International Journal of Geometric Methods in Modern Physics

We discuss the proposal of a new method to transform an $f(R)$ metric gravity theory into a general relativistic theory including an auxiliary scalar field, recently introduced by Cotsakis et al. We argue that (i) the fact that the fourth-order equations of $f(R)$ metric gravity can be recast (via a Legendre transformation) into Einstein equations without any conformal rescaling has been thoroughly clarified in the previous literature, and (ii) the newly proposed method produces a set of equations that are not equivalent to the original theory. In the conclusion, a comment is added on another aspect of the Legendre transformation which seems to be generally overlooked.

Our work deals with the dynamical system analysis of quintessence dark energy scalar field model with exponential potential. A dynamical system analysis has been applied at the background level. Using suitable transformation of variables, the evolution equations are reduced to an autonomous system for exponential form of the scalar potential. The critical points are analyzed with center manifold theory and stability has been discussed by using Schwarzian derivative. Finally, cosmological implications of the critical points are discussed and it is found that the stability of the late-time attractor changes for quintessence dark energy model.

Primordial Black Holes (PBHs) could dominate in the early universe and, evaporating before Big bang Nucleosynthesis, provide new freeze in mechanism of dark matter (DM) production. The proposed scenario is considered for two possible mechanisms of PBH formation and the corresponding continuous PBH mass spectra so that the effect of non-single PBH mass spectrum is taken into account in the results of PBH evaporation, by which PBH dominance in the early universe ends. We specify the conditions under which the proposed scenario can explain production of dark matter in very early universe.

This study examines the structure of the universe during its high-energy era. In this context, we examine Barrow’s proposal of entropy, which forecasts a fractal configuration for the geometric properties of the black hole’s horizon. By applying the principles of thermodynamics, we get the equation of motion using the Barrow entropic consideration. A quasi-de Sitter phase, in which fractal and non-fractal states may coexist, is determined. This is an uncertainty state (coexistence of fractal and non-fractal states), and so a deformation for de Sitter spacetime, which contains a high-energy scale in the framework of the COBE normalization. With the kinetic energy term of the inflation field, we remove this ambiguity owing to the field oscillations. However, the oscillatory behavior of the inflation field continues and causes the universe to enter a new phase state. At this point, in which the inflation field oscillations are over, we obtain a solution that provides the universe transition to the radiation and dust phases.

In this paper, we study the relativistic dynamics of quantum oscillator fields within the context of a position-dependent mass (PDM) system in the background of a curved space-time. The chosen curved space-time is generated by a magnetic field incorporating a non-zero cosmological constant called Einstein–Maxwell–Lambda solution. To analyze PDM quantum oscillator fields, we introduce a modification into the Klein–Gordon equation by substituting the four-momentum vector $p_{\mu }\to (p_{\mu }+i\eta X_{\mu }+i{{ℱ}}_{\mu })$, where various four-vectors are defined by $X_{\mu }=(0,r,0,0)$, ${{ℱ}}_{\mu }=(0,{{ℱ}}_r,0,0)$ with ${{ℱ}}_r=\frac{f^{\prime }(r)}{4f(r)}$, and $\eta$ is the mass oscillator frequency. The radial wave equation for the modified Klein–Gordon equation is derived and subsequently solve for two distinct scalar multipliers: (i) $f(r)=e^{\frac{1}{2}\alpha r^2}$ and (ii) $f(r)=r^{\beta }$, where $\alpha \ge 0$ and $\beta \ge 0$. The resultant approximate energy levels and wave function for quantum oscillator fields are demonstrated to be influenced by the cosmological constant and the geometrical topology parameter which breaks the degeneracy of the energy spectrum. Furthermore, we observed noteworthy modifications in the approximate energy levels and wave function when compared to the results derived in the flat space.

$f(R,L_m)$ gravity provides an alternative explanation for the late-time acceleration of the universe without introducing the concept of dark energy. In this paper, the non-static Bianchi type-III spacetimes have been derived by considering $f(R,L_m)=f_1(R)+f_2(R)H(L_m)$. The solutions of field equations of $f(R,L_m)$ theory of gravity and Bianchi type-III metric have been obtained for some special cases by taking particular metric components. Noether symmetries have been discussed for the derived spacetimes and it was observed that all of the discussed spacetimes must have at least five Noether symmetries.

In this paper, we define a ruled surface obtained by an infinitesimal bending of a curve in Minkowski $3$-space. We also characterize such ruled surfaces according to causal characters of the curve. Furthermore, we find appropriate conditions of ruled surfaces obtained by infinitesimal bendings of curves to be (lightlike) developable, in particular, cylindrical or conical in Minkowski $3$-space, with examples presented.

The impact of deformation space on the physical characteristics of diverse physics systems has been thoroughly investigated in research papers. In this work, we study the deformed Klein–Gordon equation (DKGE) in the three-dimensional relativistic non-commutative quantum space (3D-RNCQS) regime by using the improved Hua plus modified Eckart potential (IHPMEP) model. For this consideration, the DKGE in the 3D-RNCQS regime is solved using the standard perturbation theory and the well-known Bopp’s shifts method with the Greene–Aldrich approximation to the centrifugal barrier. The new relativistic energy equation and eigenfunction for the IHPMEP in the presence of deformation space-space for the heterogeneous (CO, HF, and NO) and homogeneous (N₂, H₂, and Li₂) diatomic molecules are obtained to be sensitive to the atomic quantum numbers ($j,l,s,$ and $m$), the mixed potential depths ($V_0,V_1,V_2$, and $V_3$), the inverse of the screening parameter $\alpha$, and non-commutativity parameters ($\mathrm{\Phi }$, $\chi$, and $\zeta$). Analysis is performed on the non-relativistic limit of new energy spectra. By appropriately adjusting the combined potential parameters, we analyze the obtained new bound state eigenvalues of the DKGE and deformed Schrödinger equation with the IHPMEP in 3D-NCQS symmetries and obtain the new modified Eckart potential, the modified Hua potential, the modified Morse potential, and the modified Pöschl–Teller potential. Within the framework of the 3D-NRNCQS regime, the homogeneous and heterogeneous composite systems under IHPMEP models are examined. A thorough investigation is carried out into the impact of space-space deformation on the thermal parameters of the IHPMEP, including the partition function, mean energy, free energy, specific heat, and entropy. This work is of a fundamental absorbability nature and pedagogical interest in atomic and molecular physics.

In this paper, we give some classification of conformally harmonic $Z$-recurrent spacetime in the context of $F(R)$-gravity theory based on its scalar curvature and obtained the expression for energy density and isotopic pressure. Their behaviors along with EoS parameter are analyzed by considering two $F(R)$-gravity models. Moreover, the energy conditions for the considered models are studied and found to support the accelerated expansion of the universe.

We introduce the notion of conformal trajectories in three-dimensional Riemannian manifolds $M^3$. Given a conformal vector field $V\in 𝔛(M^3)$, a conformal trajectory of $V$ is a regular curve $\gamma$ in $M^3$ satisfying ${\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }=qV\times {\gamma }^{\prime }$, for some fixed nonzero constant $q\in ℝ$. In this paper, we study the conformal trajectories in the space forms ${ℝ}^3$, ${𝕊}^3$ and ${ℍ}^3$. For (non-Killing) conformal vector fields in ${𝕊}^3$ (respectively, in ${ℍ}^3$), we prove that conformal trajectories have constant curvature and its torsion is a linear combination of trigonometric (respectively, hyperbolic) functions on the arc-length parameter. In the case of Euclidean space ${ℝ}^3$, we obtain the same result for the radial vector field and characterizing all conformal trajectories.

In this study, our primary focus lies in meticulously exploring the intricacies of the universe by employing a flat Friedmann–Lemaître–Robertson–Walker (FLRW) model within the framework of $f(R,G)$ gravity. In our analysis, the $f(R,G)$ function is delineated as the sum of two distinct components, commencing with a quadratic correction of the geometric term denoted by $f(R)$, structured as $f(R)=R+\xi R^2$, alongside a matter term denoted by $f(G)=\lambda G^2$, where $R$ and $G$ symbolize the Ricci scalar and Gauss–Bonnet invariant, respectively. In the pursuit of solutions to the gravitational field equations within the $f(R,G)$ formalism, we embrace a specific expression for the scale factor, represented as $a={sinh}^{\frac{1}{\alpha }}(\beta t)$ [D. Rabha and R. R. Baruah, The dynamics of a hyperbolic solution in $f(R,G)$ gravity, Astron. Comput. 45 (2023) 100761]. In this context, the parameters $\alpha$ and $\beta$ intricately shape the scale factor’s behavior. The model posits the intriguing prospect of perpetual cosmic acceleration when $0<\alpha <1.19$, signifying a continuous expansion of the universe. Conversely, for $\alpha \ge 1.19$, the model proposes a pivotal transition from an early deceleration phase to the present accelerated epoch, a transformative shift in line with our understanding of cosmic evolution. Furthermore, the model demonstrates its credibility by satisfying Jean’s instability condition during the shift from a radiation-dominated era to a matter-dominated era, substantiating the formation of cosmic structures. In our analysis, a central focus is directed toward scrutinizing the equation of state parameter $\omega$ within our model. We delve into a comprehensive examination of the scalar field and meticulously assess the energy conditions surrounding the derived solution. To establish the robustness of our model, we deploy an array of diagnostic tools, including the Jerk, Snap, and Lerk parameters, along with the Om diagnostic, Classical stability of the model, and statefinder diagnostic tools, Observational Constraints on the Model Parameters. The outcomes, intertwined with a detailed analysis of both the results and the inherent intricacies of the model, are diligently clarified and presented.

It is known that a simply connected Riemann surface satisfies the isoperimetric equality if and only if it has constant Gaussian curvature. In this paper, we show that the circles centered at origin in the Randers Poincaré disc satisfy the isoperimetric equality with respect to different volume forms however, these Randers metrics do not necessarily have constant (negative) flag curvature. In particular, we show that Osserman’s result [12] of the Riemannian case cannot be extended to the Finsler geometry as such.

In this paper, we study the dynamical stability analysis by considering the torsion field $\varphi$ which is proportional to the Hubble parameter and barotropic equation-of-state parameter in the framework of the homogenous and isotropic FLRW metric with non-zero torsion in the scenario of dark matter and dark energy interacting terms. We discuss the stability of the critical points corresponding to the eigenvalues in the presence of dust and radiation at quintessence, $\mathrm{\Lambda }$CDM and phantom regimes by utilizing the different linear and nonlinear interaction models which depend on the energy density. As a result, we observe that the first linear model shows the stable behavior throughout the universe for dust and radiation phase. The other linear and nonlinear models show the stable critical points, computing with the eigenvalues at phantom and $\mathrm{\Lambda }$CDM era with the best fit value of interaction strength and coupling constant for dust and radiation.

First, we prove that a Ricci symmetric spacetime is a perfect fluid spacetime if and only if it is a generalized Robertson–Walker spacetime. Consequently, we establish that under certain constraints on the associated scalar, such a spacetime turns into a static spacetime and is classified under Petrov classification $I$, $D$, or $O$. Also, we notice that a semi-symmetric as well as pseudo-Ricci symmetric generalized Robertson–Walker spacetime is a perfect fluid spacetime. We examine conformally flat generalized Robertson–Walker spacetime as a solution of $f({{ℛ}}^{\ast })$-gravity theory and describe the physical relevance. For the model $f({{ℛ}}^{\ast })=ln({{ℛ}}^{\ast })-exp(-{{ℛ}}^{\ast })$, numerous energy conditions in terms of associated scalars are investigated. This model agrees with the current investigations which states that the Cosmos is in an accelerating phase.

This paper is devoted to investigating the behavior of charged compact stars in the $f(R,\varphi ,X)$ theory of gravity, where $R$ denotes the Ricci Scalar, $\varphi$ is a scalar potential, and $X$ is a kinetic term. For this purpose, we consider spherically symmetric spacetime with Bardeen geometry as exterior spacetime to investigate various properties of compact stars, including energy density and pressure components. Matching conditions are used to find model parameters. We present a detailed analysis, including a discussion of equations of state parameters, and stability criteria. In this analysis, we discuss the energy conditions to check the viability of our model. Furthermore, an equilibrium condition can be visualized through the modified Tolman–Oppenheimer–Volkov equation. It is concluded that Bardeen’s geometry supports the existence of compact stars in the background of the $f(R,\varphi ,X)$ theory of gravity.

The complete title of this paper should be “The geometry of the constant gravitational field or how to switch from the mechanical Lagrangian to a geometric Lagrangian” and in this abstract we sketch why. To begin with, let us say that Einstein managed to integrate the constant gravitational field in the special theory of relativity, obtaining a metric whose geodesics are locally parabolas. The existence of a similar metric deduced from the considerations of Newtonian mechanics is somehow obstructed by the Lagrangian involved in the description of the constant gravitational field. Therefore, we need to explain how the mechanical Lagrangian above which is not suitable for a metric description of the constant gravitational field can be switched into a geometrical Lagrangian having the same geodesics and suitable for the metric description of the constant gravitational field. The geometric method found by us is related to a parabolic type of transformations. A consequence of the previous method can be applied for the study of bending of light rays in a constant gravitational field.

In this work, static traversable wormholes are investigated in the context of $f(R,L_m)$ gravity theory. Considering $f(R,L_m)=a{R}^2+cR+d{L}_m$ model (in which $a,c$ and $d$ are constants) and assuming $2\rho +p_r+2p_t=0$, a new shape function is obtained which is not asymptotically flat, but satisfies all the conditions of a shape function. Also in this case, all energy conditions (except strong energy conditions) are satisfied. Assuming $f(R,L_m)=L_m{R}^2$ and shape function $b(r)=\frac{r_0^2}{r}$, a new energy density function is obtained. Finally, a comparative study of ECs is investigated for different shape functions under the assumption $\rho =q{(\frac{r}{r_0})}^{-n}$ and $f(R,L_m)=k{L}_m{R}^{1+1.5w}$ (where $k$, $w$ and $q$ are constants) and we have seen that the presence of exotic matter is not required in most of the cases.

The nonlinear Semilinear Dispersive-Fisher (SDF) model is an important equation describing wave dynamics in dispersive media influenced by nonlinear and diffusive effects. This study enhances both analytical and numerical methods to solve the SDF model and validate the solutions. We utilize the Modified Khater (MKhat) and Unified (UF) methods for analytical solutions, and He’s Variational Iteration (HVI) scheme for numerical approximations. Our investigation connects the SDF model to other nonlinear equations like the Korteweg–de Vries (KdV) and Fisher–Kolmogorov (FK) equations, demonstrating the consistency between analytical and numerical solutions. This research contributes to the understanding and modeling of wave phenomena in dispersive media with nonlinear and diffusive dynamics, offering refined analytical and numerical techniques specific to the SDF model. Key contributions include introducing the MKhat and UF methods for analytical solutions and employing the HVI scheme for numerical approximations, which are less represented in current literature. This work is positioned in mathematical physics, focusing on nonlinear evolution equations.