Killing and Noether Symmetries of Plane Symmetric Spacetime

Abstract

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 Minkwoski 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 [1].

Classification of Variational Conservation Laws of General Plane Symmetric Spacetimes

“…In fact, since the plane symmetric static spacetime is a special case of our considered spacetime metric (1), one can recover all the results obtained in Refs. [16] and [18] by first solving the system of differential constraints we have provided in Sec. 3 to determine the unknown functions then substitute these functions in the solution of the determining system given by Eqs.…”

“…[16] Furthermore, the isometries of plane symmetric spacetimes have been investigated by Feroze et al [17] Additionally, Shamir et al, studied Minkowski, the Taub and the anti-deSitter spacetimes via killing and Noether symmetries. [18] In this work, we present an extension of some work of Feroze et al, and others by giving a complete classification of the plane symmetric non-static spacetimes in terms of Noether symmetries admitted by them. This classification is obtained by first finding the general solution for the determining equations resulting from the invariance criterion for Noether symmetries.…”

A note on some perfect fluid Kantowski–Sachs and Bianchi type III spacetimes and their conformal vector fields in f(R) theory of gravity

“…These conservation laws play a vital role in physics as the invariance of physical quantities under a given transformation help physicists to simplify their problems. For example the existence of time like symmetry (also known as time-like Killing vector field) ensures conservation of energy of a particle defined by the equation , E vp  where v and p represents time-like Killing vector field and momentum of the particle respectively [1]. Moreover, symmetry restrictions are found to be extremely useful for the investigation of gravitational waves as well as to address the problem of localization of energy momentum in general relativity (GR).…”

Hojman symmetry in f ( T ) $f(T)$ theory

“…One of the well-known symmetries in physics is Noether symmetry. In fact, Noether symmetry has been extensively used in cosmology and gravity theories, for instance, scalar field cosmology [1,2], f (R) theory [3][4][5]26], scalar-tensor theory [6,7], f (T ) theory [8,25], Gauss-Bonnet gravity [9], non-minimally coupled cosmology [10], and others [11,27,28]. It is worth noting that a (point-like) Lagrangian should be given a priori when one uses Noether symmetry in these models.…”