Excision boundary conditions for black-hole initial data

We define and extensively test a set of boundary conditions that can be applied at black-hole excision surfaces when the Hamiltonian and momentum constraints of general relativity are solved within the conformal thin-sandwich formalism. These boundary conditions have been designed to result in black holes that are in quasiequilibrium and are completely general in the sense that they can be applied with any conformal three-geometry and slicing condition. Furthermore, we show that they retain precisely the freedom to specify an arbitrary spin on each black hole. Interestingly, we have been unable to find a boundary condition on the lapse that can be derived from a quasiequilibrium condition. Rather, we find evidence that the lapse boundary condition is part of the initial temporal gauge choice. To test these boundary conditions, we have extensively explored the case of a single black hole and the case of a binary system of equal-mass black holes, including the computation of quasicircular orbits and the determination of the innermost stable circular orbit. Our tests show that the boundary conditions work well.

Figure 1

Solution of the quasiequilibrium equations recovering the Kerr-Schild slicing of Schwarzschild. Plotted are the maximum values of Hamiltonian and momentum constraints and of time derivatives, as well as the deviation of $M_{\mathrm{i}\mathrm{r}\mathrm{r}}$, $E_{\mathrm{A}\mathrm{D}\mathrm{M}}$, and $M_K$ from the analytical answer $M$. $N_r$ is the radial number of collocation points in each of the two spherical shells.

Figure 2

Spinning single black hole initial-data sets.

Figure 3

Single rotating black hole initial-data sets: Deviations from the exact Kerr metric.

Figure 4

Single boosted black hole initial-data sets.

Figure 5

Single boosted black hole initial-data sets: Deviations from exact time-independence.

Figure 6

Convergence of the elliptic solver for the binary black-hole configurations. Plotted are the constraint violations in Hamiltonian and momentum constraint, and differences to the highest resolution solved. $N$ is the cube-root of the total number of grid-points.

Figure 7

Constant $M_{\mathrm{i}\mathrm{r}\mathrm{r}}$ sequence of corotating equal-mass black holes. Maximal slicing is used in these cases, and three different excision boundary conditions for the lapse are used.

Figure 8

Constant $M_{\mathrm{i}\mathrm{r}\mathrm{r}}$ sequence of corotating equal-mass black holes. Comparison of post-Newtonian EOB sequences with numerical maximal slicing results from HKV and this paper.

Figure 9

Plot of the irreducible mass of one black hole when the sequence of solutions is normalized to maintain $\mathrm{d}E=\Omega \mathrm{d}J$. Three different choices for the lapse boundary condition are shown for both corotating (upper plot) and irrotational (lower plot) black holes in an equal-mass binary. The vertical dashed line shows the approximate location of the ISCO defined by a minimum in $E_b$. The vertical dotted line shows the approximate location of the ISCO defined by a minimum in $E_{\mathrm{A}\mathrm{D}\mathrm{M}}$. In the irrotational case, this latter ISCO line is off the plot to the right.

Figure 10

ISCO configuration for three different choices of the lapse boundary condition for equal-mass corotating (CO: QE) and irrotational (IR: EQ) black holes computed with the maximal slicing condition. For comparison, results of Refs. 18, 35, 39, 41 are included. For post-Newtonian calculations the size of the symbol indicates the order, the largest symbol being 3PN. “PN standard” [41] represents a PN-expansion in the standard form without use of the EOB-technique (only 2PN and 3PN are plotted).

Figure 11

Constant $M_{\mathrm{i}\mathrm{r}\mathrm{r}}$ sequence of corotating equal-mass black holes. Comparison of post-Newtonian EOB sequences with numerical maximal slicing results from HKV and this paper.

Figure 12

ISCO configuration for three different choices of the lapse boundary condition for equal-mass corotating and irrotational black holes computed with the maximal slicing condition. Symbols as in Fig. 10.

Figure 13

Constant $M_{\mathrm{i}\mathrm{r}\mathrm{r}}$ sequence of corotating equal-mass black holes. Comparison of post-Newtonian EOB sequences with numerical maximal slicing results from HKV and this paper.

Figure 14

ISCO configuration for three different choices of the lapse boundary condition for equal-mass corotating and irrotational black holes computed with the maximal slicing condition. Symbols as in Fig. 10.

Figure 15

Constant $M_{\mathrm{i}\mathrm{r}\mathrm{r}}$ sequence of irrotational equal-mass black holes. Maximal slicing is used in these cases, and three different excision boundary conditions for the lapse are used.

Figure 16

Constant $M_{\mathrm{i}\mathrm{r}\mathrm{r}}$ sequence of irrotational equal-mass black holes. Comparison of post-Newtonian EOB sequences with numerical maximal slicing results from Conformal Image and this paper.

Figure 17

Constant $M_{\mathrm{i}\mathrm{r}\mathrm{r}}$ sequence of irrotational equal-mass black holes. Comparison of post-Newtonian EOB sequences with numerical maximal slicing results from Conformal Image and this paper.

Figure 18

Constant $M_{\mathrm{i}\mathrm{r}\mathrm{r}}$ sequence of irrotational equal-mass black holes. Comparison of post-Newtonian EOB sequences with numerical maximal slicing results from Conformal Image and this paper.

Figure 19

Constant $M_{\mathrm{i}\mathrm{r}\mathrm{r}}$ sequence of corotating equal-mass black holes. Eddington-Finkelstein slicing is used in these cases, and three different excision boundary conditions for the lapse are used.

Figure 20

Constant $M_{\mathrm{i}\mathrm{r}\mathrm{r}}$ sequence of irrotational equal-mass black holes. Eddington-Finkelstein slicing is used in these cases, and three different excision boundary conditions for the lapse are used.

Figure 21

Plot of the residual of the quasiequilibrium boundary condition on the lapse (51) of one black hole. Three different choices for the lapse boundary condition are shown for corotating black holes in an equal-mass binary. The upper plot shows the average of the residual over the boundary surface. The lower plot shows the $L_2$-norm of the residual. The vertical dashed line shows the approximate location of the ISCO defined by a minimum in $E_b$. The vertical dotted line shows the approximate location of the ISCO defined by a minimum in $E_{\mathrm{A}\mathrm{D}\mathrm{M}}$.

Figure 22

Plot of the residual of the quasiequilibrium boundary condition on the lapse (51) of one black hole. Three different choices for the lapse boundary condition are shown for irrotational black holes in an equal-mass binary. The upper plot shows the average of the residual over the boundary surface. The lower plot shows the $L_2$-norm of the residual. The vertical dashed line shows the approximate location of the ISCO defined by a minimum in $E_b$.

Figure 23

Plot of the $L_2$-norm of ${\partial }_t\ln \psi$ as evaluated on the excision boundary of one black hole. The upper half of the figure shows the results for corotating black holes in an equal-mass binary, while the lower half shows the results for irrotational black holes. For both cases, three different choices for the lapse boundary condition are shown. In the upper half of the figure, the vertical dashed line shows the approximate location of the ISCO defined by a minimum in $E_b$. The vertical dotted line shows the approximate location of the ISCO defined by a minimum in $E_{\mathrm{A}\mathrm{D}\mathrm{M}}$. In the lower half of the figure, the vertical dashed line shows the approximate location of the ISCO defined by a minimum in $E_b$.