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Twistor spaces

Part of the Fundamental Theories of Physics book series (FTPH, volume 69)

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 pS, 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 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.

Keywords

Twistor Space Null Geodesic Spinor Field Complex Vector Space Stein Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Kluwer Academic Publishers 2002

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