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The paper "Measuring Propagation Speed of Coulomb Fields" by R. de Sangro et al points out that:

...the Lienard-Weichert retarded potential leads to the same formula as the one obtained assuming that the electric field propagates with infinite velocity

Moreover, they conduct an experiment which purportedly measures a "rigid" (aka infinite velocity) $-\nabla \phi $ ("Coulomb Field") and further claim:

The Feynman interpretation of the L.W. formula for uniformly moving charges does not show consistency with our experimental data.

This violates special relativity because information can be transmitted at infinite speed simply by changing the position of a charge and measuring, at a large distance, the associated changes in $-\nabla \phi $.

Where did they go wrong?

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    $\begingroup$ I'm no expert, but I just read the paper and they definitely didn't provide evidence to support the second statement you quoted in the question. They collected some (very imprecise because of their lack of knowledge about the electron beam's shape) data which was consistent with all of the theories in question. It was a non-result, essentially. $\endgroup$ – Duncan Harris May 1 '16 at 23:31
  • $\begingroup$ Also note how their sensor results are mostly oscillations at the natural frequency of their sensors. They got only a little bit of information about the shape of the electric field "wake" of the beam, and what little they got was made useless by their lack of knowledge of the shape of the beam itself. $\endgroup$ – Duncan Harris May 1 '16 at 23:36
  • $\begingroup$ THe most important bit of the paper: $\endgroup$ – Duncan Harris May 1 '16 at 23:37
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    $\begingroup$ "One issue common to all our measurements stands out clearly: in the same experimental conditions (sensor position, trigger timing, cable lengths, DAQ settings) the two distributions are different. We attribute this difference to less than perfect reliability in the beam delivery conditions (launch angles, total beam length, charge distribution in the beam pulse length, stray magnetic fields, etc.), over which we had little control." $\endgroup$ – Duncan Harris May 1 '16 at 23:38
  • $\begingroup$ Why even publish this failure? $\endgroup$ – CuriousOne May 2 '16 at 0:20
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There exists a different interpretation of the measurements in the paper:

I copy from the concluding paragraph:

Our conclusion is that most probably the signal registered in the Frascati experiment 1 originates from the radiation due to the acceleration of the beam, and does not belong to the Coulomb disk of the charge. As for the latter, it should be sought for hundreds of meters ahead, already outside of the laboratory, after the beam itself is absorbed by its concrete wall, unless, certainly, it is screened by other possible objects

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The experiment is correct but it only proves that field around electrons is static. The same result would be when replacing the electron beam with collimated microwave beam... Electric field in the direction perpendicular to the wave's propagation is static...

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