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 $v_B(T\gg \frac{1}{\ell })=\sqrt{\frac{d}{2(d-1)}}$.

v1:20 pages, 4 figures; v2: 22 pages, title changed, major changes in section 4, small changes in App. A, references added. v3: clarifications and references added, matches the version to be published in JHEP

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