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.
