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., [67, 64, 35]) and its connection to the goodcut equation and the study of asymptotically shearfree NGCs at ; for a more in depth exposition of this connection, see [51, 6].
Let be any asymptoticallyflat spacetime manifold, with conformal future null infinity , coordinatized by . We can consider the complexification of , referred to as , which is in turn coordinatized by , where now and is different, but close to . Assuming analytic asymptotic Bondi shear , it can then be analytically continued to , i.e., we can consider . We have seen in Section 4 that solutions to the goodcut equation
yield a four complex parameter family of solutions, given by In our prior discussions, we interpreted these solutions as defining a four (complex) parameter family of surfaces on corresponding to each choice of the parameters . In order to force agreement with the conventional description of Penrose’s asymptotic twistor theory we must use the complex conjugate goodcut equation whose properties are identical to that of the goodcut equation. Its solutions, written as define complex twosurfaces in for fixed . If we fix , then Eq. (A.3) becomes an ordinary secondorder differential equation with solutions describing curves in space. Hence, each solution to this ODE is given by specifying initial conditions for and at some arbitrary initial point, .Note that it is not necessary that on . However, we chose this initial point to be the complex conjugate of the constant , i.e., we take and its first derivative at as the initial conditions. Then the initial conditions for Eq. (A.3) can be written as [51]
with . Asymptotic projective twistor space, denoted , is the space of all curves in generated by initial condition triplets [67]: an asymptotic projective twistor is the curve corresponding to . A particular subspace of , called null asymptotic projective twistor space (), is the family of curves generated by initial conditions, which lie on (real) ; that is, at the initial point, , the curve should cross the real , i.e., should be real, . Equivalently, an element of can be said to intersect its dual curve (the solution generated by the complex conjugate initial conditions) at . The effect of this is to reduce the threedimensional complex twistor space to five real dimensions. In standard notation, asymptotic projective twistors are defined in terms of their three complex twistor coordinates, [67]. These twistor coordinates may be reexpressed in terms of the asymptotic twistor curves byBy only considering the twistor initial conditions , we can drop the initial value notation, and just let and . The connection of twistor theory with shearfree NGCs takes the form of the flatspace Kerr theorem [67, 51]:
Theorem (Kerr Theorem). Any analytic function on (projective twistor space) generates a shearfree NGC in Minkowski space.
Any analytic function on projective twistor space generates a shearfree NGC in Minkowski space by obtaining the , which defines the congruence via solving the algebraic equation

It automatically satisfies the complex conjugate shearfree condition

We are interested in a version of the Kerr theorem that yields the regular asymptotically shearfree NGCs. Starting with the general fourparameter solution to Eq. (A.3), i.e., , we chose an arbitrary world line , so that we have
By inserting these into the twistor coordinates, Eq. (A.6), we find The and are now functions of and , while the is now to be treated as a fixed quantity, the complex conjugate of , and not as an independent variable. By eliminating in Eqs. (A.8) and (A.9), we obtain a single function of , , and : namely, . Thus, the regular asymptotically shearfree 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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Living Rev. Relativity 15, (2012), 1
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