In previous chapters, we have discussed twistors for flat (or conformally flat) space-time. The question arises: can one define twistors for curved space-time? In general, there is no completely satisfactory way of doing so. The reason why difficulties occur can be illustrated by studying the scattering of twistors through an impulsive gravitational wave (Penrose and MacCallum 1973): the complex structure of twistor space gets ‘shifted’, so that there is no unique complex structure. Another way of seeing this involves hypersurface twistors (Penrose 1975, Penrose and Ward 1980, Penrose 1983). Here one associates a ‘curved’ twistor space with each hypersurface in space-time. But the complex structure of this twistor space depends, in general, on the choice of the hypersurface, and shifts as one moves from one hypersurface to another.

An approach which bypasses these problems is to use ‘ambitwistors’, which correspond to the complex null geodesies in a complex space-time. The ambitwistor space does have a natural complex structure: it is a five-dimensional complex manifold (LeBrun 1983). But some of the power of twistor theory is lost. We shall return to this subject in Chapter 10. In the present chapter, we shall deal with anti-self-dual space-times (i.e., those in which the Weyl tensor is anti-self-dual, see §6.2).