How do we think at the perspective of negatively charged wire in special relativity that causes magnetism?

Question

I am currently try to learn special relativity, and then i crossed on some video about how special relativity can cause magnetism here:

https://youtu.be/1TKSfAkWWN0

At timestamp 01:56 - 02:24 he explains that on positively charged cat and negatively charged wire frame of reference, the positively charged wire would seem the one to be moving and causing it to become looks more dense, causing the wire to be more positive and push the cat away. At timestamp 02:25 - 02:44 he explain in the other perspective which is positively charged wire, that the magnetic force are the cause of the pushed positively charged cat, but isn't that what we want to know in the first place? Also, when we try to think from positively charged wire again, isn't the negatively charged wire and positively charged cat looks more denser, so why doesn't the wire attract the cat?

Answer 1

That veritatasium video isn't particularly good, unfortunately.

You may want to try this youtube video: "The hidden link between electricity and magnetism" (Youtube channel: 'STEM Cell') (I haven't watched all of it but the author seems thorough.)

About the version presented in the Veritasium video: it's not just that the switch is to another perspective, the switch is to another case:
The two cases are:
-the cat is stationary with respect to the physical wire
-the cat is in motion relative to the physical wire.

Each of those two cases can be described in terms of a coordinate system that is stationary with respect to the wire, or in terms of a coordinate system that has a velocity with respect to the physical wire.

Only people who are already quite familiar with relativistic physics will be able to follow the narration in the Veritasium video, which of course defeats the purpose.

Derek Muller is generally very good, but when it comes to relativistic physics: in several different videos I have seen him fumble.

Answer 2

There is no negatively charged wire frame, because the wire never has a negative charge density. In Derek's frame (at rest with respect to the positively charged lattice), the linear charge density is zero because the density of protons and electrons are equal, with opposite charges:

$$\lambda_{\rm Derek} = \lambda_+ + \lambda_- =\lambda(1-1)=0 $$

In Henry's frame (at rest with respect to the electrons), the proton are Lorentz contracted and the electron are Lorentz dilated:

$$\lambda_{\rm Henry} = \gamma\lambda_+ + \frac 1 {\gamma}\lambda_- =(\gamma-\frac 1{\gamma})\lambda $$

(If you don't understand why the electrons are dilated, review Bell's Spaceship paradox, and replace the spaceships with electrons).

A little algebra shows:

$$\lambda_{\rm Henry}=\gamma(1-\frac 1{\gamma^2})\lambda= \gamma\beta^2\lambda$$

showing there is a net positive charge.

It's tempting to say, "Thats because Lorentz contraction", but as with all relativity problem, that is half the story. From Henry's frame, the relativity of simultaneity supplies a factor of $\gamma^2\beta^2$ (relative to Derek), and the Lorentz contraction/dilation supplies a factor of $1/\gamma$...to the extent there is any physical meaning to separating the slope and intercept in a Lorentz transformation.