We study real-time propagation of a massive scalar field on the extremal BTZ black hole spacetime, focusing on the Aretakis instability of the event horizon. We obtain a simple time-domain expression for the AdS 3 retarded Green function with Dirichlet boundary conditions and construct the corresponding time-domain BTZ retarded Green function using the method of images. The field decays at different rates on and off the horizon, indicating that transverse derivatives grow with time on the horizon (Aretakis instability). We solve the null geodesic equation in full generality and show that the instability is associated with a class of null geodesics that orbit near the event horizon arbitrarily many times before falling in. In an appendix we also treat the problem in the frequency domain, finding consistency between the methods.

We show that the Aretakis instability of compact extremal horizons persists in the planar case of interest to holography and discuss its connection with the emergence of "semi-local quantum criticality" in the field theory dual. In particular, the spatially localized power-law decay of this critical phase corresponds to spatially localized power-law growth of stress-energy on the horizon. For near-extremal black holes these phenomena occur transiently over times of order the inverse temperature. The boundary critical phase is characterized by an emergent temporal conformal symmetry, and the bulk instability seems to be essential to preserving the symmetry in the presence of interactions. We work primarily in the solvable example of charged scalar perturbations of fivedimensional (near-)extremal planar Reissner-Nordström anti-de Sitter spacetime and argue that the conclusions hold more generally. a

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