Is Gauss's law still valid near the speed of light?
I finally have a moment to write out something.
Intro
First off, it's better if I say a little something about Gauss's Law -- it's not just one law. There are two important versions, one for electricity and one for magnetism, and together they form up half of Maxwell's equations (which have a differential form and an integral form.) These two laws have very similar form and describe the net flux out of any closed surface.
The electricity version says that if you take any arbitrary volume shape and break it up into tiny areas small enough that you can treat them as &quot;uncurved and flat&quot;, and if you then measure the magnitude of the electric flux perpendicular to that area, multiplying the tiny area by this magnitude, and sum up these product terms over the entire surface of the volume, then you will get a sum that is directly proportional to the total charge contained within that surface (no matter how that electric charge is arranged within the volume.) There will be a proportionality constant involved to convert the sum you get into the actual electric charge. But it's just one of those constants like G for the gravitational force acting between two particles of mass. (In this case, \$\epsilon_{_0}\$.)
The magnetism version says that if you take any arbitrary volume shape and break it up into tiny areas small enough that you can treat them as &quot;uncurved and flat&quot;, and if you then measure the magnitude of the magnetic flux perpendicular to that area, multiplying the tiny area by this magnitude, and sum up these product terms over the entire surface of the volume, then you will get a sum that is always zero. This is why people say that magnetic monopoles don't exist. If they did, then the sum wouldn't be zero if a volume enclosed even one monopole and Gauss's magnetism law would be falsified. So far, that hasn't happened and so this lack of contradicting experimental evidence has been taken as a heavy weight against them. For enclosed magnetic dipoles (a simple case), some flux will point outward and some will point inward (opposite signs) and Gauss's magnetism law says the sum of all these product terms must result in zero.
The above is a simple statement. It's more nuanced, though. Suppose the volume encloses no electric charge. Then Gauss's electricity law says that the sum must be zero. But this must be true, even if there are lots of charges outside of the volume. So the laws say something more -- not only do they talk about what's inside an arbitrary volume of your choosing, they also imply you will get the same sum without regard to charges outside of the volume, too.
Answer
I had mentioned in my comments that Gauss's Laws apply in relativistic situations. I'm going to avoid equations here and just refer to some concepts that may still be convincing without the math.
In Gauss's electricity law above, the volume is considered to be finite, with a surface vector field that is integrated. But we can shrink it to an infinitesimal volume, if we like. That process is called divergence and it generates the differential form of Gauss's electricity law. The scalar for each such infinitesimal volume will be either positive (a &quot;source&quot;) or negative (a &quot;sink.&quot;) This process of shrinking to an infinitesimal volume, or using divergence, develops what's called the &quot;local&quot; relationship between charge and field and has meaning at the same location and time, not at different locations and times, which is what avoids relativistic problems.
Put another way, divergence is co-ordinate-free, so it's computed values must be the same in any co-ordinate system. This is why I mentioned that divergence is the key to understanding why Gauss's electricity law works in relativistic situations. The differential form of Gauss's electricity law is inherently co-ordinate-free and that property makes it also relativistically correct.
(It does take a little extra work to then move from this differential form of Gauss's electricity law to the integral form, while continuing to demonstrate that the integral form must also be relativistically correct. That's the part that I didn't want to write out here and, instead, wanted you to find elsewhere. But you start by integrating the divergence with respect to the infinitesimal volumes and kind of &quot;back yourself into&quot; the integral form of Gauss's electricity law using only the rigorous application of mathematical axioms and logic to arrive there. The equations are few, but I'd need to write a paragraph or so for each, so I'd like to avoid it here.)
Hopefully, this is enough said to help you find some material that expands on what I just wrote.