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 [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 onIn 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 Then the initial conditions for Equation (362) can be written as [36
]
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 [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
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It automatically satisfies the complex conjugate shear-free condition
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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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