Cosmological consequences of gravity with spin and torsion

Abstract

We show that the Einstein-Cartan-Sciama-Kibble theory of gravity with torsion not only extends general relativity to account for the intrinsic spin of matter, but it may also eliminate major problems in gravitational physics and answer major questions in cosmology. These problems and questions include: the origin of the Universe, the existence of singularities in black holes, the nature of inflation and dark energy, the origin of the matter-antimatter asymmetry in the Universe, and the nature of dark matter.The big-bang cosmology, based on Einstein's general theory of relativity (GR), successfully describes primordial nucleosynthesis and predicts the cosmic microwave background radiation. It does not, however, address some fundamental questions. What is the origin of the rapid expansion of the Universe from an initial, extremely hot and dense state? What is the nature of dark energy? And what caused the observed asymmetry between matter and antimatter in the Universe? The validity of GR also breaks inside black holes and at the beginning of the big bang, where the matter is predicted to compress indefinitely to curvature singularities. The big-bang cosmology also requires an inflationary scenario with additional fields to explain why the present Universe appears spatially flat, homogeneous and isotropic. In this essay, we briefly describe how the solution to the above questions and problems may come from an old adaptation of GR, the Einstein-Cartan-Sciama-Kibble (ECSK) theory of gravity [1][2][3]. This classical theory naturally extends GR to account for the quantum-mechanical, intrinsic angular momentum (spin) of elementary particles that compose gravitating matter, causing spacetime to exhibit a geometric property called torsion.The ECKS gravity is based on the gravitational Lagrangian density proportional to the curvature scalar, as in GR. This theory, however, removes the GR restriction of the affine connection Γ k i j be symmetric. Instead, the antisymmetric part of the connection S k ij = Γ k [i j] (torsion tensor) is regarded as a dynamical variable like the metric tensor g ik . Varying the total action for the gravitational field and matter with respect to the metric gives the Einstein equations that relate the curvature to the dynamical energy-momentum tensor T ik = (2/ √ −g)δL/δg ik , where L is the matter Lagrangian density and g = det(g ik ) [4]. These equations can be written in a GR form as G ik = κ(T ik + U ik ), where the source is the modified energy-momentum tensor with an additional term U ik quadratic in the torsion tensor. G ik is the standard Einstein tensor and κ = 8πG/c 4 . These equations can also be written as R ik − R j j g ik /2 = κθ ik , where R ik (Γ) is the Ricci tensor of the connection and θ ik is the canonical energy-momentum tensor.Varying the total action with respect to the torsion gives the Cartan equations [1-3],

Renormalizability of the Dirac equation in torsion gravity with nonminimal coupling

“…We will prove that such non-minimal couplings will even be super-renormalizable. Moreover, for consistency with the existing literature, we will show that the presented theory preserves the results on the avoidance of cosmological singularities, which have been already established [3,[5][6][7][8][9][10]. The paper will be divided in the following way: Section II is dedicated to the description of the theory and its general properties, particularly the conservation laws; Section III is instead dedicated to some interesting aspects of the theory such as renormalizability and parity violation; Section IV is devoted to cosmological considerations; Section V is dedicated to the summary of the results, and an overall discussion as conclusion.…”

Collapse and dispersal of a homogeneous spin fluid in Einstein–Cartan theory

“…The spin effects can be negligible at the early stages of the collapse, while as the collapse proceeds, these effects would play a significant role in the final fate of the scenario of the collapse. This situation can be compared to the singularity removal for a FLRW spacetime in the very early universe, as we go backwards in time [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. We showed that in contrast to the homogeneous dust collapse which leads inevitably to the formation of a spacetime singularity, the occurrence of such a singularity is avoided and instead a bounce occurs at the end of the contracting phase.…”

“…Substituting for the torsion tensor, from Eq. (5), into these terms we get the combined field equations [38,39,53,54,[62][63][64][65][66]:…”

“…One such framework, within which the inclusion of spin effects of fermions can be worked out and thus will allow non-trivial dynamical consequences to be extracted is the Einstein-Cartan (EC) theory [35][36][37][38][39]. Within this context, many cosmological models have been found in which the unphysical big bang singularity is replaced with a bounce at a finite amount the scalar factor [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. From another perspective, the research of the recent years has shown that in the final stages of a typical scenario of the collapse where a high energy regime governs, the effects of quantum gravity would regularize the singularity that happens in the classical model [55,56].…”

Kinematics of Einstein-Cartan universes

“…When the space has torsion, the generalised Bianchi identities are also known as the Weitzenbock identities and take the form (e.g. see [20])…”