[Paper Review] Entanglement entropy of asymptotically flat non-extremal and extremal black holes with an island

Wontae Kim, Mungon Nam|arXiv (Cornell University)|Mar 30, 2021

Black Holes and Theoretical PhysicsPhysics and Astronomy51 references60 citations

TL;DR

This paper applies the island prescription to asymptotically flat non-extremal and extremal Reissner-Nordström black holes, using an off-shell generalized entropy approach to compute entanglement entropy. It finds linear growth followed by saturation (Page curve) for non-extremal black holes and logarithmic growth with a constant plateau after the Page time for extremal black holes, with finite entropy even when the island reaches the curvature singularity—resolved by applying the same method to the regular Hayward black hole.

ABSTRACT

The island rule for the entanglement entropy is applied to an eternal Reissner-Nordstr\"om black hole. The key ingredient is that the black hole is assumed to be in thermal equilibrium with a heat bath of an arbitrary temperature and so the generalized entropy is treated as being off-shell. Taking the on-shell condition to the off-shell generalized entropy, we find the generalized entropy and then obtain the entanglement entropy following the island rule. For the non-extremal black hole, the entanglement entropy grows linearly in time and can be saturated after the Page time as expected. The entanglement entropy also has a well-defined Schwarzschild limit. In the extremal black hole, the island prescription provides a logarithmically growing entanglement entropy in time and a constant entanglement entropy after the Page time. In the extremal black hole, the boundary of the island hits the curvature singularity where the semi-classical approximations appear invalid. To avoid encountering the curvature singularity, we apply this procedure to the Hayward black hole regular at the origin. Consequently, the presence of the island in extremal black holes can provide a finite entanglement entropy, which might imply non-trivial vacuum configurations of extremal black holes.

Motivation & Objective

• To resolve the information paradox in asymptotically flat black holes by applying the island prescription to both non-extremal and extremal Reissner-Nordström black holes.
• To investigate how the presence of an island affects entanglement entropy in extremal black holes, where the island boundary may reach the curvature singularity.
• To address the breakdown of semi-classical approximations at the singularity by applying the method to the regular Hayward black hole.
• To derive a consistent Page curve for entanglement entropy in extremal black holes and compare it with non-extremal and Schwarzschild limits.
• To establish a framework for computing entanglement entropy in extremal black holes using off-shell generalized entropy, followed by on-shell limit.

Proposed method

• Introduce an off-shell generalized entropy Sgen(κ) by assuming the black hole is in thermal equilibrium with a heat bath of arbitrary temperature β−1.
• Apply the on-shell limit limκ→κH Sgen(κ) to obtain the physical generalized entropy, where κH is the horizon surface gravity.
• Use the island rule: S = min{ext [A(∂I)/4GN + Smatter(R ∪ I)]} to compute fine-grained entanglement entropy, with Smatter derived from 2D CFT s-wave approximation.
• Model the matter entanglement entropy as Smatter = c/3 log d(x,y), with d(x,y) computed in Kruskal coordinates and including thermal effects.
• Apply the formalism to both Reissner-Nordström (singular at origin) and Hayward (regular at origin) black holes to avoid curvature singularity issues.
• Use the late-time approximation tb ≫ 1/κH to simplify time-dependent behavior of entanglement entropy.

Experimental results

Research questions

• RQ1 How does the entanglement entropy evolve over time in a non-extremal Reissner-Nordström black hole with an island?
• RQ2 What is the behavior of entanglement entropy in an extremal Reissner-Nordström black hole with an island, particularly near the curvature singularity?
• RQ3 Can the island prescription yield a finite and well-defined entanglement entropy in extremal black holes despite the singularity?
• RQ4 How does the entanglement entropy in the extremal case compare to the non-extremal and Schwarzschild limits?
• RQ5 Does the use of a regular Hayward black hole resolve the breakdown of semi-classical approximations in extremal black hole entropy calculations?

Key findings

• For non-extremal Reissner-Nordström black holes, the entanglement entropy grows linearly with time and saturates after the Page time, consistent with the expected Page curve.
• The entanglement entropy in the non-extremal case reduces to the Schwarzschild result when the electric charge vanishes, confirming consistency with known limits.
• In the extremal Reissner-Nordström black hole, the entanglement entropy grows logarithmically with time and becomes constant after the Page time.
• The island boundary in the extremal case reaches the curvature singularity, where semi-classical approximations break down, indicating a potential need for quantum gravity effects.
• When applied to the regular Hayward black hole, the island prescription yields a finite entanglement entropy in the extremal limit, suggesting non-trivial vacuum configurations in extremal black holes.
• The off-shell generalized entropy approach provides a consistent framework for computing entanglement entropy in both non-extremal and extremal black holes, even in singular regimes.