Puncture black hole initial data: A single domain Galerkin-collocation method for trumpet and wormhole data sets

We present a single-domain Galerkin-collocation method to calculate puncture initial data sets for single and binary black holes, either in the trumpet or wormhole geometries. The combination of aspects belonging to the Galerkin and the collocation methods together with the adoption of spherical coordinates in all cases are shown to be very effective. We propose a unified expression for the conformal factor to describe trumpet and spinning black holes. In particular, for the spinning trumpet black holes, we exhibit the deformation of the limit surface due to the spin from a sphere to an oblate spheroid. We also revisit the energy content in the trumpet and wormhole puncture data sets. The algorithm can be extended to describe binary black holes.

Figure 1

The three-dimensional spatial domain viewed as a cube described by the coordinates $-1\le x<1$, $-1<y<1$ ($y=\cos \theta$) that correspond to $0\le r<\infty$ and $0\le \theta \le \pi$, while the azimuthal angle $\varphi$ is maintained.

Figure 2

Convergence of the ADM mass for trumpet and wormhole spinning punctures (upper and lower graphs, respectively). Here $J_0=0.5m_0^2$ and $m_0=1.0$. For the trumpet data, we have included two convergence tests corresponding to $L_0=0.2$ and $L_0=2.0$ to make clear the influence of the map parameter. The exponential convergence of the ADM mass is more evident for $L_0=0.2$ than for $L_0=2.0$; the convergence is algebraic. For the wormhole spinning puncture the convergence is clearly exponential where $L_0=0.5$.

Figure 3

Convergence of the ADM mass for trumpet and wormhole boosted punctures (upper and lower graphs, respectively). Here $P_0=1.0m_0$ and $m_0=1.0$, and the map parameters are $L_0=0.1$ and $L_0=1.0$, respectively. The exponential convergence is achieved in both cases.

Figure 4

Both graphs show the eccentricity of the minimal surface versus $J_0/m_0^2$ and $J_0/M_{\mathrm{ADM}}^2$, respectively. The eccentricity tends to a limit value of about 0.439. In the inset the continuous line represents the approximate exact solution of Ref. [24] valid for small angular momentum parameter together with the numerical eccentricities.

Figure 5

Radiation content for spinning and boosted black holes (upper and lower graphs, respectively). The circles and boxes refer to trumpet and wormhole data sets, respectively.

Figure 6

The regular function $1+u(r,\theta ,\varphi )$ projected onto the plane $y=z=0$ ($\theta =0,\varphi =0,\pi$). We have fixed the boost parameter to $P_0=0.2m_0$ while varying the spin as $J_0=0.1m_0^2,0.2m_0^2,..,0.5m_0^2$ with the corresponding profiles indicated by the curves from bottom to top. We have set $Nx=40$ and $N_y=12$.

Figure 7

Convergence of the ADM mass for the binary of boosted trumpet punctures located on the $z$ axis with $a=3$, $m_1=m_2=0.5$, and $P_0=0.4m_1$. We have set $N_x=20,25,30,35,..,100$ and $N_y=14$. In the second panel we show the profile of $1+u$ ($N_x=100$, $N_y=14$) projected onto the plane $x=y=0$.

Figure 8

Illustrations in two- and three-dimensional plots (upper and lower panels, respectively) of $1+u$ for the binary of boosted wormhole punctures in which $a=3$, $m_1=m_2=0.5$, and $P_0=0.4m_1$.

Figure 9

Locations of the apparent horizon $r=h(\theta )$ for a boosted black hole in the wormhole representation for several values of the momentum parameter along the $z$ axis. The corresponding locations for the trumpet representation are similar.