Geometric entropy and time-like entanglement entropy on a rotating BTZ black hole

This paper analyzes the double Wick rotation of a rotating BTZ black hole to derive a dual transition matrix with an imaginary chemical potential, demonstrating that geometric and time-like entanglement entropies can be reproduced through specific coordinate identifications and defining a new Lorentzian entanglement growth based on the linear growth coefficient of the time-like entropy.

The Gist

Imagine you are looking at a spinning black hole. In the world of physics, this is a very complex object, but let's call it the "Cosmic Spinning Top."

This paper is about a clever trick physicists use to understand how information (specifically, "entanglement") behaves inside and around this spinning top. They use a mathematical magic spell called "Double Wick Rotation."

Here is the story of what they did, explained simply:

1. The Magic Mirror (Double Wick Rotation)

Imagine you have a photo of a spinning top. Now, imagine you have a special mirror that doesn't just flip the image left-to-right, but flips time and space into each other.

• Normally, we think of time as a river flowing forward and space as the ground you stand on.
• This "Double Wick Rotation" is like swapping the river and the ground. Suddenly, what used to be "time" acts like "space," and what used to be "space" acts like "time."

The authors found that if you look at a Rotating Black Hole through this magic mirror, it looks exactly like a different kind of black hole, but with a twist: it has an "Imaginary Chemical Potential." (Think of this as a weird, invisible dial that controls how the black hole spins, but the numbers on the dial are imaginary, not real).

2. The Two Sides of the Coin

The paper proves that these two seemingly different black holes are actually twins.

• Twin A: The real, spinning black hole we know.
• Twin B: The "Double Wick Rotated" version (the one seen in the magic mirror).

The authors showed that if you take Twin B and just rotate your coordinate system (like turning a globe), it becomes Twin A. This is a huge deal because it means we can study the weird, hard-to-understand Twin B by using the math we already know about Twin A.

3. Measuring the "Spooky Connection" (Entanglement)

In quantum physics, particles can be "entangled," meaning they are connected in a spooky way where what happens to one instantly affects the other, even if they are far apart.

• Geometric Entropy: This is like measuring how "messy" or "connected" the space is around the black hole. The authors used their magic mirror trick to calculate this for the spinning black hole. They found that the "messiness" follows a specific pattern, like a song with a very specific rhythm.
• Time-Like Entanglement: This is the really new part. Usually, we measure entanglement between two points in space (like two people standing on opposite sides of a room). But here, they measured entanglement between two points in time (like a person at 1:00 PM and the same person at 2:00 PM).

4. The New Discovery: The "Growth Rate"

When they looked at this "Time-Like Entanglement," they noticed something fascinating. As time goes on, the connection between the past and the future doesn't just stay the same; it grows.

• The Analogy: Imagine a rubber band stretching between yesterday and today. The longer you wait, the more the rubber band stretches.
• The Finding: The authors found that this stretching happens at a steady, linear speed. They defined a new number (a "growth exponent") that tells us exactly how fast this connection stretches.

Why is this important?
Usually, scientists use a number called the "Lyapunov exponent" to measure chaos (how fast things get messy). But this number disappears when a black hole is spinning at its maximum speed (the "extremal" limit).
However, this new "Time-Like Entanglement Growth" does not disappear. It keeps working even when the black hole is spinning as fast as possible. It's like a backup engine that keeps running even when the main engine seems to have stalled.

Summary

In plain English, this paper says:

1. We can swap time and space to turn a spinning black hole into a mathematically equivalent, but easier-to-study, version.
2. Using this trick, we can calculate how "connected" the universe is around the black hole.
3. We discovered a new way to measure chaos by looking at how connections grow over time rather than just across space.
4. This new measurement works even in the most extreme conditions where other measurements fail, giving us a new tool to understand the deepest secrets of black holes.

It's like finding a new pair of glasses that lets you see the "heartbeat" of a black hole, even when it's spinning so fast that everything else looks frozen.

Technical Summary

1. Problem Statement

The paper addresses the theoretical challenge of defining and calculating geometric entropy and time-like entanglement entropy in the context of a rotating BTZ (Bañados-Teitelboim-Zanelli) black hole.

• Context: Geometric entropy is the double Wick-rotated (DWR) version of standard entanglement entropy, typically studied in Euclidean CFTs. However, its physical interpretation in time-dependent systems or dual to non-static backgrounds (like rotating black holes) remains unclear.
• The Issue: The DWR of a rotating BTZ black hole introduces a closed time-like curve (CTC) and an imaginary chemical potential, making the standard density matrix formalism insufficient. The authors aim to clarify the equivalence between the DWR geometry and the physical rotating BTZ black hole to compute these entropies rigorously.
• Goal: To derive the transition matrix dual to the DWR BTZ black hole, establish the equivalence between the DWR metric and the rotating BTZ metric via coordinate transformations, and define a new "Lorentzian entanglement growth exponent" based on time-like entanglement entropy.

2. Methodology

The authors employ a combination of Holographic duality (AdS$_3$/CFT$_2$), conformal field theory (CFT) techniques, and gravitational path integrals.

• Double Wick Rotation (DWR): The authors perform a double Wick rotation on the rotating BTZ metric ($z \to -iz'$), effectively exchanging spatial and Euclidean time coordinates. This transforms the thermal density matrix $\rho$ into a transition matrix $\rho'$.
• Transition Matrix Construction: They construct the transition matrix for the DWR theory, showing it takes the form of a thermal ensemble with an imaginary chemical potential ($\Omega_E = -i\Omega$).
• Modular Transformations & Quotients:
  • They analyze the DWR metric as a quotient of $AdS_3$ ($SL(2, \mathbb{C})/SU(2)$).
  • They utilize modular transformations ($\tau \to -1/\tau'$) to relate the thermal AdS background to the DWR BTZ black hole.
• Coordinate Equivalence: A key methodological step involves demonstrating that the DWR metric (with imaginary parameters) is mathematically equivalent to a standard rotating BTZ black hole via a specific coordinate transformation and parameter exchange ($r_+ \leftrightarrow \tilde{r}_-$).
• Stress Tensor Analysis: They compute the expectation values of the stress-energy tensor in the dual CFT using the Schwarzian derivative and Virasoro zero modes to verify the energy and momentum of the system.
• Analytic Continuation: For time-like entanglement entropy, they perform an analytic continuation ($x \to i\tilde{t}, t \to \tilde{x}$) to map spacelike intervals to timelike ones, leading to a non-Hermitian Hamiltonian structure that is subsequently regularized.

3. Key Contributions

A. Derivation of the Transition Matrix

The authors derive the transition matrix $\rho'$ dual to the DWR BTZ black hole:
$$\rho' = e^{-i\beta \tilde{P} - \beta \Omega_E \tilde{H}}$$
They show that this ensemble, characterized by an imaginary chemical potential, correctly reproduces the periodicity of the DWR geometry. This provides a formal framework to calculate geometric entropy using standard CFT tools on a torus with exchanged cycles.

B. Equivalence of Geometries

The paper establishes a rigorous equivalence between the Double Wick Rotated BTZ and the Rotating BTZ black hole.

• By performing a coordinate transformation ($r^2 = u^2 - r_+^2 + \tilde{r}_-^2$) and exchanging parameters ($r_+ \leftrightarrow \tilde{r}_-$), the DWR metric is shown to be identical to the rotating BTZ metric.
• This implies that observables in the DWR background can be calculated by simply swapping the inverse temperature ($\beta$) and the chemical potential term ($\beta \Omega_E$) in the standard rotating BTZ formulas.

C. Geometric Entropy Formula

Using the equivalence principle, the authors derive the geometric entropy ($S_G$) for a general interval. By applying the double Wick rotation to the standard entanglement entropy formula (which involves $\sinh$ functions), they obtain a formula involving $\sin$ functions:
$$S_G = \frac{c}{6} \log \left( \frac{\beta^2(1 + \Omega_E^2)}{\pi^2 \epsilon^2} \sin \left( \frac{\pi \Delta w'}{\beta(1 - i\Omega_E)} \right) \sin \left( \frac{\pi \Delta \bar{w}'}{\beta(1 + i\Omega_E)} \right) \right)$$
This result confirms that geometric entropy is the analytic continuation of entanglement entropy.

D. Time-like Entanglement Entropy and Growth Exponent

The authors derive the time-like entanglement entropy ($S_{TL}$) by analytically continuing the interval to the time direction.

• They find that $S_{TL}$ grows linearly at late times.
• New Definition: They propose a Lorentzian entanglement growth exponent ($\lambda_{TL}$) defined by the coefficient of this linear growth:
  $$\lambda_{TL} = \frac{c}{6 r_+}$$
  where $r_+$ is the outer horizon radius.

4. Key Results

1. Formal Equivalence: The DWR BTZ black hole is not a distinct physical object but is equivalent to a rotating BTZ black hole with exchanged thermodynamic parameters ($\beta$ and $\beta \Omega_E$).
2. Geometric Entropy: The geometric entropy is successfully reproduced as the double Wick rotation of the standard entanglement entropy, validating the use of twisted operators in this context.
3. Time-like Entropy Behavior: The time-like entanglement entropy exhibits linear growth at late times, $S_{TL}(t \to \infty) \propto t$.
4. Extremal Limit Robustness: Unlike the standard Lyapunov exponent ($\lambda_L = 2\pi T$), which vanishes in the extremal limit ($T \to 0$), the new exponent $\lambda_{TL}$ remains finite. This suggests that time-like entanglement entropy captures dynamical information about holographic chaos that persists even in extremal black holes.

5. Significance

• Clarifying Geometric Entropy: The paper resolves ambiguities regarding geometric entropy in non-static, rotating backgrounds by establishing a concrete duality with the rotating BTZ black hole.
• New Diagnostic for Chaos: The introduction of $\lambda_{TL}$ provides a new tool for probing holographic chaos. Its survival in the extremal limit suggests it is a more robust indicator of quantum information scrambling in near-horizon geometries than traditional thermal chaos indicators.
• Holographic Interpretation of CTCs: The work offers insights into how physical laws (specifically entanglement growth) behave in geometries containing closed time-like curves, showing they can be mapped to standard rotating black holes via analytic continuation.
• Future Directions: The methodology opens the door to studying massive fermions and general moduli parameters on tori, potentially extending these results to higher-dimensional holographic systems.

In summary, this paper successfully bridges the gap between Euclidean geometric entropy and Lorentzian time-like entanglement entropy in rotating black hole backgrounds, providing a unified framework and a new metric for quantum chaos.