Semi-teleparallel theories of gravitation (Q5931046)

In various papers, for different physical reasons, their authors discussed the possibility to generalize the theory of general relativity by going over from Riemannian to geometrically more rich geometries. In particular, teleparallelized Riemannian, Riemann-Cartan, metric-affine, and purely affine geometries were considered as promising candidates. As far as the teleparallelized Riemannian geometry is concerned, it was shown that it can be regarded as constrained Riemann-Cartan geometry. Referring to some of these papers, the present article is dealing with mathematical aspects of metric-compatible geometries, where the results are partially formulated terms of fiber bundles. It is shown that, in the spectra of these geometries, there is a further candidate (called semi-teleparallel geometry) that is formally a constrained Riemann-Cartan geometry, too, that, however, should be seen as a separate geometry since it incorporates preferred frames. For a special action vacuum field equations are derived and the coupling to the Dirac field is discussed, where, however, the physical meaning of this theory remains as long undetermined as there is no principle introduced that specifies the assumed frames.