I remember when I took E&M my teacher said that special relativity was built int

Question

I remember when I took E&M my teacher said that special relativity was built into Maxwells equations.
Lets look at a line charge moving at non-relativistic speeds first.
We use amperes Law to find the B field.
frame 1: B=?0v?2?r

frame 2: relativistic speed B?=?0?v??2?r
in frame 2 the line charge becomes length contracted so the gamma factor takes care of that.
v and v' are the speeds relative to the moving charge.
Now if relativity was built into maxwells equations why did I need to correct for frame 2.
Or if I knew the charge per length in frame 2 at speed v' then would I calculate it just like frame 1.
If I know the correct charge density at any speed do I just calculate it normally with amperes law and there is no need for a relativistic correction.
I know there is an E field in each frame but Im just looking at the B field.

Explanation / Answer

Changing to units in which ?0 ?0 and c are 1, we may combine the two 3-vectors E and B into the 6-component Faraday 2-form (E;B), or its dual, the Maxwell 2-form (E;B)?.

And we may define the current 4-vector J as (Qf,jf).

Then the differential versions of Gauss' Law and the Amp

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