3.2 Real cuts from the complex good cuts, I

Though our discussion of shear-free NGCs has relied, in an essential manner, on the use of the complexification of ℑ+ and the complex world lines in complex Minkowski space, it is the real structures that are of main interest to us. We want to find the intersection of the complex GCF with real ℑ+, i.e., what are the real points and real cuts of -- uB = G (τ,ζ, ζ), (&tidle; -- ζ = ζ), and what are the values of τ that yield real uB.

To construct an associated family of real cuts from a GCF, we begin with

-- √ -- -- uB = G (τ,ζ,ζ) = --2-ξ0(τ ) − 1ξi(τ)Y0(ζ,ζ ) (90 ) 2 2 1i
and write
τ = s + iλ (91 )
with s and λ real. The cut function can then be rewritten
-- -- uB = G (τ, ζ,ζ) = G (s + iλ,ζ,ζ ) (92 ) -- -- = GR (s,λ,ζ,ζ) + iGI (s,λ, ζ,ζ),
with real -- GR (s,λ,ζ,ζ) and -- GI (s,λ,ζ,ζ ). The -- GR (s,λ,ζ,ζ) and -- GI (s,λ,ζ,ζ ) are easily calculated from -- G(τ,ζ,ζ ) by
{ --------------} G (s,λ,ζ,ζ) = 1- G(s + iλ,ζ,ζ) + G (s + iλ, ζ,ζ) , (93 ) R 2 -- 1 { -- --------------} GI (s,λ,ζ,ζ) = 2- G(s + iλ,ζ,ζ) − G (s + iλ, ζ,ζ) . (94 )

By setting

-- GI (s, λ,ζ,ζ) = 0 (95 )
and solving for
-- λ = Λ(s,ζ,ζ ) (96 )
we obtain the associated real slicing,
u (R) = G (s,Λ (s,ζ,ζ),ζ,ζ). (97 ) B R
Thus, the values of τ that yield real values of uB are given by
-- τ = s + iΛ(s,ζ,ζ ). (98 )

As an example, using Equation (90View Equation), we find to first order in λ

√2-- √2-- 1[ ] -- uB = ---ξ0R(s) − ---ξ0I(s)′λ − -- ξiR(s) − ξiI(s)′λ Y 01i(ζ,ζ) (99 ) 2 -- 2 -- 2 [√ 2 √ 2 ] 1[ ] -- +i ----ξ0I(s) +----ξ0R(s)′λ − i--ξiI(s) + ξiR(s)′λ Y10i(ζ, ζ), 2 -- 2 2 u(R)= GR (s, Λ,ζ,ζ) (100 ) B √ -- √ -- [ ] --2-0 --2-0 ′ 1- i i ′ 0 -- = 2 ξR(s) − 2 ξI(s) λ − 2 ξR(s) − ξI(s) λ Y 1i(ζ,ζ), √ --0 i 0 -- λ = Λ(ζ,ζ) = − -√--2ξI(s) +-ξI(s)Y1i(ζ,ζ)-. (101 ) [ 2ξ0R(s)′ − ξiR (s)′Y10i(ζ, ζ)]
An Important Remark: We saw earlier that the shear-free angle field was given by
L (uB,ζ, ¯ζ) = ∂ (τ)G (τ, ζ, ¯ζ), (102 ) uB = G (τ,ζ, ¯ζ) ⇔ τ = T (uB, ζ, ¯ζ), (103 )
where real values of uB should be used. If the real cuts, ( -- -) uB = GR s,Λ (s,ζ,ζ),ζ,ζ, were used instead to calculate L (uB,ζ,ζ¯), the results would be wrong. The restriction of τ to yield real uB, does not commute with the application of the ∂ operator, i.e.,
¯ L(uB, ζ,ζ) ⁄= ∂GR.

The ∂ differentiation must be done first, holding τ constant, before the reality of u B is used. In other words, though we are interested in real + ℑ, it is essential that we consider its (local) complexification.


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