How far away is far enough for extracting numerical waveforms, and how much do they depend on the extraction method?

We present a method for extracting gravitational waves from numerical spacetimes which generalizes and refines one of the standard methods based on the Regge–Wheeler–Zerilli perturbation formalism. At the analytical level, this generalization allows a much more general class of slicing conditions for the background geometry, and is thus not restricted to Schwarzschild-like coordinates. At the numerical level, our approach uses high-order multi-block methods, which improve both the accuracy of our simulations and of our extraction procedure. In particular, the latter is simplified since there is no need for interpolation, and we can afford to extract accurate waves at large radii with only little additional computational effort. We then present fully nonlinear three-dimensional numerical evolutions of a distorted Schwarzschild black hole in Kerr–Schild coordinates with an odd parity perturbation and analyse the improvement that we gain from our generalized wave extraction, comparing our new method to the standard one. In particular, we analyse in detail the quasinormal frequencies of the extracted waves, using both methods. We do so by comparing the extracted waves with one-dimensional high resolution solutions of the corresponding generalized Regge–Wheeler equation. We explicitly see that the errors in the waveforms extracted with the standard method at fixed, finite extraction radii do not converge to zero with increasing resolution. We find that even with observers as far out as R = 80M—which is larger than what is commonly used in state-of-the-art simulations—the assumption in the standard method that the background is close to having Schwarzschild-like coordinates increases the error in the extracted waves considerably. Furthermore, those errors are dominated by the extraction method itself and not by the accuracy of our simulations. For extraction radii between 20M and 80M and for the resolutions that we use in this paper, our new method decreases the errors in the extracted waves, compared to the standard method, by between one and three orders of magnitude. In a general scenario, for example a collision of compact objects, there is no precise definition of gravitational radiation at a finite distance, and gravitational wave extraction methods at such distances are thus inherently approximate. The results of this paper bring up the possibility that different choices in the wave extraction procedure at a fixed and finite distance may result in relative differences in the waveforms which are actually larger than the numerical errors in the solution.