Numerical study of the general-relativistic gravitational collapse of a perfect fluid

In this dissertation I study the critical behavior of a perfect fluid collapsing under its own gravity, with a linear equation of state, by solving the full set of nonlinear partial differential equations resulting from Einstein's theory of general relativity. To solve these equations accurately, I have developed a highly elaborate "numerical laboratory" specifically designed to handle the extreme behavior of this problem. The accuracy of my laboratory was examined extensively with two important test models. The results of the current work extend the previous works of M. Choptuik, C. R. Evans and J. S. Coleman, D. Maison, and many others. The first half of my results are derived from studying the precisely-critical behavior of the fluid. Maison's semi-analytical work had suggested that the critical solution for such a collapsing fluid might exhibit continuous self-similarity (CSS) for values of k less than approximately 0.89 [where the equation of state is pressure equals k times energy density]. Above this value, Maison conjectured that CSS solutions do not exist. However, owing to the assumptions of his approach, his results only suggest what may occur in full collapse simulations. In the current work, I have solved the full set of Einstein's equations---which had previously only been done for k = 1/3---to demonstrate the existence of CSS critical solutions for k values ranging from 0.1 to 0.999. My results thus contradict the above conjecture by Maison. The second half of my results are derived from studying the slightly super-critical behavior of a collapsing perfect fluid. Maison extended Evans and Coleman's work by using a semi-analytical perturbation approach to calculate possible critical exponents for k values less than 0.89. In the current work, I solved the full set of Einstein's equations to demonstrate the existence of mass-scaling laws for k values ranging from 0.1 to 0.999. The values of the critical exponents that I determined are in complete agreement with Maison's results (for k values less than 0.89). This agreement provides significant validation of his semi-analytical approach, as well as further confirmation of the extreme accuracy of my numerical laboratory.