Our modern understanding of gravity was first introduced in the early 20th Century by the pioneering work of Albert Einstein, who formulated the General Theory of Relativity. This began in 1905 with him reworking classical notions of space and time: from his two postulates that (1) the laws of electrodynamics and optics hold in all inertial reference frames, and (2) the speed of light in a vacuum is always the same, he showed it followed that these measures which were previously held to be absolute are actually relative to a particular observer or coordinate system [26]. While this original theory was only valid in the absence of gravity, and so would later be termed the Special Theory of Relativity, it was soon generalized in 1915, with the publication of the Einstein Field Equations (EFEs) [27] [29], to take this additional layer of complexity into account, and so to be capable in principle of explaining all the phenomena covered by Newtonian mechanics, in addition to providing new testable predictions, such as regarding the perihelion precession of Mercury, on which it was found to be in greater agreement with observations than earlier theories were. In the General Theory, space-time is described by the 10 components of a symmetrical 4-dimensional tensor, known as the metric, which describes how distances between points are determined. Gravity is understood then not as a true force, as it is in Newtonian gravity, but instead using the tools provided by differential geometry, in which gravity can instead be interpreted a property of the geometry of space-time, which is curved by mass- energy, with particles merely following geodesic paths through this geometry. While the most straightforward gravitational phenomenon is that of the curving of space-time by the presence of a massive body, as early as 1916 [28] Einstein additionally proposed the existence of gravitational waves: the radiation of energy from a system due to the periodic change in space-time curvature caused by the motion of massive bodies relative to each other.

In most cases, these gravitational waves are exceptionally small, and indeed entirely negligible, however, for some of the most extreme astrophysical events, such as in the collision of black holes, this ceases to be true, with multiple solar masses worth of energy being radiated away. Interferometers have been built around the world to attempt to detect the gravitational waves emitted from such events, including LIGO in the United States of America [41], KAGRA in Japan [54], and VIRGO in Italy [57]. LIGO first observed a confirmed signal from a black hole-black hole merger in 2015 [42], and in the following years dozens more detections have been made, including of a neutron star-neutron star merger that was also observed in the electromagnetic spectrum by the Fermi Gamma-ray Burst Monitor [43].

Such gravitational wave astronomy provides a wide range of possibilities, both as a new way of testing the limits of General Relativity, and also of observing distant astrophysical events and objects, which previously could only be investigated via electromagnetic radiation. While currently this is limited to certain classes of black hole and neutron star mergers, the next generation of gravitational wave detectors, such as the space-based interferometer LISA that is planned to be launched in the 2030s [44], will not only increase the rate at which mergers are detected by orders of magnitude, but will also be able to detect new gravitational wave sources, including extreme mass-ratio inspirals and the gravitational background that originated in the early universe.

Yet, while this field shows a lot of promise, even just making predictions out of Einstein’s theory is exceptionally challenging. The first full analytical solution to the EFEs, the Schwarzschild solution describing the curvature around a single non-rotating and uncharged point mass, was published in 1916 [52], only a short time after Einstein’s original proposal of General Relativity, yet it would take until 1963 for this to be extended to the case of a rotating mass [40]. To avoid these analytical complexities, a range of numerical approaches have been developed, yet these too are far from trivial - the biggest additional difficulty of numerical relativity, when compared to classical mechanics, is that time is not an absolute universal variable over which a simulation can straightforwardly be evolved, but is instead itself part of the variable space-time geometry.

In this thesis I will be focusing on two of these numerical approaches in particular that are used in the context of binary black-hole simulations, as is needed for interpreting interferometer data. The first of these is BSSN [5] [53], one example of a 3+1 approach in which the 4-dimensions of space-time are ’foliated’ into 3-dimensional sheets between which it is possible to evolve (albeit with added complications, when compared with a classical simulation, as to how one moves between these sheets), which has found substantial success in simulating black holes - being used, for example, for one of the first full simulations of a binary black-hole merger [20]. Yet BSSN comes with certain drawbacks, in that it is computationally expensive when simulating large regions of space-time, as for example is needed in order to investigate the transmission of a gravitational wave from a distant source to a detector on Earth. The second approach I will look at, meanwhile, has the opposite problem. This is the characteristic formulation, which can easily be written in ’compactified’ coordinates, which allow for one to simulate across even an infinite distance in a finite computation time, but which is not capable of accurately evolving a simulation close to a source. To take advantage of the strengths of both of these approaches, while avoiding their weaknesses, it was proposed by Bishop in the 90s to take a mixed approach, combining the usage of both into a single simulation. Two possible ways of accomplishing this, termed Cauchy-Characteristic Extraction (CCE) [12] and Cauchy-Characteristic Matching (CCM) [11], were suggested, of which CCE has seen successful implementation in determining wave- forms from black-hole mergers [48]. CCM, however, has yet to see such success, with it’s viability even having been questioned recently [33], and it is this, along with the component BSSN and characteristic parts, which I aim to investigate in this thesis.

In chapter 2, I will provide a literature review, covering the key past developments regarding CCM. Then, in chapter 3, I will describe my own numerical implementation. Here I begin in section 3.1 by briefly discuss the numerical tools I used for my Python implementation, highlighting in particular my usage of the COFFEE package. This is followed in section 3.2 by an in-depth treatment of the Cauchy side of CCM, building up from ADM to Cartesian BSSN and then to BSSN in spherical polar coordinates, as well as a discussion of my choices in gauge and initial data for the BSSN system, and the SAT method that I implemented as an approach to provide boundary data. Then, in 3.3, I will move on to the characteristic formulation, here building up from the Bondi-Sachs metric and providing an overview of the usage of spherical harmonics, before again looking at a possible choice for the initial data. Finally, in 3.4, I will turn to CCM itself, describing one suggestion for a gauge choice as well as the requisite coordinate transformation for moving between the BSSN and characteristic systems. On to chapter 4, I will give the results of my implementation, explaining my success at implementing both the BSSN and characteristic systems, and highlighting in particular the new results derived from my use of the SAT method for the BSSN system, but ultimate failure to fully develop a working CCM implementation. I will then summarise what I have accomplished in chapter 5, and also discuss future work that could be done on CCM based on my results.