Revisiting Twistors

In twistor theory, α-planes are the building blocks of classical field theory in complexified compactified Minkowski space-time. The α-planes are totally null two-surfaces S in that, if p is any point on S, and if v and w are any two null tangent vectors at p ∈ S, the complexified Minkowski metric η satisfies the identity η(v,w) = v_aw^a = 0. By definition, their null tangent vectors have the two-component spinor form λ^Aπ^A′, where λ^A is varying and π^A′ is fixed. Therefore, the induced metric vanishes identically since η(v,w) = λ^Aπ^A′ μ_Aπ_A^′ = 0 = η(v, v) = λ^Aπ^A′ λ_Aπ_A^′ . One thus obtains a conformally invariant characterization of flat space-times. This definition can be generalized to complex or real Riemannian space-times with non-vanishing curvature, provided the Weyl curvature is anti-self-dual. One then finds that the curved metric g is such that g(v,w) = 0 on S, and the spinor field π_A^′ is covariantly constant on S. The corresponding holomorphic two-surfaces are called α-surfaces, and they form a three-complex-dimensional family. Twistor space is the space of all α-surfaces, and depends only on the conformal structure of complex space-time.

Projective twistor space PT is isomorphic to complex projective space CP³. The correspondence between flat space-time and twistor space shows that complex α-planes correspond to points in PT, and real null geodesics to points in PN, i.e. the space of null twistors. Moreover, a complex space-time point corresponds to a sphere in PT, and a real space-time point to a sphere in PN. Remarkably, the points x and y are null-separated iff the corresponding spheres in PT intersect. This is the twistor description of the light-cone structure of Minkowski space-time.

A conformally invariant isomorphism exists between the complex vector space of holomorphic solutions of  ◻φ = 0 on the forward tube of flat space-time, and the complex vector space of arbitrary complex-analytic functions of three variables, not subject to any differential equation. Moreover, when curvature is non-vanishing, there is a one-to-one correspondence between complex space-times with anti-self-dual Weyl curvature and scalar curvature R = 24Λ, and sufficiently small deformations of flat projective twistor space PT which preserve a one-form τ homogeneous of degree 2 and a three-form ρ homogeneous of degree 4, with τ ∧ dτ = 2Λρ. Thus, to solve the anti-self-dual Einstein equations, one has to study a geometric problem, i.e. finding the holomorphic curves in deformed projective twistor space.

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