Geometrical/Physical Interpretation of the Conserved Quantities Corresponding to Noether Symmetries of Plane Symmetric Space-Times

Abstract

The aim of this paper is to give the geometrical/physical interpretation of the conserved quantities corresponding to each Noether symmetry of the geodetic Lagrangian of plane symmetric space-times. For this purpose, we present a complete list of plane symmetric nonstatic space-times along with the generators of all Noether symmetries of the geodetic Lagrangian. Additionally, the structure constants of the associated Lie algebras, the Riemann curvature tensors, and the energy-momentum tensors are obtained for each case. It is worth mentioning that the list contains all classes of solutions that have been obtained earlier during the classification of plane symmetric space-times by isometries and homotheties.

Physical Significance of Noether Symmetries

“…It was further extended to homotheties and collineations. Classification by Noether symmetries [41][42][43][44][45][46][47] has yielded solutions of the Einstein equations along with their conserved quantities. This line is also worth pursuing.…”

“…This has been conducted for static plane, static spherical, and static cylindrically symmetric spacetimes by Feroze and her collaborators [40][41][42][43][44][45]. The complete classification of non-static plane and non-static spherically symmetric spacetimes via Noether symmetry worked by Jamil et al [46,47]. The Lie and Noether symmetries of geodesic equations have been studied for the Friedmann metrics by Tsamparlis and Paliathanasis [48].…”

Physical Significance of Noether Symmetries

“…This has been done for static plane, static spherical, and static cylindrically symmetric spacetimes by Feroze and her collaborators [40,41,42,43,44,45]. The complete classification of non-static plane and non-static spherically symmetric spacetimes via Noether symmetry worked by Jamil et al [46,47]. The Lie and Noether symmetries of geodesic equations have been studied for the Friedmann metrics by Tsamparlis and Paliathanasis [48].…”

Positive Energy Condition and Conservation Laws in Kantowski-Sachs Spacetime via Noether Symmetries

“…Ali et al [16][17][18] worked on the classification of different spacetimes via NS including static plane, static spherical, and static cylindrically-symmetric spacetimes. Jamil et al [19,20] worked on the complete classification of non-static plane and non-static spherically-symmetric spacetimes via NS. Paliathanasis et al [21] established a relation between the Lie symmetries of the Klein-Gordon equation and conformal Killing vectors of the underlying geometry, where they also stated that the resulting Lie symmetries of the conformal algebra are also NS.…”