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In most of textbooks I am familiar with (with one exception - Matveev’s course) it is tacitly assumed that electric charge is independent of an inertial frame.

I am looking for a reference to a more detailed discussion of this law.

Qmechanic
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MKO
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  • There is no such reference, nor a detailed discussion. It is simply noted that by assuming that the electric charge are of point particle electrons and protons, then the relativistic length contraction and other stuff will predict correct results. – naturallyInconsistent Feb 01 '24 at 08:39
  • In the theory electric charge is a scalar and that takes care of that. This has to be established with experiments, so what you probably would have to do is to search for precision experiments about relativistic charge conservation if you want to know "how we know this". That's actually a pretty good question and very fundamental for all kinds of things. – FlatterMann Feb 01 '24 at 08:50
  • It's an interesting question though. Rest mass is invariant because it's the modulus of a four-vector. Is there a similar property for charge i.e. is charge the modulus of any four-vector? There is a related question here though I don't think it is a duplicate. – John Rennie Feb 01 '24 at 08:56
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    We also had answers about the theoretical repercussions here: https://physics.stackexchange.com/q/410918/ – FlatterMann Feb 01 '24 at 08:58
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    See Procyons answer here: https://physics.stackexchange.com/q/248688/. It's a semi-classical argument. One would have to check if it's actually correct for the ground state of a quantum mechanical system like an atom since that does not really have "a velocity". It is as "stationary" as things come. – FlatterMann Feb 01 '24 at 09:00
  • The most "direct" experiment I can think of at the moment are the beam position and beam current monitors at LHC. The major error source would probably be the beam loss, which is controlled by other beam loss monitoring techniques. So if you don't believe any indirect argument from atomic physics, we have a (very crude) direct measurement up to 7TeV for protons. – FlatterMann Feb 01 '24 at 09:11
  • See also the discussion in the chat here. – John Rennie Feb 01 '24 at 10:17

1 Answers1

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As a consequence of the local $U(1)$ symmetry of the electromagnetic interaction, the associated conserved charge is defined by $Q=\int_{\mathbb{R}^3} d^3x \, j^{\, 0}(x)$, where $j^{\, 0}$ is the time component of the electromagnetic 4-current density $j^\mu(x)$ (transforming as a 4-vector field) obeying the continuity equation $\partial_ \mu j^\mu(x)=0$ (following from the Noether theorem). As a consequence, the conserved quantity $Q$ is independent of the chosen reference frame. More detailed discussions can be found in text books presenting Maxwell's theory as a relativistic field theory and, of course, in text books on quantum electrodynamics.

Hyperon
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  • I think this is a purely theoretical argument based on the theory which implicitly postulates the charge invariance law. So there might be a circled logic here. I am rather looking for a reference to an experimental evidence of the law. – MKO Feb 01 '24 at 10:26
  • The derivation is a bit more complicated than in this answer. Derivations are given in Jackson and in my EM textbook. It is not an assumption, but has to be derived. – Jerrold Franklin Feb 02 '24 at 17:41