Teukolsky equation for near-extremal black holes beyond general relativity: near-horizon analysis

Abstract: We study gravitational perturbations on the near-horizon region of extremal and near-extremal rotating black holes in a general higher-derivative extension of Einstein gravity. We find a decoupled modified Teukolsky equation that rules the gravitational perturbations and that separates into an angular and a radial equation. The angular equation leads to a deformation of the spin-weighted spheroidal harmonics, while the radial equation takes the same form as in Kerr except for a modification of the angular separation constants. We provide a detailed analysis of the corrections to these angular separation constants and find analytic results for axisymmetric modes as well as in the eikonal limit. As an application, we reproduce recent results that show that extremal Kerr black holes in higher-derivative gravity become singular under certain deformations and extend them by including parity-breaking corrections, which we show lead to the same effect. Finally, we obtain constraints on the form of the full modified Teukolsky radial equation by demanding that it has the right near-horizon limit. These results serve as an stepping stone towards the study of quasinormal modes of near-extremal black holes in higher-derivative extensions of GR.