However, I think the fact that the Faraday cage is not symmetric when it comes to static electric fields is very interesting. Ask yourself this: how does the cage know what "inside" versus "outside" means? You know the old joke where a mathematician has to enclose the greatest area with a given fence and so she takes the fence and wraps it around herself and declares "I am outside the fence." Well, why can't I declare myself to be inside the Faraday cage and suddenly get the benefits of its shielding?
You might say that the difference is that the outside has the "boundary" of the universe and that makes it distinct from the inside. [1] But we don't yet know that the universe isn't finite and wrapped up on itself spatially, so if that were the difference, then isn't a Faraday cage experimental evidence in favor of the universe not being finite? I think therein lies the answer: if we wanted to actually use this as an experiment, we would have to place a charge outside of the Faraday cage and wait for sufficiently long time that the electric field is effectively static. However, for our experiment to work, we would have to wait so long that the electric field becomes static on the scale of the entire universe. And so no, the Faraday cage is not an experiment we can practically perform to measure the shape of the universe and the difference is the obvious one: the exterior is the larger one and is simply just "big enough" that we can't ever wait for the EM fields to settle down in the exterior like we can on the interior.
[1] Mathematically, this come in as the fact that the Poincare lemma https://en.wikipedia.org/wiki/Closed_and_exact_differential_... requires a "contractible" domain. The exterior of the cage is not contractible, so we are not guaranteed to be able to use the integral forms of Maxwell's equations over the exterior of the cage. However, the interior is contractible.
