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 quasi-equilibrium. 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 firstly establish the consistency of including in the set of boundary conditions a constant surface gravity prescription, interpretable as a lapse boundary condition, and secondly we assess how the prescriptions presented recently by Dain et al. for guaranteeing the well-posedness of the Conformal Transverse Traceless equations with quasi-equilibrium 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.