International Journal of Geometric Methods in Modern Physics

In this paper, we review how an algebraic formulation for the dynamics of a physical system allows to describe a reduction procedure for both classical and quantum evolutions.

The new tetrads introduced previously for non-null electromagnetic fields in Einstein–Maxwell spacetimes enable a direct link to the local electromagnetic gauge group of transformations. Due to the peculiar elements in the construction of these new tetrads, a direct connection can be established between the local group of electromagnetic gauge transformations and local groups of tetrad transformations on two different local and orthogonal planes of eigenvectors of the Einstein–Maxwell stress–energy tensor. These tetrad vectors are gauge dependent. It is an interesting and relevant problem to study if there are local gauge transformations that can map on the timelike-spacelike plane, the timelike and the spacelike vectors into the intersection of the local light cone and the plane itself. How many of these local gauge transformations exist and how the mathematics and the geometry of these particular transformations play out. These local gauge transformations would be singular and it is important to identify them.

In this paper, we investigate a feasible holography with the Kitaev model using dilatonic gravity in AdS₂. We propose a generic dual theory of gravity in the AdS₂ and suggest that this bulk action is a suitable toy model in studying quantum mechanics in Kitaev model using gauge/gravity duality. This gives a possible equivalent description for the Kitaev model in the dual gravity bulk. Scalar and tensor perturbations are investigated in details. In the case of near AdS perturbation, we show that the geometry still “freezes” as is AdS, while the dilation perturbation decays at the AdS boundary safely. The time-dependent part of the perturbation is an oscillatory model. We discover that the dual gravity induces an effective and renormalizable quantum action. The entanglement entropy for bulk theory is computed using extremal surfaces. We prove that these surfaces have a fold bifurcation regime of criticality.

This study explores the interior geometry of static relativistic charged spheres in the background of a recently proposed modified $f(G,T)$ gravity, where $G$ and $T$ are the Gauss–Bonnet (GB) invariant and trace of energy–momentum tensor, respectively. The GB gravity is the low-energy limit of superstring theories. The structures of specific relativistic charged spheres Vela $X-1$, $SAXJ1808.4-3658$, and $4U1820-30$ are studied in this theory of gravity. We analyzed several physical behaviors of these relativistic charged spheres with the help of observational data and investigated the various aspects like density profile, stresses, the distribution of charges, stability, etc.

In this paper, a model of 3D Helmholtz resonator with two close point-like windows is considered. The Dirichlet condition is assumed at the boundary. The model is based on the theory of self-adjoint extensions of symmetric operators in Pontryagin space. The model is explicitly solvable and allows one to obtain the equation for resonances (quasi-eigenvalues) in an explicit form. A proper choice of the model parameter leads to the coincidence of the model solution with the main term of the asymptotics (in the window width) of the realistic solution, corresponding to small windows. A regularization is suggested to obtain a realistic limiting result for two merging windows.

In this paper, we show how the quasinormal modes (QNMs) arise from the perturbations of massive scalar fields propagating in the curved background by using the artificial neural networks. To this end, we architect a special algorithm for the feedforward neural network method (FNNM) to compute the QNMs complying with the certain types of boundary conditions. To test the reliability of the method, we consider two black hole spacetimes whose QNMs are well known: $4D$ pure de Sitter (dS) and five-dimensional Schwarzschild anti-de Sitter (AdS) black holes. Using the FNNM, the QNMs of are computed numerically. It is shown that the obtained QNMs via the FNNM are in good agreement with their former QNM results resulting from the other methods. Therefore, our method of finding the QNMs can be used for other curved spacetimes that obey the same boundary conditions.

Recently, the inverse electrodynamics model (IEM) was introduced and applied to find Reissner–Nordström black holes in the context of the general relativity coupled minimally with the nonlinear electrodynamics. The solution consists of both electric and magnetic fields as of the dyonic solutions. Here, in this note, we show that the IEM model belongs to a more general class of the nonlinear electrodynamics with $T=T_{\mu }^{\mu }=0$. Here, $T_{\mu }^{\nu }$ is the energy momentum tensor of the nonlinear electrodynamic Lagrangian. Naturally, such a dyonic RN black hole solution is the solution for this general class.

In this paper, we first introduce a nonisospectral problem associate with a loop algebra. Based on the nonisospectral problem, we deduce a nonisospectral integrable hierarchy by solving a nonisospectral zero curvature equation. It follows that the standard AKNS hierarchy and KN hierarchy are obtained by reducing the resulting nonisospectral hierarchy. Then, the Hamiltonian system of the resulting nonisospectral hierarchy is investigated based on the trace identity. Additionally, an extended integrable system of the resulting nonisospectral hierarchy is worked out based on an expanded higher-dimensional Loop algebra.

This paper studies the nature of fractional linear transformations in a general relativity context as well as in a quantum theoretical framework. Two features are found to deserve special attention: the first is the possibility of separating the limit-point condition at infinity into loxodromic, hyperbolic, parabolic and elliptic cases. This is useful in a context in which one wants to look for a correspondence between essentially self-adjoint spherically symmetric Hamiltonians of quantum physics and the theory of Bondi–Metzner–Sachs transformations in general relativity. The analogy therefore arising suggests that further investigations might be performed for a theory in which the role of fractional linear maps is viewed as a bridge between the quantum theory and general relativity. The second aspect to point out is the possibility of interpreting the limit-point condition at both ends of the positive real line, for a second-order singular differential operator, which occurs frequently in applied quantum mechanics, as the limiting procedure arising from a very particular Kleinian group which is the hyperbolic cyclic group. In this framework, this work finds that a consistent system of equations can be derived and studied. Hence, one is led to consider the entire transcendental functions, from which it is possible to construct a fundamental system of solutions of a second-order differential equation with singular behavior at both ends of the positive real line, which in turn satisfy the limit-point conditions. Further developments in this direction might also be obtained by constructing a fundamental system of solutions and then deriving the differential equation whose solutions are the independent system first obtained. This guarantees two important properties at the same time: the essential self-adjointness of a second-order differential operator and the existence of a conserved quantity which is an automorphic function for the cyclic group chosen.

In this contribution, we consider the well-known equivalence between $f(R)$ gravity and Brans–Dicke-type scalar–tensor theories to study the evolution of scalar cosmological perturbations for a class of shear-free cosmological dust models with irrotational fluid flows. We use the $1+3$ covariant formalism to present the covariant linearized evolution and constraint equations. We then derive the integrability conditions describing a consistent evolution of the linearized field equations of quasi-Newtonian universes in the modified (scalar–tensor) theory of gravity. Finally, we derive the evolution equations for the density and velocity perturbations of the quasi-Newtonian universe. We apply the harmonic decomposition and explore the behavior of the matter density contrast by considering $R^n$ toy models. The growth of the matter density contrast for both short- and long-wavelength modes has been examined by applying certain assumptions of the initial conditions. We then apply the so-called quasi-static approximation to obtain exact solutions on small scales, but the results show that this approximation is not applicable here. Moreover, any small deviation from general relativity and any small change in the initial conditions of the perturbations causes huge orders-of-magnitude deviations from limiting general relativistic results, potentially putting constraints on the modified theory in the quasi-Newtonian cosmologies treatment. Our current work differs from other works in the literature, in that it is the first such work to show quasi-Newtonian cosmologies are unstable to linearized perturbations in modified gravity.

Xu et al. (Eur. Phys. J. C 79 (2019) 708) have anticipated the theory of Gravity. The modified study of $f(Q,T)$ is elucidated here as Cosmological model. In it the action holds a role as a capricious arbitrary function $f(Q,T)$. At this juncture $Q$ functions as non-metricity and for matter fluid, $T$ outlines as energy-momentum tensor. The function $f(Q,T)$ quadratic in $Q$ and linear in $T$ as $f(Q,T)=\alpha Q+\beta Q^2+\gamma T$ has been taken as our research in which $\alpha$, $\beta$ and $\gamma$ stand as model parameters, induced by $f(R,T)$ gravity. A range of cosmological parameters have been attained by us such as in Universe viz. Hubble parameter $H$, Friedmann–Lemaitre–Robertson–Walker (FLRW), deceleration parameter $q$, etc. in terms of scale-factor and in terms of redshift $z$ by confining to the law of energy-conservation. The fittest values of the model parameters have been acquired by us as the observational constrictions on the model, by utilizing the accessible data sets like Hubble data sets $H(z)$, union $2.1$ compilation of SNe Ia data sets and Joint Light Curve Analysis (JLA) data sets. We have applied $R^2$-test formula. The values of various observational parameters have been premeditated by us viz. $H_0$, $q_0$, $t_0$ and state finder parameters $(s,r)$. They are absolutely very close to the standard cosmological models. It has also been observed by us that the deceleration parameter $q(z)$ exhibits signature-flipping (transition) point within the range $0.423\le z_t\le 0.668$. It is observed that it changes its phase from decelerated to accelerated expanding universe with equation of state (EoS) $-1.071\le \omega -0.96$ for $0\le z\le 3$.

This work discusses about the existence of compact star model in the context of $f(R,T)$ gravity with $R$ as the Ricci scalar and $T$ as the trace of energy–momentum tensor $T_{\mu \nu }$. The model has been developed by considering the spherically symmetric spacetime consisting of isotropic fluid with $f(R,T)=R+2\beta T$ with $\beta$ be the coupling parameter. The corresponding field equations are solved by choosing the well-known Finch–Skea ansatz [M. R. Finch and J. E. F. Skea, A realistic stellar model based on an ansatz of Duorah and Ray, Class. Quantum Gravity 6(4) (1989) 467–476]. For spacetime continuity, we elaborate the boundary conditions by considering the exterior region as Schwarzschild metric. The unknown constants appearing in the solution are evaluated for the compact star PSR J 1614-2230 for different values of coupling constant. The physical properties of the model, e.g. matter density, pressure, stability, etc. have been discussed both analytically and graphically. This analysis showed that the geometry and matter are compatible with each other as well as the model is in stable equilibrium in the context of $f(R,T)$ modified gravity.

In this paper, first, we deduce various classes of Bianchi type-I perfect fluid solutions in the $f(T)$ gravity. In order to obtain the above-mentioned classes, we use some algebraic techniques that help to formulate 18 cases. As an application, we bifurcate the obtained classes according to their conformal vector fields (CVFs). Inspecting each class using the process of direct integration, we come to know that in one case, the space-time admits proper CVFs whereas in rest of the cases, obtained metrics either become conformally flat or admit underlying symmetries of the CVFs i.e. homothetic vector fields (HVFs) or Killing vector fields (KVFs). The overall dimension of CVFs for the Bianchi type-I perfect fluid space-times in $f(T)$ gravity turns out to be three, four, five, six and fifteen.

We study a relation between certain extensions of the Clifford bundle and Finsler type structures that naturally generalize the standard Clifford relation between (pseudo)-Riemannian metric structures and Dirac matrices. We show for flat metrics that there is a triangle map between Finsler structures constructed from a (pseudo)-Riemannian metric and $1$-forms on $M$, the extension of the Clifford bundle and relevant Dirac type operators.

In this paper, we discuss a new way to get a quantum holonomy around topological defects in $C_{60}$ fullerenes. For this, we use a Kaluza–Klein extra dimension approach. Furthermore, we discuss how an extra dimension could promote the formation of new freedom degrees which would open a discussion about a possible qubits computation.

In this paper, we attempt to introduce a new approach to determine the magnetic helicity and energy dissipation of the normal electromagnetic vortex filaments in magnetohydrodynamics (MHD). We successfully compute the vector potential and magnetic helicity of Frenet–Serret vectors and associated electromagnetic fields in the existence of the Lorentz force. We further describe a set of differential equations relating the magnetic vector potential of the electromagnetic vector fields and the Godbillon–Vey helicity model. Then, we focus on the time evolution of the normal electromagnetic field lines in MHD to evaluate the evolution of the magnetic helicity and energy dissipation theoretically. As a result, we obtain analytical solutions and numerical simulations of the time evolution equations of both geometric quantities as well as the magnetic and electric field vectors.