Numerical implementation of isolated horizon boundary conditions

We study the numerical implementation of a set of boundary conditions derived from the isolated horizon formalism, and which characterize a black hole whose horizon is in quasiequilibrium. More precisely, we enforce these geometrical prescriptions as inner boundary conditions on an excised sphere, in the numerical resolution of the conformal thin sandwich equations. As main results, we first establish the consistency of including in the set of boundary conditions a constant surface gravity prescription, interpretable as a lapse boundary condition, and second we assess how the prescriptions presented recently by Dain et al. for guaranteeing the well-posedness of the conformal transverse traceless equations with quasiequilibrium horizon conditions extend to the conformal thin sandwich elliptic system. As a consequence of the latter analysis, we discuss the freedom of prescribing the expansion associated with the ingoing null normal at the horizon.

Figure 1

Relative difference between ${\kappa }_{\mathrm{Kerr}}(a,J)$ and ${\kappa }_{\mathrm{o}}$, as a function of ${\kappa }_{\mathrm{o}}$, for solutions to ($b=N$, $\kappa ={\kappa }_{\mathrm{o}}$) for different values of ${\Omega }_{\mathrm{o}}$ in the rotating case.

Figure 2

Numerical analysis of condition (15). Values of ${\theta }_{(\widehat{k})}/\tilde{H}=K_{ij}{s}^i{s}^j/\tilde{H}$ and ${\Psi }^6·K_{ij}{s}^j{s}^j/\tilde{H}$ in the spherically symmetric case, after compactification by means of the $\arctan $ function.

Figure 3

Values of ${\Psi }^6·K_{ij}{s}^i{s}^j/\tilde{H}$ and ${\theta }_{(\widehat{k})}/\tilde{H}$, for boundary conditions $K_{ij}{s}^i{s}^j=\lambda$, $b=N$ and ${\Omega }_{\mathrm{o}}=0,0.10,0.11,0.12,\dots ,0.20$. Curves depart from the spherically symmetric case (dotted curve corresponds to ${\Omega }_{\mathrm{o}}=0$) as the rotating parameter increases. Solutions exist for every prescribed ${\theta }_{(\widehat{k})}$ and are unique.

Figure 4

Values of ${\Psi }^6·K_{ij}{s}^i{s}^j/\tilde{H}$ and ${\theta }_{(\widehat{k})}/\tilde{H}$, for boundary conditions $K_{ij}{s}^i{s}^j=\lambda$, $N=0.2$ and ${\Omega }_{\mathrm{o}}=0.00,0.01,0.02,\dots ,0.15$. If the rotating parameter is smaller than ${\Omega }_{\mathrm{o}}\approx 0.12620$, the curves reach the coordinate origin, otherwise they diverge to $-\infty$ as ${\theta }_{(\widehat{k})}/\tilde{H}\to 0$.

Figure 5

Values of ${\Psi }^6·K_{ij}{s}^i{s}^j/\tilde{H}$ and ${\theta }_{(\widehat{k})}/\tilde{H}$, for boundary conditions $K_{ij}{s}^i{s}^j=\lambda$, ${\kappa }_{\mathrm{o}}=0.3$ and ${\Omega }_{\mathrm{o}}=0.00,0.02,0.04,\dots ,0.18,0.20$. Curves with ${\Omega }_{\mathrm{o}}\ne 0$ do not pass through the origin: conclusions in Ref. 12 do not extend to the rotating case. On the other hand, ${\Omega }_{\mathrm{o}}\ne 0$ curves become indistinguishable of the spherically symmetric curve (dotted line) for sufficiently negative values of $K_{ij}{s}^i{s}^j$.