  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.
