International Journal of Geometric Methods in Modern Physics

This paper deals with the dynamical systems analysis of Friedmann–Robertson–Walker (FRW) model of the Universe in the framework of general relativity with quadratic equation of state and bulk viscosity. The evolution equations are transformed into an autonomous system of differential equations using suitable variables transformation. Stability analysis of cosmological models with quadratic equation of state parameter are discussed in detail in two different scenarios viz, first Universe filled with barotropic fluid and second filled with bulk viscous fluid. The nature of critical points is analyzed for both cases accordance with respective eigenvalues. We have also analyzed the stable attractor for both cases and examined their properties from the point of cosmological view.

Motivated by the work of Nojiri et al. [S. Nojiri, S. D. Odintsov and E. N. Saridakis, Holographic inflation, Phys. Lett. B 797 (2019) 134829], this study reports a model of inflation under the consideration that the inflationary regime is originated by a type of holographic energy density. The infrared cutoff has been selected based on the modified holographic model that is a particular case of Nojiri–Odintsov holographic dark energy [S. Nojiri and S. D. Odintsov, Unifying phantom inflation with late-time acceleration: Scalar phantom–non-phantom transition model and generalized holographic dark energy, Gen. Relativ. Gravit. 38 (2006) 1285] that unifies phantom inflation with the acceleration of the universe on late time. On getting an analytical solution for Hubble parameter we considered the presence of bulk viscosity and the effective equation of state parameter appeared to be consistent with inflationary scenario with some constraints. It has also being observed that in the inflationary scenario the contribution of bulk viscosity is not of much significance and its influence is increasing with the evolution of the universe. Inflationary observables have been computed for the model and the slow-roll parameters have been computed. Finally, it has been observed that the trajectories in $n_s-r$ are compatible with the observational bound found by Planck. It has been concluded that the tensor to scalar ratio for this model can explain the primordial fluctuation in the early universe as well.

We have obtained the approximate analytical solutions of the nonrelativistic Hydrogen-like atoms such as $({\mathrm{\text{He}}}^{+},{\mathrm{\text{Li}}}^{2+}$ and ${\mathrm{\text{Be}}}^{+})$ and neutral atoms such as (${}^{22}\mathrm{\text{Na}}{,}^{\mathrm{\text{12}}}\mathrm{\text{C}}$ and ${}^{\mathrm{\text{158}}}\mathrm{\text{Au}}$) atoms with a newly proposed generalized perturbed Yukawa potential with centrifugal barrier (GPYPCB) model using the generalized Bopp’s shift method and standard perturbation theory in the symmetries of noncommutative three-dimensional real space phase (NC: 3D-RSP). By approximating the centrifugal term through the Greene–Aldrich approximation scheme, we have obtained the energy eigenvalues and generalized Hamiltonian operator for all orbital quantum numbers $\ell$ in the symmetries of NC: 3D-RSP. The potential is a superposition of the perturbed Yukawa potential and new terms proportional with $(\frac{exp(-2ar)}{r^4},\frac{exp(-2ar)}{r^3},\frac{exp(-ar)}{r^3}\text{ and }\frac{exp(-ar)}{r^2}$) appear as a result of the effects of noncommutativity properties of space and phase on the perturbed Yukawa potential model. The obtained energy eigenvalues appear as functions of the generalized Gamma function, the discreet atomic quantum numbers $(j,n,l,s\text{ and }m)$, two infinitesimal parameters $(\mathrm{\Theta },\overline{𝜃})$, which are induced by (position–position and phase–phase). In addition, the dimensional parameters $(a,V_0,V_1,\alpha )$ of perturbed Yukawa potential with centrifugal barrier model in NC: 3D-RSP. Furthermore, we have shown that the corresponding Hamiltonian operator in (NC: 3D-RSP) symmetries is the sum of the Hamiltonian operator of perturbed Yukawa potential model and the two operators are modified spin–orbit interaction and the modified Zeeman operator for the previous Hydrogenic and neutral atoms.

We define a Weyl-type curvature tensor of $(1,2)$-type to provide a characterization for Finsler metrics of constant flag curvature. This Weyl-type curvature tensor is projective invariant only to projective factors that are Hamel functions. Based on this aspect, we construct new families of projectively related Finsler metrics that have constant flag curvature.

The goal of this paper is to study SdS space-time and its gravitational field with consideration of the canonical form and invariants of the curvature tensor. The characteristic of $\lambda$-tensor identifies the type of gravitational field. Gaussian curvature quantities enunciated in terms of curvature invariants.

In this paper, we study almost cosymplectic manifolds admitting almost quasi-Yamabe solitons $(g,V,m,\lambda )$. First, we prove that an almost cosymplectic $(\kappa ,\mu )$-manifold is locally isomorphic to a Lie group if $(g,V,m,\lambda )$ is a nontrivial closed quasi-Yamabe soliton. Next, we consider an almost $\alpha$-cosymplectic manifold admitting a nontrivial almost quasi-Yamabe soliton and prove that it is locally the Riemannian product of an almost Kähler manifold with the real line if the potential vector field $V$ is collinear with the Reeb vector filed. For the potential vector field $V$ being orthogonal to the Reeb vector filed, we also obtain two results. Finally, for a closed almost quasi-Yamabe soliton on compact $K$-cosymplectic manifolds, we prove that it is trivial if $r-\lambda$ is nonnegative, where $r$ is the scalar curvature.

It is shown that the algebra $J_3[C\otimes O]\otimes Cl(4,C)$ based on the complexified Exceptional Jordan, and the complex Clifford algebra in 4D, is rich enough to describe all the spinorial degrees of freedom of three generations of fermions in 4D, and include additional fermionic dark matter candidates. Furthermore, the model described in this paper can account also for the Standard Model gauge symmetries. We extend these results to the Magic Star algebras of Exceptional Periodicity developed by Marrani–Rios–Truini and based on the Vinberg cubic $T$ algebras which are generalizations of exceptional Jordan algebras. It is found that there is a one-to-one correspondence among the real spinorial degrees of freedom of four generations of fermions in 4D with the off-diagonal entries of the spinorial elements of the $pair$ $T_3^{8,n},({\bar{T}}_3^{8,n})$ of Vinberg matrices at level $n=2$. These results can be generalized to higher levels $n>2$ leading to a higher number of generations beyond 4. Three $pairs$ of $T$ algebras and their conjugates $\bar{T}$ were essential in the Magic Star construction of Exceptional Periodicity that extends the $e_8$ algebra to $e_8^{(n)}$ with $n$ integer.

In this work, an attempt is made to study the thermodynamical analysis at the apparent horizon in the framework of fractal universe. We consider the Bekenstein entropy to examine validity of the generalized second law of thermodynamics (GSLT) and thermal equilibrium for the four different cases which are developed with the utilization of different forms of squared speed of sound. In each case, we explore the behavior of total entropy through the graphical variation of its first- and second-order derivatives with respect to redshift parameter ($z$). It is found that generalized second law of thermodynamics holds for Cases 1 and 2 for $z>-0.1$ and $z>-0.2$, respectively and it holds in late times as well. However, for Cases $3$ and $4$, this law is satisfied in early, present and future epochs. Furthermore, for Cases 1 and 2, instability of thermodynamic equilibrium is observed, but for Cases 3 and 4, it holds in the specific intervals $z<-0.5$ and $z<-0.9$, respectively.

Let $M$ be a three-dimensional real hypersurface in a nonflat complex space form of complex dimension two. In this paper, we prove that $M$ is $\eta$-parallel with two distinct principal curvatures at each point if and only if it is locally congruent to a geodesic sphere in $ℂP^2$ or a horosphere, a geodesic sphere or a tube over totally geodesic complex hyperbolic plane in $ℂH^2$. Moreover, $\eta$-parallel real hypersurfaces in $ℂP^2$ and $ℂH^2$ under some other conditions are classified and these results extend Suh’s in [Characterizations of real hypersurfaces in complex space forms in terms of Weingarten map, Nihonkai Math. J. 6 (1995) 63–79] and Kon–Loo’s in [On characterizations of real hypersurfaces in a complex space form with $\eta$-parallel shape operator, Canad. Math. Bull. 55 (2012) 114–126].

In this paper, we study the possibility of obtaining a stable flat dark energy-dominated universe in a good agreement with observations in the framework of Swiss-cheese brane-world cosmology. Two different brane-world cosmologies with black strings have been introduced for any cosmological constant $\mathrm{\Lambda }$ using two empirical forms of the scale factor. In both models, we have performed a fine-tuning between the brane tension and the cosmological constant so that the Equation of state (EoS) parameter $\omega (t)\to -1$ for the current epoch, where the redshift $z\simeq 0$. We then used these fine–tuned values to calculate and plot all parameters and energy conditions. The deceleration–acceleration cosmic transition is allowed in both models, and the jerk parameter $j\to 1$ at late-times. Both solutions predict a future dark energy-dominated universe in which $\omega =-1$ with no crossing to the phantom divide line. While the pressure in the first solution is always negative, the second solution predicts a better behavior of cosmic pressure where the pressure is negative only in the late-time accelerating era but positive in the early-time decelerating era. Such a positive-to-negative transition in the evolution of pressure helps to explain the cosmic deceleration–acceleration transition. Since black strings have been proved to be unstable by some authors, this instability can actually reflect doubts on the stability of cosmological models with black strings (Swiss-cheese type brane-worlds cosmological models). For this reason, we have carefully investigated the stability through energy conditions and sound speed. Because of the presence of quadratic energy terms in Swiss-cheese type brane-world cosmology, we have tested the new nonlinear energy conditions in addition to the classical energy conditions. We have also found that a negative tension brane is not allowed in both models of the current work as the energy density will no longer be well defined.

Nonstatic plane symmetric spacetimes are considered to study conformal vector fields (VFs) in the $f(R)$ theory of gravity. Firstly, we investigate some proper nonstatic plane symmetric spacetimes by solving the Einstein field equations (EFEs) in the $f(R)$ theory of gravity using algebraic techniques. Secondly, we find CVFs of the obtained spacetimes by means of the direct integration approach. There exist seven cases. Studying each case in detail, we find that the CVFs are of dimension three, five, six and fifteen.

From a study of an oscillator in a 4D noncommutative (NC) spacetime, we establish the Hamilton equations of motion. The formers are solved to give the oscillator position and momentum coordinates. These coordinates are used to build a metric similar to that describing a cosmic string. On this basis, Dirac and Klein–Gordon oscillators are investigated. Their spectrum and dynamics are analyzed giving rise to novel interesting properties.

In this paper, differential properties of the one-parameter hyperbolic dual spherical kinematics are developed with explicit expressions independent of coordinates systems. We calculate Euler–Savary equations of spherical kinematics in the dual Lorentzian 3-space ${𝔻}_1^3$. Then from E. Study’s map new proofs are directly attained for the Disteli’s formulae and their spatial equivalents are examined in detail. Lastly for spherical and planar motions, the point trajectories theoretical expressions of the point trajectories are investigated with a certain value of acceleration and velocity, which are regarded as different forms of Euler–Savary equation form.

In modern physics, the general theory of relativity (GTR) successfully predicts the vital structure of gravity. The GTR confirmed that gravity is the distortion of four-dimensional spacetime $(x,y,z,t)$ by massive bodies. Such a prediction of the GTR is one of the vital successes towards the development goals of modern physics. Though, the central foundation for all the calculations of the GR is the hypothesis of continuum spacetime. In this paper, the author introduces a theorem of the quantum mechanical theory of time (QMT) to test whether the metric time is discrete or continuous. To prove the theorem, the author applied the set theory of cardinal numbers on the energy–time uncertainty relationship. The proof of the theorem confirmed that the metric time is discrete, and has an intrinsic quantum nature. The result implied that the continuum spacetime assumption of the GTR is found fundamentally erroneous. Therefore, spacetime is discrete and needs to be analyzed by the principles of quantum mechanics (QM).