## 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., [67, 64, 35]) and its connection to the good-cut equation and the study of asymptotically shear-free NGCs at ; for a more in depth exposition of this connection, see [51, 6].

Let be any asymptotically-flat 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 good-cut 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 good-cut equation
whose properties are identical to that of the good-cut equation. Its solutions, written as
define complex two-surfaces in for fixed . If we fix , then Eq. (A.3) becomes an ordinary second-order 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 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,  [67]. These twistor coordinates may be re-expressed in terms of the asymptotic twistor curves by

By only considering the twistor initial conditions , we can drop the initial value notation, and just let and . The connection of twistor theory with shear-free NGCs takes the form of the flat-space Kerr theorem [67, 51]:

Theorem (Kerr Theorem). Any analytic function on (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 by obtaining the , which defines the congruence via solving the algebraic equation

It automatically satisfies the complex conjugate shear-free condition

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.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 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].