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. . .terms
One could also (as I often prefer to do for conceptual purposes) put the
time derivative terms in with the curl terms leaving the true source
on the RHSs of these equations completely absent. The composite
expressions involving the mixed curls and the partial time derivatives
along with the other Maxwell equations (involving the divergences of E &
B) are themselves just the various components of the exterior derivative
(generalized curl) and 4-divergence of the EM field strength tensor when
they are written in the relativistically covariant notation of 4-tensors
and forms on Minkowski spacetime (where time has been demoted from a
global parameter to another coordinate direction in spacetime). As such
these, now homogeneous, equations do *not need* (or have in regions free
of real charges and real currents) any source terms. The propagating
wave-like motions do not require such source terms since the propagating
nature of the solutions of the equations is supplied by the (hyperbolic)
effects of the indefinite signature of the metric for spacetime.
David Bowman
dbowman@georgetowncollege.edu