Post-Newtonian methods and the gravito-electromagnetic analogy

Abstract

In theoretical physics there is a well known analogy between general relativity and electrodynamics which is based on a linearization of Einstein's field equation. Such a kind of analogy is interesting since Einstein's equation (on component form) is extremely complicated mathematically and hard to gain physical insight into. It turns out though, that a lot of papers in the rich literature on the topic are in lack of a systematic method. The main purpose of this thesis is to study the analogy in a more consistent way using state of the art perturbative methods. The analysis is based on the socalled post-Newtonian approximation-scheme which provides a systematic way to expand any metric theory of gravity, and which also takes account for non-linear effects. By choosing suitable variables and gauge (coordinate) conditions, the post-Newtonian limit of general relativity is reformulated in a way which is appropriate for the discussion. The same kind of systematic expansion is also applied to electrodynamics. In this way the theories are compared in a consistent way beyond their lowest order approximations. This work, described in chapter 4, is basically just a comparison of the mathematical structure of the considered approximations of the theories. In the following chapter the perspective is extended by exploring the huge conceptual difference between the theories. Based on calculations the geometric significance of curvature in the post-Newtonian approximation of general relativity is investigated. The most interesting finding from this chapter is that the analogy turns out being stronger when gravitational phenomena are evaluated in a so-called local proper reference frame, which is quite interesting since in that case the theories are treated on a more equal footing conceptually.