Gravity as a Gauge Theory of Translations

Abstract

The Poincaré group can be interpreted as the group of isometries of a Minkowskian space. This point of view suggests to consider the group of isometries of a given space as the suitable group to construct a gauge theory of gravity. We extend these ideas to the case of maximally symmetric spaces to reach a realistic theory including the presence of a cosmological constant. Introducing the concept of "minimal tetrads" we deduce Einstein gravity in the vacuum as a gauge theory of translations.

Higgs and Goldstone Modes in Crystalline Solids

“…D. and G. Grensing came to similar conclusions, obtaining the gauging of the Poincaré group in a form that allowed to express General Relativity as a gauge theory of this symmetry group. More recently, the Yang–Mills theory of the affine group (the semidirect product of translations and general linear transformations ) was formulated, where tetrads have been identified with nonlinear translational connections, for which the given expression is a simplified yet correct version of the general formulation.…”

Dynamical Variables in Gauge-Translational Gravity

“…Strictly speaking a similar question arises with the definition of g(0) ρσ present in (19), and we are to show that the structure and properties of this minimal metric tensor can be derived from general integrability conditions. In a previous work ( [27]) we have seen that the field equations of gravity in the vacuum can be interpreted as a gauge theory of translations defined in the metric of a maximally symmetric background space. Now we are going to show that this result holds without making recourse to the field equations even in the presence of matter, or, in other words, as a consequence of the underlying gauge structure of the theory which is previous to any dynamics.…”

“…Using Hausdorff-Campbell formulas to deal with exponentials, after a little algebra (the details can be found in [25] [26]) we obtain:…”

“…In a previous work [25] we have seen that the field equations of gravity in the vacuum can be interpreted as a gauge theory of translations defined in the metric of a maximally symmetric background space. Now we are going to show that this result holds without making recourse to the field equations even in the presence of matter, or, in other words, as a consequence of the underlying gauge structure of the theory which is previous to any dynamics.…”

“…Thus g(0) µν is determined from first principles and previously to any dynamics. As a particular case, for k → 0 equation ( 32) is replaced by [25][26]…”