Dynamical Surface Gravity in Spherically Symmetric Black Hole Formation

TL;DR

This paper investigates various definitions of dynamical surface gravity in spherically symmetric black hole formation, comparing local and non-local approaches through analytical examples and numerical simulations.

Contribution

It introduces two new definitions of surface gravity in PG coordinates and analyzes their properties, highlighting differences from existing definitions in dynamical spacetimes.

Findings

Local definitions can differ significantly from non-local ones in dynamical settings.

Numerical results show variations in surface gravity during black hole formation.

The study clarifies the roles of covariance, extremality, and static limits in defining surface gravity.

Abstract

We study dynamical surface gravity in a general spherically symmetric setting using Painlev\'{e}-Gullstrand (PG) coordinates. Our analysis includes several definitions that have been proposed in the past as well as two new definitions adapted to PG 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 "non-local" 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 non-local definitions of surface gravity.