About the Symmetry of General Relativity

Abstract

In this work we use generalized deformed gauge groups for investigation of symmetry of general relativity (GR). GR is formulated in generalized reference frames, which are represented by (anholonomic in general case) affine frame fields. The general principle of relativity is extended to the requirement of invariance of the theory with respect to transitions between generalized reference frames, that is, with respect to the group $GL^g$ of local linear transformations of affine frame fields. GR is interpreted as the gauge theory of the gauge group of translations $T^g_M$, and therefore is invariant under the space-time diffeomorphisms. The groups $GL^g$ and $T^g_M$ are united into group $S^g_M$, which is their semidirect product and is the complete symmetry group of the general relativity in an affine frame (GRAF). The consequence of $GL^g$-invariance of GRAF is the Palatini equation, which in the absence of torsion goes into the metricity condition, and vice versa, that is, is fulfilled identically in the Riemannian space. The consequence of the $T^g_M$-invariance of GRAF is representation of the Einstein equation in the superpotential form, that is, in the form of dynamic Maxwell equations (or Young-Mills equations). Deformation of the group $S^g_M$ leads to renormalisation of energy-momentum of the gravitation field. At the end we show that by limiting admissible reference frames (by $GL^g$-gauge fixing) from GRAF, in addition to Einstein gravity, one can obtain other local equivalent formulations of GR: general relativity in an orthonormal frame or teleparallel equivalent of general relativity, dilaton gravity, unimodular gravity, etc.