Conjuncted: Twistor Spaces

The α-planes can be generalized to a suitable class of curved complex space-times. By a complex space-time (M,g) we mean a four-dimensional Hausdorff manifold M with holomorphic metric g. Thus, with respect to a holomorphic coordinate basis x^a, g is a 4×4 matrix of holomorphic functions of x^a, and its determinant is nowhere-vanishing. Remarkably, g determines a unique holomorphic connection ∇, and a holomorphic curvature tensor R^a_bcd. Moreover, the Ricci tensor R_ab becomes complex-valued, and the Weyl tensor C^a_bcd may be split into independent holomorphic tensors, i.e. its self-dual and anti-self-dual parts, respectively. With our two-spinor notation, one has

C_abcd = ψ_ABCD ε_A′B′ ε_C′D′ + ψ^~_{A′B′C′D′} ε_AB ε_CD

where ψ_ABCD = ψ_(ABCD), ψ_{A′B′C′D′} = ψ^~_{(A′B′C′D′)}. The spinors ψ and ψ^~ are the anti-self-dual and self-dual Weyl spinors, respectively.

ψ^~_{A′B′C′D′} = 0, R_ab = 0, are called right-flat or anti-self-dual, whereas complex vacuum space-times such that

ψ_ABCD = 0, R_ab = 0,

are called left-flat or self-dual. This definition only makes sense if space-time is complex or real Riemannian, since in this case no complex conjugation relates primed to unprimed spinors (i.e. the corresponding spin-spaces are no longer anti-isomorphic). Hence, for example, the self-dual Weyl spinor ψ^~_{A′B′C′D′} may vanish without its anti-self-dual counterpart ψ_ABCD having to vanish as well, or the converse may hold.

By definition, α-surfaces are complex two-surfaces S in a complex space-time (M, g) whose tangent vectors v have the two-spinor form, where λ^A is varying, and π^A′ is a fixed primed spinor field on S. From this definition, the following properties can be derived:

(i) tangent vectors to α-surfaces are null;

(ii) any two null tangent vectors v and u to an α-surface are orthogonal to one another;

(iii) the holomorphic metric g vanishes on S in that g(v, u) = g(v, v) = 0, ∀ v, u, so that α-surfaces are totally null;

(iv) α-surfaces are self-dual, in that F ≡ v ⊗ u − u ⊗ v takes the two-spinor form;

(v) α-surfaces exist in (M,g) iff the self-dual Weyl spinor vanishes, so that (M, g) is anti-self-dual.

The properties (i)–(iv), are the same as in the flat-space-time case, provided we replace the flat metric η with the curved metric g. Condition (v), however, is a peculiarity of curved space-times.

We want to prove that, if (M,g) is anti-self-dual, it admits a three-complex- parameter family of self-dual α-surfaces. Indeed, given any point p ∈ M and a spinor μ_A^′ at p, one can find a spinor field π_A^′ on M, satisfying the equation

π^A′ ∇_AA^′ π_B^′ = ξ_Aπ_B^′,

and such that

π_A^′(p) = μ_A^′(p)

Hence π_A^′ defines a holomorphic two-dimensional distribution, spanned by the vector fields of the form λ^Aπ^A′, which is integrable. Thus, in particular, there exists a self-dual α-surface through p, with tangent vectors of the form λ^Aμ^A′ at p. Since p is arbitrary, this argument may be repeated ∀p ∈ M. The space P of all self-dual α-surfaces in (M,g) is three-complex-dimensional, and is called twistor space of (M, g).