Apparent horizon as a membrane

TL;DR

This work investigates physical black holes that form horizons in finite time ($\Phi$BHs) and differ from eternal black holes in their near-horizon geometry. It develops an observation-facing framework in spherical symmetry, deriving an approximate near-horizon metric and treating the timelike apparent horizon as a membrane carrying a 2D viscous fluid via Israel junction conditions. The analysis identifies two near-horizon solution classes (dynamical $\Phi$BHs with $k=0$ and static MBHs with $k=1$), establishes a York–Frolov separatrix as a baseline for connecting to Rindler geometry, and shows that a Kodama–Hayward-type surface gravity provides a robust, dynamical generalisation. The results furnish key redshift, acceleration, and extrinsic-curvature data that enable predictions of quasinormal modes, light rings, and potential gravitational echoes, offering observable signatures distinguishing $\Phi$BHs from classical MBHs.

Abstract

The requirement that a trapped spacetime domain forms in finite time for distant observers is logically possible and sometimes unavoidable, but its consequences are not yet fully understood. In spherical symmetry, the characterization of the near-horizon geometry of these physical black holes is complete and shows marked differences from their eternal counterparts. Whether these differences lead to observable signatures remains unclear. We construct an approximate near-horizon metric that encapsulates them and is suitable for modeling. The timelike apparent horizon of physical black holes provides a natural surface for a consistent membrane description: we obtain closed-form expressions for the redshift, proper acceleration, and extrinsic curvature, and assign a two-dimensional viscous-fluid stress tensor via junction conditions. These results also provide an additional perspective on the relation between Rindler and near-horizon geometries. Among dynamical generalizations of surface gravity, only a subset applies to these models. We complete their analysis and recover the intuitive definition of surface gravity -- the acceleration in the frame of a near-horizon observer, redshifted to infinity -- directly from the membrane acceleration.