Invariant Relativistic Electrodynamics. Clifford Algebra Approach
  In the usual Clifford algebra formulation of electrodynamics the Faraday bivector field F is decomposed into the observer dependent sum of a relative vector E and a relative bivector e_5 B by making a space-time split, which depends on the observer velocity. (E corresponds to the three-dimensional electric field vector, B corresponds to the three-dimensional magnetic field vector and e_5 is the (grade-4) pseudoscalar.) In this paper it is proved that the space-time split and the relative vectors are not relativistically correct, which means that the ordinary Maxwell equations with E and B and the field equations (FE) with F are not physically equivalent. Therefore we present the observer independent decomposition of F by using the 1-vectors of electric E and magnetic B fields. The equivalent, invariant, formulations of relativistic electrodynamics (independent of the reference frame and of the chosen coordinatization for that frame) which use F, E and B, the real multivector Psi = E - e_5 cB and the complex 1-vector Psi = E - icB are developed and presented here. The new observer independent FE are presented in formulations with E and B, with real and complex Psi. When the sources are absent the FE with real and complex Psi become Dirac like relativistic wave equations for the free photon. The expressions for the observer independent stress-energy vector T(v) (1-vector), energy density U (scalar), the Poynting vector S and the momentum density g (1-vectors), the angular momentum density M (bivector) and the Lorentz force K (1-vector) are directly derived from the FE. The local conservation laws are also directly derived from the FE and written in an invariant way. 
Tomislav Ivezic