B CR Structures

A CR manifold š’© is a differentiable manifold endowed with an additional structure called its ‘CR structure’; formally this is a complex distribution (i.e., a sub-bundle L ⊂ T š’© ⊗ ā„‚) which is formally integrable and almost Lagrangian [22Jump To The Next Citation Point]. More concretely, the CR structure can be described by a set of vectors or 1-forms on š’© defined up to a particular gauge freedom. In the context of this review, we are interested in the case where š’© is a real three-manifold.

A CR structure on a real three manifold š’©, with local coordinates xa, can be given intrinsically by equivalence classes of one-forms, one real, one complex and its complex conjugate [44Jump To The Next Citation Point]. If we denote the real one-form by l and the complex one-form by m, then these are defined up to the transformations:

l → a(xa)l, (B.1 ) m → f(xa )m + g(xa)l.

The (a,f,g ) are functions on š’©: a is nonvanishing and real, f and g are complex function with f nonvanishing. We further require that there be a three-fold linear-independence relation between these one-forms [44]:

l ∧ m ∧ ¯m ā„= 0. (B.2 )

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 2 ā„‚. We show how the choice of any specific asymptotically shear-free NGC induces an embeddable CR structure on ℑ+. 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 (l∗a, m ∗a, m-∗a, n ∗a) of Eq. (6.14View Equation). The associated dual one-forms, restricted to + ℑ (after a conformal rescaling of m), become (with a slight notational dishonesty),

L L¯ l∗ = duB − -------dζ − ------d ¯ζ, (B.3 ) -- 1 + ζ¯ζ 1 + ζ¯ζ ∗ dζ --∗ dζ m = -----¯, m = -----¯, 1 + ζζ 1 + ζζ
with L = L(uB, ζ, ¯ζ), satisfying the shear-free condition. (This same result could have been obtained by manipulating the exterior derivatives of the twistor coordinates, Eq. (A.6View Equation).) The dual vectors – also describing the CR structure – are
--- -∂- -∂-- -∂-- š” = P ∂ζ + L ∂u = ∂ (uB) + L ∂u , (B.4 ) -- B -- B š” = P -∂-+ L -∂-- = ∂ (u ) + L -∂--, ∂ζ ∂uB B ∂uB ∂ š” = ----. ∂uB

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 + ℑ. To see how our three manifold, + ℑ, can be embedded into 2 ā„‚ we introduce the CR equation [45]

--- -∂-- š”K ≡ ∂(uB)K + L ∂uB K = 0

and seek two independent (complex) solutions, -- -- K1 = K1(uB, ζ,ζ),K2 = K2 (uB, ζ,ζ) that define the embedding of + ℑ into 2 ā„‚ with coordinates (K1, K2 ). We have immediately that -- K1 = ζ = x − iy is a solution. The second solution is also easily found; we see directly from Eq. (4.13View Equation[54],

∂ T + L TĖ™= 0, (B.5 ) (uB)
that
-- τ = T (uB,ζ,ζ),

the inverse to -- uB = X (τ,ζ,ζ), is a CR function and that we can consider + ℑ to be embedded in the 2 ā„‚ of -- (τ,ζ).

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, ℑ+ 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 ℑ+ which is generated by a congruence with its source on a real world line ξa(s) ∈ š•„ is Levi-flat [7Jump To The Next Citation Point].

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.


  Go to previous page Go up Go to next page