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.

**References:**

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)**

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