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Monday, 6 June 2011

What are general covariant transformations?

Based on the relativity principle, Einstein’s General Relativity and its contemporary generalizations are invariant under general covariant transformations. But what are general covariant transformations? Theoreticians write general covariant transformations of tensor fields in an explicit form, but do not define their class. For instance, what is a difference between general covariant transformations and gauge transformations in gauge theory? Is gravitation theory a gauge theory, and is a gravitation field a gauge field similarly to an electromagnetic field and gluons?

From the mathematical viewpoint, general covariant transformations characterize the category of so-called natural bundles S->X. 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 S so that the corresponding map T: V(X)->V(S) of the Lie algebra V(X) of vector fields on X to the Lie algebra V(S) of vector fields on a natural bundle S is the Lie algebra morphism, i. e.,
For instance, the above mentioned tensor bundles, e. g., tangent and cotangent bundles of X are natural bundles.

It follows from this definition, that one should develop gravitation theory as classical field theory on natural bundles. Furthermore, in accordance with the geometric equivalence principle, the structure group of these bundles must be reducible to the Lorentz group and, thus, gravitation theory is classical field theory on natural bundles with spontaneous symmetry breaking. The corresponding Higgs field just is a gravitational field. In contrast with Higgs fields in gauge theory of internal symmetries, this Higgs field is dynamic because it is not brought into the constant Minkowski metric by general covariant transformations. 


G. Sardanashvily, Gauge gravitation theory from geometric viewpoint, arXiv: gr-qc/0512115
G. Giachetta, L. Mangiarotti, G. Sardanashvily, Advanced Classical Field Theory (WS, 2009)

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