E Shear-Free Congruences from Complex World Lines

In this appendix, we show that the family of complex light-cones with apex on a complex world line in complex Minkowski space 𝕄 ℂ have null generators that form a real shear-free null geodesic congruence in real Minkowski space [7].

Theorem. There exists a mapping from the arbitrary complex-analytic world line a ξ (τ ) ∈ 𝕄 ℂ to the real shear-free NGC in 𝕄 given by complex null displacements.

Proof: We first recall from Section 3 that regular real shear-free NGCs in 𝕄 are parametrically given by

--a xa = uB (ˆla + ˆna ) − L ˆm − L¯ˆma + (r∗ − r0)ˆla (E.1 )
u = ξb(τ)^l , τ = T (u ,ζ, ¯ζ) (E.2 ) B b B L (uB,ζ, ¯ζ) = ξa(τ)mˆa (ζ, ¯ζ), L-(u ,ζ, ¯ζ) = ξa(τ)mˆ (ζ, ¯ζ). B a
The τ is taken so that uB is real via Eq. (3.27View Equation): τ = s + iΛ(s,ζ,ζ¯), and r0 is the arbitrary origin for the affine parameter along each geodesic of the congruence.

Beginning with the world line, ξa(τ), we add to it a specific complex null displacement (to be constructed)

La = La0(uB,ζ, ¯ζ) + rLa1(uB, ζ, ¯ζ), LaL = La L = LaL = 0 a 1 1a 0 0a
parametrized by the real variable r. We will show that the curve (complex null geodesic) given by
a a ¯ a ¯ a ¯ x = ξ (T(uB, ζ,ζ)) + L0(uB, ζ,ζ) + rL 1(uB,ζ, ζ)

with fixed (uB,ζ, ¯ζ) but varying r, is identical to that given by Eq. (E.1View Equation).

This is demonstrated by taking the world line, ξa(τ), written in terms of its components (b b b b-- ξ lb,ξ nb,ξ mb, ξ mb) as

a b ˆ a b ˆa b --a b -- a ξ (τ) = ξ (τ )lbˆn + ξ (τ)^nbl − ξ(τ)mˆb mˆ − ξ (τ)ˆmbmˆ (E.3 )
and replacing the na by the identity
√ -- ˆna = 2ta − ˆla. a a t = δ0.
This yields
ξa(τ) = √2-ξb(τ)ˆlta + √2-ξb(τ)tˆla − 2ξb(τ)ˆl^la − ξb(τ )mˆ mˆa − ξb(τ)ˆm- ˆma. (E.4 ) b b b b b
Using the relations, Eq. (3.34View Equation), etc.,
b ˆ ¯ uB = ξ (τ)lb, τ = T (uB, ζ,ζ) L (uB,ζ, ¯ζ) = ξb(τ)ˆmb ^ ¯ a -- ¯ L (uB,ζ,ζ) = ξ (τ)mˆa (ζ,ζ),
Eq. (E.4View Equation) becomes
-- -- ξa(τ ) = √2u ta + √ 2ξ0(τ)ˆla − 2u ˆla − L ˆma − L&tidle;ˆma. (E.5 ) B B

By adding the complex null vector (displacement),

a &tidle; -- a √ --0 ˆa L = (L − L)mˆ + (r − r0 + 2uB − 2ξ (τ))l

to both sides of Eq. (E.5View Equation), we obtain

a a a x ≡ ξ (τ) + L ( ) xa = ξa(τ) + (&tidle;L − L)mˆa + r − r + 2u − √2-ξ0(τ) ˆla 0 B xa = u ta − Lmˆa − L-ˆma + (r − r )ˆla B 0
To complete our task we now restrict the values of τ to those that produce a real u, namely
τ → τ(R) = s + iΛ (s,ζ, ¯ζ)

and restrict r to the real.

We see that by adding a null ray, combinations of a ˆm and ˆa l, directly to the complex world line ξa(τ), we obtain a mapping of the complex world line directly to the real shear-free NGC, Eq. (E.1View Equation). Note that when the affine parameter, r, is chosen (complex) as √ -- r = r0 − 2u + 2ξ0(τ) the ˆla term drops out and we have the ‘point’ ξa(τ) surrounded by the embedded complex sphere, za = ξa(τ) + La (u, ζ, ¯ζ) = ξa(τ) + (&tidle;L − L)mˆa 0. The ray can be thought of as having its origin on this surface.

We thus have the explicit relationship between the complex world line and the shear-free NGC, completing the proof.

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