The Upper Bound of Event Horizon Formation Time in Generalized Oppenheimer-Snyder Collapse

TL;DR

The paper proves that among asymptotically flat, static, spherically symmetric exteriors with the same ADM mass $m$ satisfying the weak energy condition, the Schwarzschild exterior maximizes the horizon-formation time ${\Delta T_{\text{eh}}}$ in generalized Oppenheimer–Snyder collapse, giving ${\Delta T_{\text{eh}}\le\tfrac{19}{6}m}$. The authors reformulate the collapse using the Misner–Sharp mass $M(R)$ and introduce dimensionless variables to express the key constraints as a probability measure problem, proving that Schwarzschild yields the largest horizon-formation time via first-order stochastic dominance. The bound is saturated by Schwarzschild, and the analysis also yields ${R_{0,{\min}}\le\tfrac{9}{2}m}$, with equality for Schwarzschild. This work extends temporal bounds on black holes beyond traditional spatial inequalities (like the Penrose bound) and provides a probabilistic framework that could inform horizon formation in other collapse settings and theories.

Abstract

We prove that, in the framework of the Oppenheimer-Snyder collapse, the Schwarzschild exterior maximizes the event horizon formation time $ΔT_{\text{eh}}=\frac{19}{6}m$ among all asymptotically flat, static, spherically-symmetric black holes with the same ADM mass $m$ that satisfy the weak energy condition. This bound extends the typical black hole inequalities--such as the Penrose inequality, which constrains spatial geometry--to temporal setting.