Gravitation as a gauge theory

Abstract

We present theories of gravitation based, respectively, on the general linear group $\mathrm{GL}(n, R)$ and its inhomogeneous extension $\mathrm{IGL}(n, R)$ [($\mathrm{SO}(n\ensuremath{-}1, 1)$ and [$\mathrm{ISO}(n\ensuremath{-}1, 1)$ for torsion-free manifolds]. Noting that the geometry of the conventional gauge theories can be described in terms of a fiber bundle, and that their action is a scalar in such a superspace, we construct principal fiber bundles based on the above gauge groups and propose to describe gravitation in terms of their corresponding scalar curvatures. To ensure that these manifolds do indeed have close ties with the space-time of general relativity, we make use of the notion of the parallel transport of vector fields in space-time to uniquely relate the connections in space-time to the gauge potentials in fiber bundles. The relations turn out to be similar to that suggested earlier by Yang. The actions we obtain are related to those of Einstein and Yang but are distinct from both and have an Einstein limit. The inclusion of internal symmetry leads to the analogs of Einstein-Yang-Mills equations. A number of variations and less attractive alternatives based on the above groups or their subgroups are also discussed.