Towards a wave-extraction method for numerical relativity. II. The quasi-Kinnersley frame

Abstract

The Newman-Penrose formalism may be used in numerical relativity to extract coordinateinvariant information about gravitational radiation emitted in strong-field dynamical scenarios. The main challenge in doing so is to identify a null tetrad appropriately adapted to the simulated geometry such that Newman-Penrose quantities computed relative to it have an invariant physical meaning. In black hole perturbation theory, the Teukolsky formalism uses such adapted tetrads, those which differ only perturbatively from the background Kinnersley tetrad. At late times, numerical simulations of astrophysical processes producing isolated black holes ought to admit descriptions in the Teukolsky formalism. However, adapted tetrads in this context must be identified using only the numerically computed metric, since no background Kerr geometry is known a priori. To do this, this paper introduces the notion of a quasi-Kinnersley frame. This frame, when space-time is perturbatively close to Kerr, approximates the background Kinnersley frame. However, it remains calculable much more generally, in space-times non-perturbatively different from Kerr. We give an explicit solution for the tetrad transformation which is required in order to find this frame in a general space-time.

Extrapolating gravitational-wave data from numerical simulations

“…Because of gauge effects, the areal-radius of a coordinate sphere changes as a function of time, so we measure this as a function of time. Finally, we measure the average value of g TT as a function of coordinate time on the extraction spheres to correct for (14).…”

Degeneracy measures for the algebraic classification of numerical spacetimes

“…As described in Ref. [12], at any point where the Weyl tensor is Type I, there are precisely three distinct families of tetrads in which two particular Weyl scalars vanish, Ψ 1 = Ψ 3 = 0 (they each amount to families of tetrads, rather than three particular tetrads, because this condition is preserved by the spin-boost freedom). A particular tetrad field, chosen from these three families to coincide with the conventional Kinnersley tetrad near infinity, is often referred to as a quasi-Kinnersley tetrad.…”

“…In Sec. III we will emphasize the fact that a tetrad well-suited to gravitational wave extraction, in particular the quasi-Kinnersley tetrad [12], may be particularly ill-suited to measuring the nearness to Petrov Type D using ∆ ij . In Sec.…”

Learning about compact binary merger: The interplay between numerical relativity and gravitational-wave astronomy

“…This issue can be alleviated in part by a judicial choice of the tetrad used to calculate Ψ 4 [36,37,38].…”

“…One is based on perturbative methods [28,29,30,31] which relies on a suitable identification of a background spacetime in particular coordinates and extracting specific quantities from the simulation. A second approach, which has become the most common one, makes use of the infrastructure developed to calculate the radiation at I + but applied at a finite distance from the source (see for instance [11,32,33,34,35,36,37,38] ). While in principle this approach can be used beyond the perturbative level and is less sensitive to identifying a correct background, quantities defined at I + need to be translated to finite distances where they may not be well or unambiguously defined.…”