The fundamental role of the retarded potential in the electrodynamics of superluminal sources

Abstract

We calculate the gradient of the radiation field generated by a polarization current with a superluminally rotating distribution pattern and show that the absolute value of this gradient increases as R^(7/2) with distance R within the sharply focused subbeams constituting the overall radiation beam. This result not only supports the earlier finding that the azimuthal and polar widths of these subbeams narrow with distance (as R^(-3) and R^(-1), respectively), but also implies that the boundary contribution to the solution of the wave equation governing the radiation field does not always vanish in the limit where the boundary tends to infinity. There is a fundamental difference between the classical expressions for the retarded potential and field: while the boundary contribution for the potential can always be made zero via a gauge transformation preserving the Lorenz condition, that for the field may be neglected only if it diminishes with distance faster than the contribution of the source density in the far zone. In the case of a rotating superluminal source, however, the boundary term in the retarded solution for the field is by a factor of order R^(1/2) larger than the source term of this solution in the limit, which explains why an argument based on the solution of the wave equation governing the field that neglects the boundary term (such as that presented by J. H. Hannay) misses the nonspherical decay of the field. Given that the distribution of the radiation field of an accelerated superluminal source in the far zone is not known a priori, the only way to calculate the free-space radiation field of such sources is via the retarded solution for the potential. Finally, we apply these findings to pulsar observational data: the more distant a pulsar, the narrower and brighter its giant pulses should be