Black hole initial data on hyperboloidal slices

We generalize Bowen-York black hole initial data to hyperboloidal constant mean curvature slices which extend to future null infinity. We solve this initial value problem numerically for several cases, including unequal mass binary black holes with spins and boosts. The singularity at null infinity in the Hamiltonian constraint associated with a constant mean curvature hypersurface does not pose any particular difficulties. The inner boundaries of our slices are minimal surfaces. Trumpet configurations are explored both analytically and numerically.

Figure 1

Radial function $f(r)$ for different values of $K$ and $C$. Plotted is the square $f^2$ for $KM=1/10$ and several different choices for $C$. Each curve is labeled by its value of $C/M^2$ and its value for $K^{2}C$. The bottom axis shows radius in units of $r/M$, and the top axis in units of $Kr$.

Figure 2

Parameter choices that result in hypersurfaces ${\Sigma }_t$ containing a minimal surface. The two panels correspond to two different choices of dimensionless variables. Parameter values on the blue solid lines represent trumpet configurations.

Figure 3

Properties of the trumpet hypersurfaces, parametrized by the dimensionless parameter $K{C}^{1/2}$.

Figure 4

Properties of CMC hypersurfaces viewed in the $R_{\mathrm{ms}}/R_{+}$ vs $K{C}^{1/2}$ plane. CMC hypersurfaces with a minimal surface must lie below the thick black line, and as this line is approached, the minimal surface approaches the black hole horizon. The dashed red lines are lines of constant $MK$, with values given by the red numbers next to the lines. The thin blue lines represent constant values of $\gamma =r_{\mathrm{ms}}/r_H$. The shaded areas are contours of constant value of $R_{\mathrm{ms}}/R_{\mathrm{AH}}$; the shade of grey changes, from top to bottom, at values 0.9, 0.5, 0.1, 0.01, and 0.001. Trumpet hypersurfaces represent the limit $R_{\mathrm{ms}}/R_{+}\to 0$.

Figure 5

The functions $g_C$ and $g_R$, which are the asymptotic values of $C/M^2$ and $R_{\mathrm{ms}}/(R_{+}KM)$ in the limit $KM\to 0$. These functions are useful for choosing excision radius and $C$ when constructing CMC initial data for black holes with given $K$, $M$, and outer boundary radius $R_{+}$.

Figure 6

Convergence of the numerical solution of the Hamiltonian constraint for a Schwarzschild black hole. Plotted are five different resolutions $N_R$, where $N_R$ is the number of radial collocation points. Solid lines show the difference from the analytic solution, dotted lines the difference from the numerical solution at the next higher resolution. The highest resolution has $N_R=104$.

Figure 7

Numerical solution for a Schwarzschild black hole with the inner boundary very close to the trumpet limit. The dashed red line shows the conformal factor $\Omega$, and the solid blue line shows the Schwarzschild radius (calculated from the proper area of coordinate spheres) as a function of the conformal radius. The dotted black line is $U_{0}R$, which equals $\Omega$ for a trumpet, with $U_0$ given in Eq. (51). The vertical dashed line locates the apparent horizon.

Figure 8

Irreducible mass versus dimensionless spin for a single black hole spinning around the z-axis.

Figure 9

Dimensionless scalar curvature of the apparent horizon versus spin for a single black hole. The solid lines are the maximum and minimum numerical values computed on CMC slices. The dashed lines are the maximum and minimum analytic values for a Kerr black hole (these lines continue to the maximal Kerr value $\zeta =1$, where the maximum and minimum are 2 and $-1/2$, respectively).

Figure 10

Coordinate locations of the apparent horizons and minimal surfaces, cut through the $x\mathrm{\text{−}}z$ plane, for a nonspinning, unboosted black hole (solid black circles), and for a nonspinning black hole boosted in the $z$ direction (dashed red circles).

Figure 11

2D Ricci scalar on the apparent horizon surface of a black hole with ${\Omega }_{\max }P/M_{\mathrm{irr}}=1.77$. In this view, a wedge-shaped region has been removed from the front.

Figure 12

Dimensionless intrinsic geometry of the apparent horizon (2D Ricci scalar, times the irreducible mass squared) versus $P{\Omega }_{\max }/M_{\mathrm{irr}}$ for a single nonspinning black hole, with the ratio $R_{\mathrm{AH}}/R_{\mathrm{ms}}$ kept fairly constant. Shown are the maximum and minimum numerical values computed on CMC slices.

Figure 13

Convergence of the elliptic solver for the unequal mass binary black hole example shown in Fig. 14. Shown is the volume L2-norm of the residual of $\Omega$ as $N$ is increased, where $N$ is the cube root of the total number of collocation points.

Figure 14

Conformal factor $\Omega$ for a spinning, boosted binary black hole, with mass ratio $2:1$. The inner boundaries are the minimal surfaces of the two black holes, and the outer boundary is null infinity. The maximum of $\Omega$ is red, and equals 2.38. The minimum is dark blue, which is zero to machine precision.