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 u B. These reality structures were first observed in [7Jump To The Next Citation Point] and recently there have been attempts to study them in the framework of holographic dualities (cf. [8Jump To The Next Citation Point] and Section 8).

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 ) (3.19 ) 2 2 1i
and write
τ = s + iλ (3.20 )
with s and λ real. The cut function can then be rewritten
-- -- uB = G (τ, ζ,ζ) = G (s + iλ,ζ,ζ ) (3.21 ) -- -- = 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) , (3.22 ) R 2 -- i{ -- --------------} GI (s,λ,ζ,ζ ) = − 2 G(s + iλ,ζ,ζ) − G (s + iλ, ζ,ζ) . (3.23 )

By setting

-- GI (s, λ,ζ,ζ) = 0 (3.24 )
and solving for
-- λ = Λ(s,ζ,ζ ) (3.25 )
we obtain the associated one-parameter, s, family of real slicings,
-- -- ( --) -- u (RB )= GR (s,Λ (s,ζ,ζ),ζ,ζ) = ξa s + iΛ (s,ζ,ζ) la(ζ,ζ). (3.26 )
Thus, the values of τ that yield real values of uB are given by
τ = s + iΛ(s,ζ,ζ-). (3.27 )

Perturbatively, using Eq. (3.19View Equation) and writing ξa(s) = ξa(s) + iξa(s) R I, we find λ to first order:

√ -- √ -- [ ] u = --2ξ0(s) − --2ξ0(s)′λ − 1- ξi(s) − ξi(s)′λ Y 0(ζ,ζ) (3.28 ) B 2 R 2 I 2 R I 1i [√ -- √ -- ] [ ] -- +i --2-ξ0(s) +--2-ξ0(s)′λ − i1-ξi(s) + ξi(s)′λ Y 0(ζ, ζ), 2 I 2 R 2 I R 1i (R) -- uB = G√R-(s, Λ,ζ,ζ)√ -- [ ] (3.29 ) 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), √ -- -- -- ----2ξ0I(s) +-ξiI(s)Y10i(ζ,ζ)- λ = Λ(s,ζ,ζ) = − [√2-ξ0(s)′ − ξi (s)′Y 0(ζ,ζ)]. (3.30 ) R R 1i

Continuing, with small values for the imaginary part of ξa(τ) = ξaR (τ ) + iξaI(τ), (ξaR(τ), ξaI(τ ) both real analytic functions) and hence small Λ(s,ζ,ζ-), it is easy to see that Λ (s,ζ,ζ) (for fixed value of s) is a bounded smooth function on the -- (ζ, ζ) sphere, with maximum and minimum values, -- λmax = Λ(s,ζmax,ζmax ) and -- λmin = Λ(s,ζmin,ζmin). Furthermore on the (-- ζ,ζ) sphere, there are a finite line-segments worth of curves (circles) that lie between -- (ζmin,ζmin) and -- (ζmax,ζmax) such that -- Λ (s,ζ,ζ) is a monotonically increasing function on the family of curves. Hence there will be a family of circles on the (ζ,ζ)-sphere where the value of λ is a constant, ranging between λ max and λmin.

Summarizing, we have the result that in the complex τ-plane there is a ribbon or strip given by all values of s and line segments parametrized by λ between λmin and λmax such that the complex light-cones from each of the associated points, ξa(s + iλ ), all have some null geodesics that intersect real + ℑ. More specifically, for each of the allowed values of τ = s + iλ there will be a circle’s worth of complex null geodesics leaving the point a ξ (s + iλ) reaching real + ℑ. It is the union of these null geodesics, corresponding to the circles on the -- (ζ,ζ)-sphere from the line segment, that produces the real family of cuts, Eq. (3.26View Equation).

The real structure associated with a complex world line is then this one-parameter family of slices (cuts) Eq. (3.26View Equation).

Remark 7. We saw earlier that the shear-free angle field was given by

¯ ¯ L (uB,ζ,ζ) = ∂ (τ)G (τ, ζ,ζ), (3.31 ) uB = G (τ,ζ, ¯ζ) ⇔ τ = T (uB,ζ, ¯ζ), (3.32 )
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 uB is used. In other words, though we are interested in real ℑ+, it is essential that we consider its (local) complexification.

There are a pair of important (dual) results that arise from the considerations of the good cuts [7Jump To The Next Citation Point, 8Jump To The Next Citation Point]. From the stereographic angle field, i.e., L from Eqs. (3.31View Equation) and (3.32View Equation), one can form two different conjugate fields, (1) the complex conjugate of L:

-- ---- -a ---- L = ∂τG = ξ (τ)ˆma (3.33 )
and (2) the holomorphic conjugate, ^L, given by
-- -- ^L = ∂ τG = ξa(τ )ˆma. (3.34 )

The two different pairs, the complex conjugate pair (-- L,L) and the holomorphic pair (L, ^L) determine two different null vector direction fields at ℑ+, the real vector field, l∗a, and the complex field, l∗a C, via the relations

a ∗a a L¯ a L- a −2 l → l = l − r m − r ¯m + O (r ), (3.35 )
and
L^ L la → l∗Ca= la −--ma − --¯ma + O(r− 2). (3.36 ) r r
Both generate, in the spacetime interior, shear-free null geodesic congruences: the first is a real twisting shear-free congruence while the latter is a complex twist-free congruence that consists of the light-cones from the world line, a a z = ξ (τ ), i.e., they focus on a ξ (τ). It is this fact that they focus on the world line, a a z = ξ (τ) that is of most relevance to us.

The twist of the real congruence, -- Σ (uB, ζ,ζ), which comes from the complex divergence,

ρ = − --1---- (3.37 ) r + iΣ 2iΣ = ∂L-+ L (L)⋅ − ∂L − LL˙. (3.38 ) a -a -- = (ξ (τ ) − ξ (τ )) (na − la).
is proportional to the imaginary part of the complex world line and consequently we have the real structure associated with the complex world line coming from two (dual) places, the real cuts, Eq. (3.26View Equation) and the twist.

It is the complex point of view of the complex light-cones coming from the complex world line that dominates our discussion.


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