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Linear perturbations of extremal black holes exhibit the Aretakisinstability, in which higher derivatives of a scalar field grow polynomiallywith time along the event horizon. This suggests that higher derivativecorrections to the classical equations of motion may become large, indicating abreakdown of effective field theory at late time on the event horizon. Weinvestigate whether or not this happens. For extremal Reissner-Nordstrom weargue that, for a large class of theories, general covariance ensures that thehigher derivative corrections to the equations of motion appear only incombinations that remain small compared to two derivative terms so effectivefield theory remains valid. For extremal Kerr, the situation is morecomplicated since backreaction of the scalar field is not understood even inthe two derivative theory. Nevertheless we argue that the effects of the higherderivative terms will be small compared to the two derivative terms as long asthe spacetime remains close to extremal Kerr.

Is there a breakdown of effective field theory at the horizon of an extremal black hole?