Accidental Symmetry Near Extreme Spinning Black Holes

Near the horizon of extremal charged black holes, an accidental symmetry is known to act on zero-temperature perturbations, transforming them into finite-temperature ones. In this paper, we uncover the corresponding accidental symmetry of the vacuum Einstein equation near the horizon of extremal spinning black holes. To do so, we first devise a new method of deriving the symmetry near the ${\rm AdS}_2 \times S^2$ near-horizon geometry of extreme Reissner-Nordstr\"om, using a new scaling coordinate transformation that unifies the near-horizon limits of extremal and near-extremal black holes in a way that is regular at zero temperature. We use our new method to obtain the accidental symmetry in the near-horizon of extreme Kerr (NHEK). We show that accidental symmetries combine neatly with the near-horizon isometries inside a Virasoro algebra.