4.2 ℋ-space and the good-cut equation

Eq. (4.15View Equation), written in earlier literature as
∂2Z = σ0(Z, ζ, &tidle;ζ), (4.17 )
is a well-known and well-studied partial differential equation, often referred to as the “good-cut equation” [31, 32Jump To The Next Citation Point]. For sufficiently regular 0 σ (uB,ζ,ζ¯) (which is assumed here) it has been proven [32Jump To The Next Citation Point] that the solutions are determined by points in a complex four-dimensional space, za, referred to as ℋ-space, i.e., solutions are given as
a uB = Z(z ,ζ,ζ&tidle;). (4.18 )

Later in this section, by choosing an arbitrary complex analytic world line in ℋ-space, a a z = ξ (τ), we describe how to construct the shear-free angle field, L (uB,ζ, &tidle;ζ). First, however, we discuss properties and the origin of Eq. (4.18View Equation).

Roughly or intuitively one can see how the four complex parameters enter the solution from the following argument. We can write Eq. (4.17View Equation) as the integral equation

∮ Z = zaˆla(ζ, &tidle;ζ) + σ0 (Z, η, &tidle;η)K+0,− 2(η, &tidle;η,ζ, &tidle;ζ)dS η (4.19 )
1 (1 + &tidle;ζη )2(η − ζ) K+0,− 2(ζ, &tidle;ζ,η, &tidle;η) ≡ −--------------------------, 4π (1 + ζ&tidle;ζ)(1 + η&tidle;η)(&tidle;η − &tidle;ζ) dη ∧ d&tidle;η dSη = 4i(1-+-η&tidle;η)2,
where zaˆla(ζ,ζ&tidle;) is the kernel of the ∂2 operator (the solution to the homogeneous good-cut equation) and K+0,−2(ζ, &tidle;ζ,η, &tidle;η) is the Green’s function for the ∂2 operator [36]. By iterating this equation, with the kernel being the zeroth iterate, i.e.,
∫ Zn (ζ, &tidle;ζ) = zaˆla(ζ,ζ&tidle;) + K+0,−2(ζ, &tidle;ζ,η, &tidle;η)σ(Zn− 1,η, &tidle;η)dSη, (4.20 ) S2 Z0 (ζ, &tidle;ζ) = zaˆla(ζ,ζ&tidle;), (4.21 )
one easily sees how the four za enter the solution. Basically, the za come from the solution to the homogeneous equation.

It should be noted again that the aˆ &tidle; z la(ζ,ζ) is composed of the l = (0,1) harmonics,

zaˆla(ζ, &tidle;ζ) = √1-z0 − 1-ziY0(ζ,ζ&tidle;). (4.22 ) 2 2 1i
Furthermore, the integral term does not contribute to these lowest harmonics. This means that solutions can be written
a a a uB = Z (z ,ζ, &tidle;ζ) ≡ z ˆla(ζ, &tidle;ζ) + Zl≥2 (z ,ζ, &tidle;ζ), (4.23 )
with Z l≥2 containing spherical harmonics l = 2 and higher.

We note that using this form of the solution implies that we have set stringent coordinate conditions on the ℋ-space by requiring that the first four spherical harmonic coefficients be the four ℋ-space coordinates. Arbitrary coordinates would just mean that these four coefficients were arbitrary functions of other coordinates. How these special coordinates change under the BMS group is discussed later.

Remark 10. It is of considerable interest that on ℋ-space there is a natural quadratic complex metric – as constructed in Appendix D – that is given by the surprising relationship [49, 32Jump To The Next Citation Point]

( ∫ )− 1 ds2 = g dzadzb ≡ -1- -dS--- , (4.24 ) (ℋ ) (ℋ )ab 8π S2(dZ )2 dZ ≡ ∇ Zdza, (4.25 ) a d-ζ ∧-dζ&tidle; dS = 4i (1 + ζζ&tidle;)2. (4.26 )
Remarkably this turns out to be a Ricci-flat metric with a nonvanishing anti-self-dual Weyl tensor and vanishing self-dual Weyl tenor, i.e., it is intrinsically a complex anti-self-dual vacuum metric. For vanishing Bondi shear, ℋ-space reduces to complex Minkowski space (i.e., g | 0 = η (ℋ )ab σ=0 ab).

4.2.1 Solutions to the shear-free equation

Returning to the issue of the solutions to the shear-free condition, i.e., Eq. (4.12View Equation), &tidle; L (uB, ζ,ζ), we see that they are easily constructed from the solutions to the good-cut equation, a &tidle; uB = Z(z ,ζ,ζ ). By choosing an arbitrary complex world line in the ℋ-space, i.e.,

za = ξa(τ ), (4.27 )
we write the GCF as
&tidle; a &tidle; uB = G (τ,ζ,ζ) ≡ Z (ξ (τ ),ζ, ζ), (4.28 )
or, from Eq. (4.23View Equation),
1 1 uB = G(τ,ζ,ζ&tidle;) = √--ξ0(τ) − -ξi(τ)Y10i(ζ, &tidle;ζ) + ξij(τ)Y20ij(ζ, &tidle;ζ) + .... (4.29 ) 2 2
This leads immediately, via Eqs. (4.16View Equation) and (4.29View Equation), to the parametric description of the shear-free stereographic angle field L(u ,ζ, &tidle;ζ) B, as well as the Bondi shear σ0(u ,ζ,ζ&tidle;) B:
-1-- 0 1-i 0 &tidle; ij 0 &tidle; uB = √2--ξ (τ ) − 2ξ (τ)Y1i(ζ,ζ ) + ξ (τ)Y2ij(ζ,ζ) + ..., (4.30 ) i 1 ij 1 L (uB, ζ, &tidle;ζ) = ξ (τ)Y1i(ζ, &tidle;ζ) − 6 ξ (τ)Y2ij(ζ, &tidle;ζ) + ..., (4.31 ) 0 &tidle; ij 2 σ (uB, ζ,ζ) = 24ξ (τ)Y2ij + .... (4.32 )
We denote the inverse to Eq. (4.29View Equation) by
τ = T(u ,ζ, &tidle;ζ), (4.33 ) B
and refer to the complex world line ξa(τ) as the ‘virtual’ source of the congruence. The asymptotic twist of the asymptotically shear-free NGC is exactly as in the flat-space case,
1-{ -- -˙ -- --˙} iΣ = 2 ∂L + LL − ∂L − L L . (4.34 )
As in the flat-space case, the derived quantity
V (τ,ζ,ζ&tidle;) ≡ ∂ G = G ′ (4.35 ) τ
plays a large role in applications. (In the case of the Robinson–Trautman metrics [71Jump To The Next Citation Point, 41Jump To The Next Citation Point] V is the basic variable for the construction of the metric.)

Using the gauge freedom, τ → τ∗ = Φ (τ), as in the Minkowski-space case, we impose the simple condition

0 ξ = τ. (4.36 )

A Brief Summary: The description and analysis of the asymptotically shear-free NGCs in asymptotically-flat spacetimes is remarkably similar to that of the flat-space regular shear-free NGCs. We have seen that all regular shear-free NGCs in Minkowski space and asymptotically-flat spaces are generated by solutions to the good-cut equation, with each solution determined by the choice of an arbitrary complex analytic world line in complex Minkowski space or ℋ-space. The basic governing variables are the complex GCF, &tidle; uB = G(τ,ζ, ζ), and the stereographic angle field on + ℑℂ, &tidle; L(uB, ζ,ζ), restricted to real + ℑ. In every sense, the flat-space case can be considered as a special case of the asymptotically-flat case.

In Sections 5 and 6, we will show that in every asymptotically flat spacetime a special complex-world line (along with its associated NGC and GCF) can be singled out using physical considerations. This special GCF is referred to as the (gravitational) UCF, and is denoted by

uB = X (τ,ζ, &tidle;ζ). (4.37 )

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