Colliding Black Holes: How Far Can the Close Approximation Go?

We study the head-on collision of two equal-mass momentarily stationary black holes, using black hole perturbation theory up to second order. Compared to first-order results, this significantly improves agreement with numerically computed wave forms and energy. Much more important, second-order results correctly indicate the range of validity of perturbation theory. This use of second-order corrections to provide ``error bars'' to the first-order results makes perturbation theory a viable tool for providing benchmarks for numerical relativity in more generic collisions and, in some range of collision parameters, for supplying wave form templates for gravitational wave detection.