Gravitational waves on Kerr: Perturbations of extremal and near-extremal black holes

As the Kerr black hole approaches extremality, its near-horizon region develops a throat of divergent proper length that effectively decouples from the asymptotically flat exterior. The geometry of this throat, described by the Near-Horizon Extreme Kerr (NHEK) metric, is itself an exact vacuum solution to the Einstein equations, and its presence changes the analytic structure of the equations of motion, splitting perturbations into distinct near-horizon and far sectors. In this work, we explicitly reconstruct the linearized metric perturbations of both the NHEK and exterior regions of an extremal (and near-extremal) Kerr black hole. We then connect these regions using matched asymptotic expansions to obtain a global solution valid across the entire spacetime. This construction also reveals how the Mano–Suzuki–Takasugi (MST) formalism simplifies in the extremal limit: in both the near and far regimes, the infinite series of hypergeometric functions collapses to a single term. The resulting framework unifies the description of metric perturbations across all spin regimes and provides analytic control over gravitational perturbations near extremality, laying groundwork for future work on self-force and higher-order effects.