International Journal of Geometric Methods in Modern Physics

This paper is intended to review the consequences of assuming tachyon scalar field as a candidate for dark energy from dynamical system technique and observational cosmology point of views. In the contexts of general relativity and teleparallel gravity, a variety of non-minimally coupled tachyon fields have been taken into consideration. These models provide us subclasses of scalar-tensor and scalar-torsion theories. This review covers the tachyonic chameleon, the non-minimally coupled tachyon field to the Gauss–Bonnet invariant and the tachyonic teleparallel dark energy models. Type Ia supernovae and baryonic acoustic oscillations datasets are the observational data that have been used to constrain the model parameters.

To make a comparison of energy conditions in the theory of general relativity and in the modified theories, we have considered $f(R)$, $f(R,T)$ and $f(R,T,Q)$ theories (where $R$ and $T$ are the Ricci scalar and trace of the energy–momentum tensor, respectively, while $Q=T^{ab}{R}_{ab}$) to test the validity of all the four energy conditions. These energy conditions had been derived to check the viability of cosmological as well as astrophysical models. In this paper, we consider the standard Friedmann–Robertson–Walker spacetime representing the homogeneous and isotropic universe, and investigate the available literature on testing energy conditions as well as calculate these conditions ourselves. In order to provide the comparative results, we test these conditions analytically as well as graphically and present how energy bounds in modified theories depend on the values of the involved parameters and its validity within certain limits. We discuss here, that which of the modified theories comply with certain energy conditions, and hence provide a supporting argument on the existence of modified theories.

In this expository paper, we want to give a brief introduction, with few key references for further reading, to the inner functioning of the new and successful algorithms of Deep Learning and Geometric Deep Learning with a focus on Graph Neural Networks. We go over the key ingredients for these algorithms: the score and loss function and we explain the main steps for the training of a model. We do not aim to give a complete and exhaustive treatment, but we isolate few concepts to give a fast introduction to the subject. We provide some appendices to complement our treatment discussing Kullback–Leibler divergence, regression, Multi-layer Perceptrons and the Universal Approximation theorem.

We study the dynamics of Wigner function, quantum discord, entanglement and quantum dissonance for bipartite Bell-diagonal states independently interacting with dephasing reservoirs. We analyze comparatively their behavior under the decoherence effect. Our main goal is to test the ability of Wigner function negativity to measure all kinds of quantum correlations in such system. Particularly, we find that this negativity can only capture entanglement and discord. Of course it succeeded in predicting the behavior of quantum dissonance, but nevertheless, it failed to detect its presence. As a matter of fact, we conclude that Wigner function negativity cannot be used as measure of quantum dissonance in the two qubit open system.

It is well known that the Noether symmetry approach is beneficial for searching physically realistic models that are solutions of the Einstein field equations (EFEs). Particularly, the generators obtained via Noether symmetry help to extract conservation laws of physics. The main goal of this study is to investigate the Noether symmetries for a class of static plane symmetric solutions in the $f(T)$ gravity. The $f(T)$ gravity, where $T$ being torsion is one of the potential candidates of modified theories of gravitation. The theory of $f(T)$ gravity seems to provide a theoretical interpretation of the late-time universe acceleration as well as to explain inflationary senecio. With the help of Noether symmetry, we also classified the solutions in terms of conserved quantities that the associated spacetime can admit. The physical and geometrical interpretations of each of the symmetry generators are also presented. It turns out that a class of static plane symmetric solutions in the $f(T)$ gravity admits 5, 6, 8, 9, 11 and 17 Noether symmetries.

In this work, we calculate new concept for their Fermi–Walker conformable derivatives with spherical timelike magnetic fiber in spherical de-Sitter space ${𝕊}_1^2$. The description is advanced for timelike magnetic trajectory. First, we find conformable fractional derivatives of spherical magnetic Lorentz fields. Moreover, we compute the Fermi–Walker conformable derivatives of the normalization and recursion operators for these spherical vector fields. Furthermore, we give some characterizations of Lorentz fields of electromagnetic fields of associated with spherical magnetic conformable particle. Finally, we obtain the energy of the conformable derivative of the normalization function of the Lorentz forces and we have energy relations of some vector fields associated with spherical frame in the de-Sitter space ${𝕊}_1^2$.

We investigate the role of bulk and shear viscosity in the spatially homogeneous anisotropic spacetime, in particular, the Kantowski–Sachs (KS) spacetime. General conditions for the bouncing evolution of universe in anisotropic background have been obtained by using the derived propagation equations of expansion scalar, shear scalar and spatial 3-curvature. We show that the presence of shear viscosity in the model prohibits the energy density to attain its extremum in the bouncing model. We explore the qualitative behavior of KS cosmologies by formulating the Einstein’s field equations into a plane-autonomous system of equations by taking dimensionless equation of state. The stability of the system has been investigated by evaluating and analyzing the eigenvalues at the critical points. The stable solutions exist for the system composed of bulk and shear viscosity. The present analysis through dynamical system method illustrates that the universe does not exhibit synchronous bounce with perfect fluid and/or viscous fluids in the KS spacetime.

A constructive procedure is given to determine all ideals of a finite-dimensional solvable Lie algebra. This is used in determining all conjugacy classes of subalgebras of a given finite-dimensional solvable Lie algebra.

This template retrieves M-shaped rational solitons and their interactions with kink waves, homoclinic breathers, multiwave, Grey-black optical solitons, periodic cross-rational solitons and kink cross-rational solitons propagation in optical fibers where self-phase modulation (SPM) is negligibly minor and therefore removed. The proposed equation contains spatio-temporal dispersions (STD), of second and third orders, to recompense for small group velocity dispersion (GVD). All new analytical solutions are found by utilizing the symbolic computation with logarithmic transformation and ansatz functions approach. Moreover, stability characteristics of all solutions are found.

We consider the Little Rip (LR), Pseudo Rip (PR) and bounce cosmological models in the Friedmann–Robertson–Walker (FRW) metric with nonzero spatial curvature. We describe the evolution of the universe using a generalized equation of state (EoS) in the presence of a viscous fluid. The conditions of the occurrence of the LR, PR and bounce were obtained from the point of view of the parameters of the generalized EoS for the cosmic dark fluid, taking into account the spatial curvature. The analytical expressions for the spatial curvature were obtained. Asymptotic cases of the early and late universe are considered. A method of Darboux transformation (DT) was proposed in the case of models of an accelerating universe with viscosity.

By applying the Keeton and Petters technique, we find the deflection angle as a series expansion with a specific factor of mass. We investigate the shadow structure of a regular black hole with cosmic string with a non-magnetized, pressure-free plasma around it. In order to separate the Hamilton equation and locate the photon areas, certain plasma distributions are taken into consideration. Moreover, by utilizing a new ray-tracing technique, we analyze the photon’s path around a regular black hole with cosmic string in the presence of plasma whose density depends on the radius coordinate. We also analyze the plasma effects on the shadow of black hole. We obtain an analytical formula that describes the boundary curve of the shadow for such a black hole in an expanding universe for an observer at any finite location outside the horizon. In the presence and absence of plasma, we detect deviations that are further investigated by analyzing the geometry of shadow angle at a particular value of plasma frequency.

The movement of a thin vortex in a thin viscous fluid by the motion of a curve propagating in Lorentz–Minkowski space $E_1^3$ is described by the vortex filament or smoke ring equation and can be viewed as a dynamical system on the space curves in $E_1^3$. This paper investigates the harmonic evolute surfaces of time-like Hasimoto surfaces in $E_1^3$. Also, we discuss the geometric properties of these surfaces, namely, we obtain the Gaussian and mean curvatures of the first and second fundamental forms. As a verification, we construct a concrete example for the meant surfaces to demonstrate our theoretical results.

At present, we are aware that some recent changes in the cosmos cannot be explained by the standard interpretation of general relativity. In order to do so, we used the reconstruction scheme for recently proposed $f(T)$ gravity to look into the universe’s accelerated expansion. Here, we define the transit scale factor (TSF), a scale factor used to explain several geometrical and physical aspects. Then, using the Markov Chain Monte Carlo (MCMC) method, we estimate the best fit values for the model parameters imposed from data from Hubble’s, Standard candles and Uncorrelated BAO. The cosmos is moving from the deceleration phase into the acceleration phase, according to the evolution of the deceleration parameter. Also, we review the statefinder’s diagnostic elements $(r,s)$. We came to the conclusion that the reconstructed $f(T)$ models indicate that the universe is in an accelerating phase at $z=0$ and acts like quintessence models, and that it approaches $\mathrm{\Lambda }$CDM models at $z=-1$ which seem to be in good accord with the observations.

We have studied quantum systems on finite-dimensional Hilbert spaces and found that all these systems are connected through local transformations. Actually, we have shown that these transformations give rise to a gauge group that connects the Hamiltonian operators associated with each quantum system. This bridge allows us to connect different quantum systems, in such a way that studying one of them allows to understand the other through a gauge transformation. Furthermore, we included the case where the Hamiltonian operator can be time-dependent. An application for this construction will be achieved in the theory of control quantum systems.

In this article, we examine the universe’s dynamical behaviour in the context of the $f(R,G)$ theory of gravity, where $R$ and $G$ represent the Ricci scalar and Gauss-Bonnet invariant, respectively. The modified field equations are solved for the selection of $f(R,G)$ function as $f(R,G)=R^{\beta }{G}^{1-\beta }$ and of the deceleration parameter as a linear function of Hubble parameter, i.e., $q=n+mH$. We predict the best fit values of model parameters that would be in agreement with the recent observational datasets. We use the CC, Pantheon and BAO datasets as well as the Bayesian analysis and likelihood function together with the MCMC method. Further, we examine the physical behavior of cosmographic parameters corresponding to the constrained values of the model parameters as well as the energy density and pressure. The model obtained exhibits a transition from decelerating to accelerating expansion phases of the universe. We show that our $f(R,G)$ model can explain the late accelerating expansion of the universe without calling any dark energy term in the energy-momentum tensor.

Using the attractive properties of octonion algebra, an alternative formulation has been proposed for the Maxwell-type equations of compressible fluids. Although the origins of electromagnetic theory and fluid mechanics are completely different, a series of suitable and elegant 8-dimensional equations have been derived in a form similar to electromagnetic, gravitational counterparts previously given in relevant literature. Moreover, the corresponding matrix representations of derived expressions have also been presented after describing the connections between the algebraic properties of Cayley–Dickson octonions and some pseudo-real matrices.

We have investigated LRS Bianchi type-I cosmological model in $f(Q,T)$ theory of gravity with observational constraints. In $f(Q,T)$ theory of gravity $Q$ is non-metricity and T is a trace of energy–momentum tensor. Two different functional forms of $f(Q,T)=AQ+BT$ and $f(Q,T)=A{Q}^n+BT$ are considered in this work where $A$, $B$ and $n$ are model parameters. We have obtained the best-fit values of constant parameters using least square method for observational constraints using the available datasets like Hubble dataset $H(z)$, JLA, Pantheon, etc., by applying Root Mean Squared Error (RMSE) formula.

In the last decade Planck PR4 data together with ground-based experimental data such as BK18, BAO and CMB lensing tightened constraint of the tensor-to-scalar ratio, starting form $r<0.14$ to $r<0.032$, while the spectral index lies within the range $0.9631\le n_s\le 0.9705$. Viability of modified gravity theories, proposed as alternatives to the dark energy issue, should therefore be tested in the light of such new result. Here, we explore $F(R,{𝒢})$ gravity theory in regard to the early universe and show that it is not compatible with newly released constraints on $r$ and $n_s$ simultaneously. Further, it also fails to produce a feasible radiation-dominated era. It therefore questions the justification of using the model for resolving the cosmic puzzle.

In this work, we construct two new wormhole solutions in the theory dealing with non-minimal coupling between curvature and matter. We take into account an explicitly non-minimal coupling between an arbitrary function of scalar curvature $R$ and the Lagrangian density of matter. For this purpose, we discuss the Wormhole geometries inspired by non-minimal curvature coupling in $f(R)$ gravity for linear model in $R$ as well as nonlinear model in $R$. To derive these solutions, we choose the Gaussian and Lorentzian density distributions. To check the viability of these solutions, we plot the graphs for energy conditions and wormhole parameters. It is found that obtained wormhole solutions in both distributions satisfy the energy condition. The resulting wormhole solutions for both non-commutative distributions are determined to be physically stable when we evaluate the stability of these wormhole solutions graphically. It is concluded that wormhole solutions exist with viable physical properties in the non-minimal curvature–matter coupling of $f(R)$ gravity with Gaussian and Lorentzian distributions.

In this paper, we consider the generalization of Gross Pitaevskii equation for condensate of bosons with nonextensive statistics. First, we use the non-additive methods and formalism to obtain the well-known Schrödinger equation. Using a suitable Hamiltonian for condensate phase and minimizing the free energy of the system by non-additive formalism, we work out the nonextensive Gross Pitaevskii equation.

In this paper, the tunneling of fermions near the event horizon of Kerr–Newman–de Sitter (KNdS) black hole is investigated in frame dragging coordinate systems, Eddington coordinate system and Painleve coordinate system by using Dirac equation with Lorentz violation theory, Feynman prescription and WKB approximation. The Hawking temperature, heat capacity and change in black hole entropy of the black hole are modified due to the presence of Lorentz violation theory. The modified Hawking temperatures, heat capacities and change in black hole entropies at the event horizon of KNdS black hole would increase or decrease depending upon the choices of ether like vectors $u^{\alpha }$. In the absence of Lorentz violation theory, the original Hawking temperature, entropy and heat capacity are recovered.

Herein, we have discussed about the accretion of two extremely interesting forms of dark energies, namely “Modified Chaplygin–Jacobi Gas” (MCJG) and “Modified Chaplygin–Abel Gas” (MCAG) onto a Schwarzschild black hole of dimension 4. First, considering the above-mentioned dark energies, we have deduced the rate of change of mass and then evolved the required mass from it in terms of the redshift $z$. We have also plotted the mass versus redshift graphs for different values of the elliptic modulus $k$ in the case of MCJG and different values of $\alpha$ in the case of MCAG, respectively. Also, mass versus redshift graphs have been portrayed in the different stages of the FRW universe. We have found that in the case of both MCJG and MCAG accretion, the mass of the black holes follow an increasing pattern. But as expected in the case of the phantom energy-filled universe, the mass of the black hole shows decreasing nature. Again in the case of the $\mathrm{\Lambda }$CDM stage of the universe, accretion of MCAG primarily leads to an increase in mass, and then it gradually diminishes.

In this study, the bouncing cosmological models have been presented in the non-metricity-based gravitational theory, the $f(Q)$ gravity, where $Q$ be the non-metricity scalar. The two bouncing cosmological models, one in which the Lagrangian $f(Q)$ is assumed to have a linear dependence on $Q$ and the other in which it has a polynomial functional form have been shown. It has been obtained that the parameters of the models largely depend on the behavior of the models. The equation of state (EoS) parameter shows the bouncing behavior of the Universe. It should be highlighted that the built-in cosmological models go against the energy requirements. The kinematical and physical characteristics of the models are also analyzed.

We explicitly study and report the effects of the gravitational fields on the spectroscopic structure of Klein–Gordon (KG) particles in Som–Raychaudhuri cosmic string spacetime with space-like dislocation and a magnetic field. We extend this study to include the effects of the gravitational fields on some position-dependent mass (PDM) KG-particles. As effects of the gravitational fields, we report new spacetime associated degeneracies, vorticity-energy correlations and charge-energy correlations. Such degeneracies and correlations are observed in the spectroscopic structure of both constant mass KG-particles and PDM KG-particles at hand.