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 π◦Φ = 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.,
[T(u),T(u')]=T([u,u']).
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.
References
G. Sardanashvily:
Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Phys. 8 (2011)
1869-1895 (#); arXiv: 1110.1176
