Wave zone extraction of gravitational radiation in three-dimensional numerical relativity

TL;DR

This work addresses the challenge of extracting accurate gravitational radiation from three-dimensional numerical relativity simulations in the wave zone with causally disconnected boundaries. It develops a pipeline based on the gauge-invariant Weyl scalar $\Psi_4$ within a $3+1$ framework, employs Misner's spherical-harmonic decomposition, and uses fixed mesh refinement to resolve nonlinear sources and the wave zone. The authors validate the approach with a linear Teukolsky wave and a nonlinear equal-mass head-on black-hole collision, achieving second-order convergence and peak errors of about 0.4%–2% across multiple extraction radii. These results provide high-quality, radius-robust waveforms and establish practical testbeds for wave-extraction schemes that can support future comparisons and astrophysical studies.

Abstract

We present convergent gravitational waveforms extracted from three-dimensional, numerical simulations in the wave zone and with causally disconnected boundaries. These waveforms last for multiple periods and are very accurate, showing a peak error to peak amplitude ratio of 2% or better. Our approach includes defining the Weyl scalar Psi_4 in terms of a three-plus-one decomposition of the Einstein equations; applying, for the first time, a novel algorithm due to Misner for computing spherical harmonic components of our wave data; and using fixed mesh refinement to focus resolution on non-linear sources while simultaneously resolving the wave zone and maintaining a causally disconnected computational boundary. We apply our techniques to a (linear) Teukolsky wave, and then to an equal mass, head-on collision of two black holes. We argue both for the quality of our results and for the value of these problems as standard test cases for wave extraction techniques.

Figures (5)

• Figure 1: Selected extraction maps. Each map shows the computational grid in a coordinate plane and is labeled with a triple of numbers $(N,R,\Delta)$, which indicate the number of (cell-centered) grid points across one coordinate direction in the coarsest region, the extraction radius, and the half-thickness of the shell, respectively. Lengths are measured in units of $\lambda$. Note that the shells generically pass through multiple refinement regions, especially since, in three dimensions, the corners of the cubic refinement regions tend to poke through the spherical extraction shells. The three circles drawn on each graph show the extraction sphere (red) and the edges of the finite-thickness shell (blue) around the sphere used in the Misner algorithm.
• Figure 2: The $l=2$, $m=0$ (spin-weight $-2$) component of the Teukolsky wave, as computed at five different radii for the highest resolution run. The waveform is preserved up to the leading order $1/r$ scaling. Panel A shows the raw data. Panel B shows the same data, scaled-up by $r_{ext}$ and shifted in time to $t=3\lambda$, the location of the inner-most extraction radius.
• Figure 3: A convergence plot for the Teukolsky wave simulation. The lines are the errors in the $l=2$, $m=0$ (spin-weight $-2$) component of the simulations at the two resolutions as compared to the analytic solution.
• Figure 4: A convergence plot of the $l=2$, $m=0$ (spin-weight $-2$) component of $\Psi_4$ for the equal mass black hole head-on collision problem. Panel A shows the convergence at the $r=20M$ extraction radius, and Panel B shows the convergence at the $r=40M$ extraction radius. Note that, consistent with the coarser resolution at larger radii, there is a slight decrease in the agreement between the scaled errors in Panel B, but that the agreement is still excellent.
• Figure 5: A comparison between the $l=2$, $m=0$ components of $\Psi_4$ as computed at distinct radii. The waveforms have been scaled-up by $r$ and shifted to the innermost extraction radius, $r=20M$. In order to account for the fact that our lapse is not geodesic, we scale the waveforms by an approximate Schwarzschild radius $R = r(1+M/(2r))^2$ rather than the simulation's radial coordinate $r$, and shift in time by the light-travel time in a Schwarzschild background.