On the use of projection operators in electrodynamics

Abstract

In classical electrodynamics all the measurable quantities can be derived from the gauge invariant Faraday tensor $F_{\alpha \beta }$. Nevertheless, it is often advantageous to work with gauge dependent variables. In [4],[2] and [8], and in the present note too, the transformation of the vector potential in Lorenz gauge to that in Coulomb gauge is considered. This transformation can be done by applying a projection operator that extracts the transverse part of spatial vectors. In many circumstances the proper projection operator is replaced by a simplified transverse one. It is widely held that such a replacement does not affect the result in the radiation zone. In this paper the action of the proper and simplified transverse projections will be compared by making use of specific examples of a moving point charge. It will be demonstrated that whenever the interminable spatial motion of the source is unbounded with respect to the reference frame of the observer the replacement of the proper projection operator by the simplified transverse one yields, even in the radiation zone, an erroneous result with error which is of the same order as the proper Coulomb gauge vector potential itself.

arXiv comment15 pages, no figures, matched to the published version

References in corpus (4)

• From Lorenz to Coulomb and other explicit gauge transformations
• Sources, Potentials and Fields in Lorenz and Coulomb Gauge: Cancellation of Instantaneous Interactions for Moving Point Charges
• Gravitational radiation and isotropic change of the spatial geometry
• Resolving a gravitational wave memory paradox

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• An introduction to gravitational waves through electrodynamics: a quadrupole comparison