International Journal of Geometric Methods in Modern Physics

I give a geometrical description of conformal gauge gravitational theory (CGTG) from the viewpoint of symmetries and affine structure. In the frames of the CGTG incorporating gravitation with torsion space-time into Standard Model of electro-weak interaction (EWI) the multi-muon events produced at the Fermilab Tevatron collider were studied. The CGTG gives the value of the torsion pseudotrace–spinor (muon) universal coupling f_T = 4.388 ⋅ 10^-17 G_F, and with limits from known experiments — torsion mass m_T = 0.4700 ⋅ 10^-7 eV or m_T = 0.445 ⋅ 10^-15 muon mass. So the value of the constant of effective four-fermions interaction f_T/m_T = 0.988, indeed may lead to multi-muon events production. The model of interaction of quantum oscillator with the tensor potential W_μν of traceless part of the torsion lead to 2 cm displacement of quark–lepton system as a whole in the magnetic field of collider in accordance with a significant sample of events related to production and decay in which at least one of the muon candidates is produced outside of the beam pipe of radius 1.5 cm. A traceless part of the torsion in CGTG does not vanish in the Newtonian limit of nonzero mass. Torsion gravity potential W_μν gives conservation of a special conformal current and may be produced in the condition of a spontaneous breaking of gauge symmetry where the gravitation mass M_X defect is 1–3 Tev ⋅ c^-2 or 10^-13M_X. This effect may be possible at known effects on top pair asymmetries at the Tevatron and LHC and takes place as the known energy dissipation above 3 TeV of the Galaxy gamma-ray and neutrino spectrum from two bubbles outside the Galaxy plane.

Extended f(R) theories of gravity have been investigated from the symmetry point of view. We briefly has been investigated Noether symmetry of two types of extended f(R) theories: f(R, T) theory, in which curvature is coupled non-minimally to the trace of energy–momentum tensor T_μν and mimetic f(R) gravity, a theory with a scalar field degree of freedom, but ghost-free and with internal conformal symmetry. In both cases we write point-like Lagrangian for flat Friedmann–Lemaitre–Robertson–Walker (FLRW) cosmological background in the presence of ordinary matter. We have been shown that some classes of models existed with Noether symmetry in these viable extensions of f(R) gravity. As a motivated idea, we have been investigating the stability of the solutions and the bouncing and ΛCDM models using the Noether symmetries. We have been shown that in mimetic f(R) gravity bouncing and ΛCDM solutions are possible. Also a class of solutions with future singularities has been investigated.

We study an effective Einstein–Finsler theory on tangent Lorentz bundle constructed as a "minimal" extension of general relativity. Black ring and Kerr-like ellipsoid exact solutions and solitonic configurations are presented. In this endeavor the relevant metric depends not only on four-dimensional spacetime coordinates and also on velocity type variables that can be interpreted as additional coordinates in the space of "extra dimensions".

In General Relativity (GR), the analysis of electric and magnetic Weyl tensors has been studied by various authors. The present study deals with cylindrically symmetric relativistic fluids in GR characterized by the vanishing of magnetic Weyl tensor-purely electric (PE) fields. A very new assumption has been adapted to solve the Einstein's equations and the obtained solution is shearing at all. We signified the importance of PE fields in the context of expansion scalar, energy density, shear and acceleration.

A new local gauge invariant method is introduced in order to maximally simplify the expression for a SU(2) non-Abelian field strength. The new tetrads introduced in previous works are going to play a fundamental role in the algorithm presented in this paper. Three new local gauge invariant objects are going to guide us through the process of making a field strength block diagonal. The process is also covariant. Any nontrivial isospace field strength projection will become block diagonal through this gauge invariant algorithm. As an application we will find new local observables in Yang–Mills theories.

In this paper, we study the Ricci directional curvature defined by H. Akbar-Zadeh in Finsler geometry and obtain the formula of Ricci directional curvature for Randers metrics. Let F = α + β be a Randers metric on a manifold M, where is a Riemannian metric and β = b_iyⁱ is a closed 1-form on M. We prove that F is a generalized Einstein metric if and only if it is a Berwald metric.

We examine the Kaluza–Klein (KK) dimensional reduction from higher dimensional space-time and the properties of the resultant Bergmann–Wagoner general action of scalar–tensor theories. With the analysis of the perturbations, we also investigate the stability of the anti-de Sitter (AdS) space-time in the (D ∈ 𝒩)-dimensional Einstein gravity with the negative cosmological constant. Furthermore, we derive the conditions for the dimensional reduction to successfully be executed and present the KK compactification mechanism.

Tensor operators as the irreducible submodules corresponding to the adjoint representation of the quantum algebra Ŭ_q(su₂) are equipped with q-analogue of the Hilbert–Schmidt scalar product by using the Wigner–Eckart theorem. Then, it is used to show that the adjoint representation of the quantum algebra Ŭ_q(su₂) is a *-representation.

In this paper, we consider rotating black hole in three dimensions with a scalar charge and discuss about the jet-quenching parameter. We find that the effect of scalar charge is decreasing and rotation parameter is increasing the value of the jet-quenching parameter.

The main purpose of this paper is to investigate parallel frames of non-lightlike curves in Minkowski space-time. For this purpose, the relations between Frenet–Serret frame and parallel frame of a given non-lightlike curve are discussed in four different cases. Then, the curvature, first and second torsion functions κ(s), τ(s), σ(s) and principal curvature functions k₁(s), k₂(s), k₃(s) are expressed in terms of the angles α, β and θ.

Semiclassical systems being symmetric under Lie group are studied. A state of a semiclassical system may be viewed as a set (X, f) of a classical state X and a quantum state f in the external classical background X. Therefore, the set of all semiclassical states may be considered as a bundle (semiclassical bundle). Its base {X} is the set of all classical states, while a fiber is a Hilbert space ℱ_X of quantum states in the external background X. Symmetry transformation of a semiclassical system may be viewed as an automorphism of the semiclassical bundle. Automorphism groups can be investigated with the help of sections of the bundle: to any automorphism of the bundle one assigns a transformation of section of the bundle. Infinitesimal properties of transformations of sections are investigated; correspondence between Lie groups and Lie algebras is discussed. For gauge theories, some points of the semiclassical bundle are identified: a gauge group acts on the bundle. For this case, only gauge-invariant sections of the semiclassical bundle are taken into account.

Fix ξ a unitary vector field on a Riemannian manifold M and γ a non-geodesic Frenet curve on M satisfying the Rytov law of polarization optics. We prove in these conditions that γ is a Legendre curve for ξ if and only if the γ-Fermi–Walker covariant derivative of ξ vanishes. The cases when γ is circle or helix as well as ξ is (conformal) Killing vector filed or potential vector field of a Ricci soliton are analyzed and an example involving a three-dimensional warped metric is provided. We discuss also K-(para)contact, particularly (para)Sasakian, manifolds and hypersurfaces in complex space forms.

We derive the Gauss–Codazzi equation in the holonomy and plane-angle representations and we use the result to write a Gauss–Codazzi equation for a discrete (2 + 1)-dimensional manifold, triangulated by isosceles tetrahedra. This allows us to write operators acting on spin network states in (2 + 1)-dimensional loop quantum gravity, representing the 3-dimensional intrinsic, 2-dimensional intrinsic, and 2-dimensional extrinsic curvatures.

In this paper for a given Banach, possibly infinite dimensional, manifold M we focus on the geometry of its iterated tangent bundle T^rM, r ∈ ℕ ∪ {∞}. First we endow T^rM with a canonical atlas using that of M. Then the concepts of vertical and complete lifts for functions and vector fields on T^rM are defined which they will play a pivotal role in our next studies i.e. complete lift of (semi)sprays. Afterward we supply T^∞M with a generalized Fréchet manifold structure and we will show that any vector field or (semi)spray on M, can be lifted to a vector field or (semi)spray on T^∞M. Then, despite of the natural difficulties with non-Banach modeled manifolds, we will discuss about the ordinary differential equations on T^∞M including integral curves, flows and geodesics. Finally, as an example, we apply our results to the infinite-dimensional case of manifold of closed curves.

In view of the many quantum field theoretical descriptions of graphene in 2 + 1 dimensions, we present another field theoretical feature of graphene, in the presence of defects. Particularly, we shall be interested in gapped graphene in the presence of a domain wall and also for superconducting graphene in the presence of a vortex. As we explicitly demonstrate, the gapped graphene electrons that are localized on the domain wall are associated with four N = 2 one-dimensional supersymmetries, with each pair combining to form an extended N = 4 supersymmetry with non-trivial topological charges. The case of superconducting graphene is more involved, with the electrons localized on the vortex being associated with n one-dimensional supersymmetries, which in turn combine to form an N = 2n extended supersymmetry with non-trivial topological charges. As we shall prove, all supersymmetries are unbroken, a feature closely related to the number of the localized fermions and also to the exact form of the associated operators. In addition, the corresponding Witten index is invariant under compact and odd perturbations.

Interpreting the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we reconsider the investigation of P–V critical behaviors of (1 + n)-dimensional AdS black holes in Lovelock–Born–Infeld gravity. In particular, we derive an explicit expression of the universal number in terms of the space dimension n. Then, we examine the phase transitions at the critical points of such black holes for 6 ≤ n < 11 as required by the physical condition of the thermodynamical quantities including criticality behaviors. More precisely, the Ehrenfest equations have been checked and they reveal that the black hole system undergoes a second phase transition at the critical points.

Mach's principle is surely one of those tantalizingly beautiful concepts in physics which remain elusive. Though general Relativity (GR) was conceived in the spirit of realizing it, the theory failed to fulfill this expectation. Here a study on the implications of imposing Mach's principle on GR with an insight that spacetime has no independent existence without a material background, is presented. This inclusion of the principle in GR turns out to be unexpectedly rewarding. The resulting theory solves many mysteries and averts lingering problems of the conventional forms of GR and cosmology.

A well-known recurrence relation for the 6j-symbol of the quantum group su_q(2) is realized as a tridiagonal, symmetric eigenvalue problem. This formulation can be used to implement an efficient numerical evaluation algorithm, taking advantage of existing specialized numerical packages. For convenience, all formulas relevant for such an implementation are collected in Appendix A. This realization is a byproduct of an alternative proof of the recurrence relation, which generalizes a classical (q = 1) result of Schulten and Gordon and uses the diagrammatic spin network formalism of Temperley–Lieb recoupling theory to simplify intermediate calculations.

In this paper, we explore classes of irrotational-fluid cosmological models in the context of f(R)-gravity in an attempt to put some theoretical and mathematical restrictions on the form of the f(R) gravitational Lagrangian. In particular, we investigate the consistency of linearized dust models for shear-free cases as well as in the limiting cases when either the gravito-magnetic or gravito-elecric components of the Weyl tensor vanish. We also discuss the existence and consistency of classes of non-expanding irrotational spacetimes in f(R)-gravity.

We obtain interacting holographic dark energy density in the framework of vector field cosmology (LIV). We consider possible modification of equation of state for the holographic energy density in lorentz invariance violation cosmology. In this case we select Jeans length as the IR cut-off in the holographic model. Then we consider the interaction between holographic energy densities ρ_Λ and ρ_m in this framework.

The Noether gauge symmetries of geodesic Lagrangian for the pp-wave spacetimes are determined in each of the Noether gauge symmetry classes of the pp-wave spacetimes. It is shown that a type N pp-wave spacetime can admit at most three Noether gauge symmetry, and furthermore the number of Noether gauge symmetries turn out to be four, five, six, seven and eight. We found that all conformally flat plane wave spacetimes admit the maximal, i.e. 10, Noether gauge symmetry. Also it is found that if the pp-wave spacetime is non-conformally flat plane wave, then the number of Noether gauge symmetry is nine or 10. By means of the obtained Noether constants the search of the exact solutions of the geodesic equations for the pp-wave spacetimes is considered and we found new exact solutions of the geodesic equations in some special Noether gauge symmetry classes.

In this paper, a generic counterexample to the strong cosmic censor conjecture is exhibited. More precisely — taking into account that the conjecture lacks any precise formulation yet — first we make sense of what one would mean by a "generic counterexample" by introducing the mathematically unambigous and logically stronger concept of a "robust counterexample". Then making use of Penrose' nonlinear graviton construction (i.e. twistor theory) and a Wick rotation trick we construct a smooth Ricci-flat but not flat Lorentzian metric on the largest member of the Gompf — Taubes uncountable radial family of large exotic ℝ⁴'s. We observe that this solution of the Lorentzian vacuum Einstein's equations with vanishing cosmological constant provides us with a sort of counterexample which is weaker than a "robust counterexample" but still reasonable to consider as a "generic counterexample". It is interesting that this kind of counterexample exists only in four dimensions.

An extension of the general coordinate transformations algebra is constructed by means geometrical consistency conditions. A class of infinite invariants is derived. In particular we construct the consistent extension of the gravitational anomaly for each even dimension. The new contributions for these anomalies allow to define an improved Ward operator for which the symmetry is restored.

Four-dimensional pseudo-Riemannian homogeneous spaces whose isotropy is non-trivial with commuting curvature operators have been studied. The only example of homogeneous Einstein four-manifold which is curvature-Ricci commuting but not semi-symmetric has been presented. Non-trivial examples of semi-symmetric homogeneous four-manifolds which are not locally symmetric, also Jacobi–Jacobi commuting manifolds which are not flat have been presented.

Noether symmetries from geodetic Lagrangian for time-conformal plane symmetric spacetime are presented. Here, time-conformal factor is used to find the approximate Noether symmetries. This is a generalization of the idea discussed,^5–6 where they obtained approximate Noether symmetries from Lagrangian for a particular plane symmetric static spacetime. In the present paper, the most general plane symmetric static spacetime is considered and perturbed it by introducing a general time-conformal factor e^ϵf(t), where ϵ is very small which causes the perturbation in the spacetime. Taking the perturbation up to the first-order, we find all Lagrangian for plane symmetric spacetimes for which approximate Noether symmetries exist.

In a new approach, by using the exact sequences, semisprays on the prolongation of a Lie algebroids are introduced and many important results on the semisprays and sprays are obtained. Also, the horizontal endomorphisms, almost complex structures, vertical, horizontal and complete lifts on the prolongation of a Lie algebroid are considered. Then the distinguished connections on the prolongation of a Lie algebroid are introduced and the torsion and curvature tensors of these connections are considered. In particular, the Berwald-type and Yano-type connections are studied.

We solve the Freidmann equations by considering a universe media as a bulk viscosity described by a modified equation of state (EOS) of the form p = (γ - 1)ρc² + Λ(t). A completely integrable dynamical equation to the scale factor is obtained and gives out the exact solution by assuming that the time-dependent parameter Λ and the bulk viscosity are linear combination of two and three terms, respectively and is expressed as: and , where R is a scale factor and Λ₀, Λ₁, ζ₀, ζ₁, ζ₂, are constants. For a special choice of the parameters, we discuss the acceleration expansion of the universe evolution and future singularities in the framework of variable speed of light (VSL) theory.

Timelike ruled surfaces are studied in dual Lorentzian space by considering E. Study Mapping and Blaschke frame. A reference timelike ruled surface is considered and associated surfaces are defined. First, it is shown that the surface generated by the instantaneous screw axis (ISA) is a Mannheim offset of reference surface. Later, the kinematic interpretations between these surfaces are introduced by means of Blaschke invariants.