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Monday, 11 January 2016

Foundations of Modern Physics 10: Relativity Principle

Studying gravitation theory, one conventionally requires that it incorporates Einstein’s General Relativity based on Relativity and Equivalence Principles reformulated in the fibre bundle terms.

Relativity Principle states that gauge symmetries of classical gravitation theory are general covariant transformations. Fibre bundles possessing general covariant transformations constitute the category of so called natural bundles. Let π: Y → X be a smooth fibre bundle. Any automorphism (Φ, f) of Y, by definition, is projected as πΦ = f ◦π onto a diffeomorphism f of its base X. The converse need not be true. A fibre bundle Y → X is called the natural bundle if there exists a group monomorphism of a group Diff(X) of diffeomorphisms of X to a group Aut(Y) of bundle automorphisms of Y → X. This functorial lift of Diff(X) to Aut(Y) are called general covariant transformations of Y.

Let us consider one-parameter groups of general covariant transformations and their infinitesimal generators. These are defined as the functorial lift T(u) of vector fields u on a base X onto Y so that the corresponding map T: V(X) V(Y) of the Lie algebra V(X) of vector fields on X to the Lie algebra V(Y) of vector fields on a natural bundle Y is the Lie algebra morphism, i. e.,

The tangent bundle TX of X exemplifies a natural bundle. Any diffeomorphism f of X gives rise to the tangent automorphisms Tf of TX which is a general covariant transformation of TX. The associated principal bundle is a fibre bundle LX of linear frames in tangent spaces to X. It also is a natural bundle. Moreover, all fibre bundles associated to LX are natural bundles. For instance, tensor bundles are natural bundles.

Following Relativity Principle, one thus should develop gravitation theory as a field theory on natural bundles.


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

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