  In the first part of the present work, we focus on the theory of
gravitoelectromagnetism (GEM), and we derive the full set of equations and
constraints that the GEM scalar and vector potentials ought to satisfy. We
discuss important aspects of the theory, such as the presence of additional
constraints resulting from the field equations and gauge condition, the
requirement of the time-independence of the vector potential and the emergence
of additional terms in the expression of the Lorentz force. We also propose an
alternative ansatz for the metric perturbations that is found to be compatible
only with a vacuum configuration but evades several of the aforementioned
obstacles. In the second part of this work, we pose the question of whether a
tensorial theory using the formalism of General Relativity could re-produce the
theory of Electromagnetism. We demonstrate that the full set of Maxwell's
equations can be exactly re-produced for a large class of models, but the
framework has several weak points common with those found in GEM.
