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
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
]
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
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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
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Living Rev. Relativity 15, (2012), 1
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