Narrow Results

In this paper, we investigate the kinetic energy and angular momentum of free particles in a class of rotating cylindrical gravitational wave spacetimes, using the Noether symmetry approach. For that purpose, we determine the conserved quantities and subsequently analyze them for the velocities of free particles in the spacetimes, thereby simplifying the relationship between velocities and coordinates. Our analysis demonstrates that arbitrary choices of metric coefficients lead to fluctuations in the velocity of free particles, thereby causing variations in their kinetic energy under the influence of rotating cylindrical gravitational waves. Furthermore, we derive expressions for the angular momentum imparted to free particles by these waves, considering various scenarios for metric coefficients. Interestingly, we observe that the angular momentum of free particles initially experiences an increase over time but eventually starts to decrease after a certain duration.

It is known that corresponding to each isometry there exists a conserved quantity. It is also known that the Lagrangian of the line element of a space is conserved. Here we investigate the possibility of the existence of "new" conserved quantities, i.e. other than the Lagrangian and associated with the isometries, for spaces of different curvatures. It is found that there exist new conserved quantities only for the spaces of zero curvature or having a section of zero curvature.

We discuss the relation between Noether (point) symmetries and discrete symmetries for a class of minisuperspace cosmological models. We show that when a Noether symmetry exists for the gravitational Lagrangian, then there exists a coordinate system in which a reversal symmetry exists. Moreover, as far as concerns, the scale-factor duality symmetry of the dilaton field, we show that it is related to the existence of a Noether symmetry for the field equations, and the reversal symmetry in the normal coordinates of the symmetry vector becomes scale-factor duality symmetry in the original coordinates. In particular, the same point symmetry as also the same reversal symmetry exists for the Brans–Dicke scalar field with linear potential while now the discrete symmetry in the original coordinates of the system depends on the Brans–Dicke parameter and it is a scale-factor duality when ${\omega }_{\mathrm{\text{BD}}}=1$. Furthermore, in the context of the O’Hanlon theory for f(R)-gravity, it is possible to show how a duality transformation in the minisuperspace can be used to relate different gravitational models.

This paper is devoted to investigate the recently proposed modified Gauss–Bonnet $f({𝒢},T)$ gravity, with ${𝒢}$, the Gauss–Bonnet term, coupled with $T$, the trace of energy–momentum tensor. We have used the Noether symmetry methodology to discuss some cosmologically important $f({𝒢},T)$ gravity models with anisotropic background. In particular, the Noether symmetry equations for modified $f({𝒢},T)$ gravity are reported for locally rotationally symmetric Bianchi type I universe. Explicitly, two models have been proposed to explore the exact solutions and the conserved quantities. It is concluded that the specific models of modified Gauss–Bonnet gravity may be used to reconstruct $\mathrm{\Lambda }$CDM cosmology without involving any cosmological constant.

We study the invariance properties of five-dimensional metrics and their corresponding geodesic equations of motion. In this context a number of five-dimensional models of the Einstein–Gauss–Bonnet (EGB) theory leading to black holes, wormholes and spacetime horns arising in a variety of situations are discussed in the context of variational symmetries of which each vector field, via Noether’s theorem (NT), provides a nontrivial conservation law. In particular, it is shown that algebraic structure of isometries and the variational conservation laws of the five-dimensional Einstein–Bonnet metric extend consistently from the well-known Minkowski, de-Sitter and Schwarzschild four-dimensional spacetimes to the considered five-dimensional ones. In the equivalent five-dimensional case, the maximal algebra of kvs is fifteen with eight additional Noether symmetries. Also, whereas the constant curvature five-dimensional case leads to fifteen kvs and one additional Noether symmetry and seven plus one in the minimal case, a number of metrics of the EGB theory in five dimensions give rise to algebras isomorphic a seven-dimensional algebra of kvs and a single additional Noether symmetry.

This paper investigates the geometry of compact stellar objects via Noether symmetry strategy in the framework of curvature-matter coupled gravity. For this purpose, we assume the specific model of this theory to evaluate Noether equations, symmetry generators and corresponding conserved parameters. We use conserved parameters to examine some fascinating attributes of the compact objects for suitable values of the model parameters. It is analyzed that compact objects in this theory depend on the conserved quantities and model parameters. We find that the obtained solutions provide the viability of this process as they are compatible with the astrophysical data.

The study of particle dynamics in the vicinity of a black hole environed by dark energy and magnetic field has attracted researchers for their importance in astrophysics and cosmology. In this paper, we study the dynamics of neutral and charged particles in the vicinity of de Sitter–Schwarzschild black hole (DS–Sch-BH) surrounded by quintessence. The effect of the DS part is explored by virtue of effective potential, effective force, and escape velocity of the particle (neutral and charged) moving around DS–Sch-BH. Finally, a comparative study is also investigated.

We review the Noether symmetry analysis for the Chameleon cosmology presented in R. Bhaumik, S. Dutta and S. Chakraborty, Int. J. Mod. Phys. A 37, 2250018 (2022). We show that the classification problem for the field equations in Chameleon cosmology is still open.

In this paper, a Noether symmetry analysis is carried out for an Euler–Bernoulli beam equation via the standard Lagrangian of its reduced scalar second-order equation which arises from the standard Lagrangian of the fourth-order beam equation via its Noether integrals. The Noether symmetries corresponding to the reduced equation is shown to be the inherited Noether symmetries of the standard Lagrangian of the beam equation. The corresponding Noether integrals of the reduced Euler–Lagrange equations are deduced which remarkably allows for three families of new exact solutions of the static beam equation. These are shown to contain all the previous solutions obtained from the standard Lie analysis and more.

In the present paper, we perform Lie and Noether symmetries of the generalized Klein–Gordon–Fock equation. It is shown that the principal Lie algebra, which is one-dimensional, has several possible extensions. It is further shown that several cases arise for which Noether symmetries exist. Exact solutions for some cases are also obtained from the invariant solutions of the investigated equation.

We show that the recent results of [S. Dutta and S. Chakraborty, Int. J. Mod. Phys. D25 (2016) 1650051] on the application of Lie/Noether symmetries in scalar field cosmology are well-known in the literature while the problem could have been solved easily under a coordinate transformation. That follows from the property, that the admitted group of invariant transformations of dynamical system is independent on the coordinate system.

The time conformal regular black hole (RBH) solutions which are admitting the time conformal factor $e^{𝜖g(t)}$, where $g(t)$ is an arbitrary function of time and $𝜖$ is the perturbation parameter are being considered. The approximate Noether symmetries technique is being used for finding the function $g(t)$ which leads to $\frac{t}{\alpha }$. The dynamics of particles around RBHs are also being discussed through symmetry generators which provide approximate energy as well as angular momentum of the particles. In addition, we analyze the motion of neutral and charged particles around two well known RBHs such as charged RBH using Fermi–Dirac distribution and Kehagias–Sftesos asymptotically flat RBH. We obtain the innermost stable circular orbit and corresponding approximate energy and angular momentum. The behavior of effective potential, effective force and escape velocity of the particles in the presence/absence of magnetic field for different values of angular momentum near horizons are also being analyzed. The stable and unstable regions of particle near horizons due to the effect of angular momentum and magnetic field are also explained.

We discuss nonminimally coupled cosmologies involving different geometric invariants. Specifically, actions containing a nonminimally coupled scalar field to gravity described, in turn, by curvature, torsion and Gauss–Bonnet scalars are considered. We show that couplings, potentials and kinetic terms are determined by the existence of Noether symmetries which, moreover, allows to reduce and solve dynamics. The main finding of the paper is that different nonminimally coupled theories, presenting the same Noether symmetries, are dynamically equivalent. In other words, Noether symmetries are a selection criterion to compare different theories of gravity.

We discuss some main aspects of theories of gravity containing nonlocal terms in view of cosmological applications. In particular, we consider various extensions of general relativity based on geometrical invariants as $f(R,{\square }^{-1}R)$, $f({𝒢},{\square }^{-1}{𝒢})$ and $f(T,{\square }^{-1}T)$ gravity where $R$ is the Ricci curvature scalar, ${𝒢}$ is the Gauss–Bonnet topological invariant, $T$ the torsion scalar and the operator ${\square }^{-1}$ gives rise to nonlocality. After selecting their functional form by using Noether symmetries, we find out exact solutions in a cosmological background. It is possible to reduce the dynamics of selected models and to find analytic solutions for the equations of motion. As a general feature of the approach, it is possible to address the accelerated expansion of the Hubble flow at various epochs, in particular the dark energy issues, by taking into account nonlocality corrections to the gravitational Lagrangian. On the other hand, it is possible to search for gravitational nonlocal effects also at astrophysical scales. In this perspective, we search for symmetries of $f(R,{\square }^{-1}R)$ gravity also in a spherically symmetric background and constrain the free parameters, Specifically, by taking into account the S2 star orbiting around the Galactic Center SgrA${}^{\ast }$, it is possible to study how nonlocality affects stellar orbits around such a massive self-gravitating object.

We determine the Lie point symmetries of the Schrödinger and the Klein–Gordon equations in a general Riemannian space. It is shown that these symmetries are related with the homothetic and the conformal algebra of the metric of the space, respectively. We consider the kinematic metric defined by the classical Lagrangian and show how the Lie point symmetries of the Schrödinger equation and the Klein–Gordon equation are related with the Noether point symmetries of this Lagrangian. The general results are applied to two practical problems: (a) The classification of all two- and three-dimensional potentials in a Euclidean space for which the Schrödinger equation and the Klein–Gordon equation admit Lie point symmetries; and (b) The application of Lie point symmetries of the Klein–Gordon equation in the exterior Schwarzschild spacetime and the determination of the metric by means of conformally related Lagrangians.

We summarize the use of Noether symmetries in Minisuperspace Quantum Cosmology. In particular, we consider minisuperspace models, showing that the existence of conserved quantities gives selection rules that allow to recover classical behaviors in cosmic evolution according to the so-called Hartle criterion. Such a criterion selects correlated regions in the configuration space of dynamical variables whose meaning is related to the emergence of classical observable universes. Some minisuperspace models are worked out starting from Extended Gravity, in particular coming from scalar-tensor, f(R) and f(T) theories. Exact cosmological solutions are derived.

We prove a theorem concerning the Noether symmetries for the area minimizing Lagrangian under the constraint of a constant volume in an n-dimensional Riemannian space. We illustrate the application of the theorem by a number of examples.

We consider the Noether Symmetry Approach for a cosmological model derived from a tachyon scalar field T with a Dirac–Born–Infeld Lagrangian and a potential V(T). Furthermore, we assume a coupled canonical scalar field ϕ with an arbitrary interaction potential B(T, ϕ). Exact solutions are derived consistent with the accelerated behavior of cosmic fluid.

We review the Noether Symmetry Approach as a geometric criterion to select theories of gravity. Specifically, we deal with Noether Symmetries to solve the field equations of given gravity theories. The method allows to find out exact solutions, but also to constrain arbitrary functions in the action. Specific cosmological models are taken into account.

The general form of Noether symmetries admitted by Lagrangians corresponding to a diagonal metric are determined. We apply this general result in order to classify different metric functions for the determination of Noether generators for the equations of motion. For the two broad cases considered, we identify symmetry algebras up to dimension thirteen.