Coulombic contribution to angular momentum flux in general relativity

Angular momentum flux in electromagnetism cannot be expressed entirely in terms of the field's radiative degrees of freedom. Its expression also involves Coulombic parts of the field, in the form of a charge aspect $q(\theta,\phi)$. Guided by the strong analogy between radiative processes in electromagnetism and gravitation, one might expect that the angular momentum flux in general relativity also involves 'Coulombic pieces' that may have been missed by specializing the flux to periodic sources at rest with respect to the frame in which the flux is evaluated. To test this, we bring together the Landau-Lifshitz formalism, which provides specific definitions for angular momentum and its associated flux, and the Bondi formalism, which provides a systematic expansion of the metric of any asymptotically flat spacetime in inverse powers of the distance away from the matter distribution. We obtain a new expression for the flux of angular momentum, which is not restricted to sources at rest nor to periodic sources. This new expression is equivalent to the standard formula when these restrictions are put in place. Contrary to expectations based on the analogy between electromagnetism and gravitation, no Coulombic information appears in the flux of angular momentum in general relativity.