Numerical evolution of plane gravitational waves

Unlike electromagnetic waves, gravitational waves self interact. This interaction is non-linear and can have very interesting properties which effect the curvature of space-time. A gravitational plane wave collider, implemented in the Python package COFFEE [20] that been developed in recent years by the Otago relativity group and implements the method of lines, can be reliably used to study this self-interaction. This was shown to work well numerically as profounded by Frauendiener, Stevens and Whale in 2014 [24]. For this reason, COFFEE will be used to study these gravitational wave propagations and subsequently collisions. The Einstein field equations are formulated as a well-posed initial boundary value problem (IBVP) in the Friedrich-Nagy gauge [26] and due to the large class of boundary conditions admitted by this framework, a variety of investigations into the propagation of plane gravitational waves could be carried out. This study focuses on the propagation of plane gravitational waves in the de Sitter (dS) space-time, which is the maximally symmetric solution of the Einstein’s vacuum field equations with a positive cosmological constant λ. There is substantial cosmological evidence that our universe is asymptotically de Sitter, yet no work, analytical nor numerical, has been done on gravitational plane waves propagating on such a space-time, mainly due to the increased complexity from the non-vanishing λ. Firstly, it is found analytically that with an arbitrary cosmological constant λ and a non-vanishing energy momentum tensor, the constraints will propagate. This means that we still have a wellposed IBVP, which is nontrivial since the Friedrich-Nagy gauge has only been shown to lead to a wellposed IBVP without matter [26]. Using this system, we consider one ingoing wave propagating on said space-time in vacuum. The area of the ingoing wave profile is varied and inferences are made about the different phenomena that arise in the curvature of space-time during the evolution. It is found that there exists a critical value of the wave’s area, ac, whereby taking the area below this value the system asymptotes to its initial state, and above the system diverges, indicating the presence of a singularity. Furthermore, we define an expansion parameter H to measure how the gravitational waves influence the accelerated expansion, generalising (numerically) results of Tsamis and Woodard.Thesis (MSc) -- Faculty of Science, Mathematics, 202