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

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 $\mathrm{\ell }$, which is dual to a $d-$dimensional conformal field theory (CFT) in hyperbolic space with temperature $T=1\mathrm{/}(2\pi \mathrm{\ell })$. 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 $v_B(T=\frac{1}{2\pi \mathrm{\ell }})=\frac{1}{d-1}$ and the planar result $v_B(T\gg \frac{1}{\mathrm{\ell }})=\sqrt{\frac{d}{2(d-1)}}$.