Applications of, and Extensions to, Selected Exact Solutions in General Relativity

TL;DR

This thesis explores exact solutions in general relativity, developing new formalisms for gravitational waves and limits leading to Bianchi type I spacetimes, with implications for understanding spacetime structures.

Contribution

It introduces a formalism for arbitrary polarization of gravitational waves in Rosen form, extends it to higher dimensions, and constructs a new ultra-local limit resulting in Bianchi type I spacetimes.

Findings

Developed a polarization formalism for Rosen gravitational waves

Extended the formalism to arbitrary dimensions and circular polarization

Constructed an ultra-local limit leading to Bianchi type I spacetimes

Abstract

In this thesis we consider several aspects of general relativity relating to exact solutions of the Einstein equations. In the first part gravitational plane waves in the Rosen form are investigated, and we develop a formalism for writing down any arbitrary polarisation in this form. In addition to this we have extended this algorithm to an arbitrary number of dimensions, and have written down an explicit solution for a circularly polarized Rosen wave. In the second part a particular, ultra-local limit along an arbitrary timelike geodesic in any spacetime is constructed, in close analogy with the well-known lightlike Penrose limit. This limit results in a Bianchi type I spacetime. The properties of these spacetimes are examined in the context of this limit, including the Einstein equations, stress-energy conservation and Raychaudhuri equation. Furthermore the conditions for the Bianchi type I spacetime to be diagonal are explicitly set forward, and the effect of the limit on the matter content of a spacetime are examined.