Colliding Black Holes from a Null Point of View: the Close Limit

Recent years have shown a vital synergism between numerical and perturbative approaches in the context of Cauchy evolution. In this work we try to make these benefits also accessible in the context of null-surface-based evolution. We treat the “close-limit” approximation to the post-merger phase of a binary black hole spacetime in terms of the characteristic initial value problem. We outline a framework to treat the Teukolsky equation 2 in a null evolution setting, and our numerical strategy to solve the perturbative problem in the time domain. A detailed discussion and results for the outgoing radiation problem are given in , the details of the ingoing radiation problem will be treated in a forthcoming paper . We apply our method to evolve “close-limit” horizon data, which we obtain from linearizing the data discussed in . There we use Bondi-Saschs coordinates, which simplify to Israel coordinates in the spherical case, which are based on an affine parameter u along the white hole horizon, normalized to u = 0 at the bifurcation sphere, and an affine parameter λ along the outgoing null cones, fixed to λ = 0 at the white hole horizon and (∂u)(∂aλ) = −1). These coordinates cover the entire Kruskal manifold r > 0 with remarkably simple analytic behavior, as first discovered by Israel . Using these convenient coordinates as a starting point, it is straightforward to write the Teukolsky equations explicitly in any background coordinate system. In order to treat the radiation near I it is advantageous to consider a “boosted tetrad” (l̃, ñ,m, m̄) with l̃ = −∇ũ, ñ satisfying l̃ña = −1 and a dyad m = 1 √