Constant charge in special relativity We had special relativity introduced in our course and used it to derive some facts. At one place we used the fact that if in frame of reference $U$ the electron has charge $e$, then it also has charge $e$ in any frame of reference.
The justification of this fact was done more by waving the hands, though: if the charge did transform from a frame of reference to another, it would be a disaster, because there'd be no state of equilibrium
Can the fact that the electric charge doesn't depend on the frame of reference be justified in a little more rigorous manner?
 A: You can determine the charge of an electron from a static measurement in one frame.
Another frame could determine the charge of an electron from a static measurement in their frame.
And they might agree or disagree. We postulate they agree, but we had three options:


*

*We could postulate that whether or not something is an electron depends on your frame (this would not be 100% totally out of line, for instance in quantum field theory in curved spacetime whether something is a vacuum does depend on whether two frames accelerate relative towards each other, and it predicts that a Geiger counter that clicks in one frame might not click if it accelerates and vice versa).

*We could postulate that the charge of an electron is not a fundamental universal constant. This basically makes charge not a property of the electron but instead a property of how things interact. Again, in QED this might seem less weird when you consider running constants and the differences between bare and dressed electrons.

*We could postulate that the charge and identity are the same in two frames. Effectively this means that instead of charge you can just count the number of positrons and such and then subtract the number of electrons and such and then you basically have the charge and really just multiply by $e$ to change unit systems from a natural system where the fundamental charge is one to another unit system where the fundamental charge isn't one.
Whether charge is conserved is a separate issue (that depends on a continuity equation, which comes from Maxwell). This is about the constant $e$ the charge per electron/positron. And the fact that we call it a constant and the fact that we don't expect something to stop being an electron in a different frame means we expect that third option.
The constant $e$ is just a unit conversion telling you the charge of an electron/positron relative to the unit of charge in your unit system. And like most systems of units it doesn't depend on your frame.

To be clear since people seem confused. We postulate that whether you have some number of electrons or positrons or muons and such is Lorentz invariant. This postulate is separate from Maxwell. Then we postulate that there is Lorentz scalar $q_i$ for each species type (one that depends on your system of units). Then we postulate that the charge four current $J$ is related to the number currents $n_e,$ $n_\mu,$ $n_\tau$ etcetera through the charges via $J=q_en_e+q_\mu n_\mu+\dots+ q_\tau n_\tau.$
And none of those subscripts are related to four vectors you could have $J^a=\sum_{\text{species}_i}q_in^a_i.$


But I still don't understand why a moving electron couldn't have a different charge, thus react differently with our "static" electrons in our frame of reference.

Moving charges do interact differently than stationary charges. But the whole point of charge is to say that a moving or stationary electron exchanges momentum exactly negatively to the way an identically moving positron does. And we use the word charge to express this fact and we express it by saying they have exactly opposite charge.
And then we express the fact that a moving or stationary proton exchanges momentum exactly the way an identically moving positron does. And we use the word charge to express this fact and we express it by saying they have exactly identical charge.
And so on. For quarks you can say a moving or stationary up quark exchanges momentum exactly 1/3 the way an identically moving positron does. And we use the word charge to express this fact and we express it by saying the up quark has exactly 1/3 the charge.
It's these differences between particles with different charge that is physically meaningful and needs to be expressed. Maxwell explains how the motion of the charges affects what it does to other charges and what it feels from other charges. But it does so in a way where you can express having the opposite effect or having 1/3 the effect and we need the concept of charge to express the difference between different particle species.
You want to say things like two electrons both at rest a certain distance $d$ away feel forces away from each other exactly as strong as an electron and a positron both at rest a the same distance $d$ away feel forces towards each other exactly as strong as the two electrons did. And by using a signed charge you can express this like $\vec F_{ab}=kq_aq_b(\vec a-\vec b)/|\vec a-\vec b|^3$ and then just say that positrons and electrons have opposite charges.
You are trying to describe the differences between species when you talk about charge. So if you want species to be Lorentz invariant and you want to talk about the different properties of two species then you need Lorentz invariant properties to talk about the differences between the species.
