Twistors and multipole moments

Abstract

The 2 n multipole moment of a stationary electromagnetic or linearized gravitational field on Minkowski space may be described by a spinor field of rank 2 n on Minkowski space. The rank 2 n and 2 n — 2 spinor fields are connected by first order differential equations. In particular, each field satisfies the twistor equation. The multipoles up to the 2 n level may be coded into a 2 n index symmetric twistor satisfying certain algebraic constraints. It is possible to obtain the multipole spinors as elements of the tensor algebra over the ten-dimensional vector space of spinors α A´B´ satisfying the twistor equation. A sequence of spin 1 fields containing arbitrary parameters is constructed. On performing the relevant charge integrals the independent components are the multipole spinors. It is easy to examine this construction from the twistor point of view. The multipole twistors are obtained as contour integrals of the holomorphic function describing the field. These give rise to two-surface integrals at future null infinity for the multipole spinors. There is an underlying Lie algebra structure which provides a connection with the work of Geroch on multipoles defined at space-like infinity. It is conjectured that the two types of multipole are identical. The multipole twistors for Kerr type linearized gravitational fields are computed. These agree with the usual results. The formalism suggests ways in which it might be possible to extend the multipole definitions to non-stationary fields and curved asymptotically flat backgrounds.