Classical gauge gravitation theory
The author presents an interpretation of a different version of classical theory of gravitation (mostly Metric-Affine Theory and General Relativity as its special case) as a gauge theory on natural bundles.NEWLINENEWLINELet \(M\) be a 4-manifold, \(\pi : LM \to M\) the principal \(GL_4(R)\)-bundle of frames and \(J^1LM \to M\) the bundle of \(1\)-jets of \(\pi\), i.e., the bundle of horizontal subspaces in the tangent bundle \(TLM\). The space of connections is defined as the total space of the quotient bundle \(\pi_C: CM := J^1LM/GL_4(R) \to M\). Sections of \(\pi_C\) are linear connections in \(M\). The curvature of a connection is described in terms of the manifold \(J^1CM\) of 1-jets of \(\pi_C\). Following the theory of \(G\)-structure, the author defines a Lorentz structure on \(M\) (which is equivalent to a Lorentz metric) as a section of the bundle \( \Sigma M:=LM/SO_{1,3}^0 \to M\) where \(SO_{1,3}^0\) is the Lorentz group or, equivalently, as a principal \(SO_{1,3}^0\)-subbundle of the frame bundle \(LM\).NEWLINENEWLINEThe product bundle \(\pi_Y :Y = \Sigma M \times CM \to M\) is considered as the bundle of dynamical variables of the Metric-Affine Gravitation Theory. The bundle \(Y\) is a natural bundle that is the group \(\mathrm{Diff}(M)\) of diffeomorphisms acts on \(Y\) as a group of automorphisms. The author demands that the Lagrangian \(L\) of the gravitation theory is a \(\pi_Y\)-horizontal 4-form on \(Y\) which is invariant with respect to \(\mathrm{Diff}(M)\). The problem of a choice of such Lagrangian is discussed as well as the energy-momentum conservation law associated with the symmetries of the theory with respect to (lifted to \(Y\)) vector fields on \(M\). The General Relativity defined by the Hilbert-Einstein Lagrangian is considered as a special case of the theory. At the end, the author considers a BRST extension of the Lagrangian \(L\) and the problem of introduction spinor fields in the theory.
scientific article; zbMATH DE number 6046015