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Are there propagating charged waves admitted in Maxwell’s equations?

  1. Jan 10, 2008 #1
    Are there propagating charged waves admitted in Maxwell’s equations?

    (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,
  2. jcsd
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