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It is said that if, say, the electric charge is not a Lorentz invariant, neutral atoms are no longer neutral, which is not experimentally valid. I want to know to what degree of precision atoms are measured to be electrically neutral and what would happen if, say, we assume that an oxygen atom has a superfluous charge of $10^{-30}$ C or $10^{-50}$ C.

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    $\begingroup$ Why the downvote? This is a great question on fundamental physics. $\endgroup$
    – John Doty
    Commented Jul 26, 2023 at 15:59
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    $\begingroup$ Wikipedia has a table here. Varies if we're talking about differences for electron-vs.-{ proton, positron, or anti-proton }. $\endgroup$
    – Nat
    Commented Jul 27, 2023 at 0:28
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    $\begingroup$ if, say, we assume that an oxygen atom has a superfluous charge of 10^-30C or 10^−50C Without looking up the relevant constants, I'd guess that might lead to electrical repulsion between the earth and the moon stronger than gravity (both have large amounts of oxygen). So solar systems and planets would never have formed, assuming a similar net charge difference scaling with atomic number or atomic mass. (Both gravity and electrostatic forces vary with 1/r^2; gravity wins over large scales only because most matter is neutral but there are no negative masses). $\endgroup$ Commented Jul 27, 2023 at 2:15
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    $\begingroup$ So, hypothetically, if electrons and protons had slightly-different charge-magnitudes, you'd expect for astronomical-bodies to simply not form at all, rather than with a slightly-off-from-parity ratio of electrons/protons? $\endgroup$
    – Nat
    Commented Jul 27, 2023 at 16:42
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    $\begingroup$ Does this answer your question? Why are atoms electrically neutral? Also: Why do atomic charges balance? $\endgroup$
    – Roger V.
    Commented Jul 28, 2023 at 7:18

2 Answers 2

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See

Bressi, G., et al. "Testing the neutrality of matter by acoustic means in a spherical resonator," Physical Review A 83, (2011): 052101 (DOI: http://dx.doi.org/10.1103/PhysRevA.83.052101)

as one of the more recent experiments on this topic. The hypothesis has been tested since the 1920s on different materials using $4$ basic methods, and results using one specific methods are reported above. A difference between the proton and electron charges would produce sound in a SF$_6$ gas trapped in a spherical resonator.

Apparently water has not been tested explicitly but other systems, including high-$Z$ systems where relativistic effects could be important, have been investigated. In no case is there evidence that “neutral” matter is not in fact neutral: recent measurements have fractional uncertainties in the range of one part in $10^{-21}$, so roughly $10^{-40}$ C.

(There is also cosmological evidence but I cannot find the reference to this right now.)

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    $\begingroup$ And what about the second part of the question? What if the charge was non zero but really really small? $\endgroup$ Commented Jul 27, 2023 at 17:42
  • $\begingroup$ @TomášZato: It becomes cosmologically significant first; perhaps best asked on astronomy. $\endgroup$
    – Joshua
    Commented Jul 27, 2023 at 20:54
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    $\begingroup$ Are these experiments saying that every single atom is neutral or only that a collection of atoms is neutral on average? A priori individual atoms could flucture around neutrality but this cancels out of average so there would be no effect on a cosmological level. $\endgroup$
    – quarague
    Commented Jul 28, 2023 at 6:35
  • $\begingroup$ @quarague this is for the bulk but it’s difficult to conceive how different atoms subject to the same conditions would have sometimes a surplus sometimes a deficit of charge, which is a scalar (unlike velocities which can average to $0$ because they they are vectors). $\endgroup$ Commented Jul 28, 2023 at 13:12
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The best bounds I am aware of come from cosmology. (Unfortunately, I know from experience that this is not something that the Particle Data Group tracks very assiduously.) If there were charge imbalances in the early universe (before recombination, when it was an opaque plasma), that would leave a strong impact on the cosmic microwave background. This is discussed in this 2005 paper [C. Caprini, P. G. Ferreira, JCAP 02, 006 (2005)] (arXiv version).

The bounds turn out to somewhat model dependent, but the general scale quoted in the paper for the electron-proton charge difference is $|q_{e-p}|\lesssim 10^{-26}e$, or about $10^{-45}$ C. My own interest in this comes from the related problem of bounds on the photon charge, which are even more model dependent—as discussed, for example, here and here.

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