Comments on Bona-Masso type slicing conditions in long-term black hole evolutions

We review in generality why time-independent endstates can be reached in black hole and collapse simulations, with and without excision. We characterise the Killing states of the Bona-Masso slicing condition with time derivative along the normals to the slice ("BMn") as solutions of a mixed elliptic/hyperbolic differential equation on the slice. We show numerically that these steady states can be reached as end states from typical initial data with excision but can be reached with the puncture method only if the puncture is not numerically well resolved. During the evolution, BMn slicings often form gauge shocks. It may be that these are not seen in current 3D simulations only through lack of resolution, although we expect that they can be avoided with some care. Finally we point out that excision with BMn as currently implemented is ill-posed and therefore not expected to converge; this can be cured. In technical appendixes, we derive the equations of pure gauge systems on a fixed spacetime, and bring the BSSN/NOR equations into 3-dimensional tensor form suitable for multiple coordinate patches or spherical polar coordinates.