The Lazarus project: A pragmatic approach to binary black hole evolutions

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 non-linear interaction before the close limit becomes applicable for the late time dynamics. 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 non-linear treatments near the interface. 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 $\psi_4$ and its time derivative $\partial_t\psi_4$ 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.Comment: New typos found in the version appeared in PRD. (Mostly found and collected by Bernard Kelly

The growth rate over trees of any family of set defined by a monadic second order formula is semi-computable

Monadic second order logic can be used to express many classical notions of sets of vertices of a graph as for instance: dominating sets, induced matchings, perfect codes, independent sets or irredundant sets. Bounds on the number of sets of any such family of sets are interesting from a combinatorial point of view and have algorithmic applications. Many such bounds on different families of sets over different classes of graphs are already provided in the literature. In particular, Rote recently showed that the number of minimal dominating sets in trees of order $n$ is at most $95^{\frac{n}{13}}$ and that this bound is asymptotically sharp up to a multiplicative constant. We build on his work to show that what he did for minimal dominating sets can be done for any family of sets definable by a monadic second order formula. We first show that, for any monadic second order formula over graphs that characterizes a given kind of subset of its vertices, the maximal number of such sets in a tree can be expressed as the \textit{growth rate of a bilinear system}. This mostly relies on well known links between monadic second order logic over trees and tree automata and basic tree automata manipulations. Then we show that this "growth rate" of a bilinear system can be approximated from above.We then use our implementation of this result to provide bounds on the number of independent dominating sets, total perfect dominating sets, induced matchings, maximal induced matchings, minimal perfect dominating sets, perfect codes and maximal irredundant sets on trees. We also solve a question from D. Y. Kang et al. regarding $r$-matchings and improve a bound from G\'orska and Skupie\'n on the number of maximal matchings on trees. Remark that this approach is easily generalizable to graphs of bounded tree width or clique width (or any similar class of graphs where tree automata are meaningful)