Page curves and Entanglement Islands for the Step-Function Vaidya Model of Evaporating Black Holes

Abstract

It was proposed recently that the fine-grained entropy of the Hawking radiation can be expressed by the semiclassical island formula, which reproduces the unitary Page curve. In this paper, we choose the ``in'' vacuum state and apply the quantum extremal surface construction to study the Page curve for the step-function Vaidya model of evaporating black holes in four dimensions, which is produced by the spherical null shells. Metrics of the three regions of this spacetimes are obtained. In addition, the entanglement islands for the step-function Vaidya model of evaporating black holes at very late times are studied. When cutoff surface $A$ is located in Minkowski region III with $u_A<u_H$ at very late times, we find that the location of the boundary of island $\partial I$ depends on the value of $8M-v_A+v_I$. Specifically, $\partial I$ is inside, at or outside the horizon when $8M-v_A+v_I$ is less than, equal to or larger than zero respectively. Moreover, when cutoff surface $A$ is located in Minkowski region III with $u_A>u_H$ after the black hole evaporates completely, we find that entanglement island still exists and $\partial I$ is located on an equal-time Cauchy surface of the observer $A$ when $r_{(A)}^2\ge 64G_N\kappa c$.

arXiv comment29 pages, 12 figures, improvements to presentation, references added, and new content added in appendix to comment about the stress-energy tensor of the matter field, published version in JHEP

References in corpus (23)

• Holographic Derivation of Entanglement Entropy from AdS/CFT
• A Covariant Holographic Entanglement Entropy Proposal
• Entanglement entropy and conformal field theory
• Information in Black Hole Radiation
• Replica Wormholes and the Entropy of Hawking Radiation
• Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime
• The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole
• The Page curve of Hawking radiation from semiclassical geometry
• The entropy of Hawking radiation
• Entanglement entropy in free quantum field theory
• Jerusalem Lectures on Black Holes and Quantum Information
• Islands in Schwarzschild black holes
• Islands in Asymptotically Flat 2D Gravity
• Notes on islands in asymptotically flat 2d dilaton black holes
• A Self-consistent Model of the Black Hole Evaporation
• Island in Charged Black Holes
• Islands and Page curves of Reissner-Nordström black holes
• The universality of islands outside the horizon
• Entanglement entropy of asymptotically flat non-extremal and extremal black holes with an island
• Island, Page Curve and Superradiance of Rotating BTZ Black Holes
• Mutual information, islands in black holes and the Page curve
• Unitary Constraints on Semiclassical Schwarzschild Black Holes in the Presence of Island
• Page Curves of Reissner-Nordström Black Hole in HD Gravity

Cited by in corpus (12)

• Island formula in Planck brane
• Mutual information of subsystems and the Page curve for Schwarzschild de-Sitter black hole
• Islands in Non-Minimal Dilaton Gravity: Exploring Effective Theories for Black Hole Evaporation
• Island of Reissner-Nordstr$\ddot{\mathbf{o}}$m anti-de Sitter black holes in the large $d$ limit
• Page Curve of AdS-Vaidya Model for Evaporating Black Holes
• Towards bit threads in general gravitational spacetimes
• Effect of backreaction on Island, Page curve and Mutual Information
• Entanglement island and Page curve for one-sided charged black hole
• Island rules for the noncommutative black hole
• Investigating the role of mutual information in the Page curve for a functional renormalization group improved Schwarzschild black hole
• Island of acoustic black hole in Schwarzschild spacetime
• New insights on mutual information in the island approach to the Page curve