A generalised Kerr-Robinson theorem

Abstract

The Kerr and Robinson theorems in four-dimensional spacetime together provide the general null solution of Maxwell's equations. Robinson's theorem reduces the problem to that of obtaining certain null foliations. The Kerr theorem shows how to represent such foliations in terms of analytic varieties in complex projective 3-space. The authors generalise these results to spinor fields of higher valence in spacetimes of arbitrary even dimension. They first review the theory of spinors and twistors for these higher dimensions. They define the appropriate generalisations of Maxwell's equations, and null solutions thereof. It is then proved that the Kerr and Robinson theorems generalise to all even dimensions. The authors discuss various applications, examples and further generalisations. The generalised Robinson theorem can be seen to extend to curved spaces which admit such null foliations. In the case of Euclidean reality conditions, the generalised Kerr theorem determines all complex structures compatible with the flat metric in terms of freely specified complex analytic varieties in twistor space. Interpretations of the generalised Kerr theorem are also provided for Lorentzian and ultrahyperbolic signatures.