Harnessing symmetries in deriving general Electromagnetic-field transformations in special relativity

We demonstrate how manifest symmetry of the general oblique Lorentz-boost matrix in conjunction with the antisymmetry of the Electromagnetic (EM)-field tensor enable one to derive the classic EM-field transformations in special theory of relativity (STR) in a broader, rigorous, yet remarkably lucid perspective, under (i) General Lorentz-boosts; (ii) Arbitrary, instantaneous, proper 3-D rotations.  Standard texts on electrodynamics ‘derive’ the EM-field transformations only by contemplating certain ideal, special static-charge and steady-current distributions with preferred symmetries, chosen geometries, invoking ‘longitudinal’ and ‘transverse’ components of the static fields so generated.  Gratifyingly, our approach manifestly encompasses time varying fields and yields the transformation relations in a coordinate-free form.  Further, the EM-field transformation under proper, instantaneous 3-D rotations is demonstrated to transform as a passive rotation of the coordinate axes.  The present didactical exercise, amenable to be incorporated in graduate courses on relativistic electrodynamics, should serve as an excellent illustration on how cogent symmetry arguments simplify otherwise tedious calculations in STR, while endowing complete generality to the connections obtained.