I have a question about special relativity and Lorentz force which bugs me:
Consider an infinitely long, initially uncharged, wire (lying along the x-axis) resting in a inertial frame S. Now let's assume a current is flowing through the wire into the positive x-direction, this means that the electrons in the wire have a velocity in negative x-direction. The current leads to a B-field around the wire.
Now imagine a single electron in a certain distance to the wire, travelling with the same velocity into the same direction as the electrons in the conductor. Since this is a charged particle moving in a magnetic field of the conductor, the electron is subject to the Lorentz force and gets attracted towards the conductor. This is the description in the inertial system S in which the conductor itself is at rest.
Now I want to describe the same situtation from a coordinate system S' which is travelling with the same velocity as the electrons. In this intertial frame the velocity of the electrons is zero, however, the (positively charged) bodies of the atoms forming the conductor are now moving. Since the electron outside the conductor is now at rest, there is no Lorentz force acting on the particle anymore, however, since the atoms of the conductor are now moving, they are subject to the Lorentz contraction. Since their charge is the same in both intertial frames, but their length is contracted, their charge density is increased. At the same time the charge densitiy of the electrons is decreased because they are now at rest and therefore not subject to the Lorentz contraction anymore (compared to the situation in S). This means that the conductor is now exhibiting a positive charge density and the particle is now attracted by the Coulomb force instead of the Lorentz force. So both descriptions lead to the same result (particle getting attracted by the wire) which is actually nice. This is also the explanation which I found around the web and I also got from physics lectures years ago.
The problem I have: If I look at the problem in S', the speed of the charged particles of the wire (atomic bodies of the wire) is leading to a Lorentz contraction and an increase in charge density. However, if I look at the same problem in the intertial frame S, the wire is electrically neutral. However, in this intertial frame the electrons in the wire are moving. Therefore they should also undergo Lorentz contraction, leading to a charged wire (and therefore an electric field and a coulomb force of the wire). But for some reason I don't understand this doesn't seem to happen. Why? What am I missing?