FrankJ777 said:
So why it that when the charged cat is stationary, and there is current of electrons moving relative to the stationary cat, the electron stream does not contract causing the negative charge density to increase and create a negative electric field?
That is a very good question and probably the hardest to understand about this scenario. I'm going to go outside my comfort zone and attempt to explain this. Hopefully I don't mess it up! (I'm sure someone will slap me if I do

) The following is based off of
Bell's Spaceship Paradox:
The key lies in the fact that the electrons are not rigid objects.
Consider a long line of spaceships attached to a frictionless rail that let's them accelerate to any velocity they want. Let's also say that we've placed a light bulb at the starting location of every ship and that there is one ship every 100 meters and that every spaceship is exactly 10 meters long. Now, let's say that each ship accelerates at the same time, at the same rate, until they are all at 0.9c. Crucially, while each spaceship has length contracted to 4.36 meters, the distance between each adjacent ship is still 100 meters! (As long as you measure the distance between each pair of adjacent ships at the same point as you did prior to acceleration.)
Now, let's say that your friend Tom is passing by at 0.9c. What will he see? He will find that each spaceship is 10 meters in length, but the distance between each adjacent ships is now over 229 meters! (The exact reason for this is both conceptually and mathematically complicated and I don't have a chance of explaining it in detail) In addition, he will find that the distance between adjacent light bulbs on the rail is 4.36 meters since he is moving relative to the rail.
Now, let's replace the ships and rail with electrons moving in a copper wire, and replace Tom with a moving positive charge.
From our lab frame, which is at rest relative to our circuit, neither the wire nor the distance between electrons is length contracted. The electrons are like the ships. Even though they are moving and their fields may be length contracted, the distance between them has not changed when viewed from the lab's frame. This is a requirement of our thought experiment since our circuit is not electrically charged. If the electrons were closer together then we would see an electric field and the circuit would indeed be electrically charged.
Now, when we transform our frame of reference to the moving positive charge, we see the wire contract but the distance between electrons increases, meaning that the positive charge density increases and the negative charge density decreases, which add together to push the positive charge away from the wire. From the lab's frame of reference this occurs because of the magnetic field generated by the current, but in the positive charge's frame of reference the wire is now electrically charged.
Also remember that since we have two distinct frames of reference, both viewing two other sets of moving/non-moving objects (negative and positive charges in the wire), it makes this example much more complicated than your basic special relativity problem. I re-wrote this post about 4 times because I myself kept getting it wrong. There are several very confusing concepts here. For one, how can the distance between electrons increase while the wire length contracts? I'm afraid I can't answer that, as it's well outside of my knowledge in this area.
As always, someone please correct me if I'm wrong.