Abstract

We present a detailed description of techniques developed to combine 3D numerical simulations and, subsequently, a single black hole close-limit approximation. This method has made it possible to compute the first complete waveforms covering the post-orbital dynamics of a binary–black-hole system with the numerical simulation covering the essential nonlinear interaction before the close limit becomes applicable for the late time dynamics. In order to couple full numerical and perturbative methods we must address several questions. To determine when close-limit perturbation theory is applicable we apply a combination of invariant a priori estimates and a posteriori consistency checks of the robustness of our results against exchange of linear and nonlinear treatments near the interface. Our method begins with a specialized application of standard numerical techniques adapted to the presently realistic goal of brief, but accurate simulations. Once the numerically modeled binary system reaches a regime that can be treated as perturbations of the Kerr spacetime, we must approximately relate the numerical coordinates to the perturbative background coordinates. We also perform a rotation of a numerically defined tetrad to asymptotically reproduce the tetrad required in the perturbative treatment. We can then produce numerical Cauchy data for the close-limit evolution in the form of the Weyl scalar ψ4 and its time derivative ∂tψ4 with both objects being first order coordinate and tetrad invariant. The Teukolsky equation in Boyer-Lindquist coordinates is adopted to further continue the evolution. To illustrate the application of these techniques we evolve a single Kerr hole and compute the spurious radiation as a measure of the error of the whole procedure. We also briefly discuss the extension of the project to make use of improved full numerical evolutions and outline the approach to a full understanding of astrophysical black-hole–binary systems which we can now pursue.