Alternative approach to solving the Hamiltonian constraint

Solving Einstein’s constraint equations for the construction of black hole initial data requires handling the black hole singularity. Typically, this is done either with the excision method, in which the black hole interior is excised from the numerical grid, or with the puncture method, in which the singular part of the conformal factor is expressed in terms of an analytical background solution, and the Hamiltonian constraint is then solved for a correction to the background solution that, usually, is assumed to be regular everywhere. We discuss an alternative approach in which the Hamiltonian constraint is solved for an inverse power of the conformal factor. This new function remains finite everywhere, so that this approach requires neither excision nor a split into background and correction. In particular, this method can be used without modification even when the correction to the conformal factor is singular itself. We demonstrate this feature for rotating black holes in the trumpet topology.

Figure 1

Numerical solutions $\Omega$ for a Schwarzschild black hole, for $n=2$ and $n=4$. Here we imposed the outer boundary at $R_{\mathrm{out}}=16M$ and used $100N+1$ grid points. The upper panels shows the solution for different values of $N$. As expected, $\Omega$ scales with $r^{n/2}$ at the center. The lower panel shows the rescaled errors $N^2\Delta \Omega$, demonstrating second-order convergence to the analytical solution, even in the neighborhood of the singularity.

Figure 2

Numerical solutions $w$ for boosted black holes with momentum $P^z/M=1.0$. The lines in the large plot show results for $w$, obtained for $n=2$, on three different numerical grids with increasing grid resolution and more distant outer boundaries. For $N=65$, the outer boundaries were imposed at $X_{\mathrm{out}}=16M$, for $N=129$ at $X_{\mathrm{out}}=24M$, and for $N=257$ at $X_{\mathrm{out}}=32M$. Also included, as crosses, are the results as computed from the Hamiltonian constraint in its original from (1) with $N=257$. The insert shows higher-resolution results (obtained with $N=257$ and $X_{\mathrm{out}}=2M$) of the region around the black hole center. This graph also includes the leading-order analytical result (13) as a solid line.

Figure 3

Solutions $w$ for a black hole spinning with angular momentum $J^z/M^2=0.1$, using $n=2$. We show numerical results (dots) in the vicinity of the singularity, together with the leading-order analytical result (16) (solid lines), both along the direction of the spin ($\cos \theta =1$) and a direction orthogonal to the spin ($\cos \theta =0$). The numerical results were obtained with $N=257$ and $X_{\mathrm{out}}=2M$.

Figure 4

Numerical results for $w$ for a black hole spinning with angular momentum $J^z/M^2=1.0$ in the $x\mathrm{\text{−}}z$ plane, obtained with $N=129$, $X_{\mathrm{out}}=4M$ and $n=2$.