universe Review Fundamental Theory of Torsion Gravity Luca Fabbri DIME, Metodi e Modelli Matematici, Università di Genova, Via all’Opera Pia 15, 16145 Genova, Italy; [email protected] Abstract: In this work, we present the general differential geometry of a background in which the space–time has both torsion and curvature with internal symmetries being described by gauge fields, and that is equipped to couple spinorial matter fields having spin and energy as well as gauge currents: torsion will turn out to be equivalent to an axial-vector massive Proca field and, because the spinor can be decomposed in its two chiral projections, torsion can be thought as the mediator that keeps spinors in stable configurations; we will justify this claim by studying some limiting situations. We will then proceed with a second chapter, where the material presented in the first chapter will be applied to specific systems in order to solve problems that seems to affect theories without torsion: hence the problem of gravitational singularity formation and positivity of the energy are the most important, and they will also lead the way for a discussion about the Pauli exclusion principle and the concept of macroscopic approximation. In a third and final chapter, we are going to investigate, in the light of torsion dynamics, some of the open problems in the standard models of particles and cosmology which would not be easily solvable otherwise. Keywords: torsion-gravity; electrodynamics; spinors Citation: Fabbri, L. Fundamental 1. Introduction Theory of Torsion Gravity. Universe In fundamental theoretical physics, there are a number of principles that are assumed, 2021, 7, 305. https://doi.org/ and, among them, one of the most important is the principle of covariance, stating that the 10.3390/universe7080305 form of physical laws must be independent from the coordinate system employed to write them. Covariance is mathematically translated into the instruction that such physical laws Academic Editors: Antonino Del have to be written in tensorial forms. Popolo and Matteo Luca Ruggiero On the other hand, because physical laws describe the shape and evolution of fields, differential operators must be used; because of covariance, all derivatives in the field Received: 8 July 2021 Accepted: 15 August 2021 equations have to be covariant: thus covariant derivatives must be defined. In its most Published: 18 August 2021 general form, the covariant derivative of, say, a vector, is given by n n s n DaV = ¶aV +V G Publisher’s Note: MDPI stays neutral sa with regard to jurisdictional claims in a where the object Gns is called connection, it is defined in terms of the transformation law published maps and institutional affil- needed for the derivative to be fully covariant, and it has three indices: the upper index iations. and the lower index on the left are the indices involved in the shuffling of the components of the vector, whereas the lower index on the right is the index related to the coordinate with respect to which the derivative is calculated eventually. Hence, there appears to be a clear distinction in the roles played by the left and the right of the lower indices, Copyright: c 2021 by the author. and therefore the connection cannot be taken to have any kind of symmetry property for Licensee MDPI, Basel, Switzerland. indices transposition involving the two lower indices at all. This article is an open access article The fact that, in the most general case, the connection has no specific symmetry implies distributed under the terms and that the antisymmetric part of the connection is not zero, and it turns out to be a tensor: conditions of the Creative Commons this is what is known as torsion tensor. Attribution (CC BY) license (https:// The circumstance for which the torsion tensor is not zero does not follow from ar- creativecommons.org/licenses/by/ guments of generality alone, but also from explicit examples: for instance, torsion does 4.0/). Universe 2021, 7, 305. https://doi.org/10.3390/universe7080305 https://www.mdpi.com/journal/universe Universe 2021, 7, 305 2 of 63 describe some essential properties of Lie groups, as it was discussed by Cartan [1–4]. Car- tan has been the first who pioneered into the study of torsion, and this is the reason why today torsion is also known as the Cartan tensor. When back at the end of the 19th century, Ricci-Curbastro and Levi–Civita developed absolute differential calculus, or tensor calculus, they did it by assuming zero torsion to simplify computations, and the geometry they eventually obtained was entirely based on the existence of a Riemann metric: this is what we call Riemann geometry. Nothing in this geometry is spoiled by letting torsion take its place in it, the only difference being that now the metric would be accompanied by torsion as the fundamental objects of the geometry: the final setting is what is called Riemann–Cartan geometry. Granted that, from a general mathematical perspective, torsion is present, one may wonder if there can be physical reasons for torsion to be zero. Physical arguments to prove that torsion must equal zero were indeed proposed in the past. However, none of them appeared to be free of fallacies or logical inconsistencies. A complete list with detailed reasons for their failure can be found in [5]. That torsion should not be equal to zero even in physical contexts is again quite general. In fact, by writing the RC geometry in anholonomic bases, the torsion can be seen as the strength of the potential arising from gauging the translation group, much in the same way in which the curvature is the strength of the potential arising from gauging the rotation group, as shown by Sciama and Kibble [6,7]. What Sciama and Kibble proved was that torsion is not just a tensor that could be added, but a tensor that must be added, besides curvature, in order to have the possibility to completely describe translations, besides rotations, in a full Poincaré gauge theory of physics [8]. At the beginning of the 20th century, when Einstein developed his theory of gravity, he did it by assuming zero torsion because, when torsion vanishes, the Ricci tensor is symmetric and therefore it can be consistently coupled to the symmetric energy tensor, realizing the identification between the space–time curvature, and its energy content expressed by Einstein field equations: this is the basic spirit of Einstein gravity. Today, we know that in physics there is also another quantity of interest called spin, and that, in its presence, the energy is no longer symmetric, so nowadays having a non-symmetric Ricci tensor besides a Cartan tensor would allow for a more exhaustive coupling in gravity, where the curvature would still be coupled to the energy but now torsion would be coupled to the spin: such a scheme would realize the identification between the space–time curvature and its energy content expressed by Einstein field equations and the identification between space–time torsion and its spin content expressed by the Sciama–Kibble field equations as the Einstein–Sciama–Kibble torsion gravity. The ESK theory of gravity is thus the most complete theory describing the dynamics of the space–time, and, because torsion is coupled to the spin in the same spirit in which curvature is coupled to energy, then it is the theory of space–time in which the coupling to its matter content is achieved most exhaustively. The central point of the situation is therefore brought to the question asking whether there actually exists something possessing both spin and energy as a form of matter, which can profit from the setting that is provided by the ESK gravity. As a matter of fact, such a theory not only exists, but it is also very well known, the Dirac spinorial field theory. With so much insight, it is an odd circumstance that there be still such a controversy about the role of torsion besides that of curvature in gravity, and there may actually be several reasons for it. The single most important one may be that Einstein gravity was first published in the year 1916 when no spin was known and, despite being then insightful to set the torsion tensor to zero, when Dirac came with a theory of spinors comprising an intrinsic spin in 1928, the successes of Einstein theory of gravity were already too great to make anyone wonder about the possibility of modifying it. Of course, this is no scientific reason to hinder research, but, sociologically, it can be easy to understand why one would not lightly go to look beyond something good, Universe 2021, 7, 305 3 of 63 especially today that the successes of the Einstein theory of gravitation have become practically complete. In the present report, we would like to change this tendency by considering torsion in gravity coupled to spinor fields and showing all advantages that we can get, from theoreti- cal consistency, to phenomenological applications. Thus, in a second chapter, we will investigate the theoretical advantages obtained from the torsion–spin interactions. These will span from the revision of the Hawking–Penrose theorem about the inevitability of gravitational singularity formation to some discussion about the positivity of the energy, passing through the Pauli exclusion principle and the concept of macroscopic approximation. In a third and final chapter, we are going to employ the presented theory to assess some of the known open problems in the standard models of particles and cosmology. One: Fundamental Theory The first chapter will be about presenting the fundamental theory, and it will be divided into three sections: in the first section, we will define all the kinematic quantities and see how they can be dynamically coupled.