International Journal of Geometric Methods in Modern Physics

Doubly special relativity has been studied for the last 20 years as a way to go beyond the special relativistic kinematics, trying to capture residual effects of a quantum gravity theory. In particular, in doubly special relativity, the Einstenian relativity principle is generalized, adding to the speed of light another relativistic invariant, the Planck energy. There are several papers in the literature showing a connection between this deformed kinematics and a curved momentum space. Here we review how such kinematics can be derived from geometrical ingredients in a rigorous way, and how they can be generalized when regarding a curved spacetime. For the last aim, it is mandatory to consider a particular geometry for all phase-space variables, the so-called generalized Hamilton spaces. This construction allows us to define a spacetime in these theories, which in fact depends on the momenta. Then, starting from such a momentum dependent metric, we also revise several concepts of general relativity, with the final aim of establishing a self-consistent geometrical structure from which quantum gravity phenomenology can be explored.

Recently, presence of gravitational and axial anomalies in Riemann–Cartan (RC) spacetime [Garcia de Andrade, Class. Quantum Gravity 38 (2021)] indicates that topological densities expressed in terms of torsion may be very useful in understanding the physics involved. Pontryagin and Euler density may be presented in terms of torsion in teleparallelism. In this paper, computation of these topological torsional densities in Kerr black holes is given. The geometric quantisation of torsion is also discussed in terms of the teleparallel metric. Actually, the recent work of Del Grosso and Poplawski [arXiv: 2107.06112] showed that the torsion quantization appears when torsion is the generator of momentum quantum mechanical commutator. Moreover, Poplawski et al. showed that use of quantum torsion in quantum electrodynamic (QED) may avoid the divergences in Feynmann integrals, process called, a torsion regularization. Here, we compute the momenta components in terms of teleparallel Kerr black hole torsion generator, like a Casimir tensor. The quantum generator, while not yet related by torsion, is seen to be naturally associated to axial torsion skew-symmetric in the teleparallel geometry. An important fact is that from torsion quantization, torsion chirality appears naturally in the Einstein–Cartan spin-torsion analogy, where now spin-angular momentum of the black hole is connected to torsion. Torsional chiral anomalies are shown to diverge at black hole singularity, and then they cannot be canceled at singularity.

In this paper, spin polarized domain walls in Einstein–Cartan (EC) gravity have been recently investigated [L. C. Garcia de Andrade, Einstein–Cartan non-supersymmetric dynamo walls, Ann. Phys. 432 (2021).], where the spin polarizes along the direction orthogonal to planar domain wall. In this case, dynamo effects may appear on the wall. Moreover, dilatonic domain walls (DDWs) have shown to be responsible for inflating DWs in EC gravity, where DWs were non-magnetized [L. Garcia de Andrade, Axial and chiral anomalies in Riemann–Cartan spacetime black holes and electrodynamics, Class. Quantum Grav. 16 (1999) 2097–2103.]. In this paper, our goal is to show that when we consider a magnetized DW in the EC–Maxwell cosmology, the dilaton potential in the form of a DDW potential expressed in terms of torsion and the magnetic energy density may drive dynamo effects. Analytical solution of the dynamo equation shows that depending on the handness of DDW metric, the magnetic fields are amplified by dynamo action. From the coupled system of ECM and dilaton field equations, we find a relation between the magnetic energy density of $10^{-38}{\text{G}}^2$ and the DDW torsion of $10^{-17}{\text{cm}}^{-1}=10^{-31}\text{GeV}$ which leads to a thin DDW thickness of $1\text{fm}$. For a galactic magnetic field, energy density of $10^{-12}{\text{G}}^2$ leads to a thickness of $10^{18}\text{fm}$, a much thinner DDW. For galactic dynamo amplification, the ratio r between the magnetic energy density and photon energy density is around $10^{-34}$. From this ratio, a relation between photon mass density and torsion is obtained. Chiral Magnetic effect (CME), torsion and magnetic discontinuities along the wall are investigated. Magnetic field discontinuity across the wall allows us to understand why the magnetic fields at DDW must be located along the wall since the orthogonal magnetic field may vanish.

The commitment behind this work is to find the solutions of compact stellar models by using the spherically symmetric static space-time and anisotropic matter distribution in the framework of $f(T,{𝒯})$ gravity. In this study, we used the new type of metric functions $\nu (r)=2log(c_1+r^2)+log(c_2)$ and evaluated other functions $\lambda (r)=log(1+16c_{2}F{r}^2)$ by the well-known Karmarkar condition. In this work, we use the rotated and spinning tetrad named as off-diagonal and most generic function of $f(T,{𝒯})$ gravity. We find the strange stellar solutions by incorporating the MIT bag equation of state $p_r=\frac{1}{3}(\rho -4B_g)$ to evaluate cosmological constant. Aiming at the simplicity of solutions we utilize the given data for masses and radii of compact objects PSR J1416-2230, 4 U 1608-52, Cen X-3, EXO 1785-248, SMC X-1. By graphical presentation of the core properties of compact models, we show by concluding that our solutions are graphically fit and physically admissible and interesting by having the compatibility with the compact stars study.

In this paper, we investigate a nonlinear evolution equation which is derived from plane curve by Lie symmetry analysis and theory of planar dynamical system. The one-dimensional optimal system of the equation is given and all bifurcations of the system in different parametric regions are given from which we can guarantee the existence of traveling wave solutions including solitary wave solutions, periodic wave solutions and many uncountable infinite traveling wave solutions. Besides, we provide exact expressions of traveling wave solutions under different parametric conditions.

In this paper, we explore the generalized (3 + 1)-dimensional modified Kadomtsev–Petviashvili equations with variable coefficients, which are usually used in the fields of ferromagnetism, magneto-optics, plasma physics and fluid mechanics. For the sake of uncovering more physical phenomena related to this system, we consider the single-traveling-wave polynomial solutions in the light of the unified method and the double-wave polynomial solutions in line with the generalized unified method. Remarkably, the solitary-, soliton- as well as elliptic-type solutions are all discussed in these two kinds of solutions. Furthermore, the physical explanations of the solutions are given graphically and analytically for different choices of the free parameters (especially of the nonlinear coefficients of the equations that related to the physical insights). By discussing the wave propagation of each solution that we procured from the perspectives of amplitude, shape, symmetry or periodicity, we are capable of realizing the inherent characteristics of this equation commendably and discovering the correlative physical world more efficiently.

In this paper, we investigate the influence of the Aharonov–Casher (A–C) effect on the generalized Dirac oscillator containing the Coulomb-type potential function related to a relativistic neutral particle having a permanent magnetic dipole moment interacting with the external electromagnetic fields in (1 + 2)-dimensional cosmic string space-time. The eigenfunctions and energy eigenvalues of such a Dirac oscillator are derived by using the Nikiforov–Uvarov method. We indicate that implementing the scenario gives the relativistic modified exact analytical solutions. In this way, we can see that the degeneracy of the relevant relativistic energy eigenvalues is broken by depending on the Coulomb strength parameter under the influence of the curvature effect and the A–C effect.

In this paper, it is shown that energy-momentum tensors defined by Einstein and Tolman in the general theory of relativity are same for any spacetime. It is also shown that in general theory of relativity, the Einstein super-potential ${𝔈}_{\alpha }^{0\nu }$ and Tolman super-potential ${𝔗}_{\alpha }^{0\nu }$ are different in general. Four different examples are also given to ensure the validity of our established statements. In all these examples, we observed that the values of the Einstein ${𝔈}_{\alpha }^{0\nu }$ and Tolman ${𝔗}_{\alpha }^{0\nu }$ super-potentials are different but the Einstein and Tolman energy-momentum distributions come out as same. All the results obtained in these four examples are consistent with the already obtained results in the literature.

This paper is devoted to explore bouncing cosmology in $f(T,{𝒯})$ modified gravity. In the background of $f(T,{𝒯})$ modified theory with $T$ being Torsion scalar and ${𝒯}$ being the trace of energy–momentum tensor, the Hubble parameter is considered for the isotropic, flat, and homogenous universe. The standard bouncing scale factor has employed the exponential term to unify bounce with late-time acceleration. Phase portrait analysis reveals us that Minkowskian origin transfers to de Sitter origin in $(H,Ḣ)$-plane and as past infinite time $t\to -\infty$$\Rightarrow$$a(t)\to 0$ and $(H,Ḣ)\to$ constant which corresponds to some physical cosmological scenarios. Since field equations of $f(T,{𝒯})$ gravity are of second order, a one-dimensional autonomous system has been extracted. Finally, outcomes have been plotted graphically, and any type of singular behavior has not been experienced. The study of the equation of state parameter against cosmic time reveals that phantom phase is quite significant for both models. The inertial force in terms of Hubble parameter and cosmic time gives rise to pseudo Rip. The study of cosmographic parameters tells us that the present model corresponds to $\mathrm{\Lambda }$CDM at a large value of cosmic time.

It has been proposed that f(R) gravity is a viable proxy for General Relativity and a means for an alternative to dark energy. Within this context, we present the analytical solutions to the Bianchi type I universe. For this spacetime, the Killing and homothetic vector fields categorize the space into fourteen classes. Noether’s theorem is used to determine the first integrals associated with each isometry for every class and we use the geometry of the Noether point symmetries of the space to establish potential functions. We also quantize the Lagrangian of the spacetime to derive the Wheeler–DeWitt equation and successively determine invariant solutions for the wave function of the universe from the Lie point symmetries admitted by the equation. Finally, for all cases, exact classical solutions to the field equations are found.

In this paper, the idea of the Ricci flow is introduced and its significance and importance to related problems in mathematics had been discussed. Several functionals are defined and their behavior is studied under Ricci flow. A unique minimizer is shown to exist for one of the functionals. This functional evaluated at the minimizer is strictly increasing. The results for the first functional considered are extended to manifold with boundary. Finally, two physically motivated examples are presented.

Based on our previous works concerning pseudo differential operators and their supersymmetric versions, we investigate to present an approach to interpret some results concerning the mapping between two superconformal field theories. As a direct result, we have reduced the number of super fields which parametrizes N = 2 super W₃-algebra theory. Such a decomposition can be interpreted physically as the interaction between a fermionic particle of conformal spin $3/2$ and a bosonic particle of spin $1$ ($\gamma$, $W^{\pm }$ or $Z^0$ for example) given rise to supersymmetric particles of total conformal spin $5/2$.

The main purpose of this paper is to investigate the cohomology of semi-invariant submanifolds in metallic Riemannian manifolds. We show that there exist well-defined canonical de Rham cohomology classes for such type of submanifolds under some necessary conditions. We also discuss the non-triviality of these cohomology classes. Finally, we provide two examples in order to support the results obtained from the paper.

In this paper, the Higgs-like approach is used to analyze the quantum dynamics of a harmonic oscillator constrained on a circle. We obtain the Hamiltonian of this system as a function of the Cartesian coordinate of the tangent line through the gnomonic projection and then quantize it in the standard way. We then recast the Hamiltonian in a shape-invariant form and derive the spectrum energy of the confined harmonic oscillator on the circle. With help of the f-deformed oscillator algebra, we construct the coherent states on the circle and investigate their quantum statistical properties. We find that such states show nonclassical features like squeezing and sub-Poissonian statistics even in small curvatures of the circle.

In this paper, we will obtain the basic parameters of primordial inflation such as power spectrum, scalar spectral index, and tensor-to-scalar ratio as a function of Hankel index $\nu$ for asymptotic-de Sitter inflation and then, based on the observational constraints on these parameters from Planck data, we obtain the allowable range for index $\nu$. We will show that only the non-phantom phase is acceptable for quasi-de Sitter inflation at early universe. As a significant result, both of the primordial and late accelerating expansion are always in a quasi-vacuum phase (with $w\approx -1$) and it has never been in a pure-vacuum phase (with $w=-1$), i.e. cosmological constant. Therefore, based on our recent finding [E. Yusofi, M. Khanpour, B. Khanpour, M. A. Ramzanpour and M. Mohsenzadeh, Surface tension of cosmic voids as a possible source for dark energy, Mon. Not. R. Astron. Soc. 511 (2022) L82], we offer the surface tension of cosmic voids as a suitable alternative to the cosmological constant $\mathrm{\Lambda }$, for accelerating universe.

Motivated by the recent interest in Lie algebraic and geometric structures arising from tensor products of division algebras and their relevance to high energy theoretical physics, we analyze generalized bioctonionic projective and hyperbolic planes. After giving a Veronese representation of the complexification of the Cayley plane $𝕆P_{ℂ}^2$, we present a novel, explicit construction of the bioctonionic Cayley–Rosenfeld plane $(ℂ\otimes 𝕆)P^2$, again by exploiting Veronese coordinates. We discuss the isometry groups of all generalized bioctonionic planes, recovering all complex and real forms of the exceptional groups $F_4$ and $E_6$, and characterizing such planes as symmetric and Hermitian symmetric spaces. We conclude by discussing some possible physical applications.

In this study, our main purpose is to study a model which is homogenous and anisotropic describing early stage of Universe in non-minimally coupled scalar theory. It is quite interesting to study a model describing early stage in modified theory which is related to scalar field. In this context, we introduced Kantowski–Sachs metric for perfect fluid in non-minimally coupled scalar theory. Exact solution of field equations is attained for $f(R,\varphi )=(1+\xi {\eta }^2{\varphi }^2)R$ model. Matter distributions for radiation, cosmic string and cosmological constant are investigated for constructed model. For different selections of arbitrary constants, Kantowski–Sachs model shows three different behaviors related with Hubble parameter and deceleration parameter. Also, some observational parameters, such as luminosity distance, angular diameter distance and distance modulus, are obtained for Kantowski–Sachs Universe. Lastly, we get Statefinder parameters for the model. All properties of constructed model are discussed from a physical and geometric point of view.

In this paper, we have investigated the physical behavior of cosmological models in modified teleparallel gravity with a linear function $f(T)=\alpha T+\beta$ where $\alpha$ and $\beta$ are model parameters and $T$ is the torsion scalar. We have considered a homogeneous and isotropic Friedman universe filled with bulk viscosity fluid. We have solved the field equations for the scale factor $a(\tau )$ and found $a(\tau )=c_1{[\mathrm{\text{sinh}}(n_2\tau )]}^{n_3}$ where $n_2=\sqrt{\frac{\beta (1+\omega )[{\kappa }^2{\xi }_0-3(1+\omega )(1+\alpha )]}{8}}$ and $n_3=\frac{2}{[{\kappa }^2{\xi }_0-3(1+\omega )(1+\alpha )]}$ and $c_1$ is an integrating constant, $\omega$ is the equation of state (EoS) for normal matter and ${\xi }_0$ is generated from bulk viscosity fluid. We have calculated the several cosmological parameters for this scale factor and studied their physical and geometrical behavior along with the observational data sets $H(z)$ and Union $2.1$ compilation of SNe Ia data sets. We have observed that the $\beta$ factor reveals the presence of cosmological constant and for $\beta =0$, the acceleration drives by the bulk viscosity of the fluid and it behaves just like dark energy model without cosmological constant. We have studied the effective EoS ${\omega }_{\mathrm{\text{eff}}}$ and found $-0.2807\le {\omega }_{\mathrm{\text{eff}}}\le -0.1760$. We have evaluated the age of the present universe as $13.87$ Gyrs. Also, we have studied the nature of deceleration parameter with the signature-flipping point at $z_t=0.5,1.14$ and the present value of deceleration parameter $q_0$ is obtained as $-0.3303,-0.4152$, respectively, for both observational datasets.