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The E field due to charge motion has zero divergence, just as the current density vector.

There is no E field that is due to the motion of the charge. None whatsoever. I don't know what put that notion into your head.

You said the MM apparatus has measured the Lorentz contraction.

Yes. You measure the speed of light and time dilation in each frame with respect to the other. You predict from that a certain shifting of the interference fringes. You observe no such shift, so you can calculate how much the device must have shrunk to compensate.

I'm just trying to figure out how and contraction from what?

The contraction is due to the perspective from different inertial frames. A person looking at a yardstick from the side will see it as a long object, while a person looking at it from the end will see it as a short object. The Lorentz transformation is analogous to this effect of rotation, except that it describes a transformation between space and time, rather than the rotation of one space axis into another.

One thing to consider about the moving electron is that the E field lines do not bunch at the point perpindicular to its motion and continue off into space in straight lines. If this were true, then the field from an electron would "move out" instantaneously. Rather the field lines resemble fluid streamlines, they curve opposite the velocity vector. The curvature is proportional to the speed.
Now, if they curve opposite the velocity, what would be the resultant E vector? Its divergence.
If the moving electron is considered a current of density J and it is moving at a high velocity so almost all the E field lines are perpindicular to the motion, then the work done on this particle: Integral E dot J is zero. The work done decreases as the velocity gets increases. Reliable model isn't it? Perhaps the field lines are confused with the dimensional Lorentz contraction of the electron.

If an electric field causes charge to move (the charge moves to establish an electric field of its own that counters the impressed field), then why is this field of the charge absent when the particle moves due to forces other then electric fields? The field exists only in the vicinity of the charge.

I'm trying to find a reference for you on this that is not proprietary. There may be something in electrohydrodynamics, frictional electrification or in the atmospheric sciences.

If you apply the Hiesenberg Uncertainty Principle to the MM experiment you get a surprise. Look at each fringe pattern as the interaction of two discrete photons. Assume the observation length is one wavelength so the observation time delta t is 1/f. f = frequency of light. We know E = hf ----> delta E = h delta f.
Now delta t times delta E = 1/f x h x delta f.
The principle states:
delta t x delta E ~ h, by substitution:
(delta f)/f ~1; This says the uncertainty of frequency is the frequency itself if one observes single fringe shifts. Since multiple fringe shifts are the products of many different photons (each having their own uncertainty) doesn't Hiesenberg UP say the null result, or any result for that matter is invalid?
To get a proper result the interference pattern of two photons must be observed over very many fringes. Am I off base here? How do I properly apply the HUP to MM?

The contraction is due to the perspective from different inertial frames. A person looking at a yardstick from the side will see it as a long object, while a person looking at it from the end will see it as a short object. The Lorentz transformation is analogous to this effect of rotation, except that it describes a transformation between space and time, rather than the rotation of one space axis into another.

Very nicely said. May I borrow this? :-))