  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 $\textrm{AdS}_3$
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.
