That the current laws of electromagnetism can be encapsulated within a syntactically dense equation is delightful, and also that this tensor fits within the paradigm of special relativity. I have commented that a larger tensor should be constructed with which to account for ∃magnetic monopole, especially within Black Hole. Today I will focus mainly on the rank-2 version that is more commonly understood.
This can be thought of as a matrix, however it is with the additional structure such that both indices transform under the rotation of a Lorentz. One must transform both ranks to perform a change into a different observer frame. In this sense, although seperately, the electric field E and magnetic field B are subjective to the observer, when combined as to E⊗B, this combined object is then one field which is preserved via flat changes in velocity and position (or a pure change of observer).
By this I mean that an observer is changed sufficiently slowly such that the acceleration be negligible upon this curvature such that we might remain within the special relativity paradigm (It has been effectively argued that acceleration can be accounted for within S.R., however this is not central to my discussion at this moment).
Mathematically, one can perform a pure change of observer from one frame to another without disrupting the physical system. However, such a physical change of observation frame would likely disrupt the curvature, thus causing a ripple in the purely flat approximation and falling under general relativity and more (a new theory or everything containing the observer effect). The Maxwell equations (here the permittivity and permeability are set to unity):
One sees that one could construct a certain something that would make ∇.(E⊗B)=0 and ∇x(E⊗B)=±∂(B⊗E)/∂t . This certain something is represented as the vector potential:
Such that under the Einstein summation convention:
Therefore the flat electrodynamics reduces to [*], a single symbol for an entire theory.