Dynamics of extended bodies in general relativity - II. Moments of the charge-current vector

Abstract The problem is considered of defining multipole moments for a tensor field given on a curved spacetime, with the aim of applying this to the energy-momentum tensor and charge-current vector of an extended body. Consequently, it is assumed that the support of the tensor field is bounded in spacelike directions. A definition is proposed for ‘a set of multipole moments’ of such a tensor field relative to an arbitrary bitensor propagator. This definition is not fully determinate, but any such set of moments completely determines the original tensor field. By imposing additional conditions on the moments in two different ways, two uniquely determined sets of moments are obtained for a vector field Jα. The first set, the complete moments, always exists and agrees with moments defined less explicitly by Mathisson. If VαJα = 0, as is the case for the charge-current vector, these moments are interrelated by an infinite set of corresponding restrictions. The second set, the reduced moments, exists if and only if VαJα = 0. These avoid such an infinite set of interrelations, there being instead only one such restriction, the constancy of the total charge of the body. The energy-momentum tensor will be treated in a subsequent paper.