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(I know many of you guys posting in

__Tensor Analysis & Differential Geometry__are up to this question. I've seen your posts!)

Put again, are Maxwell’s equations not as robust as is commonly believed, that they should allow for unit velocity (v=c) charged waves?

This would be surprising odd to find true, but my math, and some inference, so far seems to support it.

Fundamentally this is a applied math problem best at home in differential topology.

So I’ve left this problem in differential forms on a pseudo-Riemann Manifold of Lorentz Metric where I found it, and where it seems to be notationally simplest.

Some physics reminders:

d*F = *J, where F = dA

d^2F = 0, completes the set of 4 maxwell equations

F is the 2-form of the electric and magnetic fields.

-J is the 4-current 1-form.

A is the covariant form of the 4-vector potential.

Applying the Laplace-De Rham operator, (d +/- *d*)^2 on F, you obtain the wave equation:

d*d*F = dJ ,

where dJ=0 for the homogeneous solutions of interest,

and where dJ=0 must be over some finite region of space-time rather than a single point, I think. Stop me, if I’m wrong.

Following the same program, the 4-current wave equation I’ve obtained is:

*d*dJ = (*d)^4 A ,

where (*d)^4 A = 0 over a region

I don’t see any other velocities associated with a solution, should it exist, other than the unit velocity, c. But a formal argument would be far better than speculation.

To get propagating waves of charge, this all boils down to asking if the amplitude of J_{0} may be a other than zero over a region, and under the given constraints, I think—and if I haven’t made any errors of course.

But I don’t know how to solve this!

Thanks for any guidance,

-phrak