Towards finer estimations of the threshold of black hole formation

Essential to gravitational research are the comprehension of singularities and the investigation of strong-field dynamics. Reliable numerical simulations require a thorough control of all non-physical dynamics that can emerge, such as constraint violations and coordinate singularities. These become particularly challenging near the threshold of collapse, where, in the absence of a horizon, arbitrarily large curvatures cannot be excised. The phenomena emerging at this threshold exhibit a remarkable simplicity in spherical symmetry, referred to as critical phenomena. The potential challenge to the cosmic censorship conjectures makes it an urgent task to examine the threshold beyond spherical symmetry. Departing from sphericity, there still is a lack of agreement among the different groups studying this issue. Clarification on the subject can only be achieved with a finer estimation of the threshold of black hole formation. In this dissertation, we present the steps we have taken to clarify the status of critical collapse beyond spherical symmetry. A successful comparison of results from three independent codes has provided an agreement regarding the collapse of gravitational waves in vacuum. Evidence against universality and a lack of evidence regarding exact discrete self-similarity depict the current understanding of the threshold of collapse in vacuum. Since a finer estimation is needed to keep exploring this and other scenarios beyond sphericity, we turn to adapting the generalised harmonic formulation of general relativity to the study of collapsing spacetimes. Adjusting damping factors to resist a collapsing lapse, we achieve the finest estimation of the threshold to date with this formulation in the collapse of massless scalar fields in spherical symmetry. Finally, we propose a necessary condition for gauges to be compatible with discrete self-similarity and present results of a gauge satisfying this condition, which performs comparably to commonly used gauges.