Dynamical surface gravity in spherically symmetric black hole formation

We study dynamical surface gravity in a general spherically symmetric setting using Painlevé-Gullstrand coordinates. Our analysis includes several definitions that have been proposed in the past as well as two new definitions adapted to Painlevé-Gullstrand coordinates. Various properties are considered, including general covariance, value at extremality, locality and static limit. We illustrate with specific examples of “dirty” black holes that even for spacetimes possessing a global timelike Killing vector, local definitions of surface gravity can differ substantially from “nonlocal” ones that require an asymptotic normalization condition. Finally, we present numerical calculations of dynamical surface gravity for black hole formation via spherically symmetric scalar field collapse. Our results highlight the differences between the various definitions in a dynamical setting and provide further insight into the distinction between local and nonlocal definitions of surface gravity.