2.3 The optical equations
Since this work concerns NGCs and, in particular, shear-free and asymptotically shear-free NGCs, it is
necessary to first define them and then study their properties.
Given a Lorentzian manifold with local coordinates,
, and a NGC, i.e., a foliation by a three
parameter family of null geodesics,
with
the affine parametrization and the (three)
labeling the geodesics, the tangent vector field
satisfies the geodesic equation
The two complex optical scalars (spin coefficients),
and
, are defined by
with
an complex (spacelike) vector that is parallel propagated along the null geodesic field and
satisfies
.
The
and
satisfy the optical equations of Sachs [72], namely,
where
and
are, respectively, a Ricci and a Weyl tensor tetrad component (see below). In flat
space, i.e., with
, excluding the degenerate case of
, plane (
) and
cylindrical fronts, the general solution is
The complex
(referred to as the asymptotic shear) and the real
(called the twist) are determined
from the original congruence, Eq. (2.19). Both are functions just of the parameters,
. Their behavior
for large
is given by
From this,
gets its name as the asymptotic shear. In Section 3, we return to the issue of the explicit
construction of NGCs in Minkowski space and in particular to the construction and detailed properties of
regular shear-free congruences.
Note the important point that, in
, the vanishing of the asymptotic shear forces the shear to vanish.
The same is not true for asymptotically-flat spacetimes. Specifically, for future null asymptotically-flat
spaces described in a Bondi tetrad and coordinate system, we have, from other considerations, that
which leads to the asymptotic behavior of
and
,
with the two order symbols explicitly depending on the leading terms in
and
. The
vanishing of
does not, in this nonflat case, imply that
vanishes. This case, referred to as
asymptotically shear-free, plays the major role later. It will be returned to in greater detail in
Section 4.