Perturbations of Test Mass Motion in the Schwarzschild Spacetime Using the Method of Hamilton and Jacobi.

Abstract

This thesis develops and applies Hamilton-Jacobi perturbation theory to describe relativistic test mass motion in the Schwarzschild spacetime. The form of Hamiltonian used is rigorously derived from the relativistic Lagrangian. A justification for other choices of Hamiltonians is also given. The results of all the intermediate calculations needed to consider arbitrary perturbations of test mass motion in the Schwarzschild geometry are presented. These results are then used to find the perturbed motion of a test mass due to a slowly spinning, spherically symmetric body. The motion is thoroughly examined for a variety of special orbits. Of particular interest are the results in the presence of a weak gravitational field. The results for the special case of a weak gravitational field should reduce to the Lense-Thirring results, but they do not. It is explicitly shown that Lense and Thirring are in error and a semi-classical derivation of the correct results is given. In the process it is shown that the Lagrange Planetary equations are invalid if the perturbing Hamiltonian depends on momentum. A generalized set of classical Lagrange Planetary equations for arbitrary perturbations is presented.