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

Observational constraints in metric-affine gravity

“…In particular, the presence of nonvanishing torsion and nonmetricity components in the affine connection can be related to the intrinsic hypermomentum of matter and provide significant corrections to the standard approach of GR, even in the Riemannian sector governed a e-mails: sbahamonde@ut.ee; sebastian.beltran.14@ucl.ac.uk (corresponding author) b e-mail: jorge.gigante.valcarcel@ut.ee by the metric structure. From a theoretical point of view, the resulting geometry can be related to the existence of a new fundamental symmetry by applying the gauge principles, which leads to the appearance of the Metric-Affine Gauge theory of gravity (MAG) [4][5][6].…”

Cosmology of Quadratic Metric-Affine Gravity

“…Under the perspective described above, MAG theories result to be rather general and powerful [11,12,[15][16][17][18][19][20][21][22][23], and can possibly lead to diverse scenarios beyond GR, which may have relevant consequences, for instance, in the cosmological context. For some relevant results already obtained in MAG cosmology we refer the reader to, e.g., [24][25][26][27][28][29][30][31].…”

The Full Quadratic Metric-Affine Gravity (Including Parity Odd Terms): Exact solutions for the Affine-Connection

“…There are many reasons that make MAG an appealing gravity Theory 1 but probably the most astonishing one is the link it provides in relation to the microstructure of matter [6,7]. In recent years, there is an ever-increasing interest in the MAG formulation [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] with a special emphasis given in Cosmological applications [22,[25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] 2 . Let us also mention that MAG has some promising features in regards to the Quantization of Gravity (see ref [41]).…”