Lorentz force and Lorentz contraction on a current carrying wire

Question

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?

Answer 1

However, in this intertial frame the electrons in the wire are moving. Therefore they should also undergo Lorentz contraction,

They do but you've already stipulated that the wire, moving electrons and all, has zero charge density in the S coordinate system (otherwise, the external electron in S would have both an electric and magnetic force acting on it).

Remember, the atoms in the lattice are not free to adjust their spacing but the mobile electrons are.

That is, it's perfectly fine to stipulate both that (1) mobile electrons are flowing in the wire in the coordinate system in which the wire is at rest and (2) the wire has zero charge density in this coordinate system.

Answer 2

Imagine an infinitely long wire. Assume that there is a current in the wire caused by electrons moving with uniform speed $v$.

Now, place a negative test charge at distance $r$ from the wire moving parallel to the flow of electrons with the same speed $v$. We know that there is a force on the test charge in the direction of the wire because the test charge is passing through the magnetic field induced by the current in the wire.

Move to the reference frame of the electrons. Now, there cannot be a force due to magnetism because the electrons are all at rest. But there still must be a force on the test charge! Where does that come from?

Well, as you observed, in the rest frame of the electrons in the wire, there is a positive charge density due to length contraction of the separation of nuclei in the wire. This excess of positive charge results in a net force on the test charge in the direction of the wire. So, the build of charge density does happen.

I'm a bit less familiar with this route, but I think you can also just transform the electromagnetic tensor $F_{\mu\nu}$. Let's pick some coordinates. Positive $z$ is the direction of the electron flow. The test charge's displacement from the wire is along positive $y$.

Then the electromagnetic field tensor is: $$F_{\mu\nu}=\left| \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -B \\ 0 & 0 & B & 0 \end{matrix}\right|$$

In the moving frame, $$F_{\mu^{\prime}\nu^{\prime}}={\Lambda^{\mu^{\prime}}}_{\mu}{\Lambda^{\nu^{\prime}}}_{\nu}F_{\mu\nu} = \left| \begin{matrix} 0 & 0 & \beta\gamma B & 0 \\ 0 & 0 & 0 & 0 \\ -\beta\gamma B & 0 & 0 & -\gamma B \\ 0 & 0 & \gamma B & 0 \end{matrix}\right| $$

We see that there is now a component of the electric field in the positive-$y$ direction, implying a force on a negative test charge in the negative $y$ direction.

Now, this only tells us that there is a component of the electric field in towards the wire, but it doesn't say where comes from. But in your example, it's reasonable to say that it comes from charge density that wasn't observed in the rest frame.