Perhaps I haven't thought through this as long or as deeply as I should, but at the moment it's got me stumped. How can the absolute charge on a particle be measured remotely? This derives from pondering how it might be possible to prove experimentally the invariance of charge on a particle vs its location relative to a black hole. I know that charge is considered to be invariant in special relativity and general relativity, but I'm wondering how the invariance might be tested empirically.
Gauss's law can be used to determine the charge within a volume, but it requires measurements of E at every point on a surface that encloses the volume. That's not a very practical remote measurement. If we know that a first, very massive, particle has an invariant charge of $Q$, then presumably we can remotely observe a second, very low-mass particle as it passes near the first particle, and deduce the charge on the second particle from its trajectory.
However, if the charge on the second particle changes due to its proximity to a black hole, the charge of the first particle will change similarly. Moreover, the gravitational mass (and the inertia) of each particle presumably will be a function of its height in the gravitational well of the black hole.
We know that the emission spectrum of, e.g., a hydrogen atom is red-shifted (relative to a distant observer) as the atom goes lower in a gravitational well. Of course that red shift can be calculated using general relativity. Could that spectrum be used to prove that the value of e is not dependent on gravitational potential? Maybe, but I suspect that unless we make assumptions about how gravitational mass is affected by height in a gravitational potential, we cannot measure charge remotely.
Millikan directly measured the charge on an oil drop in his famous experiment, but really included measurements of two quantities: the voltage required to suspend an oil drop, and the mass of the oil drop. In the case of extreme gravitational potential differences, though, we cannot assume that a particle's mass is invariant. Can we even remotely measure the charge-to-mass ratio of the particle without making untested assumptions?
That a free-falling observer will always observe the same laws of physics as any other free-falling observer is an acceptable assumption. However, I don't think it would work to drop a lab apparatus to the charged particle and measure its charge as the apparatus flies past, then have the apparatus report back to a remote observer. The charged particle, suspended at a fixed height above the black hole, will be accelerated relative to the apparatus. Moreover, I suspect the experiment could at best only prove that the apparatus and the particle both change in such a way that the results appear invariant from the perspective of the apparatus.
Maybe if I'd had a better course in general relativity this would seem trivial- but no. So to repeat the question: How can the absolute charge on a particle be measured remotely?
