A Twistor Theory

Throughout this review, the study of the asymptotic gravitational field has been at the heart of all our investigations. Here we make contact with Penrose’s asymptotic twistor theory (see, e.g., [51Jump To The Next Citation Point4822]). We give here a brief overview of asymptotic twistor theory 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 [36Jump 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) (360 )
yield a four complex parameter family of solutions, given by
uB = G(za;ζ,ζ&tidle;). (361 )
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), (362 )
whose properties are identical to that of the good-cut equation. Its solutions, written as
-- -a &tidle; uB = G (z ;ζ,ζ), (363 )
define complex two-surfaces in ℑ+ ℂ for fixed za. If, in addition to fixing the za, we fix ζ = ζ0 ∈ ℂ, then Equation (362View Equation) becomes an ordinary second-order differential equation with solutions describing curves in &tidle; (uB, ζ) space. Hence, each solution to this ODE is given by specifying initial conditions for G&tidle; and ∂ &tidle;ζ &tidle;G at some arbitrary initial point, &tidle;ζ = &tidle;ζ0. Note that it is not necessary that &tidle;ζ0 = ζ¯0 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 Equation (362View Equation) can be written as [36Jump To The Next Citation Point]

-- &tidle; uB0 = G (ζ0,ζ0), -- (364 ) -- --- ∂G L0 = ∂G (ζ0, &tidle;ζ0) = P0-&tidle;-(ζ0, &tidle;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 ) [51Jump To The Next Citation Point]: an asymptotic projective twistor is the curve corresponding to -- (uB0, ζ0,L0).

A particular subspace of ℙ 𝔗, called null asymptotic projective twistor space (ℙ 𝔑), is the family of curves generated by initial conditions, which lie on (real) ℑ+; that is, at the initial point, &tidle;ζ0 = ¯ζ0, 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, (μ0,μ1,ζ) [51Jump To The Next Citation Point]. These twistor coordinates may be re-expressed in terms of the asymptotic twistor curves by

0 ¯ ¯ μ = uB0 − L0ζ0, (365 ) μ1 = L¯0 + ζ0uB0, ζ = ζ0.

By only considering the twistor initial conditions &tidle;ζ = 0 ζ¯ 0, we can drop the initial value notation, and just let uB0 = uB and &tidle; ζ = ¯ ζ.

The connection of twistor theory with shear-free NGCs takes the form of the flat-space Kerr theorem [51Jump To The Next Citation Point36Jump To The Next Citation Point]:

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

Any analytic function on projective twistor space generates a shear-free NGC in Minkowski space, i.e., from 0 1 ----- F(μ ,μ ,ζ) ≡ F(uB − Lζ,L + ζuB,ζ) = 0, one can construct a shear-free NGC in Minkowski space. The -- -- -- L = L (uB, ζ,ζ), which defines the congruence, is obtained by solving the algebraic equation

--- -- F (uB − Lζ, L + ζuB, ζ) = 0.

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 Equation (362View Equation), i.e., -- uB = G(za;ζ,ζ&tidle;), we chose an arbitrary world line -a a z = ξ (τ), so that we have

-- a ¯ --- ¯ -- uB = G-(ξ-(τ),ζ,ζ) = X (τ,ζ,ζ ), (366 ) L(τ,ζ,ζ¯) = ∂ (τ)X (τ,ζ, ¯ζ).

By inserting these into the twistor coordinates, Equation (365View Equation), we find

--- --- μ0(τ,ζ, ¯ζ) = uB − L¯¯ζ = X − ¯ζ¯∂(τ)X, (367 ) μ1(τ,ζ, ¯ζ) = ¯L + ζu = ¯∂ X-+ ζX.- (368 ) B (τ)

The μ0 and μ1 are now functions of τ and ζ: 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 Equations (367View Equation) and (368View Equation), we obtain a single function of 0 μ, 1 μ, and ζ: namely, 0 1 F (μ ,μ ,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 [5136].

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