζ,ζ) = ∂(τ)G (τ,ζ,ζ) ⋅T,
using Equation (78View Equation), the shear-free condition (76View Equation) becomes
2 ¯ ∂ (τ)G (τ,ζ,ζ) = 0. (85 )
This equation will be referred to as the homogeneous Good-Cut Equation and its solutions as flat-space 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 Equation (85View 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., za = ξa(τ), then we can express any regular solution G in terms of the complex world line ξa(τ) [26Jump To The Next Citation Point27Jump To The Next Citation Point]:

√ -- ¯ a ˆ ¯ --2ξ0- 1- i 0 uB = G (τ,ζ,ζ) = ξ (τ)la(ζ, ζ) ≡ 2 − 2ξ Y 1i. (86 )
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), (87 ) uB = ξa (τ )ˆla(ζ, ¯ζ). (88 )

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 Equation (70View Equation) for the connection between the l = (0, 1) harmonics in Equation (86View Equation) and the Poincaré translations. We see in the next Section 4 how this result generalizes to regular asymptotically shear-free NGCs.

Remark: We point out that this construction of regular shear-free NGCs in Minkowski space is a special example of the Kerr theorem (cf. [51Jump To The Next Citation Point]). Writing Equations (87View Equation) and (88View Equation) as

-- - -- u = a-+-bζ-+-bζ-+-cζζ, B 1 + ζζ - -- -- -- L = (b-+-cζ) −-ζ(a +-bζ), 1 + ζζ
where the (a(τ),b(τ),c(τ ),d(τ)) are simple combinations of the a ξ (τ ), we then find that
L + uζ-= b + cζ, -- u − 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 + uζ, u − Lζ,ζ) = 0.

This is a special case of the general solution to Equation (76View 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 uB as τ changes, i.e.,

V (τ,ζ, &tidle;ζ) ≡ ∂τG = G ′. (89 )

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