Noether Symmetries Versus Killing Vectors and Isometries of Spacetimes

Abstract

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.

Killing and Noether Symmetries of Plane Symmetric Spacetime

“…Thus we conclude that the Killing symmetries of static plane symmetric spacetime always form a sub algebra of the Noether symmetries. These examples not only provides Killing and Noether symmetries but also validate the conjecture given by [1].…”

“…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]. …”

“…Later on, Hussain [21] gave a counter example to prove that the conjecture was not true in general for any spacetime with non-zero curvature. Bokhari et al [1] studied the symmetry generators of Killing vectors and Noether symmetries. It was concluded that the Noether symmetries contained the set of Killing symmetries.…”

Lie and Noether symmetries of geodesic equations and collineations

“…Recent work in this direction has been by Bokhari et all ( [11], [12]) in which the relation of the Noether symmetries with the KVs of some special spacetimes it is discussed.…”

Energy Content of Colliding Plane Waves Using Approximate Noether Symmetries

“…Finding an exact solution to these field equations is one of the major problems in GR. The Noether symmetries approach has been proved to be fruitful in this regard due to their wide range of applications in the field of DEs [22,23]. It is worth mentioning here that most of physically interesting solutions to the field equations, available in literature, possess some kind of symmetry [24].…”