⚛️ high-energy theory

Entanglement surfaces for rotating cylindrical black holes

This paper constructs and analyzes entanglement surfaces, including islands and Hartman-Maldacena surfaces, for rotating cylindrical black holes in double holographic setups, revealing a new critical parameter associated with the extremal limit that governs the behavior of island surfaces in both bottom-up and top-down models.

Original authors: Fabio Billiato, Alessandra Gnecchi

Published 2026-02-16

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CC BY 4.0

Original authors: Fabio Billiato, Alessandra Gnecchi

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

Imagine you are trying to understand how a black hole "evaporates" and loses information over time. This is a huge puzzle in physics because, according to old rules, black holes should destroy information, which breaks the fundamental laws of quantum mechanics.

To solve this, physicists use a clever trick called Holography. Think of it like a 3D movie projected onto a 2D screen. The black hole lives in a 3D "bulk" universe, but its behavior is encoded on a 2D "screen" (the boundary) that acts like a giant radiation detector.

This paper explores what happens when that black hole isn't just sitting still, but is spinning (rotating) and shaped like a cylinder (like a long tube) instead of a sphere.

Here is the breakdown of their discovery using simple analogies:

1. The Setup: The Spinning Tube and the "Bath"

Usually, black holes in these theories are stuck in a box where they can't lose heat. To make them evaporate, the authors connect the black hole to a "bath" (a non-gravitating environment) that sucks up the radiation.

The Analogy: Imagine a hot, spinning washing machine (the black hole) sitting in a giant, empty room (the bath). The authors are studying how the "heat" (information) leaks from the machine into the room.
The Twist: They made the washing machine spin faster and faster, eventually reaching a limit where it spins so fast it becomes "extremal" (a special, frozen state).

2. The Mystery: Where is the Information? (The "Islands")

In the old "static" (non-spinning) models, physicists found that the information doesn't just float away. Instead, it hides in a secret zone called an "Island."

The Analogy: Imagine you drop a secret note into a river (the radiation). You expect the note to float downstream forever. But, the "Island" theory says the note actually gets caught in a hidden eddy behind a waterfall (the black hole horizon). To an observer downstream, the note seems lost, but it's actually safe in a hidden pocket of space connected to the river.

3. The Discovery: Spinning Changes the Rules

The authors asked: What happens to these secret "Islands" when the black hole spins really fast?

They found that rotation introduces a new critical speed (a new "tipping point"). Before, there was only one rule for when islands could exist. Now, there are three distinct regimes (zones) depending on how the "bath" is connected to the black hole:

Zone 1 (Slow Spin / Wide Connection): Islands exist easily, just like in the non-spinning case. The secret notes are safe in the hidden eddies.
Zone 2 (Medium Spin / Medium Connection): This is the new, interesting zone. The spinning creates a "repulsion" effect. The islands are pushed away from the black hole's center. It's like the spinning creates a centrifugal force that pushes the secret notes further out, changing where they can hide.
Zone 3 (Fast Spin / Tight Connection): This is the most dramatic finding. If the black hole spins too fast and the connection is too tight, the islands disappear entirely.
- The Analogy: Imagine the river spinning so violently that the hidden eddies (islands) are torn apart. The secret notes can no longer hide; they are forced to merge with the main flow or hit a "topological singularity" (a weird knot in space-time where the rules break down).

4. The "Top-Down" vs. "Bottom-Up" Check

The authors didn't just guess; they checked this using two different methods:

Bottom-Up (The Lego Model): They built a simplified 5D universe using general relativity rules.
Top-Down (The Blueprint): They built the model using the full, complex machinery of String Theory (10 dimensions).

The Result: Even though the math is wildly different between the two, they both told the same story. The spinning black hole creates these three zones, and the "Islands" vanish in the third zone. This gives them high confidence that this isn't just a mathematical quirk, but a real physical feature of rotating black holes.

5. Why Does This Matter?

This paper helps us understand the Page Curve, which is the graph that proves black holes preserve information (solving the paradox).

The Takeaway: Rotation adds a new layer of complexity. It creates a new "danger zone" (the extremal limit) where the way information is stored changes drastically.
The Metaphor: Think of the black hole as a spinning top. As it spins faster, the way it balances on its tip changes. At a certain speed, the "islands" of information that usually keep the universe safe simply can't exist anymore, forcing the universe to find a new way to balance the books.

Summary in One Sentence

By studying a spinning, cylindrical black hole, the authors discovered that rotation creates a new "speed limit" where the hidden pockets of information (islands) that usually save the universe from paradoxes are pushed out of existence, a finding that holds true whether you look at it with simple math or complex string theory.

1. Problem Statement

The study of Hawking evaporation and the information paradox within the AdS/CFT framework often relies on "double holography" setups, where a black hole on a lower-dimensional brane is coupled to a non-gravitating bath. While the entanglement structure (specifically the emergence of "islands" that resolve the Page curve) has been extensively studied for static black branes, the effects of rotation on these extremal surfaces remain largely unexplored.

The authors address the challenge of constructing extremal surfaces for rotating cylindrical black holes. Unlike rotating Kerr-AdS black holes, which involve complex angular fibrations making analytical and numerical solutions difficult, the authors utilize a solution generated by boosting a planar AdS black hole with one compactified direction ( $S^1$ ). This creates a rotating cylindrical black hole that admits a tractable double holographic description in both bottom-up (5D braneworld) and top-down (10D Type IIB string theory) models.

2. Methodology

The authors employ a dual approach to analyze the system:

Bottom-Up Model (5D Braneworld):
- They utilize the Karch-Randall (KR) model, embedding a 4D cylindrical black hole on a subcritical End-of-the-World (ETW) brane within a 5D AdS bulk.
- The bulk metric is a "black string" geometry where the 4D slices are the rotating cylindrical black holes.
- They apply the HRT (Hubeny-Rangamani-Takayanagi) prescription to find codimension-2 extremal surfaces in the 5D bulk.
- Two classes of surfaces are analyzed:
  1. Island Surfaces: Connect the bath boundary to the ETW brane.
  2. Hartman-Maldacena (HM) Surfaces: Connect the bath boundary to the black hole horizon (crossing the Einstein-Rosen bridge).
- The authors solve the resulting non-linear ordinary differential equations (ODEs) numerically to determine the shape and area of these surfaces.
Top-Down Model (10D Type IIB String Theory):
- They embed the solution into a 10D Type IIB supergravity background dual to 3D conformal interfaces in $\mathcal{N}=4$ Super Yang-Mills.
- The geometry involves an $AdS_4 \times S^2 \times S^2$ fibration over a Riemann surface.
- They derive the induced area functional for extremal surfaces wrapping the internal spheres and the cylinder coordinates.
- They solve the resulting partial differential equations (PDEs) using a relaxation method to find numerical solutions for island and HM surfaces.

3. Key Contributions

Extension to Stationary Backgrounds: The paper successfully extends the double holographic construction from static to stationary (rotating) backgrounds, proving that the "island" mechanism persists in the presence of rotation.
Discovery of a New Critical Regime: The authors identify a third regime in the entanglement phase space, bounded by a new critical parameter related to the extremal limit of the rotating black hole.
Qualitative Agreement between Models: They demonstrate that the qualitative behavior of extremal surfaces (existence of islands, critical angles, and phase transitions) is consistent between the bottom-up 5D effective theory and the top-down 10D string theory construction, despite quantitative differences due to internal geometry.
Generalization to Higher Dimensions: The analysis is extended to arbitrary dimensions $d$ , showing that the critical behaviors are intrinsic properties of the braneworld model governed by the rotation parameter.

4. Key Results

A. Three Distinct Regimes

The phase space of the system is divided into three regimes based on the brane angle $\mu_0$ (in 5D) or the tension parameter $\alpha$ (in 10D):

Regime I ( $\mu_0 > \mu_c$ ):
- Both empty AdS and the extremal rotating black hole support island surfaces.
- Islands remain far from the horizon even as extremality is approached.
- The "atoll" (a region on the brane where islands cannot land) does not exist.
Regime II ( $\mu_e < \mu_0 < \mu_c$ ):
- Empty AdS does not support islands, but the extremal rotating black hole does.
- An "atoll" forms on the brane.
- As rotation increases, the critical anchor point ( $r_A$ ) moves, and the size of the atoll increases.
Regime III ( $\mu_0 < \mu_e$ ):
- New Finding: Neither empty AdS nor the extremal rotating black hole supports island surfaces.
- As the system approaches extremality, island surfaces develop a singularity (a cusp) at the midpoint of the bulk and effectively merge with HM surfaces.
- This regime is bounded by a new critical angle $\mu_e$ (numerically $\approx 0.5287$ in 5D), which is distinct from the standard critical angle $\mu_c$ ( $\approx 0.9869$ ) associated with the static case.

B. Rotation Effects

Repulsion Effect: Rotation exerts a "repulsion" on island surfaces, pushing them away from the black hole horizon and closer to the defect (brane) compared to the static case.
Constant Entropy Belt: The size of the "constant entropy belt" (the region of the bath where the Page curve is constant) increases with rotation. The "Page angle" (the minimum brane angle required for islands to exist) shifts as a function of rotation.
Extremal Limit: In the extremal limit ( $\Omega \to 1$ ), the horizon moves toward the singularity. In Regime III, this causes the island surfaces to become singular, indicating a breakdown of the island phase for extremal rotating black holes in this regime.

C. Top-Down vs. Bottom-Up

The 10D string theory model reproduces the three-regime structure found in the 5D model.
New critical values are found in 10D ( $\alpha_c \approx 4.0$ and $\alpha_e \approx 7.9$ ).
While the qualitative physics (existence of the atoll, the three regimes, and the repulsion effect) matches, a quantitative match is impossible due to the dependence of the extremal surfaces on the specific internal geometry (the "fat-brane" nature of the top-down model).

5. Significance

Robustness of Islands: The results confirm that the island mechanism, crucial for resolving the black hole information paradox, is robust against rotation.
Extremal Physics: The discovery of the new critical parameter $\mu_e$ highlights that the extremal limit of rotating black holes plays a fundamental role in determining the entanglement phase structure, acting as a distinct phase boundary separate from the static critical angle.
Holographic Dictionary: The work strengthens the connection between bottom-up effective field theories and top-down string theory constructions, showing that complex features like rotation-induced phase transitions are captured consistently across different levels of holographic description.
Future Directions: The paper sets the stage for studying more complex configurations, such as charged black holes or those with scalar fields, and investigating the quantum backreaction of the CFT on the extremal metric.

In summary, this paper provides a comprehensive analysis of how rotation modifies the entanglement structure of black holes in double holography, revealing a rich phase structure governed by both the standard critical angle and a new critical angle associated with the extremal limit.