Advances in Computational General Relativity

Abstract

Einstein's theory of gravity, general relativity, describes space and time together as one entity-spacetime. The governing equations couple together the evolution of spacetime and matter in a highly nonlinear fashion, making closed-form analytic solutions only available for simple systems with a high degree of symmetry. Computer simulations are used to study general relativity and the dynamics of matter in strongly gravitating systems. Numerical simulations of binary neutron star inspiral and coalescence are very challenging and current codes will be unable to produce accurate models for understanding observations as gravitational wave and electromagnetic experiments improve over the next few years. In the first two chapters of this thesis we develop and implement new numerical methods that we expect will reduce the cost of binary neutron star simulations by a factor of three to ten. This reduction in computational cost will no doubt be used to increase the accuracy of the simulations in order to meet the demands of new and ongoing experiments. In the second part of this thesis we study the formation of microscopic black holes. In the late 80’s and early 90’s Choptuik used numerical relativity to answer the question "What happens at the threshold of black hole formation?" Studying black hole formation requires resolving six orders of magnitude in space and time. This is why before the present work there were no detailed studies of critical behavior in 3d. This work contributes to understanding how the symmetry of the spacetime affects the threshold of black hole formation. In the last two chapters of this thesis we turn our attention to studying the stability of anti-de Sitter space against the formation of black holes. Anti-de Sitter spacetime has seen a lot of interest recently thanks to the anti-de Sitter/conformal field theory conjecture, which relates black hole formation in anti-de Sitter space to thermalization of the conformal field theory on the boundary of the spacetime. We study massive scalar fields in anti-de Sitter space by solving the Einstein equations numerically, finding new evidence for chaos and rich dynamics compared to the massless scalar field case. Finally, we use perturbation theory to probe arbitrarily small perturbations, which cannot be done by numerically solving the Einstein equations. We present evidence that anti-de Sitter space in more than four spacetime dimensions is unstable against black hole formation.