International Journal of Geometric Methods in Modern Physics

In this work, the directional spherical indicatrices of a timelike space curve using tangent, quasi-normal and quasi-binormal vectors with q-frame are introduced. Then we work on the condition, that a timelike space curve to be slant helix, by using the geodesic curvature of the directional normal spherical indicatrix. Finally, an application of the results is given.

In this work, we explore the Tsallis holographic dark energy (THDE) model with IR cutoff as Granda–Oliveros horizon describing the Universe experiencing an accelerating expansion phase in the framework of flat Friedmann–Lemaître–Robertson–Walker (FLRW) Universe. The Universe evolution from earlier decelerated to the current accelerated phase is exhibited by the deceleration parameter acquired in the THDE model. By the value of the Tsallis parameter $\delta$, the equation of state (EoS) parameter for the THDE model represents the rich behavior of the Cosmos as, the quintessence era (${\omega }_T\ge -1$), crossing the phantom divide line and phantom era (${\omega }_T<-1$). The squared sound speed $v_s^2$ also suggests that the THDE model is classically stable at present. Also, the correspondence with the quintessence and phantom scalar field for the THDE model is analyzed to describe the accelerated expansion of the Universe.

In this work, the solutions of traversable wormholes are investigated inside modified $f(R)$ gravity under non-commutative geometry since matter possesses Lorentzian density distribution of a particle-like gravitation source. To find the exact wormhole solutions, two different shape functions $b(r)=r_0(\frac{a^r}{a^{r_0}})$, $0<a<1$, and $b(r)=r_0{(\frac{cosh(r_0)}{cosh(r)})}^{\mu }$, $0<\mu <1$, are considered. The first shape function was proposed by Mishra and Sharma [A new shape function for wormholes in $f(R)$ gravity and General Relativity, preprint (2020), arXiv:2003.00298v1 [physics.gen-ph]], however the second is newly defined in this paper. The behaviors of both shape functions are analyzed with the throat radius $r=r_0=1$. The equation-of-state (EoS) parameter energy conditions, and anisotropy parameter are discussed with graphical point of view.

In this paper, we have investigated the stability of General Relativistic Hydrodynamics (GRHD) model in a Friedmann–Robertson–Walker space-time with the volumetric power law in teleparallel gravity. The basic equations are derived along with its thermodynamical aspects. Thermodynamic temperature and entropy density of the model are also obtained. The state finder diagnostic pair and jerk parameter are analyzed to characterize different phases of the universe and the well-known astrophysical phenomena such as look-back time, the luminosity distance with redshift are derived. The model shows an accelerated expansion with inflationary era in the early and the very late time of the cosmic evolution. The GRHD model is stable at the early phases of the universe and is unstable at late times.

In this paper, we study Einstein standard stationary space-times $(M,g)=(ℝ\times \mathrm{\Sigma },-\beta d{t}^2+dt\otimes g_{\mathrm{\Sigma }}(\xi ,.)+g_{\mathrm{\Sigma }}(\xi ,.)\otimes dt+g_{\mathrm{\Sigma }})$ of dimension $n+1\ge 3$ with $\xi$ is conformal on the space-like hypersurface $(\mathrm{\Sigma },g_{\mathrm{\Sigma }})$. When $\mathrm{\Sigma }$ is assumed to be connected and complete, we give a criterion so that $(M,g)$ is Einstein. In particular, we show that if $(M,g)$ is Einstein and $\mathrm{\Sigma }$ is complete and connected, then the scalar curvature of $M$ and that of $\mathrm{\Sigma }$ are non-positive and $\xi$ is Killing with respect to $g_{\mathrm{\Sigma }}$.

The stability of two different bounce scenarios from $F(R,\mathcal{𝒢})$ modified gravity at later times is studied, namely a hyperbolic cosine bounce model and a matter-dominated one. After describing the main characteristics of $F(R,\mathcal{𝒢})$ modified gravity, the two different bounce scenarios stemming from this theory are reconstructed and their stability at late stages is discussed. The stability of the hyperbolic cosine model is proven, while the concrete matter-bounce model here chosen does not seem to accomplish the necessary conditions to be stable at later times.

This paper investigates the Noether symmetry approach in modified $f({ℛ},\phi ,\chi )$ theory of gravity, where ${ℛ}$ is the Ricci scalar, $\phi$ is a scalar field and $\chi$ is the kinetic energy term. For this purpose, we consider Locally Rotationally Symmetric (LRS) Bianchi Type-I spacetimes. The cosmological solutions are developed through Noether symmetry approach. The determining equations are computed in the context of a point-like Lagrangian for $f({ℛ},\phi ,\chi )$ gravity. In particular, three different cases are taken into account for the LRS Bianchi Type-I spacetimes. Some important conserved quantities for the Lagrangian in this modified theory of gravity are worked out through the determining equations. In this scenario, the graphical behavior of energy density, pressure component and equation of state parameter are reported and analyzed with the help of first integrals of motion. The negative trends of the strong energy conditions actually suggest that the said theory supports expanding Universe with dark energy.

In this paper, we first define a supersymmetric Gelfand–Dickey hierarchy using the Sato equation. Then we introduce some time variables and derivations involving those that from a non-abelian Lie superalgebra. Next, we construct additional symmetries for the supersymmetric Gelfand–Dickey hierarchy with the aid of the Orlov–Schulman operators which involve the above time variables and dressing operators. We give the additional flow equations of the supersymmetric Gelfand–Dickey hierarchy, and prove that the additional flows are symmetries of the supersymmetric Gelfand–Dickey hierarchy. In this way, an isomorphism between the additional symmetries of the supersymmetric Gelfand–Dickey hierarchy and the super $W_{\infty }$ algebra is constructed. Finally, the $B$ type supersymmetric Gelfand–Dickey hierarchy is constructed and we consider the additional symmetries for the supersymmetric Gelfand–Dickey hierarchy with the $B$ type condition which is compatible with the flows of the $B$ type supersymmetric Gelfand–Dickey hierarchy.

Within $R^2$ gravity, we study the linear stability of strongly gravitating spherically symmetric configurations supported by a polytropic fluid. All calculations are carried out in the Jordan frame. It is demonstrated that, as in general relativity, the transition from stable to unstable systems occurs at the maximum of the curve mass-central density of the fluid.

Motivated by some well-known results in the phase space description of quantum optics and quantum information theory, we aim to describe the formalism of quantum field theory by explicitly considering the holomorphic representation for a scalar field within the deformation quantization program. Notably, the symbol of a symmetric ordered operator in terms of holomorphic variables may be straightforwardly obtained by the quantum field analogue of the Husimi distribution associated with a normal ordered operator. This relation also allows to establish a $c$-equivalence between the Moyal and the normal star-products. In addition, by writing the density operator in terms of coherent states we are able to directly introduce a series representation of the Wigner functional distribution, which may be convenient in order to calculate probability distributions of quantum field observables without performing formal phase space integrals at all.

Quantum and classical physical states are represented in a unified way when they aredescribed by symplectic tomography. Therefore this representation allows us to study directly the necessary conditions for a classical universe to emerge from a quantum state. In a previous work on the de Sitter universe this was done by comparing the classical limit of the quantum tomograms with those resulting from the classical cosmological equations.

In this paper, we first review these results and extend them to all the de Sitter models. We show further that these tomograms can be obtained directly from transposing the Wheeler–De Witt equation to the tomographic variables. Subsequently, because the classic limits of the quantum tomograms are identified with their asymptotic expressions, we find the necessary conditions to extend the previous results by taking the tomograms of the WKB solutions of the Wheeler–DeWitt equation with an any potential. Furthermore, in the previous works, we found that the de Sitter models undergo the quantum-to-classical transition when the cosmological constant decays to its present value, we discuss at the end how far we can extend this result to more general models.

In the conclusions, after discussing any improvements and developments of the results of this work, we sketch a phenomenological approach from which to extract information about the initial states of the universe.

Within the solar system, approximate realizations of the three-body problem occur when a comet approaches a planet while being affected mainly by such a planet and the Sun, and this configuration was investigated by Tisserand within the framework of Newtonian gravity. The exact relativistic treatment of the problem is not an easy task, but this paper develops an approximate calculational scheme which computes for the first time the tiny effective-gravity correction to the equation of the surface for all points of which it is equally advantageous to regard the heliocentric motion as being perturbed by the attraction of Jupiter, or the jovicentric motion as being perturbed by the attraction of the Sun. This analysis completes the previous theoretical investigations of effective-gravity corrections to the Newtonian analysis of three-body systems, and represents an intermediate step towards relativistic effects on cometary motions.

We investigate phase transitions and critical behaviors of the Kerr–Sen black hole in four dimensions. Computing the involved thermodynamical quantities including the specific heat and using the Ehrenfest scheme, we show that such a black hole undergoes a second-order phase transition. Adopting a new metric form derived from the Gibss free energy scaled by a conformal factor associated with extremal solutions, we calculate the geothermodynamical scalar curvature recovering similar phase transitions. Then, we obtain the scaling laws and the critical exponents, matching perfectly with mean field theory.

The purpose of this work is to study the Tsallis, Rényi and Sharma–Mittal holographic dark energy models in order to evaluate the accelerated expansion of the Universe. In this regard, we consider the modified field equations for logarithmic and power law versions of entropy corrected models in FRW Universe filled with interacting dark energy and cold dark matter within the framework of Hořava–Lifshitz gravity. Employing the Nojiri and Odintsov (NO) cut-off as infrared cutoff, we explore the nature of the different cosmological quantities like the equation of state parameter, squared speed of sound and ${\omega }_d$–${\omega }_d^{\prime }$ cosmological plane during the cosmic evolution. The equation of state parameter shows the different stages of the evolution of the Universe for the considered models. By analyzing the cosmological plane ${\omega }_d$–${\omega }_d^{\prime }$, we obtain the freezing region for these models. Also, due to the study of squared speed of sound, we show the classically stable behavior of the considered models.

In this paper, we construct Darboux transformations of the noncommutative BKP (ncBKP) hierarchy and the noncommutative CKP (ncCKP) hierarchy. From these Darboux transformations, the explicit differences between the BKP (CKP) hierarchy and the noncommutative BKP (ncCKP) hierarchy can be seen clearly in this paper. These studies might be useful in the M theory and string theory in future.

The main purpose of this work is to focus on a discussion of Lie symmetries admitted by de Sitter–Schwarzschild spacetime metric, and the corresponding wave or Klein–Gordon equations constructed in the de Sitter–Schwarzschild geometry. The obtained symmetries are classified and the variational (Noether) conservation laws associated with these symmetries via the natural Lagrangians are obtained. In the case of the metric, we obtain additional variational ones when compared with the Killing vectors leading to additional conservation laws and for the wave and Klein–Gordon equations, the variational symmetries involve less tedious calculations as far as invariance studies are concerned.