You've got a Freund--Rubin in Me: The Aretakis Instability of Extremal Black Branes

We investigate how the Aretakis instability affects non-dilatonic extremal black $p$-branes by focusing on their near-horizon geometry. Crucially, the strength of the instability, $\mathit{\text{i.e.}}$ the number of transverse derivatives needed to see non-decay/blow-up of fields on the horizon at late null time, is given by the scaling dimensions with respect to the near-horizon ${\mathrm{A}\mathrm{d}\mathrm{S}}_{p+2}$-factor. This renders the problem of determining the severity of the Aretakis instability equivalent to computing the Kaluza–Klein spectrum of fields on Freund–Rubin spaces. We use this to argue that non-dilatonic extremal black branes suffer from the Aretakis instability even in the absence of additional fields – we find that this is weaker than for extremal black holes. We also argue that the scaling dimensions determine the smoothness of stationary deformations to the original black brane background – here, our findings indicate that generically more modes can lead to worse curvature singularities compared to extremal black holes.