International Journal of Geometric Methods in Modern Physics

In this study, we obtained generalized Cayley formula, Rodrigues equation and Euler parameters of an orthogonal matrix in 3-dimensional generalized space $E^3(\alpha ,\beta )$. It is shown that unit generalized quaternion, which is defined by the generalized Euler parameters, corresponds to a rotation in $E^3(\alpha ,\beta )$ space.We found that the rotation in matrix equation forms using matrix form of the generalized quaternion product. Besides, in $E^3(\alpha ,\beta )$ space, we obtained the rotations determined by the unit quaternions and unit split quaternions, which are special cases of generalized quaternions for $\alpha ,\beta \in ℝ,$ in 3-dimensional Eulidean space $E^3,$ in 3-dimensional Lorentzian space $E_1^3,$ respectively.

In this paper, we classify parabolic revolution surfaces in the three-dimensional simply isotropic space ${𝕀}^3$ under the condition

Using the loop quantum gravity, based on polymer quantization, we will argue that the polymer length (like string length) can be several orders larger than the Planck length, and this can have low energy consequences. We will demonstrate that a short distance modification of a quantum system by polymer quantization and by string theoretical considerations can produce similar behavior. Moreover, it will be demonstrated that a family of different deformed Heisenberg algebras can produce similar low energy effects. We will analyze such polymer correction to a degenerate Fermi gases in a harmonic trap, and its polymer corrected thermodynamics.

All equivalence classes for electromagnetic potentials and space-time metrics of Stackel spaces provided that Hamilton–Jacobi equation and Klein–Gordon–Fock equation for a charged test particle can be integrated by the method of complete separation of variables are found. The separation is carried out using the complete sets of mutually commuting integrals of motion of type (1.1) whereby in a privileged coordinate system, the given equations turn into parabolic type equations. Hence, these metrics can be used as models for describing plane gravitational waves.

We are interested in defining new energy functionals and solving them by using the variational approach method and Darboux equations. That is, we aim to define a new class of elastic curves on the regular surface $\mathrm{\Lambda }$. We further improve an alternative method to find critical points of the bending energy functionals acting on a class of magnetic curves on $\mathrm{\Lambda }$. As a result, we classify these critical curves as elastic magnetic curves of the Darboux vector family.

Several complementary approaches to investigate knotted solutions of Maxwell’s equations in vacuum are now available in literature. However, only partial results towards a unified description of them have been achieved. In order to shed some light on the debate, we inspect new ways to view relations between existing formalisms. In particular, combining the machinery of quaternions and spacetime foliations, we show how the Bateman pairs and Rañada complex maps are tied together in the case of the electromagnetic Hopfion. It is still an open question whether a similar approach applies to more complex solutions.

We discuss the generalization of phenomenological equations for electromagnetic field in superconductor based on algebra of space-time sedeons. It is shown that the combined system of London and Maxwell equations can be reformulated as a single sedeonic wave equation for the field with nonzero mass of quantum, in which additional conditions are imposed on the scalar and vector potentials, relating them to the deviation of charge density and currents in the superconducting phase. Also, we considered inhomogeneous equations including external sources in the form of charges and currents of the normal phase. In particular, a screening of the Coulomb interaction of external charges in a superconducting media is discussed.

In this paper, the multifluid equations of a plasma are reformulated in terms of conic sedenions in order to better reflect the analogies between multifluid plasma equations and Maxwell equations of classical electromagnetism. This formalism also provides us an efficient mathematical tool for unification of equations of fluid dynamics and electromagnetism in a compact and elegant way. Although the presented formulation enables us to express all of the field equations related to different disciplines, a set of Maxwell equations for multifluid plasma is combined into a single sedenionic equation. Moreover, the wave equation with source terms is generalized in a form similar to gravi-electromagnetism counterpart previously derived using this type sedenion.

In this study, we shall explore conformal symmetry to examine the wormhole models by considering traceless fluid. In this regard, we shall take anisotropic fluid with spherically symmetric space-time. Further, we shall calculate the properties of shape-functions, which are necessary for the existence of wormhole geometry. The presence of exotic matter is confirmed in all the cases through the violation of the Null Energy Condition. Furthermore, we have discussed the stability of wormhole solutions through the Tolman–Oppenheimer–Volkoff (TOV) equation. It is observed that our acquired solutions are stable under the particular values of involved parameters in different cases in conformal symmetry.

In this paper, we have explored the non-static anisotropic gravitational collapse and expansion solutions in Rastall theory of gravity. The field equations have been formulated for the non-static and inhomogeneous gravitating source. The Misner–Sharp mass function, auxiliary solution and trapped condition have been used to obtained a trapped surface. The auxiliary solutions have been used to obtain the expansion and collapse solutions; these solutions depend on $C(r,t)$ and parameter $\beta$ (which appears due to parametric form of metric components); also the range of parameter $\beta$ has been examined. The expansion scalar $\mathrm{\Theta }$ depends on parameter $\beta$, in the case of expansion $\mathrm{\Theta }>0$ for $\beta >-2$, while for collapse $\mathrm{\Theta }<0$ with $\beta <-2$. Also, the dynamics of the gravitating spherical source has been discussed graphically with the effects of Rastall parameter $\lambda$. For the physically reasonable fluid, the validity of energy conditions has been discussed for expansion and collapse solutions with the various values of $\lambda$.

This paper examines the cosmic evolution and current expansion for flat FRW universe through generalized ghost pilgrim dark energy model in $f(R)$ gravity. For this purpose, we reconstruct the models in terms of red-shift parameter with respect to three scale factors, i.e. power-law, intermediate and a scale factor defining two unified phases. We study compatibility of reconstructed $f(R)$ models through $r$-$s$ planes whereas graphical analysis of squared speed of sound leads to investigate their stability. We also explore the behavior of cosmological parameters such as equation of state and deceleration parameters. Moreover, we study the existence of freezing/thawing regions via $\omega$-${\omega }^{\prime }$ planes. We conclude that squared speed of sound identifies both stability/instability of reconstructed $f(R)$ models while equation of state and deceleration parameters specifies transition from decelerated to accelerated expanding cosmos. The $\omega$-$\acute{\omega }$ planes discover thawing as well as freezing regions for particular values of pilgrim parameter.

This paper investigates stability of thin-shell developed from the matching of interior traversable wormhole with exterior Ayon–Beato–Garcia–de Sitter regular black hole through cut and paste approach. We employ Israel formalism and Lanczos equations to obtain the components of surface stress-energy tensor at thin-shell. These surface stresses violate null and weak energy conditions that suggest the presence of exotic matter at thin-shell. The surface pressure explains collapse as well as expanding behavior of the developed geometry. We explore stability of the constructed thin-shell through both perturbations along shell radius as well as barotropic equation of state for three appropriate values of the shape function $b(x)=a{x}^{\alpha }+(1-a)$. It is concluded that stability of thin-shell depends on the shape function, charge and cosmological constant.

In this paper, we have employed the time-dependent Karmarkar condition to the shear-free nonstatic radiative model undergoing dissipative gravitational collapse (GC) with anisotropic radial heat flux distribution in the metric $f(R)$ formalism. We have obtained the final solutions of the $f(R)$ field equations under the consideration of three viable models $f(R)=R^2$, $f(R)=R+\lambda R(R+\beta R^2)$ and $f(R)=R+\xi R(e^{-\frac{R}{\tilde{R}}}$-$1)$, which are physically realistic at both the early and late-time cosmological epochs. In this relevance, the Karmarkar condition together with boundary condition presents a complete radiating star model which are experienced by the initial static configuration. The matching conditions have been derived between the exterior Vaidya metric and interior line-element across the boundary three surface which permits to study the physical credibility of our radiating star model. Afterwards, we have comprehensively discussed the physical significance of the matter variables and causal thermodynamical profile by graphical representation for our radiating star object. Moreover, we have also studied that the GC proceeds without the appearance of the horizon, i.e. the model is free of horizon.

Starting from the suitable action of four-dimensional Einstein gravity, which is coupled to a scalar dilaton field and logarithmic nonlinear electrodynamics, the related field equations have been solved in a spherically symmetric and energy-dependent spacetime. It has been found that the scalar potential, as the exact solution of the scalar field equations, must be written as the linear combination of two Liouville-type potentials. Three families of novel exact black holes have been introduced as the exact solutions to the gravitational equations, which are asymptotically non-flat and non-AdS. It has been illustrated that our solutions, recover their corresponding quantities in the Einstein–Maxwell-dilaton gravity theory when the nonlinearity parameter of electrodynamics is chosen very large. The thermodynamic and conserved quantities of the black holes have been calculated under the influence of the rainbow function. Then, through calculation of the Smarr mass formula, we have shown that the first law of black hole thermodynamics remain valid for all of the new dilatonic black holes. Thermal stability or phase transition of the black holes have been investigated by use of the canonical ensemble and thermodynamic geometry approaches, separately. By comparison of the results of these two alternative approaches, it has been found that they lead to the same results provided that the thermodynamic metric of HPEM is used.

We show that the scalar curvature of a $K$-contact Ricci soliton is constant and satisfies sharp bounds. Next we show that the scalar curvature of a ($2n+1$)-dimensional $K$-contact Ricci almost soliton is equal to $2n(2n+1)$ plus the divergence of a global vector field. Finally, we show that, if a complete connected Sasakian or $\eta$-Einstein $K$-contact manifold of dimension $>3$ is a proper Ricci almost soliton, then it is isometric to a unit sphere.

A time-dependent strain induces non-Abelian gauge field in a sheet of graphene. We discuss the effective field theory of this system of graphene subjected to a general form of time-dependent strain. We study the modifications to this effective field theory as a result of breaking Lorentz symmetry down to its sub-group, $SIM(1)$, employing the VSR formalism. As the effective theory describing the graphene system has gauge symmetry; to quantize this theory, we add a suitable ghost and gauge fixing terms to the original action. The resultant action is observed to be invariant under a BRST symmetry. We have studied the BRST symmetry of the resultant theory, and explicitly constructed the BRST transformations.