The smallest scale arises by defining the hole radius as unity. Then, another scale is the wavelength associated with ring-down oscillations of the black hole, about 20 times as large. Adequate extraction of the radiation waveforms requires an extraction zone of hundreds of times the hole radius, setting a large outer scale for the problem. Large computers are needed to run these problems. The surface of a black hole is a causal boundary, analogous to a supersonic point in fluid mechanics.
Signals can propagate in only one direction across it.
Thus it is sensible and beneficial to consider differencing that computes only the outside domain of the problem, which can radiate to infinity. The study of the gravitational field can be cast in one of two equivalent ways that differ greatly in the way they describe the gravitational field. One way treats the gravitational field as a configuration with given derivatives that is evolved by hyperbolic equations into the future of the problem. The second method treats the three-dimensional characteristics of the problem, which are the loci of the signal flow from events in the central region out to infinity, and evolves the system from one to the next of these characteristic hypersurfaces. Techniques are needed for joining two substantially different descriptions at some boundary surrounding the central, dynamical part of the simulation.