Horizons, hyperbolic systems, and inner boundary conditions in numerical relativity

We discuss several closely connected open questions central to Numerical Relativity. These involve computational, numerical, theoretical, and physical aspects of evolving and understanding dynamical black hole spacetimes. We discuss finding and understanding the event horizon (EH), which is the causal boundary separating the black hole interior from its exterior, in dynamical black hole spacetimes. In the EH studies, we formulate a set of tools for analyzing the behavior of the EH, including proposing a construction of the membrane paradigm suitable for numerical relativity. Moreover, we probe the geometry of the two black hole collision in detail, and investigate limits of the distortion of a single black hole EH interacting with a gravitational wave. We discuss both standard and hyperbolic formulations of the Einstein equations and how these are amenable to numerical treatment with modern parallel and adaptive computational techniques. In the course of this discussion, we present a three-dimensional implementation of the Bona-Masso hyperbolic system, and a three dimensional code we call "Cactus," which we apply to several spacetimes. Finally, exploiting certain mathematical properties of the Bona-Masso system and the causal structure of horizons, we discuss and compare several methods to implement a (preliminary) apparent horizon boundary condition for evolving black hole spacetimes using the full three-dimensional Cactus code.