Twistor spaces

Abstract

In twistor theory, α-planes are the building blocks of classical field theory in complexified compactified Minkowski space-time. 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 _a w^a=0. By definition, their null tangent vectors have the two-component spinor form λ^Aπ^A’, where λ^A Ais varying and π^A’ is fixed. Therefore, the induced metric vanishes identically since η(v,w)=(λ^Aπ^A’)(μ _A π _A’ )=0=η(v,v)=(λ^Aπ^A’)(λ _A π _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 □-surfaces, and they form a three-complex-dimensional family. Twistor space is the space of all □-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 □-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 if and only if the corresponding spheres in PT intersect. This is the twistor description of the light-cone structure of Minkowski space-time.