Gravitational wave extraction from an inspiraling configuration of merging black holes

TL;DR

The approach follows the "puncture" treatment of black holes, but utilizing a new gauge condition which allows the black holes to move successfully through the computational domain, and applies these techniques to an inspiraling binary, modeling the radiation generated during the final plunge and ringdown.

Abstract

We present new techniqes for evolving binary black hole systems which allow the accurate determination of gravitational waveforms directly from the wave zone region of the numerical simulations. Rather than excising the black hole interiors, our approach follows the "puncture" treatment of black holes, but utilzing a new gauge condition which allows the black holes to move successfully through the computational domain. We apply these techniques to an inspiraling binary, modeling the radiation generated during the final plunge and ringdown. We demonstrate convergence of the waveforms and good conservation of mass-energy, with just over 3% of the system's mass converted to gravitional radiation.

Figures (5)

• Figure 1: Hamiltonian constraint error $C_H$ for $h_f=M/24$ and $M/32$, at two times when a puncture is near to crossing the positive $x$-axis. The data are scaled such that the lines should superpose in the case of perfect $2^{\rm nd}$-order convergence. The inset shows that $C_H$ is well-behaved in the region near the punctures. The horizontal lines indicate the approximate location of the apparent horizons; at the later time a common horizon has formed.
• Figure 2: The positions of the apparent horizons at times $t=0,5,10,15$, and $20M$ for our $M/16$ run. The curve shows the trajectories of centroids of the individual apparent horizons.
• Figure 3: Real part of $r \Psi_4$ extracted from the numerical simulation on spheres of radii $r_{EX}=20$, and $40M$ for the medium and high resolution runs. The waveforms extracted at different radii have been rescaled by $1/r_{EX}$ and shifted in time to account for the wave propagation time between the extraction spheres. At high resolution ($h_f = M/32$) there is no discernible dependence on extraction radius. For comparison, we show Lazarus waveforms from Ref. Baker:2002qf.
• Figure 4: Differences of the real part of $r \Psi_4$ for resolutions of $h_f=M/16, M/24$, and $M/32$ appropriately scaled such that for perfect $2^{\rm nd}$-order convergence the lines would lay on top of each other.
• Figure 5: Conservation of mass-energy for the highest resolution case, $h_f = M/32$. We compare the ADM mass $M_{\rm ADM}$ with the mass remaining, $M - E$, after gravitational radiation energy loss $E$. The good agreement, based on extraction spheres at $r_{EX}=40$ and $50M$, indicates conservation of energy in the simulation.