A Twistor Theory

Throughout this review, the study of the asymptotic gravitational field has been at the heart of all our investigations, and there is a natural connection between asymptotic gravitational fields and twistor theory. We give here a brief overview of Penrose’s asymptotic twistor theory (see, e.g., [67Jump To The Next Citation Point, 64, 35]) and its connection to the good-cut equation and the study of asymptotically shear-free NGCs at ℑ+; for a more in depth exposition of this connection, see [51Jump To The Next Citation Point, 6Jump To The Next Citation Point].

Let ℳ be any asymptotically-flat spacetime manifold, with conformal future null infinity + ℑ, coordinatized by (uB, ζ, ¯ζ). We can consider the complexification of ℑ+, referred to as ℑ+ℂ, which is in turn coordinatized by (uB, ζ, &tidle;ζ), where now uB ∈ ℂ and &tidle;ζ is different, but close to ¯ζ. Assuming analytic asymptotic Bondi shear σ0(uB,ζ,ζ¯), it can then be analytically continued to ℑ+ ℂ, i.e., we can consider 0 &tidle; σ (uB, ζ,ζ). We have seen in Section 4 that solutions to the good-cut equation

∂2G = σ0 (G,ζ, &tidle;ζ) (A.1 )
yield a four complex parameter family of solutions, given by
uB = Z(za;ζ,ζ&tidle;). (A.2 )
In our prior discussions, we interpreted these solutions as defining a four (complex) parameter family of surfaces on ℑ+ ℂ corresponding to each choice of the parameters za. In order to force agreement with the conventional description of Penrose’s asymptotic twistor theory we must use the complex conjugate good-cut equation
-2-- -- -- ∂ G = σ0(G, ζ, &tidle;ζ), (A.3 )
whose properties are identical to that of the good-cut equation. Its solutions, written as
-- -a &tidle; uB = Z (z ;ζ,ζ), (A.4 )
define complex two-surfaces in ℑ+ ℂ for fixed za. If we fix ζ = ζ0 ∈ ℂ, then Eq. (A.3View Equation) becomes an ordinary second-order differential equation with solutions describing curves in (u , &tidle;ζ) B space. Hence, each solution to this ODE is given by specifying initial conditions for &tidle; G and &tidle; ∂ &tidle;ζG at some arbitrary initial point, &tidle;ζ = &tidle;ζ0.

Note that it is not necessary that &tidle;ζ0 = ¯ζ0 on ℑ+ ℂ. However, we chose this initial point to be the complex conjugate of the constant ζ0, i.e., we take -- G and its first ^ ζ derivative at -- ^ζ = ζ0 as the initial conditions. Then the initial conditions for Eq. (A.3View Equation) can be written as [51Jump To The Next Citation Point]

-- ¯ uB0 = G (ζ0,ζ0), -- (A.5 ) -- --- ∂G L0 = ∂G (ζ0, ¯ζ0) = P0-¯-(ζ0, ¯ζ0), ∂ζ0
with P0 = 1 + ζ0&tidle;ζ0. Asymptotic projective twistor space, denoted ℙ 𝔗, is the space of all curves in ℑ+ ℂ generated by initial condition triplets -- (uB0,ζ0,L0) [67Jump To The Next Citation Point]: an asymptotic projective twistor is the curve corresponding to (u ,ζ,L- ) B0 0 0. A particular subspace of ℙ𝔗, called null asymptotic projective twistor space (ℙ𝔑), is the family of curves generated by initial conditions, which lie on (real) + ℑ; that is, at the initial point, &tidle;ζ0 = ζ¯0, the curve should cross the real ℑ+, i.e., should be real, uB0 = ¯uB0. Equivalently, an element of ℙ𝔑 can be said to intersect its dual curve (the solution generated by the complex conjugate initial conditions) at &tidle;ζ0 = ¯ζ0. The effect of this is to reduce the three-dimensional complex twistor space to five real dimensions. In standard notation, asymptotic projective twistors are defined in terms of their three complex twistor coordinates, 0 1 (μ ,μ ,ζ) [67Jump To The Next Citation Point]. These twistor coordinates may be re-expressed in terms of the asymptotic twistor curves by
μ0 = uB0 − ¯L0¯ζ0, (A.6 ) 1 ¯ μ = L0 + ζ0uB0, ζ = ζ0.

By only considering the twistor initial conditions &tidle;ζ0 = ζ¯0, we can drop the initial value notation, and just let uB0 = uB and ζ&tidle;= ¯ζ. The connection of twistor theory with shear-free NGCs takes the form of the flat-space Kerr theorem [67Jump To The Next Citation Point, 51Jump To The Next Citation Point]:

Theorem (Kerr Theorem). Any analytic function on ℙ𝕋 (projective twistor space) generates a shear-free NGC in Minkowski space.

Any analytic function F(μ0, μ1,ζ) on projective twistor space generates a shear-free NGC in Minkowski space by obtaining the -- -- -- L = L(uB, ζ,ζ), which defines the congruence via solving the algebraic equation

F (μ0,μ1,ζ) = F (u − L-ζ,L-+ ζu ,ζ) = 0. B B

It automatically satisfies the complex conjugate shear-free condition

--- -˙- ∂L + LL = 0.

We are interested in a version of the Kerr theorem that yields the regular asymptotically shear-free NGCs. Starting with the general four-parameter solution to Eq. (A.3View Equation), i.e., -- uB = Z (za; ζ, &tidle;ζ), we chose an arbitrary world line -a a z = ξ (τ), so that we have

-- a ¯ -- ¯ -- uB = Z(ξ-(τ),ζ,ζ ) = G (τ,ζ,ζ), (A.7 ) L(τ,ζ, ¯ζ) = ∂(τ)G (τ,ζ, ¯ζ).
By inserting these into the twistor coordinates, Eq. (A.6View Equation), we find
0 ¯ ¯¯ -- ¯¯ -- μ (τ,ζ,ζ) = uB − Lζ = G −-ζ∂(τ)G, (A.8 ) μ1(τ,ζ,ζ¯) = L¯+ ζuB = ¯∂ (τ)G + ζG. (A.9 )
The 0 μ and 1 μ are now functions of τ and ζ, while the ¯ζ is now to be treated as a fixed quantity, the complex conjugate of ζ, and not as an independent variable. By eliminating τ in Eqs. (A.8View Equation) and (A.9View Equation), we obtain a single function of μ0, μ1, and ζ: namely, F (μ0,μ1,ζ ) = 0. Thus, the regular asymptotically shear-free NGCs are described by a special class of twistor functions. This is a special case of a generalized version of the Kerr theorem [67, 51].

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