Scrambling in Hyperbolic Black Holes: shock waves and pole-skipping

TL;DR

The paper analyzes scrambling in $(d+1)$-dimensional hyperbolic black holes using the holographic duality. It computes OTOCs via the eikonal (shock-wave) approach in a Rindler-AdS background and validates the results against known CFT calculations, while also deriving the butterfly velocity through both shock-wave methods and a pole-skipping analysis. A universal Lyapunov exponent λ_L = 2πT emerges, and v_B(T) smoothly interpolates between the Rindler-AdS value 1/(d-1) at T = 1/(2πℓ) and the planar value √(d/[2(d-1)]) at high T, with explicit expressions depending on the horizon scale r_0/ℓ. The pole-skipping analysis corroborates the shock-wave results and reveals a consistent relation between chaos and horizon data even in hyperbolic geometries, hinting at bounds on v_B beyond planar setups.

Abstract

We study the scrambling properties of $(d+1)$-dimensional hyperbolic black holes. Using the eikonal approximation, we calculate out-of-time-order correlators (OTOCs) for a Rindler-AdS geometry with AdS radius $\ell$, which is dual to a $d-$dimensional conformal field theory (CFT) in hyperbolic space with temperature $T = 1/(2 π\ell)$. We find agreement between our results for OTOCs and previously reported CFT calculations. For more generic hyperbolic black holes, we compute the butterfly velocity in two different ways, namely: from shock waves and from a pole-skipping analysis, finding perfect agreement between the two methods. The butterfly velocity $v_B(T)$ nicely interpolates between the Rindler-AdS result $v_B(T=\frac{1}{2π\ell})=\frac{1}{d-1}$ and the planar result $v_B(T \gg \frac{1}{\ell})=\sqrt{\frac{d}{2(d-1)}}$.

Figures (4)

• Figure 1: Penrose diagram for two-sided black holes with asymptotically AdS geometry.
• Figure 2: Left: representation of the 'in' state ${\color{red}{V_{\bf x_3}(t_3)}} {\color{blue}{W_{\bf x_4}(t_4)}}| \text{TFD}\rangle$ on a bulk slice that touches the right boundary at time $t_3$. Right: representation of the 'out' state ${\color{blue}{W_{\bf x_2}(t_2)^{\dagger}}} {\color{red}{W_{\bf x_1}(t_1)^{\dagger}}}| \text{TFD}\rangle$ on a bulk slice that touches the right boundary at time $t_2$.
• Figure 3: Temperature dependence of the butterfly velocity for hyperbolic black holes. The Rindler-AdS$_{d+1}$ result $(T,v_B)=(\frac{1}{2\pi \ell},\frac{1}{d-1})$ is indicated by the blue dot. Here we set $d=4$ and $\ell=1$.
• Figure 4: Sound channel quasinormal modes of the hyperbolic black hole defined in (\ref{['eq-hypBH3']}). The blue dots represent the zeros of $a(\omega,k)$ in \ref{['qnm1']}. They form a line that passes though the special point $(\omega_{\star}, k_{\star}) =i (2 \pi T, 2 \pi T / v_{B})$ with $v_{B}$ given in \ref{['result123']}. Here, $k :=-i L/\ell$, and we set $z_{0}=1$. For simplicity, we used the asymptotic form of the Legendre functions in the ansatz for the metric perturbations.