Noether Gauge Symmetries Research Articles

Noether symmetry approach in Eddington-inspired Born\u2013Infeld gravity

In this work, we take a short recap of a formal framework of the Eddington-inspired Born–Infeld (EiBI) theory of gravity and derive the point-like Lagrangian for underlying theory based on the use of Noether gauge symmetries (NGS). We study a Hessian matrix and quantify Euler–Lagrange equations of EiBI universe. We discuss the NGS approach for the Eddington-inspired Born–Infeld theory and show that there exists the de Sitter solution in this gravity model.

Isotropic exact solutions in F(R,Y,phi ) gravity via Noether symmetries

The present article investigates the existence of Noether and Noether gauge symmetries of flat Friedman–Robertson–Walker universe model with perfect fluid matter ingredients in a generalized scalar field formulation namely f(R,Y,phi ) gravity, where R is the Ricci scalar and Y denotes the curvature invariant term defined by Y=R_{alpha beta }R^{alpha beta }, while phi represents scalar field. For this purpose, we assume different general cases of generic f(R,Y,phi ) function and explore its possible forms along with field potential V(phi ) by taking constant and variable coupling function of scalar field omega (phi ). In each case, we find non-trivial symmetry generator and its related first integrals of motion (conserved quantities). It is seen that due to complexity of the resulting system of Lagrange dynamical equations, it is difficult to find exact cosmological solutions except for few simple cases. It is found that in each case, the existence of Noether symmetries leads to power law form of scalar field potential and different new types of generic function. For the acquired exact solutions, we discuss the cosmology generated by these solutions graphically and discuss their physical significance which favors the accelerated expanding eras of cosmic evolution.

Exact solutions and conserved quantities in f(R, T) Gravity

This paper explores Noether and Noether gauge symmetries of anisotropic universe model in $f(R,T)$ gravity. We consider two particular models of this gravity and evaluate their symmetry generators as well as associated conserved quantities. We also find exact solution by using cyclic variable and investigate its behavior via cosmological parameters. The behavior of cosmological parameters turns out to be consistent with recent observations which indicates accelerated expansion of the universe. Next we study Noether gauge symmetry and corresponding conserved quantities for both isotropic and anisotropic universe models. We conclude that symmetry generators and the associated conserved quantities appear in all cases.

Classification of static plane symmetric spacetime via Noether gauge symmetries

In this paper, general static plane symmetric spacetime is classified with respect to Noether operators. For this purpose, Noether theorem is used which yields a set of linear partial differential equations (PDEs) with unknown radial functions [Formula: see text], [Formula: see text] and [Formula: see text]. Further, these PDEs are solved by taking different possibilities of radial functions. In the first case, all radial functions are considered same, while two functions are taken proportional to each other in second case, which further discussed by taking four different relationships between [Formula: see text], [Formula: see text] and [Formula: see text]. For all cases, different forms of unknown functions of radial factor [Formula: see text] are reported for nontrivial Noether operators with non-zero gauge term. At the end, a list of conserved quantities for each Noether operator Tables 1–4 is presented.

Noether Gauge Symmetries for Petrov Type D-Levi-Civita Space-Time in Spherical and Cylindrical Coordinates

Petrov Type D-Levi-Civita (DLC) space-time is considered in two different coordinates, that is, spherical and cylindrical. Noether gauge symmetries and their corresponding conserved quantities for respective metric with the restricted range of parameters and coordinates are discussed.

New exact solutions of Bianchi I, Bianchi III and Kantowski–Sachs spacetimes in scalar-coupled gravity theories via Noether gauge symmetries

The Noether symmetry approach is useful tool to restrict the arbitrariness in a gravity theory when the equations of motion are underdetermined due to the high number of functions to be determined in the ansatz. We consider two scalar-coupled theories of gravity, one motivated by induced gravity, the other more standard; in Bianchi I, Bianchi III and Kantowski–Sachs cosmological models. For these models, we present a full set of Noether gauge symmetries, which are more general than those obtained by the strict Noether symmetry approach in our recent work. Some exact solutions are derived using the first integrals corresponding to the obtained Noether gauge symmetries.

Comment on “Classification of Cosmic Scale Factor via Noether Gauge Symmetries” [Int. J. Theor. Phys. 54, 2343 (2015)

We discuss the relationship between the Noether point symmetries of the geodesic Lagrangian, in a (pseudo)Riemannian manifold, with the elements of the Homothetic algebra of the space. We observe that the classification problem of the Noether symmetries for the geodesic Lagrangian is equivalent with the classification of the Homothetic algebra of the space, which in the case of a Friedmann-Lema\^{\i}tre-Robertson-Walker spacetime is a well-known result in the literature.

Noether gauge symmetry classes for pp-wave spacetimes

The Noether gauge symmetries of geodesic Lagrangian for the pp-wave spacetimes are determined in each of the Noether gauge symmetry classes of the pp-wave spacetimes. It is shown that a type N pp-wave spacetime can admit at most three Noether gauge symmetry, and furthermore the number of Noether gauge symmetries turn out to be four, five, six, seven and eight. We found that all conformally flat plane wave spacetimes admit the maximal, i.e. 10, Noether gauge symmetry. Also it is found that if the pp-wave spacetime is non-conformally flat plane wave, then the number of Noether gauge symmetry is nine or 10. By means of the obtained Noether constants the search of the exact solutions of the geodesic equations for the pp-wave spacetimes is considered and we found new exact solutions of the geodesic equations in some special Noether gauge symmetry classes.

Noether gauge symmetries of geodesic motion in stationary and nonstatic Gödel-type spacetimes

In this study, we obtain Noether gauge symmetries of geodesic motion for geodesic Lagrangian of stationary and nonstatic Gödel-type spacetimes, and find the first integrals of corresponding spacetimes to derive a complete characterization of the geodesic motion. Using the obtained expressions for [Formula: see text] of each spacetimes, we explicitly integrate the geodesic equations of motion for the corresponding stationary and nonstatic Gödel-type spacetimes.

Classification of Cosmic Scale Factor via Noether Gauge Symmetries

In this paper, a complete classification of Friedmann-Robertson-Walker (FRW) spacetime by using approximate Noether approach is presented. Considered spacetime is discussed for three different types of universe i.e. flat, open and closed. Different forms of cosmic scale factor a with respect to the nature of the universe, which posses the nontrivial Noether gauge symmetries (NGS) are reported. The perturbed Lagrangian corresponding to FRW metric in the Noether equation is used to get Noether operators. For different types of universe minimal and maximal set of Noether operators are reported. A list of Noether operators is also computed which is not only independent from the choice of the cosmic scale factor but also from the type of universe. Further, corresponding energy type first integral of motions are also calculated.

Approximate Noether gauge symmetries of the Bardeen model

We investigate the approximate Noether gauge symmetries of the geodesic Lagrangian for the Bardeen spacetime model. This is accommodated by a set of new approximate Noether gauge symmetry relations for the perturbed geodesic Lagrangian in the spacetime. A detailed analysis to the spacetime of Bardeen model up to third-order approximate Noether gauge symmetries is presented.

Lie and Noether symmetries in some classes of pp-wave spacetimes

We derive the Lie point symmetries of geodesic equations and and Noether gauge symmetries of the geodesic Lagrangian for some classes of pp-wave spacetimes. The considered classes of pp-wave spacetimes are the plane-wave spacetimes (i) , (ii) and (iii) , and a pp-wave spacetime (iv) , where H is the scale factor in pp-wave spacetime. We discuss the relations between the obtained Lie point and Noether gauge symmetry generators. We also calculate the first integrals in each of the cases, and give a solution of geodesic equations using those first integrals in case (i) as an example.

Symmetries of geodesic motion in Gödel-type spacetimes

In this paper, we study Noether gauge symmetries of geodesic motion for geodesic Lagrangian of four classes of metrics of Gödel-type spacetimes for which we calculated the Noether gauge symmetries for all classes I-IV, and find the first integrals of corresponding classes to derive a complete characterization of the geodesic motion. Using the obtained expressions for ṫ,ṙ,ϕ̇ and ż of each classes I-IV which depends essentially on two independent parameters m and w, we explicitly integrated the geodesic equations of motion for the corresponding Gödel-type spacetimes.

Noether gauge symmetry approach in quintom cosmology

In literature usual point like symmetries of the Lagrangian have been introduced to study the symmetries and the structure of the fields. This kind of Noether symmetry is a subclass of a more general family of symmetries, called Noether Gauge Symmetries (NGS). Motivated by this mathematical tool, in this article, we discuss the generalized Noether symmetry of Quintom model of dark energy, which is a two component fluid model of quintessence and phantom fields. Our model is a generalization of the Noether symmetries of a single and multiple components which have been investigated in detail before. We found the general form of the quintom potential in which the whole dynamical system has a point like symmetry. We investigated different possible solutions of the system for diverse family of gauge function. Specially, we discovered two family of potentials, one corresponds to a free quintessence (phantom) and the second is in the form of quadratic interaction between two components. These two families of potential functions are proposed from the symmetry point of view, but in the quintom models they are used as phenomenological models without clear mathematical justification. From integrability point of view, we found two forms of the scale factor: one is power law and second is de-Sitter. Some cosmological implications of the solutions have been investigated

GRADED LAGRANGIAN FORMALISM

Graded Lagrangian formalism in terms of a Grassmann-graded variational bicomplex on graded manifolds is developed in a very general setting. This formalism provides the comprehensive description of reducible degenerate Lagrangian systems, characterized by hierarchies of non-trivial higher-order Noether identities and gauge symmetries. This is a general case of classical field theory and Lagrangian non-relativistic mechanics.

Noether symmetries of some homogeneous universe models incurvature corrected scalar-tensor gravity

We explore Noether gauge symmetries of FRW and Bianchi I universemodels for perfect fluid in scalar-tensor gravity with extra termR−1 as curvature correction. Noether symmetry approach can beused to fix the form of coupling function ω(ϕ) and thefield potential V(ϕ). It is shown that for both models, theNoether symmetries, the gauge function as well as the conservedquantity, i.e., the integral of motion exist for the respectivepoint like Lagrangians. We determine the form of coupling functionas well as the field potential in each case. Finally, we investigatesolutions through scaling or dilatational symmetries for Bianchi Iuniverse model without curvature correction and discuss itscosmological implications.

Noether gauge symmetry of modified teleparallel gravity minimally coupled with a canonical scalar field

This paper is devoted to the study of Noether gauge symmetries of f(T) gravity minimally coupled with a canonical scalar field. We explicitly determine the unknown functions of the theory: f(T), V(ϕ), and W(ϕ). We have shown that there are two invariants for this model, one of which defines the Hamiltonian, H, under time invariance (energy conservation) and the other is related to scaling invariance. We show that the equation of state parameter in the present model can cross the cosmological constant boundary. The behavior of the Hubble parameter in our model closely matches to that of the ΛCDM model, thus our model is an alternative to the latter.

Noether gauge symmetry approach in Gauss–Bonnet dilatonic theory of gravity

The approach of Noether symmetries with gauge term in the Gauss–Bonnet dilatonic theory of gravity is used for different values of the state parameter, ω, of the background matter. It is found that for ω = –1 (dark energy Universe), ω = 0 (matter-dominated Universe), and ω = 1 (the case of stiff matter), for the existence of Noether gauge symmetries, the potential V([Formula: see text]) and the coupling of the Gauss–Bonnet term with gravity Λ([Formula: see text]) are obtained as exponential functions of the scalar field [Formula: see text]. For the case of arbitrary ω the coupling parameter Λ([Formula: see text]) is found as a linear function of the scalar field [Formula: see text] and the potential is a constant, for the existence of Noether gauge symmetries. Here it is observed that both the potential and coupling parameter depend on the evolutionary history of the Universe.

Fluctuations around classical solutions for gauge theories in Lagrangian and Hamiltonian approach

We analyze the dynamics of gauge theories and constrained systems in general under small perturbations around a classical solution (background) in both Lagrangian and Hamiltonian formalisms. We prove that a fluctuations theory, described by a quadratic Lagrangian, has the same constraint structure and number of physical degrees of freedom as the original non-perturbed theory, assuming the non-degenerate solution has been chosen. We show that the number of Noether gauge symmetries is the same in both theories, but that the gauge algebra in the fluctuations theory becomes Abelianized. We also show that the fluctuations theory inherits all functionally independent rigid symmetries from the original theory, and that these symmetries are generated by linear or quadratic generators according to whether the original symmetry is preserved by the background, or is broken by it. We illustrate these results with the examples.

RIGID AND GAUGE NOETHER SYMMETRIES FOR CONSTRAINED SYSTEMS

We develop the general theory of Noether symmetries for constrained systems, that is, systems that are described by singular Lagrangians. In our derivation, the Dirac bracket structure with respect to the primary constraints appears naturally and plays an important role in the characterization of the conserved quantities associated to these Noether symmetries. The issue of projectability of these symmetries from tangent space to phase space is fully analyzed, and we give a geometrical interpretation of the projectability conditions in terms of a relation between the Noether conserved quantity in tangent space and the presymplectic form defined on it. We also examine the enlarged formalism that results from taking the Lagrange multipliers as new dynamical variables; we find the equation that characterizes the Noether symmetries in this formalism, and we also prove that the standard formulation is a particular case of the enlarged one. The algebra of generators for Noether symmetries is discussed in both the Hamiltonian and Lagrangian formalisms. We find that a frequent source for the appearance of open algebras is the fact that the transformations of momenta in phase space and tangent space only coincide on shell. Our results apply with no distinction to rigid and gauge symmetries; for the latter case we give a general proof of the existence of Noether gauge symmetries for theories with first and second class constraints that do not exhibit tertiary constraints in the stabilization algorithm. Among some examples that illustrate our results, we study the Noether gauge symmetries of the Abelian Chern–Simons theory in 2n+1 dimensions. An interesting feature of this example is that its primary first class constraints can only be identified after the determination of the secondary constraint. The example is worked out retaining all the original set of variables.