Numerical treatment of hyperbolic systems in black-hole spacetimes

A new system of more sensitive gravitational wave detectors will be up and running in the near future. Theoretical predictions are needed to compare to the detected signals in order to identify and pinpoint the source. The source which is thought to contribute to a large portion of the detected gravitational radiation is the coalescing binary black hole system. Computation of the templates will involve solving the full nonlinear Einstein equations in 3 + 1 dimensions. This is a problem which is too difficult to solve analytically and therefore must be solved numerically. Numerically, this is also a very difficult problem. There are many techniques that must be used which we know very little about such as how to excise the black holes from the computational grid to avoid the singularities, how to apply boundary conditions, how to handle coordinate singularities, etc. Also at question is the manner in which the Einstein equations are posed. In the past, the formulations primarily used were of a mixed elliptic-hyperbolic type. Well-posedness cannot in general be proven for such a system and therefore much attention as of late has been given to certain hyperbolic formulations. One such formulation, the Einstein-Christoffel system, is thought to be the most promising. It has only thirty variables and contains only physical characteristic speeds. There is also a similarity of the Einstein-Christoffel system to six copies of a scalar field equation. Therefore, if one could master the numerical techniques in evolving a system of similar mathematical structure, a generalization to the full Einstein-Christoffel system could potentially be made. This author examines just that. The evolution of a scalar field on different black hole backgrounds is carried out using black hole excision with an operator splitting, characteristic numerical technique. The result is a stable, consistent method for evolving hyperbolic systems in black hole spacetimes in three dimensions using black hole excision.