International Journal of Geometric Methods in Modern Physics

We investigate the generalized second law of thermodynamics by assuming the interaction of dark energy and dark matter in Chern–Simons modified gravity. We consider a family of holographic dark energy models by assuming its various cutoffs such as Hubble horizon, event horizon, their combination, Ricci scalar and its generalized form. The general criteria of generalized second law of thermodynamics in terms of coincidence parameter is being developed. This criterion is being applied for the above-mentioned holographic dark energy models to check the validity of the generalized second law of thermodynamics and the constraints where it is respected are referred.

We discuss the interacting modified QCD ghost dark energy and generalized ghost pilgrim dark energy with cold dark matter in the framework of dynamical Chern–Simons modified gravity. We investigate the cosmological parameters such as Hubble parameter, deceleration parameter and equation of state. We also discuss the physical significance of various cosmological planes like ${\omega }_{𝜗}-{\omega }_{𝜗}^{\prime }$ and statefinders. It is found that the results of cosmological parameters as well as planes explain the accelerated expansion of the Universe and are compatible with observational data.

Starting with the study of the geometry on the cotangent bundle (phase space), it is shown that the maximal proper force condition, in the case of a uniformly accelerated observer of mass $m$ along the $x$ axis, leads to a minimum value of $x$ lying $inside$ the Rindler wedge and given by the black hole horizon radius $2Gm$. Whereas in the uniform circular motion case, we find that the maximal proper force condition implies that the radius of the circle cannot exceed the value of the horizon radius $2Gm$. A correspondence is found between the black hole horizon radius and a singularity in the curvature of momentum space. The fact that the geometry (metric) in phase spaces is observer-dependent (on the momentum of the massive particle/observer) indicates further that the matter stress energy tensor and vacuum energy in the underlying space-time may admit an interpretation in terms of the curvature in momentum spaces. Some final comments are made pertaining to the Asymptotic Safety program in gravity and why phase space geometry seems to be a proper arena for a space–time–matter unification.

In this paper, the dynamical behavior of the anisotropic universe has been investigated in $f(R,T)$ theory of gravity. The functional $f(R,T)$ has been rescaled in the form $f(R,T)=\mu R+\mu T$, where $R$ is the Ricci scalar and $T$ is the trace of energy momentum tensor. Three cosmological models are constructed using the power law expansion in Bianchi type $V{I}_h$ ($BV{I}_h$) universe for three different values of $h=-1,0,1$, where the matter field is considered to be of bulk viscous fluid. It is observed that the anisotropic $BV{I}_h$ model in the modified theory of gravity is in agreement with a quintessence phase for the value of $h=-1$ and $0$. We could not obtain a viable cosmological model in accordance with the present day observations for $h=1$. The bulk viscous coefficient in both the cases are found to be positive and remain constant at late time. The physical behaviors of the models along with the energy conditions are also studied.

In this paper, the Gödel-type space-times have been considered. We are interested in the investigation of relativistic free bosons in the Gödel-type space-times. Therefore, the Klein–Gordon equation considering the Gödel-type space-times has been constructed in a general manner. Then to accomplish our goal, two well-known special cases have been considered for the unknown function in the tensor metric of the Gödel-type space-times. The explicit forms of the wave functions for each case as well as the energy eigenvalue equations have been derived in detail.

In this paper, we investigate the behavior of quantum correlation for Gisin states based on the bipartite superposed coherent states (SCS). Moreover, we analyze the basis for minimizing the quantum discord (or equivalently basis maximizing the classical correlation) as function of the mixing parameter and the coherent amplitude. We found that for our quantum system, the quantum discord behavior is not smooth and could experience a sudden change (S.C).

In this work, we obtain the Raychaudhuri equations for various types of Finsler spaces as the Finsler–Randers (FR) space-time and in a generalized geometrical structure of the space-time manifold which contains two fibers that represent two scalar fields ${\varphi }^{(1)},{\varphi }^{(2)}$. We also derive the Klein–Gordon equation for this model. In addition, the energy conditions are studied in a FR cosmology and are correlated with FRW model. Finally, we apply the Raychaudhuri equation for the model $M\times \{{\varphi }^{(1)}\}\times \{{\varphi }^{(2)}\}$, where M is a FRW-space-time.

In this paper, the backward and forward Neumann type systems are generalized to deduce the quasi-periodic solutions for a negative-order integrable system of 2-component KdV equation. The 2-component negative-order KdV (2-nKdV) equation is depicted as the zero-curvature representation of two spectral problems. It follows from a symmetric constraint that the 2-nKdV equation is reduced to a pair of backward and forward Neumann type systems, where the involutive solutions of Neumann type systems yield the finite parametric solutions of 2-nKdV equation. The negative-order Novikov equation is given to specify a finite-dimensional invariant subspace for the 2-nKdV flow. With a spectral curve given by the Lax matrix, the 2-nKdV flow is linearized on the Jacobi variety of a Riemann surface, which leads to the quasi-periodic solutions of 2-nKdV equation by using the Riemann-Jacobi inversion.

We classify all warped product space-times in three categories as (i) generalized twisted product structures, (ii) base conformal warped product structures and (iii) generalized static space-times and then we obtain the Einstein equations with the corresponding cosmological constant by which we can determine uniquely the warp functions in these warped product space-times.

In this work, we focus on the gravitationally influenced adiabatic particle creation process, a mechanism that does not need any dark energy or modified gravity models to explain the current accelerating phase of the universe. Introducing some particle creation models that generalize some previous models in the literature, we constrain the cosmological scenarios using the latest compilation of the Type Ia Supernovae data only, the first indicator of the accelerating universe. Aside from the observational constraints on the models, we examine the models using two model independent diagnoses, namely the cosmography and $Om$. Further, we establish the general conditions to test the thermodynamic viabilities of any particle creation model. Our analysis shows that at late-time, the models have close resemblance to that of the $\mathrm{\Lambda }$CDM cosmology, and the models always satisfy the generalized second law of thermodynamics under certain conditions.

In this paper, the formation of cylindrical wormhole during evolution of manifolds is studied. It is shown that this type of wormholes may be produced at two stages and then disappeared very fast at the third stage. First, one $N$-dimensional is formed by joining point-like manifolds. Then, this manifold is torn and two child manifolds plus one Chern–Simons manifold appeared. Our universe is born on one of the child manifolds and connected to the other one by Chern–Simons manifold. At the third stage, this Chern–Simons manifold-which plays the role of cylindrical wormhole, dissolves into universes and gives its energy to them and causes inflation. Thus, the Chern–Simons cylindrical wormhole is unstable and dissolves in our four-dimensional universes and another universe very fast.

This paper is devoted to the construction of a unified formalism for Palatini and unimodular gravity. The idea is to employ a relationship between unified formalism for a Griffiths variational problem and its classical Lepage-equivalent variational problem. The main geometrical tools involved in these constructions are canonical forms living on the first jet of the frame bundle for the spacetime manifold. These forms play an essential role in providing a global version of the Palatini Lagrangian and expressing the metricity condition in an invariant form. With them, we were able to find the associated equations of motion in invariant terms and, by using previous results from the literature, to prove their involutivity. As a bonus, we showed how this construction can be used to provide a unified formalism for the so-called unimodular gravity by employing a reduction of the structure group of the frame bundle to the special linear group.

The relationship between gauge and gravity amounts to understanding the underlying new geometrical local structures. These structures are new tetrads specially devised for Yang–Mills theories, Abelian and non-Abelian in four-dimensional Lorentzian curved spacetimes. In the present paper, a new tetrad is introduced for the Yang–Mills $SU(3)\times SU(2)\times U(1)$ formulation. These new tetrads establish a link between local groups of gauge transformations and local groups of spacetime transformations that we previously called LB1 and LB2. New theorems are proved regarding isomorphisms between local internal $SU(3)\times SU(2)\times U(1)$ groups and local tensor products of spacetime LB1 and LB2 groups of transformations. These new tetrads define at every point in spacetime two orthogonal planes that we called blades or planes one and two. These are the local planes of covariant diagonalization of the stress–energy tensor. These tetrads are gauge dependent. Tetrad local gauge transformations leave the tetrads inside the local original planes without leaving them. These local tetrad gauge transformations enable the possibility to connect local gauge groups Abelian or non-Abelian with local groups of tetrad transformations. On the local plane one, the Abelian group $U(1)$ of gauge transformations was already proved to be isomorphic to the tetrad local group of transformations LB1, for example. LB1 is $SO(1,1)$ plus two different kinds of discrete transformations. On the local orthogonal plane two $U(1)$ is isomorphic to LB2 which is just $SO(2)$. That is, we proved that LB1 is isomorphic to $SO(2)$ which is a remarkable result since a noncompact group plus two discrete transformations is isomorphic to a compact group. These new tetrads have displayed manifestly and nontrivially the coupling between Yang–Mills fields and gravity. The new tetrads and the stress–energy tensor allow for the introduction of three new local gauge invariant objects. Using these new gauge invariant objects and in addition a new general local duality transformation, a new algorithm for the gauge invariant diagonalization of the Yang–Mills stress–energy tensor is developed as an application. This is a paper about grand Standard Model gauge theories — General Relativity gravity unification and grand group unification in four-dimensional curved Lorentzian spacetimes.

In metric formalism, Noether symmetry of $F(R)$ theory of gravity in vacuum and in the presence of pressureless dust yields $F(R)\propto R^{\frac{3}{2}}$ along with the conserved current $\frac{d}{dt}(a\sqrt{R})$ in Robertson–Walker metric and nothing else. However, Roshan and Shojai have claimed in [Palatini $f(R)$ gravity and Noether symmetry, Phys. Lett. B 668 (2008) 238–240] that Noether symmetry in the context of Palatini $F(\Re )$ theory of gravity admits $F(\Re )\propto {\Re }^s$, (where $s$ is arbitrary) in matter domain era in Friedmann–Robertson–Walker universe. But, it has been shown that the conserved current obtained under the process does not satisfy the field equations in general. Here, it is shown that Noether Symmetry admits $F(\Re )\propto {\Re }^{\frac{3}{2}}$ along with a conserved current ${\mathrm{\Sigma }}_{0}a[\frac{ȧ}{a}+\frac{\dot{\mathrm{\Phi }}}{2\mathrm{\Phi }}]$ in Palatini gravity. Thus, their claim is not right.

One of the important quantities in cosmology and astrophysics is the 3-velocity of an object. Specifically, when the gravitational fields are strong, one should require the employment of general relativity both in its definition and measurement. Looking into the literature for GR-based definitions of 3-velocity, one usually finds different ad hoc definitions applied according to the case under consideration. Here, we introduce and analyze systematically the two principal definitions of 3-velocity assigned to a test particle following the timelike trajectories in stationary spacetimes. These definitions are based on the $1+3$ (threading) and $3+1$ (slicing) spacetime decomposition formalisms and defined relative to two different sets of observers. After showing that Synge’s definition of spatial distance and 3-velocity is equivalent to those defined in the $1+3$ (threading) formalism, we exemplify the differences between these two definitions by calculating them for particles in circular orbits in axially symmetric stationary spacetimes. Illustrating its geometric nature, the relative linear velocity between the corresponding observers is obtained in terms of the spacetime metric components. Circular particle orbits in the Kerr spacetime, as the prototype and the most well known of stationary spacetimes, are examined with respect to these definitions to highlight their observer-dependent nature.

We also examine the Kerr-NUT spacetime in which the NUT parameter, contributing to the off-diagonal terms in the metric, is mainly interpreted not as a rotation parameter but as a gravitomagnetic monopole charge. Finally, in a specific astrophysical setup which includes rotating black holes, it is shown how the local velocity of an orbiting star could be related to its spectral line shifts measured by distant observers.

Using the curvature theory for the ruled surfaces a technique for robot trajectory planning is presented. This technique ensures the calculation of robot’s next path. The positional variation of the Tool Center Point (TCP), linear velocity, angular velocity are required in the work area of the robot. In some circumstances, it may not be physically achievable and a re-computation of the robot trajectory might be necessary. This technique is suitable for re-computation of the robot trajectory. We obtain different robot trajectories which change depending on the darboux angle function and define trajectory ruled surface family with a common trajectory curve with the rotation trihedron. Also, the motion of robot end effector is illustrated with examples.

In this paper, we have studied flat Friedmann–Lemaître–Robertson–Walker (FLRW) model with modified Chaplygin gas (MCG) having equation of state $p_m=A\rho -\frac{B}{{\rho }^{\gamma }}$, where $0\le A\le 1$, $0\le \gamma \le 1$ and $B$ is any positive constant in $f(R,T)$ gravity with particle creation. We have considered a simple parametrization of the Hubble parameter $H$ in order to solve the field equations and discussed the time evolution of different cosmological parameters for some obtained models showing unique behavior of scale factor. We have also discussed the statefinder diagnostic pair $\{r,s\}$ that characterizes the evolution of obtained models and explore their stability. The physical consequences of the models and their kinematic behaviors have also been scrutinized here in some detail.

The self-interaction for a static scalar charge in the space-time of extreme charged anti-dilatonic wormhole is calculated. We assume that the scalar charge is the source of massless scalar field with minimal coupling of the scalar field to the curvature of spacetime.