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 Poincaré 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 Poincaré 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.
Highlights
The success of Einstein’s General Theory of Relativity (GR) to describe the behaviour of the gravitational interaction continues to amaze us
It was discovered that the electromagnetic field itself is intimately related to local internal symmetries, under the U (1) group that acts on the four-spinor fields of charged matter [10]
The gauge approach to Yang-Mills fields follows from two major steps, the rigid symmetries of a physical system described by a matter Lagrangian, and the localization of those symmetries
Summary
The success of Einstein’s General Theory of Relativity (GR) to describe the behaviour of the gravitational interaction continues to amaze us. The main aim of this work is to review and discuss the basics and recent developments within gauge theories of gravity and post-Riemann geometries in connection to their perspectives for achieving an unified field theory This is a vital part of the effort to understand the nature of spacetime and gravity in those regimes where the standard picture provided by GR may break down, challenging our current ideas on the spacetime paradigm. We illustrate it with the case of PGTG, consider the field equations, the Noether conserved currents (sources of gravity), and include discussions on several topics in gauge theories of gravity, such as quadratic Poincaré gauge models, the ECSK theory, the teleparallel equivalent of GR (an example of a translational gauge model), quadratic metric-affine Lagrangians, the breaking of Lorentz invariance in the presence of non-metricity, the nature of the hypermomentum currents and the probing of post-Riemann geometries with test matter.
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