I must say Veritasium isn't my favourite after confusing radio wave radiation/ coupling with power transfer through conductive wires, his notorious light bulb/switch experiment. It is true the EM coupling between the wires 1m apart takes only 1/c seconds, but the power, what will actually power the light has to travel the full length of said wires.
If you pay a bit more attention, he was also dealing with a hypothetical "magic" light bulb that would turn on if it saw any current; even the negligible current from radiation coupling. As I understood (been a while since I watched the videos), he never claimed that you'd see "full voltage" that quickly.
Oh here we go again...

"Negligible current" is also what's carrying your data along Ethernet cables. It's just a basic damned transmission line, with some hundreds of ohms impedance (in the scenario). Current is proportional to voltage and the car battery in the demo would easily light an LED transiently (if dimly; it would have to be done at night).
You can work such a problem with full fields, but ignoring transient/waveguide modes, it reduces to a linear TL. Radio waves, in the propagating far-field sense, do not apply at all here.
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As you can see, the amount of misinterpretation, or misinformation, generated on this topic, as a result of that one video, is immense. It's hard to say if it was actually worth presenting; is it better to be ignorant of a topic entirely, than for some (many? most? how many actually "get it" is hard to say; I would guess they're probably the silent majority, but who knows) to gain understanding, and many to gain an incorrect understanding (or worse yet, be turned off entirely by the controversy and reject any understanding on the topic).
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ANYWAY, regarding relativity and magnetism, yes, it's very simple, trivial really, how to express Maxwell's laws in 4-space; it's one (or two??) equation(s) in a simple relation. All the complexity is encoded into spacetime itself, the definitions of the variables and tensors used to describe it. Considering relativity is derived from E&M (Lorentz invariance), this is no accident, and indeed one might say the simplicity is required so as to simplify working with the equations in other terms.
And, conversely -- one can define a gravitomagnetic effect just the same. It's an apparent or emergent effect, much as magnetism from the relative motion of charges in EM -- which, I forget the particulars, if it's because it arises when you assume proper time in 3-space and thus there has to be some effect corresponding basically to the lack of simultaneity, and this is the result; or if it's a speed-of-light thing where if you assume gravity is instantaneous (it's not) you get unstable orbits but including such an effect puts things back into place; or, these might be wholly equivalent perspectives. Again, I don't remember exactly.
Tim