Edth-a differential operator on the sphere

Abstract

Introduction. In (9) Newman and Penrose introduced a differential operator which they denoted ð, the phonetic symboledth. This operator acts onspin weighted, orspinandconformally weightedfunctions 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 anangular momentum lowering operatorfor 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 thegood cut equationdescribing Newman'sℋ-spacefor anasymptotically flatspace-time (10). This development is closely related to Penrose's theory oftwistorsand, in particular, toasymptotic twistors(14).