A CR manifold is a differentiable manifold endowed with an additional structure called its ‘CR
structure’; formally this is a complex distribution (i.e., a sub-bundle
) which is formally
integrable and almost Lagrangian [22
]. More concretely, the CR structure can be described by a set of
vectors or 1-forms on
defined up to a particular gauge freedom. In the context of this review, we are
interested in the case where
is a real three-manifold.
A CR structure on a real three manifold , with local coordinates
, can be given intrinsically by
equivalence classes of one-forms, one real, one complex and its complex conjugate [44
]. If we denote the real
one-form by
and the complex one-form by
, then these are defined up to the transformations:
The are functions on
:
is nonvanishing and real,
and
are complex function
with
nonvanishing. We further require that there be a three-fold linear-independence relation between
these one-forms [44]:
Any three-manifold with a CR structure is referred to as a three-dimensional CR manifold. There are
special classes (referred to as embeddable) of three-dimensional CR manifolds that can be directly
embedded into . We show how the choice of any specific asymptotically shear-free NGC induces an
embeddable CR structure on
. Though there are several ways of arriving at this CR structure, the
simplest way is to look at the asymptotic null tetrad system associated with the asymptotically
shear-free NGC, i.e., look at the (
,
,
,
) of Eq. (6.14
). The associated dual
one-forms, restricted to
(after a conformal rescaling of
), become (with a slight notational
dishonesty),
Therefore, for the situation discussed here, where we have singled out a unique asymptotically shear-free
NGC and associated complex world line, we have a uniquely chosen CR structure induced on
. To see how our three manifold,
, can be embedded into
we introduce the CR
equation [45]
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and seek two independent (complex) solutions, that define the
embedding of
into
with coordinates
. We have immediately that
is a solution. The second solution is also easily found; we see directly from Eq. (4.13
) [54],
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the inverse to , is a CR function and that we can consider
to be embedded in the
of
.
An important structure associated to any embeddable CR manifold of codimension one is its Levi form;
this determines a metric on the CR structure as a bundle on the manifold [22]. As we have just discussed,
is just such a CR manifold, and one can show that its Levi form (in the CR structure induced by an
asymptotically shear-free NGC) is proportional to the twist of the congruence. Hence, any CR structure on
which is generated by a congruence with its source on a real world line
is
Levi-flat [7
].
In the context of this review, the important observation is that the physical content of asymptotically shear-free NGCs is encoded in the corresponding CR structure. This gives a physical interpretation for CR structures in the setting of asymptotically flat spacetimes. It would be interesting for future research to study the relationship between our findings and those of [34], which demonstrates how the Einstein equations for algebraically special spacetimes can be realized in terms of the embeddable CR structures associated with their PNDs.
http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
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