The evolution of single black holes merely serves as a benchmark test in numerical relativity. Single black hole spacetimes can be described in terms of analytic expressions and their mathematical analysis has yielded a wealth of insight into the general theory of relativity.
It has turned out to be extremely difficult, however, to evolve such spacetimes numerically in 3-dimensional simulations. The key difficulties arise from the complexity of Einstein's equations and the fact, that black hole spacetimes contain singularities, that is, points where physical quantities take on infinite values. We illustrate this feature in the following graphic.
![[traceK]](Images/trk_ini.jpg)
The figure shows the trace of the Extrinsic Curvature of a single static black hole (in so-called Kerr Schild coordinates) as a function of cartesian coordinates x,y (the z-direction is suppressed). One clearly sees, how the curvature increases near the black hole center. In fact, it becomes infinite and the spikes in the image are merely an artifact of the visualization program which was instructed to ignore points, where the curvature exceeds a threshold value.
State of the art computer codes are now able to evolve such spacetimes for essentially arbitrarily long times without encountering difficulties. A convenient measure for achieving this goal is to monitor the time derivative of the extrinsic curvature averaged over the whole grid. Because single black hole spacetimes do not, normally, depend on time, the time derivative should be zero within numerical precision. The following graphic shows this averaged time derivative we have obtained with the black hole excision technique.
![[rhs_tracek]](Images/dKdt_psu.gif)
Clearly, the time variation drops exponentially until variations are only visible at the round-off level of the representation of numbers in the computer.