⚛️ high-energy theory

Island of acoustic black hole in Schwarzschild spacetime

This study investigates the entanglement entropy of analogue Hawking radiation in acoustic black holes surrounding a Schwarzschild black hole, demonstrating that the emergence of islands restores unitarity and yields a Page curve in the non-extremal case, whereas the extremal case results in divergent entropy.

Original authors: Yu-Ye Cheng, Jia-Rui Sun

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yu-Ye Cheng, Jia-Rui Sun

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A "Sound" Black Hole

Imagine you are standing in a very thick, super-dense fog (a superfluid). Now, imagine a real black hole is sitting in the middle of this fog. Because the fog is so dense and moving, it creates a special kind of barrier for sound waves.

In this paper, the authors study an "Acoustic Black Hole." This isn’t a black hole that sucks in light; it’s a region where sound (specifically, tiny vibrations called phonons) cannot escape. It’s like a "dumb hole"—if you shout inside it, your voice never gets out.

The scientists wanted to answer a famous puzzle in physics called the Information Paradox. The paradox asks: If a black hole evaporates over time by emitting radiation, does the information about what fell into it disappear forever? Quantum mechanics says "No, information must be preserved." But standard calculations suggested it does disappear.

To solve this, the authors used a mathematical tool called the "Island Formula." Think of an "Island" as a hidden pocket of space inside the black hole that actually belongs to the outside world’s information. By including this "Island" in their calculations, they checked if the information is saved.

The Two Scenarios: Normal vs. Extreme

The authors looked at two different types of these sound-black holes, depending on how fast the fog is flowing.

1. The Non-Extremal Case (The "Normal" Flow)

Imagine the fog is flowing fast enough to create two distinct barriers for sound: an inner wall and an outer wall.

Without the Island: If you just look at the sound waves escaping, it looks like information is leaking away forever. The "messiness" (entropy) of the sound keeps growing and growing, like a room getting messier every second without anyone cleaning it.
With the Island: When the authors added the "Island" to the math, something magical happened. The "messiness" stopped growing. It rose for a while, hit a peak, and then leveled off. This pattern is called the Page Curve.
The Analogy: Imagine a library where books are being burned (the black hole evaporating). Without the Island, the knowledge in the books is lost forever. With the Island, it’s like discovering that every burned book was secretly copied onto a microfilm hidden in the basement (the Island). The total amount of knowledge in the universe stays constant. The "Island" is the basement archive.

In this normal case, the math works perfectly. The information is saved, and the system remains "unitary" (meaning no information is lost).

2. The Extremal Case (The "Critical" Flow)

Now, imagine the fog is flowing at a very specific, critical speed where the inner and outer sound barriers merge into one single, tight barrier. This is the "extremal" case.

The Problem: Here, the math gets weird. The "surface gravity" (a measure of how strongly the black hole pulls) drops to zero.
Without the Island: The "messiness" (entropy) shoots up to infinity. It’s like the room doesn’t just get messy; it explodes into chaos. Because it goes to infinity, you can’t define a clear "Page Time" (the moment when the information starts being saved).
With the Island: Even though the initial calculation blows up, when you include the Island, the total messiness becomes finite again. It settles at a stable number, roughly equal to the size (area) of the sound barrier.
The Analogy: Imagine trying to balance a pencil on its tip. In the normal case, it falls over predictably. In the extremal case, the pencil is perfectly balanced but infinitely unstable. Without the Island, the pencil’s position is undefined (infinite uncertainty). With the Island, it’s like placing the pencil in a protective glass case. The uncertainty is contained, and the system stabilizes.

Why Does This Matter?

The authors conclude that this "Acoustic Black Hole" model behaves just like a real black hole should regarding information.

Unitarity is Saved: In the normal case, the information is preserved. The "Island" ensures that the sound waves (phonons) carry the information out, rather than losing it.
The Island is Visible: In real black holes, the "Island" is hidden deep inside, behind the event horizon, so we can never see it. But in this acoustic model, the "Island" sits outside the light-blocking horizon (though inside the sound-blocking horizon). This means, in theory, we could build a lab experiment to "see" the Island, which is impossible with real black holes.
Universal Rule: This suggests that the "Island Formula" isn’t just a trick for one specific type of black hole. It might be a universal rule for any system with a horizon (a boundary nothing can cross), whether it’s made of gravity, sound, or something else.

Summary in a Nutshell

The Setup: A black hole surrounded by a superfluid creates a "sound trap" (Acoustic Black Hole).
The Question: Does information get lost when sound escapes this trap?
The Tool: They used the "Island Formula," which adds a hidden region (the Island) to the calculation.
The Result:
- For normal sound traps, the Island saves the information, creating a stable "Page Curve."
- For extreme sound traps, the math gets chaotic without the Island, but the Island tames the chaos and saves the information.
The Takeaway: Information is not lost. The "Island" is the key to keeping the universe’s information ledger balanced.

Technical Summary: Island of Acoustic Black Hole in Schwarzschild Spacetime

Problem and Motivation
The paper addresses the black hole information paradox within the framework of analogue gravity, specifically focusing on acoustic black holes formed by a superfluid surrounding a Schwarzschild black hole. The central problem is to determine whether the unitarity of quantum mechanics is preserved in this analogue system during the evaporation process via analogue Hawking radiation (phonons). While the island formula has successfully resolved the information paradox in various gravitational contexts by reproducing the Page curve, its application to acoustic black holes in curved spacetime—particularly distinguishing between non-extremal and extremal cases—remains to be fully explored. The authors aim to calculate the entanglement entropy of the phonon radiation using the island formula to verify unitarity and investigate the connection between entanglement entropy and the causal structure of spacetime.

Methodology
The study constructs the acoustic black hole metric using the relativistic Gross-Pitaevskii (GP) theory for a complex scalar field in a fixed Schwarzschild background. By analyzing perturbations of the fluid phase, an effective acoustic metric is derived, which contains both an optical event horizon ( $r_h$ ) and acoustic horizons ( $r_\pm$ ). The system is classified into two cases based on the tuning parameter $\xi$ :

Non-extremal case ( $\xi > 4$ ): Distinct inner and outer acoustic horizons exist.
Extremal case ( $\xi = 4$ ): The acoustic horizons coincide ( $r_+ = r_- = 2r_h$ ), resulting in vanishing surface gravity.

The authors employ the island formula to compute the generalized entanglement entropy ( $S_{gen}$ ) of the radiation region $R$ . The calculation involves minimizing the generalized entropy over all possible island configurations $I$ :
$S_R = \min \left\{ \text{ext} \left[ \frac{\text{Area}(\partial I)}{4 \tilde{G}_N} + S_{\text{finite}}^{\text{matter}}(I \cup R) \right] \right\}$
The analysis considers two scenarios for the radiation region: one close to the acoustic horizon and one far from it. The entanglement entropy of the phonons is calculated using conformal field theory techniques, including the s-wave approximation and Kruskal coordinates, while accounting for the geometric contributions of the island boundaries.

Key Contributions and Results

Non-Extremal Case:
- Without Islands: In the early-time phase, the entanglement entropy of the phonons grows linearly with time, diverging at late times, consistent with Hawking’s original calculation.
- With Islands: The emergence of islands modifies the entropy behavior. For radiation regions both near and far from the horizon, the island contribution dominates after a critical "Page time." The resulting entanglement entropy stabilizes to a finite value approximately proportional to the area of the outer acoustic horizon ( $2\pi r_+^2 / G_N$ ).
- Page Curve: The transition from the no-island phase to the island phase successfully reproduces the Page curve, indicating that the entanglement entropy increases, saturates, and remains stable, thereby preserving unitarity. The Page time is explicitly calculated and found to be finite.
Extremal Case:
- Without Islands: Due to the vanishing surface gravity ( $\kappa_+ \to 0$ ), the entanglement entropy diverges. This divergence is attributed to the causal structure where the Cauchy horizon terminates at a timelike singularity, leading to an ill-defined Page time.
- With Islands: Despite the divergence in the no-island phase, the inclusion of islands yields a finite entanglement entropy. The entropy stabilizes to a value equal to twice the area entropy of the acoustic horizon. The position of the island is determined to be slightly outside the horizon.
- Distinction from Previous Works: Unlike previous studies on extremal Reissner-Nordström black holes where divergence was linked to the black hole singularity, this paper attributes the divergence in the no-island phase to the specific causal structure of the acoustic horizon in the extremal limit.

Significance and Conclusion
The paper claims that its results verify the unitarity of the analogue gravity system, demonstrating that the island formula is effective in resolving the information paradox for acoustic black holes in Schwarzschild spacetime. The study provides further insight into the relationship between entanglement entropy and the causal structure of spacetime, particularly highlighting how the extremal limit affects the definition of the Page time.

The authors emphasize that this work suggests the island formula may serve as a universal mechanism for restoring unitarity in any effective theory with a horizon, not just in the context of AdS/CFT correspondence. Additionally, the model offers a theoretical advantage: since the acoustic horizon lies outside the optical event horizon, the associated island region is potentially observable without accessing the black hole interior, providing a conceptual bridge for understanding information recovery in real black holes. The paper concludes by noting open questions regarding the isolation of photon and phonon contributions and the comparison of island structures with standard Schwarzschild black holes.