This is Introduction of our article: D. Ivanenko, G. Sardanashvily, The gauge treatment of gravity, Physics Reports, 94 (1983) 1-45.
At present Einstein's General
Relativity (GR) still remains the most satisfactory theory of classical
gravitation for all now observable gravitational fields. GR successfully passed
the test of recent experiments on the radiolocation of planets and on the
laser-location of the Moon, which have put the end to some other versions of
gravitation theory, e.g., the scalar-tensor theory.
At the same time the conventional description
of gravity by Einstein's GR obviously faced a number of serious problems [55],
and even some corner-stones of gravitation theory still remain disputable up to
our day. This is reflected also in the rather curious uninterrupted flow of
proposals for new designations of this theory.
Really, it is difficult to
find in physics another example of continuous discussions about naming a
well-established theory like Einstein's theory of gravitation. The author's own
proposal "Allgemeine Relativitatstheorie" as pointing on the
generalization of the Special Relativity (SR) is still not admitted by some
scientists. A. Friedmann wrote about "small" and "great"
principles. V. Fock was a supporter of Fokker's expression
"chronogeometry", insisting that GR does not possess any
"relativity" (in the sense of SR), and believing with Kretschmann and
P. Havas that "general covariance" requirement is a trivial one.
H.-J. Treder speaks about "geochronometrical gravity". J.A. Wheeler
repeatedly writes about "geometrodynamics", though previously it
designated "already unified" theory of gravitation and
electromagnetism of Rainich. One would speak now about Newtonian
"gravistatics" and Einsteinian "gravidynamics" (after B. de
Witt and D. Ivanenko) or "gravitodynamics" (A. Mercier).
It is well known that
Einstein repeatedly insisted that the Relativity Principle (RP) is not a priori
necessary, i.e., it is not physically trivial. At the same time formulation of
this principle is known to be directly connected with establishing the notion
of reference frames in gravitation theory, which itself still remains under
discussion.
The problem of reference
frame definition in GR was not paid sufficient attention up to about the
mid-1950s. As was stressed when founding GR, e.g., in the
Einstein-Kretschmann-Kottler discussions, revived later by V. Fock, P. Havas
and also by H.-J. Treder, the important possibility of a general covariant
formulation means that coordinates are only auxiliary quantities losing their
immediate physical sense of observable objects, in contrast with, e.g., the
coordinates in Minkowski space. Then there arose the necessity to distinguish
coordinate systems from reference frames. Einstein, though emphasizing for the
first time the role of reference frames, however, did not give himself any
formal definition of them. Moreover many contemporary authors mixing
coordinates and reference terms simply ignore this problem. But more precise
reference system determination is necessary for correct experimentation, for
stating the Cauchy problem for gravitational field equations, for describing
spinors in GR, and for other problems of gravitation theory.
Most deeply the reference
frame problem is examined in the tetrad version of GR in combination with the
technique of 3+1) decomposition [76, 91, 47], where tetrads, thought to define
local reference frames, are erected in all space-time points. But the dilemma
to make up these reference tetrads by a certain choice of physical devices is
as yet far from a final solution. All the more in the general case of a curved
space-time
there may not exist any
continuous tetrad distribution, but only up to admitting SO(3)-transformations
of tetrads.
The Equivalence Principle
(EP) being another corner-stone of GR is also open to question, e.g., one
separates "weak", "middle-strong", and "strong"
equivalence principles [112, 108].
In GR the Equivalence
Principle supplements RP and must establish the existence of a certain
reference frame, where all physical laws would take the known special relativistic
form; and it seems naturally for some authors to establish the disappearance of
a gravitational field as the criterion of transition from GR to SR in some
reference frame.
All existent formulations of
EP are based upon the empirical equality of inertial mass, gravitational active
and passive charges. In the case of a uniform gravitational field it provides
the existence of a reference frame searched for, in which the motion law of
probe particles is viewed in the same way as free motion in SR. In the general
case EP is formulated as a sui generis
localization of this equivalence for uniform gravitational fields, i.e., a
local inertial frame must exist, where a metric field becomes the Minkowski
one, and its Christoffel connection disappears in a given space-time point. But
gravitation curvature does generally not vanish in such a reference frame. Does
this mean that the Equivalence Principle in gravitation holds only in the
"weak" variant, i.e., only for laws, e.g., of probe particle motion,
which do not contain more than first-order derivatives of a gravitational
field?
Then to what degree is it
correct to speak about such a special relativistic attribute as energy-momentum
of a gravitational field itself? Maybe it is the cause of the known problem of
gravitation energy, which led to vivid disputes, starting with Einstein-Grossman
pre-GR works up to recent days, and presenting a broad display of opinions, as,
e.g., in the case of gravitational waves: positive energy of waves, or no
energy at all!
Note also the widely
discussed singularity problem in GR, which shows that either we are unable to
gain insight into the nature of singularities as yet, or that GR (at least in
its classical version) is incapable to describe extremal gravitational fields.
These and some other
difficulties of the GR picture of gravity motivate one's attempts to
reformulate gravitation theory from non-conventional standpoints extending the
framework of Einstein's GR.
But why gauge gravity? Can
the gauge treatment of gravity really solve the above-mentioned problems? Beforehand
nobody knows. But today many of these problems seem to be put in the shadow of
the urgent goal of the gravity unification with the elementary particle world.
Just this goal stimulated by the grand unification program in contemporary
particle physics puts the gauge version in the forefront of modern gravitation
research.
Today, gauge theory provides
the theoretical base of all modern unification attempts in particle physics. It
has become clear that weak and electromagnetic interactions can be successfully
unified by the Weinberg-Salam gauge model, and there is growing evidence that
strong interaction is also mediated by gauge particles or gluons within the
framework of chromodynamics. In field theory gauge potentials become a standard
tool for describing interactions with very different symmetries. And apparently
the single gap in the modern gauge picture still remains gauging the external
or space-time ymmetries of fields and particles, that includes the gauge
gravity also.
Moreover, gauge theory using
the mathematical formalism of fiber bundles realizes in fact the known program
of the 1920s to build the geometric unified picture of various interactions.
And it is strange enough that just the gravitation theory, being the first
example of field geometrization, has still not any recognized gauge version.
Although the first gauge treatment of gravity was suggested immediately after
the gauge theory birth itself [109, 8, 62].
The main dilemma which during
25 years has been confronting the establishment of the gauge gravitation
theory, is that gauge potentials represent connections on fiber bundles, while
gravitational fields in GR are only metric or tetrad (vierbein) fields.
Connections as fundamental
quantities appeared together with the metric in Weyl's and Eddington's
generalizations of GR on gravity with nonmetricity and torsion, and in this
quality were again recognized by Einstein in his last scientific paper [25].
But even in the gauge gravitation theory connections cannot at all substitute the
metric, because there are no groups, whose gauging would lead to the purely
gravitational part of space-time connections (Christoffel symbols or
Fock-Ivanenko spinorial coefficients). To separate such gravitational
components from gauge fields, e.g., of the Lorentz group, metric or tetrad
fields have to be introduced.
At the present time the
gravitation theory is viewed actually as the affine-metric theory possessing
two independent potentials, namely, metric and connection, and just this
constitutes the peculiarity of the gauge approach to the gravitation theory in
comparison with the gauge models of the Yang-Mills type for internal
symmetries.
This article aims to match
the gravitation theory and gauge theory within the framework of gauge theory of
external symmetries. Because both gravitation and gauge theories have the
geometric formulations in terms of the fiber bundle formalism, we shall use the
fiber bundle language (for necessary mathematics see [100, 63, 102, 103]).
References:
