Numerical simulations of gravitational collapse

In this thesis we present a numerical study of gravitational collapse, within the framework of Einstein's theory of general relativity. We restrict our attention to spacetimes possessing axial symmetry, and incorporate a massless scalar field as the matter source.

Our primary objectives are the study of critical phenomena at the threshold of black hole formation, and the stable simulation of black hole spacetimes. An integral part of the thesis is concerned with developing the necessary numerical tools and techniques to successfully solve these problems. To that end, we have implemented a variant of Berger and Oliger's adaptive mesh refinement (AMR) algorithm, with enhancements that allow us to incorporate elliptic equations into the AMR framework. Using this code, we simulate critical collapse of axisymmetric distributions of scalar field energy, which is the first time this has been done in the non-perturbative regime. For several classes of initial data, our results are consistent with a hypothesized universal critical solution. However, from the collapse of prolate initial data, we find indications that there may be an additional, non-spherical unstable mode. This putative instability eventually causes a near-critical echoing solution to bifurcate into two, causally disconnected solutions that each resemble the spherical critical solution. Furthermore, we speculate that this bifurcation process would continue indefinitely at threshold, resulting in an infinite cascade of near-spherical solutions. However, the evidence for this second unstable mode is not conclusive, and more work will be needed, possibly with an enhanced code, to answer this question.

To numerically study spacetimes containing black holes, one needs to avoid the singularities that occur inside of the holes. The technique that we have implemented to accomplish this is called black hole excision . This aspect of the code is still work-in-progress, for we have not yet incorporated excision into the AMR-based code, and the class of excision boundary conditions we currently employ are inconsistent with the complete set of field equations. However, we are able to obtain stable simulations using a constrained evolution scheme, and we present two preliminary examples, one showing black hole formation, the other a head-on black hole collision.