International Journal of Geometric Methods in Modern Physics

A new theory, named G(General)-theory in 14 dimensions, has been proposed that is reduced to $F(R)$-gravity and produces the metric of FRW universe in four dimensions. In this theory, the Universe is born in six stages. First, there is nothing in the world. Then, two strings, one with positive energy and one with negative energy in 14th, dimension are created such that the sum over their energies is zero. These strings are excited and for flowing their energies, other dimensions are produced. Second, these strings decay to G0-branes. Third, these branes join each other and construct Gp-branes which tensor fields live on. The rank of these fields can change from zero to p for $p\le 6$ and from zero to six for $p>6$. Four, by compacting Gp-branes on three circles, supersymmetry is born which contains the equal number of degrees of freedom for fermions and bosons. Six, by reducing G-theory to four dimensions, FRW universe is emerges and initial tensor fields produce the predicted shape of $F(R)$-gravity.

In this paper, we propose a new time-varying deceleration parameter for the description of a unified cosmological evolution (early inflation, deceleration and present acceleration) using Padé approximation. Within the background of Einstein’s general relativity and for a spatially flat homogeneous and isotropic model, the proposed deceleration parameter leads to a variable equation of state of effective matter that serves all matter within EoS $-1\le \omega \le 1$ (quintessence, relativistic and non-relativistic baryonic matter, stiff matter and cosmological constant), exactly in the same order as it required for various evolutionary phases of the universe. We consider a minimally coupled scalar field with self-interacting potential as the only matter source in the universe. The adopted scheme allows us to determine the potential for the scalar field that drives the cosmological evolution proposed by the deceleration parameter. We perform an explicit numerical computation where analytical solutions are not possible. We examine the validity of the model by depicting the statefinder parameters and comparing them with the recent observational outcomes of cosmography.

In this paper, we examined the viability bounds of a higher derivative $f(R,\square R,T)$ theory through analyzing energy conditions (where $R,\square R$ and $T$ are the Ricci scalar, d’Alemberts operator and trace of energy–momentum tensor, respectively). We take flat Friedmann–Lemaître–Robertson–Walker spacetime coupled with ideal configurations of matter content. We consider three different realistic models of this gravity, that could be utilized to understand the stability of cosmological solutions. After constructing certain bounds mediated by energy conditions, more specifically the weak energy condition, we discuss viable zones of the under considered modified models in an environment of recent estimated numerical choices of the cosmic parameters.

Quantum splines are curves in a Hilbert space or, equivalently, in the corresponding Hilbert projective space, which generalize the notion of Riemannian cubic splines to the quantum domain. In this paper, we present a generalization of this concept to general density matrices with a Hamiltonian approach and using a geometrical formulation of quantum mechanics. Our main goal is to formulate an optimal control problem for a nonlinear system on ${𝔲}^{\ast }(n)$ which corresponds to the variational problem of quantum splines. The corresponding Hamiltonian equations and interpolation conditions are derived. The results are illustrated with some examples and the corresponding quantum splines are computed with the implementation of a suitable iterative algorithm.

Recently, the role of energy conditions in $f(R)$-cosmology has been investigated. We generalize these results to BIonic systems in accelerating systems and show that the energy conditions can be changed in this system. We show that $f(R)$-gravity produces a wormhole between two universes and forms a BIonic system. On the other hand, the acceleration creates two regions in the Rindler space-time which BIon in each region acts reverse to other in another region. This means that by the expansion of universes of the BIon in one region, universes of another BIon in other regions contract. Also, in this model, by increasing the order of curvatures in $f(R)$-gravity, the energy and the entropy of system in one region increases and in other region decreases. Amount of this growth or decrease in the energy depends on the acceleration of universes. Finally, by calculating the dark energy equation of state, we observe that one universe enters to phantom phase and goes toward the big rip singularity and another goes out of phantom state.

This paper deals with the study of invariant submanifolds of generalized Sasakian-space-forms with respect to Levi-Civita connection as well as semi-symmetric metric connection. We provide an example of such submanifolds and obtain many new results including the necessary and sufficient conditions under which the submanifolds are totally geodesic. The Ricci solitons of such submanifolds are also studied.

We propose to describe the acceleration of the universe by introducing a model of two coupled fluids. We focus on the accelerated expansion at the early stages. The inflationary expansion is described in terms of a van der Waals equation of state for the cosmic fluid, when account is taken of bulk viscosity. We assume that there is a weak interaction between the van der Waals fluid and the second component (matter). The gravitational equations for the energy densities of the two components are solved for a homogeneous and isotropic Friedmann–Robertson–Walker (FRW) universe, and analytic expressions for the Hubble parameter are obtained. The slow-roll parameters, the spectral index, and the tensor-to-scalar ratio are calculated and compared with the most recent astronomical data from the Planck satellite. Given reasonable restriction on the parameters, the agreement with observations is favorable.

In the framework of teleparallel gravity, the Friedman–Robertson–Walker (FRW) cosmological model with scalar tensor theory, where scalar field is non-minimally coupled to both the torsion scalar and boundary term, is studied. Utilizing the Noether symmetry approach in such a theory, we obtain the explicit forms of the couplings and potential as a function of the scalar field. We present some important cosmological solutions for the modified field equations using these functions gotten via the Noether symmetry approach. Finally, the interesting cosmological properties of these solutions are discussed in detail, and it is shown that they can describe a universe leading to the late-time accelerating expansion.

In this paper, we will consider a subclass of models of Horndeski theories of gravity and we will check for several Static Spherically Symmetric solutions. We will find a model which admits an exact black hole (BH) solution and we will study its thermodynamics by using the Euclidean Action. We will see that, in analogy with the case of General Relativity (GR), the integration constant of the solution can be identified with the mass of the BH itself. Other solutions will be discussed, by posing special attention on the possibility of reproducing the observed profiles of the rotation curves of galaxies.

A non-spherical solution of Einstein’s field equations, possessing a naked curvature singularity on the symmetry axis, satisfying the strong curvature condition, is presented. The spacetime admits closed timelike curves which appear after a certain instant of time in a causally well-behaved manner. The matter–energy content radiation fields, coupled with anisotropic fluids, obeying the energy conditions, diverge on the symmetry axis.

In this paper, we study the mimetic theory and derive a new spherically symmetric black hole solution. The asymptotic behavior of this solution behaves as a flat spacetime. This black hole is characterized by the fact that it has different components of $g_{tt}$ and $g^{rr}$. Nevertheless, both of these components have a coinciding Killing and event horizons. Furthermore, this black hole has curvature singularities which are stronger than those of the known black hole solutions in general relativity. This feature can be shown by calculating some invariants of curvature. We study the stability of the perturbation and the related anti-evaporation of the Nariai spacetime.

In this paper, a cosmological model involving interacting varying Chaplygin gas of a specific type and a tachyonic matter is considered to address the problem known as the accelerated expansion of the universe. The study shows that a specific form of the potential and appropriate values of the parameters allows a transition from early tachyonic to late-time tachyonic dark matter with ${\omega }_T\approx 0$. This point, up to our knowledge, makes the considered cosmological scenario unique in the literature. Moreover, discussion on future finite-time singularities for the considered models finalizes the paper. The study shows that in case of non-interacting model, the fate of universe will be either Type II (The Sudden Singularity), or Type IV (“Generalized Sudden Singularity”) singularity. On the other hand, the consideration of $Q=3bH{\rho }_{de}$ non-gravitational interaction in addition to Type II and Type IV singularities will induce also Type V (“$\omega$-singularity”) singularity, while with $Q=3bqH{\rho }_{de}$ only Type IV singularity will be observed. Moreover, two forms of scale factor have been considered allowing to analytically obtain the behavior of the cosmological parameters without specifying an explicit form of the potential for tachyonic matter.

This study deals with spatially homogeneous and anisotropic locally rotationally symmetric (LRS) Bianchi type-I universe with cosmic string source in the framework of $f(R)$ theory of gravity [S. Capozziello, S. Carloni and A. Troisi, Quintessence without scalar fields, Recent Res. Dev. Astron. Astrophys. 1 (2003), 625; S. Nojiri and S. D. Odintsov, Modified gravity with negative and positive powers of curvature: Unification of inflation and cosmic acceleration, Phys. Rev. D 68 (2003) 123512]. Solving the field equations using (i) relation between metric potentials, (ii) power law relation between $F(R)$ and average scale factor $a(t)$ and (iii) equations of state for string models we have presented Takabayasi [T. Takabayasi, Quantum Mechanics Determinism, Causality, and Particles (Springer, Berlin, 1976)], Nambu [P. S. Letelier, String cosmologies, Phys. Rev. D 28 (1983) 2414–2419] and Reddy [D. R. K. Reddy, A string cosmological model in a scalar–Tensor theory of gravitation, Astrophys. Space Sci. 286 (2003) 359–363] string cosmological models. The dynamical parameters of our models are determined and their physical behavior is discussed. The most interesting result about the models is that the anisotropic effects are wiped out at late times.

The projective curvature tensor is an invariant under geodesic preserving transformations on semi-Riemannian manifolds. It possesses different geometric properties than other generalized curvature tensors. The main object of the present paper is to study some semisymmetric type and pseudosymmetric type curvature restricted geometric structures due to projective curvature tensor. The reduced pseudosymmetric type structures for various Walker type conditions are deduced and the existence of Venzi space is ensured. It is shown that the geometric structures formed by imposing projective operator on a (0,4)-tensor is different from that for the corresponding (1,3)-tensor. Characterization of various semisymmetric type and pseudosymmetric type curvature restricted geometric structures due to projective curvature tensor are obtained on semi-Riemannian manifolds, and it is shown that some of them reduce to Einstein manifolds for the Riemannian case. Finally, to support our theorems, four suitable examples are presented.

In this paper, we study the non-Gaussian character in generalized coherent states in the framework of a noncommutative space by adding photons to deformed coherent states, which is called the photon-added nonlinear coherent states (PANCSs). We find that the non-Gaussianity of PANCSs is enhanced with the increase of the photon-added number and it increases monotonically with the amplitude of the coherent states. Interestingly, we obtain that the maximal value of non-Gaussianity measure occurs at a certain critical value of the coherent state amplitude, where this critical value is determined by the kind of the deformation. Using the Mandel’s parameter, we examine the statistical properties for the PANCSs and show that the Mandel’s parameter may take positive and negative values depending on the choice of the amplitude and deformation parameter, exhibiting sub-Poissonian distribution and super-Poissonian distribution.

We interpret, in the realm of relativistic quantum field theory, the tangential operator given by Coleman and Mandula [All possible symmetries of the $s$ matrix, Phys. Rev. 159 (1967) 1251–1256] (see also [Much, Pottel and Sibold, Preconjugate variables in quantum field theory and their applications, Phys. Rev. D 94(6) (2016) 065007]) as an appropriate coordinate operator. The investigation shows that the operator generates a Snyder-like noncommutative spacetime with a minimal length that is given by the mass. By using this operator to define a noncommutative spacetime, we obtain a Poincaré invariant noncommutative spacetime and in addition solve the soccer-ball problem. Moreover, from recent progress in deformation theory we extract the idea of how to obtain, in a physical and mathematically well-defined manner, an emerging noncommutative spacetime. This is done by a strict deformation quantization known as Rieffel deformation (or warped convolutions). The result is a noncommutative spacetime combining a Snyder and a Moyal-Weyl type of noncommutativity that in addition behaves covariant under transformations of the whole Poincaré group.

The aim of this paper is to investigate the influence of expansion-free condition and cosmological constant in the modeling of planar relativistic self-gravitating objects. For this purpose, we have considered a non-static diagonally oriented planar geometry that is coupled with locally anisotropic non-radiating matter configurations. Firstly, we have explored the factor responsible for maintaining conformal flat state by relating tidal forces with matter dynamical variables. Then, we have smoothly joined our relativistic interior with the exterior one through Darmois junction conditions in the $\mathrm{\Lambda }$-dominated epoch. We have also shown that the well-known perfect fluid Skripkin model has disobeyed matching conditions for the case of planar system with cosmological constant. Lastly, we have inspected family of exact analytical solutions of Einstein-$\mathrm{\Lambda }$ field equations with various backgrounds, like, an/isotropic matter, separable energy density and barotropic equation of state.

In this study, we investigate an anisotropic Bianchi type-$III$ space-time in the presence of matter and holographic dark energy components within the framework of general relativity. We obtained the solution of the field equations by assuming (i) the expansion scalar $(𝜃)$ in the model is proportional to shear scalar $(\sigma )$, (ii) hybrid expansion law for average scale factor (keeping an eye on the recent scenario of accelerating nature of the universe). We develop cosmological parameters like deceleration and equation of state parameters. These parameters are plotted versus redshift $z$ for different values of power component of average scale factor $l$. We observe that the equation of state varies in quintessence region ($-1<{\omega }_{\mathrm{\Lambda }}<-1/3$) and ultimately tends to $\mathrm{\Lambda }$CDM model (${\omega }_{\mathrm{\Lambda }}=-1$). The deceleration parameter exhibits a smooth transition from early decelerated epoch to present accelerated era. In addition, we establish the correspondence between our holographic dark energy model and quintessence scalar field. We also express the self-interacting potential $V(\phi )$ and scalar field $\phi$ of quintessence model as functions of cosmic time $t$, which describes the accelerated expansion of the universe.

In the literature, Holditch theorem was obtained under periodic rotation and translation motions in [H. Holditch, Geometrical theorem, Q. J. Pure Appl. Math. 2 (1858) 38] or periodic shear and translation motions in [O. Röschel, Der satz von Holditch in der isotropen ebene. Abh. Braunschweig. Wiss. Ges. 36 (1984) 27–32]. In this paper, by introducing the projection of a vector onto a plane, scalar area, area vector of a surface, we investigate Holditch theorem under periodic rotation, translation and shear motions. We give two interpretations for the Holditch type theorem in Galilean space.