4.3 Real cuts from the complex good cuts, II

The construction of real structures from the complex structures, i.e., finding the complex values of τ that yield real values of uB and the associated real cuts, is virtually identical to the flat-space construction of Section 3. The real structure associated with the complex Minkowski space complex world lines is easily extended to the ℋ-space world lines associated with asymptotically flat spacetimes. The only difference is that we start with the GCF
-- -- uB = G (τ, ζ,ζ) = ξa(τ)ˆla(ζ,ζ¯) + Gl≥2 (τ, ζ,ζ) (4.38 )
rather than the flat-space
u = G (τ,ζ,ζ) = ξa(τ)ˆl (ζ, ¯ζ). B a

Again assuming that the Bondi shear is sufficiently small and the ℋ-space complex world line is not too far from the “real”, the solution to the good-cut equation (4.15View Equation), i.e., Eq. (4.38View Equation), with τ = s + iλ, is decomposed into real and imaginary parts,

-- 1 ( -- -- --) 1 ( -- -- --) G (τ,ζ,ζ) = -- G (s + iλ,ζ, ζ) + G (s − iλ,ζ,ζ) + -- G (s + iλ,ζ, ζ) − G (s − iλ,ζ,ζ) . (4.39 ) 2 2
Setting the imaginary part to zero and solving for λ we obtain an expression of the form,
-- λ = Λ (s,ζ,ζ).

As in the flat case, for fixed s = s0, Λ has values on a line segment bounded between some λmin and λ max. The allowed values of τ are again on a ribbon in the τ-plane (i.e., region which is topologically ℝ × I for an interval I); all values of s and allowed values on the λ-line segments.

Each level curve of the function -- λ = Λ (s0,ζ,ζ ) = constant on the -- (ζ,ζ)-sphere (closed curves or isolated points) determines a specific subset of the null directions and associated null geodesics on the light-cone of the complex point a -- ξ (s0 + iΛ(s0,ζ,ζ )) that intersect the real + ℑ. These geodesics will be referred to as ‘real’ geodesics. As λ moves over all allowed values of its segment, we obtain the set of ℋ-space points, -- ξa(s0 + iΛ (s0,ζ,ζ)) and their collection of ‘real’ geodesics. From Eq. (4.39View Equation), these ‘real’ geodesics intersect ℑ+ on the real cut

u(R) = G (s + iΛ (s ,ζ,ζ),ζ,ζ). (4.40 ) B R 0 0

As s varies we obtain a one-parameter family of cuts. If these cuts do not intersect with each other we say that the complex world line ξa(τ) is by definition ‘timelike.’ This occurs when the time component of the real part of the complex velocity vector, a a v (τ ) = dξ (τ)∕dτ, is sufficiently large.

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