USE OF APPROXIMATE SYMMETRY METHODS TO DEFINE ENERGY OF GRAVITATIONAL WAVES

Abstract

In this thesis approximate Lie symmetry methods for differential equations are used to investigate the problem of energy in general relativity and in particular in gravitational waves. For this purpose second-order approximate symmetries of the system of geodesic equations for the Reissner-Nordström (RN) spacetime are studied. It is shown that in the second-order approximation, energy must be rescaled for the RN spacetime.

Then the approximate symmetries of a Lagrangian for the geodesic equations in the Kerr spacetime are investigated. Taking the Minkowski spacetime as the exact case, it is shown that the symmetry algebra of the Lagrangian is 17 dimensional. This algebra is related to the 15 dimensional algebra of conformal isometries of the Minkowski spacetime. First introducing spin angular momentum per unit mass as a small parameter first-order approximate symmetries of the Kerr spacetime as a first perturbation of the Schwarzschild spacetime are considered. We then investigate the second-order approximate symmetries of the Kerr spacetime as a second perturbation of the Minkowski spacetime.

Next, second-order approximate symmetries of the system of geodesic equations for the charged-Kerr spacetime are investigated. A rescaling of the arc length parameter for consistency of the trivial second-order approximate symmetries of the geodesic equations indicates that the energy in the charged-Kerr spacetime has to be rescaled.

Since gravitational wave spacetimes are time-varying vacuum solutions of Einstein's field equations, there is no unambiguous means to define their energy content. Here a definition, using slightly broken Noether symmetries is proposed. A problem is noted with the use of the proposal for plane-fronted gravitational waves. To attain a better understanding of the implications of this proposal we also use an artificially constructed time-varying non-vacuum plane symmetric metric and evaluate its Weyl and stress-energy tensors so as to obtain the gravitational and matter components separately and compare them with the energy content obtained by our proposal. The procedure is also used for cylindrical gravitational wave solutions. The usefulness of the definition is demonstrated by the fact that it leads to a result on whether gravitational waves suffer self-damping.