The Equivalence principle is treated as one of the corner-stones of gravitation theory. However, there exist its different formulations. One separates “weakest”, “weak”, “middle-strong” and “strong” Equivalence principles. All of them are based on the empirical equality of inertial mass, gravitational active and passive charges, and they establish the existence of a certain reference frame, where physical laws would take the known special relativistic form, i. e., a gravitational field effectively disappears.
The “weakest” Equivalence principle is restricted to the motion law of a probe point mass in a uniform gravitational field.
Its sui generis localization is the “weak” Equivalence principle that states the existence of a desired local inertial frame at a given world point. This is the case of equations depending on a gravitational field and its first order derivatives, e. g., the equations of mechanics of probe point masses, and the equations of electromagnetic and Dirac fermion fields.
The “middle-strong” Equivalence principle is concerned with any matter, except a gravitational field, while the “strong” one is applied to all physical laws.
Apparently, only the “weakest” and “weak” Equivalence principles are true. It is the “weak” Equivalence principle that the identification of a gravitational field to a pseudo-Riemannian metric satisfies to. However, the “weak” Equivalence principle provides a necessary, but not sufficient condition of such identification. Moreover, it does not explain the existence of a gravitational field itself.
To overcome these difficulties, we have reformulated the Equivalence principle as follows.
In geometric terms Special Relativity can be characterized as the geometry of Lorentz invariants. Then the Equivalence principle can be formulated to require the existence of Lorentz invariants on a world manifold X. We agree to call it the geometric Equivalence principle. Its requirement holds if and only if the tangent bundle TX of X admits an atlas with Lorentz transition functions, i. e., a structure group of the associated principal bundle LX of frames in TX is reduced to the Lorentz group SO(1,3). By virtue of the well known theorem, this reduction takes place if and only if the quotient bundle LX/SO(1,3)->X admits a global section, which is a pseudo-Riemannian metric on X.
Thus the geometric Equivalence principle provides the necessary and sufficient conditions of the existence of a pseudo-Riemannian metric on a world manifold that we observe as a gravitational field.
Moreover, if a structure group of the frame bundle LX is reduced to the Lorentz group, it always is reduced to the spatial rotation group SO(3). In accordance with the above mentioned theorem, this reduction defines a space-time decomposition of the tangent bundle TX and, thus, makes a world manifold X into a space-time.
The geometric Equivalence principle also provides the necessary condition of the existence of Dirac’s spinor fields, possessing Lorentz symmetries, on a world manifold.
Thus, one can think of an observable Dirac fermion matter as being the underlying physical reason of the geometric Equivalence principle and, consequently, the existence of a pseudo-Riemannian gravitational field.
In gravitation theory, the geometric Equivalence principle characterizes spontaneous symmetry breaking of space-time symmetries and, thus, clarifies the physical nature of a gravitational field as a Higgs field responsible for this symmetry breakdown.
D. Ivanenko, G. Sardanashvily, The gauge treatment of gravity, Physics Reports 94 (1983) 1-45.
G. Sardanashvily, Gauge gravitation theory from the geometric viewpoint, Int. J. Geom. Methods Mod. Phys. 3 (2006) N1 v-xx; arXiv: gr-qc/0512115
WikipediA: Equivalence principle (geometric)