  Riemannian and teleparallel geometrical approaches to the investigation of
Maxwell electrodynamics shown that a unified field theory of gravitation and
electromagnetism a la Einstein can be obtained from a stationary metric. This
idea contrasts with the recently proposed pre-metric electrodynamics by Hehl
and Obukhov. In the teleparallel case the definition of the electric field is
obtained straightforward from the spacetime metric and the orthonormal basis
frame of teleparallelism. In this case the only nonvanishing component of
Cartan torsion is defined as the effective electric field. In this approach the
gravitational potentials or metric coefficients are expressed in terms of the
effective or analogous electric and magnetic potentials. Thefore the Maxwell
equations in vacuum can be obtained by derivation of this electric field
definition as usual. In the Riemannian case we consider an electrostatic
spacetime where the Einstein equations in vacuum in the approximation of linear
fields. The constraint of Einstein equations in vacuum are shown to lead or to
the Coulomb equation or to a singular behaviour on the metric which would
represent a kind of effective electrodynamic black hole event horizon.
