Entanglement Surfaces for Rotating Cylindrical BHs

• The paper demonstrates that rotation modifies entanglement surface morphology and phase structure, influencing island formation and RT/HRT entropy calculations.
• It details a formalism using cylindrical metrics and boundary-anchored extremal surfaces to compute von Neumann and reflected entropies in AdS settings.
• The study highlights that horizon topology and angular momentum critically affect quantum information measures and phase transitions in holographic models.

Entanglement surfaces for rotating cylindrical black holes precisely characterize the extremal co-dimension–2 surfaces anchored at boundary subregions in asymptotically AdS, stationary black hole backgrounds with nonzero angular momentum and cylindrical horizon topology. Such surfaces play a central role in computing von Neumann entropy, reflected entropy, and entanglement wedge cross-sections in both bottom-up and top-down holographic models, including applications to island formulae and information paradox diagnostics. Rotation drives both qualitative and quantitative modifications to the entanglement phase structure and the morphology of extremal (RT/HRT) surfaces.

1. Spacetime Structure and Rotating Cylindrical Black Holes

Rotating cylindrical black holes are stationary solutions of Einstein–AdS gravity with cylindrical horizon topology. In four dimensions, the canonical line element (in Lemos–Zanchin form) is: $$ds_4^2 = \ell_4^2 \frac{dr^2}{b(r)} + e^{2r} \left[ - \frac{b(r) - \Omega^2}{1 - \Omega^2} dt^2 + \frac{2\Omega(b(r)-1)}{1-\Omega^2} \frac{\ell_\phi}{\ell_4} d\phi dt + \frac{1 - \Omega^2 b(r)}{1-\Omega^2} \frac{\ell_\phi^2}{\ell_4^2} d\phi^2 + dz^2 \right]$$ where $b(r) = 1 - e^{-3(r - r_h)}$, $\Omega \in [0,1)$ encodes the angular velocity of the horizon, and $\phi \sim \phi + 2\pi$, $z \in \mathbb{R}$. The horizon at $r_h$ satisfies $b(r_h)=0$. The extremal ($T=0$) limit $\Omega \to 1$ pushes the horizon to $r \to -\infty$ with a fixed combination

$e^{3r_h} / (1 - \Omega^2) = \mathrm{const}.$

In higher dimensions, generalizations employ the Kerr–AdS–cylinder solution with multiple independent rotation parameters $\{a_i\}$ and corresponding angular coordinates, see (Nadi et al., 2019).

Table 1: Key Parameters in Rotating Cylindrical Black Holes

Parameter	Physical Role	Range/Notes
$\ell_4$	AdS radius	Fixed by cosmology/embedding
$\Omega$	Rotation/boost parameter	$0 \leq \Omega < 1$
$r_h$	Horizon location	$b(r_h) = 0$
$\ell_\phi$	Circumference of $S^1$	Topological parameter
$m, J$	Mass and angular momentum	Densities, set by $\Omega, r_h$

(Billiato et al., 3 Dec 2025, Nadi et al., 2019)

2. Extremal-Surface Formalism: RT/HRT Prescription

The computation of entanglement entropy proceeds via finding extremal area (in 4d/5d, area; in 3d, length) surfaces homologous to a chosen boundary region $\mathcal{R}$. The standard Ryu–Takayanagi (RT) prescription in static, and Hubeny–Rangamani–Takayanagi (HRT) in stationary, backgrounds applies. For the 4d/5d rotating cylindrical solution (Billiato et al., 3 Dec 2025), extremal surfaces ("island surfaces") in the AdS$_5$ braneworld setup are parametrized as

$t = t(\mu),\quad r = r(\mu),$

and embedded at fixed $(\phi, z)$ wrapping $S^1 \times \mathbb{R}$. The induced metric along the surface reduces (after gauge fixing) to an effective area functional

$$A_{\text{isl}} = V_{\text{cyl}} \ell_5^3 \int_{\mu_0}^{\pi} d\mu\, \frac{e^{2r}}{\sin^3\mu} \sqrt{\Delta(\mu)}$$

with

$$\Delta(\mu) = \frac{1 - \Omega^2 b(r)}{1 - \Omega^2} \left[ 1 + b(r) (r')^2 \right] - e^{2r} b(r) (t')^2.$$

Neumann boundary conditions at the brane-environment interface ($\mu = \mu_0$) ensure $t'(\mu_0) = 0$, so the minimal surfaces are always at fixed boundary time.

An analogous formalism governs the BTZ (3d) setting, where the codimension–2 extremal surfaces reduce to spacelike geodesics (curve of extremal length) parametrized by the radial coordinate (Iizuka et al., 2014, Basu et al., 8 Oct 2024).

3. Effects of Rotation on Entanglement Surfaces

Angular momentum fundamentally alters the embedding and area of entanglement surfaces. Key effects:

• The radial effective potential is "squeezed" by the factor $(r^2 - r_+^2) (r^2 - r_-^2)$, altering how deeply geodesics probe the bulk. In particular, in the BTZ context,

$$\frac{dr}{dx} = \pm \sqrt{ (r^2 - r_+^2)(r^2 - r_-^2) } \frac{ \sqrt{ r^2 - r_*^2 } }{ r_* }$$

• The proper length (interval width in the boundary CFT) is rescaled by $\sqrt{1 - \Omega^2}$, thus modifying effective temperature $T \to T \sqrt{1 - \Omega^2}$.
• Presence of the $r_-$ (inner horizon) brings a left–right asymmetry, replaced at the CFT level by a dependence on chiral interval coordinates $\Delta\phi \pm \Omega \Delta t$.
• In multiboundary settings, rotation splits chiral temperatures, thereby delaying entanglement saturation and modifying the location of phase transitions between "connected" and "disconnected" extremal surfaces (Iizuka et al., 2014).

4. Entanglement Entropy, Islands, and Phase Structure

The area of the minimal entanglement surface yields the leading contribution to von Neumann entropy (suitably regularized) via the standard RT/HRT formula

$S_{\text{EE}} = \frac{A_{\min}}{4G}$

(for length in 3d; area in higher dimensions). For the BTZ geometry, the regulated geodesic length between boundary points yields: $$L(P_1,P_2) = \ln\left\{ \sinh[\pi T_- (\Delta\phi - \Delta t)]\,\sinh[\pi T_+ (\Delta\phi + \Delta t)] \right\} - \ln(\pi^2 T_+ T_-) + 2\ln r_\infty$$ where $T_\pm = (r_+ \pm r_-)/(2\pi)$.

In the context of 5d double holography (Billiato et al., 3 Dec 2025), the computation distinguishes between "island" surfaces (anchored on the brane) and Hartman–Maldacena (HM) surfaces (traversing the wormhole). The system exhibits three regimes, characterized by two numerically-determined critical parameters ($\mu_c, \mu_e$ in the braneworld, or $\alpha_c, \alpha_e$ in the top-down model):

Regime	Range	Island Existence	Surface Behavior
I	$\mu_0 > \mu_c$	Both AdS and extremal BH admit islands	Plateaux $\Delta r^\pm$
II	$\mu_e < \mu_0 < \mu_c$	Only extremal BH supports islands	Atoll region appears
III	$\mu_0 < \mu_e$	No islands for any case	Island/HM coalesce

Rotation "repels" island surfaces from the horizon, broadening the regime of constant-entropy, and introduces an additional, extremal-controlled critical parameter that signals a new phase boundary.

(Billiato et al., 3 Dec 2025)

5. Entanglement Wedge Cross-Section and Reflected Entropy

In deformed holographic CFT$_2$ duals with angular momentum, the entanglement wedge cross-section (EWCS) and reflected entropy are accessible via minimal cross-sections in the bulk. For a single interval, EWCS for the rotating BTZ at finite cutoff is given by

$$\tilde{E}_W(A : A^c) = \frac{1}{2} \left[ S(A) - S_{\text{th}}(A) \right] + \frac{\ln 2}{4G_N}$$

where $S_{\text{th}}(A)$ denotes the thermal entropy associated with a geodesic wrapping the horizon. The reflected entropy satisfies $S_R(A:A^c) = 2\,E_W$. The rotational dependence enters the segment lengths and modifies the explicit surface embedding (Basu et al., 8 Oct 2024).

6. First Law of Entanglement and Response to Angular Momentum

For small deformations around AdS or static black holes, the change in entanglement entropy admits a "first law" structure incorporating the excitation energy and angular momentum in the boundary region. For the rotating BTZ (Ghosh et al., 2016), the variation, to second order in small interval size $l$, is

$$\Delta S_{\text{HEE}} = \frac{l^2(r_+^2 + r_-^2)}{48G} - \frac{l^4[(r_+^2 + r_-^2)^2 + 4r_+^2 r_-^2]}{1440G} + O(l^6 r^6)$$

with the first law taking the form

$$d(\Delta S_{\text{HEE}}) = \frac{1}{T_E} d(\Delta E) - \frac{\Omega_E}{T_E} d(\Delta J)$$

where the entanglement temperature $T_E$ and entanglement angular velocity $\Omega_E$ interpolate between the short-interval regime and the large-interval thermodynamic result. In higher-dimensional rotating cylinders, the first law generalizes accordingly in terms of all independent rotation planes (Nadi et al., 2019).

Rotation generically "tilts" the extremal surfaces in the bulk. For BTZ, the geodesic acquires a nontrivial $t$-profile, reflecting the net angular momentum and the imbalance between left/right movers (Ghosh et al., 2016).

7. Implications for Quantum Information and Holography

The combined presence of angular momentum and nontrivial horizon topology in rotating cylindrical black holes generates richer phase structures in the entanglement diagram, introduces additional critical points controlling island existence, and modifies the approach to Page transitions. In extremal limits, sufficiently rapid rotation can preclude the formation of quantum extremal islands, obstructing "island" saddles needed for restoring unitarity in the information paradox. This demonstrates that the quantum information-theoretic properties of black holes in AdS/CFT are exquisitely sensitive to both spacetime topology and global charges such as angular momentum (Billiato et al., 3 Dec 2025).

Rotation alters both the entanglement bottleneck geometries (captured by the RT/HRT surfaces and associated wedge cross-sections) and the real-time dynamics of entropy transfer between sectors in multiboundary or doubled CFT constructions (Iizuka et al., 2014). This sensitivity establishes entanglement surfaces for rotating cylindrical black holes as a precision probe of quantum gravity, holography, and information flow in strongly-correlated systems.