Excised black hole spacetimes: Quasilocal horizon formalism applied to the Kerr example

We present a numerical work aiming at the computation of excised initial data for black hole spacetimes in full general relativity, using the Dirac gauge in the context of a constrained formalism for the Einstein equations. Introducing the isolated horizon formalism for black hole excision, we especially solve the conformal metric part of the equations, and assess the boundary condition problem for it. In the stationary single black hole case, we present and justify a no-boundary treatment on the black hole horizon. We compare the data obtained with the well-known analytic Kerr solution in Kerr-Schild coordinates, and assess the widely used conformally flat approximation for simulating axisymmetric black hole spacetimes. Our method shows good concordance on physical and geometrical issues, with the particular application of the isolated horizon multipolar analysis to confirm that the solution obtained is indeed the Kerr spacetime. Finally, we discuss a previous suggestion in the literature for the boundary conditions for the conformal geometry on the horizon.

Figure 1

Accuracy for Einstein equations resolution (in the original ($3+1$) version of 39) as a function of dimensionless parameter $M_{\mathcal{H}}\Omega$ (see Sec. 5c for a definition of $M_{\mathcal{H}}$). Data are absolute maximum error values for both constraint equations, and dynamical equations in both cases. Data are taken with $N_r=33$, $N_{\theta }=17$, $N_{\phi }=1$. Lapse on the horizon is $N_{\mathcal{H}}=0.55$.

Figure 2

Dependence on the parameter $M_{\mathcal{H}}\Omega$ of the ADM mass, the Komar angular momentum at infinity $J_K$ and the angular momentum parameter $\frac{a}{M}$ (both defined in Sec. 4a) for both cases. The value of the lapse on the horizon is here fixed at $N_{\mathcal{H}}=0.55$.

Figure 3

Different diagnostics for stationarity in both cases, comparing physical quantities at the horizon and at infinity. The virial error computes the difference between ADM and Komar masses at infinity, rescaled with the ADM mass, and using asymptotic behaviors of the lapse and conformal factor. The radiation energy content outside the black hole (respectively, outer angular momentum content) is the absolute difference between the horizon mass $M_{\mathcal{H}}$ (respectively, angular momentum on the horizon $J_{\mathcal{H}}$) and the ADM mass $M_{\mathrm{ADM}}$ (respectively, Komar angular momentum at infinity $J_K$), rescaled with the ADM mass (respectively, Komar angular momentum at infinity).

Figure 4

Value of $1-{ϵ}_A$ for both data sets.

Figure 5

Computations of the rescaled second-order mass multipole $(\frac{M_2}{M^3})$ and the rescaled third-order angular multipole $(\frac{J_3}{M^4})$. Relative differences with respect to values for a Kerr-Schild analytical horizon are displayed.