  Linear perturbations of extremal black holes exhibit the Aretakis
instability, in which higher derivatives of a scalar field grow polynomially
with time along the event horizon. This suggests that higher derivative
corrections to the classical equations of motion may become large, indicating a
breakdown of effective field theory at late time on the event horizon. We
investigate whether or not this happens. For extremal Reissner-Nordstrom we
argue that, for a large class of theories, general covariance ensures that the
higher derivative corrections to the equations of motion appear only in
combinations that remain small compared to two derivative terms so effective
field theory remains valid. For extremal Kerr, the situation is more
complicated since backreaction of the scalar field is not understood even in
the two derivative theory. Nevertheless we argue that the effects of the higher
derivative terms will be small compared to the two derivative terms as long as
the spacetime remains close to extremal Kerr.
