Introduction

Whilst black holes in four dimensions are well mannered, being spherically symmetric or having special algebraic properties which enable them to be found analytically, moving beyond four dimensions many solutions of interest appear to have no manners whatsoever. The problem of finding these unruly black holes becomes that of solving a nonlinear coupled set of partial differential equations (PDEs) for the metric components given by the Einstein equations. In general it is unlikely that closed-form analytic solutions will be found for many of the exotic black holes discussed earlier in this book. If we are to understand their properties then we must turn to numerical techniques to tackle the PDEs that describe them. It is the purpose of this chapter to develop general numerical methods to address the problem of finding static and stationary black holes.

Surely the phrase “the devil is in the detail” could not have a truer application than to numerics. The emphasis of this chapter will be to provide a road map in which we formulate the problem in as unified, elegant and geometric a way as possible. We will also discuss concrete algorithms for solving the resulting formulation, but the extensive details of implementation will not be addressed, probably much to the reader's relief. Such details can be found in the various articles cited in this chapter.