Amplification of metric perturbations by extremal horizons
Under normal circumstances, an observer falling into a black hole feels nothing special upon crossing the horizon. In this paper we find an intriguing exception: For most of the parameter range of the extremal Kerr-Newman (KN) spacetime, horizon co-rotating perturbations are enhanced near the horizon, such that an infalling observer experiences an anomalously large tidal deformation as they enter the black hole. Such perturbations arise when there is a persistent source outside the black hole that is either co-rotating itself or has discrete Fourier support at the associated frequencies $ω=mΩ_H$ (where $m$ is the azimuthal number and $Ω_H$ is the horizon frequency). The enhancement is formally infinite at precise co-rotation, and we work with a nearly co-rotating mode to keep perturbation theory under control. The underlying physics is the emergence of discrete self-similarity in the extremal limit, with complex scaling weights of the form $-1/2\pm iα$ for real $α$. It is analogous to the black hole Meissner effect, except that the near-horizon field is enhanced rather than screened. We numerically calculate the scaling exponents for coupled gravitoelectromagnetic (GEM) perturbations of extremal KN black holes and show that the complex exponents arise in the parameter range $Q < Q_*$ with $Q_*\approx.93M$. These exponents also predict the decay and growth rates (Aretakis effect) of generic GEM perturbations of the KN spacetime, both on and off the horizon. The enhancement of co-rotating perturbations can be viewed as a driven Aretakis instability.