Black Hole Excision for Dynamic Black Holes

Abstract

We extend previous work on 3D black hole excision to the case of distorted black holes, with a variety of dynamic gauge conditions that are able to respond naturally to the spacetime dynamics. We show that the combination of excision and gauge conditions we use is able to drive highly distorted, rotating black holes to an almost static state at late times, with well behaved metric functions, without the need for any special initial conditions or analytically prescribed gauge functions. Further, we show for the first time that one can extract accurate waveforms from these simulations, with the full machinery of excision and dynamic gauge conditions. The evolutions can be carried out for long times, far exceeding the longevity and accuracy of even better resolved 2D codes. While traditional 2D codes show errors in quantities such as apparent horizon mass of over 100% by t = 100M, and crash by t = 150M, with our new techniques the same systems can be evolved for hundreds of M's in full 3D with errors of only a few percent.

Figures (7)

• Figure 1: We show the evolution of the radial metric function $g_{rr}/\Psi^{4}$ for a Schwarzschild BH along the $x-$axis, constructed from the cartesian components. The upper panel shows the grid stretching in the metric for singularity avoiding slicing with vanishing shift and no excision, while the lower panel shows the metric for the new gauge conditions with an excision box inside a sphere of radius $1M$. Note the difference in the vertical scales. Without shift and excision the metric grows out of control, while with shift and excision a peak begins to form initially but later freezes in as the shift drives the metric to a static configuration (note the time labels).
• Figure 2: We show the lapse and shift for the excision evolution of a Schwarzschild BH. After around 10M, the lapse and shift freeze in as the metric is driven to a static configuration. The size of the excision box was allowed to grow with the change in the coordinate location of the AH.
• Figure 3: The solid line shows the development of the AH mass $M_{AH}$, determined through a 3D AH finder, for the simulation of a Schwarzschild BH shown above, while the dashed lines show the AH mass obtained using 2D and 3D codes with no shift and no excision. The 2D code crashes at $t \simeq 150M$, the 3D run without shift crashes at $t \simeq 50M$, while the 3D run with shift and excision reaches an effectively static state and the error remains less than a few percent even after $t=200M$.
• Figure 4: We show the AH masses $M_{AH}$ for a BH with even-parity distortion for the 2D (no excision, no shift) and 3D (excision, shift) cases. The 3D result continues well past $150M$, while the 2D result becomes very inaccurate and crashes by $t=100M$.
• Figure 5: We show the ratio of the polar and equatorial circumferences as a measure of the dynamics of the AH for the BH with even-parity distortion .