Neutrino Anomalies and Chiral Flipping in Anisotropic Einstein-Cartan Cosmology

Recently Palle has investigated the chiral vorticity and Cartan torsion in neutrino asymmetries. In his case he addressed this problem in Goedel s like anisotropic Einstein-Catan cosmology. In this paper we discusse how these ideas applied to sheared Bianchi types I Einstein-Cartan (EC) neutrino amisotropic cosmology, affect the handness of neutrinos in the universe. Actually here a novel concept of the chiral metric is introduced where metric functions also possess two distinct signs as in neutrino flipping or helicity. The axial anomaly equation for neutrinos in the presence of torsion and metric chirality is shown to produce left-handed neutrinos from right-handed torsion. Metric chirality is shown to be able to define how the metric would behave far away of neutrino density. Chiral flipping of the chiral neutrinos in the presence of torsion is also investigated. It is shown that when the chiral torsion is left-handed the chemical chiral potential vanishes as the universe expands.

On generalized P-reducible Finsler manifolds

The class of generalized P-reducible manifolds (briefly GP-reducible manifolds) was first introduced by Tayebi and his collaborates [1]. This class of Finsler manifolds contains the classes of P-reducible manifolds, C-reducible manifolds and Landsberg manifolds. We prove that every compact GP-reducible manifold with positive or negative character is a Randers manifold. The norm of Cartan torsion plays an important role for studying immersion theory in Finsler geometry. We find the relation between the norm of Cartan torsion, mean Cartan torsion, Landsberg and mean Landsberg curvatures of the class of GP-reducible manifolds. Finally, we prove that every GP-reducible manifold admitting a concurrent vector field reduces to a weakly Landsberg manifold.

On a Yamabe Type Problem in Finsler Geometry

In this paper, a newnotion of scalar curvature for a Finslermetric F is introduced, and two conformal invariants Y(M, F) and C(M, F) are deûned. We prove that there exists a Finslermetric with constant scalar curvature in the conformal class of F if the Cartan torsion of F is suõciently small and Y(M, F)C(M, F) < Y(Sn) where Y(Sn) is the Yamabe constant of the standard sphere.

The connections of pseudo-Finsler spaces

We give an introduction to (pseudo-)Finsler geometry and its connections. For most results we provide short and self-contained proofs. Our study of the Berwald nonlinear connection is framed into the theory of connections over general fibered spaces pioneered by Mangiarotti, Modugno and other scholars. The main identities for the linear Finsler connection are presented in the general case, and then specialized to some notable cases like Berwald's, Cartan's or Chern–Rund's. In this way it becomes easy to compare them and see the advantages of one connection over the other. Since we introduce two soldering forms we are able to characterize the notable Finsler connections in terms of their torsion properties. As an application, the curvature symmetries implied by the compatibility with a metric suggest that in Finslerian generalizations of general relativity the mean Cartan torsion vanishes. This observation allows us to obtain dynamical equations which imply a satisfactory conservation law. The work ends with a discussion of yet another Finsler connection which has some advantages over Cartan's and Chern–Rund's.

A Class of Finsler Metrics with Bounded Cartan Torsion

In this paper, we find a class of (α, β) metrics which have a bounded Cartan torsion. This class contains all Randers metrics. Furthermore, we give some applications and obtain two corollaries about curvature of this metrics.

On Strongly Convex Indicatrices in Minkowski Geometry

The geometry of indicatrices is the foundation of Minkowski geometry. A strongly convex indicatrix in a vector space is a strongly convex hypersurface. It admits a Riemannian metric and has a distinguished invariant—(Cartan) torsion. We prove the existence of non-trivial strongly convex indicatrices with vanishing mean torsion and discuss the relationship between the mean torsion and the Riemannian curvature tensor for indicatrices of Randers type.

SPIN-ROTATION COUPLING EFFECTS ON THE STRUCTURE FORMATION IN GÖDEL UNIVERSE AND COBE DATA

A rotating universe represented by the Gödel metric in spacetimes with Cartan torsion is investigated. The spin–torsion coupling is shown to contribute to the structure formation in an apppreciable way only at the early stages of the universe. To demonstrate these conjectures, we have made use of a spinning fluid model in Einstein–Cartan gravity. We also show that autoparallel equation in Riemann–Cartan spacetime leads to the evolution equation of the cosmological perturbation where the spin-rotation coupling contributes to the growth of inhomogeneities through a constant term. From this idea, one is able to estimate that torsion in the Planck era is around 1015 s -1 where COBE constraint on temperature fluctuations is used in this computation. A spin density of the universe of 1053 g · cm -1 s -1 is also obtained in our framework.

COSMIC RELIC TORSION FROM INFLATIONARY COSMOLOGY

An inflationary de Sitter solution of Teleparallel Equivalent of General Relativity (TERG) is obtained. In this model Cartan torsion is shown to be a cosmological relic in the sense that it decays from earlier epochs of the Universe until extremely small values at the present epoch. This would be the reason why it is very difficult to measure cosmological torsion at the present epoch and only extremely small relic torsion would be left. Torsion plays a role similar to the inflaton field in its interaction with the scalar field. The torsion mass is determined from the teleparallel action in terms of the Planck mass. The value of the torsion mass is of the order of Planck mass. An upper limit for torsion of 10-18s-1 is obtained for the de Sitter phase. By considering the Friedmann phase it is possible to show that torsion induces an oscillation on the Universe.

DISTRIBUTIONAL TORSION OF HYBRID DEFECTS

Distributional sources of cosmic walls crossed by cosmic strings are obtained from Riemann–Cartan (RC) geometry. The matter density of the planar wall is maximum at the point where the cosmic string crosses the cosmic wall. Cartan torsion has support on the cosmic string given by the Dirac δ-function. Off the sources we are left with a torsionless vacuum. It is suggested that this hybrid defect geometry with torsion may serve as a model for a spinning string from a quark which ends on an axionic domain wall.