Einstein-Cartan Theory Research Articles

MULTI-COMPONENT BIANCHI I COSMOLOGIES WITH TORSION

The Bianchi I cosmological models containing radiation, stiff fluid and a nonminimally coupled material scalar field with polynomial potentials of the fourth degree are considered in the framework of the Einstein–Cartan theory (ECT). Exact partial solutions are obtained for arbitrary positive values of the coupling constant ξ. The restrictions on ξ are found from the integrability conditions. It is demonstrated that singular models with the asymptotical isotropization by the de Sitter type and by the power-law (t4/3) type at late-times are possible. It is shown that the bouncing models with the isotropization by the de Sitter law of expansion at late-times are admissible. The role of sources in the evolution of models is elucidated.

On the energy-momentum and spin tensors in the Riemann–Cartan space

General classical theories of material fields in an arbitrary Riemann-Cartan space are considered. For these theories, with the help of equations of balance, new non-trivially generalized, manifestly generally covariant expressions for canonical energy-momentum and spin tensors are constructed in the cases when a Lagrangian contains (a) an arbitrary set of tensorial material fields and their covariant derivatives up to the second order, as well as (b) the curvature tensor and (c) the torsion tensor with its covariant derivatives up to the second order. A non-trivial manifestly generally covariant generalization of the Belinfante symmetrization procedure, suitable for an arbitrary Riemann-Cartan space, is carried out. A covariant symmetrized energy-momentum tensor is constructed in a general form.

The energy and momentum of the Universe

The Einstein–Cartan–Sciama–Kibble theory of gravity naturally extends general relativity to include quantum-mechanical, intrinsic angular momentum of matter by equipping spacetime with torsion. Using the Einstein energy–momentum pseudotensor for the gravitational field in this theory, we show that the energy and momentum of the closed Universe are equal to zero. Since the positive energy from mass and motion of the observed matter in the Universe exceeds in magnitude the negative energy from gravity, the Universe must contain another form of matter whose energy is negative. This form, which cannot be composed from particles, might be the observed dark matter.

Cosserat-Cartan Modification of Einstein-Riemann Relativity and Cosmic Dark Energy Density

Based on pioneering works by Sciama and Kibble to extend Einstein-Cartan theory of gravity we give a new derivation for the cosmic energy density. It is argued that the ‘t Hooft-Veltman and Wilson method of renormalization implies the relativity of fractal spacetime at the quantum scale and a dark energy density of E(D) = 95.5 percent. It is further revealed that similar conclusions could be made using A.C. Eringen’s nonlocal elasticity. Finally the wider philosophical implication of the theory is discussed.

Fermion condensate from torsion in the reheating era after inflation

The inclusion of Dirac fermions in Einstein-Cartan gravity leads to a four-fermion interaction mediated by nonpropagating torsion, which can allow for the formation of a Bardeen-Cooper-Schrieffer condensate. By considering a simplified model in $2+1$ spacetime dimensions, we show that even without an excess of fermions over antifermions, the nonthermal distribution arising from preheating after inflation can give rise to a fermion condensate generated by torsion. We derive the effective Lagrangian for the spacetime-dependent pair field describing the condensate in the extreme cases of nonrelativistic and massless fermions, and show that it satisfies the Gross-Pitaevski equation for a gapless, propagating mode.

Nonlocal integral conservation laws and tetradic currents in the Einstein-Cartan theory. II. Problem of the gravitational energy and tetradic currents

Einstein-Cartan gravitational field equations are formed as Maxwell equations: the codifferential of the tetrad differential is equal to the conserved tetradic current. This yields local integral conservation laws for any tetrad field and, in particular, allows solving the old problem of the gravitational energy in the general relativity and Einstein-Cartan theories.

Covariant differential identities and conservation laws in metric-torsion theories of gravitation. II. Manifestly generally covariant theories

The present paper continues the work of Lompay and Petrov [J. Math. Phys. 54, 062504 (2013)] where manifestly covariant differential identities and conserved quantities in generally covariant metric-torsion theories of gravity of the most general type have been constructed. Here, we study these theories presented more concretely, setting that their Lagrangians \documentclass[12pt]{minimal}\begin{document}$\mathscr {L}$\end{document}L are manifestly generally covariant scalars: algebraic functions of contractions of tensor functions and their covariant derivatives. It is assumed that Lagrangians depend on metric tensor g, curvature tensor R, torsion tensor T and its first \documentclass[12pt]{minimal}\begin{document}${\bm {\nabla }}\mathbf {T}$\end{document}∇T and second \documentclass[12pt]{minimal}\begin{document}${\bm {\nabla }}{\bm {\nabla }}\mathbf {T}$\end{document}∇∇T covariant derivatives, besides, on an arbitrary set of other tensor (matter) fields \documentclass[12pt]{minimal}\begin{document}$\bm {\varphi }$\end{document}φ and their first \documentclass[12pt]{minimal}\begin{document}${\bm {\nabla }}\bm {\varphi }$\end{document}∇φ and second \documentclass[12pt]{minimal}\begin{document}${\bm {\nabla }}{\bm {\nabla }}\bm {\varphi }$\end{document}∇∇φ covariant derivatives: \documentclass[12pt]{minimal}\begin{document}$\mathscr {L}= \mathscr {L}(\mathbf {g},\mathbf {R}; \;\mathbf {T},{\bm {\nabla }}\mathbf {T},{\bm {\nabla }}{\bm {\nabla }}\mathbf {T}; \;\bm {\varphi },{\bm {\nabla }}\bm {\varphi },{\bm {\nabla }}{\bm {\nabla }}\bm {\varphi })$\end{document}L=L(g,R;T,∇T,∇∇T;φ,∇φ,∇∇φ). Thus, both the standard minimal coupling with the Riemann-Cartan geometry and non-minimal coupling with the curvature and torsion tensors are considered. The studies and results are as follow: (a) A physical interpretation of the Noether and Klein identities is examined. It was found that they are the basis for constructing equations of balance of energy-momentum tensors of various types (canonical, metrical, and Belinfante symmetrized). The equations of balance are presented. (b) Using the generalized equations of balance, new (generalized) manifestly generally covariant expressions for canonical energy-momentum and spin tensors of the matter fields are constructed. In the cases, when the matter Lagrangian contains both the higher derivatives and non-minimal coupling with curvature and torsion, such generalizations are non-trivial. (c) The Belinfante procedure is generalized for an arbitrary Riemann-Cartan space. (d) A more convenient in applications generalized expression for the canonical superpotential is obtained. (e) A total system of equations for the gravitational fields and matter sources are presented in the form more naturally generalizing the Einstein-Cartan equations with matter. This result, being a one of the more important results itself, is to be also a basis for constructing physically sensible conservation laws and their applications.

Flag-dipole spinor fields in ESK gravities

We consider the Riemann-Cartan geometry as a basis for the Einstein-Sciama-Kibble theory coupled to spinor fields: we focus on f(R) and conformal gravities, regarding the flag-dipole spinor fields, type-(4) spinor fields under the Lounesto classification. We study such theories in specific cases given, for instance, by cosmological scenarios: we find that in such background the Dirac equation admits solutions that are not Dirac spinor fields, but in fact the aforementioned flag-dipoles ones. These solutions are important from a theoretical perspective, as they evince that spinor fields are not necessarily determined by their dynamics, but also a discussion on their structural (algebraic) properties must be carried off. Furthermore, the phenomenological point of view is shown to be also relevant, since for isotropic Universes they circumvent the question whether spinor fields do undergo the Cosmological Principle.

Is Einstein-Cartan Theory Coupled to Light Fermions Asymptotically Safe?

The difference between Einstein's general relativity and its Cartan extension is analyzed within the scenario of asymptotic safety of quantum gravity. In particular, we focus on the four-fermion interaction which distinguishes the Einstein-Cartan theory from its Riemannian limit.

Nonlocal integral conservation laws and tetradic currents in the Einstein-Cartan theory I. Nonlocal conservation laws

Nonlocal integral conservation laws in the Einstein-Cartan theory for the energy-momentum tensor of general-form sources, in essence being an integral equivalent of convoluted Bianchi identities, are obtained.

Signature transition in Einstein–Cartan cosmology

In the context of Einstein–Cartan theory of gravity, we consider a Friedmann–Lemaitre–Robertson–Walker (FLRW) cosmological model with Weyssenhoff perfect fluid. We focus attention on those classical solutions that admit a degenerate metric in which the scale factor has smooth behavior in the transition from a Euclidean to a Lorentzian domain. It is shown that the spin–spin contact interaction enables one to obtain such a signature changing solutions due to the Riemann–Cartan (U4) structure of space–time.

A study of plane torsion waves in the Poincaré gauge theory of gravity

The variational equations of the gravitational field in Riemann-Cartan space are derived in the external forms formalism using the method of Lagrange indefinite multipliers for the Poincare gauge theory of gravity with quadratic Lagrangians in the general form. The structure of the irreducible parts of torsion propagating in the form of plane waves in Riemann-Cartan spacetime is investigated.

Scalar torsion and a new symmetry of general relativity

We reformulate the general theory of relativity in the language of Riemann-Cartan geometry. We start from the assumption that the space-time can be described as a non-Riemannian manifold, which, in addition to the metric field, is endowed with torsion. In this new framework, the gravitational field is represented not only by the metric, but also by the torsion, which is completely determined by a geometric scalar field. We show that in this formulation general relativity has a new kind of invariance, whose invariance group consists of a set of conformal and gauge transformations, called Cartan transformations. These involve both the metric tensor and the torsion vector field, and are similar to the well known Weyl gauge transformations. By making use of the concept of Cartan gauges, we show that, under Cartan transformations, the new formalism leads to different pictures of the same gravitational phenomena. We show that in an arbitrary Cartan gauge general relativity has the form of a scalar-tensor theory. In this approach, the Riemann-Cartan geometry appears as the natural geometrical setting of the general relativity theory when the latter is viewed in an arbitrary Cartan gauge. We illustrate this fact by looking at the one of the classical tests of general relativity theory, namely the gravitational spectral shift. Finally, we extend the concept of space-time symmetry to the more general case of Riemann-Cartan space-times endowed with scalar torsion. As an example, we obtain the conservation laws for auto-parallel motion in a static spherically symmetric vacuum space-time in a Cartan gauge, whose orbits are identical to Schwarzschild orbits in general relativity.

Covariant differential identities and conservation laws in metric-torsion theories of gravitation. I. General consideration

Arbitrary diffeomorphically invariant metric-torsion theories of gravity are considered. It is assumed that Lagrangians of such theories contain derivatives of field variables (tensor densities of arbitrary ranks and weights) up to a second order only. The generalized Klein-Noether methods for constructing manifestly covariant identities and conserved quantities are developed. Manifestly covariant expressions are constructed without including auxiliary structures like a background metric. In the Riemann-Cartan space, the following manifestly generally covariant results are presented: (a) The complete generalized system of differential identities (the Klein-Noether identities) is obtained. (b) The generalized currents of three types depending on an arbitrary vector field displacements are constructed: they are the canonical Noether current, symmetrized Belinfante current, and identically conserved Hilbert-Bergmann current. In particular, it is stated that the symmetrized Belinfante current does not depend on divergences in the Lagrangian. (c) The generalized boundary Klein theorem (third Noether theorem) is proved. (d) The construction of the generalized superpotential is presented in detail, and questions related to its ambiguities are analyzed.

THE RELATION BETWEEN THE MODEL OF A CRYSTAL WITH DEFECTS AND PLEBANSKI'S THEORY OF GRAVITY

In the present investigation, we show that there exists a close analogy of geometry of space–time in general relativity (GR) with a structure of defects in a crystal. We present the relation between the Kleinert's model of a crystal with defects and Plebanski's theory of gravity. We have considered the translational defects — dislocations and the rotational defects — disclinations — in the three- and four-dimensional crystals. The four-dimensional crystalline defects present the Riemann–Cartan space–time which has an additional geometric property — "torsion" — connected with dislocations. The world crystal is a model for the gravitation which has a new type of gauge symmetry: the Einstein's gravitation has a zero torsion as a special gauge, while a zero connection is another equivalent gauge with nonzero torsion which corresponds to the Einstein's theory of "teleparallelism". Any intermediate choice of the gauge with nonzero connection [Formula: see text] is also allowed. In the present investigation, we show that in the Plebanski formulation the phase of gravity with torsion is equivalent to the ordinary or topological gravity, and we can exclude a torsion as a separate dynamical variable.

GRAVITY AND MIRROR GRAVITY IN PLEBANSKI FORMULATION

We present several theories of four-dimensional gravity in the Plebanski formulation, in which the tetrads and the connections are the independent dynamical variables. We consider the relation between different versions of gravitational theories: Einsteinian, "topological," "mirror" gravities and gravity with torsion. We assume that our world, in which we live, is described by the self-dual left-handed gravity, and propose that if the Mirror World exists in Nature, then the "mirror gravity" is the right-handed antiself-dual gravity. In this connection, we give a brief review of gravi-weak unification models. In accordance with cosmological measurements, we consider the Universe with broken mirror parity. We also discuss the problems of cosmological constant and communication between visible and mirror worlds. Investigating a special version of the Riemann–Cartan space–time, which has torsion as an additional geometric property, we have shown that in the Plebanski formulation the ordinary and dual "topological" sectors of gravity, as well as the gravity with torsion, are equivalent. Equations of motion are obtained. In this context, we have also discussed a "pure connection gravity" — a diffeomorphism-invariant gauge theory of gravity. Loop Quantum Gravity is also briefly reviewed.

On Poincaré gauge theory of gravity, its equations of motion, and Gravity Probe B

Ever since E.Cartan in the 1920s enriched the geometric framework of general relativity (GR) by introducing a {\it torsion} of spacetime, the question arose whether one could find a measurement technique for detecting the presence of a torsion field. Mao et al.(2007) claimed that the rotating quartz balls in the gyroscopes of the Gravity Probe B experiment, falling freely on an orbit around the Earth, should "feel" the torsion. Similarly, March et al.(2011) argue with the precession of the Moon and the Mercury and extend later their considerations to the Lageos satellite.--- A consistent theory of gravity with torsion emerged during the early 1960's as gauge theory of the Poincar\'e group. This Poincar\'e gauge theory of gravity incorporates as simplest viable cases the Einstein-Cartan(-Sciama-Kibble) theory (EC), the teleparallel equivalent GR|| of GR, and GR itself. So far, PG and, in particular, the existence of torsion have {\it not} been experimentally confirmed. However, PG is to be considered as the standard theory of gravity with torsion because of its very convincing gauge structure.--- Since the early 1970s up to today, different groups have shown more or less independently that torsion couples only to the {\it elementary particle spin} and under no circumstances to the orbital angular momentum of test particles. This is established knowledge and we reconfirm this conclusion by discussing the energy-momentum law of PG, which has same form for all versions of PG. Therefore, we conclude that, unfortunately, the investigations of Mao et al. and March et al. do not yield any information on torsion.

Einstein–Cartan gravity, Asymptotic Safety, and the running Immirzi parameter

In this paper we analyze the functional renormalization group flow of quantum gravity on the Einstein–Cartan theory space. The latter consists of all action functionals depending on the spin connection and the vielbein field (co-frame) which are invariant under both spacetime diffeomorphisms and local frame rotations. In the first part of the paper we develop a general methodology and corresponding calculational tools which can be used to analyze the flow equation for the pertinent effective average action for any truncation of this theory space. In the second part we apply it to a specific three-dimensional truncated theory space which is parametrized by Newton’s constant, the cosmological constant, and the Immirzi parameter. A comprehensive analysis of their scale dependences is performed, and the possibility of defining an asymptotically safe theory on this hitherto unexplored theory space is investigated. In principle Asymptotic Safety of metric gravity (at least at the level of the effective average action) is neither necessary nor sufficient for Asymptotic Safety on the Einstein–Cartan theory space which might accommodate different “universality classes” of microscopic quantum gravity theories. Nevertheless, we do find evidence for the existence of at least one non-Gaussian renormalization group fixed point which seems suitable for the Asymptotic Safety construction in a setting where the spin connection and the vielbein are the fundamental field variables.

Einstein–Cartan–Dirac theory in (1+2)-dimensions

Einstein–Cartan theory is formulated in (1+2) dimensions using the algebra of exterior differential forms. A Dirac spinor is coupled to gravity and the field equations are obtained by a variational principle. The space–time torsion is found to be given algebraically in terms of a quadratic spinor condensate field. Circularly symmetric, exact solutions that collapse to AdS 3 geometry in the absence of the Dirac condensate are found.

Galactic dynamo seeds from f-meson dominance gravity with torsion and early universe dynamos from Einstein-Cartan gravity

Galactic dynamo seeds from f-meson dominance gravity with torsion and early universe dynamos from Einstein-Cartan gravity