  For internal gauge forces, the result of locally gauging, i.e., of performing
the substitution $\partial \rightarrow D$, is physically the same whether
performed on the action or on the corresponding Euler-Lagrange equations of
motion. Rather unsettling, though, such commutativity fails for the standard
way of coupling a Dirac fermion to the gravitational field in the setting of a
local Lorentz gauge theory of general relativity in the vierbein formalism, the
equivalence principle thus seemingly being here violated. This paper will
present a formalism in which commutativity holds for the gravitational force as
well, the action for the gravitational field itself being still the
Einstein-Hilbert one. Notably, in this formalism, the spinor field will carry a
world/coordinate index, rather than a Lorentz spinor index as it does
standardly. More generally, no Lorentz indices will figure, neither vector
indices nor spinor indices, which from a parsimonious point of view seems quite
satisfactory.
