Complex General Relativity pp 43-60 | Cite as

# 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^{ A }π^{ A’ }, where λ^{ A } Ais varying and π^{ A’ } is fixed. Therefore, the induced metric vanishes identically since η(*v,w*)=(λ^{ A }π^{ A’ })(μ_{ A }π_{ A’ })=0=η(*v,v*)=(λ^{ A }π^{ 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 *CP*^{3}. 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.

## Keywords

Twistor Space Null Geodesic Spinor Field Complex Vector Space Stein Manifold## Preview

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