Radial gauge in Poincaré gauge field theories

The generalization of the Fock–Schwinger or ‘‘radial’’ gauge condition xiAi =0 to the gauge theories of the Poincaré group, which describe the gravitational field, is treated. It is shown that the choice of a radial gauge is equivalent to the use of normal coordinates and of tetrads parallel transported along autoparallel lines starting at the origin. The formulas that give the fields in the radial gauge starting from an arbitrary gauge and the formulas that give the gauge potentials in terms of the gauge field strengths are derived. The residual gauge freedom, which consists of the arbitrariness in the choice of the origin of the coordinate system and a tetrad of orthonormal vectors at this point is discussed in detail. It is the analog of the usual Poincaré invariance in flat space-time theories. The whole treatment can be extended to gauge theories of the affine and Euclidean groups. As an application, some properties of the homogeneous and isotropic states with random geometric fields are found.