Why to gauge gravity?

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]). 

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