Metric perturbations and their slow evolution for modelling extreme mass ratio inspirals via the gravitational self force approach

In 2015, gravitational waves (GWs) were observed by direct detection for the very first time, over one-hundred years since the publication of Einstein's theory of general relativity (GR). Since then, GWs produced by a variety of systems have been detected. The laser interferometer space antenna (LISA), due to be launched in 2037 by the European Space Agency, will be sensitive to a new frequency of the GW spectrum than we are currently capable of detecting with ground based interferometry. One of the most highly anticipated sources of GWs detectable to LISA, that we have so far been blind to, are extreme mass ratio inspirals (EMRIs). These are binary systems comprised of a massive black hole that is at least ten-thousand times more massive than its satellite. Provided our models are accurate enough, matched filtering between real and theoretical GW signals can provide a measure of precisely how well GR describes our Universe. To achieve this scientific goal, we must calculate the phase of GWs sourced by EMRIs to post-adiabatic order, which in turn requires knowledge of the gravitational self-force (GSF) and metric perturbation through second-order in the small mass ratio. This thesis aims to further our understanding of the evolution of EMRI spacetimes, by determining the phase and amplitude of the GWs they admit. Within the framework of GR, black hole perturbation theory (BHPT), gravitational self-force (GSF) theory, and the two-timescale approximation, this work presents a number of novel calculations as tools for modelling EMRI waveforms. In particular, the MST package was developed for the Black Hole Perturbation Toolkit (BHPToolkit), which solves the Regge-Wheeler (RW) and Teukolsky equations via the Mano-Suzuki-Takasugi method. Another major result in this thesis is the Lorenz gauge calculation of the slowly-evolving first-order metric perturbation for quasicircular, equatorial orbits on a Schwarzschild background during inspiral. This provides a key ingredient to the source of the second-order metric perturbation, and is already being used to generate post adiabatic EMRI waveforms via the GSF approach. Post-adiabatic waveforms presented in this thesis are also found to describe intermediate mass ratio inspirals (IMRIs) to a high degree of accuracy, systems which are already being detected by interferometers on the ground. Thus work presented here is deemed applicable for GW science now and in the future. The transition to plunge is also examined in detail, and waveforms are computed during the transition regime to adiabatic order, again for quasicircular, equatorial orbits around a Schwarzschild black hole. Perturbations to a Kerr black hole will also explored, and a final output of this work is the `pure gauge' contribution to the first-order Lorenz gauge metric perturbation, generated by a gauge vector