Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Edth-a differential operator on the sphere

Michael Eastwooda1 and Paul Toda2

a1 Institute for Advanced Study, Princeton

a2 Mathematical Institute, Oxford University

Introduction. In (9) Newman and Penrose introduced a differential operator which they denoted ð, the phonetic symbol edth. This operator acts on spin weighted, or spin and conformally weighted functions on the two-sphere. It turns out to be very useful in the theory of relativity via the isomorphism of the conformal group of the sphere and the proper inhomogeneous Lorentz group (11, 4). In particular, it can be viewed (2) as an angular momentum lowering operator for a suitable representation of SO(3) and can be used to investigate the representations of the Lorentz group (4). More recently, edth has appeared in the good cut equation describing Newman's xs210B-space for an asymptotically flat space-time (10). This development is closely related to Penrose's theory of twistors and, in particular, to asymptotic twistors (14).

(Received December 22 1980)

(Revised February 16 1981)