At this point we might suspect that we are all set to carry out dynamical simulations involving relativistic gravitational fields in three spatial dimensions, i.e., dynamical simulations in full 3 + 1 dimensional spacetimes. After all, we have derived the 3 + 1 evolution equations for the gravitational fields in Chapter 2, developed techniques for the construction of initial field data in Chapter 3, and discussed strategies for imposing suitable coordinate conditions in Chapter 4. Should the spacetime in which we are interested contain the most common matter sources, we can consult Chapter 5 for the matter source terms and the matter equations of motion. Beyond assembling all of the relevant equations, we have also sketched algorithms for solving them numerically in Chapter 6. If black holes are present, we have derived methods to locate and measure their horizons in Chapter 7; if gravitational waves are generated we have discussed how to extract radiation wave forms numerically in Chapter 9. Finally, we have evolved relativistic systems in spherical symmetry (1 + 1 dimensions) in Chapter 8 and in axisymmetry (2 + 1 dimensions) in Chapter 10. What else is left to do before going on to perform simulations in full 3 + 1 dimensions?

Suppose then we were to plunge ahead with the tools currently at our disposal to build a dynamical spacetime in 3 + 1 dimensions. We could gain some confidence and computational experience by choosing to explore as our first case a simple dynamical system with a known analytic solution.