The main purpose of this talk is to describe, by way of concrete examples, how the field of numerical relativity now contributes to our understanding of open questions in gravitational collapse, heavy-ion physics, and turbulence. I will begin by motivating these studies in terms of the physical systems they are intended to clarify,then provide specific examples of how to describe these systems with numerical simulations of asymptotically AdS spacetimes in the fully non-linear regime of general relativity.

The collision of black holes is a classic problem in the strong-field regime of general relativity, with no known closed-form solution. When exact solutions are not known, or perturbative expansions about known solutions are inadequate to capture the non-linear dynamics, the numerical solution of the Einstein field equations is required. The main purpose of this talk is to describe, in detail, the necessary ingredients for achieving stable Cauchy evolution of black hole collisions in asymptotically anti-de Sitter (AdS) spacetimes. I will begin by motivating this program in terms of the heavy-ion physics it is intended to clarify. I will then give an overview of anti-de Sitter space, the mapping to its dual boundary conformal field theory, and the method we use to numerically solve the fully non-linear Einstein field equations with AdS boundary conditions. As a concrete example of these ideas, I will describe simulations of stable AdS black hole mergers in 5 dimensions.