The
equivalence principle is one of the corner-stones of
gravitation theory. Different formulations of the equivalence principle are labeled ''weakest'', ''weak'', ''middle-strong'' and ''strong.'' All of these formulations are based on the empirical equality of inertial mass, gravitational active and passive charges.
The ''weakest'' equivalence principle is restricted to the motion law of a probe point mass in a uniform
gravitational field. Its 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.
The above-mentioned variants of the equivalence principle aim to guarantee the transition of
General Relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
to
Special Relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
in a certain
reference frame
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathe ...
. However, only the particular ''weakest'' and ''weak'' equivalence principles are true.
To overcome this difficulty, the equivalence principle can be formulated in geometric terms as follows.
In the spirit of
Felix Klein's Erlanger program
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is nam ...
, Special Relativity can be characterized as the
Klein geometry
In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space ''X'' together with a transitive action on ''X'' by a Lie group ''G'', which acts as ...
of
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
invariants. Then the geometric equivalence principle is formulated to require the existence of Lorentz invariants on a
world manifold . This requirement holds if the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of
admits an atlas with Lorentz transition functions, i.e., a structure group of the associated
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
of linear tangent frames in
is reduced to the Lorentz group
. By virtue of the well known theorem on
structure group reduction, this reduction takes place if and only if the quotient bundle
possesses a global section, which is a
pseudo-Riemannian metric
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
on
.
Thus the geometric equivalence principle provides the necessary and sufficient conditions of the existence of a pseudo-Riemannian metric, i.e., a
gravitational field on a world manifold.
Based on the geometric equivalence principle, gravitation theory is formulated as
gauge theory where a gravitational field is described as a
classical Higgs field responsible for spontaneous breakdown of space-time symmetries.
See also
*
Equivalence principle
*
Gauge gravitation theory
*
Reduction of the structure group
In differential geometry, a ''G''-structure on an ''n''-manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''.
The notion of ''G''-structures includes vario ...
References
* H.-J. Treder, ''Gravitationstheorie und Äquivalenzprinzip'', Akademie-Verlag, Berlin, 1971.
*
S. Weinberg, ''Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity'', J. Wiley and Sons Inc., N.Y., 1972.
*
D.Ivanenko,
G.Sardanashvily, The gauge treatment of gravity, Physics Reports 94 (1983) 1. {{doi, 10.1016/0370-1573(83)90046-7
Gravity
General relativity
Principles