Showing papers in "International Journal of Geometric Methods in Modern Physics in 2016"

On the theory and applications of modern cosmography

Abstract: Cosmography represents an important branch of cosmology which aims to describe the universe without the need of postulating a priori any particular cosmological model. All quantities of interest are expanded as a Taylor series around here and now, providing in principle, a way of directly matching with cosmological data. In this way, cosmography can be regarded a model-independent technique, able to fix cosmic bounds, although several issues limit its use in various model reconstructions. The main purpose of this review is to focus on the key features of cosmography, emphasizing both the strategy for obtaining the observable cosmographic series and pointing out any drawbacks which might plague the standard cosmographic treatment. In doing so, we relate cosmography to the most relevant cosmological quantities and to several dark energy models. We also investigate whether cosmography is able to provide information about the form of the cosmological expansion history, discussing how to reproduce the dark fluid from the cosmographic sound speed. Following this, we discuss limits on cosmographic priors and focus on how to experimentally treat cosmographic expansions. Finally, we present some of the latest developments of the cosmographic method, reviewing the use of rational approximations, based on cosmographic Pade polynomials. Future prospects leading to more accurate cosmographic results, able to better reproduce the expansion history of the universe, are also discussed in detail.

A generally relativistic gauge classification of the Dirac fields

Abstract: We consider generally relativistic gauge transformations for the spinorial fields finding two mutually exclusive but together exhaustive classes in which fermions are placed adding supplementary information to the results obtained by Lounesto, and identifying quantities analogous to the momentum vector and the Pauli–Lubanski axial vector. We discuss how our results are similar to those obtained by Wigner by taking into account the system of Dirac field equations. We will investigate the consequences for the dynamics and in particular we shall address the problem of getting the nonrelativistic approximation in a consistent way. We are going to comment on extensions.

Pais–Uhlenbeck oscillator and negative energies

Abstract: We review the occurrence of negative energies in Pais–Uhlenbeck oscillator. We point out that in the absence of interactions, negative energies are not problematic neither in the classical nor in the quantized theory. However, in the presence of interactions that couple positive and negative energy degrees of freedom, the system is unstable, unless the potential is bounded from below and above. We review some approaches in the literature that attempt to avoid the problem of negative energies in the Pais–Uhlenbeck oscillator.

A note on generalized Robertson–Walker space-times

Abstract: A generalized Robertson–Walker (GRW) space-time is the generalization of the classical Robertson–Walker space-time. In the present paper, we show that a Ricci simple manifold with vanishing divergence of the conformal curvature tensor admits a proper concircular vector field and it is necessarily a GRW space-time. Further, we show that a stiff matter perfect fluid space-time or a mass-less scalar field with time-like gradient and with divergence-free Weyl tensor are GRW space-times.

Cosmological constant Λ in f(R,T) modified gravity

Abstract: In this paper, we have studied the Bianchi type-III cosmological model in the presence of cosmological constant in the context of f(R,T) modified theory of gravity. Here, we have discussed two classes of f(R,T) gravity, i.e. f(R,T) = R + f(T) and f(R,T) = f1(R) + f2(T). In both classes, the modified field equations are solved by the relation expansion scalar θ that is proportional to shear scalar σ which gives A = Cn, where A and C are metric potentials. Also we have discussed some physical and kinematical properties of the models.

Thermodynamic properties of modified gravity theories

Abstract: We review thermodynamic properties of modified gravity theories, such as F(R) gravity and f(T) gravity, where R is the scalar curvature and T is the torsion scalar in teleparallelism. In particular, we explore the equivalence between the equations of motion for modified gravity theories and the Clausius relation in thermodynamics. In addition, thermodynamics of the cosmological apparent horizon is investigated in f(T) gravity. We show both equilibrium and nonequilibrium descriptions of thermodynamics. It is demonstrated that the second law of thermodynamics in the universe can be met, when the temperature of the outside of the apparent horizon is equivalent to that of the inside of it.

A Dynamical System Analysis of $f(R,T)$ Gravity

Abstract: We investigate equations of motion and future singularities of f(R,T) gravity where R is the Ricci scalar and T is the trace of stress-energy tensor. Future singularities for two kinds of equation of state (barotropic perfect fluid and generalized form of equation of state) are studied. While no future singularity is found for the first case, some kind of singularity is found to be possible for the second. We also investigate f(R,T) gravity by the method of dynamical systems and obtain some fixed points. Finally, the effect of the Noether symmetry on f(R,T) is studied and the consistent form of f(R,T) function is found using the symmetry and the conserved charge.

Curvature properties of some class of warped product manifolds

Abstract: We prove that warped product manifolds with p-dimensional base, p = 1, 2, satisfy some pseudosymmetry type curvature conditions. These conditions are formed from the metric tensor g, the Riemann–Christoffel curvature tensor R, the Ricci tensor S and the Weyl conformal curvature C of the considered manifolds. The main result of the paper states that if p = 2 and the fiber is a semi-Riemannian space of constant curvature (when n is greater or equal to 5) then the (0, 6)-tensors R ⋅ R − Q(S,R) and C ⋅ C of such warped products are proportional to the (0, 6)-tensor Q(g,C) and the tensor C is a linear combination of some Kulkarni–Nomizu products formed from the tensors g and S. We also obtain curvature properties of this kind of quasi-Einstein and 2-quasi-Einstein manifolds, and in particular, of the Goedel metric, generalized spherically symmetric metrics and generalized Vaidya metrics.

Constraints on singular evolution from gravitational baryogenesis

Abstract: We investigate how the gravitational baryogenesis mechanism can potentially constrain the form of a Type IV singularity. Specifically, we study two different models with interesting phenomenology, that realize two distinct Type IV singularities, one occurring at the end of inflation and one during the radiation domination era or during the matter domination era. As we demonstrate, the Type IV singularities occurring at the matter domination era or during the radiation domination era are constrained by the gravitational baryogenesis, in such a way so that these do not render the baryon to entropy ratio singular. Both the cosmological models we study cannot be realized in the context of ordinary Einstein–Hilbert gravity, and hence our work can only be realized in the context of F(R) gravity and more generally in the context of modified gravity only.

Standard static finsler spacetimes

Abstract: We introduce the notion of a standard static Finsler spacetime ℝ × M where the base M is a Finsler manifold. We prove some results which connect causality with the Finslerian geometry of the base extending analogous ones for static and stationary Lorentzian spacetimes.

Darboux transformation and exact solutions of the integrable Heisenberg ferromagnetic equation with self-consistent potentials

Abstract: Integrable Heisenberg ferromagnetic equations are an important subclass of integrable systems. The M-XCIX equation is one of a generalizations of the Heisenberg ferromagnetic equation and are integrable. In this paper, the Darboux transformation of the M-XCIX equation is constructed. Using the DT, a 1-soliton solution of the M-XCIX equation is presented.

Forces in the complex octonion curved space

Abstract: The paper aims to extend major equations in the electromagnetic and gravitational theories from the flat space into the complex octonion curved space. Maxwell applied simultaneously the quaternion analysis and vector terminology to describe the electromagnetic theory. It inspires subsequent scholars to study the electromagnetic and gravitational theories with the complex quaternions/octonions. Furthermore Einstein was the first to depict the gravitational theory by means of tensor analysis and curved four-space–time. Nowadays some scholars investigate the electromagnetic and gravitational properties making use of the complex quaternion/octonion curved space. From the orthogonality of two complex quaternions, it is possible to define the covariant derivative of the complex quaternion curved space, describing the gravitational properties in the complex quaternion curved space. Further it is possible to define the covariant derivative of the complex octonion curved space by means of the orthogonality of two ...

Nonlinear ordinary differential equations : a discussion on symmetries and singularities

Abstract: Two essential methods, the symmetry analysis and the singularity analysis, for the study of the integrability of nonlinear ordinary differential equations is the purpose of this work. The main similarities and the differences of these two different methods are discussed.

Selected topics on the topology of ideal fluid flows

Abstract: This is a survey of certain geometric aspects of inviscid and incompressible fluid flows, which are described by the solutions to the Euler equations. We will review Arnold’s theorem on the topological structure of stationary fluids in compact manifolds, and Moffatt’s theorem on the topological interpretation of helicity in terms of knot invariants. The recent realization theorem by Enciso and Peralta-Salas of vortex lines of arbitrarily complicated topology for stationary solutions to the Euler equations will also be introduced. The aim of this paper is not to provide detailed proofs of all the stated results but to introduce the main ideas and methods behind certain selected topics of the subject known as Topological Fluid Mechanics. This is the set of lecture notes, the author gave at the XXIV International Fall Workshop on Geometry and Physics held in Zaragoza (Spain) during September 2015.

The W2-curvature tensor on warped product manifolds and applications

Abstract: The purpose of this paper is to study the W2-curvature tensor on (singly) warped product manifolds as well as on generalized Robertson–Walker and standard static space-times. Some different expressions of the W2-curvature tensor on a warped product manifold in terms of its relation with W2-curvature tensor on the base and fiber manifolds are obtained. Furthermore, we investigate W2-curvature flat warped product manifolds. Many interesting results describing the geometry of the base and fiber manifolds of a W2-curvature flat warped product manifold are derived. Finally, we study the W2-curvature tensor on generalized Robertson–Walker and standard static space-times; we explore the geometry of the fiber of these warped product space-time models that are W2-curvature flat.

Jacobi stability analysis of Rikitake system

Abstract: We study the Rikitake system through the method of differential geometry, i.e. Kosambi–Cartan–Chern (KCC) theory for Jacobi stability analysis. For applying KCC theory we reformulate the Rikitake system as two second-order nonlinear differential equations. The five KCC invariants are obtained which express the intrinsic properties of nonlinear dynamical system. The deviation curvature tensor and its eigenvalues are obtained which determine the stability of the system. Jacobi stability of the equilibrium points is studied and obtain the conditions for stability. We study the dynamics of Rikitake system which shows the chaotic behaviour near the equilibrium points.

Holographic superconductors with Weyl corrections

Abstract: A quick review on the analytical aspects of holographic superconductors (HSCs) with Weyl corrections has been presented. Mainly, we focus on matching method and variational approaches. Different types of such HSC have been investigated — s-wave, p-wave and Stuckelberg ones. We also review the fundamental construction of a p-wave type, in which the non-Abelian gauge field is coupled to the Weyl tensor. The results are compared from numerics to analytical results.

Kosambi–Cartan–Chern (KCC) theory for higher-order dynamical systems

Abstract: The Kosambi–Cartan–Chern (KCC) theory represents a powerful mathematical method for the investigation of the properties of dynamical systems. The KCC theory introduces a geometric description of the time evolution of a dynamical system, with the solution curves of the dynamical system described by methods inspired by the theory of geodesics in a Finsler spaces. The evolution of a dynamical system is geometrized by introducing a nonlinear connection, which allows the construction of the KCC covariant derivative, and of the deviation curvature tensor. In the KCC theory, the properties of any dynamical system are described in terms of five geometrical invariants, with the second one giving the Jacobi stability of the system. Usually, the KCC theory is formulated by reducing the dynamical evolution equations to a set of second-order differential equations. In this paper, we introduce and develop the KCC approach for dynamical systems described by systems of arbitrary n-dimensional first-order differential equations. We investigate in detail the properties of the n-dimensional autonomous dynamical systems, as well as the relationship between the linear stability and the Jacobi stability. As a main result we find that only even-dimensional dynamical systems can exhibit both Jacobi stability and instability behaviors, while odd-dimensional dynamical systems are always Jacobi unstable, no matter their Lyapunov stability. As applications of the developed formalism we consider the geometrization and the study of the Jacobi stability of the complex dynamical networks, and of the Λ-Cold Dark Matter (ΛCDM) cosmological models, respectively.

The role of Lagrange multiplier in Gauss–Bonnet dark energy

Abstract: We review accelerating cosmology in Gauss–Bonnet gravity with Lagrange multiplier constraint studied in [S. Capozziello, A. N. Makarenko and S. D. Odintsov, Phys. Rev. D 87 (2013) 084037, arXiv: 1302.0093 [gr-qc], S. Capozziello, M. Francaviglia and A. N. Makarenko, Astrophys. Space Sci. 349 (2014) 603–609, arXiv: 1304.5440 [gr-qc]. Several examples of dark energy universes are presented. We can get new dark energy solutions (with additional scalar) as well as certain limits to earlier found accelerating solutions.

Surface family with a common involute asymptotic curve

Abstract: We construct a surface family possessing an involute of a given curve as an asymptotic curve. We express necessary and sufficient conditions for that curve with the above property. We also present natural results for such ruled surfaces. Finally, we illustrate the method with some examples, e.g. circles and helices as given curves.

Hamiltonian dynamics on matched pairs

Abstract: The cotangent bundle of a matched pair Lie group, and its trivialization, are shown to be a matched pair Lie group. The explicit matched pair decomposition on the trivialized bundle is presented. On the trivialized space, the canonical symplectic two-form and the canonical Poisson bracket are explicitly written. Various symplectic and Poisson reductions are perfomed. The Lie–Poisson bracket is derived. As an example, Lie–Poisson equations on 𝔰𝔩(2, ℂ)∗ are obtained.

Pseudo-Z symmetric space-times with divergence-free Weyl tensor and pp-waves

Abstract: In this paper we present some new results about n(≥ 4)-dimensional pseudo-Z symmetric space-times. First we show that if the tensor Z satisfies the Codazzi condition then its rank is one, the space-time is a quasi-Einstein manifold, and the associated 1-form results to be null and recurrent. In the case in which such covector can be rescaled to a covariantly constant we obtain a Brinkmann-wave. Anyway the metric results to be a subclass of the Kundt metric. Next we investigate pseudo-Z symmetric space-times with harmonic conformal curvature tensor: a complete classification of such spaces is obtained. They are necessarily quasi-Einstein and represent a perfect fluid space-time in the case of time-like associated covector; in the case of null associated covector they represent a pure radiation field. Further if the associated covector is locally a gradient we get a Brinkmann-wave space-time for n > 4 and a pp-wave space-time in n = 4. In all cases an algebraic classification for the Weyl tensor is provided for n = 4 and higher dimensions. Then conformally flat pseudo-Z symmetric space-times are investigated. In the case of null associated covector the space-time reduces to a plane wave and results to be generalized quasi-Einstein. In the case of time-like associated covector we show that under the condition of divergence-free Weyl tensor the space-time admits a proper concircular vector that can be rescaled to a time like vector of concurrent form and is a conformal Killing vector. A recent result then shows that the metric is necessarily a generalized Robertson–Walker space-time. In particular we show that a conformally flat (PZS)n, n ≥ 4, space-time is conformal to the Robertson–Walker space-time.

Stefan–Sussmann singular foliations, singular subalgebroids and their associated sheaves

Abstract: We explain and motivate Stefan–Sussmann singular foliations, and by replacing the tangent bundle of a manifold with an arbitrary Lie algebroid, we introduce singular subalgebroids. Both notions are defined using compactly supported sections. The main results of this note are an equivalent characterization, in which the compact support condition is removed, and an explicit description of the sheaf associated to any Stefan–Sussmann singular foliation or singular subalgebroid.

Chaplygin-gas Solutions of f(R) Gravity

Abstract: We explore exact f(R) gravity solutions that mimic Chaplygin-gas inspired ΛCDM cosmology. Starting with the original, generalized and modified Chaplygin-gas (MCG) equations of state (EoS), we reconstruct the forms of f(R) Lagrangians. The resulting solutions are generally quadratic in the Ricci scalar, but have appropriate ΛCDM solutions in limiting cases. These solutions, given appropriate initial conditions, can be potential candidates for scalar field-driven early universe expansion (inflation) and dark energy-driven late-time cosmic acceleration.

Translation surfaces in the three-dimensional simply isotropic space 31

Abstract: In this paper, we classify translation surfaces in the three-dimensional simply isotropic space 𝕀31 under the condition Δix i = λixi where Δ is the Laplace operator with respect to the first and second fundamental forms and λ is a real number. We also give explicit forms of these surfaces.

Universality of the Einstein theory of gravitation

Abstract: We show that generalizations of general relativity theory, which consist in replacing the Hilbert Lagrangian LHilbert = 1 16π|g|R by a generic scalar density L = L(gμν,Rμνκλ) depending upon the metric gμν and the curvature tensor Rμνκλ, are equivalent to the conventional Einstein theory for a (possibly) different metric tensor gμν and (possibly) a different set of matter fields. The simple proof of this theorem relies on a new approach to variational problems containing metric and connection.

Noether symmetries of vacuum classes of pp-waves and the wave equation

Abstract: The Noether symmetry algebras admitted by wave equations on plane-fronted gravitational waves with parallel rays are determined. We apply the classification of different metric functions to determine generators for the wave equation, and also adopt Noether's theorem to derive conserved forms. For the possible cases considered, there exist symmetry groups with dimensions two, three, five, six and eight. These symmetry groups contain the homothetic symmetries of the spacetime.

A new approach on electromagnetism with dual number coefficient octonion algebra

Abstract: Dual number coefficient octonion (DNCO) is one of the kind of octonion, it has 16 components with an additional dual unit e. Starting with DNCO algebra, we develop the generalized electromagnetic field equations of dyons regarding the DNCOS spaces, which has two octonionic space-times namely the octonionic internal space-time and the octonionic external space-time. Besides, the generalized four-potential components of dyons have been expressed with respect to the dual octonion form. Furthermore, we obtain the symmetrical form of Dirac–Maxwell equations, and the generalized potential wave equations for dyons in terms of the dual octonion. Finally, we conclude that dual octonion formulation is compact and simpler like octonion formulation.

Ricci inheritance collineations in Kantowski–Sachs spacetimes

Abstract: In this paper, we investigate Ricci Inheritance Collineations (RICs) in Kantowski–Sachs spacetimes. RICs are discussed in detail when Ricci tensor is degenerate and nondegenerate. In both the cases, RICs are obtained and it turns out that the dimension of Lie algebra of RICs is finite when Ricci tensor is nondegenerate. In the case when Ricci tensor is degenerate, we get finite as well as infinite dimensional group of RICs.

Cosmology with higher-derivative matter fields

Abstract: We investigate the cosmological implications of a new class of modified gravity, where the field equations generically include higher-order derivatives of the matter fields, arising from the introduction of non-dynamical auxiliary fields in the action. Imposing a flat, homogeneous and isotropic geometry, we extract the Friedmann equations, obtaining an effective dark-energy sector containing higher-derivatives of the matter energy density and pressure. For the cases of dust, radiation and stiff matter, we analyze the cosmological behavior, finding accelerating, de Sitter and non-accelerating phases, dominated by matter or dark-energy. Additionally, the effective dark-energy equation-of-state parameter can be quintessence-like, cosmological-constant-like or even phantom-like. The detailed study of these scenarios may provide signatures, that could distinguish them from other candidates of modified gravity.