International Journal of Geometric Methods in Modern Physics

Coherently with the principle of analogy suggested by Dirac, we describe a general setting for reducing a classical dynamics, and the role of the Noether theorem — connecting symmetries with constants of the motion — within a reduction. This is the first of two papers, and it focuses on the reduction within the Poisson and the symplectic formalism.

In this paper, some analogies between the Shapiro effect in the solar gravitational field and the Sagnac phase shift have been found. Starting from Einstein equivalence principle (EEP), which states the equivalence between the gravitational force and the pseudo-force experienced by an observer in a noninertial frame of reference, we imagine an observer on a rotating platform immersed in a gravitational field. In the Shapiro effect, for example, we know that the speed of an electromagnetic signal, calculated from the Earth, is less than $c$, but, if we calculate the speed using a clock at rest in the solar gravitational field, where the photon is passing, we get that the speed of light is $c$. Similarly, by considering the fictitious gravitational field of the rotating platform, if we look for a clock with respect to which the signal speed is $c$, we can interpret the time delay as a gravitational effect.

In this paper, we investigate some feasible regions for the existence of wormholes by introducing non-commutative geometry in terms of Gaussian and Lorentzian distributions in $f(R,G)$ modified theory of gravity. We explore wormhole solutions by assuming a viable model $f(R,G)=f_1(R)+f_2(G)$, where $f_1(R)$ is assumed to be a linear function of Ricci scalar and $f_2(G)$ is chosen to be a power law model. For $f(R,G)$ model under discussion, we select suitable form of redshift and shape functions, which is necessary for the existence of wormholes. We discuss mainly two types of solutions corresponding to different values of free parameters and obtain numerical results. The stability condition for numerical solutions is discussed via TOV equations and it is proved that gravitational and hydrostatic forces show opposite behavior to anisotropic force and hence cancel each other’s effect, which provides a stable wormhole configuration. By using graphical evolution, it has been found that null energy conditions (NEC) are violated for non-commutative Gaussian and Lorentzian distributions. However, some feasible regions have been found for the existence of wormhole solutions with Gaussian and Lorentzian distributions in the context of $f(R,G)$ gravity.

We explore the possibility to extend the Heisenberg’s uncertainty principle to a nonlinear extension of the quantum algebra related to a functional operator of the momenta as ${ℱ}(P)$. We show that such an extension of quantum mechanics is intimately connected to the non-commutative space-time algebra and the Lorentz symmetry deformations. We show that a large class of ${ℱ}(P)$ models can introduce superluminal modes in the quantized theories. We also show that the Hořava–Lifshitz theory is related to a large class of ${ℱ}(P)$ Quantum Mechanics.

A geometrical construction for a global dynamical time for binary point-like particle systems, modeled by relativistic equations of motions, is presented. Thus, we provide a convenient tool for the calculation of the dynamics of recent models for the dynamics of black holes that use individual proper times. The construction is naturally based on the local Lorentzian geometry of the spacetime considered. Although in this presentation we are dealing with the Minkowskian spacetime, the construction is useful for gravitational models that have as a seed Minkowski spacetime. We present the discussion for a binary system, but the construction is obviously generalizable to multiple particle systems. The calculations are organized in terms of the order of the corresponding relativistic forces. In particular, we improve on the Darwin and Landau–Lifshitz approaches, for the case of electromagnetic systems. We discuss the possibility of a Lagrangian treatment of the retarded effects, depending on the nature of the relativistic forces. The higher-order contractions are based on a Runge–Kutta type procedure, which is used to calculate the quantities at the required retarded time, by increasing evaluations of the forces at intermediate times. We emphasize the difference between approximation orders in field equations and approximation orders in retarded effects in the equations of motion. We show this difference by applying our construction to the binary electromagnetic case.

In this study, we investigated scalar field in $f(G)$-gravity by using LRS Bianchi type-I universe. Massless and massive scalar field models are separately constructed in $f(G)$-gravity. Massless scalar field models are examined in the cases of constant and exponential potential fields. For all models, solutions of field equations are obtained under the consideration of $A=B^n$. $f(G)$ functions for each model are separately attained in theory. It is shown that constructed models in the presence of massless scalar field permit quintessence scalar field. Also, it is observed that each model indicates expanding universe with deceleration. Also, kinematical quantities are analyzed in the light of obtained solutions. All models are concluded with a geometric and physical perspective.

Here, we have investigated the interaction of Bianchi type-I anisotropic cloud string cosmological model universe with electromagnetic field in the context of general relativity. In this paper, the energy-momentum tensor is assumed to be the sum of the rest energy density and string tension density with an electromagnetic field. To obtain exact solution of Einstein’s field equations, we take the average scale factor as an integrating function of time. Also, the dynamics and significance of various physical parameters of model are discussed.

In the context of cubic gravity for flat FRW metric we discuss the behavior of cosmological parameters (equation of state (EoS) parameter and square speed of sound) at Hubble horizon with the four different models of Hubble parameter. We observe the validity of generalized second law of thermodynamics (GSLT) and thermal equilibrium condition. It is found that cosmological parameters lie within the observational constraints. Also, GSLT and thermal equilibrium condition holds in almost all cases of Hubble parameter.

In this paper, we consider the particle on curved graphene space-time. In that case, we calculate the geometric form of potential which is known as Gaussian function. Here, we introduce the metric background which completely corresponds to curved graphene space-times. This metric leads us to obtain the geometry potential and we make the Laplace Beltrami equation in the mentioned metric background. We also rearrange such relation in terms of the second-order equation. By using the known polynomial, we solve the particle equation of motion in graphene background. In that case, we arrive the energy spectrum which has three terms. We take advantage from energy spectrum and investigate the thermal properties of system. The additional terms give us an opportunity to obtain the corrected entropy and free energy. So, we show that the additional term comes from geometry potential. This correction is important for the large scale. Hence, we show that correction term is logarithmic as well as small scale corrections.

In this paper, the thermodynamical properties and the phase transitions of the charged accelerating anti-de Sitter (AdS) black holes are investigated in the framework of the $f(R)$ gravity. By studying the conditions for the phase transitions, it has been shown that the $P-V$ criticality and the van der Waals like phase transitions can be achieved for $T\approx T_c$. The Joule–Thomson expansion effects are also examined for the charged accelerating AdS black holes of the $f(R)$ gravity. Here, we derive the inversion temperatures as well as the inversion curves. Then, we determine the position of the reverse point for different values of mass $M$ and parameter $b$ for the corresponding black hole. At this point, the Joule–Thompson coefficient is zero. So, in such case, we can say that such point is very important for the finding of cooling–heating regions. Finally, we calculate the ratio of minimum inversion temperature and critical temperature for such black hole.

Klein–Gordon oscillator in the background space-time generated by a rotating cosmic string subject to a Cornell-type scalar and Coulomb-type vector potentials including an internal magnetic flux is studied. We obtain the relativistic energy eigenvalues and the corresponding eigenfunctions and analyze a relativistic analogue of the Aharonov–Bohm effect for bound states.

In this paper, we have investigated some cosmological consequences of a quintessence dark energy model. In particular, we have obtained the forms of the equation of state parameter, the deceleration parameter and the field potential by considering a simple relation between the scale factor and the time derivative of the scalar field, instead of assuming any functional form for the scalar field potential or the scale factor or the equation of state parameter. We have found that the model provides the desired early deceleration followed by present acceleration of the universe. The potential derived numerically in this work in the form $V(\varphi )\simeq V_1{\varphi }^2+V_2{e}^{\varphi }+V_3$, where $V_1$, $V_2$ and $V_3$ are real constant parameters. It has also been found that our model mimics as the standard $\mathrm{\Lambda }$CDM model in future. Finally, we have also shown the evolution of the normalized Hubble parameter for our model and the $\mathrm{\Lambda }$CDM model and compared that with the latest Hubble parameter data.

In this paper, we have presented the Einstein–Maxwell equations which are described by the spherically symmetric spacetime in the presence of charge by exploiting the Tolman–Kuchowicz spacetime. The corresponding field equations are constructed and the form of charge distribution is chosen to be $Q_0(r)=Q{(\frac{r}{R})}^3=\xi r^3$, where $\xi$ is a constant quantity. We also find the values of unknown constants from junction conditions and discuss the behavior of effective energy density, effective radial and tangential pressure and anisotropic factor with two viable $f(R)$ models. We examine the physical stability of charged stellar structure through energy conditions, causality and stability condition. We use modified form of TOV equation for anisotropic charged fluid sphere to analyze the equilibrium condition. In this work, we model the compact star candidate SAXJ 1808.4 – 3658 and study the compactness level and anisotropic behavior corresponding to the variation of physical parameters which are involved in $f(R)$ models. Further, we evaluate some important properties such as mass-radius ratio compactness factor and surface redshift. It is depicted from this study that the obtained solutions provide strong evidences for more realistic and viable stellar model.

We reconstruct the Cartan Equations of null Killing magnetic curve $\gamma (s)$ in ${ℝ}_1^3$ with Killing magnetic vector field $V$ under the new Cartan frame $\{{\gamma }^{\prime }(s),V\wedge {\gamma }^{\prime }(s),V^{\prime }\wedge {\gamma }^{\prime }(s)-V\}$, which describe some new geometrical properties of $\gamma (s)$. The singularity properties of the rectifying surfaces and the binormal osculating surfaces of null Killing magnetic curves are given. As an application, two examples are given to explain the main results, where the singular loci of null Killing magnetic curves are obtained.

In this work, the Dirac equation is studied in the $z=0$ Lifshitz black hole ($Z0$LBH) spacetime. The set of equations representing the Dirac equation in the Newman–Penrose (NP) formalism is decoupled into a radial set and an angular set. The separation constant is obtained with the aid of the spin weighted spheroidal harmonics. The radial set of equations, which are independent of mass, is reduced to Zerilli equations (ZEs) with their associated potentials. In the near horizon (NH) region, these equations are solved in terms of the Bessel functions of the first and second kinds arising from the fermionic perturbation on the background geometry. For computing the boxed quasinormal modes (BQNMs) instead of the ordinary quasinormal modes (QNMs), we first impose the purely ingoing wave condition at the event horizon. Then, Dirichlet boundary condition (DBC) and Newmann boundary condition (NBC) are applied in order to get the resonance conditions. For solving the resonance conditions, we follow the Hod’s iteration method. Finally, Maggiore’s method (MM) is employed to derive the entropy/area spectra of the $Z0$LBH which are shown to be equidistant.

As quantum optical phenomena are based on Maxwell’s equations, and it is becoming important to understand quantum optical phenomena at short distances, so it is important to analyze quantum optics using short distance corrected Maxwell’s equation. Maxwell’s action can be obtained from quantum electrodynamics using the framework of effective field theory, and so the leading order short distance corrections to Maxwell’s action can also be obtained from the derivative expansion of the same effective field theory. Such short distance corrections will be universal for all quantum optical systems, and they will affect all short distance quantum optical phenomena. In this paper, we will analyze the form of such corrections, and demonstrate the standard formalism of quantum optics can still be used (with suitable modifications) to analyze quantum optical phenomena from this short distance corrected Maxwell’s actions.