Conformal thin-sandwich puncture initial data for boosted black holes

We apply the puncture approach to conformal thin-sandwich black-hole initial data. We solve numerically the conformal thin-sandwich puncture (CTSP) equations for a single black hole with nonzero linear momentum. We show that conformally-flat solutions for a boosted black hole have the same maximum gravitational radiation content as the corresponding Bowen-York solution in the conformal transverse-traceless decomposition. We find that the physical properties of these data are independent of the free slicing parameter.

Figure 1

Convergence behavior of the errors in the momentum constraint solution along the $z$-axis (${\beta }^x$) and the 3D diagonal (${\beta }^y$ and ${\beta }^z$), with outer boundary at $2m$.

Figure 2

Convergence behavior of the momentum constraint along the $z$-axis (${\beta }^x$) and the 3D diagonal (${\beta }^y$ and ${\beta }^z$), with outer boundary at $0.02m$.

Figure 3

Convergence of ${\beta }^x$ (along $z$-axis) when an inner boundary condition is used, and the outer boundary is at $1.0m$. Differences were scaled assuming second-order convergence.

Figure 4

Convergence of the full CTSP system for one boosted black hole with outer boundary at 16.0$m$, using scalar Robin outer boundary conditions on the shift vector. The differences are scaled assuming second-order convergence. The base grid spacing is $h=0.125m$.

Figure 5

Convergence of the full CTSP system for one boosted black hole with outer boundary at 16$m$, using vector Robin outer boundary conditions on the shift vector. The differences are scaled assuming second-order convergence. The base grid spacing is $h=0.125m$.

Figure 6

Solutions of the full CTSP system for one boosted black hole with outer boundary at 16$m$, with $h=m/24$. The solutions for $u$, $v$ and ${\beta }^x$ did not differ substantially when either scalar or vector Robin outer boundary conditions were applied to the shift vector. The solutions of ${\beta }^y$ and ${\beta }^z$ are shown for both types of outer boundary condition.

Figure 7

The percentage error in $P^x$ for different choices of resolution and outer boundary for the Laguna case, with $c=m=1$, ${\beta }_0^i=-0.75$, and analytic $P^x=0.5m$.

Figure 8

The linear momentum $P_x$ of one CTSP black hole as a function of the value of ${\beta }_0^x$ at the puncture.

Figure 9

The disagreement in linear momentum $P_x$ between the Laguna and full CTSP solutions for one black hole, as a function of the value of ${\beta }_0^x$ at the puncture.

Figure 10

Radiation content of single black-hole initial-data sets, with $m=c=1$. Comparison of Laguna case with full CTSP data.

Figure 11

Comparison of radiation content of single black hole initial-data sets, for various choices of the lapse parameter $c>0$. The choice of $c$ has a minimal effect on the physical properties of the initial data.