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Imagine a stationary negative charge sitting beside an electrically neutral wire which has a current flowing through it. From the reference frame of the negative charge, the nuclei inside of the wire are at rest and the electrons are moving to the side at some drift velocity. Due to Lorentz contraction, the electrons appear closer together than the nuclei do. Since the electrons appear closer together, there appears to be a net negative charge on the wire from the viewpoint of the stationary negative charge. Thus, the charge is repelled from a wire, despite the fact that there is no net charge on the wire itself.
I have not looked at what was written about the wire and charge in detail (it has been covered before on this site btw) through lack of time, however I would venture to say that the bit about the Lorentz contraction and the electrons being closer together is nonsense. The electron drift velocity is nothing like fast enough to have any noticeable relativistic effect.