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According to tparker at Why charge is Lorentz invariant but relativistic mass is not?

So there are two different ways to generalize the mathematical form of Coulomb's law to make it relativistic. If you choose to do so in a way such that the charge is Lorentz invariant, you naturally get classical elecromagnetism. If you choose to do so in way such that the "charge" acquires a factor of γ under Lorentz boosts, you naturally get general relativity. Remarkably, both choices come up in nature.

If the charge that scaled by a factor of γ would acquire a "relativistic mass" in proportion to γ^2, wouldn't its motion be essentially indistinguishable from what we see in experiments, as far as its acceleration due to external electric and magnetic fields is concerned, especially in particle accelerators, where we assume an invariant charge and a "relativistic mass" that scales with γ? If so, then have there been any attempts to measure this theoretical non-Lorentz invariant charge by measuring the fields of the charge having relative motion to a field sensor? Also, where in physics would this be theoretically convenient? Is it relevant to the Ehrenfest paradox, for example?

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Without getting too far into details, these ideas get overthought and skewed depending on the media source or paper that you're reading. Consider yourself as an observer in an interial reference frame $A$, which moves at some speed $v\neq0$ relative to a different inertial reference frame $B$. Now make the further consideration that $v$ is not negligible compared to the speed of light $c$: $A$ and $B$ are in relative motion at relativistic speeds. Now consider an observer in $B$ shooting charged electrons through an electric field. We will assume that both you and the observer in $B$ have a way of detecting the motion of these electrons. You will both see the motion of the electrons as different, and you certainly could think of the electrons as having a smaller or larger charge. However, the same effect comes from a disagreement in the electric field. If we assume that the charge on an electron is a constant, then the difference in force between the two frames can be attributed to a disagreement in the strength of the electric field. This is beyond my reach, but you'd need to devise an experiment where you can measure the charge of an electron without the use of electric fields or magnetic fields. If you cannot, then it is always the case that we can attribute the disagreement between observers in $A$ and $B$ to disagreements in the electric and magnetic field strengths.

In the book Spacetime Physics by Edwin F. Taylor and John Archibald Wheeler, they state that the charge on an electron is a constant, because if it was not then you could measure the "absolute speed" of your reference frame just by measuring the charge on an electron: as your "absolute speed" increases, the charge you measure would change. This violates laws of relativity: there is no "absolute speed", just a relative one. In the same book the authors make the same argument for the mass of an object, and thus we do not think of the mass as changing. Relativistic mass "exists", but it is defined by an observer not in the reference frame of an object. In that sense it is worthless, because the observer will not handle the object ever, and the value of the relativistic mass means very little to anyone that is not in the same reference frame as our inertial observer.

Personally I much prefer to think of mass and charge not changing (which to my understanding is the prefered view), I think it makes things much less ambiguous. Some sources are not very credible, and like to elaborate improperly on ideas in relativity because it is fun and interesting. However, often these elaborations go too far and statements are made that conflict with the true principles. I think this is one of those cases where people really want to attribute a mass increase because it's cool to think of mass changing just because speed does, while in reality we only really care about the proper mass.

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We know that electric charge is invariant because electrons in atoms move at relativistic speeds but protons do not, yet the electrical neutrality of atoms and molecules has been verified to about the level of one part in $10^{20}$ to $10^{21}$. Similar experiments have been carried out for the neutron, to similar levels of precision. See http://arxiv.org/abs/hep-ph/9209259 . These results don't depend on any assumption about the behavior of mass.

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