Fundamental Symmetries and Spacetime Geometries in Gauge Theories of Gravity: Prospects for Unified Field Theories

TL;DR

This paper explores gauge theories of gravity, emphasizing their geometric structures, symmetry principles, and potential for unifying fundamental interactions within a new spacetime framework.

Contribution

It provides a comprehensive analysis of gauge gravity theories, including Poincaré gauge models and their implications for unified field theories and spacetime paradigms.

Findings

Analysis of Poincaré gauge theories and their field equations

Discussion on non-Riemannian geometries and Lorentz symmetry breaking

Proposals for using gauge principles to develop unified theories

Abstract

Gravity can be formulated as a gauge theory by combining symmetry principles and geometrical methods in a consistent mathematical framework. The gauge approach to gravity leads directly to non-Euclidean, post-Riemannian spacetime geometries, providing the adequate formalism for metric-affine theories of gravity with curvature, torsion and non-metricity. In this paper, we analyze the structure of gauge theories of gravity and consider the relation between fundamental geometrical objects and symmetry principles as well as different spacetime paradigms. Special attention is given to Poincar\'{e} gauge theories of gravity, their field equations and Noether conserved currents, which are the sources of gravity. We then discuss several topics of the gauge approach to gravitational phenomena, namely, quadratic Poincar\'{e} gauge models, the~Einstein-Cartan-Sciama-Kibble theory, the teleparallel equivalent of general relativity, quadratic metric-affine Lagrangians, non-Lorentzian connections, and the breaking of Lorentz invariance in the presence of non-metricity. We also highlight the probing of post-Riemannian geometries with test matter. Finally, we briefly discuss some perspectives regarding the role of both geometrical methods and symmetry principles towards unified field theories and a new spacetime paradigm, motivated from the gauge approach to gravity.