International Journal of Geometric Methods in Modern Physics

This work is a pedagogical review dedicated to a modern description of the Bondi–Metzner–Sachs (BMS) group. Minkowski space-time has an interesting and useful group of isometries, but, for a generic space-time, the isometry group is simply the identity and hence provides no significant informations. Yet symmetry groups have important role to play in physics; in particular, the Poincaré group describing the isometries of Minkowski space-time plays a role in the standard definitions of energy-momentum and angular-momentum. For this reason alone it would seem to be important to look for a generalization of the concept of isometry group that can apply in a useful way to suitable curved space-times. The curved space-times that will be taken into account are the ones that suitably approach, at infinity, Minkowski space-time. In particular we will focus on asymptotically flat space-times. In this work, the concept of asymptotic symmetry group of those space-times will be studied. In the first two sections we derive the asymptotic group following the classical approach which was basically developed by Bondi, van den Burg, Metzner and Sachs. This is essentially the group of transformations between coordinate systems of a certain type in asymptotically flat space-times. In the third section the conformal method and the notion of “asymptotic simplicity” are introduced, following mainly the works of Penrose. This section prepares us for another derivation of the BMS group which will involve the conformal structure, and is thus more geometrical and fundamental. In the subsequent sections we discuss the properties of the BMS group, e.g. its algebra and the possibility to obtain as its subgroup the Poincaré group, as we may expect. The paper ends with a review of the BMS invariance properties of classical gravitational scattering discovered by Strominger, that are finding application to black hole physics and quantum gravity in the literature.

In this paper, using the elegant language of differential forms, we provide a covariant formulation of the equilibrium statistical mechanics of non-Hamiltonian systems. The key idea of the paper is to focus on the structure of phase space and its kinematical and dynamical roles. While in the case of Hamiltonian systems, the structure of the phase space is a symplectic structure (a nondegenerate closed two-form), we consider an almost symplectic structure for the more general case of non-Hamiltonian systems. An almost symplectic structure is a nondegenerate but not necessarily closed two-form structure. Consequently, the dynamics becomes non-Hamiltonian and based on the fact that the structure is nondegenerate, we can also define a volume element. With a well-defined volume in hand, we derive the Liouville equation and find an invariant statistical state. Recasting non-Hamiltonian systems in terms of the almost symplectic geometry has at least two advantages: the formalism is covariant and therefore does not depend on coordinates and there is no confusion in the determination of the natural volume element of the system. For clarification, we investigate the application of the formalism in two examples in which the underlying geometry of the phase space is locally conformal symplectic geometry.

Recently, it has been claimed by Chinaglia and Zerbini that the curvature singularity is present even in the so-called regular black hole solutions of the Einstein equations. In this brief note, we show that this criticism is devoid of any physical content.

In the present paper, we have been investigated a new dark energy model in anisotropic Bianchi-type-I (B-I) space-time with redshift-dependent equation of state (EoS) parameter. The Einstein’s field equations have been solved by applying a variation-law for hyperbolic scale factor $a(t)={[\mathrm{\text{sin}}h(\alpha t)]}^{\frac{1}{n}}$ which provides a time-dependent deceleration parameter and time-dependent EoS parameter. We also have been found the redshift-dependent EoS parameter. The existing range of the dark energy EoS parameter $\omega$ for derived model is found to be in good agreement with the recent observations. The cosmological constant $\mathrm{\Lambda }$ is found to be a decreasing function of time and it approaches a small positive value at the present epoch which is collaborated by results from recent supernovae Ia observations. It has also been suggested that the dark energy that explains the observed accelerating universe may arise due to the contribution to the vacuum energy of the EoS in a time-dependent background. Geometric and Kinematic properties of the model and the behavior of the anisotropy of the dark energy have been discussed.

In this study, we investigate the special type of magnetic trajectories associated with a magnetic field ${ℬ}$ defined on a 3D Riemannian manifold. First, we consider a moving charged particle under the action of a frictional force, $f$, in the magnetic field ${ℬ}$. Then, we assume that trajectories of the particle associated with the magnetic field ${ℬ}$ correspond to frictional magnetic curves ($f$-magnetic curves$)$ of magnetic vector field ${ℬ}$ on the 3D Riemannian manifold. Thus, we are able to investigate some geometrical properties and physical consequences of the particle under the action of frictional force in the magnetic field ${ℬ}$ on the 3D Riemannian manifold.

In this paper, we study spontaneous symmetry breaking of gauge theories in a Lorentz violating background. Here, Lorentz symmetry will be broken down to its subgroup using the formalism of very special relativity. The breaking of the Lorentz symmetry will modify the gauge theories, and this will in turn modify the Higgs mechanisms for such theories. We will explicitly demonstrate that the different Hilbert spaces in various gauges of this theory can be related to each other through the gaugeon formalism. We will also discuss the FFBRST transformation for this theory, and observe that gaugeon formalism can be obtained from same. Thus, by making the BRST parameter finite and field dependent, we can relate different Hilbert spaces in different gauges for a gauge theory with spontaneous symmetry breaking in VSR.

The requirement of reflection positivity (RP) for Euclidean field theories is considered. This is done for the cases of a scalar field, a higher derivative scalar field theory and the scalar field theory defined on a non-integer dimensional space (NIDS). It is shown that regarding RP, the analytical structure of the corresponding Schwinger functions plays an important role. For the higher derivative scalar field theory, RP does not hold. However for the scalar field theory on a NIDS, RP holds in a certain range of dimensions where the corresponding Minkowskian field is defined on a Hilbert space with a positive definite scalar product that provides a unitary representation of the Poincaré group. In addition, and motivated by the last example, it is shown that, under certain conditions, one can construct non-local reflection positive Euclidean field theories starting from the corrected two point functions of interacting local field theories.

In this paper, we present three classes of non-lightlike curve evolution according to the Darboux frame and we give three geometric phases according to three classes in Minkowski 3-space.

We study radical screen transversal ($ST$)-lightlike submanifolds of an indefinite Kaehler manifold admitting a quarter-symmetric non-metric connection and obtain a necessary and sufficient condition for the screen distribution of a radical $ST$-lightlike submanifold to be integrable. We also study totally umbilical radical $ST$-lightlike submanifolds and obtain some characterization theorems for a radical $ST$-lightlike submanifold to be a lightlike product manifold. Finally, we establish some results regarding the vanishes of null sectional curvature.

The present study reports a reconstruction scheme for $f(T)$ gravity considering the scale factor in the power law form. The equation of state parameter has been studied for this reconstructed model along with the deceleration parameter and the statefinder pair $\{r,s\}$. The statefinder trajectory has been found to interpolate between dust and $\mathrm{\Lambda }$CDM phase of the universe. Cosmological evolution of primordial perturbations has been studied through scalar metric fluctuations and finally the reconstructed $f(T)$ model has been tested for its consistency with the generic expansion of the universe.

In this paper, we search the existence of invariant solutions of Bianchi type I space-time in the context of $f(R,T)$ gravity with special case $f(R,T)=R+2f(T)$. The exact solution of the Einstein’s field equations are derived by using Lie point symmetry analysis method that yield two models of invariant universe for symmetries $X^{(1)}$ and $X^{(3)}$. The model with symmetries $X^{(1)}$ begins with big bang singularity while the model with symmetries $X^{(3)}$ does not favor the big bang singularity. Under this specification, we find out at set of singular and nonsingular solution of Bianchi type I model which present several other physically valid features within the framework of $f(R,T)$ gravity.

We present a reconstruction technique for models of $f(R)$ gravity from the Chaplygin scalar field in flat de Sitter spacetimes. Exploiting the equivalence between $f(R)$ gravity and scalar–tensor (ST) theories, and treating the Chaplygin gas (CG) as a scalar field model in a universe without conventional matter forms, the Lagrangian densities for the $f(R)$ action are derived. Exact $f(R)$ models and corresponding scalar field potentials are obtained for asymptotically de Sitter spacetimes in early and late cosmological expansion histories. It is shown that the reconstructed $f(R)$ models all have General Relativity (GR) as a limiting solution.

In this paper, we investigate the evolution of the equation of state of the interacting viscous dark energy in $f(R)$ gravity. We first consider the case when the dark energy does not interact with the dark matter and after, the case where there is a coupling between these dark components. The viscosity and the interaction between the two fluids are parameterized by constants $\xi$ and $\delta$ respectively for which a detailed investigation on the cosmological implications has been done. In the later part of the paper, we explore some bulk viscosity models with Little and Pseudo Rip infinite time singularities within $f(R)$ modified gravity. We obtain analytic expressions for characteristic properties of these cosmological models.

It was shown that both Lovelock gravity and Born–Infeld (BI) electrodynamics can be obtained from low effective limit of string theory. Motivated by the mentioned unique origin of the gauge-gravity theories, we are going to find a close relation between them. In this research, we start from the Lagrangian of a BI-type nonlinear electrodynamics with an exponential form to extract the action of Lovelock gravity. We investigate the origin of Lovelock gravity in a system of branes which are connected with each other by different wormholes through a BIonic system. These wormholes are produced as due to the nonlinear electrodynamics which are emerged on the interacting branes. By approaching branes, wormholes dissolve into branes and Lovelock gravity is generated. Also, throats of some wormholes become smaller than their horizons and they transit to black holes. Generalizing calculations to M-theory, it is found that by compacting Mp-branes, Lovelock gravity changes to nonlinear electrodynamics and thus both of them have the same origin. This result is consistent with the prediction of BIonic model in string theory.

We describe how it is possible to define a Hodge–de Rham Dirac operator associated to a suitable Cartan-Killing metric form upon the exterior algebra over the quantum spheres ${\mathrm{\text{SU}}}_q(2)$ equipped with a three-dimensional left covariant calculus.

Let $M$ be an almost coKähler $3$-manifold such that the vertical Ricci curvatures are invariant along the Reeb vector field. In this paper, we prove that $M$ is semi-symmetric if and only if it is coKähler.

We obtain a necessary condition for homology group to be zero on CR-warped product submanifold in Euclidean spaces in terms of second fundamental form of the submanifold and warping function. By using this condition, we show that such CR-warped product submanifold is a homotopy sphere.