International Journal of Geometric Methods in Modern Physics

We prove that an almost pseudo ${𝒵}$ symmetric spacetime with Codazzi type of ${𝒵}$ tensor is perfect fluid whose velocity vector field $B^i$ is parallel. The energy density $\sigma$ of such perfect fluid spacetime is constant and the state equation is obtained. Also, this spacetime is shown to be a static spacetime. Next, it is shown that such spacetime is a GRW spacetime with Einstein fiber. This kind of spacetime is investigated in f (R) theory of gravity; in this case, we find the forms of the isotropic pressure p and the energy density $\sigma$ which are constant. Further, some energy conditions are studied in this spacetime. Finally, a concrete example of almost pseudo ${𝒵}$ symmetric spacetimes is considered.

I propose using the group $\text{SL}(4,ℝ)$ as a generalization of the Dirac group $\text{SL}(2,ℂ)$ used in quantum mechanics, in an attempt to match the symmetries of the ‘internal’ spacetime of elementary particles to the symmetries of the ‘external’ spacetime of general relativity. While my conclusions do not amount to a new model or theory, they do highlight some specific points where there are apparent conflicts between the mathematics and the physics of the standard model, that might repay closer investigation. As evidence that this mathematical approach may have interesting things to say about physics, I propose two mass equations that seem not to have been observed previously.

This paper concerns with the similarity solutions using Lie group theoretic method for one-dimensional unsteady isothermal flow behind the shock wave in the mixture of non-ideal gas and small solid particles in rotating medium. Group theoretic technique brings the different possible cases of potential solutions with the power law, particular case of power law and exponential law shock paths with the different choices of constants present in the generators of the Lie group of transformations. Similarity solutions for non-ideal dusty gas (a mixture of non-ideal gas and small solid dust particles) exist if the density of the ambient medium is constant. Solutions are obtained for both exponential law and power law shock paths. The effects of the variation of the physical parameters on the flow variables distribution behind the shock wave front, and on the shock are discussed. The shock strength increases with an increase in the ratio of the specific heat of the solid particles to the specific heat of the gas at constant volume ${\beta }^{\prime }$, the initial azimuthal velocity variation index ${𝜃}_1$, and the ratio of the density of solid particles to the initial density of the gas $G_0$. The shock wave decays with an increase in the value of gas non-idealness parameter $\overline{b}$.

By introducing a new coordinate condition we divide the cylindrical spacetime of Levi-Civita (LC) into a sourceful inner ($\rho \le {\rho }_0=\mathrm{\text{const.}}$) and vacuum outer ($\rho \ge {\rho }_0$) regions. The emergent fluid with zero radial pressure confined to the finely-tuned inner region serves to make the spacetime regular in a natural way. At $\rho =0$ the LC spacetime becomes ‘Cartesian flat’ so that no deficit angle arises.

In this contribution, we investigate hierarchical nature of large-scale structure clustering through the oscillatory nature of the solutions of the Schrödinger-like Friedmann equation in a modified gravitational background described by the f (R) gravity theory. We find the cosmological solutions to the Schrödinger equation for different ranges of n of the $R^n$ toy model for both radiation and matter-dominated epochs of the expansion history for open, flat and closed spacetimes. Our results show that for certain choices of the model parameters and initial conditions, the formation and distribution of cosmic structures might indeed be hierarchical, leading to a natural explanation for the breakdown of the cosmological principle on small scales.

A new class of conformal invariants for a given spacetime M is introduced exploiting the conformal geometry of any light ray $\mathrm{\Gamma }$. Each congruence of light rays passing through a given point p defines the sky S(p) of such point. The new conformal invariants are defined on the bundle of skies of the spacetime M, being called sky invariants accordingly. The natural conformal covariant derivative defined on a light ray and its associated covariant calculus allows us to show the existence of a natural conformal invariant differential of arc that, together with the restriction of the curvature of the conformal covariant derivative, can be used to construct a sky invariant that will be called the sky curvature. An algorithm, that can be implemented on any symbolic manipulation software system, to compute the sky curvature will be discussed and the main ideas and the explicit computation of the sky curvature are illustrated in Schwarzschild spacetime.

We compute analytically the radiative quantum corrections, up to next-to-leading loop order, to the universal critical exponents for both massless and massive O(N) $\lambda {\varphi }^4$ scalar squeezed field theories for probing the universality hypothesis. For that, we employ six distinct and independent methods. The outcomes for the universal squeezed critical exponents obtained through these methods are identical among them and reduce to the conventional ones where squeezing is absent. Although the squeezing mechanism modifies the internal properties of the field, the squeezed critical indices are not affected by the squeezing effect, thus implying the validity of the universality hypothesis, at least at the loop level considered. At the end, we present the corresponding physical interpretation for the results in terms of the geometric symmetry properties of the squeezed field.

The problem of FLRW universes can be seen in the excellent works of A. Friedmann, G. Lemaître, H. P. Robertson, A. G. Walker. The key of obtaining FLRW universes with or without cosmological constant is related to the special form of the contravariant energy–momentum tensor which describes a perfect fluid with components

The answer is yes and we prove the special role of the constant ratio $K=\frac{A^{″}(𝜃)}{A(𝜃)}$ in the creation of FLRW universes. The solutions obtained lead to FLRW universes which come from surfaces having constant Gaussian curvature related by the previous $K$ from the rule $K_G=-K$. Therefore, in the case of the 2D metrics $d{\mathrm{\Omega }}_A^2=d{𝜃}^2+A^2(𝜃)d{\varphi }^2$, only the ones having constant Gaussian curvature create FLRW universes. In fact, we find all possible FLWR-type solutions of Einstein field equations coming from constant Gaussian curvature geometries. We complete our study about possible FLRW universes with an example of a spacetime that cannot be “filled” with matter using the above FLRW conditions. This becomes a new example of universe without matter satisfying the Einstein’s field equation with cosmological constant.

In this study, we consider a time-fractional shallow water wave model which was introduced by Gear and Grimshaw. This model represents the dynamics of two-layered shallow water waves in oceans and rivers. Also it covers many shallow water wave models with special values taken to parameters. By performing the group invariant analysis approach for the governing system with Riemann–Liouville derivative, symmetry generators were found. Similarity reduction was constructed by Erdelyi–Kober sense. In addition, by adapting the Sine-Gordon expansion method, we find some exact traveling wave solutions. Furthermore, explicit solutions in form of power series are examined via the residual power series method. Exact and approximate solutions are compared for some $\alpha$ values and these are given as tables.

In this paper, we studied a higher-dimensional space and time fractional model, namely, the (3+1)-dimensional dissipative Burgers equation which can be used to describe the shallow water waves phenomena. Here, the analyzed tool is the Lie symmetry scheme in the sense of the Riemann–Liouville fractional derivative. First of all, the symmetry of this considered equation was yielded. Then, based on the above obtained symmetry, the one-parameter Lie group was obtained. Subsequently, this model can be changed into the lower-dimensional equation with the Erdélyi–Kober fractional operators. Lastly, conservation laws of this studied equation via a new conservation theorem were also received. After such a series of processing, these new results play an important role in our understanding of this higher-dimensional space and time differential equations.

This work interprets the quantum terms in a Lagrangian, and consequently of the wave equation and momentum tensor, in terms of a modified spacetime metric. Part I interprets the quantum terms in the Lagrangian of a Klein–Gordon field as scalar curvature of conformal dilation covector nm that is proportional to $\hslash$ times the gradient of wave amplitude R. Part II replaces conformal dilation with a conformal factor $\rho$ that defines a modified spacetime metric $g^{\prime }$= exp$(\rho )g$, where g is the gravitational metric. Quantum terms appear only in metric $g^{\prime }$ and its metric connection coefficients. Metric $g^{\prime }$ preserves lengths and angles in classical physics and in the domain of the quantum field itself. $g^{\prime }$ combines concepts of quantum theory and spacetime geometry in one structure. The conformal factor can be interpreted as the limit of a distribution of inclusions and voids in a lattice that cause the metric to bulge or contract. All components of all free quantum fields satisfy the Klein–Gordon equation, so this interpretation extends to all quantum fields. Measurement operations, and elements of quantum field theory are not considered.

In this paper, we obtain Euler’s and De Moivre’s formulas for the $4\times 4$ matrix representation of Pauli quaternions. Moreover, we provide De Moivre’s formula for the light-like Pauli quaternions. Additionally, we give the nth roots of the matrix representation of Pauli quaternions. Moreover, we exemplify some of the results with illustrative examples to support the main formulas.

To date, a mathematically consistent construction of effective rotating black hole models in the context of Loop Quantum Gravity (LQG) is still lacking. In this work, we start with the assumption that rotating LQG black hole metrics can be effectively obtained using Newman–Janis Algorithm. Then, based on a few extra fair assumptions on the seed metric functions, we make a conjecture on what a rotating LQG black hole would generically look like. Our general arguments and conclusions can be supported by some known specific examples in the literature.

By considering the back-reaction of the spacetime through the spherically symmetric quantum fluctuations of the background metric, Kazakov and Solodukhin removed the singularity of the Schwarzschild black hole. This regular Schwarzschild black hole has a spherical central region with a radius of the order of the Planck length. On the other hand, due to the positively accelerating expansion of the Universe, it seems that there exists a universal repulsive force known as dark energy. In the framework of quantum field theories, the quintessence field is a candidate model for investigating and modeling dark energy. Accordingly, by taking into account the quintessential matter field in the background of the Schwarzschild black hole, Kiselev gained the metric of this black hole surrounded by quintessence. By combining these two above ideas, in this study, we consider the quantum-corrected Schwarzschild black hole surrounded by quintessence to investigate null and time-like geodesics structure. Generally, this study points out that black holes are quantum-gravitational objects. We will show that the accelerated expansion of the Universe, instead of dark energy, happens because of the presence of quantum effects in this setup. Also, due to the presence of the central Planck-size sphere, the regular black hole has been possessed a shifting over radial coordinate in its inner structure.

The contact geometric structure of the thermodynamic phase space is used to introduce a novel symplectic structure on the tangent bundle of the equilibrium space. Moreover, it turns out that the equilibrium space can be interpreted as a Lagrange submanifold of the corresponding tangent bundle, if the fundamental equation is known explicitly. As a consequence, Hamiltonians can be defined that describe thermodynamic processes.

This work reports the effect of thermal radiation on singularities in the presence of holographic Ricci dark energy (HRDE) with generalized IR cut-off. The Eckart approach has been considered to initiate the study. The density of HRDE, which is a just particular cases of general Nojiri–Odintsov (NO) HDE [S. Nojiri and S. D. Odintsov, Unifying phantom inflation with late-time acceleration: Scalar phantom-non-phantom transition model and generalized holographic dark energy, Gen. Relativ. Gravit.  38(8) (2006) 1285–1304], in non-interacting and non-viscous case has been reconstructed and it is found that with the passage of cosmic time t, it is decreasing. The Equation of State (EoS) parameter of HRDE in non-interacting and viscous scenario, the EoS parameter of modified Chaplygin gas (MCG) in a non-interacting and non-viscous scenario and the EoS parameter of MCG in an interacting and non-viscous scenario have been reconstructed and studied. There is a chance of Type I, Type II, Type III, Type IV and $w$-singularities for the conditions we obtained. For the effect of thermal radiation of inhomogeneous viscous fluid in an interacting scenario, there is a scope of avoidance of Big-Rip Singularity.

The aim of this paper is to investigate and analyze the behavior of charged compact stars in the modified $f(R,\varphi ,\chi )$ theory of gravity. When solving the field equations, we assume that Krori– Barua space-time describes the geometry of the inner space [K. Krori and J. Barua, A singularity-free solution for a charged fluid sphere in general relativity, J. Phys. A Math. Gen.  8 (1975) 508]. We further analyzed various physical characteristics for our model, including energy density, pressure components, anisotropy parameter, mass function, compactness element, and redshift through their graphical representations. Additionally, we have checked the equilibrium position by taking Tolman–Oppenheimer–Volkoff equation and stability by using Abreu’s criterion [H. Abreu, H. Hernandez, and L. Nunez, Sound speeds, cracking and the stability of self-gravitating anisotropic compact objects, Class. Quantum Grav.  24 (2007) 4631] and relativistic adiabatic index. For our study, we choose Hercules XI, Centaurus XIII, EXO 1785-248, LMC XIV stars for analysis.

In this paper, we consider certain analytical features of a stochastic model that can explain competition among species and simultaneous predation on the competing species from a geometric perspective. This allows us to build a systematic description of models admitting singular Lagrangians. The model equations are shown to admit a Jacobi Last Multiplier which allows us to construct an appropriate Lagrangian. Due to the singular nature of the Lagrangian, the Hamiltonian formalism may be shown to exist in a submanifold of the cotangent space under certain minimal regularity conditions. In this communication, the Hamiltonian description of the model is constructed via the introduction of Dirac brackets and explicit results for the “Kill the winner” model and its reductions are presented.