The Noether Symmetries of the Lagrangians of Spacetimes

Abstract

In this thesis Noether symmetries are used for the classi cation of plane symmetric, cylindrically symmetric and spherically symmetric static spacetimes. We consider general metrics for these spacetimes and use their general arc length minimizing Lagrangian densities for the classi cation purpose. The coe cients of the metric in case of plane symmetric static spacetime are general functions of x while the coe cients of cylindrically symmetric and spherically symmetric static spacetimes are general functions of the radial coordinate r. The famous Noether symmetry equation is used for the arc length minimizing Lagrangian densities of these spacetimes. Noether symmetries and particular arc length minimizing Lagrangian densities of plane symmetric, cylindrically symmetric and spherically symmetric static spacetimes are obtained. Once we get the particular Lagrangian densities, we can obtain the corresponding particular spacetimes easily. This thesis not only provides classi cation of the spacetimes but we can also obtain rst integrals corresponding to each Noether symmetry. These rst integrals can be used to de ne conservation laws in each spacetime. By using general arc length minimizing Lagrangian for plane symmetric, cylindrically symmetric and spherically symmetric static spacetimes in the Noether symmetry equation a system of 19 partial di erential equations is obtained in each case. The solution of the system in each case provides us three important things; the classi cation of the spacetimes, the Noether symmetries and the corresponding rst integrals which can be used for the conservation laws relative to each spacetime. Energy and momentum, the de nitions of which are the focus of many investigations in general relativity, are important quantities in physics. Since there is no invariant de - nitions of energy and momentum in general relativity to de ne these quantities we use the ii iii approximate Noether symmetries of the general geodesic Lagrangian density of the general time conformal plane symmetric spacetime. We use approximate Noether symmetry condition for this purpose to calculate the approximate Noether symmetries of the action of the Lagrangian density of time conformal plane symmetric spacetime. From this approach, those spacetimes are obtained the actions of which admit the rst order approximation. The corresponding spacetimes are the approximate gravitational wave spacetimes which give us information and insights for the exact gravitational wave spacetimes. Some of the Noether symmetries obtained here carry approximate parts. These approximate Noether symmetries can further be used to nd the corresponding rst integrals which describe the conservation laws in the respective spacetimes. Some of the vacuum solutions of Einstein eld equations for plane symmetric, cylindrically symmetric and spherically symmetric static spacetimes have also been explored.