Geodesic Geometry Of Some Static Axisymmetric Vacuum Spacetimes

The method of separation of variables has been a longstanding method for the solution of linear partial differential equations.This method has a long history of giving a method for solving difficult mathematical problems and simplifying them.In this work knowledge of the separable solutions of Laplace's equation in three dimensions that have no azimuthal angular dependence is used to investigate a variety of axisymmetric static spacetimes.In so doing the geodesic geometry of these space times is investigated.This lays the foundation for matching these solutions to interior solutions of a static axisymmetric body.In addition the study of the geodesic geometry enables a study of the motion of small bodies in these exterior vacuum spacetimes to be elucidated.Much of this work is new and the presentation is self contained.