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., [51, 48, 22]). 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 .
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
In order to force agreement with the conventional description of Penrose’s asymptotic twistor theory we must use the complex conjugate good-cut equationnot 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 : 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, . 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 [51, 36]:
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 [51, 36].
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