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

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