Abstract
The main purposes of this paper are (i) to illustrate explicitly by a number
of examples the gauge functions chi(x, t) whose spatial and temporal
derivatives transform one set of electromagnetic potentials into another
equivalent set; and (ii) to show that, whatever propagation or non-propagation
characteristics are exhibited by the potentials in a particular gauge, the
electric and magnetic fields are always the same and display the experimentally
verified properties of causality and propagation at the speed of light. The
example of the transformation from the Lorenz gauge (retarded solutions for
both scalar and vector potential) to the Coulomb gauge (instantaneous,
action-at-a-distance, scalar potential) is treated in detail. A transparent
expression is obtained for the vector potential in the Coulomb gauge, with a
finite nonlocality in time replacing the expected spatial nonlocality of the
transverse current. A class of gauges (v-gauge) is described in which the
scalar potential propagates at an arbitrary speed v relative to the speed of
light. The Lorenz and Coulomb gauges are special cases of the v-gauge. The last
examples of gauges and explicit gauge transformation functions are the
Hamiltonian or temporal gauge, the nonrelativistic Poincare or multipolar
gauge, and the relativistic Fock-Schwinger gauge.
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