Infrared Regularization and Finite Size Dynamics of Entanglement Entropy in Schwarzschild Black Hole

TL;DR

This paper investigates how entanglement entropy behaves in Schwarzschild black holes with finite regions, proposing a regularization method and analyzing island formation, revealing limitations of the island prescription in resolving the information paradox.

Contribution

It introduces a new infrared regularization preserving key properties and classifies finite entangling regions, analyzing their entanglement dynamics and limitations of island-based solutions.

Findings

Discontinuous entanglement entropy evolution for mirror-symmetric regions.

Finite size regions have critical thresholds for island emergence.

Island prescription does not resolve the information paradox for some finite regions.

Abstract

In this paper, infrared regularization of semi-infinite entangling regions and island formation for regions of finite size in the eternal Schwarzschild black hole are considered. We analyze whether the complementarity property and pure state condition of entanglement entropy can be preserved in the given approximation. We propose a special regularization that satisfies these two properties. With regard to entangling regions of finite size, we derive two fundamental types of them, which we call "mirror-symmetric" (MS) and "asymmetric" (AS). For MS regions, we discover a discontinuous evolution of the entanglement entropy of Hawking radiation due to finite lifetime of the island. The entanglement entropy of matter for semi-infinite regions in two-sided Schwarzschild black hole does not follow the Page curve. The lifetime of AS regions is bounded from above due to the phenomenon that we call "Cauchy surface breaking". Shortly before this breaking, the island configuration becomes non-symmetric. For both types of finite regions, there is a critical size, below which the island never dominates. For regions smaller than some other critical size, the island does not emerge. Finally, we show that the island prescription does not help to solve the information paradox for certain finite regions.