On the Symmetry Structures of the Minkowski Metric and a Weyl Re-Scaled Metric

Symmetries of spacetime manifolds which are given by Killing vectors are compared with the symmetries of a Lagrangian constructed from a Weyl re-scaled metric used in discussing disorder operators in Gauge theories. 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). It is shown that the Noether symmetries obtained by considering the Lagrangian provide additional symmetries which are not provided by the Killing vectors. New conservation law/s are determined.     

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.     

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.     

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).     

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.     

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.     

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.     

The geometric nature of Lie and Noether symmetries

It is shown that the Lie and the Noether symmetries of the equations of motion of a dynamical system whose equations of motion in a Riemannian space are of the form [(x)\ddot]ⁱ+G_jkⁱ[(x)\dot]^j[(x)\dot] ^k+f(xⁱ)=0{\ddot{x}^{i}+\Gamma_{jk}^{i}\dot{x}^{j}\dot{x} ^{k}+f(x^{i})=0} where f(x ⁱ ) is an arbitrary function of its argument, are generated from the Lie algebra of special projective collineations and the homothetic algebra of the space respectively. Therefore the computation of Lie and Noether symmetries of a given dynamical system in these cases is reduced to the problem of computation of the special projective algebra of the space. It is noted that the Lie and Noether symmetry vectors are common to all dynamical systems moving in the same background space. The selection of the vectors which are Lie/Noether symmetries for a given dynamical system is done by means of a set of differential conditions involving the vectors and the potential function defining the dynamical system. The general results are applied to a number of different applications concerning (a) The motion in Euclidean space under the action of a general central potential (b) The motion in a space of constant curvature (c) The determination of the Lie and the Noether symmetries of class A Bianchi type hypersurface orthogonal spacetimes filled with a scalar field minimally coupled to gravity (d) The analytic computation of the Bianchi I metric when the scalar field has an exponential potential.     

The Noether Conservation Laws of Some Vaidiya Metrics

In this paper, we show that a large amount information can be extracted from a knowledge of the vector fields that leave the action integral invariant, viz., Noether symmetries. In addition to a larger class of conservation laws than those given by the isometries or Killing vectors, we may conclude what the isometries are and that these form a Lie subalgebra of the Noether symmetry algebra. We perform our analysis on versions of the Vaidiya metric yielding some previously unknown information regarding the corresponding manifold. Lastly, with particular reference to this metric, we show that the only variations on m(u) that occur are m=0, m=constant, m=u and m=m(u).     

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.     

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.     

Noether conserved quantities and Lie point symmetries for difference nonholonomic Hamiltonian systems in irregular lattices

The Noether conserved quantities and the Lie point symmetries for difference nonholonomic Hamiltonian systems in irregular lattices are studied. The generalized Hamiltonian equations of the systems are given on the basis of the transformation operators in the space of discrete Hamiltonians. The Lie transformations acting on the lattice, as well as the equations and the determining equations of the Lie symmetries are obtained for the nonholonomic Hamiltonian systems. The discrete analogue of the Noether conserved quantity is constructed by using the Lie point symmetries. An example is discussed to illustrate the results.     

Symmetries and motions in manifolds

In these lectures the relations between symmetries, Lie algebras, Killing vectors and Noether's theorem are reviewed. A generalisation of the basic ideas to include velocity-dependent co-ordinate transformations naturally leads to the concept of Killing tensors. Via their Poisson brackets these tensors generate an a priori infinite-dimensional Lie algebra. The nature of such infinite algebras is clarified using the example of flat space-time. Next the formalism is extended to spinning space, which in addition to the standard real co-ordinates is parametrised also by Grassmann-valued vector variables. The equations for extremal trajectories (“geodesics”) of these spaces describe the pseudo-classical mechanics of a Dirac fermion. We apply the formalism to solve for the motion of a pseudo-classical electron in Schwarzschild space-time.     

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.     

The Lie symmetries and Noether conserved quantities of discrete mechanical systems with variable mass

This paper studies the Lie symmetries and Noether conserved quantities of discrete mechanical systems with variable mass. The discrete Euler-Lagrange equation and energy evolution equation are derived by using a total variational principle. The invariance of discrete equations under infinitesimal transformation groups is defined to be Lie symmetry. The condition of obtaining the Noether conserved quantities from the Lie symmetries is also presented. An example is discussed for applications of the results.     

The Lie symmetries and Noether conserved quantities of discrete non-conservative mechanical systems

This paper investigates the Lie symmetries and Noether conserved quantities of discrete non-conservative mechanical systems. The variational principle of discrete mechanics, from which discrete motion equations of systems are deduced, is generalized to the case of including the time variational. The requirement for an invariant group transformation is defined to be the Lie symmetry and the criterion when the Noether conserved quantities may be obtained from Lie symmetries is also presented. An example is discussed for applications of the results.     

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.     

Symmetries in a very special relativity and isometric group of Finsler space

We present an explicit connection between the symmetries in a Very Special Relativity (VSR) and isometric group of a specific Finsler space. It is shown that the line element that is invariant under the VSR symmetric group is a Finslerian one. The Killing vectors in Finsler space are constructed in a systematic way. The Lie algebras corresponding to the symmetries of VSR are obtained from a geometric famework. The dispersion relation and the Lorentz invariance violation effect in the VSR are discussed.     

Symmetries in a very special relativity and isometric group of Finsler space

We present an explicit connection between the symmetries in a Very Special Relativity (VSR) and isometric group of a specific Finsler space. It is shown that the line element that is invariant under the VSR symmetric group is a Finslerian one. The Killing vectors in Finsler space are constructed in a systematic way. The Lie algebras corresponding to the symmetries of VSR are obtained from a geometric famework. The dispersion relation and the Lorentz invariance violation effect in the VSR are discussed.     

The reduction of the Laplace equation in certain Riemannian spaces and the resulting Type II hidden symmetries

We prove a general theorem which allows the determination of Lie symmetries of the Laplace equation in a general Riemannian space using the conformal group of the space. Algebraic computing is not necessary. We apply the theorem in the study of the reduction of the Laplace equation in certain classes of Riemannian spaces which admit a gradient Killing vector, a gradient Homothetic vector and a special Conformal Killing vector. In each reduction we identify the source of Type II hidden symmetries. We find that in general the Type II hidden symmetries of the Laplace equation are directly related to the transition of the CKVs from the space where the original equation is defined to the space where the reduced equation resides. In particular we consider the reduction of the Laplace equation (i.e., the wave equation) in the Minkowski space and obtain the results of all previous studies in a straightforward manner. We consider the reduction of Laplace equation in spaces which admit Lie point symmetries generated from a non-gradient HV and a proper CKV and we show that the reduction with these vectors does not produce Type II hidden symmetries. We apply the results to general relativity and consider the reduction of Laplace equation in locally rotational symmetric space times (LRS) and in algebraically special vacuum solutions of Einstein’s equations which admit a homothetic algebra acting simply transitively. In each case we determine the Type II hidden symmetries.