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 with distance R, within the sharply focused subbeams that constitute the overall radiation beam from such a source. In addition to supporting the earlier finding that the azimuthal and polar widths of these subbeams become narrower (as and , respectively) with distance from the source, this result 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 (as is commonly assumed in textbooks and the published literature). While the boundary contribution to the retarded solution for the potential can always be rendered equal to zero by means of a gauge transformation that preserves the Lorenz condition, the boundary contribution to the retarded solution of the wave equation for the field may be neglected only if it diminishes with distance faster than the contribution of the source density. In the case of a rotating superluminal source, however, the boundary term in the retarded solution for the field is by a factor of the order of larger than the source term of this solution, in the limit where the boundary tends to infinity. This result explains why an argument based on the solution of the wave equation governing the field in which the boundary term is neglected [such as that presented by Hannay, J. Opt. Soc. A 23, 1530 (2006)] misses the nonspherical decay of the field that is generated by a rotating superluminal source. The only way one can calculate the free-space radiation field of an accelerated superluminal source is via the retarded solution for the potential. Our findings have implications also for the observations of the pulsar emission: The more distant a pulsar, the narrower and brighter its giant pulses should be.
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