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.
Electromagnetism – To What Degree of Precision Are Atoms Electrically Neutral
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Take a simple analogy: if you take a bar of metal and put it in a potential gradient (e.g. bring a charge near, but not touching, one end) then you will attract electrons to the positive end. So one end of the bar acquires a negative charge and the other acquires a positive charge. If you now cut the bar in half you have two pieces of metal, one with a negative charge and the other with an equal and opposite positive charge - so it's a sort of capacitor.
The same thing happens with your KCl solution. If you polarise it then split the solution you end up with a negatively charged solution with an excess of Cl$^-$ and a positively charged solution with an excess of K$^+$. There's nothing especially mysterious about this, and more than it's mysterious that a lump of metal can have a net charge.
However if you connect the two solutions then, just like a capacitor, a current will flow between the two solutions. One solution will end up with KCl and potassium metal and the other will end up with KCL and chlorine (you won't actually get potassium and chlorine because they'll react with the water and you'll end up generating hydrogen and oxygen instead).
Any textbook will tell you that solutions are electrically neutral because in practice it's very difficult to maintain more thn a very small charge. However there's no reason a solution can't carry a (small) charge just like anything else.
Yes, the atom will be torn apart, and eventually in a Black Hole (BH) get ripped apart radially (to the BH) and get compressed into nothingness perpendicularly.
Even before that happens the gravitational tidal force will rip off the electRons and have the nucleus break up and have most of it converted to neutrons, and then rip those off and get to the quarks, and eventually fall into the singularity. That's why we say that a BH forms when there is too much gravity, nothing can withstand the gravitational effects. Not electron pressure (which holds up white dwarf stars), not nuclear forces (neutron stars), and not strong forces (quark stars, or some parts of the cores of neutron stars).
Yes, the equivalence principle says everything will be accelerated (i.e. pulled) the same way, but only until the force differential between two objects in the atom see different accelerations - that's the gravitational tidal effects, caused by very strong curvatures of the spacetime due to gravity.
Now, that all according to classical GR. As they get closer to the singularity in the BH quantum gravity effect will increase, and we don't know what eventually happens because we still don't have a theory of quantum gravity. But the atomic components all get obliterated as explained above. Quantum gravity effects enter in after that, as the gravitational effects become even stronger, and smaller distance scales get affected. The characteristic distances at which this happens is on the order of the Planck length, about $10^{-33}$ cms, much less than the nuclear or quark scales.
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See
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.)