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 l, which is dual to a d-dimensional conformal field theory (CFT) in hyperbolic space with temperature T = 1/(2 pi 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 v(B) (T) nicely interpolates between the Rindler-AdS result v(B) (T = 1/2 pi l) = 1/d-1 and the planar result v(B) (T >> 1/l) = root d/2(d-1).