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

We review, without assuming symmetry, why time-independent endstates can be reached in black hole and collapse simulations, with and without excision. Generalizing earlier work in spherical symmetry, we characterize 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 in spherical symmetry 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. As pointed out in [19] the moving puncture method typically uses a shift that systematically underresolves the puncture, and thus this method becomes a type of excision. During the evolution, BMn slicings often form gauge shocks. It may be that these are not seen in current 3D simulations only through the systematic underresolution provided by the shift, 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 appendices, we derive the equations of pure gauge systems on a fixed spacetime, and bring the BSSN/NOR equations into three-dimensional tensor form suitable for multiple coordinate patches or spherical polar coordinates.