Numerical Simulation of Orbiting Black Holes

We present numerical simulations of binary black hole systems which for the first time last for about one orbital period for close but still separate black holes as indicated by the absence of a common apparent horizon. An important part of the method is the construction of comoving coordinates, in which both the angular and the radial motion are minimized through a dynamically adjusted shift condition. We use fixed mesh refinement for computational efficiency.

Figure 1

Evolution of the AH mass for the black hole binary with ${\rho }_0=3.0M$. The evolution lasts longer than one orbital period of $114M$ defined by the initial data. The squares mark a run with seven nested levels with coarsest resolution $2M$ and finest resolution $h=0.03125M$, and with the spherical outer boundary at about $48M$, which crashes around $145M$. Also plotted are results from seven control runs with the outer boundary at $24M$ and $96M$, with a cubical outer boundary, and with the AH extracted on a coarser grid to check its convergence. There is little difference in the results, except that the runs with the boundary at $24M$ last somewhat longer.

Figure 2

The panel on the left shows convergence of the AH mass. The number and size of the refinement levels was not changed, but the overall resolution was rescaled by a constant factor. There is a linear downward drift in the mass which becomes smaller with increasing resolution. The panel on the right displays the mass at infinity estimated on a sphere of radius $20M$ assuming a Schwarzschild background, showing fluctuations of about 20% to 40%. The lower and upper lines for a given resolution correspond to a cubical outer boundary at $24M$ and $48M$, respectively.

Figure 3

Evolution of the $x$ and $y$ components of the shift vector along the $y$ axis. The punctures are located on the $y$ axis at $y=\pm 3M$.

Figure 4

Evolution of the AH of one of the black holes in the $x\mathrm{\text{−}}y$ plane. The dashed line shows the AH at $t-24M$ in each panel. Initially, the AH moves outward quickly while the gauge adjusts itself near the black hole. It then slowly shrinks toward the center while being deformed slightly until eventually it drifts out of shape before the run fails around $145M$. Note that the proper area changes linearly and only on the order of 10% during the entire run; see Fig. 1.