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

Abstract: In this paper, we analyze the double Wick rotation of a rotating BTZ black hole and the entanglement entropy. We derive the transition matrix dual to the double Wick-rotated BTZ black hole, which has the usual shape at an imaginary chemical potential. In the dual gravity side, the double Wick rotated BTZ black hole, which is obtained as a quotient, is equal to a rotating BTZ black hole after the coordinate transformation and the identification of periodicity. The geometric entropy and time-like entanglement entropy are reproduced by the identification. New Lorentzian entanglement growth is defined by the coefficient of linear growth of time-like entanglement entropy.

• The paper derives a duality between the double Wick rotated BTZ geometry and a boundary CFT with imaginary chemical potential, clarifying entropy measures.
• It employs explicit transition matrices and modular properties to map Lorentzian to rotating BTZ metrics, ensuring consistency with gravitational thermodynamics.
• A novel formulation of time-like entanglement entropy reveals linear late-time growth, offering a new probe into quantum chaos and non-static holographic dynamics.

Geometric and Time-like Entanglement Entropy in Rotating BTZ Black Holes

Introduction

This work provides a detailed analysis of geometric entropy and time-like entanglement entropy in the context of the rotating BTZ black hole, focusing on their holographic description via the AdS$_3$/CFT$_2$ correspondence. The paper establishes a duality between the double Wick-rotated BTZ geometry and a boundary CFT at imaginary chemical potential, develops the requisite transition matrices, and systematically explores their impact on various quantum information measures, particularly in time-dependent and non-static backgrounds. The results clarify several ambiguities in the interpretation of geometric entropy in time-dependent settings and systematically identify new invariants associated with Lorentzian entanglement dynamics.

Transition Matrices and Double Wick Rotation

The double Wick rotation is implemented on the rotating BTZ background, resulting in a geometry with nontrivial closed time-like curves for real angular momentum. The authors rigorously derive the transition matrix dual to this geometry, showing its equivalence to a rotating BTZ black hole after a specific coordinate transformation and identification of periodicity. This identification is encoded in a transition matrix with imaginary chemical potential, given by:

$$\rho' = e^{-i\beta \tilde P + i\beta \Omega \tilde H}$$

where the period exchange realizes a formal ensemble analogous to conventional thermal CFT, albeit with analytic continuation in the chemical potential.

The analysis further exploits the modular properties of the BTZ geometry, demonstrating that the double Wick rotated background sits in the $SL(2,\mathbb{Z})$ family of quotients of $AdS_3$, with specific consequences for the associated modular parameter and identifications on the boundary torus. The equivalence of free energies and other thermodynamic potentials is established, highlighting the robustness of these dualities under analytic continuation.

Stress Tensor Expectation Values and Virasoro Zero Modes

Using complex coordinate transformations and the conformal mapping between the cylinder and the plane, the expectation value of the boundary stress energy tensor is evaluated explicitly in the dual CFT. The Virasoro zero modes, corresponding to the BTZ mass and angular momentum, are directly related to the modular parameter, with precise dependence on the central charge. The calculations exhibit consistency with the gravitational side, including the behavior in the extremal ($r_+ = r_-$) and double Wick rotated limits. The double Wick rotated identification yields nonzero vacuum energy density, matching the gravitational calculations for the associated quotient geometry.

Equivalence of Rotating and Double Wick-Rotated Geometries

A key technical result of the paper is the explicit coordinate transformation that maps the Lorentzian sector of the double Wick rotated BTZ metric to the standard rotating BTZ black hole, provided the appropriate periodicity is imposed:

$$O^{DWR}(r_+, \tilde{r}_-) = O^{BTZ}(\tilde{r}_-, r_+)$$

This facilitates a unified treatment of observables, including the stress tensor, partition function, and correlators, across both backgrounds. Notably, geometric entropy and entanglement entropy are shown to be related by the exchange of the thermal and spatial cycles, realizing the double Wick rotation at the level of the von Neumann entropy.

Geometric Entropy and Its Holographic Dual

The geometric entropy is recovered by tracing over a subsystem in the doubled, analytically continued ensemble. The resulting expression,

$$S_G = \frac{c}{6} \log \left[\frac{\beta^2(1+\Omega_E^2)}{\pi^2\epsilon^2} \sin\left(\frac{\pi \Delta w'}{\beta(1-i\Omega_E)}\right) \sin\left(\frac{\pi \Delta \bar{w}'}{\beta(1+i\Omega_E)}\right)\right]$$

is obtained by analytic continuation from the standard interval entanglement entropy on the BTZ geometry and matches prior results for geometric entropy in pure AdS and charged AdS black holes. This formula captures the leading CFT contribution and is sensitive to both the modular parameter and chemical potential.

Time-Like Entanglement Entropy and Lorentzian Growth

A novel contribution is the analytic continuation of entanglement entropy computations to timelike intervals, achieved by $x = i \tilde{t}$ and $t = \tilde{x}$, resulting in a transition matrix with complex evolution generators. This leads to a time-like entanglement entropy,

$_2$0

which exhibits linear late-time growth:

$_2$1

The rate,

$_2$2

is non-vanishing even in the extremal limit, in contrast to the Lyapunov exponent extracted from conventional OTOCs. This feature provides a new diagnostic for quantum chaos and the propagation of correlations in Lorentzian signature, robust to low temperature and extremality.

Theoretical and Practical Implications

These results enhance the understanding of geometric entropy in non-static, time-dependent holographic settings and provide a systematic framework for addressing analytic continuations (double Wick rotations) in both boundary CFT and bulk gravity. The introduction of the Lorentzian entanglement growth rate extends the suite of quantum information-theoretic diagnostics available for probing black hole interiors and chaos, with prospective implications for the study of quantum quenches, phase transitions, and the dynamical connectivity of near-horizon AdS regions.

Generalizations to higher dimensions, other black hole backgrounds, and modular deformations on $_2$3 CFT tori are immediate directions. The explicit mapping between geometric and entanglement entropies also invites further study in theories with less symmetry or matter content, particularly for massive fermions and non-conformal field theories.

Conclusion

This paper rigorously establishes the equivalence of geometric entropy and time-like entanglement entropy via double Wick rotation in the rotating BTZ black hole and its dual CFT. All relevant transition matrices, modular structures, and entropy functionals are analytically constructed. The findings reveal new universal features of Lorentzian entanglement dynamics and identify entropic growth rates that persist in the extremal limit, enriching the toolkit for probing quantum gravity and holography in dynamical, non-static contexts (2604.15720).