[Paper Review] On the Entanglement of Multiple CFTs via Rotating Black Hole Interior

Norihiro Iizuka, Noriaki Ogawa|arXiv (Cornell University)|Feb 19, 2014

Black Holes and Theoretical Physics3 citations

TL;DR

This paper investigates entanglement between multiple boundary conformal field theories (CFTs) in a 3D rotating eternal black hole via the AdS/CFT correspondence. Using the Ryu-Takayanagi formula, it computes geodesic lengths between different asymptotic boundaries—arising from analytic continuation of the rotating BTZ geometry—and shows that entanglement entropy depends non-trivially on the choice of boundary pair, revealing how bulk interior geometry influences boundary quantum correlations.

ABSTRACT

We study the minimal surfaces between two of the multiple boundaries of 3d maximally extended rotating eternal black hole. Via AdS/CFT, this corresponds to investigating the behavior of entanglements of the boundary CFT with multiple sectors. Non-trivial time evolutions of such entanglements detect the geometry inside the horizon, and behave differently depending on the choice of the two boundaries.

Motivation & Objective

• To understand how quantum entanglement between multiple CFT sectors is encoded in the geometry of a rotating BTZ black hole.
• To investigate whether the 8 disconnected boundary regions of the maximally extended rotating BTZ black hole are truly decoupled or exhibit non-trivial entanglement.
• To explore how the time evolution of entanglement entropy detects the structure of the black hole interior via bulk geodesics.
• To compute and compare geodesic lengths between different boundary pairs using analytic continuation and the Ryu-Takayanagi formula.
• To clarify the role of the inner and outer horizons and the orbifold structure in determining boundary entanglement patterns.

Proposed method

• Derives the embedding of the rotating BTZ black hole in R^{2,2} using coordinates (x₀, x₁, x₂, x₃) with specific hyperbolic and trigonometric functions of r, t, φ.
• Applies analytic continuation to map between different boundary regions (1_{++}, 1_{--}, 1_{+-}, etc.) by shifting u^± = φ ± t into complex planes with imaginary shifts.
• Computes geodesic lengths between points on different boundaries using the formula L = log(X_n) − log(π²T₊T₋) + log(r_∞²), where X_n involves sinh and cosh of T₊ and T₋ times δu^±.
• Uses the Ryu-Takayanagi formula to compute entanglement entropy as a function of geodesic length between boundary regions.
• Analyzes the winding number n ∈ ℤ to find the minimal geodesic length, ensuring spacelike separation on the boundary.
• Considers the full set of 12 boundary regions (3 types × 4 sign combinations) and their analytic continuations to map entanglement across horizons.

Experimental results

Research questions

• RQ1 How does the entanglement entropy between two CFT sectors depend on which pair of boundaries is chosen in the rotating BTZ black hole?
• RQ2 What is the role of the black hole’s inner and outer horizons in mediating entanglement between distinct boundary regions?
• RQ3 How do analytic continuations of time and angular coordinates connect different boundary regions and affect geodesic lengths?
• RQ4 Can the time evolution of entanglement entropy reveal the presence of a black hole interior, even in the absence of a classical horizon?
• RQ5 Why do the entanglement patterns differ between regions 1, 2, and 3 in the maximally extended rotating BTZ geometry?

Key findings

• The rotating BTZ black hole has 12 disconnected boundary regions (3 types × 4 sign combinations), not 8, due to the complex analytic structure of the embedding.
• Geodesic lengths between boundary pairs depend on the choice of region pair and exhibit different functional forms (sinh vs. cosh) depending on the analytic continuation path.
• Entanglement entropy computed via the Ryu-Takayanagi formula shows non-trivial time evolution that depends on the specific pair of boundaries, indicating that the bulk interior geometry influences boundary correlations.
• The minimal geodesic length is determined by the winding number n ∈ ℤ, and the minimum X_n is positive only if the boundary points are spacelike separated.
• Region 3, which has a t ≃ t + 2π identification, features a conical singularity at r = √(r₊² + r₋²), indicating a breakdown of smoothness in the bulk.
• The Penrose diagram and boundary map in global coordinates (θ, τ) show that the 12 regions are distributed non-uniformly, with region 2 appearing as straight lines on the boundary, and the allowed parameter ranges restricted by |tanh(πT₊u₊)| ≤ 1 and |tanh(πT₋u₋)| ≤ 1.