The God has created a man in order that he creates that the God fails to do

Friday, 11 December 2015

Foundations of Modern Physics 9: Gauge gravitation theory

Classical gravitation theory on a world manifold X is formulated as gauge theory on natural bundles over X which admit general covariant transformations as the canonical functorial lift of diffeomorphisms of their base X. Natural bundles are exemplified by a principal linear frame bundle LX -> X and the associated, (e.g., tensor) bundles. This is metric-affine gravitation theory whose dynamic variables are general linear connections (principal connections on LX) and a metric (tetrad) gravitational field. The latter is represented by a global section of the quotient bundle W=LX/SO(3,1) and, thus, it is treated as a classical Higgs field responsible for the reduction of a structure group GL(4,R) of LX to a Lorentz group SO(1,3). The underlying physical reason of this reduction is both the geometric Equivalence Principle and the existence of Dirac spinor fields. Herewith, a structure Lorentz group of LX always is reducible to its maximal compact subgroup SO(3) that provides a world manifold X with a space-time structure. The physical nature of gravity as a Higgs field is characterized by the fact that, given different tetrad gravitational fields h, the representations of holonomic coframes {dx} on a world manifold X by Dirac's gamma-matrices are non-equivalent. Consequently, the Dirac operators in the presence of different gravitational fields fails to be equivalent, too. To solve this problem, we describe Dirac spinor fields in terms of a composite spinor bundle S -> W -> X where S -> W is a spinor bundle associated with a SO(1,3)-principal bundle LX -> W. A key point is that, given a global section h of W -> X, the pull-back bundle h*S of S -> W describes Dirac spinor fields in the presence of a gravitational field h. At the same time, W -> X is a natural bundle which admits general covariant transformations. As a result, we obtain a total Lagrangian of a metric-affine gravity and Dirac spinor fields, whose gauge invariance under general covariant transformations implies an energy-momentum conservation law. Our physical conjecture is that a metric gravitational field as the Higgs one is non-quantized, but it is classical in principle.


D. Ivanenko, G. Sardanashvily: The gauge treatment of gravity, Phys. Rep. 94 (1983) 1-45 (#).

G. Sardanashvily: Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Phys. 8 (2011) 1869-1895 (#)arXiv: 1110.1176.

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