International Journal of Geometric Methods in Modern Physics

In this paper, we derive the exact solution of traversable wormholes illustrating spherically symmetric geometry with anisotropic matter distribution entering the throat in the formalism of $f({ℛ},{𝒯})$ gravity, where ${ℛ}$ is Ricci scalar and ${𝒯}$ is trace of energy–momentum tensor. For this purpose, we assume a power law-type generic function $f({ℛ},{𝒯})={ℛ}+\gamma {{ℛ}}^2+\alpha {𝒯}$, where $\gamma$ and $\alpha$ are being constants, with two different choices of shape functions $a(r)=r_0(\frac{b^r}{b^{r_0}}),0<a(r)<1$ and $a(r)=r_0{(\frac{cosh(r_0)}{cosh(r)})}^{\mu },0<\mu <1$. For each approach, we find the exact solution and studied the existence of wormhole solutions in the presence of exotic and non-exotic matter. The graphical behavior of equation of state (EoS) parameter and energy condition bounds is also investigated for each shape function. The realistic wormhole solutions are obtained for the shape functions which satisfy necessary conditions at the throat radius $r=r_0=1$. Finally, we observe that there is small deviation in the results obtained by General Relativity and $f({{ℛ}}^2,{𝒯})$ gravity.

In this work, we study some parameterizations of the deceleration parameter and investigate the accretion of the fluid in these parameterized models upon the higher-dimensional black hole (BH) and wormholes. For the undergoing analysis, $(n+2)$-dimensional wormhole (proposed by Morris and Thorne) and Schwarzschild BH were chosen in the backdrop of higher-dimensional Friedmann–Robertson–Walker (FRW) space-time. For these parameterized models, we analyze the change of masses of BH and wormhole in different dimensions.

In this paper, we investigate the gravitational lensing for power-Maxwell charged quintessence black hole (BH) without cosmological constant in Rastall’s theory of gravity in both the strong-field limit and weak-field limit. Furthermore, we discuss the gravitational lensing by the linear electrodynamic (LE) and nonlinear electrodynamic (NLE) Maxwell field with the equation of state (EoS) parameter of the quintessence ($\omega$) and compared to the case with Einstein’s gravity. We discuss the behaviors of the photon sphere radius $r_c$, impact parameter $u_c$, strong lensing coefficients $b_1$, $b_2$ and the deflection angle $\alpha (u)$ on the equatorial plane $(𝜃=\frac{\pi }{2})$ with the different values of Rastall parameter $\lambda$ for the case of LE and NLE Maxwell field in the strong field limit. We also investigate the angular separation $X$, angular position ${𝜃}_{\infty }$ and the magnification of the relativistic images. Further, we also study about the weak lensing BHs in Rastall gravity. We investigate the behaviors of weak deflection angle with the different values of Rastall parameter $\lambda$ for the case of LE and NLE Maxwell field in the weak-field limit.

Quantum fields are considered as generators of infinite-dimensional Clifford algebra $\mathrm{\text{Cl}}(\infty )$, which can be either orthogonal (in the case of fermions) or symplectic (in the case of bosons). A generic quantum state can be expressed as a superposition of the basis elements of $\mathrm{\text{Cl}}(\infty )$, with the superposition coefficients being multiparticle complex-valued wavefunctions. The basis elements, that are products of the generators of $\mathrm{\text{Cl}}(\infty )$ in the Witt basis, act as creation and annihilation operators. They create positive and negative energy states that include the bare and the Dirac vacuum as special cases. It is shown that the nonvanishing electric charge arises from an extra dimension or from doubling the number of creation and annihilation operators, which brings an extra imaginary unit $ī$ into the description. A further extension is to consider $ī$ as one of the quaternionic imaginary units and consider a generic state as having values in the quaternionic algebra or, equivalently, in the complexified two-dimensional Clifford algebra, $\mathrm{\text{Cl}}(2)\otimes C$. It contains two distinct fundamental representations of $\mathrm{\text{SU}}(2)$, one associated with the weak isospin doublet $({\nu }_e,e^{-})$ and the other one with the doublet of new leptons, denoted by $({𝜖}^{+},{\nu }_{𝜖})$, that together with the new quarks $(u^{\prime },d^{\prime })$ can be identified with dark matter.

This work will proceed to examine the dynamics of collapsing charged string cloud in a spherically symmetric rainbow metric. This is the framework that allows new stars and compact objects to emerge. The dynamical variables (mass density, string tension) and their physical existence as energy conditions are computed and visually depicted. We also calculated the duration and radius of development of the collapsing string cloud’s apparent horizons. The mass density, string tension, expected time and radius of apparent horizon creation are dependent on the Rainbow parameter $\eta .$ At the center of the configuration $R=0,$ the mass density, string tension and electric field intensity are at their maximum leading to the formation of a compact object (black hole). The fluid’s mass density and string tension are reduced by the electric field. Furthermore, the presence of an electric field delays the formation of a compact object (black hole).

In this work, we model two event horizons associated with a null Cartan curve as two lightlike hypersurfaces, respectively. We define a lightlike surface and a spacelike surface whose images coincide with the sets of critical values of two event horizons, and meanwhile we present two curves whose images coincide with the sets of critical values of these two surfaces, respectively. Using the singularity theory, we characterize the local topological structures of two event horizons, two surfaces and two curves at their singularities by means of two new invariants. Moreover, we also present a spacelike braneworld model along the particle as a spacelike surface in hyperbolic 3-space. An important fact shows that from the viewpoint of Legendrian dualities, this surface is ${\mathrm{\Delta }}_2$-dual to the tangent trajectory $L(t)$ of the null Cartan curve in Lorentz–Minkowski space-time. Meanwhile, we also consider a curve whose image is the set of critical values of this surface in hyperbolic 3-space. The third invariant of the null Cartan curve characterizes the singularities of the surface $\mathcal{ℒ}\mathcal{𝒮}$ and the curve $\mathcal{𝒫}$ in hyperbolic 3-space. A result indicates that surface $\mathcal{ℒ}\mathcal{𝒮}$ is locally diffeomorphic to the swallowtail $SW$ or cuspidal edge $CE$ and $\mathcal{𝒫}$ is locally diffeomorphic to the $(2,3,4)$-cusp at certain a singular point. It is also shown that there exist deep relationships between the singularities of the surface $\mathcal{ℒ}\mathcal{𝒮}$ and the curve $\mathcal{𝒫}$ and the order of contact between $L(t)$ and elliptic quadric ${ℰ}{𝒬}(v_0,-1)$ or the order of contact between $L(t)$ and spacelike hyperplane $\mathrm{\text{HP}}(v_0,-1)$. Finally, we present several examples to describe the main results.

Ever since a new symmetry was found for the imperfect fluid with vorticity, the question of the effect of perturbations on the symmetry itself has been raised. This new symmetry arose when realizing that local four-velocity gauge-like transformations would render the left-hand side of the Einstein equations invariant because the metric tensor would be invariant under this new kind of local transformations. Then the point was raised about the invariance of such a kind of transformations of the stress–energy tensor on the right-hand side of the Einstein equations in curved four-dimensional Lorentz spacetimes. It was verified that these invariances do not work with plain perfect fluid but they do work for imperfect fluids. The imperfect fluid stress–energy tensor will be invariant under local four-velocity gauge-like transformations when additional transformations are introduced for several variables included in the stress–energy tensor itself. This local invariance was also the criteria introduced in order to present a new stress–energy tensor for vorticity as well. New tetrads are at the core of the realization of the existence of this new symmetry because it is through these new tetrads that this new symmetry is realized. It is through the local transformation of these new tetrad vectors that we can prove that the metric tensor is invariant. This new kind of symmetry has its origins in a similar tetrad formulation as to the Einstein–Maxwell spacetimes formalism presented in previous manuscripts. In this paper, we will introduce local perturbations by external agents to the relevant objects in the imperfect fluid geometry. We will demonstrate a theorem that proves that the symmetries under four-velocity gauge-like transformations will be instantaneously broken but at the same time transformed into new symmetries. Because the local orthogonal planes determined by these new tetrads, which happen to be the local planes of symmetry will tilt under local perturbations. There will be a symmetry evolution under perturbations.

Conformal symmetries act as a source to investigate and classify exact solutions of the Einstein field equations (EFEs) via conformal vector fields (CVFs). It is well known that such classification leads to an important class of symmetries known as Killing symmetry which is the source of generating conservation laws of physics. In this paper, first we explore various classes of Kantowski–Sachs (KS) and Bianchi type III solutions in $f(T)$ gravity by adopting some algebraic techniques. Utilizing the above-mentioned technique, we come to know that there exist 30 cases where the KS and Bianchi type III space-times admit solutions in $f(T)$ gravity. Inspecting all the classes precisely, we familiarize that 25 solutions formulate non-conformally flat metrics whereas rest of the five solutions tend to formulate conformally flat metrics. We utilize the resulting solutions in finding the CVFs via direct integration approach. After a detailed study, we found that in two cases, the space-times admit proper CVFs, whereas in rest of the cases, the space-times either become conformally flat or it admit homothetic vector fields (HVFs) or Killing vector fields (KVFs). The overall dimension of CVFs for the space-times under consideration has turned out to be four, five, six or fifteen.

In this paper, we study the perturbative regime in the static patch of de Sitter metric in the Regge–Wheeler formalism. After realizing that perturbative regime in a de Sitter spacetime depicted in terms of usual spherical coordinates cannot be extended up to the cosmological horizon, we study perturbative equations, in particular the axial ones, in terms of the tortoise coordinate $r_{\ast }$. We show that perturbative regime can be extended up to the cosmological horizon, provided that suitable boundary conditions are chosen. As an application, we explore the Regge–Wheeler equation at short distances by performing a Taylor expansion. In order to study some possible quantum effects at short distances, we impose to the equation so-obtained the same boundary conditions suitable for a quantum 3D harmonic oscillator. As a result, a discrete spectrum can be obtained. The aforementioned spectrum is analyzed and a relation with possible effects denoting quantum behavior of gravitons is suggested.

This paper explores the aspects of compact stars in $f(T,{𝒯})$ gravity, where $T$ and ${𝒯}$ express the torsion and trace of the energy–momentum tensor. To achieve this goal, we consider a spherically symmetric spacetime with an anisotropic source of fluid. In particular, the Krori–Barua (KB) spacetime is considered to explore the solution of compact stars. In addition, we select three specific types of compact stars models, namely J1416-2230, 4U 1608-52 and Cen X-3. We develop the field equations using the KB spacetime with some specific form of $f(T,{𝒯})$ gravity. In addition, we consider the Schwarzschild geometry for the matching conditions at the boundary. Then, we calculate the values of all the relevant parameters by imposing matching conditions. We provide detailed graphical analysis to discuss the physical acceptability of different parameters, namely energy density, pressure, anisotropy and gradient. We also examine the stability of compact stars by exploring energy conditions, equations of state, conditions of causality, redshift functions, mass functions and compactness functions. Finally, we find that the obtained solutions are physically viable and also having good properties for the compact star models.

In the paper, on account of the Kummer confluent hypergeometric series of first type, we suggest the Turań-type inequalities for the hypergeometric supersine and supercosine functions by using the Cauchy–Bunyakovsky–Schwarz inequality.

The main goal of this paper is to study the weak gravitational lensing by Horndeski black hole in weak field approximation. In order to do so, we exploit the Gibbons–Werner method to the optical geometry of Horndeski black hole and implement the Gauss–Bonnet theorem to accomplish the deflection angle of light in weak field region. Furthermore, we have endeavored to extend the scale of our work by comprising the impact of plasma medium on the deflection angle as properly. Later, the graphical influence of the deflection angle of photon on Horndeski black hole in plasma and nonplasma medium is examined.

In this paper, in an extended theory of gravity, we have presented bouncing cosmological model at the backdrop of an isotropic, homogeneous space-time, in the presence of general relativistic hydrodynamics (GRH). The scale factor has been chosen in such a manner that with appropriate normalization, the quintom bouncing scenario can be assessed. Accordingly, the bounce occurs at $t=0$ and the corresponding Hubble parameter vanishes at the bounce epoch. The equation of state (EoS) parameter and the energy conditions of the model have been analyzed. The violation of strong energy condition further supports the behavior of extended gravity. As the bouncing cosmology suffers with instability, this model also shows the similar behavior.

Quantum hydrodynamics described by Madelung equations is analyzed in the framework of symplectic geometry i.e. in covariant phase space approach to geometric field theory. The pre-symplectic manifolds providing the phase spaces describing the Hamiltonian dynamics of quantum fluid are constructed from the set of all solutions of Madelung equations and their corresponding Lagrangian densities. The Madelung equations under consideration are the Madelung equations associated to Schroedinger equations (in the nonrelativistic case) and Madelung equations associated to Klein–Gordon equations (in the relativistic case). The cases where the coupling with electromagnetic fields is present are also considered here. Our symplectic formulation is different from that of [M. Spera, Moment map and gauge geometric aspects of the Schrodinger and Pauli equations, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1–36] in the choice of fundamental fields or variables. Here we regard density function $\rho$ and phase function $S$ not as canonical pair but as the fundamental fields of the theory. The Hamiltonian vector fields corresponding to an observable are obtained from the Hamiltonian equation generated by the observable. The Poisson bracket of two observables then is determined by the Hamiltonian vector fields associated to each observable. In general, the Poisson bracket of two observables is not unique due to the fact that every observable has more than one corresponding Hamiltonian vector field. It is pointed out that the Poisson bracket has a unique value over a certain subset of the set of all observables defined on the pre-symplectic manifold of the Madelung equation under consideration.

In this paper, we present a comprehensive study on the $(4+1)$-Fokas equation that includes new solutions and conservation laws of the equation. Specifically, for the first time, new shrunken-periodic and fascinating interactions on the bright soliton background have been investigated. Furthermore, a bright, dark, dipole soliton, tripole soliton, multipole soliton, periodic, doubly-periodic, damped-periodic, kink, lump, rogue wave, breather and their interactions have been attained and graphically illustrated. In addition, a significant interaction between four parabolic-periodic solitary waves is first detected and emphasized by harnessing 3D and contour plots. On the other hand, the latter has been compared to an experimentally-induced Moiré effect from a monitor screen camera capture. These outstanding results are achieved using the Lie symmetry approach. The multiple reduction process of the $(4+1)$-Fokas equation through the Lie algebra spanned elements, and the corresponding optimal system guarantees 23 solutions in general forms. To determine the physical meaning of the latter, an appropriate ansatz for the arbitrary functions and parameters has been proposed and listed in the corresponding tables, which leads to the various types of solitary wave solutions. Finally, the conserved vectors for the underlying equation are derived using the multiplier method.

In this paper, the novel exact solitary wave solutions for the generalized nonlinear Schrödinger equation with parabolic nonlinear (NL) law employing the improved $cosh(\mathrm{\Gamma }(\varpi ))-\mathrm{\text{sech}}(\mathrm{\Gamma }(\varpi ))$ function scheme and the combined $cos(\mathrm{\Gamma }(\varpi ))-sec(\mathrm{\Gamma }(\varpi ))$ function scheme are found. Diverse collections of hyperbolic and trigonometric function solutions acquired rely on a map between the considered equation and an auxiliary ODE. Received solutions are recast in several hyperbolic, rational and trigonometric forms based on different restrictions between parameters involved in equations and integration constants that appear in the solution. A few significant ones among the reported solutions are pictured to perceive the physical utility and peculiarity of the considered model utilizing mathematical software. The main subject of this work is that one can visualize and update the knowledge to overcome the most common techniques and defeat to solve the ODEs and PDEs. We demonstrated that these solutions validated the program using Maple and found them correct. The proposed methodology for solving the metamaterials model has been designed to be effectual, unpretentious, expedient and manageable. Applications of the solutions by the mentioned techniques will be useful to investigate the signals properties of optical fibers, plasma physics phenomena, electromagnetic fields occurrences and various types of nonlinear metamaterials models.

In this paper, we have investigated a spatially homogeneous and anisotropic accelerating Universe in Lyra’s geometry. Einstein’s field equations are solved explicitly by assuming dependent relation among the directional scale factors. In this paper, we use observational $H(z)$ data (OHD) and Pantheon compilation of recent type Ia supernovae (SN Ia) data to constrain the model parameters of the Universe in derived model. We constrained the present value of Hubble parameter $H_0$ and deceleration parameter $q_0$ as $H_0=69.39_{-0.78}^{+0.90}$ km/s/Mpc and $q_0=-0.583_{-0.009}^{+0.010}$ together with transition redshift $z_t=0.66_{-0.04}^{+0.05}$. We have also constrained the present age of the Universe in close vicinity of its observational value. Furthermore, we diagnose the statefinder parameters to know a geometrical discrimination of this model with standard $\mathrm{\Lambda }$CDM model. Some physical aspects of the Universe are also discussed.

This paper deals with an anisotropic cosmological model in the modified theory of $f(Q)$-gravity filled with bulk-viscosity fluid. We investigated the modified $f(Q)$-gravity with quadratic form $f(Q)=-\alpha Q^2$, where $Q$ is the non-metricity scalar and $\alpha$ is a positive constant, in Locally-Rotationally Symmetric (LRS) Bianchi type-$I$ space time universe. We have solved the modified Einstein’s field equations with bulk-viscosity factor $\xi (t)={\xi }_{0}H+{\xi }_1{H}^2$ and obtained average scale factor $a(t)=k_0{[e^{k_{1}t}\mathrm{\text{sech}}(k_{2}t)]}^3$ where $k_0$ is a constant and $k_1=\frac{{\xi }_1{(m+2)}^2}{108\alpha (2m+1)}$, and $k_2=-\frac{(m+2)\sqrt{{\xi }_1{(m+2)}^2+48\alpha {\xi }_0(2m+1)}}{108\alpha (2m+1)}$. We have investigated the model in two cases ${\xi }_0=0$ and ${\xi }_1=0$. We have found the best-fit values of model parameters with observational datasets like Union 2.1 compilation and Joint Light Curve Analysis (JLA) datasets by applying $R^2$-test formula in curve-fitting. Using these best-fit values of model parameters, we have studied the cosmological parameter like Hubble parameter $H$ and deceleration parameter $q$, equation of state (EoS) parameter, age of the present universe $t_0$, etc. We have obtained a transit model (i.e. decelerating to accelerating in current) in the first case ${\xi }_0=0$ and a transit phase model from accelerating to decelerating in the second case ${\xi }_1=0$. We have investigated the EoS parameter ${\omega }_v$ for bulk-viscosity and obtained the present values as $-1.2<{\omega }_v<2.5$, which is supported by the observational results. We have also analyzed energy conditions and statefinder parameters and found that our model approaches to LCDM model in the future.

In this work, we identify a smooth autonomous dynamical on a two-dimensional manifold with an exterior differential system $(\mathrm{\Sigma },{ℐ})$, where $\mathrm{\Sigma }$ is a three-dimensional Riemannian manifold and ${ℐ}$ is the differential ideal generated by the contact forms on $\mathrm{\Sigma }$. We investigate the intrinsic and the extrinsic geometry of a surface in $\mathrm{\Sigma }$ and show that for a particular dynamical system $\mathrm{\Sigma }$ admits a totally geodesic surface determined by a constant value of a coordinate function. We also exhibit that such a surface may define intrinsically nonflat minimal surface which is not necessarily totally geodesic.