Accurate Evolution of Orbiting Binary Black Holes

We present a detailed analysis of binary black hole evolutions in the last orbit and demonstrate consistent and convergent results for the trajectories of the individual bodies. The gauge choice can significantly affect the overall accuracy of the evolution. It is possible to reconcile certain gauge-dependent discrepancies by examining the convergence limit. We illustrate these results using an initial data set recently evolved by Brügmann et al. [Phys. Rev. Lett. 92, 211101 (2004)]. For our highest resolution and most accurate gauge, we estimate the duration of this data set’s last orbit to be approximately $59M_{\mathrm{ADM}}$.

Figure 1

The minimal proper distance between individual AHs as a function of time. Lines show two representative resolutions $h=0.025M$ and $h=0.015M$ for each gauge choice GC1, GC2, and GC3 discussed in the text and an $h=0.0125$ evolution for GC1. Points show Richardson extrapolations (RE) for resolutions $h\in \{0.018,0.015,0.0125\}M$ for GC1, $h\in \{0.020,0.018,0.015\}M$ for GC2, and $h\in \{0.025,0.020,0.015\}M$ for GC3.

Figure 2

Angular motion of individual AHs as a function of time (lower scale) and angle (upper scale, in radians measured backwards along the orbit from the first appearance of a common AH). The curve shows the angular velocity $\Omega$ of the AH centroid for GC3, as measured by integrating the corrections applied to the gauge via (1). The horizontal line corresponds to the ${\Omega }_0=0.055$ estimated for the initial data.

Figure 3

Schematic showing the motion of one of the BHs with time, for the GC3 gauge choice at the highest resolution $h=0.015M$. At intervals of $t=5M$, the AH cross sections in the $xy$ plane are plotted by transforming the corotating coordinate system by the specified angle and distance. The apparent growth of the AHs with time is a nonphysical coordinate effect. The first appearance of a common AH at $t=124M$ and the corresponding final single AH are shown superposed on the figure as dotted lines.