A note on instabilities of extremal black holes under scalar perturbations from afar

In a previous work of the author, it was shown that instabilities of solutions to the wave equation develop asymptotically along the event horizon of extremal Kerr provided a certain expression H 0 of the initial data is non-trivial on the horizon. In this note, we remove this restriction by showing that instabilities develop even from the initial data supported arbitrarily far away from the horizon (for which, in particular, H 0 = 0). The latter instabilities concern one order higher derivatives compared to the case where H 0 ≠ 0. The result also applies to extremal Reissner–Nordstrom. This note was motivated by the numerical analysis of Lucietti, Murata, Reall and Tanahashi.