Einstein-Cartan Theory Research Articles

Comparison of String Fluid Dynamical Models in General Relativity and Einstein–Cartan theory

Dynamical models of string fluids areconstructed from the general energy-momentum tensor forstring fluids in general relativity and theEinstein-Cartan theories obtained from the Ray-Hilbertvariational principle. Examples of solutions to the fieldequations for general relativistic spacetimes are givenand compared with solutions obtained from the postulatedenergy-momentum tensor of Letelier. Solutions to the field equations in Riemann-Cartanspacetimes are compared with an extended Leteliersolution. All calculations are given for both thestandard and the extended thermodynamics versions inwhich the latter includes the string as thermodynamicvariables. In general relativity, it is shown for blackhole solutions that the general feature of strings(through the string vector) is to produce a shrinkage of the black hole horizon. In RiemannCartanspacetimes, the torsion field equation shows that stringvector can be identified with the torsion vector. Themost striking feature of strings in Riemann-Cartan spacetimes is that in the Reissner-Nordstromsolution, the addition of torsional strings produces thecorrect asympototic behavior of the metric necessary tomatch the experimental galactic rotationcurves.

String fluid dynamical models in the Einstein-Cartan theory

Using the general, energy-momentum tensor for a dynamical, string fluid developed from the Ray-Hilbert variational principle, physically motivated applications to Riemann-Cartan spacetime are given within the framework of the Einstein-Cartan theory. The string density is considered for the cases when it is or is not a thermodynamical variable of the fluid. Solutions are given for both cases and compared with other models in both general relativity and the Einstein-Cartan theory. The use of a string fluid as a contributer to galactic halo density is also discussed.

Torsion dependence of the covariant Taylor expansion in Riemann-Cartan space

We clarify the dependence on the torsion tensor of all coefficients in the covariant Taylor expansion of the HMDS coefficient in the heat kernel for the fermion of spin in Riemann-Cartan space. The dependence on the torsion tensor of the quantities in the expansion coefficients is determined by the matrices in the quantities, except for that of the matrix potential X itself. The quantities with matrices contain the torsion tensor only in a peculiar form, and the others are independent of the torsion tensor.

Plane torsion waves in quadratic gravitational theories in Riemann-Cartan space

The definition of the Riemann-Cartan space of plane-wave type is given. The condition under which the torsion plane waves exist is found. It is expressed in the form of the restriction imposed on the coupling constants of the 10-parametric quadratic gravitational Lagrangian. In the mathematical appendix the master formula for the commutator of the variation operator and Hodge operator is proved. This formula is applied for the variational procedure when the gravitational field equations are obtained in terms of the exterior differential forms.

SPHERICALLY SYMMETRIC SOLUTION FOR TORSION AND THE DIRAC EQUATION IN 5D SPACETIME

Torsion in a 5D spacetime is considered. In this case, gravitation is defined by the 5D metric and the torsion. It is conjectured that torsion is connected with a spinor field. In this case, Dirac's equation becomes the nonlinear Heisenberg equation. It is shown that this equation has a discrete spectrum of solutions with each solution being regular on the whole space and having finite energy. Every solution is concentrated on the Planck region and hence we can say that torsion should play an important role in quantum gravity in the formation of bubbles of spacetime foam. On the basis of the algebraic relation between torsion and the classical spinor field in Einstein–Cartan gravity, the geometrical interpretation of the spinor field is considered as "the square root" of torsion.

Can Torsion Play a Role in Angular Momentum Conservation Law?

In Einstein-Cartan theory, by the use of thegeneral Noether theorem, the general covariantangular-momentum conservation law is obtained withrespect to the local Lorentz transformations. Thecorresponding conservative Noether current is interpreted asthe angular momentum tensor of the gravity-matter systemincluding the spin density. It is pointed out that,assuming the tetrad transformation given by eq. (15), then torsion does not play a role in theconservation law of angular momentum.

CHIRAL ANOMALY IN ASHTEKAR'S APPROACH TO CANONICAL GRAVITY

The Dirac equation in Riemann–Cartan spacetimes with torsion is reconsidered. As is well-known, only the axial covector torsion A, a one-form, couples to massive Dirac fields. Using diagrammatic techniques, we show that besides the familiar Riemannian term only the Pontrjagin type four-form dA ∧ dA does arise additionally in the chiral anomaly, but not the Nieh–Yan term d* A, as has been claimed recently. Implications for cosmic strings in Einstein–Cartan theory as well as for Ashtekar's canonical approach to quantum gravity are discussed.

Curvature-free asymmetric metric connections and Lanczos potentials in Kerr–Schild spacetimes

The set of spinor equations linking the curvature spinors of a Riemann–Cartan space with the curvature spinors of the corresponding Riemann space is derived, and these are used to establish a simple relationship between Ψ-flat connections of the Riemann–Cartan space and Lanczos potentials of the Riemann space. This not only yields, very easily, a recent result of Bergqvist for the Kerr metric, but also enables Bergqvist’s result to be generalized; specifically we show that a curvature-free connection, associated with a class of Kerr–Schild metrics, can be identified as a Lanczos potential for the Weyl conformal curvature spinor of these spaces.

INDUCED MASSES AND SPIN SELF-INTERACTION FROM FIVE-DIMENSIONAL EINSTEIN–CARTAN–DIRAC THEORY

The Einstein–Cartan theory with a dilaton field coupling arising from five-dimensional principal fiber bundle structure of the metric is investigated. A Dirac type spinor field is coupled to the metric and to translational gauge fields. After the dimensional reduction, the theory introduces a gravitational spin–spin contact interaction coming from the translational degrees of freedom, which is repulsive between Dirac particles whose spins are aligned and attractive when spins are opposed. A classically stable ground state for the dilaton field exists, which is infinitely degenerate with respect to a rescaling of the metric. The induced mass term in the Dirac equation can be interpreted, e.g. as the bare electron mass, as long as we regard this theory as an effective "medium" energy model coming from finite string field theories.

Inflation in Bianchi type-IX Einstein-Cartan cosmological model

Within the framework of Einstein-Cartan theory with Weyssenhoff fluid, we investigate an inflation model for an anisotropic Bianchi type-IX cosmological model. The system of field equations is solved numerically and an inflation epoch is achieved. The general condition for the inflation to occur is also discussed. This anisotropic model evolves quickly towards to an isotropic one.

LETTER: String Theory and an Integrable Model in Two-Dimensional Gravity with Dynamical Torsion

The integrability character of an integrable model of two-dimensional gravity with bosonic string coupling in Riemann-Cartan space is studied in the present letter. The equations of motion in the model are reduced to a nonlinear integrable equation. The general numerical solutions of this equation are found. In addition, the exact solution of scalar curvature is obtained.

Non-Riemannian geometry of macroscopic spin distributions

The non-Riemannian geometry of macroscopic spin distributions in thermodynamics and ferromagnetism is obtained from the respective partition functions. An expression for the Cartan torsion in terms of the chemical potential is obtained. Analogies with the Einstein-Cartan theory of gravitation are discussed. From the partition function of ferromagnetism a spin-torsion relation analogous to the one obtained in Einstein-Cartan theory is given where piezomagnetic effects are taken into account.

Construction of Real Space{Time from Complex Linear Metric Connections

A construction of real space-time based on metric linear connections in a complex manifold is described. The construction works only in two or four dimensions. The four-dimensional case based on a connection reducible to group U(2, 2) can generate Riemann-Cartan geometry on the real submanifold of the original complex manifold. The possibility of connecting the appearance of Dirac fields with anholonomic complex frames is discussed.

On certain relationships between cosmological observables in the Einstein-Cartan gravity

We show that if one takes into account Penrose’s conformai technique in the Einstein-Cartan gravity, it is possible to obtain a relation between Hubble’s expansion and Birch’s rotation of the Universe at space-like infinity. This relation also leads to the final reduction of the gravitational coupling to a dimensionless quantity of order unity at space-like infinity. Initial-value equations at space-like infinity in the Einstein-Cartan gravity can be used as a starting point to study the structure formation and evolution of the Universe with Ω = 1. Current measurements of the expansion and rotation of the Universe favour the massive spinning particle as a candidate particle for dark matter.

Kaluza-Klein theory and Dirac equation in higher dimensional Riemann-Cartan space

The higher dimensional Kaluza-Klein theory in Riemann-Cartan space is discussed. To clarify its implications, we investigate the simplest five-dimensional case of the theory in detail. The Einstein-like, Maxwell, and Dirac equations in four-dimensional space-time are obtained by reducing the corresponding five-dimensional field equations. The effect of spin-spin interaction induced by torsion is revealed by analyzing the Dirac equation in this case.

A generalized Lichnerowicz formula, the Wodzicki residue and gravity

We prove a generalized version of the well-known Lichnerowicz formula for the square of the most general Dirac operator ovtilde| D on an even-dimensional spin manifold associated to a metric connection ovtilde|▿. We use this formula to compute the subleading term Φ 1(x, x, D 2) of the heat-kernel expansion of ovtilde| D 2. The trace of this term plays a key role in the definition of a (euclidian) gravity action in the context of non-commutative geometry. We show that this gravity action can be interpreted as defining a modified euclidian Einstein-Cartan theory.

New Approach to the U(2,2)-Symmetry in Spinor and Gravitation Theory

Generally-relativistic Dirac equation is reinterpreted as an approximation to the more fundamental model based on the quadruplet of Klein-Gordon particles with the internal hermitian metric of the neutral signature (++−−). Compensating field of the local group U(2,2) of internal symmetries splits under the GL(2, C)-reduction into gravitational co-tetrad, Einstein-Cartan-Weyl affine connection, and an auxiliary co-tetrad-like object. Equation for the matter field shows under the SL(2,C)-reduction a reasonable correspondence with the generally relativistic Dirac theory. Masses of fermions may be dynamically generated by non vanishing “vacuum” values of geometrodynamical quantities. There is also a convincing correspondence between Yang-Mills equations for the U(2,2)-gauge field and the Einstein-Cartan theory, or more general metric-affine theories of gravitation. In our approach the tetrad field is neither a reference frame nor a compensating field of translations in originally Minkowskian space, but a part of the U(2,2)-gauge multiplet.

Rodichev condition on torsion, spinors, and conservation laws in Riemann-Cartan space

We examine the consequences of the Rodichev condition of complete antisymmetry of the torsion tensor in an orthonormal reference frame, which is equivalent to the condition of the existence of a pair of mutual spinor connectivities. It is shown that it makes possible a reformulation of general theorems about the relationship between first integrals of the geodesic equations and the Killing vectors, conservation laws for the symmetric energy—momentum tensor, and the Lagrangian formalism for a spinor field in the case of Riemann-Cartan space in the effective Riemann spacetime with an external torsion field.

WEYL NON-ABELIAN GAUGE FIELD

It is shown that the congruent transference introduced by Weyl in 1921 defines a non-Abelian gauge field. The simplest gauge-invariant equations are proposed for this field. Its relation with the Riemann–Cartan geometry is also discussed.

Phase transition in conformally induced gravity with torsion.

We have considered the quantum behavior of a conformally induced gravity in the minimal Riemann-Cartan space. The regularized one-loop effective potential considering the quantum fluctuations of the dilaton and the torsion fields in the Coleman-Weinberg sector gives a sensible phase transition for an inflationary phase in De Sitter space. For this effective potential, we have analyzed the semi-classical equation of motion of the dilaton field in the slow-rolling regime.