  A new tetrad introduced within the framework of geometrodynamics for non-null
electromagnetic fields allows for the geometrical analysis of the Lorentz force
equation and its solutions in curved spacetimes. When expressed in terms of
this new tetrad, the electromagnetic field displays explicitly maximum
simplification, and the degrees of freedom are manifestly revealed. In our
manuscript we are deducing the Lorentz force equation on purely Riemannian
geometrical grounds. The equation arises on the basis of Frenet-Serret analysis
through the use of our new tetrads and gauge invariance arguments only. The
force is deduced through a geometrical construction that precludes any other
mathematical form other than the one already accepted. Therefore, a significant
and fundamental result such as the first geometrical proof on the necessity of
the force in the equation to have the structure already accepted in physics and
not any other, is given. Through the use of the Frenet-Serret formulae and
gauge invariance arguments we are also able to express in terms of the new
tetrad vectors the Lorentz force equation as a generalized form of Fermi-Walker
transport.
