Noether Symmetries, Corresponding Conservation Laws and Applications

Abstract

This thesis is based on a geometrical/physical analysis of the conserved quantities/forms related to each Noether symmetry of the geodetic Lagrangian of plane symmetric and spherically symmetric spacetimes. We present a complete list of such metrics along with their Noether symmetries of the geodetic Lagrangian. The conserved quantities corresponding to each Noether symmetry are obtained. Thereafter, a detailed discussion of the geometrical and physical interpretation of these quantities is given. Additionally, the structure constants of the associated Lie algebras are obtained for each case. Furthermore, we find the Ricci tensors to see which metrics are gravitational wave solutions and the scalar curvatures are obtained in each case to analyze the essential singularities. The stress-energy tensors and their traces are obtained in each case as these are the sources of spacetime curvature. The last part of this thesis is to use the symmetries to obtain the invariant solutions whenever possible. The problem of constructing the optimal system has been be used to classify invariant solutions. We intend to find the one-dimensional optimal systems of the Lie subalgebras for the system of geodesic equations by using Noether symmetries. Further, we find the invariants corresponding to each element of the optimal system. These invariants enable one to reduce the system of geodesic equations (nonlinear system of 2nd order ordinary differential equations (ODEs)) to a system of first order ODEs. The resulting systems are solved via known methods (e.g., separation of variables, integrating factor etc). In some cases, we are able to get exact solutions of the system of geodesic equations.