Noether Symmetries Versus Killing Vectors and Isometries of Spacetimes

Symmetries of spacetime manifolds which are given by Killing vectors are compared with the symmetries of the Lagrangians of the respective spacetimes. We find the point generators of the one parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian (Noether symmetries). In the examples considered, it is shown that the Noether symmetries obtained by considering the Larangians provide additional symmetries which are not provided by the Killing vectors. It is conjectured that these symmetries would always provide a larger Lie algebra of which the KV symmetres will form a subalgebra. PACS: 04.25.-g, 02.20.Sv, 11.30.-j     

Noether versus Killing symmetry of conformally flat Friedmann metric

In a recent study Noether symmetries of some static spacetime metrics in comparison with Killing vectors of corresponding spacetimes were studied. It was shown that Noether symmetries provide additional conservation laws that are not given by Killing vectors. In an attempt to understand how Noether symmetries compare with conformal Killing vectors, we find the Noether symmetries of the flat Friedmann cosmological model. We show that the conformally transformed flat Friedman model admits additional conservation laws not given by the Killing or conformal Killing vectors. Inter alia, these additional conserved quantities provide a mechanism to twice reduce the geodesic equations via the associated Noether symmetries.     

Energy conditions and conservation laws in LTB metric via Noether symmetries

In this paper, we have investigated Noether symmetries in Lemaitre–Tolman–Bondi (LTB) metric. Using the Lagrangian associated with the LTB metric, the set of determining equations for Noether symmetries is obtained and then integrated in several cases. It is shown that the LTB metric can be classified in to eight distinct classes corresponding to Noether algebra of dimension 4, 5, 6, 7, 8, 9, 11 and 17. The obtained Noether symmetries are compared with Killing and homothetic vectors. The well known Noether’s theorem is used to find the expressions for conservation laws in each case. Moreover, it is shown that most of the obtained metrics are anisotropic or perfect fluid models which satisfy certain energy conditions and the equation of state.     

Killing and Noether Symmetries of Plane Symmetric Spacetime

This paper is devoted to investigate the Killing and Noether symmetries of static plane symmetric spacetime. For this purpose, five different cases have been discussed. The Killing and Noether symmetries of Minkowski spacetime in cartesian coordinates are calculated as a special case and it is found that Lie algebra of the Lagrangian is 10 and 17 dimensional respectively. The symmetries of Taub’s universe, anti-deSitter universe, self similar solutions of infinite kind for parallel perfect fluid case and self similar solutions of infinite kind for parallel dust case are also explored. In all the cases, the Noether generators are calculated in the presence of gauge term. All these examples justify the conjecture that Killing symmetries form a subalgebra of Noether symmetries (Bokhari et al. in Int. J. Theor. Phys. 45:1063, 2006).     

Lie and Noether symmetries of geodesic equations and collineations

The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric. The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. We apply the results in the case of Einstein spaces, the Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each case the Lie and the Noether symmetries are computed explicitly together with the corresponding linear and quadratic first integrals.     

Classification of Static Spherically Symmetric Spacetimes by Noether Symmetries

Noether symmetries of some of the well known spherically symmetric static solutions of the Einstein’s field equations are classified. The resulting Noether symmetries in each case are compared with conservation laws given by Killing vectors and collineations of the Ricci and Riemann tensors for corresponding solutions.     

Noether symmetries of Bianchi type II spacetimes

This paper is devoted to investigate Noether symmetries of Bianchi type II spacetimes. We use the reduced involutive form of the determining equations to classify their possible algebras. We show that Noether symmetries contain both Killing vectors and homothetic motions.     

Energy Conditions and Conservation Laws in LRS Bianchi Type I Spacetimes via Noether Symmetries

In this paper, we have completely classified the locally rotationally symmetric (LRS) Bianchi type I spacetimes via Noether symmetries (NS). The usual Lagrangian corresponding to LRS Bianchi type I metric is used to find the set of determining equations. To achieve a complete classification, these determining equations are generally integrated to find the components of NS vector field and the metric coefficients. During this procedure, several cases arise which give different Noether algebras of dimension 5,..., 9, 11, and 17. A comparison is established between the obtained NS and the Killing and homothetic vectors. Corresponding to all NS generators, the conservation laws are stated by using Noether's theorem. The metrics which we have obtained as a result of our classification are shown to be anisotropic or perfect fluids which satisfy certain energy conditions.     

Classification of Variational Conservation Laws of General Plane Symmetric Spacetimes

In this article we discuss Noether conservation laws admitted by a Lagrangian L = gab(dx~a/ds)(dx~b/ds)of a test particle moving in the field of a general plane symmetric non-static spacetime metric. In this context, we first present a general solution representing a Noether symmetry vector subject to differential constraints satisfied by the general plane symmetric non-static metric. We then use a class of plane symmetric non-static metrics obtained by Feroze et al. and discuss, in each case, Noether conservation laws in comparison with Killing symmetries.     

n<Superscript><Emphasis Type="Italic">th</Emphasis></Superscript>-Order Approximate Lagrangians Induced by Perturbative Geometries

A family of perturbative Lagrangians that describe approximate and multidimensional Klein-Gordon equations are studied. We probe the existence of approximate Noether symmetries via generalized geometric conditions for a perturbation of any order. The knowledge of the geometric conditions uncovers that unlike exact symmetries, the approximate Noether symmetries of the Lagrangian which describes the motion of a particle in n-dimensional space under the action of an autonomous force, is inequivalent to the Noether symmetries admitted by the Klein-Gordon Lagrangian in general.     

APPROXIMATE PARTIAL NOETHER OPERATORS OF THE SCHWARZSCHILD SPACETIME

The objective of this paper is twofold: (a) to find a natural example of a perturbed Lagrangian that has different partial Noether operators with symmetries different from those of the underlying Lagrangian. First we regard the Schwarzschild spacetime as a perturbation of the Minkowski spacetime and investigate the approximate partial Noether operators for this perturbed spacetime. It is shown that the Minkowski spacetime has 12 partial Noether operators, 10 of which are different from the 17 Noether symmetries for this spacetime. It is found that for the perturbed Schwarzschild spacetime we recover the exact partial Noether operators as trivial first-order approximate partial Noether operators and there is no non-trivial approximate partial Noether operator as for the Noether case. As a consequence we state a conjecture. (b) Then we prove a conjecture that the approximate symmetries of a perturbed Lagrangian form a subalgebra of the approximate symmetries of the corresponding perturbed Euler–Lagrange equations and illustrate it by our examples. This is in contrast to approximate partial Noether operators.     

Lagrangian mechanics and the geometry of configuration spacetime

Holonomic rheonomic systems having a finite number of degrees of freedom are considered in classical nonrelativistic mechanics. It is shown that the configuration spacetime manifold $\text{M}$ of such a system can be furnished with a linear symmetric connection (called the “dynamical connection”) in such a way that the worldline of the system is a geodesic on $\text{M}$. The connection is based upon a degenerate metric structure (called a “generalized Galilei structure”) which in turn is uniquely determined by the system and the forces acting on it. The connection is compatible with the generalized Galilei structure in the sense that the covariant derivatives of the latter vanish. Systems which can be described in terms of a Lagrangian give rise to a particularly interesting class of dynamical connections, called “Lagrange connections,” whose geometry is studied in some detail. Within the class of generalized Galilei connections they are characterized by a geometrical condition imposed on the affine curvature tensor. Noether symmetries of the dynamical system turn out to be equivalent to “isometries” of the generalized Galilei structure together with collineations of the Lagrange connection. They form a Lie group. Spacelike generators of Noether symmetries are linked to the existence of “conservors” (i.e., covectors with vanishing symmetrized covariant derivatives). Timelike generators of Noether symmetries give rise to (second rank) Killing tensors.     

Noether and some other dynamical symmetries in Kantowski–Sachs model

The forms of coupling of the scalar field with gravity, appearing in the induced theory of gravity, and that of the potential are found in the Kantowski–Sachs model under the assumption that the Lagrangian admits Noether symmetry. The form thus obtained makes the Lagrangian degenerate. The constrained dynamics thus evolved due to such degeneracy has been analysed and a solution has also been presented which is inflationary in behaviour. It has further been shown that there exists other technique to explore the dynamical symmetries of the Lagrangian simply by inspecting the field equations. Through this method Noether along with some other dynamical symmetries are found which do not make the Lagrangian degenerate.     

Coupling Parameters and the Form of the Potential via Noether Symmetry

We explore the conditions for the existence of Noether symmetries in the dynamics of FRW metric, non minimally coupled with a scalar field, in the most general situation, and with nonzero spatial curvature. When such symmetries are present we find a general exact solution for the Einstein equations. We also show that non Noether symmetries can be found. Finally, we present an extension of the procedure to the Kantowski-Sachs metric which is particularly interesting in the case of degenerate Lagrangian.     

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-Lemaître-Robertson-Walker spacetime is a well-known result in the literature.     

Noether symmetries and conserved quantities for spaces with a section of zero curvature

In an earlier paper (Feroze, 2010 [21]), the existence of new conserved quantities (Noether invariants) for spaces of different curvatures was discussed. There, it was conjectured that the number of new conserved quantities for spaces with an m$m$-dimensional section of zero curvature is m$m$. Here, along with the proof of this conjecture, the form of the new conserved quantities is also presented. For the illustration of the theorem, an example of conformally flat spacetime is constructed which also demonstrates that the conformal Killing vectors (CKVs), in general, are not symmetries of the Lagrangian for the geodesic equation.     

Complete Classification of Conformal Killing Symmetries of Plane-Symmetric Static Spacetimes in Teleparallel Theory

We have studied the conformal, homothetic and Killing vectors in the context of teleparallel theory of gravitation for plane-symmetric static spacetimes. We have solved completely the non-linear coupled teleparallel conformal Killing equations. This yields the general form of teleparallel conformal vectors along with the conformal factor for all possible cases of metric functions. We have found four solutions which are divided into one Killing symmetries and three conformal Killing symmetries. One of these teleparalel conformal vectors depends on x only and other is a function of all spacetime coordinates. The three conformal Killing symmetries contain three proper homothetic symmetries where the conformal factor is an arbitrary non-zero constant.     

Symmetries and variational calculation of discrete Hamiltonian systems

We present a numerical simulation method of Noether and Lie symmetries for discrete Hamiltonian systems. The Noether and Lie symmetries for the systems are proposed by investigating the invariance properties of discrete Lagrangian in phase space. The numerical calculations of a two-degree-of-freedom nonlinear harmonic oscillator show that the difference discrete variational method preserves the exactness and the invariant quantity.     

Conserved charges for gravity with locally anti-de sitter asymptotics

A new formula for the conserved charges in 3+1 gravity for spacetimes with local anti-de Sitter asymptotic geometry is proposed. It is shown that requiring the action to have an extremum for this class of asymptotia sets the boundary term that must be added to the Lagrangian as the Euler density with a fixed weight factor. The resulting action gives rise to the mass and angular momentum as Noether charges associated to the asymptotic Killing vectors without requiring specification of a reference background in order to have a convergent expression. A consequence of this definition is that any negative constant curvature spacetime has vanishing Noether charges. These results remain valid in the Lambda = 0 limit.     

A note on Noether symmetries and conformal Killing vectors

A conjecture was stated in Hussain et al. (Gen Relativ Grav 41:2399, 2009), that the conformal Killing vectors form a subalgebra of the symmetries of the Lagrangian that minimizes arc length, for any spacetime. Here, a counter example is constructed to demonstrate that the above statement is not true in general for spacetimes of non-zero curvature.