Gravitational collapse of K-essence matter in Painlevè-Gullstrand coordinates

We conduct numerical simulations in Painlevè-Gullstrand coordinates of a variety of K-essence-type scalar fields under spherically symmetric gravitational collapse. We write down generic conditions on the K-essence lagrangian that can be used to determine whether superluminality and Cauchy breakdown are possible. Consistent with these conditions, for specific choices of K-essence-type fields we verify the presence of superluminality during collapse, while for other type we do not. We also demonstrate that certain choices of K-essence scalar fields present issues under gravitational collapse in Painlevè-Gullstrand coordinates, such as a breakdown of the Cauchy problem.