Geometric Constraints via Page Curves: Insights from Island Rule and Quantum Focusing Conjecture

TL;DR

This paper identifies a geometric condition involving the blackening factor's second derivative near the horizon that guarantees the emergence of Page curves in black hole evaporation, supported by the quantum focusing conjecture.

Contribution

It establishes a broad geometric criterion ($f''(r)<0$ near the horizon) for the existence of islands and Page curves in static black holes, extending previous model-specific results.

Findings

Negativity of $f''(r)$ near the horizon ensures island formation.

The criterion applies to asymptotically Minkowski and (A)dS spacetimes.

Supports the quantum focusing conjecture as a basis for the result.

Abstract

Exploring the inverse problem tied to the Page curve phenomenon and island paradigm, we investigate the geometric conditions underpinning black hole evaporation where information is preserved and islands manifest, giving rise to the characteristic Page curve. Focusing on a broad class of static black hole metrics in asymptotically Minkowski or (anti-)de Sitter spacetimes, we derive a pivotal constraint on the blacken factor $f(r)$ for which the island exists and reproduce the Page curve. Specifically, we reveal that a sufficient yet not universally necessary criterion -- manifested in the negativity of the second derivative of $f(r)$, i.e. $f^{\prime \prime} (r)<0$, in proximity to the event horizon where $r \sim r_h+ {\cal O} (G_N)$, ensures the emergence of Page curves in a manner transcending specific theoretical models. This pivotal finding, supported by the tenets of the quantum focusing conjecture.