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

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 $T = 1/(2 \pi \ell)$. 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 $v_B(T=\frac{1}{2\pi \ell})=\frac{1}{d-1}$ and the planar result $v_B(T \gg \frac{1}{\ell})=\sqrt{\frac{d}{2(d-1)}}$.