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 ℓ, which is dual to a d-dimensional conformal field theory (CFT) in hyperbolic space with temperature T = 1/(2π ℓ). 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 vB (T) nicely interpolates between the Rindler-AdS result vB(T=12πℓ)=1d−1 and the planar result vB(T≫1ℓ)=d2(d−1). © 2019, The Author(s).