D ℋ-Space Metric

In the following, the derivation of the ℋ-space metric of (4.24View Equation) is given. We begin with the cut function, -- -- uB = Z(ξa(τ),ζ, ζ) = G(τ,ζ,ζ ) that satisfies the good-cut equation -- ∂2Z = σ0(Z,ζ,ζ ). The (-- ζ, ζ) are (for the time being) completely independent of each other, though -- ζ is to be treated as being ‘close’ the complex conjugate of ζ. Taking the gradient of Z (za,ζ,ζ), multiplied by an arbitrary four vector a v (i.e., a V = v Z,a), we see that it satisfies the linear good cut equation,

∂2Z,a = σ0,Z Z,a (D.1 ) 2 0 ∂ V = σ ,Z V.
Let V0 be a particular solution, and assume for the moment that the general solution can be written as
∗ Z,a = V0la (D.2 )
with the four components of l∗ a to be determined. Substituting Eq. (D.2View Equation) into the linearized good-cut equation, we have
2 ∗ 0 ∗ ∂ (V0la) = σ ,Z V0la, ∂(l∗a∂(V0) + V0∂l∗a) = σ0,Z V0l∗a, ∗ 2 ∗ 2 ∗ 0 ∗ la∂ (V0) + 2∂V0∂la + V0∂ la = σ ,Z V0la, 2∂V0∂l∗a + V0∂2l∗a = 0, 2V ∂V ∂l∗+ V 2∂2l∗= 0, 0 0 a 0 a ∂V02∂l∗a + V 20 ∂2l∗a = 0, ∂(V 2∂l∗) = 0, 0 a
which integrates immediately to
V 20 ∂l∗a = m ∗a (D.3 )
where the ∗ m a are three independent l = 1, s = 1 functions.

By taking linear combinations they can be written as

m ∗ = T bmˆb = T b∂ˆlb a a a

where ˆ la is our usual ˆ √2-( -ζ+ζ i(ζ−-ζ) 1−-ζζ) la = 2 1,− 1+ζζ,− 1+ζζ ,1+ ζζ, and the coefficients b Ta are functions only of the coordinates za. Assuming that the monopole term in V 2 is sufficiently large so that it has no zeros and then by rescaling V we can write V −2 as a monopole plus higher harmonics in the form

V −2 = 1 + ∂W, 0

where W is a spin-wt s = − 1 quantity. From Eq. (D.3View Equation), we obtain

∂l∗a = V0−2m ∗a = (1 + ∂W )m ∗a = m ∗a + ∂(W m ∗a) = Tab∂ ˆlb + Tab∂ (W ˆmb ) = Tba∂(ˆlb + W ˆmb),

which integrates to

∗ bˆ la = Ta (la + W mˆa ). (D.4 )

The general solution to the linearized good-cut equation is thus

Z,a = V0l∗a = V0T ba(ˆlb + W mˆb ), (D.5 ) a a b ˆ V = v Z,a= V0v Ta(lb + W ˆmb).

We now demonstrate that

∫ (gabvavb)−1 = (8π)−1 V −2dΩ, (D.6 ) -- dζ ∧ dζ dΩ = 4i--------2. (D.7 ) (1 + ζζ)

In the integral of (D.6View Equation), we replace the independent variables -- (ζ,ζ) by

-- -- ζ∗ = -ζ-+-W--, ζ∗ = ζ. (D.8 ) 1 − W ζ
After some algebraic manipulation we obtain
dΩ ∗ = V − 2d Ω, (D.9 ) 0
and (surprisingly)
√ --( ∗ -- -- ∗ ∗-) (ˆla + W ˆma) = L ∗≡ --2- 1, − ζ--+-ζ-,− i(ζ-−-ζ-)-, 1 −-ζ-ζ , (D.10 ) a 2 1 + ζ ∗ζ 1 + ζ ∗ζ 1 + ζ∗ζ
so that
a b ∗ V = V0v TaL b. (D.11 )

Inserting Eqs. (D.8View Equation), (D.9View Equation) and (D.11View Equation) into (D.6View Equation) we obtain

∫ (gabvavb)−1 = (8π)−1 (V0vaTabL ∗b)−2V02dΩ∗, (D.12 ) ∫ −1 a b ∗− 2 ∗ = (8π) (v TaL b) dΩ , ∫ = (8π)−1 (v∗bL ∗b)−2dΩ ∗.

Using the form Eq. (D.10View Equation) the last integral can be easily evaluated (most easily done using 𝜃 and φ) leading to

(g vavb)− 1 = (η v∗av∗b)−1 = (TcT dη vavb)− 1, (D.13 ) ab ab a b cd gab = T caTbd ηcd,
In particular, note that when σ0 = 0 (i.e., the case of an everywhere shear-free NGC) T a= δa b b and the metric on ℋ-space reduces to the complex Minkowski metric, as claimed throughout the text. We can go a step further by taking the derivative of Eq. (D.12View Equation) with respect to va to find the covariant form of v, namely
∫ ----va---- ---gabvb--- −1 a b ∗ −3 b ∗ ∗ (g vavb)2 = (g vavb)2 = (8π) (v TaL b) TaL bdΩ . ab ab


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