Turduckening black holes: An analytical and computational study

We provide a detailed analysis of several aspects of the turduckening technique for evolving black holes. At the analytical level we study the constraint propagation for a family of formulations of Einstein’s field equations and identify under what conditions the turducken procedure is rigorously justified and under what conditions constraint violations will propagate to the outside of the black holes. We present high resolution spherically symmetric studies which verify our analytical predictions. Then we present three-dimensional simulations of single distorted black holes using different variations of the turduckening method and also the puncture method. We study the effect that these different methods have on the coordinate conditions, constraint violations, and extracted gravitational waves. We find that the waves agree up to small but nonvanishing differences, caused by escaping superluminal gauge modes. These differences become smaller with increasing detector location.

Figure 1

Proper speeds as a function of proper distance from the horizon for the turduckening case TA. The curve labeled “coord” is the proper speed of the coordinate system with respect to the normal observers, $\sqrt{{\tilde{\gamma }}_{rr}}{e}^{2\varphi }{\beta }^r/\alpha$. The curve labeled “mode 1” is the proper speed ${\mu }_1=\sqrt{2/\alpha }$. The curve labeled “mode 5” is the proper speed ${\mu }_5=\sqrt{3}{e}^{2\varphi }/(2\alpha )$. The horizontal line is light speed.

Figure 2

Hamiltonian constraint versus coordinate radius $r$ at various times, in units of $M$. TA and TB denote two types of turduckening, and P is puncture data. The vertical line shows the location of the black hole horizon.

Figure 3

Hamiltonian constraint versus coordinate radius $r$ for $m=1$ and $m=1.25$, where $m$ is the parameter that controls the mixing of the momentum constraint in the equation of motion for ${\Gamma }^r$. The smoothing-type TB is used in both cases. The vertical line shows the location of the black hole horizon. As predicted by the theory, the constraint violations inside the black hole do propagate to the outside if $m>1$.

Figure 4

The difference between lapse functions plotted versus areal radius in units of $M$. The curve TA-P is the difference for turducken data with stuffing TA and puncture data. The curve TB-P is the difference for turducken data with stuffing TB and puncture data.

Figure 5

Areal radius $R$ versus proper distance from the horizon. The horizontal line at $R=2$ is the horizon. By $t=50M$ the three curves (TA, TB, and P) are indistinguishable (at this scale) from the stationary $1+\log $ slice S.

Figure 6

Common logarithm of the difference between lapse functions. The top two graphs show the difference between case TD and puncture data for low ($\Delta r=M/100$) and high ($\Delta r=M/200$) resolutions. The bottom two graphs show the difference between case TE and puncture data for low and high resolutions. [Note that the curves at time $t=3M$ appear to terminate at an areal radius of about 3.3 (case TD) or 3.5 (case TE). This occurs because, beyond these values, the lapse difference is exactly 0.0 to 13 or more decimal places. The logarithm is undefined for larger values of the areal radius.]

Figure 7

The difference between the coordinate radius for turducken data TD and puncture data, as a function of areal radius. Both low ($\Delta r=M/100$) and high ($\Delta r=M/200$) resolution results are shown, although the curves are difficult to distinguish. The vertical line is the location of the black hole horizon.

Figure 8

Convergence plot for the Hamiltonian constraint along the $x$ axis at $T=115.2M$ for the small turduckening region $r_t=0.1M$ (left plot) and for the puncture run (right plot). The medium and high resolution curves have been scaled for fourth order convergence. The vertical line shows the location of the horizon. The plots cover the range over which the waveforms were extracted (up to $R=60M$). At this time this region is still causally disconnected from the outer boundary.

Figure 9

Lapse profile along the $x$ axis for the distorted single black hole evolutions at the times $T=0M$ (top left graph), $T=3.84M$ (top right graph), $T=9.6M$ (bottom left graph), and $T=96M$ (bottom right graph) for the high resolution, comparing different turduckening radii. The vertical lines show the location of the apparent horizon. The insets in the bottom graphs enlarge a small region near the horizon. While the lapse profiles differ near the origin, they are very similar at the horizon, and their difference decreases with time. However, for the medium and large turduckening radii, a superluminal gauge mode is clearly escaping. On the other hand, the differences between the lapse profiles of the puncture and small turduckening radius cases converge to zero with resolution.

Figure 10

Effect of the turduckening radius on the $\ell =2$, $m=0$ mode of gravitational waveforms $r{\Psi }_4$, extracted at $R=30M$. The top left graph compares all three resolutions, and the top right graph shows their differences, scaled for fourth order convergence, both for the small turduckening region $r_t=0.1M$. The bottom left graph shows the scaled differences for a pure puncture run, and the fact that it looks virtually identical indicates that $r_t=0.1M$ is a good choice for these resolutions. On the other hand, the bottom right graph shows the same for the large turduckening radius $r=0.2M$ where noticeable differences are visible. We attribute this to the smaller number of grid points between the turduckening region and the initial apparent horizon (see Table ), so this case is not yet in the convergent regime.

Figure 11

Difference between the $\ell =2$, $m=0$ mode of $r{\Psi }_4$ for a puncture evolution and for three different turduckening radii, extracted at $R=30M$ (left-hand graphs) and $R=60M$ (right-hand graphs). See the main text for the discussion.

Figure 12

Difference between the $\ell =2$, $m=0$ mode of $r{\Psi }_4$ for the turducken runs with $n=6$ and $n=2$ (both with $r_t=0.1$) extracted at $R=30M$ (left plot). The right plots shows the same for the puncture run and the turducken with $r_t=0.1$ and $n=2$.