Spontaneously broken de Sitter symmetry and the gravitational holonomy group

Abstract

The local geometrical structure of general relativity is analyzed in detail from the standpoint of a formulation of gravity as a gauge theory of the de Sitter group SO(3,2). In order to reproduce the structure of the Einstein-Cartan theory, it is essential that the SO(3,2) gauge symmetry be spontaneously broken down to the Lorentz group. In the geometrical analysis of this spontaneously broken theory, the Goldstone field of the symmetry-breaking mechanism plays a central role, representing the coordinates of a point in an internal anti-de Sitter space where the motions induced by parallel transport across space-time take place. In order to establish the connection between the SO(3,2) gauge theory and the Einstein-Cartan theory, the gravitational vierbein and spin connection are derived from the original SO(3,2) gauge fields by passing over to a set of nonlinearly-transforming fields through a redefinition involving the Goldstone field. The original SO(3,2) gauge fields have a different but equally important role: they generate pseudotranslations and rotations in the internal anti-de Sitter space under a kind of parallel transport across space-time that is called development. Development maps curves in space-time into image curves in the internal space, and vector fields along the curves in space-time into image vector fields along the image curves. Considering development along infinitesimal closed curves in space-time leads to the proper interpretation of the effects of torsion and of curvature in terms of the nonclosure of image curves and of the rotation of image vectors with respect to their original values.