## 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., [514822]). 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 . For a more in depth exposition of this connection, see [36].

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, in addition to fixing the , we fix , then Equation (362) 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 Equation (362) can be written as [36]

with . Asymptotic projective twistor space, denoted , is the space of all curves in generated by initial condition triplets [51]: 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, [51]. 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 [5136]:

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, i.e., from , one can construct a shear-free NGC in Minkowski space. The , which defines the congruence, is obtained by 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 Equation (362), i.e., , we chose an arbitrary world line , so that we have

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

The and are now functions of and : the is now to be treated as a fixed quantity, the complex conjugate of , and not as an independent variable.

By eliminating in Equations (367) and (368), 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 [5136].