4.2 ℋ-space and the good-cut equation

Equation (177View Equation), written in earlier literature as
∂2Z = σ0(Z, ζ, &tidle;ζ), (179 )
is a well-known and well-studied partial differential equation, often referred to as the “good-cut equation” [1920Jump To The Next Citation Point]. For sufficiently regular 0 σ (uB,ζ,ζ¯) (which is assumed here) it has been proven [20Jump 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;). (180 )

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 Equation (180View Equation).

Roughly or intuitively one can see how the four complex parameters enter the solution from the following argument. We can write Equation (179View 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 (181 )
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 [23]. 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, (182 ) S2 Z0 (ζ, &tidle;ζ) = zaˆla(ζ,ζ&tidle;), (183 )
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 a z ˆla(ζ, &tidle;ζ) is composed of the l = (0,1) harmonics,

zaˆl (ζ, &tidle;ζ) = √1-z0 − 1-ziY0(ζ,ζ&tidle;). (184 ) a 2 2 1i
Furthermore, the integral term does not contribute to these lowest harmonics. This means that solutions can be written
uB = Z (za, ζ, &tidle;ζ) ≡ zaˆla(ζ, &tidle;ζ) + Zl≥2 (za,ζ, &tidle;ζ), (185 )
with Zl≥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: It is of considerable interest that on ℋ-space there is a natural quadratic complex metric – demonstrated in Appendix D – that is given by the surprising relationship [3420]

( ∫ )− 1 ds2 = g dzadzb ≡ -1- -dS--- , (186 ) (ℋ ) (ℋ )ab 8π S2 (dZ )2 dZ ≡ ∇ Zdza, (187 ) a d-ζ ∧-dζ&tidle; dS = 4i (1 + ζζ&tidle;)2 . (188 )
UpdateJump To The Next Update Information Remarkably this turns out to be a Ricci-flat metric with a nonvanishing anti-self-dual Weyl tensor, i.e., it is intrinsically a complex vacuum metric. For vanishing Bondi shear, ℋ-space reduces to complex Minkowski space (i.e., g(ℋ)ab|σ0=0 = ηab).

4.2.1 Solutions to the shear-free equation

Returning to the issue of the solutions to the shear-free condition, i.e., Equation (174View Equation), L (uB, ζ, &tidle;ζ), we see that they are easily constructed from the solutions to the good-cut equation, uB = Z(za,ζ, &tidle;ζ). By choosing an arbitrary complex world line in the ℋ-space, i.e.,

za = ξa(τ ), (189 )
we write the GCF as
uB = G (τ,ζ, &tidle;ζ) ≡ Z (ξa(τ ),ζ, &tidle;ζ), (190 )
or, from Equation (185View Equation),
uB = G(τ,ζ,ζ&tidle;) = √1-ξ0(τ) − 1ξi(τ)Y10i(ζ, &tidle;ζ) + ξij(τ)Y20ij(ζ, &tidle;ζ) + .... (191 ) 2 2
This leads immediately, via Equations (178View Equation) and (179View Equation), to the parametric description of the shear-free stereographic angle field &tidle; L(uB, ζ,ζ), as well as the Bondi shear 0 &tidle; σ (uB,ζ,ζ ):
1 1 uB = √---ξ0(τ ) − -ξi(τ)Y01i(ζ,ζ&tidle;) + ξij(τ)Y02ij(ζ, &tidle;ζ) + ..., (192 ) 2 2 L (uB, ζ, &tidle;ζ) = ξi(τ)Y11i(ζ, &tidle;ζ) − 6 ξij(τ)Y21ij(ζ, &tidle;ζ) + ..., (193 ) 0 ij 2 σ (uB, ζ, &tidle;ζ) = 24ξ (τ)Y2ij + .... (194 )
We denote the inverse to Equation (191View Equation) by
τ = T(u ,ζ, &tidle;ζ). (195 ) B
The asymptotic twist of the asymptotically shear-free NGC is exactly as in the flat-space case,
1 { -- -˙ -- -- } iΣ = -- ∂L + LL − ∂L − L ˙L . (196 ) 2
As in the flat-space case, the derived quantity
V (τ,ζ,ζ&tidle;) ≡ ∂ τG = G ′ (197 )
plays a large role in applications. (In the case of the Robinson–Trautman metrics [55Jump To The Next Citation Point28Jump To The Next Citation Point] V is the basic variable for the construction of the metric.)

Using the gauge freedom, τ ⇒ τ∗ = Φ (τ)), in a slightly different way than in the Minkowski-space case, we impose the simple condition

0 ξ = τ. (198 )
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, uB = G(τ,ζ, &tidle;ζ), and the stereographic angle field on ℑ+ℂ, L(uB, ζ, &tidle;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

u = X (τ,ζ, &tidle;ζ). (199 ) B

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